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abstract: 'A generalization of von Staudt’s theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least $1$ is 4-transitive.'
address: 'Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram 9190401, Jerusalem, Israel'
author:
- Yatir Halevi and Itay Kaplan
bibliography:
- 'PGammaL2.bib'
title: 'A Generalization of von Staudt’s Theorem on Cross-Ratios'
---
[^1]
Introduction
============
In his book *Geometrie der Lage* (see [@vs]), first appearing in 1847, Karl Georg Christian von Staudt wanting to establish (real) projective geometry on an axiomatic approach defined a projectivity to be a permutation of the projective line $\mathbb{P}(\mathbb{R})=\mathbb{R}\cup\{\infty\}$ preserving *harmonic quadruples*, i.e. quadruples of distinct elements having cross-ratio $-1$ (which can be defined strictly geometrically), where the cross-ratio of quadruple of distinct elements is $$[a,b;c,d]=\frac{c-a}{c-b}\cdot \frac{d-b}{d-a}=-1.$$
He proved that a projectivity is a composition of a finite number of perspectivities (which are basic geomtric maps). It was noticed later that there is a small gap in von Staudt’s reasoning, see [@coolidge; @voelke] for a detailed historical background.
Given a field $F$, a *projectivity* (also known as a homography or a fractional linear transformation) of the projective line $\mathbb{P}(F)=F\cup \{\infty\}$ is an element of the group $$PGL_2(F)=\left\{\frac{ax+b}{cx+d}: a,b,c,d\in F,\, ad-bc\neq 0\right\},$$ where the usual conventions when dealing with $\infty,0$ and fractions apply here. It is easy to see that projectivities preserve cross-ratios.
It was Schreier and Sperner who first proved in [@sperner page 191] that every permutation of $F\cup\{\infty\}$, where $F$ is a field of characteristic $\neq 2$, preserving harmonic quadruples, i.e. $$[a,b;c,d]=-1 \Longrightarrow [f(a),f(b);f(c),f(d)]=-1,$$ is an element of $$P\Gamma L_2(F)=\left\{\frac{ax^\sigma+b}{cx^\sigma+d}:a,b,c,d\in F,\, ad-bc\neq 0, \, \sigma\in Aut(F)\right\}.$$ Any result in this spirit is now called a *von Staudt theorem*.
Over the years this theorem was generalized by relaxing the assumptions on $F$, for instance for $F$ a skew-field or a ring with some additional assumptions, see the introduction in [@havlicek] for a survey of results in that direction.
In this paper, we follow a generalization in a different direction. Hoffman eliminated the restriction on the characteristic of the field and replaced $-1$ with any field element which is fixed by $Aut(F)$, see [@hoffman]. Our main result is Corollary \[C:main\] from the text:
Let $F$ be a field, ${k}$ its prime field and $\emptyset\neq O\subseteq F\setminus \{0,1\}$ which is $Aut(F)$-invariant. If
1. ${k}(O)\subsetneq F$ and
2. if ${\mathrm{char}}(F)=2$ then $F$ is perfect and $|F|>4$,
then the subgroup of permutations $f$ of $F\cup \{\infty\}$ satisfying $$[a,b;c,d]\in O\Longleftrightarrow [f(a),f(b);f(c),f(d)]\in O$$ is exactly $P\Gamma L_2(F)$.
The motivation for seeking such a generalization came from infinite symmetric groups, and some model theory. It is well known that for a cardinal $\kappa$, the closed subgroups (in the product topology) of the infinite symmetric group $S_\kappa$, for a cardinal $\kappa$, in the product topology correspond exactly to automorphisms groups of first-order structures. Thus finding closed supergroups of such groups, sheds light on the the first order theory of such structures.
In [@itaypierre], the second author and Pierre Simon proved that the affine groups $AGL_n(\mathbb{Q})$ (for $n\geq 2$) and the projective linear groups $PGL_n(\mathbb{Q})$ (for $n\geq 3$) are maximal closed in $S_\omega$. They ask whether it is true that $P\Gamma L_2(F)$ is maximal closed, for an algebraically closed field $F$ of transcendence degree greater that $1$. The aim of this paper is a step towards answering this question.
Bogomolov and Rovinsky proved that $P\Gamma L_n(F)$ is maximal closed for $n\geq 3$ and any field $F$, see [@bogrov]. The reason for the distinction between $n=2$ and $n\geq 3$ is that by the fundamental theorem of projective geometry, $P\Gamma L_n(F)$ (for $n\geq 3$) is exactly the collineation group of $\mathbb{P}^{n-1}(F)$. On the other hand, for $n=2$, since $\mathbb{P}^1(F)$ is the projective line, all the points are collinear.
If $P\Gamma L_2(F)$ were not maximal closed, a proper supergroup of it must preserve one out of a known family of relations, two of them being quaternary relations (see [@itaypierre] for details). The aim is to show that it can not preserve any member of this family of relations. In this paper, using the above theorem, we conclude that if $F$ is an algebraically closed field of transcendence degree at least $1$, then any group of permutation of $F\cup\{\infty\}$ properly containing $P\Gamma L_2(F)$ is $4$-transitive and hence does not preserve any proper quaternary relation.
### Acknowledgments {#acknowledgments .unnumbered}
This project started as a derivative from the work of the second author with Pierre Simon in [@itaypierre]. Eventually ideas from that time contributed to the proof of the main theorem here, namely in the proof of (4) implies (1) in Theorem \[T:main\]. We thank him for allowing us to use his ideas.
Proofs
======
Let $f$ be a permutation of $F\cup\{\infty\}$ and $\emptyset\neq O\subseteq F\setminus \{0,1\}$. We say that $f$ is *$O$-preserving* if $$[a,b;c,d]\in O\Longleftrightarrow [f(a),f(b);f(c),f(d)]\in O,$$ where $a,b,c,d$ are distinct elements from $F\cup\{\infty\}$.
In this paper the cross-ratio is only taken for distinct points so that it takes values in $F\setminus\{0,1\}$ (one can expand the definition to allow repetitions, but this will not be used).
Throughout we will implicitly use the following property of the cross-ratio:
> For any two quadruples of distinct elements of $F\cup\{\infty\}$, $\{A,B,C,D\}$ and $\{A,B,C,X\}$, the following holds: $$[A,B;C,D]=[A,B;C,X]\Longleftrightarrow D=X.$$
\[P:gen-perm-function\] For every $x\in F\setminus\{0,1\}$ there exists a unique function $$g_x:X\to F\cup\{\infty\},$$ where $X\subseteq (F\cup\{\infty\})^3$ is the set of triples of distinct elements, such that for every distinct $a,b,c\in F\cup\{\infty\}$
1. $g_x(a,b,c)\neq a,b,c$,
2. $\left[a,b;c,g_x(a,b,c)\right]=x$ and
3. the map $x\mapsto g_x(a,b,c)$ is injective.
Furthermore, if $f$ is an $O$-preserving permutation of $F\cup\{\infty\}$, for some $\emptyset\neq O\subseteq F\setminus\{0,1\}$, then for every distinct $a,b,c\in F\cup\{\infty\}$ there exists a permutation $\alpha:O\to O$, such for every $x\in O$ $$f\left( g_x(a,b,c)\right)=g_{\alpha (x)}\left( f(a),f(b),f(c)\right).$$
Property $(2)$ uniquely determines $g_x$, and by the definition of the cross-ratio we get the following formula: $$g_x(a,b,c)=\frac{b(c-a)-ax(c-b)}{(c-a)-x(c-b)}.$$ Properties $(1)$ and $(3)$ follow easily.
As for the furthermore, for $x\in O$, define $$\alpha(x):=\left[f(a),f(b);f(c),f(g_x(a,b,c))\right]\in O$$ and likewise $$\alpha^{-1}(x):=\left[a,b;c,f^{-1}(g_x(f(a),f(b),f(c)))\right]\in O.$$ They are both elements of $O$ since $f$ is $O$-preserving, and note that by uniqueness, the definition of $\alpha$ gives that $$f(g_x(a,b,c))=g_{\alpha (x)}(f(a),f(b),f(c)).$$
For every $a,b,c,y\in F\cup\{\infty\}$ distinct, if $[a,b;c,y]\in O$ then $$\alpha([a,b;c,y])=[f(a),f(b);f(c),f(y)]$$ and if $[f(a),f(b);f(c),y]\in O$ then $$\alpha^{-1}([f(a),f(b);f(c),y])=[a,b;c,f^{-1}(y)].$$
Assume that $[a,b;c,y]=x \in O$. By uniqueness necessarily $$y=g_x(a,b,c).$$ It now follows that $$\alpha([a,b;c,y])=\alpha(x)=[f(a),f(b);f(c),f(g_x(a,b,c))]=[f(a),f(b);f(c),f(y)],$$ as required.
The proof for $\alpha^{-1}$ is similar.
We may now compute $$(\alpha\circ\alpha^{-1})(x)=\alpha\left([a,b;c,f^{-1}(g_x(f(a),f(b),f(c)))]\right)=$$ $$[f(a),f(b);f(c),g_x(f(a),f(b),f(c))]=x.$$ Similarly we get that $(\alpha^{-1}\circ \alpha)(x)=x$, as needed.
The previous proposition is obviously also true if we permute the coordinates of the cross-ratio, e.g. consider a function $h_x$ which guarantees that $$[a,b;h_x(a,b,c),c]=x.$$
We will frequently use the following hypothesis.
\[H:hyp\] The set $O$ is a subset of $F\setminus \{0,1\}$. The function $f$ is an $O$-preserving permutation of $F\cup\{\infty\}$ which fixes $\{0,1,\infty\}$ pointwise.
The field $K=k(O)$ is the field generated by the elements of $O$, where $k$ is the prime field.
\[C:permutation-functions\] Assume Hypothesis \[H:hyp\]. For all $a\neq b\in F$ there exist permutations $\tau_{a,b},\rho_{a,b},\chi_{a,b},\alpha_{a,b},\beta_{a,b}:O\to O$, such that for every $x\in O$: $$f(ax+b(1-x))=f(a)\tau_{a,b}(x)+f(b)(1-\tau_{a,b}(x)),$$ $$f\left( \frac{a-(1-x)b}{x}\right)=\frac{f(a)-(1-\rho_{a,b}(x))f(b)}{\rho_{a,b}(x)},$$ $$f\left(\frac{a-xb}{1-x}\right)=\frac{f(a)-\chi_{a,b}(x)f(b)}{1-\chi_{a,b}(x)},$$ $$f\left(\frac{abx-bx-ab+a}{ax-x-b+1}\right)=\frac{f(a)f(b)\alpha_{a,b}(x)-f(b)\alpha_{a,b}(x)-f(a)f(b)+f(a)}{f(a)\alpha_{a,b}(x)-\alpha_{a,b}(x)-f(b)+1},$$ $$f\left(\frac{a-b-abx+bx}{a-b-ax+x}\right)=\frac{f(a)-f(b)-f(a)f(b)\beta_{a,b}(x)+f(b)\beta_{a,b}(x)}{f(a)-f(b)-f(a)\beta_{a,b}(x)+\beta_{a,b}(x)}.$$ (for $\alpha_{a,b}$ and $\beta_{a,b}$ we also require that $a,b\neq 1$.)
Moreover, $f\restriction O$ is a permutation of $O$.
We apply Proposition \[P:gen-perm-function\]. Let $a\neq b\in F$. For $\tau_{a,b}$ use the identity $$[ax+b(1-x),a;b,\infty]=x.$$
For $\rho_{a,b}$ use the identity $$\left[a,\frac{a-(1-x)b}{x};b,\infty\right]=x.$$
For $\chi_{a,b}$ use the identity $$\left[a,b;\frac{a-xb}{1-x},\infty\right]=x.$$
For $\alpha_{a,b}$ use the identity $$\left[ b,a;1,\frac{abx-bx-ab+a}{ax-x-b+1}\right]=x.$$
For $\beta_{a,b}$ use the identity $$\left[b,1;a,\frac{a-b-abx+bx}{a-b-ax+x}\right]=x.$$
In order to show that $f\restriction O$ is a permutation of $O$, note that $[a,1,0,\infty]=a$ for every $a\in F\setminus \{0,1\}$.
\[L:first-closures\] Assume Hypothesis \[H:hyp\]. For every $a,b\in K$ and $x\in O$ $$f(a)+(1-x)f(b)\in f(K) \text{ and}$$ $$xf(a)+f(b)\in f(K).$$
We start with the first assertion, so let $a,b\in K$ and $x\in O$. If $b=0$ there is nothing to show. If $a=0$ and $b\neq 0$, then since $\tau_{0,b}$ is a permutation, by Corollary \[C:permutation-functions\], $$(1-x)f(b)=(1-(\tau_{a,b}\circ\tau_{a,b}^{-1})(x))f(b)=f((1-\tau_{a,b}^{-1}(x))b)\in f(K).$$ We may thus assume that $a,b\neq 0$ and let $x_2=\rho_{a,0}^{-1}(x)$, so $f(a/x_2)=f(a)/x$. If $b=\frac{a}{x_2}$ then $$f(a)+f(b)-xf\left(\frac{a}{x_2}\right)=f(b)\in f(K).$$ Now, assume that $b\neq \frac{a}{x_2}$, and let $x_3=\tau_{a/x_2,b}^{-1}(x)$. Hence $$f\left(\frac{a}{x_2}x_3+b(1-x_3)\right)=f\left(\frac{a}{x_2}\right)\tau_{a/x_2,b}(x_3)+f(b)(1-\tau_{a/x_2,b}(x_3))$$ $$=f(a)+f(b)-xf(b).$$
Now the second assertion. If $a=0$ there is nothing to show. If $a\neq 0$ and $b=0$ then since $\tau_{a,b}$ is a permutation, $xf(a)\in f(K)$. We may thus assume that $a,b\neq 0$ and let $x_2=\chi_{b,0}^{-1}(x)$. If $a=\frac{b}{1-x_2}$ then $$xf\left(\frac{b}{1-x_2}\right)+f(b)=\frac{x}{1-x}f(b)+f(b)=f(a)\in f(K).$$ Finally, assume that $a\neq \frac{b}{1-x_2}$, and let $x_3=\tau_{a,b/(1-x_2)}^{-1}(x)$. Hence $$f\left(ax_3+\frac{b}{1-x_2}(1-x_3)\right)=f(a)\tau_{a,b/(1-x_2)}(x_3)+f\left(\frac{b}{1-x_2}\right)(1-\tau_{a,b/(1-x_2)}(x_3))$$ $$=f(a)x+f(b).$$
\[L:second-closures\] Assume Hypothesis \[H:hyp\]. For every $0\neq a\in K$ and $x\in O$, $$-f(a)^2x+f(a)x+f(a)\in f(K) \text{ and}$$ $$1+x-\frac{x}{f(a)}\in f(K).$$
Let $x\in O$ and $a\in K$ with $a\neq 0$. If $a=1$ both assertions are trivial, so assume $a\neq 1$.
We start with the first assertion. Since $\tau_{a,0}$ is a permutation, by Corollary \[C:permutation-functions\], we may define $x_1=\tau_{a,0}^{-1}(x)$ so, $f(x_1a)=xf(a)$. We aim to use the permutation $\alpha_{a,x_1a}$. Obviously, $x_1a\neq a$ and if $x_1a=1$ then $xf(a)=1$ and $-f(a)^2x+f(a)x+f(a)=0$. Thus by Corollary \[C:permutation-functions\], $\alpha_{a,x_1a}$ is a permutation. Let $x_2:=\alpha_{a,x_1a}^{-1}(x)$, so $$f\left(\frac{a(x_1a)x_2-(x_1a)x_2-a(x_1a)+a}{ax_2-x_2-(x_1a)+1}\right)=$$$$\frac{f(a)f(x_1a)x-f(x_1a)x-f(a)f(x_1a)+f(a)}{f(a)x-x-f(x_1a)+1}=-f(a)^2x+f(a)x+f(a).$$
Now for the second assertion. Since $f\restriction O$ is a permutation, we may define $x_1:=f^{-1}(x)$. We aim to use the permutation $\beta_{a,x_1}$. If $a=x_1$ then the statement is obviously true, so we may assume that $a\neq x_1$ (and both not equal to $1$). By Corollary \[C:permutation-functions\], $\beta_{a,x_1}$ is a permutation, so we may define $x_2:=\beta_{a,x_1}^{-1}(x)$ and so $$f\left(\frac{a-x_1-ax_1x_2+x_1x_2}{a-x_1-ax_2+x_2}\right)=\frac{f(a)-f(x_1)-f(a)f(x_1)x+f(x_1)x}{f(a)-f(x_1)-f(a)x+x}=$$ $$1+x-\frac{x}{f(a)}.$$
\[P:K-to-K\] Assume Hypothesis \[H:hyp\], and if when ${\mathrm{char}}(F)=2$ we assume further that $O$ is closed under taking square-roots, then $f(K)=K$.
We first show that $K\subseteq f(K)$. Note that $O\subseteq f(K)$, indeed $f \restriction O$ is a permutation by Corollary \[C:permutation-functions\].
Let $a,b\in K$ and $x\in O$, which exists since $O$ is non-empty. By Lemma \[L:first-closures\], $f(a)+f(b)-xf(b)\in f(K)$. By the same lemma $$f(a)+f(b)=xf(b)+\left( f(a)+f(b)-xf(b)\right)\in f(K).$$
In order to show that if $f(c)\in f(K)$ then also $-f(c)$, first notice that by considering $\rho_{0,c}$ in Corollary \[C:permutation-functions\] we see that $$-\frac{1-x}{x}f(c)\in f(K).$$ By considering $\chi_{d,0}$ in the same corollary, we see that $$\frac{f(d)}{1-x}\in f(K),$$ for any $d\in K$. In particular $-\frac{1}{x}f(c)\in f(K)$, for any $c\in K$. Similarly, by considering $\tau_{a,0}$, we get that $xf(c)\in f(K)$ for all $c\in K$ and together $-f(c)\in f(K)$ for all such $c$, as needed.
As for the multiplication, by Lemma \[L:second-closures\], $$-f(a)^2x+f(a)x+f(a)\in f(K),$$ for every $x\in O$. Using the above, and since $f(a),f(a)x\in f(K)$, $$-f(a)^2x\in f(K).$$ So $-f(a)^2x=f(b)$ for some $b\in K$. Since $\frac{f(b)}{-x}\in f(K)$ as well, $f(a)^2\in f(K).$
Let $a\neq 0\in K$, by Lemma \[L:second-closures\], $1+x-\frac{x}{f(a)}\in f(K)$ for any $x\in O$. Using the above, and since $1,x\in f(K)$, $$\frac{x}{f(a)}\in f(K).$$ Using a similar argument to the previous paragraph, $-\frac{1}{f(a)}\in f(K)$, so $\frac{1}{f(a)}\in f(K)$.
Finally, we first assume that ${\mathrm{char}}(F)\neq 2$. Let $a,b\in K$. Since $\left( f(a)+f(b)\right)^2\in f(K)$ we get that $$2f(a)f(b)\in f(K).$$ To get that $f(K)$ is a subfield, we need this final claim:
If $a\in K$ then $\frac{f(a)}{2}\in f(K)$.
We may assume that $a\neq 0$. Since $f(a)\in f(K)$ then $1/f(a)\in f(K)$ and so also $2/f(a)$. Take the inverse again and $f(a)/2\in f(K)$.
We conclude that if ${\mathrm{char}}(F)\neq 2$, $f(K)$ is a field and so $K\subseteq f(K)$.
Assume that ${\mathrm{char}}(F)=2$. For any $n\geq 0$, let $$(f(K))^{2^n}:=\{a^{2^n}:a\in f(K)\}.$$ Note that since $f(K)$ is closed under squares, $\{(f(K))^{2^n}:n\geq 0\}$ forms a decreasing sequence under inclusion. The following is an easy observation:
For every $n\geq 0$, $(f(K))^{2^n}$ is also closed under addition, additive and multiplicative inverses and taking square powers.
Consider $L:=\bigcap_{n\geq 0}(f(K))^{2^n}$.
$L$ is a field containing $O$. Therefore, $K\subseteq L\subseteq f(K)$.
Since $O$ is closed under square-roots it is contained in $L$ and by the previous claim $L$ is closed under addition, additive and multiplicative inverses and taking square-powers. Let $a,b\in L$ and let $n\geq 0$. We will show that $ab\in (f(K))^{2^n}$. Since $a,b\in (f(K))^{2^{n+1}}$, by the following form of Hua’s identity (first mentioned in [@hua] but we use the more manageable form from [@jacobson page 2]):
$$a-(a^{-1}+(b^{-2}-a)^{-1})^{-1}=a^2b^2,$$ and by the last claim, $a^2b^2\in (f(K))^{2^{n+1}}$. Since the Frobenius map is injective, $ab\in (f(K))^{2^n}$, as required.
Either way, $K\subseteq f(K)$, but since $f^{-1}$ is also $O$-preserving by definition, we actually have $K\subseteq f(K)\subseteq K$ as required.
For an $Aut(F)$-invariant subfield $K\subseteq F$, a *$K$-chain* is an image of $K\cup \{\infty\}$ under the action of $P\Gamma L_2 (F)$.
The term *$K$-chain* is due originally to von Staudt who introduced it for any real subline of the complex projective line. Note that the usual definition of $K$-chain is for any subfield $K\subseteq F$ and only images under the action of $PGL_2(F)$ (see [@chain Definition 2.2.2]). Naturally, these definitions are equivalent for $Aut(F)$-invariant subfields. See [@chain] for more on this subject, and in a higher level of generality.
We recall that an action of a group $G$ on a set $X$ (with $|X|\geq k$) is *$k$-transitive* if $G$ acts transitively on the set of $k$-tuples of distinct elements of $X$. For example, the action of the group $PGL_2(F)$ on $F\cup \{\infty\}$ is $3$-transitive.
\[C:sending K-lines to K-lines\]
Let $f$ be an $O$-preserving permutation of $F\cup\{\infty\}$, for some non-empty $O\subseteq F\setminus \{0,1\}$ which is $Aut(F)$-invariant. If, when ${\mathrm{char}}(F)= 2$, we further assume $O$ is closed under taking square-roots, then $f$ sends $K$-chains to $K$-chains, where $K={k}(O)$.
Let $X=T(K\cup\{\infty\})$ be a $K$-chain, where $T\in P\Gamma L_2(F)$. Since $O$ is $Aut(F)$-invariant, $f\circ T$ is also $O$-preserving. Thus we may assume that $X=K\cup\{\infty\}$. Since $PGL_2(F)$ is 3-transitive and preserves the cross-ratio, by composing with an element of $PGL_2(F)$ we may assume that $f$ fixes $\{0,1,\infty\}$ pointwise. By Proposition \[P:K-to-K\], $f(K\cup \{\infty\})=K\cup\{\infty\}$ as needed.
We recall some definitions from affine geometry. Given a $K$-vector space $V$, an *affine line* is a set of the form $Ka+b$ for some $a,b\in V$ with $a\neq 0$. A map $T:V\to V$ is called *semilinear* if there exists a field automorphism $\sigma \in Aut(K)$ such that for all $v,u\in V$ and $x,y\in K$ $$T(xv+yu)=\sigma(x)T(v)+\sigma(y)T(u).$$
[@fundamentalaffine Theorem 3.5.6]\[F:fundamentalaffine\] Let $K$ be any field and $V$ a $K$-vector space of dimension at least $2$. If $f$ a permutation of $V$ sending affine lines to affine lines then there exists a semilinear map $T:V\to V$ and $b\in V$ such that $f(x)=T(x)+b$ for all $x\in V$.
The following theorem is a generalization of the main theorem in [@hoffman].
\[T:main\] Let ${k}\subsetneq F$ be a field, where ${k}$ is the prime field of $F$, and let $f$ be a permutation of $F\cup\{\infty\}$.
If ${\mathrm{char}}(F)=2$ we assume further that
(a) $F$ is perfect and
(b) $|F|>4$.
Then the following are equivalent
1. $f\in P\Gamma L_2(F)$.
2. $f$ is $O$-preserving for all non-empty $O\subseteq F\setminus \{0,1\}$ which is $Aut(F)$-invariant and satisfies $k(O)\subsetneq F$.
3. $f$ is $O$-preserving for some non-empty $O\subseteq F\setminus \{0,1\}$ which is $Aut(F)$-invariant and satisfies $k(O)\subsetneq F$.
4. There exists an $Aut(F)$-invariant subfield $K\subsetneq F$ such that $f$ sends $K$-chains to $K$-chains.
5. For all $Aut(F)$-invariant subfields $K\subsetneq F$, $f$ sends $K$-chains to $K$-chains.
<!-- -->
1. Since we are assuming $O$ is non-empty, if we drop the assumption that $|F|>4$ then $(1)$ might be true even if $(3)$ is not. For example, for $F=\mathbb{F}_4$, the field with $4$ elements, there are no proper intermediate fields between $\mathbb{F}_2$ and $\mathbb{F}_4$, so $(3)$ is not true.
2. The implication $(4)\Rightarrow (1)$ is well known for fields $F$ with ${\mathrm{char}}(F)\neq 2$, see [@chain Theorem 9.2.5], but we provide a direct proof.
$(1)\Rightarrow (2)$. This is by the definition of $P\Gamma L_2(F)$.
$(2)\Rightarrow (3)$. If ${\mathrm{char}}(F)\neq 2$ just take $O=k\setminus \{0,1\}$ (which is non-empty). If ${\mathrm{char}}(F)=2$, take elements generating the subfield of $4$ elements.
$(3)\Rightarrow (4)$. Let $K={k}(O)$. Since $O$ is $Aut(F)$-invariant, $K$ is $Aut(F)$-invariant and if ${\mathrm{char}}(F)=2$ then $O$ is closed under taking square-roots since $F$ is perfect and hence the inverse of the Frobenius map is an automorphism. Now apply Corollary \[C:sending K-lines to K-lines\].
$(4)\Rightarrow (1)$. By composing with an element of $PGL_2(F)$, we may assume that $f$ fixes $\{0,1,\infty\}$ pointwise. We plan to use Fact \[F:fundamentalaffine\], so we must show that, in $F$ as a $K$-vector space, $f\restriction F$ sends affine lines to affine lines. Since $f$ fixes $\{\infty\}$, and sends $K$-chains to $K$-chains, it is sufficient to show the following, where by a *projective affine line* we mean a union of an affine line with $\{\infty\}$,
Recall also that any $K$-chain is equal to some $T(K\cup\{\infty\})$, where $T(x)$ is of the form $$\frac{ax^\sigma+b}{cx^\sigma+d},$$ for $a,b,c,d\in F$, with $ad-bc\neq 0$, and $\sigma\in Aut(F)$. Since $K$ is $Aut(F)$-invariant we may assume that $\sigma=id$.
A subset of $F\cup\{\infty\}$ is a $K$-chain which includes $\infty$ if and only if it is a projective affine line.
A projective affine line has the form $a(K\cup\{\infty\})+b$, for $a,b\in F$, so it is a $K$-chain. For the other direction, by translation it is enough to show that any $K$-chain containing $0$ and $\infty$ is a projective affine line.
Note that projective affine lines that contain $0$ are just of the form $aK$ (for $a\neq 0$), and that both families of projective affine lines containing $0$ and $K$-chains containing $0$ and $\infty$ are closed under scalar multiplication (by non-zero elements from $F$) and inverse ($x\mapsto 1/x$). So it is enough to show that after applying finitely many operations of the form above on a $K$-chain containing $0$ and $\infty$ gives a projective affine line (containing $0$). Assume that we are given a $K$-chain of the form $T(K\cup\{\infty\})$ for $T$ as above which contains $0$ and $\infty$.
[•]{}
If $T$ is of the form $ax+b$ with $a\neq 0$, then $b=0$ and we are done.
If $T$ is of the form $b/(cx+f)$ with $c\neq 0$, then we are done by the first bullet (after dividing by $b$ and taking inverse).
If the $T$ is of the form $\frac{ax+b}{cx+d}$ with $a,c\neq 0$, then after multiplying by $c/a$, we may assume that $a=c=1$, and then since the chain contains $0$, $b\in K$, and since the chain contains $\infty$, $d\in K$. So it is equal to $(K\cup\{\infty\})$.
As a result, $f$ preserves the system of affine lines in the $K$-vector space $F$. Since $K\subsetneq F$, $\dim_K F\geq 2$ so by the fundamental theorem of affine geometry (Fact \[F:fundamentalaffine\]) and since $f(0)=0$, $f$ must be additive and so also $f(-a)=-f(a)$ for all $a\in F$.
The conjugation of $f$ by the $PGL_2(F)$ map $x\mapsto 1/x$ also satisfies the above, so it is also additive. This translates to $$\frac{f(a)f(b)}{f(a)+f(b)}=f\left( \frac{ab}{a+b}\right),$$ for all nonzero $a,b\in F$. By setting in the equation $a=1$ and $b=t-1$ (for $t\neq 1$) we obtain, $f(t)f(t^{-1})=1$, thus $f$ commutes with inversion.
Putting in the same equation $b=1-a$, for $a\neq 0,1$, we obtain $f(a)f(1-a)=f(a(1-a))$, which gives $f(a^2)=f(a)^2$.
If ${\mathrm{char}}(F)\neq 2$ then, since $f$ is additive, $f(x/2)=f(x)/2$ for all $x\in F$. Set $a=x+y$ in the last equation to get $f(xy)=f(x)f(y)$ for all $x,y\in F$.
If ${\mathrm{char}}(F)=2$ we once again use Hua’s identity: $$f(a^2b^2)=f\left(a-(a^{-1}+(b^{-2}-a^{-1})^{-1}\right)$$ $$=f(a)-(f(a)^{-1}+(f(b)^{-2}-f(a)^{-1})^{-1}=f(a^2)f(b^2).$$ Hence $f(ab)^2=\left( f(a)f(b)\right)^2$, so $f(ab)=f(a)f(b)$.
Either way, we get that $f$ is an automorphism of $F$, an in particular $f\in P\Gamma L_2(F)$.
$1\Rightarrow 5$ is clear and $5\Rightarrow 4$ follows by taking $K=k$.
As a direct corollary of Theorem \[T:main\] we get the following.
\[C:main\] Let $F$ be a field, ${k}$ its prime field and $\emptyset\neq O\subseteq F\setminus \{0,1\}$ which is $Aut(F)$-invariant. If
1. ${k}(O)\subsetneq F$ and
2. if ${\mathrm{char}}(F)=2$ then $F$ is perfect and $|F|>4$,
then the subgroup of $O$-preserving permutations of $F\cup \{\infty\}$ is exactly $P\Gamma L_2(F)$.
It was shown by Hoffman in [@hoffman], that if $F$ is a field, $a\in F\setminus \{0,1\}$ and $f$ is an $\{a\}$-preserving permutation of $F\cup\{\infty\}$ then $f\in P\Gamma L_2(F)$. One may ask, what about if $f$ preserves a set of cardinality larger than $1$? Theorem \[T:main\] only gives a partial answer. More specifically, can the assumption $k(O)\subsetneq F$ be dropped? For example:
Does the subgroup of permutations of $\mathbb{Q}(\sqrt{2})\cup\{\infty\}$ which are $\{\pm\sqrt{2}\}$-preserving properly contain $P\Gamma L_2(\mathbb{Q}(\sqrt{2}))$?
Every proper extension of $P\Gamma L_2(F)$ is $4$-transitive
============================================================
Our final aim is to show that, as a corollary of Theorem \[T:main\], any group of permutations of $F\cup\{\infty\}$, for $F$ algebraically closed of transcendence degree at least $1$, which properly contains $P\Gamma L_2(F)$ must be $4$-transitive and as a result does not preserve any non-trivial $4$-relation.
\[L:orbits\] Let $\{O_i\}_{i\in I}$ be the orbits of $Aut(F)$ acting on $F\setminus \{0,1\}$. Then the orbits of $P\Gamma L_2(F)$ acting on quadruples of distinct elements from $F\cup \{\infty\}$ are $$\{(a,b,c,d): [a,b;c,d]\in O_i\}_{i\in I}.$$
Let $T\circ \sigma\in P\Gamma L_2(F,)$, for $T\in PGL_2(F)$ and $\sigma\in Aut(F)$. Since elements of $PGL_2(K)$ preserve the cross-ratio, and $\sigma(O_i)=O_i$ by definition, $P\Gamma L_2(F)$ preserves the orbits.
Now, let $(x,y,z,w), (a,b,c,d)$ be quadruples of distinct elements such that $$[x,y;z,w],[a,b;c,d]\in O_i.$$ By applying an element of $Aut(F)$ we may assume that $[x,y;z,w]=[a,b;c,d]$.
Since $PGL_2(F)$ is $3$-transitive, there exists $T\in PGL_2(K)$ such that $T(x)=a, T(y)=b, T(z)=c$. So we have that $$[a,b;c,T(w)]=[a,b;c,d].$$ As $a,b,c$ are distinct we have that $T(w)=d$.
Let $F$ be an algebraically closed field of transcendence degree at least $1$, and $H$ be a group of permutations of $F\cup\{\infty\}$ properly containing $P\Gamma L_2(F)$. Then $H$ is $4$-transitive.
By Lemma \[L:orbits\], the action of $P\Gamma L_2 (F)$ breaks the space of quadruples of distinct elements from $F\cup\{\infty\}$ into infinitely many finite orbits (corresponding to finite Galois orbits) and one infinite orbit (corresponding to the Galois orbit of transcendentals).
Thus it is enough to show that every orbit of the action of $H$ on the space of quadruples of distinct elements from $F\cup\{\infty\}$ intersects the orbit corresponding to the transcendentals.
Aiming for a contradiction, assume there exists an orbit $X$ of the action of $H$, which only contains orbits with algebraic cross-ratio, i.e. $$X=\bigcup_{i\in I_0} \{(a,b,c,d):[a,b;c,d]\in O_i\},$$ for some $I_0\subseteq I$ and $O_i$ finite, where $I$ and $O_i$ are as in Lemma \[L:orbits\]. Let $O=\bigcup_{i\in I_0} O_i$ be the cross-ratios arising from quadruples from $X$ and let $K={k}(O)$, where ${k}$ is the prime field. Note that $K\subsetneq F$ and that $O$ is $Aut(F)$ invariant. By assumption every element of $H$ is $O$-preserving and thus by Corollary \[C:main\], $H\subseteq P\Gamma L_2(F)$, contradiction.
What about other fields? For instance, is it true that every group of permutations of $\mathbb{Q}(\sqrt{2})\cup\{\infty\}$ properly containing $P\Gamma L_2(\mathbb{Q}(\sqrt{2}))$ must be $4$-transitive?
[^1]: The first author was partially supported by the European Research Council grant 338821. The second author would like to thank the Israel Science Foundation for their support of this research (Grant no. 1533/14).
|
---
abstract: |
We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring that the dual of the result of gluing together two pseudo-bundles can equivalently be obtained by gluing together their duals, which is not automatic in the diffeological context. We show that, assuming that the dual of the gluing map, which itself does not have to be a diffeomorphism, on the total space is one, the commutativity condition is satisfied, via a natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved.
MSC (2010): 53C15 (primary), 57R35, 57R45 (secondary).
author:
- 'Ekaterina [<span style="font-variant:small-caps;">P</span>ervova]{}'
title: 'Pseudo-bundles of exterior algebras as diffeological Clifford modules'
---
Introduction {#introduction .unnumbered}
============
This work is intended as a supplement to [@clifford-alg], dealing with some issues regarding pseudo-bundles of exterior algebras associated to finite-dimensional diffeological vector pseudo-bundles, viewed as pseudo-bundles of diffeological Clifford modules, so endowed with the action of the corresponding diffeological pseudo-bundles of Clifford algebras. Let us explain as briefly as possible what all these objects are; the precise definitions are to be found in a dedicated section, or else in the references given therein.
First of all, the *diffeology*. The notion is due to J.M. Souriau [@So1], [@So2] and is a categorical extension of the notion of a smooth structure; in essence, or maybe as an example, it is a way to consider a topological space which is in no way a manifold, as if it were one. In and of itself, a diffeology on a set is a collection of maps into this set which are declared to be smooth; this collection must satisfy certain conditions. The set is then a diffeological space, and all the basic constructions follow; there is a notion of smooth maps between two diffeological spaces, that of the underlying topology, the so-called *D-topology* (introduced in [@iglFibre], see also [@CSW_Dtopology] for a recent treatment), and so on. The notion builds on existing ones, such as those of *differentiable spaces*, *V-manifolds*, and so on (see, for instance, [@satake1957], [@chen1977], to name a few). A particularly important aspect of diffeology is that all the usual topological constructions, notably subsets and quotients, have an inherited diffeological structure (unlike the case of smooth manifolds, where subsets and quotients are quite rarely smooth manifolds themselves).
The basic object for us, though, is not just a diffeological space, but a *diffeological vector pseudo-bundle* ([@iglFibre], [@vincent], [@iglesiasBook], [@pseudobundles]). The difference with respect to the standard notion is not only in the fact that the smooth structure is replaced by a diffeological one (under some respects this would be a minor difference), but also in that it does not have to be locally trivial, although in many contexts we do add this assumption, see the discussion of *pseudo-metrics*, which replace the usual Riemannian metrics — diffeological pseudo-bundles frequently do not carry the latter.
On the other hand, as explained in the references listed above, the usual operations on vector bundles have their diffeological counterparts; in particular, direct sums, tensor products, and taking duals all apply. From this, obtaining pseudo-bundles of tensor algebras, those of exterior algebras, or defining, in the abstract, pseudo-bundles of Clifford modules is automatic.
The point of view that we take in this paper has to do with studying the behavior of these concepts under the operation of *diffeological gluing*. This procedure is one of the many possible extensions of the concept of an atlas on a smooth manifold, and the resulting spaces are among the more obvious extensions of smooth manifolds and include some well-known singular spaces; for instance, a manifold with a conical singularity can be seen as a result of gluing of a usual smooth manifold to a single-point space.
For the basic operations on diffeological vector pseudo-bundles, the behavior under diffeological gluing was considered in [@pseudobundles] (see also [@pseudometric-pseudobundle] for some details); for tensor algebras and pairs of given Clifford modules, in [@clifford-alg]. What is lacking is a study of gluing of pseudo-bundles of the exterior algebras, in particular, the covariant version. This paper aims to fill this void (another motivation for it is to provide some necessary building blocks for defining the notion of a diffeological Dirac operator and studying its behavior under gluing, see [@dirac]).
#### The content
Section 1 goes over the main definitions used and introduces notation. In Section 2 we consider the compatibility of pseudo-metrics in terms of the assumptions on the gluing map(s); in Section 3 we relate this to the gluing-dual commutativity condition, and in Section 4 we show that the compatibility of dual pseudo-metrics implies that the commutativity condition must be satisfied. In Section 5 we show that the gluing-dual commutativity diffeomorphism is an isometry. All these allow us to consider, in Sections 6-8, Clifford algebras (the covariant case), the exterior algebras, and the corresponding Clifford actions; in particular, in Section 8 we establish, where appropriate, several equivalences showing that, again under the gluing-dual commutativity assumption, everything reduces to two cases, the contravariant case and the covariant one. Section 9 contains a couple of simple (but necessarily lengthy) examples.
#### Acknowledgments
This work benefitted from some assistance, for which I would like to thank\
Prof. Riccardo Zucchi, even if he always says that he is being overvalued, some of his doctorate students (Martina and Leonardo), although they do not expect this at all, and also Prof. Mario Petrini (he most definitely will be surprised).
Main definitions and known facts
================================
We now go, as briefly as possible, over the main definitions that appear in what follows (for terms whose use is not as frequent, we will provide definitions as we go along).
Diffeology and diffeological vector spaces
------------------------------------------
Let $X$ be a set. A **diffeology** on $X$ (see [@So1], [@So2]) is a set ${{\mathcal D}}=\{p:U\to X\}$ of maps into $X$, each defined on a domain of some ${{\mathbb{R}}}^n$ (with varying $n$), that satisfies the following conditions: 1) it includes all constant maps, *i.e.*, maps of form $U\to\{x_0\}$, for all open (possibly disconnected) sets $U\subseteq{{\mathbb{R}}}^n$ and for all points $x_0\in X$; 2) for any ${{\mathcal D}}\ni p:U\to X$ and for any usual smooth map $g:V\to U$ (again defined on some domain $V\subseteq{{\mathbb{R}}}^m$) we have $p\circ g\in{{\mathcal D}}$; and 3) if a set map $p:U\to X$ is such that its domain of definition $U\subseteq{{\mathbb{R}}}^n$ has an open cover $U=\cup_{i\in I}U_i$ for which $p|_{U_i}\in{{\mathcal D}}$ then $p\in{{\mathcal D}}$.
The maps composing ${{\mathcal D}}$ are called **plots**. A standard example of diffeology/ diffeological space is a usual smooth manifold $M$, with diffeology composed of all usual smooth maps into $M$. On the other hand, any set (usually at a least a topological space) admits plenty of non-standard diffeologies, obtained via the concept of a *generated diffeology*.
#### Generated diffeologies
Given a fixed set $X$, various diffeologies on it can be compared with respect to the inclusion;[^1] for two diffeologies ${{\mathcal D}}$ and ${{\mathcal D}}'$ such that ${{\mathcal D}}\subset{{\mathcal D}}'$ one says that ${{\mathcal D}}$ is **finer** than ${{\mathcal D}}'$, whereas ${{\mathcal D}}'$ is said to be **coarser**. Frequently, for a given property $P$ which a diffeology might possess, there is the finest and/or the coarsest diffeology with the property $P$ (see [@iglesiasBook], Sect. 1.25); this fact is often used in describing concrete diffeologies, or defining a class of them. A specific example of the former is the **generated diffeology**: for a set $X$ and a set $A=\{p:U\to X\}$ of maps into $X$, the diffeology **generated by $A$** is the smallest diffeology on $X$ that contains $A$. Notice that $A$ can be *any* set; it might include non-differentiable maps, discontinuous ones, and so on.
#### Smooth maps
A map $f:X\to Y$ between two diffeological spaces $X$ and $Y$ is considered **smooth** if for every plot $p$ of $X$ the composition $f\circ p$ is a plot of $Y$. Note that it might easily happen that all $f\circ p$ are plots of $Y$, but that *vice versa* is not true: $Y$ may have plots that do not have form $f\circ p$, whatever the plot $p$ of $X$. On the other hand, if for every plot $q:U\to Y$ of $Y$ and for every point $u\in U$ there is a plot $p_u$ of $X$ such that in a neighborhood of $u$ we have $q=f\circ p_u$ then we say that the diffeology of $Y$ is the **pushforward** of the diffeology of $X$ via $f$, and conversely, the diffeology of $X$ is the **pullback** of the diffeology of $Y$ by $f$.
#### Subset, quotient, product, and disjoint union diffeologies
All typical topological constructions admit diffeological counterparts. If $X'$ is any subset of a diffeological space $X$, it carries the **subset diffeology** that consists of all plots of $X$ whose range is contained in $X'$; and if $X/\sim$ is any[^2] quotient of $X$, with $\pi:X\to X/\sim$ being the natural projection, then the standard choice of diffeology on $X/\sim$ is the **quotient diffeology**, defined as the pushforward of the diffeology of $X$ by $\pi$. As we said above, this means that locally each plot of $X/\sim$ has form $\pi\circ p$, where $p$ is a plot of $X$.
Let us now have several (a finite number of, although the definition can be stated more broadly) diffeological spaces $X_1,\ldots,X_n$. Their usual direct product also has its standard diffeology, the **product diffeology**, defined as the coarsest diffeology such that the projection on each term is smooth. Locally any plot of this diffeology is just an $n$-tuple $(p_1,\ldots,p_n)$, where each $p_i$ is a plot of the corresponding $X_i$. Finally, the disjoint union $\sqcup_{i=1}^nX_i$ of these spaces has the **disjoint union diffeology**, this being the finest diffeology such that the inclusion of each term into the disjoint union is smooth. Locally, any plot of this diffeology is a plot of precisely one of terms $X_i$.
#### Functional diffeology
The space $C^{\infty}(X,Y)$ of all smooth (in the diffeological sense) maps between two diffeological spaces $X$ and $Y$ also has its standard diffeology, called the **functional diffeology**. It consists of all possible maps $q:U\to C^{\infty}(X,Y)$ such that for every plot $p:U'\to X$ of $X$ the natural evaluation map $U\times U'\ni(u,u')\mapsto q(u)(p(u'))\in Y$ is smooth (with respect to the diffeology of $Y$; the product $U\times U'$ is still a domain, therefore asking for the evaluation map to be smooth is equivalent to asking it to be a plot of $Y$).
#### Diffeological vector spaces
This is one specific instance of a diffeological space endowed also with an algebraic structure whose operations are smooth for the diffeology involved.[^3] A (real) **diffeological vector space** is a vector space $V$ endowed with a diffeology such that the addition map $V\times V\to V$ and the scalar multiplication map ${{\mathbb{R}}}\times V\to V$ are smooth (for the product diffeology on $V\times V$ and ${{\mathbb{R}}}\times V$ respectively). For a fixed $V$ there can be many such diffeologies; any one of them is called a **vector space diffeology** (on $V$). If $V$ is finite-dimensional, and so as just a vector space is isomorphic to some ${{\mathbb{R}}}^n$, then the finest of all vector space diffeologies is the one consisting of all usual smooth maps into it.[^4] This diffeology is called the **standard diffeology**; endowed with it, $V$ is called a **standard space**.
All usual operations on vector spaces (taking subspaces, quotients, direct sums, tensor products, and duals) admit their natural diffeological counterparts ([@vincent], [@wu]; see also [@multilinear]), via the more general diffeological constructions described above. Thus, any vector subspace of a diffeological vector space $V$ is automatically endowed with the subset diffeology; every quotient space carries the quotient diffeology; the direct sum carries the product diffeology (relative to the diffeologies of its terms); and the tensor product has the quotient diffeology of the finest vector space diffeology on the free product of the factors that contains the product diffeology on their direct product. Finally, the diffeological dual $V^*$ of $V$ is defined as the space of all diffeologically smooth linear maps $L^{\infty}(V,{{\mathbb{R}}})$ (where ${{\mathbb{R}}}$ is standard) endowed with the functional diffeology. Notice that, unless a finite-dimensional $V$ is a standard space, we have $\dim(V^*)<\dim(V)$ (and in general, the space of smooth linear maps between two diffeological vector spaces is strictly smaller than the space of all linear maps).
#### Pseudo-metrics and characteristic subspaces
A finite-dimensional diffeological vector space $V$ does not admit a smooth scalar product, unless it is standard (see [@iglesiasBook]). The best possible substitute for it is any smooth symmetric semi-definite positive bilinear form of rank $\dim(V^*)$; (at least one) such a form exists on any finite-dimensional $V$ and is called a **pseudo-metric**.
A pseudo-metric $g$ on a finite-dimensional diffeological vector space $V$ allows to identify in $V$ the unique vector subspace $V_0$ which is maximal, with respect to inclusion, for the following two properties: the subset diffeology of $V_0$ is that of a standard space, and $V_0$ splits off smoothly in $V$, which means there is a usual vector space direct sum decomposition $V=V_0\oplus V_1$ such that the corresponding direct sum diffeology on $V$ relative to the subset diffeologies on $V_0$ and $V_1$ coincides with the initial diffeology of $V$.[^5] This subspace can be described as the subspace generated by all the eigenvectors of $g$ that are relative to the positive eigenvalues; however, it does not actually depend on the specific choice of a pseudo-metric and is instead an invariant of $V$ itself. It is called the **characteristic subspace** of $V$.
Diffeological vector pseudo-bundles and pseudo-metrics on them
--------------------------------------------------------------
A *diffeological vector pseudo-bundle* is a diffeological counterpart of a usual smooth vector bundle.[^6] Apart from the diffeological smoothness replacing the usual concept of smooth maps, they lack an atlas of local trivializations, although in many contexts, and in most of what follows, we do add this assumption.
#### Diffeological vector pseudo-bundles
Let $V$ and $X$ be diffeological spaces, and let $\pi:V\to X$ be a smooth surjective map. The map $\pi$, or the total space $V$, is called a **diffeological vector pseudo-bundle** if for each $x\in X$ the pre-image $\pi^{-1}(x)$ carries a vector space structure such that the following three maps are smooth: the addition map $V\times_X V\to V$ (where $V\times_X V$ is endowed with the subset diffeology as a subset of $V\times V$), the scalar multiplication map ${{\mathbb{R}}}\times V\to V$, and the zero section $X\to V$. An example of a diffeological vector pseudo-bundle which is not locally trivial, can be found in [@CWtangent] (see Example 4.3).
#### The fibrewise operations and fibrewise diffeologies
Since each fibre of a diffeological vector pseudo-bundle is a diffeological vector space, all the usual operations on vector bundles (direct sums, tensor products, dual bundles) can be performed on/with pseudo-bundles (see [@vincent], and also [@pseudobundles] for some details), although not in an entirely similar way (the lack of local trivializations prevents that). Instead, these operations are performed by first carrying out the operation in question to each fibre, defining the total space of the new pseudo-bundle as the union of the resulting diffeological vector spaces (with the obvious fibering over the base), and finally defining the diffeology of this total space as the finest that induces on each fibre its existing diffeology.[^7]
As an example, and also because this instance will be particularly important for us, let us consider dual pseudo-bundles. Let $\pi:V\to X$ be a diffeological vector pseudo-bundle with finite-dimensional fibres. The dual pseudo-bundle of $V$ is $$V^*=\cup_{x\in X}(\pi^{-1}(x))^*,$$ where $(\pi^{-1}(x))^*=L^{\infty}(\pi^{-1}(x),{{\mathbb{R}}})$ is the diffeological dual of the diffeological vector space $\pi^{-1}(x)$, the pseudo-bundle projection $\pi^*$ is given by $\pi^*((\pi^{-1}(x))^*)=\{x\}$ for all $x\in X$, and the diffeology on $V^*$ is characterized as follows: a map $q:{{\mathbb{R}}}^l\supseteq U'\to V^*$ is a plot of $V^*$ if and only if for every plot $p:{{\mathbb{R}}}^m\supseteq U\to V$ of $V$ the evaluation map $(u',u)\mapsto q(u')(p(u))\in{{\mathbb{R}}}$ is smooth for the subset diffeology on its domain of definition $\{(u',u)\,|\,\pi^*(q(u'))=\pi(p(u))\}\subseteq{{\mathbb{R}}}^{l+m}$ and the standard diffeology on ${{\mathbb{R}}}$. The collection of all possible maps $q$ satisfying this property does form a diffeology, equipped with which, $V^*$ becomes a difffeological vector pseudo-bundle, and furthermore, the corresponding subset diffeology on each fibre $(\pi^*)^{-1}(x)$ coincides with the usual (functional) diffeology on $(\pi^{-1}(x))^*$.
Finally, another useful observation (and maybe a peculiarity of diffeology) is that any collection of vector subspaces, one per fibre, in a diffeological vector pseudo-bundle is again a diffeological vector pseudo-bundle (called a **diffeological sub-bundle**), for the subset diffeology. Likewise, any collection of quotients, one of each fibre, is a diffeological vector pseudo-bundle for the quotient diffeology. We call these facts a peculiarity since they go well beyond what happens for the usual smooth vector bundles.
#### Pseudo-metrics on diffeological vector pseudo-bundles
Let $\pi:V\to X$ be a diffeological vector pseudo-bundle with finite-dimensional fibres. A **pseudo-metric** on it is a smooth section of the pseudo-bundle $V^*\otimes V^*$ such that for each $x\in X$ the bilinear form $g(x)$ is a pseudo-metric, in the sense of diffeological vector spaces, on the fibre $\pi^{-1}(x)$. Not all pseudo-bundles admit a pseudo-metric (see [@pseudobundles]; it is not quite clear yet under which conditions a pseudo-bundle admits a pseudo-metric), although if a pseudo-bundle is locally trivial with a finite atlas of local trivializations, the reasoning similar to that in the case of usual smooth vector bundles would allow to conclude its existence on any pseudo-bundle with the above two properties.
Diffeological gluing
--------------------
On the level of the underlying topological[^8] spaces, diffeological gluing (introduced in [@pseudobundles]) is just the usual topological gluing. The result is endowed with a canonical diffeology, called the gluing diffeology. It is the finest diffeology for several properties, and is usually finer than other natural diffeologies on the same space.
### Gluing of spaces, maps, and pseudo-bundles
The basic ingredient in the definition of the diffeological gluing procedure is the operation of gluing of two diffeological spaces, where we must essentially specify which diffeology is assigned to the space obtained by the usual topological gluing. This basic construction is then extended to gluing of smooth maps between diffeological spaces, a particularly important instance of which is the gluing of diffeological vector pseudo-bundles.
#### Diffeological spaces
Let $X_1$ and $X_2$ be two diffeological spaces, and let $f:X_1\supseteq Y\to X_2$ be a map defined on a subset $Y$ of $X_1$ and smooth for the subset diffeology on $Y$. Let $$X_1\cup_f X_2:=\left(X_1\sqcup X_2\right)/_{\sim},$$ where the equivalence relation $\sim$ is given by, $X_1\sqcup X_2\ni x_1\sim x_2\in X_1\sqcup X_2$ if and only if either $x_1=x_2$ or $x_1\in Y$ and $x_2=f(x_1)$. Denote by $\pi:X_1\sqcup X_2\to X_1\cup_f X_2$ the quotient projection, and define the **gluing diffeology** on $X_1\cup_f X_2$ to be the pushforward of the disjoint union diffeology on $X_1\sqcup X_2$ by $\pi$.[^9]
This construction is particularly adapted to endowing piecewise-linear objects with a diffeology (among others). An easiest example is a wedge of two lines, which can be identified with the union of the two coordinate axes in ${{\mathbb{R}}}^2$. It is interesting to notice that the gluing diffeology on this union is finer than the subset diffeology relative to its inclusion into ${{\mathbb{R}}}^2$, as demonstrated by an example due to Watts, see Example 2.67 in [@watts]. (From this, it is easy to extrapolate the existence of similar examples in other dimensions). Indeed, relative to the gluing diffeology the natural inclusions [^10] $$i_1:X_1\setminus Y\hookrightarrow X_1\cup_f X_2\mbox{ and }i_2:X_2\hookrightarrow X_1\cup_f X_2$$ are smooth, and their ranges form a disjoint cover of $X_1\cup_f X_2$. Notice in particular that $X_1$ does not in general inject into $X_1\cup_f X_2$, while $X_2$ always does; in other words, *a priori* the operation of gluing is not symmetric.[^11]
This asymmetry is demonstrated, for instance, by the following (useful in practice) description of plots of the gluing diffeology. Since the latter is a pushforward diffeology, any plot of it lifts to a plot of the covering space $X_1\sqcup X_2$. By the properties of the disjoint union diffeology, this means that if $p:U\to X_1\cup_f X_2$ is a plot and $U$ is connected then it either lifts to a plot $p_1$ of $X_1$ or a plot $p_2$ of $X_2$. By construction of $X_1\cup_f X_2$, we obtain that in the former case $$p(u)=\left\{\begin{array}{ll} i_1(p_1(u)) & \mbox{if }p_1(u)\in X_1\setminus Y,\\
i_2(f(p_1(u))) & \mbox{if }p_1(u)\in Y, \end{array}\right.$$ while in the latter case we simply have $p=i_2\circ p_2$.
#### Gluing of maps
The operation of gluing of diffeological spaces, when performed on the domains and possibly the ranges of some given smooth maps, defines a gluing of these maps, provided the maps themselves satisfy a natural compatibility condition.[^12] More precisely, suppose first that $X_1,X_2,Z$ are diffeological spaces and $\varphi_i:X_i\to Z$ for $i=1,2$ are smooth maps. Let $f:X_1\supseteq Y\to X_2$ be a smooth map, and suppose that $\varphi_2(f(y))=\varphi_1(y)$ for all $y\in Y$ (the maps $\varphi_1$ and $\varphi_2$ are then said to be **$f$-compatible**). Then the map $\varphi_1\cup_f\varphi_2:X_1\cup_f X_2\to Z$ given by $$(\varphi_1\cup_f\varphi_2)(x)=\left\{\begin{array}{ll}
\varphi_1(i_1^{-1}(x)) & \mbox{if }x\in\mbox{Range}(i_1),\\
\varphi_2(i_2^{-1}(x)) & \mbox{if }x\in\mbox{Range}(i_2)
\end{array}\right.$$ is well-defined and smooth. In fact, assigning the map $\varphi_1\cup_f\varphi_2\in C^{\infty}(X_1\cup_f X_2,Z)$ to each pair $(\varphi_1,\varphi_2)$, with $\varphi_i\in C^{\infty}(X_i,Z)$ for $i=1,2$, of $f$-compatible maps yields a map $$C^{\infty}(X_1,Z)\times_{comp}C^{\infty}(X_2,Z)\to C^{\infty}(X_1\cup_f X_2,Z)$$ that is smooth for the functional diffeology on $C^{\infty}(X_1\cup_f X_2,Z)$ and the subset diffeology[^13] on the set of $f$-compatible pairs $C^{\infty}(X_1,Z)\times_{comp}C^{\infty}(X_2,Z)$ (see [@pseudometric-pseudobundle] for details).
Finally, all of this extends to the case of two maps with distinct ranges, that is, $\varphi_1:X_1\to Z_1$ and $\varphi_2:X_2\to Z_2$, with appropriate gluings of $X_1$ to $X_2$ and $Z_1$ to $Z_2$. Specifically, let again $f:X_1\supseteq Y\to X_2$ be smooth, but consider also a smooth $g:Z_1\supseteq\varphi_1(Y)\to Z_2$; assume that $\varphi_2(f(y))=g(\varphi_1(y))$ for all $y\in Y$ (the maps $\varphi_1$ and $\varphi_2$ are said to be **$(f,g)$-compatible**). Notice that the counterparts of $i_1$ and $i_2$ for the space $Z_1\cup_g Z_2$ are the natural inclusion maps $j_1:Z_1\setminus\varphi_1(Y)\hookrightarrow Z_1\cup_g Z_2$ and $j_2:Z_2\to Z_1\cup_g Z_2$. We define the map $\varphi_1\cup_{(f,g)}\varphi_2$ by setting $$(\varphi_1\cup_{(f,g)}\varphi_2)(x)=\left\{\begin{array}{ll}
j_1(\varphi_1(i_1^{-1}(x))) & \mbox{if }x\in\mbox{Range}(i_1),\\
j_2(\varphi_2(i_2^{-1}(x))) & \mbox{if }x\in\mbox{Range}(i_2).
\end{array}\right.$$ All the analogous statements, in particular, the smoothness of the thus-defined map $C^{\infty}(X_1,Z_1)\times_{comp}C^{\infty}(X_2,Z_2)\to C^{\infty}(X_1\cup_f X_2,Z_1\cup_g Z_2)$, continue to hold (see [@pseudometric-pseudobundle]).
#### Pseudo-bundles
Let us now turn to gluing of two pseudo-bundles; this is of course a specific instance of gluing of $(f,g)$-compatible maps. We specifically indicate it in order to to fix notation and standard terminology, and also to give conditions under which the result of gluing is again a diffeological vector pseudo-bundle.
Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two such pseudo-bundles, let $f:X_1\supseteq Y\to X_2$ be a smooth map, and let $\tilde{f}:V_1\supseteq\pi_1^{-1}(Y)\to V_2$ be any smooth lift of $f$ that is linear on each fibre (where it is defined). The gluing of these pseudo-bundles consists in the already-defined operations of gluing $V_1$ to $V_2$ along $\tilde{f}$, gluing $X_1$ to $X_2$ along $f$, and $\pi_1$ to $\pi_2$ along $(\tilde{f},f)$. It is easy to check (see [@pseudobundles]) that the resulting map $\pi_1\cup_{(\tilde{f},f)}\pi_2:V_1\cup_{\tilde{f}}V_2\to X_1\cup_f X_2$ is a diffeological vector pseudo-bundle for the gluing diffeologies on $V_1\cup_{\tilde{f}}V_2$ and $X_1\cup_f X_2$; in particular, the vector space structure on its fibres is inherited from either $V_1$ or $V_2$ (more precisely, it is inherited from $V_1$ on fibres over the points in $i_1(X_1\setminus Y)$, and from $V_2$ on fibres over the points in $i_2(X_2)$).
#### Standard notation for gluing of pseudo-bundles
We now fix some standard notation that applies to pseudo-bundles specifically. We have already described the standard inclusions $$i_1:X_1\setminus Y\hookrightarrow X_1\cup_f X_2,\,\,\,i_2:X_2\hookrightarrow X_1\cup_f X_2,$$ $$j_1:V_1\setminus\pi_1^{-1}(Y)\hookrightarrow V_1\cup_{\tilde{f}}V_2,\,\,\,j_2:V_2\hookrightarrow V_1\cup_{\tilde{f}}V_2.$$ When dealing with more than one gluing at a time, we will needed a more complicated notation, which is as follows. Let $\chi_1\cup_{(\tilde{h},h)}\chi_2:W_1\cup_{\tilde{h}}W_2\to Z_1\cup_h Z_2$ be any pseudo-bundle obtained by gluing; denote by $Y'\subset Z_1$ the domain of definition of $h$. Then the counterparts of $i_1$ and $i_2$ will be denoted by[^14] $$i_1^{Z_1}:Z_1\setminus Y'\hookrightarrow Z_1\cup_h Z_2\mbox{ and }i_2^{Z_2}:Z_2\hookrightarrow Z_1\cup_h Z_2\mbox{ for }Z_1\cup_h Z_2$$ $$j_1^{W_1}:W_1\setminus\chi_1^{-1}(Y')\hookrightarrow W_1\cup_{\tilde{h}}W_2\mbox{ and }j_2^{W_2}:W_2\hookrightarrow W_1\cup_{\tilde{h}}W_2\mbox{ for }W_1\cup_{\tilde{h}}W_2.$$ Obviously, $i_1^{W_1}$ and $j_1^{W_1}$ would mean the same thing, and the same goes for $i_2^{W_2}$ and $j_2^{W_2}$; however, we use the different letters so that there always be a clear distinction between the base space and the total space. Finally, since the base space will be the same for all our pseudo-bundles, and so we will usually use the abbreviated notation $i_1,i_2$ for it.
#### The switch map
As we mentioned many times already, the operation of gluing for diffeological spaces is asymmetric. However, if we assume that gluing map $f$ is a diffeomorphism with its image then obviously, we can use its inverse to perform the gluing in the reverse order, with the two results, $X_1\cup_f X_2$ and $X_2\cup_{f^{-1}}X_1$, being canonically diffeomorphic via the so-called **switch map** $$\varphi_{X_1\leftrightarrow X_2}:X_1\cup_f X_2\to X_2\cup_{f^{-1}}X_1.$$ Using the notation just introduced, this map can be described by $$\left\{\begin{array}{ll}
\varphi_{X_1\leftrightarrow X_2}(i_1^{X_1}(x))=i_2^{X_1}(x) & \mbox{for }x\in X_1\setminus Y,\\
\varphi_{X_1\leftrightarrow X_2}(i_2^{X_2}(f(x)))=i_2^{X_1}(x) & \mbox{for }x\in Y,\\
\varphi_{X_1\leftrightarrow X_2}(i_2^{X_2}(x))=i_1^{X_2}(x) & \mbox{for }x\in X_2\setminus f(Y).\end{array}\right.$$ This is well-defined, not only because the maps $i_1^{X_1}$ and $i_2^{X_2}$ are injective with disjoint ranges covering $X_1\cup_f X_2$, but also because $f$ is a diffeomorphism with its image.
### Gluing and operations
Diffeological gluing of pseudo-bundles is relatively well-behaved with respect to the usual operations on vector bundles. More precisely, it commutes with the direct sum and the tensor product, while he situation is somewhat more complicated for the dual pseudo-bundles, see [@pseudobundles] (the facts needed are recalled below).
#### Direct sum
Gluing of diffeological vector pseudo-bundles commutes with the direct sum in the following sense. Given a gluing along $(\tilde{f},f)$ of a pseudo-bundle $\pi_1:V_1\to X_1$ to a pseudo-bundle $\pi_2:V_2\to X_2$, as well as a gluing along $(\tilde{f}',f)$ of a pseudo-bundle $\pi_1':V_1'\to X_1$ to a pseudo-bundle $\pi_2':V_2'\to X_2$, there are two natural pseudo-bundles that can be formed from these by applying the operations of gluing and direct sum. These are the pseudo-bundles $$(\pi_1\cup_{(\tilde{f},f)}\pi_2)\oplus(\pi_1'\cup_{(\tilde{f}',f)}\pi_2'):(V_1\cup_{\tilde{f}}V_2)\oplus(V_1'\cup_{\tilde{f}'}V_2')\to X_1\cup_f X_2\mbox{ and }$$ $$(\pi_1\oplus\pi_1')\cup_{(\tilde{f}\oplus\tilde{f}',f)}(\pi_2\oplus\pi_2'):(V_1\oplus V_1')\cup_{\tilde{f}\oplus\tilde{f}'}(V_2\oplus V_2')\to X_1\cup_f X_2;$$ they are diffeomorphic as pseudo-bundles, that is, there exists a fibrewise linear diffeomorphism $$\Phi_{\cup,\oplus}:(V_1\cup_{\tilde{f}}V_2)\oplus(V_1'\cup_{\tilde{f}'}V_2')\to(V_1\oplus V_1')\cup_{\tilde{f}\oplus\tilde{f}'}(V_2\oplus V_2')$$ (see below) that covers the identity map on the base $X_1\cup_f X_2$.
#### Tensor product
What has just been said about the direct sum, applies equally well to the tensor product. Specifically, the two possible pseudo-bundles are $$(\pi_1\cup_{(\tilde{f},f)}\pi_2)\otimes(\pi_1'\cup_{(\tilde{f}',f)}\pi_2'):(V_1\cup_{\tilde{f}}V_2)\otimes(V_1'\cup_{\tilde{f}'}V_2')\to X_1\cup_f X_2\mbox{ and }$$ $$(\pi_1\otimes\pi_1')\cup_{(\tilde{f}\otimes\tilde{f}',f)}(\pi_2\otimes\pi_2'):(V_1\otimes V_1')\cup_{\tilde{f}\otimes\tilde{f}'}(V_2\otimes V_2')\to X_1\cup_f X_2,$$ and again, they are pseudo-bundle-diffeomorphic via $$\Phi_{\cup,\otimes}:(V_1\cup_{\tilde{f}}V_2)\otimes(V_1'\cup_{\tilde{f}'}V_2')\to(V_1\otimes V_1')\cup_{\tilde{f}\otimes\tilde{f}'}(V_2\otimes V_2')$$ covering the identity on $X_1\cup_f X_2$.
#### The dual pseudo-bundle
The case of dual pseudo-bundles is substantially different. For one thing, to even make sense of the commutativity question, we must assume that $f$ is invertible (which in general it does not have to be). Moreover, even with this assumption, in general the operation of gluing does *not* commute with that of taking duals. The reason of this is easy to explain. Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two diffeological vector pseudo-bundles, and let $(\tilde{f},f)$ be a gluing between them; consider the pseudo-bundle $\pi_1\cup_{(\tilde{f},f)}\pi_2:V_1\cup_{\tilde{f}}V_2\to X_1\cup_f X_2$ and the corresponding dual pseudo-bundle $$(\pi_1\cup_{(\tilde{f},f)}\pi_2)^*:(V_1\cup_{\tilde{f}}V_2)^*\to X_1\cup_f X_2;$$ compare it with the result of the induced gluing (performed along the pair $(\tilde{f}^*,f)$) of $\pi_2^*:V_2^*\to X_2$ to $\pi_1^*:V_1^*\to X_1$, that is, the pseudo-bundle $$\pi_2^*\cup_{(\tilde{f}^*,f)}\pi_1^*:V_2^*\cup_{\tilde{f}^*}V_1^*\to X_2\cup_{f^{-1}}X_1.$$ It then follows from the construction itself that for any $y\in Y$ (the domain of gluing) we have $$((\pi_1\cup_{(\tilde{f},f)}\pi_2)^*)^{-1}(i_2^{X_2}(f(y)))\cong(\pi_2^{-1}(f(y)))^*\mbox{ and }(\pi_2^*\cup_{(\tilde{f}^*,f)}\pi_1^*)^{-1}(i_2^{X_1}(y))\cong(\pi_1^{-1}(y))^*;$$ since $i_2^{X_2}(f(y))$ and $i_2^{X_1}(y)$ are related by the switch map, for the two pseudo-bundles to be diffeomorphic in a natural way[^15] the two vector spaces $(\pi_2^{-1}(f(y)))^*$ and $(\pi_1^{-1}(y))^*$ must be diffeomorphic, and *a priori* they are not.[^16]
Thus, we obtain the necessary condition (for the pseudo-bundles $(V_1\cup_{\tilde{f}}V_2)^*$ and $V_2^*\cup_{\tilde{f}^*}V_1^*$ to be diffeomorphic), which is that $(\pi_2^{-1}(f(y)))^*\cong(\pi_1^{-1}(y))^*$ for all $y\in Y$. We do note right away that this condition may not be sufficient, in the sense that two pseudo-bundles over the same base may have all respective fibres diffeomorphic without being diffeomorphic themselves (this can be illustrated by the standard example of open annulus and open Möbius strip, both of which, equipped with the standard diffeology[^17] can be seen as pseudo-bundles over the circle). We will leave it at that for now, returning to the issue of the gluing-dual commutativity later in the paper.
### The commutativity diffeomorphisms
We now say more about the commutativity diffeomorphisms mentioned in the previous section.[^18]
#### The diffeomorphism $\Phi_{\cup,\oplus}$
We have already mentioned the existence of this diffeomorphism, which is a pseudo-bundle map $$\Phi_{\cup,\oplus}:(V_1\cup_{\tilde{f}}V_2)\oplus(V_1'\cup_{\tilde{f}'}V_2')\to(V_1\oplus V_1')\cup_{\tilde{f}\oplus\tilde{f}'}(V_2\oplus V_2')$$ that covers the identity map on $X_1\cup_f X_2$. We now add that this map can be described (in fact, fully defined) by the following identities: $$\Phi_{\cup,\oplus}\circ(j_1^{V_1}\oplus j_1^{V_1'})=j_1^{V_1\oplus V_1'}\,\,\mbox{ and }\,\, \Phi_{\cup,\oplus}\circ(j_2^{V_2}\oplus j_2^{V_2'})=j_2^{V_2\oplus V_2'}.$$
#### The diffeomorphism $\Phi_{\cup,\otimes}$
Once again, the case of the tensor product is very similar to that of the direct sum. The already-mentioned diffeomorphism $$\Phi_{\cup,\otimes}:(V_1\cup_{\tilde{f}}V_2)\otimes(V_1'\cup_{\tilde{f}'}V_2')\to(V_1\otimes V_1')\cup_{\tilde{f}\otimes\tilde{f}'}(V_2\otimes V_2')$$ is uniquely determined by the identities $$\Phi_{\cup,\otimes}\circ(j_1^{V_1}\otimes j_1^{V_1'})=j_1^{V_1\otimes V_1'}\,\,\mbox{ and }\,\, \Phi_{\cup,\otimes}\circ(j_2^{V_2}\otimes j_2^{V_2'})=j_2^{V_2\otimes V_2'};$$ notice that these, themselves, suffice to ensure that it covers the identity map on $X_1\cup_f X_2$.
#### The gluing-dual commutativity conditions, and diffeomorphism $\Phi_{\cup,*}$
We have already described the situation regarding the gluing-dual commutativity, first of all the fact, that it is far from being always present. At this moment we concentrate on what it actually means for the gluing to commute with taking duals (once again leaving aside the question when this does happen). Specifically, we say that the **gluing-dual commutativity condition** holds, if there exists a diffeomorphism $$\Phi_{\cup,*}:(V_1\cup_{\tilde{f}}V_2)^*\to V_2^*\cup_{\tilde{f}^*}V_1^*$$ that covers the switch map, that is, $$(\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*)\circ\Phi_{\cup,*}=\varphi_{X_1\leftrightarrow X_2}\circ(\pi_1\cup_{(\tilde{f},f)}\pi_2)^*,$$ and such that the following are true: $$\left\{\begin{array}{ll}
\Phi_{\cup,*}\circ((j_1^{V_1})^*)^{-1}=j_2^{V_1^*} & \mbox{on }(\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*)^{-1}(i_2^{X_1}(X_1\setminus Y)),\\
\Phi_{\cup,*}\circ((j_2^{V_2})^*)^{-1}=j_2^{V_1^*}\circ\tilde{f}^* & \mbox{on }(\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*)^{-1}(i_2^{X_1}(Y)),\\
\Phi_{\cup,*}\circ((j_2^{V_2})^*)^{-1}=j_1^{V_2^*} & \mbox{on }(\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*)^{-1}(i_1^{X_2}(X_2\setminus f(Y))).\end{array}\right.$$
### Gluing and pseudo-metrics
The behavior of pseudo-metrics under gluing depends significantly on whether the gluing-dual commutativity condition is satisfied. More precisely, if we glue together two pseudo-bundles carrying a pseudo-metric each, then under a certain natural compatibility condition (see below) for these pseudo-metrics, the new pseudo-bundle carries a pseudo-metric as well; but how the latter is constructed depends on the existence of the commutativity diffeomorphism $\Phi_{\cup,*}$.
#### The compatibility notion for pseudo-metrics
Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two finite-dimensional diffeological vector pseudo-bundles, endowed with pseudo-metrics $g_1$ and $g_2$ respectively, and let $(\tilde{f},f)$ be a pair of smooth maps that defines gluing of $X_1$ to $X_2$; let $Y\subseteq X_1$ be the domain of definition of $f$. The pseudo-metrics $g_1$ and $g_2$ are said to be **compatible with (the gluing along) the pair $(\tilde{f},f)$** if for all $y\in Y$ and for all $v,w\in\pi_1^{-1}(y)$ the following is true: $$g_1(y)(v,w)=g_2(f(y))(\tilde{f}(v),\tilde{f}(w)).$$ If $f$ is invertible, this means that $g_2$ and $g_1$ are $(f^{-1},\tilde{f}^*\otimes\tilde{f}^*)$-compatible as smooth maps $X_2\to V_2^*\otimes V_2^*$ and $X_1\to V_1^*\otimes V_1^*$ respectively.
As can be expected,[^19] the existence of compatible pseudo-metrics on two pseudo-bundles imposes substantial restrictions on their fibres over the domain of gluing. Later in the paper we will make precise statements to this effect.
#### The induced pseudo-metric in the presence of $\Phi_{\cup,*}$
If we assume that the gluing-dual commutativity condition is satisfied, this implies also that $f$ is invertible (with smooth inverse). In this case we can use the map $$g_2\cup_{(f^{-1},\tilde{f}^*\otimes\tilde{f}^*)}g_1:X_2\cup_{f^{-1}}X_1\to(V_2^*\otimes V_2^*)\cup_{\tilde{f}^*\otimes\tilde{f}^*}(V_1^*\otimes V_1^*)$$ to construct a pseudo-metric on $V_1\cup_{\tilde{f}}V_2$ by taking the following composition of it with the switch map and the commutativity diffeomorphisms: $$\tilde{g}=\left(\Phi_{\cup,*}^{-1}\otimes\Phi_{\cup,*}^{-1}\right)\circ\Phi_{\otimes,\cup}\circ(g_2\cup_{(f^{-1},\tilde{f}^*\otimes\tilde{f}^*)}g_1)\circ\varphi_{X_1\leftrightarrow X_2},$$ where $\varphi_{X_1\leftrightarrow X_2}$ is the switch map, $\Phi_{\cup,*}$ (of which we need the inverse) is the just-seen gluing-dual commutativity diffeomorphism, while $\Phi_{\otimes,\cup}:(V_2^*\otimes V_2^*)\cup_{\tilde{f}^*\otimes\tilde{f}^*}(V_1^*\otimes V_1^*)\to(V_2^*\cup_{\tilde{f}^*}V_1^*)\otimes(V_2^*\cup_{\tilde{f}^*}V_1^*)$ is the appropriate version of the tensor product-gluing commutativity diffeomorphism.
#### Constructing a pseudo-metric on $V_1\cup_{\tilde{f}}V_2$ when $\Phi_{\cup,*}$ does not exist
Although we will mostly deal with the cases where the gluing-dual commutativity condition is present (and so the above definition of the pseudo-metric $\tilde{g}$ on $V_1\cup_{\tilde{f}}V_2$ is sufficient), we briefly mention that even if such condition does not hold, the flexibility of diffeology allows for a direct construction of a pseudo-metric on $V_1\cup_{\tilde{f}}V_2$. This construction uses the fact that each fibre of $V_1\cup_{\tilde{f}}V_2$ is naturally identified with one of either $V_1$ or $V_2$; accordingly, $\tilde{g}$ can be defined to coincide with either $g_1$ or $g_2$ on each fibre individually. The surprising fact is that $\tilde{g}$ coming from this construction is still diffeologically smooth across the fibres; see [@pseudometric-pseudobundle] for details.
The pseudo-bundle of smooth linear maps
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The last more-or-less standard construction that we need is that of **pseudo-bundle of smooth linear maps**. Let $\pi_1:V_1\to X$ and $\pi_2:V_2\to X$ be two finite-dimensional diffeological vector pseudo-bundles with the same space $X$. For every $x\in X$ the space $L^{\infty}(\pi_1^{-1}(x),\pi_2^{-1}(x))$ of smooth linear maps $\pi_1^{-1}(x)\to\pi_2^{-1}(x)$ is a (finite-dimensional) diffeological vector space for the functional diffeology. The union $$\mathcal{L}(V_1,V_2)=\cup_{x\in X}L^{\infty}(\pi_1^{-1}(x),\pi_2^{-1}(x))$$ of all these spaces has the obvious projection (denoted $\pi^L$) to $X$, and the pre-image of each point under this projection has vector space structure. It becomes a diffeological vector pseudo-bundle when endowed with the **pseudo-bundle functional diffeology**, that is defined as the finest diffeology containing all maps $p:U\to\mathcal{L}(V_1,V_2)$, with $U\subseteq{{\mathbb{R}}}^m$ an arbitrary domain, that possess the following property: for every plot $q:{{\mathbb{R}}}^{m'}\supseteq U'\to V_1$ of $V_1$ the corresponding evaluation map $(u,u')\mapsto p(u)(q(u'))\in V_2$ defined on $Y'=\{(u,u')\,|\,\pi^L(p(u))=\pi_1(q(u'))\}\subset U\times U'$ is smooth for the subset diffeology of $Y'$. This is the type of object where the Clifford actions live.
The pseudo-bundles of Clifford algebras and Clifford modules
------------------------------------------------------------
For diffeological pseudo-bundles, these have already been described in the abstract setting (see [@clifford-alg]). We briefly summarize the main points that appear therein, noting that the main conclusions do not differ from the usual case, or are as expected anyhow.
### The pseudo-bundle $cl(V,g)$
Let $\pi:V\to X$ be a finite-dimensional diffeological pseudo-bundle endowed with a pseudo-metric $g$. The construction of the corresponding pseudo-bundle $cl(V,g)$ of Clifford algebras is the immediate one, since all the operations involved have already been described. Specifically, the **pseudo-bundle of Clifford algebras $\pi^{Cl}:cl(V,g)\to X$** is given by $$cl(V,g):=\cup_{x\in X}cl(\pi^{-1}(x),g(x))$$ and is endowed with the quotient diffeology coming from **pseudo-bundle of tensor algebras** $\pi^{T(V)}:T(V)\to X$. The latter pseudo-bundle has total space given by $T(V):=\cup_{x\in X}T(\pi^{-1}(x))$, where $T(\pi^{-1}(x)):=\bigoplus_{r}(\pi^{-1}(x))^{\otimes r}$ is the usual tensor algebra of the diffeological vector space $\pi^{-1}(x)$ (in particular, it is endowed with the vector space direct sum diffeology relative to the tensor product diffeology on each factor).
We will not make much use of the algebra structure on this pseudo-bundle, and so will actually consider all the direct sums involved to be finite (limited by the maximum of the dimensions of fibres of $V$ in question — this includes the assumption that such maximum exists), thus considering, instead of the whole $T(V)$ its finite-dimensional sub-bundle $T_{\leqslant n}(V)$, with fibre at $x$ the space $T_{\leqslant n}(\pi^{-1}(x))$ consisting of all tensors in $T(\pi^{-1}(x))$ of degree at most $n$. These fibres are not algebras, but of course each of them is a vector subspace of the corresponding fibre of $T(V)$.
Recall that the subset diffeology on each fibre of $T(V)$ is that of the tensor algebra[^20] of the individual fibre $\pi^{-1}(x)$. In each such fibre we choose the subspace $W_x$ that is the kernel of the universal map $T(\pi^{-1}(x))\to{C \kern -0.1em \ell}(\pi^{-1}(x),g(x))$. Then, as is generally the case, $W=\cup_{x\in X}W_x\subset T(V)$ endowed with the subset diffeology relative to this inclusion is a sub-bundle of $T(V)$. The fibre of the corresponding quotient pseudo-bundle at any given point $x\in X$ is $cl(\pi^{-1}(x),g(x))$, and the quotient diffeology on the fibre is that of the Clifford algebra over the vector space $\pi^{-1}(x)$. This is exactly the diffeology that we endow ${C \kern -0.1em \ell}(V,g)$ with.
### The pseudo-bundle ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ as the result of a gluing
The main result, that we immediately state and that appears in [@clifford-alg], is the following one.
\[gluing:clifford:cited:thm\] Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two diffeological vector pseudo-bundles, let $(\tilde{f},f)$ be two maps defining a gluing between these two pseudo-bundles, both of which are diffeomorphisms, and let $g_1$ and $g_2$ be pseudo-metrics on $V_1$ and $V_2$ respectively, compatible with this gluing. Let $\tilde{g}$ be the pseudo-metric on $V_1\cup_{\tilde{f}}V_2$ induced by $g_1$ and $g_2$. Then there exists a map $\tilde{F}^{{C \kern -0.1em \ell}}$ defining a gluing of the pseudo-bundles ${C \kern -0.1em \ell}(V_1,g_1)$ and ${C \kern -0.1em \ell}(V_2,g_2)$, and a diffeomorphism $\Phi^{{C \kern -0.1em \ell}}$ between the pseudo-bundles ${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$ and ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ covering the identity on $X_1\cup_f X_2$.
Let us briefly describe the maps $\tilde{F}^{{C \kern -0.1em \ell}}$ and $\Phi^{{C \kern -0.1em \ell}}$. The construction of $\tilde{F}^{{C \kern -0.1em \ell}}$ is the immediately obvious one. It is defined on each fibre over a point $y\in Y$ as the map ${C \kern -0.1em \ell}(\pi_1^{-1}(y),g_1|_{\pi_1^{-1}(y)})\to{C \kern -0.1em \ell}(\pi_2^{-1}(f(y)),g_2|_{\pi_2^{-1}(f(y))})$ induced by $\tilde{f}$ via the universal property of Clifford algebras. In practice, this means that on each fibre $\tilde{F}^{{C \kern -0.1em \ell}}$ is linear and multiplicative (with respect to the tensor product), so if $v_1\otimes\ldots\otimes v_k$ is a representative of an equivalence class in ${C \kern -0.1em \ell}(\pi_1^{-1}(y),g_1|_{\pi_1^{-1}(y)})$ (viewed as the appropriate quotient of $T(\pi_1^{-1}(y))$) then by this definition $$\tilde{F}^{{C \kern -0.1em \ell}}(v_1\otimes\ldots\otimes v_k)=\tilde{F}^{Cl}(v_1)\otimes\ldots\otimes\tilde{F}^{{C \kern -0.1em \ell}}(v_k)=\tilde{f}(v_1)\otimes\ldots\otimes\tilde{f}(v_k).$$ That this is well-defined as a map ${C \kern -0.1em \ell}(\pi_1^{-1}(y),g_1|_{\pi_1^{-1}(y)})\to{C \kern -0.1em \ell}(\pi_2^{-1}(f(y)),g_2|_{\pi_2^{-1}(f(y))})$ follows from the compatibility of pseudo-metrics $g_1$ and $g_2$. Indeed, if $v,w\in\pi_1^{-1}(y)$ then $$\tilde{F}^{{C \kern -0.1em \ell}}(v\otimes w+w\otimes v+2g_1(y)(v,w))=\tilde{f}(v)\otimes\tilde{f}(w)+\tilde{f}(w)\otimes \tilde{f}(v)+2g_1(y)(v,w)$$ by the above formula, and $2g_1(y)(v,w)=2g_2(f(y))(\tilde{f}(v),\tilde{f}(w))$ by the compatibility. Thus, $\tilde{F}^{{C \kern -0.1em \ell}}$ preserves the defining relation for Clifford algebras, so indeed it is well-defined (on each fibre; hence on the whole pseudo-bundle).
The diffeomorphism $\Phi^{{C \kern -0.1em \ell}}$, which we specify to be ${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)\to{C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$, is then the natural identification; namely, by definitions of the gluing operation and that of the induced pseudo-metric $\tilde{g}$, over a point of form $x=i_1^{X_1}(x_1)$ both the fibre of ${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$ and that of ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ are naturally identified with ${C \kern -0.1em \ell}(\pi_1^{-1}(x),g_1|_{\pi_1^{-1}(x)})$, while over any point of form $x=i_2^{X_2}(x_2)$ they are identified with ${C \kern -0.1em \ell}(\pi_2^{-1}(x),g_2|_{\pi_2^{-1}(x)})$.
### Gluing of pseudo-bundles of Clifford modules
A statement similar to that of the Theorem cited in the previous section can also be obtained for given Clifford modules over the algebras ${C \kern -0.1em \ell}(V_1,g_1)$ and ${C \kern -0.1em \ell}(V_2,g_2)$. This requires some additional assumptions on these modules, and the appropriate notion of the compatibility of the actions.
#### The two Clifford modules
Let $\pi_1:V_1\to X_1$, $\pi_2:V_2\to X_2$, $(\tilde{f},f)$, $g_1$, and $g_2$ be as in Theorem \[gluing:clifford:cited:thm\]. Recall that this yields the following pseudo-bundles, $\pi_1^{{C \kern -0.1em \ell}}:{C \kern -0.1em \ell}(V_1,g_1)\to X_1$, $\pi_2^{{C \kern -0.1em \ell}}:{C \kern -0.1em \ell}(V_2,g_2)\to X_2$, and $\pi_1^{{C \kern -0.1em \ell}}\cup_{(\tilde{F}^{{C \kern -0.1em \ell}},f)}\pi_2^{{C \kern -0.1em \ell}}: {C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)={C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})\to X_1\cup_f X_2$.
Now suppose that we are given two pseudo-bundles of Clifford modules, $\chi_1:E_1\to X_1$ and $\chi_2:E_2\to X_2$ (over ${C \kern -0.1em \ell}(V_1,g_1)$ and ${C \kern -0.1em \ell}(V_2,g_2)$ respectively), that is, there is a smooth pseudo-bundle map $c_i:cl(V_i,g_i)\to\mathcal{L}(E_i,E_i)$ that covers the identity on the bases. Suppose further that there is a smooth fibrewise linear map $\tilde{f}':\chi_1^{-1}(Y)\to\chi_2^{-1}(f(Y))$ that covers $f$. We describe the pseudo-bundle $E_1\cup_{\tilde{f}'}E_2$ as a Clifford module over ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$, with respect to an action induced by $c_1$ and $c_2$.
#### Compatibility of $c_1$ and $c_2$
Similar to how it occurs for smooth maps, there is always an action of ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ on $E_1\cup_{\tilde{f}'}E_2$ induced by $c_1$ and $c_2$, it is smooth on each fibre but in general, it might not be smooth across the fibres. For it to be so, we need a notion of compatibility for Clifford actions, which is as follows.
The actions $c_1$ and $c_2$ are **compatible** (with respect to $\tilde{F}^{{C \kern -0.1em \ell}}$ and $\tilde{f}'$) if for all $y\in Y$, for all $v\in(\pi_1^{{C \kern -0.1em \ell}})^{-1}(y)$, and for all $e_1\in\chi_1^{-1}(y)$ we have $$\tilde{f}'(c_1(v)(e_1))=c_2(\tilde{F}^{{C \kern -0.1em \ell}}(v))(\tilde{f}'(e_1)).$$
We note that the compatibility of $c_1$ and $c_2$ as it has been just defined, does not automatically translate into their $(\tilde{F}^{{C \kern -0.1em \ell}},\tilde{f}')$ compatibility as smooth maps in ${C \kern -0.1em \ell}(V_i,g_i)\to\mathcal{L}(E_i,E_i)$; see [@clifford-alg] for a discussion on this.
#### The induced action
Assuming now that the two given actions $c_1$ and $c_2$ are compatible in the sense just stated, we can define an induced action on $E_1\cup_{\tilde{f}'}E_2$, that is, a smooth homomorphism $$c:{C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})\to\mathcal{L}(E_1\cup_{\tilde{f}'}E_2,E_1\cup_{\tilde{f}'}E_2).$$ Using the already-mentioned identification, via the diffeomorphism $\Phi^{{C \kern -0.1em \ell}}$, of ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ with ${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$, the action $c$ can be described by defining first $$c'(v)(e)=\left\{\begin{array}{ll}
j_1^{E_1}\left(c_1((j_1^{{C \kern -0.1em \ell}(V_1,g_1)})^{-1}(v))((j_1^{E_1})^{-1}(e))\right) & \mbox{if }v\in\mbox{Im}(j_1^{{C \kern -0.1em \ell}(V_1,g_1)})\Rightarrow e\in\mbox{Im}(j_1^{E_1}), \\
j_2^{E_2}\left(c_2((j_2^{cl(V_2,g_2)})^{-1}(v))((j_2^{E_2})^{-1}(e))\right) & \mbox{if }v\in\mbox{Im}(j_2^{cl(V_2,g_2)})\Rightarrow e\in\mbox{Im}(j_2^{E_2}). \end{array}\right.$$ Since the images of the inductions $j_1^{{C \kern -0.1em \ell}(V_1,g_1)}$ and $j_2^{{C \kern -0.1em \ell}(V_2,g_2)}$ are disjoint and cover ${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$, and those of $j_1^{E_1}$ and $j_2^{E_2}$ cover $E_1\cup_{\tilde{f}'}E_2$ (and are disjoint as well), this is a well-defined fibrewise action of the former on the latter. From the formal point of view, we must also pre-compose it with the inverse of $\Phi^{{C \kern -0.1em \ell}}$, to obtain an action $$c=c'\circ(\Phi^{{C \kern -0.1em \ell}})^{-1}:{C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})\to\mathcal{L}(E_1\cup_{\tilde{f}'}E_2,E_1\cup_{\tilde{f}'}E_2).$$ Then, the following is true (see [@clifford-alg]).
The action $c$ is smooth as a map ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})\to\mathcal{L}(E_1\cup_{\tilde{f}'}E_2,E_1\cup_{\tilde{f}'}E_2)$.
The induced pseudo-metrics on dual pseudo-bundles
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Let $\pi:V\to X$ be a locally trivial finite-dimensional diffeological vector pseudo-bundle endowed with a pseudo-metric $g$; then its dual pseudo-bundle $\pi^*:V^*\to X$ admits an induced pseudo-metric $g^*$ (see [@pseudometric-pseudobundle]; we also recall the definition below). In this section we consider gluings of two pseudo-bundles endowed with compatible pseudo-metrics, and the corresponding dual constructions; starting from indicating which restrictions are imposed in the initial pseudo-bundles by the existence of compatible pseudo-metrics on them, we proceed to discuss when the induced pseudo-metrics on the dual pseudo-bundles are compatible in their turn, and finally, how it relates to the gluing-dual commutativity condition.
The induced pseudo-metric on $V^*$
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Let $\pi:V\to X$ be a locally trivial finite-dimensional diffeological vector pseudo-bundle, and let $g$ be a pseudo-metric on it. Under the assumption of local triviality of $V$,[^21] the dual pseudo-bundle $V^*$ carries an induced pseudo-metric $g^*$, which is obtained by what can be considered a diffeological counterpart of the usual natural pairing.
Specifically, consider the pseudo-bundle map $\Phi:V\to V^*$ defined by $$\Phi(v)=g(\pi(v))(v,\cdot)\mbox{ for all }v\in V.$$ It is not hard to show (see [@pseudometric] for the case of a single diffeological vector space, and then [@pseudometric-pseudobundle] for the case of pseudo-bundles) that $\Phi$ is surjective, smooth, and linear on each fibre.
The **induced pseudo-metric $g^*$** is given by the following equality: $$g^*(x)(\Phi(v),\Phi(w)):=g(x)(v,w)\mbox{ for all }x\in X \mbox{ and for all }v,w\in V\mbox{ such that }\pi(v)=\pi(w)=x.$$ This is well-defined, because whenever $\Phi(v)=\Phi(v')$ (which obviously can occur only for $v,v'$ belonging to the same fibre), the vectors $v$ and $v'$ differ by an element of the isotropic subspace of the fibre to which they belong. Furthermore, since each fibre of the dual pseudo-bundle (in the finite-dimensional case) carries the standard diffeology (see [@pseudometric]), $g^*(x)$ is always a scalar product. Notice that without the requirement of local triviality, we cannot guarantee that $g^*$ is indeed a pseudo-metric, and more precisely, that it is smooth (the map $\Phi$ always has a right inverse, which *a priori* may not be smooth).
Let $\pi:V\to X$ be a finite-dimensional diffeological vector pseudo-bundle endowed with a pseudo-metric $g$, let $\pi^*:V^*\to X$ be the dual pseudo-bundle, and let $g^*$ be the induced pseudo-metric. Then for all $x\in X$ the symmetric bilinear form $g^*(x)$ on $(\pi^*)^{-1}(x)$ is non-degenerate.
Existence of compatible pseudo-metrics: the case of a diffeological vector space
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Before treating various issues regarding the induced pseudo-metrics, it makes sense to consider in more detail what the compatibility of two pseudo-metrics means. We do so starting with the case of just diffeological vector spaces (we consider the duals of vector spaces in the section that immediately follows, and pseudo-bundles in the one after that).
Let $V$ and $W$ be finite-dimensional diffeological vector spaces, let $g_V$ be a pseudo-metric on $V$, and let $g_W$ be a pseudo-metric on $W$. We assume that we are given a smooth linear map $f:V\to W$, with respect to which $g_V$ and $g_W$ are compatible, $g_V(v_1,v_2)=g_W(f(v_1),f(v_2))$. We show that, quite similarly to usual vector spaces and scalar products, there are pairs of diffeological ones such that no pair of pseudo-metrics is compatible with respect to any smooth linear $f$. The similarity that we are referring to has to do with the fact that the compatibility of two pseudo-metrics with respect to $f$ essentially amounts to $f$ being a diffeological analogue of an isometry onto a subspace. As is well-known, between two usual vector spaces such isometry may not exist (it is necessary that the dimension of the domain space must be less or equal to that of the target space), and something similar happens for diffeological vector spaces; and then further conditions are added in terms of their diffeological structures.
### The characteristic subspaces of $V$ and $W$
Assuming that two given pseudo-metrics $g_V$ and $g_W$ on diffeological vector spaces $V$ and $W$ respectively are compatible with a given $f:V\to W$ has several implications for the diffeological structures of $V$ and $W$; describing these requires the following notion.
Given a pseudo-metric $g$ on a finite-dimensional diffeological vector space $V$, the subspace $V_0\leqslant V$ generated by all the eigenvectors of $g$ relative to the non-zero eigenvalues has subset diffeology that is standard; and among all subspaces of $V$ whose diffeology is standard, it has the maximal dimension, which is equal to $\dim(V^*)$. In general, $V$ contains more than one subspace of dimension $\dim(V^*)$ whose diffeology is standard. But the subspace $V_0$ is the only one that also splits off as a smooth direct summand.[^22] Thus, $V_0$ does not actually depend on the choice of a pseudo-metric and is an invariant of the space itself (see [@pseudometric]). We call this subspace the **characteristic subspace** of $V$.
Let us now return to the two diffeological vector spaces $V$ and $W$ above. Let $V_0$ and $W_0$ be their characteristic subspaces, and let $V_1\leqslant V$ and $W_1\leqslant W$ be the isotropic subspaces relative to $g_V$ and $g_W$ respectively, such that $V=V_0\oplus V_1$ and $W=W_0\oplus W_1$ with each of these decompositions being smooth. We also recall [@pseudometric] that $V_0$ not only has the same dimension as $V^*$, but for any fixed pseudo-metric is diffeomorphic to it, via (the restriction to $V_0$ of) the map $\Phi_V:v\mapsto g_V(v,\cdot)$; likewise, $W_0$ is diffeomorphic to $W^*$ via $\Phi_W:w\mapsto g_W(w,\cdot)$.
### The necessary conditions
Let us now assume that the given $g_V$, $g_W$, and $f$ satisfy the compatibility condition. The corollaries of this assumption can be described in terms of the characteristic subspaces of $V$ and $W$, and therefore in terms of their diffeological duals.
#### The kernel of $f$
The first corollary is quite trivial, and starts with a simple linear algebra argument. Let $v\in V$ belong to the kernel of $f$. Then by the compatibility assumption for $g_V$ and $g_W$ we have $$g_V(v,v')=g_W(0,f(v'))=0\mbox{ for any }v'\in V.$$ Thus, the kernel of $f$ is contained in the maximal isotropic subspace $V_1$, therefore the restriction of $f$ to $V_0$ is a bijection with its image. This restriction is of course a smooth map, and since $V_0$ splits off as a smooth direct summand, it is an induction (that is, a diffeomorphism with its image). Finally, $f$ itself is a diffeomorphism of $V_0\oplus\left(V_1/\mbox{Ker}(f)\right)$ with its image in $W$. In particular, we have the following.
\[exist:compatible:pseudo-metrics:f:necessary:lem\] Let $V$ and $W$ be finite-dimensional diffeological vector spaces, and let $f:V\to W$ be a smooth linear map. If $V$ and $W$ admit pseudo-metrics compatible with $f$ then the $\mbox{Ker}(f)\cap V_0=\{0\}$.
Notice that in the standard case $V$ and $W$ would be vector spaces, $g_V$ and $g_W$ scalar products on them, and $f$ an isometry of $V$ with its image in $W$. In particular, $f$ would be injective; Lemma \[exist:compatible:pseudo-metrics:f:necessary:lem\] is the diffeological counterpart of that.
#### The dimensions of $V$ and $W$, and those of $V^*$ and $W^*$
Continuing the analogy with the standard case, we observe that the standard inequality $\dim(V)\leqslant\dim(W)$ does not have to hold in the diffeological setting. What instead is true, is the corresponding inequality for the dimensions of their dual spaces, which follows from the lemma below.
Let $V$ and $W$ be finite-dimensional diffeological vector spaces, let $f:V\to W$ be a smooth linear map, and suppose that $V$ and $W$ carry pseudo-metrics $g_V$ and $g_W$ respectively, compatible with respect to $f$. Then the subset diffeology of $f(V_0)$ is the standard one.
Let $e_1,\ldots,e_n$ be a $g_V$-orthonormal basis of $V_0$; then by compatibility $f(e_1),\ldots,f(e_n)$ is a $g_W$-orthonormal basis of $f(V_0)$, which can be completed to a basis of eigenvectors of $g_W$. It suffices to show that the projection of $W$ on the line generated by each $f(e_i)$ is a usual smooth function. Since this projection is given by $w\mapsto g_W(f(e_i),w)$, the claim follows from the smoothness of $g_W$.
Now, the fact that $f(V_0)$ carries the standard diffeology, does not automatically imply that it is contained in $W_0$ — there are standard subspaces that are not (we will however show later on that this inclusion does hold for $f(V_0)$). However, $f(V_0)$ is still a standard subspace of $W$, and since $W_0$ has maximal dimension among such subspaces, we have $$\dim(V^*)=\dim(V_0)=\dim(f(V_0))\leqslant\dim(W_0)=\dim(W^*).$$ Therefore we have the following statement.
Let $V$ and $W$ be finite-dimensional diffeological vector spaces. If there exist a smooth linear map $f:V\to W$ and pseudo-metrics $g_V$ and $g_W$ on $V$ and $W$ respectively, compatible with respect to $f$, then $$\dim(V^*)\leqslant\dim(W^*).$$
In other words, if $\dim(V^*)>\dim(W^*)$, then no two pseudo-metrics on $V$ and $W$ are compatible, whatever the map $f$ (which obviously mimics the standard situation: there is no isometry from the space of a bigger dimension to one of smaller dimension).[^23]
#### The subspace $f(V_0)$ in $W$
We now show that the *a priori* case when $f(V_0)$ is not contained in $W_0$ is actually impossible, that is, if $f:V\to W$ is such that $V$ and $W$ admit compatible pseudo-metrics then $f$ sends the characteristic subspace of $V$ to the characteristic subspace of $W$.
\[compatible:pseudometrics:image:smooth:summand:lem\] Let $V$ and $W$ be finite-dimensional diffeological vector spaces, let $f:V\to W$ be a smooth linear map, and suppose that $V$ and $W$ admit compatible pseudo-metrics $g_V$ and $g_W$ respectively. Then $f(V_0)$ splits off smoothly in $W$.
Let $e_1,\ldots,e_n$ be a $g_V$-orthonormal basis of $V_0$. Then by assumption $f(e_1),\ldots,f(e_n)$ is a $g_W$-orthonormal basis of $f(V_0)$. This can be completed to an orthogonal basis of $W$ composed of eigenvectors of $g_W$; denote by $u_1,\ldots,u_k$ the elements added, ordered in such a way that the eigenvectors corresponding to the zero eigenvalue are the last $m$ vectors. Let us show that the usual direct sum decomposition $W=f(V_0)\oplus\mbox{Span}(u_1,\ldots,u_k)$ is a smooth one.
Let $p:U\to W$ be a plot of $W$, and let $p'$ be its composition with the projection (associated to the direct sum decomposition just mentioned) of $W$ to $f(V_0)$. It suffices to show that $p'$ is a plot of $f(V_0)$. Notice that by the choice of the basis $f(e_1),\ldots,f(e_n),u_1,\ldots,u_k$ of $W$ (more precisely, by the $g_W$-orthogonality of said basis) we have $$p'(u)=g_W(f(e_1),p(u))f(e_1)+\ldots+g_W(f(e_n),p(u))f(e_n),$$ where each coefficient $g_W(f(e_i),p(u))$ is an ordinary smooth function $U\to{{\mathbb{R}}}$ by the smoothness of the pseudo-metric $g_W$. This means precisely that $p'$ is a plot of $f(V_0)$, whence the claim.
From the lemma just proven, we can now easily draw the following conclusion.
Under the assumptions of Lemma \[compatible:pseudometrics:image:smooth:summand:lem\], there is the inclusion $f(V_0)\leqslant W_0$.
The subspace $W_0$ is the only subspace of dimension equal to that of $W^*$ that has standard diffeology and splits off smoothly. Since $f(V_0)\oplus W_0'$ has all the same properties, we obtain that $f(V_0)\oplus W_0'=W_0$.
#### The summary of necessary conditions
We collect the conclusions of this section in the following statement.
\[necessary:exist:pseudometrics:vspaces:thm\] Let $V$ and $W$ be two finite-dimensional diffeological vector spaces, and let $f:V\to W$ be a smooth linear map. If there exist pseudo-metrics $g_V$ and $g_W$ on $V$ and $W$ respectively that are compatible with respect to $f$ then the following are true:
1. $\dim(V^*)\leqslant\dim(W^*)$;
2. $\mbox{Ker}(f)\cap V_0=\{0\}$, where $V_0$ is the characteristic subspace of $V$;
3. The subset diffeology on $f(V_0)$ relative to its inclusion into $W$ is standard;
4. $f(V_0)$ splits off smoothly in $W$.
### Sufficient conditions
Suppose now that $V$ and $W$ are such that the just-mentioned necessary condition is satisfied, and let $f:V\to W$ be a smooth linear map such that $\mbox{Ker}(f)\cap V_0=\{0\}$ (where $V_0$ is the characteristic subspace of $V$). By definition of a pseudo-metric, if $V=V_0\oplus V_1$ is a smooth decomposition of $V$[^24] then $g_V$ is defined by its restriction to $V_0$ (which is a scalar product) and is extended by zero elsewhere. The same is true of $g_W$ and the corresponding smooth decomposition $W_0\oplus W_1$. In this way we obtain the following.
\[compatible:pseudometrics:vspace:st-to-st:prop\] Let $V$ and $W$ be finite-dimensional diffeological vector spaces such that $\dim(V^*)\leqslant\dim(W^*)$, and let $f:V\to W$ be a smooth linear map such that $\mbox{Ker}(f)\cap V_0=\{0\}$ and $f(V_0)\leqslant W_0$. Then $V$ and $W$ admit pseudo-metrics compatible with respect to $f$.
Let us fix smooth decompositions $V=V_0\oplus V_1$ and $W=W_0\oplus W_1$. To construct a pseudo-metric $g_V$, choose a basis $v_1,\ldots,v_k$ of $V_0$ and a basis $v_{k+1},\ldots,v_n$ of $V_1$; then set $$g_V(v_i,v_j)=\delta_{i,j}\mbox{ for }i,j=1,\ldots,k\mbox{ and }g_V(v_i,v_{k+j})=0\mbox{ for }i=1,\ldots,n,\,j=1,\ldots,n-k,$$ and extend by bilinearity and symmetricity. To define $g_W$, then, consider $f(v_1),\ldots,f(v_k)\in W_0$; notice that they are linearly independent by the assumption on $\mbox{Ker}(f)$. Add first $u_1,\ldots,u_l\in W_0$ to obtain the basis $f(v_1),\ldots,f(v_k),u_1,\ldots,u_l$ of $W_0$. Finally, choose a basis $w_1,\ldots,w_m$ of $W_1$ to obtain the basis $f(v_1),\ldots,f(v_k),u_1,\ldots,u_l,w_1,\ldots,w_m$ of the whole $W$. It then suffices to define $g_W$ to be $$g_W(f(v_i),f(v_j))=\delta_{i,j},\,\,\,g_W(u_i,u_j)=\delta_{i,j},\,\,\,g_W(f(v_i),u_j)=0,\,\,\,g_W(f(v_i),w_p)=0,\,\,\, g_W(u_i,w_p)=0$$ and extend by bilinearity and symmetry. The bilinear maps $g_V$ and $g_W$ thus obtained are smooth, because each of the characteristic subspaces $V_0$ and $W_0$ splits off as a smooth direct summand, and by construction $g_V$ and $g_W$ are zero maps outside of $V_0$ and $W_0$ respectively. Finally, that they are pseudo-metrics and are compatible with each other is immediate from their definitions, whence the conclusion.
We are ready to establish the final criterion of the existence of compatible pseudo-metrics on a pair of diffeological vector spaces, that we state in the following form.
\[criterio:exist:pseudometrics:vspaces:thm\] Let $V$ and $W$ be two finite-dimensional diffeological vector spaces, and let $f:V\to W$ be a smooth linear map. Then $V$ and $W$ admit compatible pseudo-metrics if and only if $\mbox{Ker}(f)\cap V_0=\{0\}$ and $f(V_0)\leqslant W_0$.
The fact that these two conditions are necessary follows from Lemma \[exist:compatible:pseudo-metrics:f:necessary:lem\] and Lemma \[compatible:pseudometrics:image:smooth:summand:lem\], so let us show that they are sufficient. Let $V=V_0\oplus V_1$ be a smooth decomposition, let $g_V$ be any pseudo-metric on $V$, and let $W=f(V_0)\oplus W'$ be a smooth decomposition that exists by assumption. Notice that, since $W'$ is just another instance of a finite-dimensional diffeological space, it has its own smooth decomposition of form $W_0'\oplus W_1$, where $W_0'$ is standard and $W_1$ has trivial diffeological dual; so the whole of $W$ smoothly decomposes as $W=\left(f(V_0)\oplus W_0'\right)\oplus W_1$. Notice that this implies that $W^*=\left(f(V_0)\oplus W_0'\right)^*$ (by the smoothness of the decomposition), in particular, they have the same dimension.
Let us define a pseudo-metric $g_W$, by setting it to coincide with $g_V$ (in the obvious sense) on $f(V_0)$, choosing any scalar product for its restriction on $W_0'$, while requiring $W_0'$ to be orthogonal to $f(V_0)$, and finally setting $W_1$ to be an isotropic subspace. That this is indeed a pseudo-metric follows from the considerations above, so it remains to show that $g_W$ is indeed compatible with $g_V$. This essentially follows from the construction, more precisely, from the fact that $f(V_0)$ is orthogonal to any its direct complement. Indeed, if $v'=v_0'+v_1'$ and $v''=v_0''+v_1''$ are any two elements of $V$, then $$g_V(v',v'')=g_V(v_0',v_0'')=g_W(f(v_0'),f(v_0''))=g_W(f(v_0')+f(v_1'),f(v_0'')+f(v_1''))=g_W(f(v'),f(v'')),$$ where the third equality is by the orthogonality just mentioned. This means that $g_V$ and $g_W$ are compatible with $f$, and the proof is finished.
Compatibility of the dual pseudo-metrics: diffeological vector spaces
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We now consider the induced pseudo-metrics on the duals of diffeological vector spaces; the main question that we aim to answer is, under what conditions the pair of pseudo-metrics dual to (induced by) two compatible ones is in turn compatible.
#### The induced pseudo-metric $g^*$ on $V^*$: definition
Recall ([@pseudometric]) that, given a finite-dimensional diffeological vector space $V$ endowed with a pseudo-metric $g$, the diffeological dual of $V$ carries the induced pseudo-metric $g^*$ (actually, a scalar product, since the diffeological dual of any finite-dimensional diffeological vector space is standard) defined by $$g^*(v_1^*,v_2^*):=g(v_1,v_2),$$ where $v_i\in V$ is any element such that $v_i^*(\cdot)=g(v_i,\cdot)$ for $i=1,2$. That this is well-defined, *i.e.*, the result does not depend on the choice of $v_i$ (as long as $g(v_i,\cdot)$ remains the same), and that $v_i^*$ always admits such a form, was shown in [@pseudometric].
#### The compatibility for the induced pseudo-metrics
Let $g_V$ and $g_W$ be pseudo-metrics on $V$ and $W$ respectively, compatible with respect to $f$. Let $w_1^*,w_2^*\in W^*$; then there exist $w_1,w_2\in W$, defined up to the cosets of the isotropic subspace of $g_W$, such that $w_i^*(\cdot)=g_W(w_i,\cdot)$ for $i=1,2$, by definition of the dual pseudo-metric $$g_W^*(w_1^*,w_2^*)=g_W(w_1,w_2),$$ and finally, $f^*(w_i^*)(\cdot)=w_i^*(f(\cdot))=g_W(w_i,f(\cdot))$.
The compatibility condition that we need to check is the following one: $$g_W^*(w_1^*,w_2^*)=g_V^*(f^*(w_1^*),f^*(w_2^*)).$$ Now, in order to calculate the right-hand term in this expression, we must choose $v_1$ and $v_2$, again defined up to their cosets with respect to the isotropic subspace of $g_V$, such that $g_V(v_i,v')=f^*(w_i^*)(v')=g_W(w_i,f(v'))$, for all elements of $v'\in V$ and for $w_1^*,w_2^*\in W^*$. The term on the right then becomes $g_V^*(f^*(w_1^*),f^*(w_2^*))=g_V(v_1,v_2)$.
#### The dual pseudo-metrics and compatibility
Let us now consider the pseudo-metrics on $V^*$ and $W^*$ dual to a pair of compatible pseudo-metrics on $V$ and $W$. We observe right away that in general, the induced pseudo-metrics are not compatible. This follows from Lemma \[exist:compatible:pseudo-metrics:f:necessary:lem\], as well as from the standard theory, all diffeological constructions being in fact extensions of the standard ones.
Let $V$ be the standard ${{\mathbb{R}}}^n$, with the canonical basis denoted by $e_1,\ldots,e_n$, and let $W$ be the standard ${{\mathbb{R}}}^{n+k}$, with the canonical basis denoted by $u_1,\ldots,u_n,u_{n+1},\ldots,u_{n+k}$. Let $f:V\to W$ be the embedding of $V$ via the identification of $V$ with the subspace generated by $u_1,\ldots,u_n$, given by $e_i\mapsto u_i$ for $i=1,\ldots,n$. Let $g_V$ be any scalar product on ${{\mathbb{R}}}^n$; this trivially induces a scalar product on $f(V)=\mbox{Span}(u_1,\ldots,u_n)\leqslant W_0$, and let $g_W$ be any extension of it to a scalar product on the whole $W$.
Let us consider the dual map on the dual the standard complement of the subspace $\mbox{Span}(u_1,\ldots,u_n)$, that is, on the dual of $\mbox{Span}(u_{n+1},\ldots,u_{n+k})$. This dual is the usual dual, so it is $\mbox{Span}(u^{n+1},\ldots,u^{n+k})$. Let $v$ be any element of $V$; since $f(v)\in\mbox{Span}(u_1,\ldots,u_n)$, we have $$f^*(u^{n+i})(v)=u^{n+i}(f(v))=0,$$ so in the end we obtain that $\mbox{Ker}(f^*)=\mbox{Span}(u^{n+1},\ldots,u^{n+k})$.
Finally, let us consider the compatibility condition. We observe that $$g_W^*(u^{n+i},u^{n+i})=g_W(u_{n+i},u_{n+i})>0,$$ since $g_W$ is a scalar product, while, of course, $$g_V^*(f^*(u^{n+i}),f^*(u^{n+i}))=0.$$ Quite evidently, the compatibility condition cannot be satisfied (unless $k=0$).
#### Sufficient conditions for compatibility of the induced pseudo-metrics
It can be inferred from the above example that the induced pseudo-metrics on the duals of standard spaces are compatible only if the spaces have the same dimension (which is not surprising, since in this case the notion of the induced pseudo-metric itself coincides with the standard one). This can be generalized to the following statement.
\[crit:dual:pseudometrics:comp:vspaces:thm\] Let $V$ and $W$ be two finite-dimensional diffeological vector spaces, and let $f:V\to W$ be a smooth linear map such that $\mbox{Ker}(f)\cap V_0=\{0\}$ and $f(V_0)\leqslant W_0$. Let $g_V$ and $g_W$ be compatible pseudo-metrics on $V$ and $W$ respectively. Then the induced pseudo-metrics $g_W^*$ and $g_V^*$ are compatible with $f^*$ if and only if $f^*:W^*\to V^*$ is a diffeomorphism.
The *only if* part of the statement, illustrated by the example above, follows from standard reasoning. Indeed, $g_W^*$ and $g_V^*$ are usual scalar products on standard spaces $W^*$ and $V^*$ respectively, and their compatibility means that $f^*$ is a usual isometry, whose existence implies that $W^*$ and $V^*$ have the same dimension, and being standard spaces, this means that they are diffeomorphic as diffeological vector spaces.
Let us prove the *if* part, namely, that $g_W^*$ and $g_V^*$ are compatible under the assumptions of the proposition. Let $e_1,\ldots,e_n$ be a $g_V$-orthogonal basis of $V_0$. Since $g_V$ and $g_W$ are compatible, $f(e_1),\ldots,f(e_n)$ is a $g_W$-orthogonal basis of $f(V_0)$. Now, $(e_1)^*,\ldots,(e_n)^*$ (recall here that $v^*$ for $v\in V$ stands for the map $v^*(\cdot)=g_V(v,\cdot)$) form a basis of $V^*$ (which is also orthogonal with respect to the induced pseudo-metric $g_V^*$), while $(f(e_1))^*,\ldots,(f(e_n))^*$ are linearly independent elements of $W^*$. Since $V^*$ and $W^*$ have the same dimension, $(f(e_1))^*,\ldots,(f(e_n))^*$ actually forms a basis of $W^*$; and so, $g_W^*$ is entirely determined by its values on pairs $(f(e_i))^*,(f(e_j))^*$, and moreover, we have $$g_W^*((f(e_i))^*,(f(e_j))^*)=g_W(f(e_i),f(e_j))=g_V(e_i,e_j)=g_V^*(e_i^*,e_j^*).$$
Finally, $f^*((f(e_i))^*)(v)=(f(e_i))^*(f(v))=g_W(f(e_i),f(v))=g_V(e_i,v)=e_i^*(v)$ for all $v\in V$. Therefore $$g_V^*(e_i^*,e_j^*)=g_V^*(f^*((f(e_i))^*),f^*((f(e_j))^*))=g_W^*((f(e_i))^*,(f(e_j))^*),$$ at which point the compatibility, with respect to $f^*$, of the pseudo-metrics $g_W^*$ and $g_V^*$ follows from $(f(e_1))^*,\ldots,(f(e_n))^*$ being a basis of $W^*$.
Compatibility of the dual pseudo-metrics: diffeological vector pseudo-bundles
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Let us now consider the following question: if two given pseudo-metrics $g_1$ and $g_2$ are compatible with respect to the gluing along a given pair of maps $(f,\tilde{f})$, when is it true that $g_2^*$ and $g_1^*$ are compatible with the gluing defined by $(f^{-1},\tilde{f}^*)$? (Obviously, we assume here that $f$ is invertible).
#### The compatibility condition for $g_2^*$ and $g_1^*$
Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be locally trivial finite-dimensional diffeological vector pseudo-bundles, let $(\tilde{f},f)$ be a gluing of the former to the latter such that $f$ is a diffeomorphism with its image, and let $g_1$ and $g_2$ be pseudo-metrics on $V_1$ and $V_2$ respectively, compatible with respect to the given gluing. The latter induces a well-defined gluing, along the maps $\tilde{f}^*$ and $f^{-1}$, of the dual pseudo-bundle $\pi_2^*:V_2^*\to X_2$ to the pseudo-bundle $\pi_1^*:V_1^*\to X_1$, the result of which is the pseudo-bundle $\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*:V_2^*\cup_{\tilde{f}^*}V_1^*\to X_2\cup_{f^{-1}}X_1$, while $g_2$ and $g_1$ induce pseudo-metrics $g_2^*:X_2\to(V_2^*)^*\otimes(V_2^*)^*$ and $g_1^*:X_1\to(V_1^*)^*\otimes(V_1^*)^*$ on the dual pseudo-bundles. They satisfy the usual compatibility condition if $$g_1^*(f^{-1}(y'))(\tilde{f}^*(v^*),\tilde{f}^*(w^*))=g_2^*(y')(v^*,w^*)$$ $$\mbox{for all }y'\in Y'=f(Y)\mbox{ and for all }v^*,w^*\in(\pi_2^{-1}(y'))^*.$$
#### The necessary condition
The compatibility between $g_2^*$ and $g_1^*$ implies in particular that for all $y\in Y$ the pseudo-metrics $g_2^*(f(y))$ and $g_1^*(y)$ are compatible with the smooth linear map $\tilde{f}^*|_{\pi_1^{-1}(y)}$ between diffeological vector spaces $(\pi_2^{-1}(f(y)))^*$ and $(\pi_1^{-1}(y))^*$. Thus, Theorem \[crit:dual:pseudometrics:comp:vspaces:thm\] implies that $\tilde{f}^*$ is a diffeomorphism on each fibre.
Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be diffeological vector pseudo-bundles, locally trivial and with finite-dimensional fibres, and let $(\tilde{f},f)$ be an invertible gluing between them. Suppose that $g_1$ and $g_2$ are two pseudo-metrics on these pseudo-bundles compatible with the gluing along $(\tilde{f},f)$. If the induced pseudo-metrics $g_2^*$ and $g_1^*$ are compatible with the gluing along $(\tilde{f}^*,f^{-1})$ then the restriction of $\tilde{f}^*$ on each fibre in its domain of definition is a diffeomorphism.
#### Criterion of compatibility
The statement that follows shows that, for the two induced pseudo-metrics to be compatible with the induced gluing, the map dual to the gluing map $\tilde{f}$ must satisfy a rather stringent condition (although an expected one).
\[when:dual:pseudometrics:compatible:bundles:thm\] Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two diffeological vector pseudo-bundles, locally trivial and with finite-dimensional fibres, let $(\tilde{f},f)$ be a gluing between them, and let $g_1$ and $g_2$ be compatible pseudo-metrics on $V_1$ and $V_2$ respectively. Then the induced pseudo-metrics $g_2^*$ and $g_1^*$ on the corresponding dual pseudo-bundles are compatible if and only if $\tilde{f}^*$ is a pseudo-bundle diffeomorphism of its domain with its image.
Notice that diffeological vector spaces may have diffeomorphic duals without being diffeomorphic themselves, and the same is true for diffeological vector pseudo-bundles.
By assumption, $g_1$ and $g_2$ are compatible with the gluing given by the pair $(\tilde{f},f)$, that is $$g_2(f(y))(\tilde{f}(v),\tilde{f}(w))=g_1(y)(v,w)\mbox{ for all }y\in Y\mbox{ and for all }v,w\in\pi_1^{-1}(y).$$ Suppose first that $\tilde{f}^*$ is a diffeomorphism with its image. Then, first of all, by the definition of $g_2^*$ we have $$g_2^*(y')(v^*,w^*)=g_2(y')(v,w)$$ for all $y'\in f(Y)$, for all $v^*,w^*\in(\pi_2^{-1}(y'))^*$, and for $v,w\in(\pi_2^{-1}(y'))_0$. Notice that $v$ and $w$ are uniquely defined by the latter condition; and they are such that $v^*(\cdot)=g_2(y')(v,\cdot)$ and $w^*(\cdot)=g_2(y')(w,\cdot)$. Notice also (we will need this immediately below) that this means $$\tilde{f}^*(v^*)(\cdot)=v^*(\tilde{f}(\cdot))=g_2(y')(v,\tilde{f}(\cdot))=g_2(y')(\tilde{f}(v_1),\tilde{f}(\cdot))=g_1(f^{-1}(y'))(v_1,\cdot),$$ where $v_1\in(\pi_1^{-1}(f^{-1}(y')))_0$ is such that $v=\tilde{f}(v_1)$; such an element exists and is uniquely defined because $\tilde{f}^*$ being a diffeomorphism is equivalent to $\tilde{f}$ being a diffeomorphism between each pair of subspaces $(\pi_1^{-1}(f^{-1}(y')))_0$ and $(\pi_2^{-1}(y'))_0$. Similarly, we have $$\tilde{f}^*(w^*)(\cdot)=g_2(y')(\tilde{f}(w_1),\tilde{f}(\cdot))=g_1(f^{-1})(w_1,\cdot)$$ for $w_1\in(\pi_1^{-1}(f^{-1}(y')))_0$ such that $w=\tilde{f}(w_1)$. It remains now to consider the left-hand part of the compatibility condition. We have: $$g_1^*(f^{-1}(y'))(\tilde{f}^*(v^*),\tilde{f}^*(w))=g_1(f^{-1}(y'))(v_1,w_1)=g_2(y')(\tilde{f}(v_1),\tilde{f}(w_1))=g_2(y')(v,w)=g_2^*(y')(v^*,w^*),$$ as wanted.
Let us prove the *vice versa* of the statement, that is, let us assume that $g_2^*$ and $g_1^*$ are compatible, and let us show that $\tilde{f}^*$ is a diffeomorphism. We notice first of all that it follows from the considerations made for individual vector spaces that $\tilde{f}^*$ is bijective and, as is the case for any dual map, it is smooth. Finally, the smoothness of its inverse follows from the fact that $V_1^*$ and $V_2^*$ are locally trivial and have standard fibres.
Compatibility of pseudo-metrics and the gluing-dual commutativity conditions
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In this section we consider which correlations there might be between the notion of compatible pseudo-metrics on two given pseudo-bundles (with a specified gluing), and the gluing-dual commutativity conditions. We start by taking our two usual pseudo-bundles, $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$, and a gluing along $(\tilde{f},f)$ between them. Assuming that these pseudo-bundles admit pseudo-metrics $g_1$ and $g_2$ compatible with the gluing, we consider the following questions: 1) what are the implications of the existence of compatible pseudo-metrics for the pseudo-bundles themselves? 2) if the gluing-dual commutativity condition holds for $V_1$, $V_2$, and $(\tilde{f},f)$, does it necessarily hold for $V_2^*$, $V_1^*$, and $(\tilde{f}^*,f^{-1})$? 3) if $g_2^*$ and $g_1^*$ exist, under what conditions are they compatible with $(\tilde{f}^*,f^{-1})$, in particular, is their compatibility equivalent to the gluing-dual commutativity? 4) does taking the dual pseudo-metric commute (in the notation to be introduced, this will be the equality $\tilde{g}^*=\widetilde{g^*}$) with the gluing of pseudo-metrics (as defined in [@pseudometric-pseudobundle])? We consider these questions in order, after quickly introducing a preliminary notion.
The characteristic sub-bundle of a finite-dimensional vector pseudo-bundle
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Let $\pi:V\to X$ be a finite-dimensional diffeological vector pseudo-bundle, and let $(\pi^{-1}(x))_0$ be the characteristic subspace of the fibre $\pi^{-1}(x)$. Denote by $V_0$ the sub-bundle of $V$ defined as $$V_0:=\cup_{x\in X}(\pi^{-1}(x))_0.$$ We say that $V_0$ is the **characteristic sub-bundle** of the pseudo-bundle $V$. It is evident from the construction that the characteristic sub-bundle of a locally trivial pseudo-bundle is itself locally trivial. Furthermore, every pseudo-metric on $V$ is uniquely defined by its restriction to $V_0$. Finally, the *vice versa* of the latter statement is also true, if we assume $V$ to be locally trivial.
Let $\pi:V\to X$ be a locally trivial diffeological vector pseudo-bundle, let $\pi_0:V_0\to X$ be its characteristic sub-bundle, and let $g_0$ be a pseudo-metric on $V_0$. Then there exists one, and only one, pseudo-metric $g$ on $V$ whose restriction on $V_0$ coincides with $g_0$.
It suffices to define $g(x)(v',v'')=\left\{\begin{array}{ll} g_0(x)(v',v'') & \mbox{if }v',v''\in\pi_0^{-1}(x)\\ 0 & \mbox{otherwise}; \end{array}\right.$ the conclusion then follows from the definitions of a pseudo-metric and that of the characteristic sub-bundle.
Let $\pi:V\to X$ be a locally trivial finite-dimensional diffeological vector pseudo-bundle that admits a pseudo-metric $g$. Then its characteristic sub-bundle $\pi_0:V_0\to X$ is diffeomorphic to its dual pseudo-bundle $\pi^*:V^*\to X$ via the natural pairing map associated to $g$.
Let $\psi_g:V_0\to V^*$ be the natural pairing associated to $g$, that is, $$\psi_g(v)(\cdot)=g(\pi(v))(v,\cdot).$$ That this is a bijection follows from its fibrewise nature and it being a bijection on each individual fibre (see [@pseudometric]); furthermore (see the same source), as a map on the characteristic subspace it is a diffeomorphism with the dual fibre. By the assumption of local triviality this implies that $\psi_g$, as well as its inverse, are smooth across the fibres as well, so they are smooth as a whole, whence the conclusion.[^25]
Implications of compatibility of $g_1$ and $g_2$ for $V_1$ and $V_2$
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This extends the criterion for diffeological vector spaces (Theorem \[criterio:exist:pseudometrics:vspaces:thm\]). The pseudo-bundle version is an immediate consequence and is as follows.
Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two diffeological vector pseudo-bundles, locally trivial and with finite-dimensional fibres, let $(\tilde{f},f)$ be a gluing between them, and let $g_1$ and $g_2$ be compatible pseudo-metrics. Then $\tilde{f}$ determines, over the domain of gluing, a smooth embedding of the characteristic sub-bundle of $V_1$ into the characteristic sub-bundle of $V_2$.
This follows directly from the already-mentioned Theorem \[criterio:exist:pseudometrics:vspaces:thm\], applied to the restriction of $\tilde{f}$ on each fibre in its domain of definition; the theorem affirms that such restriction is an embedding of each characteristic subspace, so the fibre of the characteristic sub-bundle, of $V_1$ into that of $V_2$. We should only add that the restriction of $\tilde{f}$ onto the intersection of its domain of definition with the characteristic sub-bundle of $V_1$ is smooth across the fibres, because $\tilde{f}$ is so.
The gluing-dual commutativity condition and gluing along a diffeomorphism
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We now recall a statement (which essentially appears in [@pseudobundles], Lemma 5.17) that (together with some results from the previous sections) will allow us to deduce the gluing-dual commutativity in a number of cases. The statement basically is that if the gluing of two pseudo-bundles is performed along a diffeomorphism, then the gluing-dual commutativity condition always holds. We also add the explicit construction of the commutativity diffeomorphism (which was not specified in the above source).
\[gluing:diffeo:implies:gluing-dual:commute:thm\] Let $\chi_1:W_1\to X_1$ and $\chi_2:W_2\to X_2$ be two diffeological vector pseudo-bundles, let $h:X_1\supseteq Y\to X_2$ be a smooth invertible map with smooth inverse, and let $\tilde{h}$ be its smooth fibrewise linear lift that is a diffeomorphism of its domain with its image. Then the map $$\Psi_{\cup,*}:(W_1\cup_{\tilde{h}}W_2)^*\to W_2^*\cup_{\tilde{h}^*}W_1^*$$ defined by $$\Psi_{\cup,*}=\left\{\begin{array}{ll}
j_2^{W_1^*}\circ(j_1^{W_1})^* & \mbox{on }((\chi_1\cup_{(\tilde{h},h)}\chi_2)^*)^{-1}(i_1^{X_1}(X_1\setminus Y)) \\
j_2^{W_1^*}\circ\tilde{h}^*\circ(j_2^{W_2})^* & \mbox{on }((\chi_1\cup_{(\tilde{h},h)}\chi_2)^*)^{-1}(i_2^{X_2}(h(Y)) \\
j_1^{W_2^*}\circ(j_2^{W_2})^* & \mbox{on }((\chi_1\cup_{(\tilde{h},h)}\chi_2)^*)^{-1}(i_2^{X_2}(X_2\setminus h(Y))
\end{array}\right.$$ is a pseudo-bundle diffeomorphism covering the switch map $\varphi_{X_1\leftrightarrow X_2}$.
It is easy to see that $\Psi_{\cup,*}$ is a bijection, so let us show that it is smooth (the proof that its inverse is smooth is then analogous). Let us first consider the general shape of an arbitrary plot $q^*$ of $(W_1\cup_{\tilde{h}}W_2)^*$, and that of an arbitrary plot $s^*$ of $W_2^*\cup_{\tilde{h}^*}W_1^*$. Let $q^*:U\to(W_1\cup_{\tilde{h}}W_2)^*$ be any plot; we can however assume that $U$ is connected, so $(\chi_1\cup_{(\tilde{h},h)}\chi_2)\circ q$, which is a plot of $X_1\cup_h X_2$, lifts to a plot of $X_1$ or to a plot of $X_2$. This means that $q^*$ acts only on fibres of $W_1\cup_{\tilde{h}}W_2$ that pullback to $W_1$ or to $W_2$, respectively. In the former case we have that there exists a plot $q_1^*$ of $W_1^*$ such that $$q^*=\left\{\begin{array}{ll} ((j_1^{W_1})^*)^{-1}\circ q_1^* & \mbox{over }i_1^{X_1}(X_1\setminus Y) \\
((j_2^{W_2})^*)^{-1}\circ(\tilde{h}^*)^{-1}\circ q_1^* & \mbox{over }i_2^{X_2}(h(Y));
\end{array}\right.$$ in the latter case there exists a plot $q_2^*$ of $W_2^*$ such that $$q^*=((j_2^{W_2})^*)^{-1}\circ q_2^*.$$ For analogous reasons, if $s^*:U'\to W_2^*\cup_{\tilde{h}^*}W_1^*$ defined on a connected domain $U'$, then either there exists a plot $s_2^*$ of $W_2^*$ such that $$s^*=\left\{\begin{array}{ll} j_1^{W_2^*}\circ s_2^* & \mbox{over }i_1^{X_2}(X_2\setminus h(Y)) \\
j_2^{W_1^*}\circ\tilde{h}^*\circ s_2^* & \mbox{over }i_2^{X_1}(h(Y)),
\end{array}\right.$$ or else there exists a plot $s_1^*$ of $W_1^*$ such that $$s^*=j_2^{W_2^*}\circ s_1^*.$$
Let us consider $\Psi_{\cup,*}\circ q^*$ for $q^*$ of the first and the second type. Assume that $q^*$ is of the first type. Then by direct calculation we obtain $$\Psi_{\cup,*}\circ q^*=\left\{\begin{array}{ll}
j_2^{W_1^*}\circ q_1^* & \mbox{over }i_2^{X_1}(X_1\setminus Y), \\
j_2^{W_1^*}\circ\tilde{h}^*\circ(j_2^{W_2})^*\circ((j_2^{W_2})^*)^{-1}\circ(\tilde{h}^*)^{-1}\circ q_1^* & \mbox{over }i_2^{X_1}(Y),
\end{array}\right.$$ that is, $\Psi_{\cup,*}\circ q^*=j_2^{W_1^*}\circ q_1^*$ over the whole of $i_2^{X_1}(X_1)$, which corresponds to a plot $s^*$ of the second type, for $s_1^*:=q_1^*$.
Similarly, if $q^*$ has its second possible form, we obtain $$\Psi_{\cup,*}\circ q^*=\left\{\begin{array}{ll}
j_2^{W_1^*}\circ\tilde{h}^*\circ q_2^* & \mbox{over }i_2^{X_1}(Y)\\
j_1^{W_2^*}\circ q_2^* & \mbox{over }i_1^{X_2}(X_2\setminus h(Y)),
\end{array}\right.$$ that is, the first possible form of a plot $s^*$, with $s_2^*:=q_2^*$.
Finally, the smoothness of $(\Psi_{\cup,*})^{-1}$, whose formula $$(\Psi_{\cup,*})^{-1}=\left\{\begin{array}{ll}
((j_1^{W_1})^*)^{-1}\circ(j_2^{W_1^*})^{-1} & \mbox{on }(\chi_2^*\cup_{(\tilde{h},h)}\chi_1^*)^{-1}(i_2^{X_1}(X_1\setminus Y)) \\
((j_2^{W_2})^*)^{-1}\circ(\tilde{h}^*)^{-1}\circ(j_2^{W_1^*})^{-1} & \mbox{on }(\chi_2^*\cup_{(\tilde{h},h)}\chi_1^*)^{-1}(i_2^{X_1}(Y) \\
((j_2^{W_2})^*)^{-1}\circ(j_1^{W_2^*})^{-1} & \mbox{on }(\chi_2^*\cup_{(\tilde{h},h)}\chi_1^*)^{-1}(i_1^{X_2}(X_2\setminus h(Y))
\end{array}\right.$$ is given by the inverses of the three maps that $\Psi_{\cup,*}$ itself, is established in a completely analogous fashion.
Gluing-dual commutativity conditions for $(\tilde{f},f)$ and $(\tilde{f}^*,f^{-1})$
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We now consider the gluing-dual commutativity condition for $V_2^*$, $V_1^*$, and $(\tilde{f}^*,f^{-1})$, under the assumption that such condition holds for $V_1$, $V_2$, and $(\tilde{f},f)$. For the duals, this condition takes form of the existence of a diffeomorphism $$\Phi_{\cup,*}^{(*)}:(V_2^*\cup_{\tilde{f}^*}V_1^*)^*\to V_1^{**}\cup_{\tilde{f}^{**}}V_2^{**}$$ covering the inverse of the switch map $\varphi_{X_1\leftrightarrow X_2}$ and satisfying the following: $$\left\{\begin{array}{ll}
\Phi_{\cup,*}^{(*)}\circ((j_2^{V_2^*})^*)^{-1}=j_1^{V_1^{**}} & \mbox{on }(\pi_1^{**}\cup_{(\tilde{f}^{**},f)}\pi_2^{**})^{-1}(i_1^{X_1}(X_1\setminus Y)),\\
\Phi_{\cup,*}^{(*)}\circ((j_1^{V_1^*})^*)^{-1}=j_2^{V_2^{**}}\circ\tilde{f}^* & \mbox{on }(\pi_1^{**}\cup_{(\tilde{f}^{**},f)}\pi_2^{**})^{-1}(i_2^{X_2}(f(Y))),\\
\Phi_{\cup,*}^{(*)}\circ((j_2^{V_1^*})^*)^{-1}=j_2^{V_2^{**}} & \mbox{on }(\pi_1^{**}\cup_{(\tilde{f}^{**},f)}\pi_2^{**})^{-1}(i_2^{X_2}(X_2\setminus f(Y))).\end{array}\right.$$ Notice that this formula can serve as a definition of a certain map $\Phi_{\cup,*}^{(*)}$ between the domain and the range indicated; what we really need to do is to show that it is a diffeomorphism.
Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two locally trivial finite-dimensional diffeological vector pseudo-bundles, let $(\tilde{f},f)$ be a pair of smooth maps that defines a gluing of the former pseudo-bundle to the latter, and let $\Phi_{\cup,*}:(V_1\cup_{\tilde{f}}V_2)^*\to V_2^*\cup_{\tilde{f}^*}V_1^*$ be the canonical gluing-dual commutativity diffeomorphism. Let $g_1$ and $g_2$ be pseudo-metrics on $V_1$ and $V_2$ respectively, compatible with respect to the gluing. Then there exists a diffeomorphism $$\Phi_{\cup,*}^{(*)}:(V_2^*\cup_{\tilde{f}^*}V_1^*)^*\to V_1^{**}\cup_{\tilde{f}^{**}}V_2^{**}$$ covering the map $(\varphi_{X_1\leftrightarrow X_2})^{-1}:X_2\cup_{f^{-1}}X_1\to X_1\cup_f X_2$.
The claim of the theorem could be restated by saying that the dual pseudo-bundles $V_2^*$ and $V_1^*$ satisfy the gluing-dual commutativity condition for the gluing pair $(\tilde{f}^*,f^{-1})$.
Recall that the gluing-dual commutativity condition for the initial pseudo-bundles, that is, for $V_1$ and $V_2$, implies that $\tilde{f}^*$ is a diffeomorphism of its domain with its image. It is then a direct consequence of Theorem \[gluing:diffeo:implies:gluing-dual:commute:thm\] that the following map $$\Phi_{\cup,*}^{(*)}=\left\{\begin{array}{ll}
j_2^{V_2^{**}}\circ\left(j_1^{V_2^*}\right)^* & \mbox{on }\left((\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*)^*\right)^{-1}(i_1^{X_2}(X_2\setminus f(Y))),\\
j_2^{V_2^{**}}\circ\tilde{f}^*\circ\left(j_2^{V_1^*}\right)^* & \mbox{on }\left((\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*)^*\right)^{-1}(i_2^{X_1}(Y)),\\
j_1^{V_1^{**}}\circ\left(j_2^{V_1^*}\right)^* & \mbox{on }\left((\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*)^*\right)^{-1}(i_2^{X_1}(X_1\setminus Y))
\end{array}\right.$$ is the desired gluing-dual commutativity diffeomorphism.
Compatibility of $g_2^*$ and $g_1^*$ implies the gluing-dual commutativity condition for $V_1$ and $V_2$
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So far we have spoken of the gluing-dual commutativity condition as a prerequisite to obtaining a canonical construction of the induced pseudo-metric on the pseudo-bundle obtained by gluing. In principle, it is not a necessary condition (a pseudo-metric on $V_1\cup_{\tilde{f}}V_2$ can be constructed directly out of compatible pseudo-metrics on $V_1$ and $V_2$, using the flexibility of diffeology in piecing together smooth maps); however, if we want at the same time to consider the dual pseudo-bundles $V_2^*$ and $V_1^*$, and to ensure that the induced pseudo-metrics on them are again compatible, the reasoning involved starts to come rather close to the gluing-dual commutativity. Indeed, in this section we show that there is essentially an equivalence between the compatibility of $g_2^*$ with $g_1^*$, and the existence of a gluing-dual commutativity diffeomorphism $\Phi_{\cup,*}$.
From $\Phi_{\cup,*}$ to the compatibility of $g_2^*$, $g_1^*$, and $(\tilde{f}^*,f^{-1})$
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It is not difficult to show that if we assume the gluing-dual commutativity, and more precisely, the existence of the specific diffeomorphism $\Phi_{\cup,*}:(V_1\cup_{\tilde{f}}V_2)^*\to
V_2^*\cup_{\tilde{f}^*}V_1^*$ given by $$\Phi_{\cup,*}=\left\{\begin{array}{ll}
j_2^{V_1^*}\circ(j_1^{V_1})^* & \mbox{on }((\pi_1\cup_{(\tilde{f},f)}\pi_2)^*)^{-1}(i_1^{X_1}(X_1\setminus Y)) \\
j_2^{V_1^*}\circ\tilde{f}^*\circ(j_2^{V_2})^* & \mbox{on }((\pi_1\cup_{(\tilde{f},f)}\pi_2)^*)^{-1}(i_2^{X_2}(f(Y)))\\
j_1^{V_2^*}\circ(j_2^{V_2})^* & \mbox{on }((\pi_1\cup_{(\tilde{f},f)}\pi_2)^*)^{-1}(i_2^{X_2}(X_2\setminus f(Y)))
\end{array}\right.$$ then the dual pseudo-metrics, if they exist, are compatible.
\[gluing-dual:commute:dual:pseudometrics:compatible:thm\] Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two finite-dimensional locally trivial diffeological vector pseudo-bundles, let $(\tilde{f},f)$ be a pair of smooth maps defining a gluing of $V_1$ to $V_2$, and let $g_1$ and $g_2$ be pseudo-metrics on $V_1$ and $V_2$ respectively, compatible with this gluing; assume that $V_1$, $V_2$, and $(\tilde{f},f)$ satisfy the gluing-dual commutativity condition. Then $g_2^*$ and $g_1^*$ are compatible with the gluing of $V_2^*$ to $V_1^*$ along the pair $(\tilde{f}^*,f^{-1})$.
By Theorem \[when:dual:pseudometrics:compatible:bundles:thm\] it suffices to show that $\tilde{f}^*$ is a diffeomorphism of its domain with its image, and this is a trivial consequence of the form in which we stated the gluing-dual commutativity condition, namely, as the smoothness of the specific diffeomorphism $\Phi_{\cup,*}$. Indeed, denoting for brevity $Z_2^*:=((\pi_1\cup_{(\tilde{f},f)}\pi_2)^*)^{-1}(i_2^{X_2}(f(Y)))$, we immediately obtain $$\tilde{f}^*=(j_2^{V_1^*})^{-1}\circ\Phi_{\cup,*}|_{Z_2^*}\circ((j_2^{V_2})^*)^{-1};$$ thus, $\tilde{f}^*$ is a composition of three diffeomorphisms, so a diffeomorphism itself.
The *vice versa*: the smoothness of $\Phi_{\cup,*}$ out of compatibility of $g_2^*$ and $g_1^*$
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The reverse implication, that is, obtaining a smooth $\Phi_{\cup,*}$ assuming the compatibility of $g_2^*$ and $g_1^*$, is now easily obtained by applying Theorem \[gluing:diffeo:implies:gluing-dual:commute:thm\], and the criteria for compatibility of the dual pseudo-metrics. Do note that the application of Theorem \[gluing:diffeo:implies:gluing-dual:commute:thm\] is not straightforward; indeed, it speaks of gluing along a diffeomorphism, and the assumptions formulated in terms of various pseudo-metrics do not provide for $\tilde{f}$ being one. Therefore we need some preliminary considerations.
### Characteristic sub-bundles and the respective dual pseudo-bundles
In order to obtain our desired conclusion, namely, that the compatibility of pseudo-metrics dual to a given pair of compatible pseudo-metrics provides for the gluing-dual commutativity, we need several preliminary statements. We collect them in this section; they are more or less of independent interest.
#### The pseudo-metric $\tilde{g}$ on the pseudo-bundle $W_1\cup_{\tilde{h}}W_2$
Assuming that $W_1$ and $W_2$ admit compatible pseudo-metrics $g_1$ and $g_2$ allows (without any further assumptions on $\tilde{h}$) for a direct construction of a pseudo-metric $\tilde{g}$ on $W_1\cup_{\tilde{h}}W_2$, which fibrewise coincides with either $g_1$ or $g_2$, as appropriate. This pseudo-metric is defined by the following formula: $$\tilde{g}(x)=\left\{\begin{array}{ll}
g_1((i_1^{X_1})^{-1}(x))\circ((j_1^{W_1})^{-1}\otimes(j_1^{W_1})^{-1}) & \mbox{for }x\in i_1^{X_1}(X_1\setminus Y) \\
g_2((i_2^{X_2})^{-1}(x))\circ((j_2^{W_2})^{-1}\otimes(j_2^{W_2})^{-1}) & \mbox{for }x\in i_2^{X_2}(X_2).
\end{array}\right.$$
#### The switch map for characteristic sub-bundles
As we have already commented, a pseudo-metric on a pseudo-bundle is essentially determined by its behavior on the characteristic sub-bundle; and furthermore, assuming the local triviality and the existence of a pseudo-metric, the characteristic sub-bundle is diffeomorphic to the dual pseudo-bundle. Thus, we can expect significant correlations between these three notions; and in this paragraph we specify some of them, as needed to reach the final aim of this section.
\[switch:characteristic:is:diffeo:lem\] Let $\chi_1:W_1\to X_1$ and $\chi_2:W_2\to X_2$ be two locally trivial diffeological vector pseudo-bundles, let $W_1^0$ and $W_2^0$ be their characteristic sub-bundles, let $h:X_1\supseteq Y\to X_2$ be a smooth invertible map with smooth inverse, and let $\tilde{h}$ be its smooth fibrewise linear lift such that its restriction $\tilde{h}_0$ on $\mbox{Domain}(\tilde{h})\cap W_1^0$ is a diffeomorphism. Then there is a canonical diffeomorphism $$\varphi_{W_1^0\leftrightarrow W_2^0}:W_1^0\cup_{\tilde{h}_0}W_2^0\to W_2^0\cup_{\tilde{h}_0^{-1}}W_1^0$$ covering the switch map $\varphi_{X_1\leftrightarrow X_2}$.
The diffeomorphism in question is in fact the same concept as the switch map (which we implied is a diffeomorphism, but did not prove that). Indeed, we denote the map obtained by analogy with $\varphi_{X_1\leftrightarrow X_2}$, by $\varphi_{W_1^0\leftrightarrow W_2^0}$ and define it to be $$\varphi_{W_1^0\leftrightarrow W_2^0}=\left\{\begin{array}{ll}
j_2^{W_1^0}\circ(j_1^{W_1^0})^{-1} & \mbox{on }(\chi_1^0\cup_{(\tilde{h}_0,h)}\chi_2^0)^{-1}(i_1^{X_1}(X_1\setminus Y)) \\
j_2^{W_1^0}\circ(\tilde{h}_0)^{-1}\circ(j_2^{W_2^0})^{-1} & \mbox{on }(\chi_1^0\cup_{(\tilde{h}_0,h)}\chi_2^0)^{-1}(i_2^{X_2}(h(Y))) \\
j_1^{W_2^0}\circ(j_2^{W_2^0})^{-1} & \mbox{on }(\chi_1^0\cup_{(\tilde{h}_0,h)}\chi_2^0)^{-1}(i_2^{X_2}(X_2\setminus f(Y))),
\end{array}\right.$$ where by $\chi_i^0$ we denote the restriction of $\chi_i$ to $W_i^0$. Notice that it is its own inverse; let us show that it is smooth.
Let $p:U\to W_1^0\cup_{\tilde{h}_0}W_2^0$; it suffices to consider the case when $U$ is connected. Under such assumption, $p$ lifts to either a plot $p_1$ of $W_1^0$ or to a plot $p_2$ of $W_2^0$, therefore $p$ itself either has form $$p=\left\{\begin{array}{ll} j_1^{W_1^0}\circ p_1 & \mbox{on }p_1^{-1}(W_1^0\setminus\mbox{Domain}(\tilde{h}_0)) \\
j_2^{W_2^0}\circ\tilde{h}_0\circ p_1 & \mbox{on }p_1^{-1}(\mbox{Domain}(\tilde{h}_0))
\end{array}\right.$$ in the former case, or it has form $p=j_2^{W_2^0}\circ p_2$ in the latter case. Accordingly, by direct calculation we obtain that $\varphi_{W_1^0\leftrightarrow W_2^0}\circ
p=j_2^{W_2^0}\circ p_1$ for $p$ that lifts to $p_1$ and $$\varphi_{W_1^0\leftrightarrow W_2^0}\circ p=\left\{\begin{array}{ll}
j_2^{W_1^0}\circ(\tilde{h}_0)^{-1}\circ p_2 & \mbox{on }p_2^{-1}(\mbox{Range}(\tilde{h}_0)) \\
j_1^{W_2^0}\circ p_2 & \mbox{on }p_2^{-1}(W_2^0\setminus\mbox{Range}(\tilde{h}_0))
\end{array}\right.$$ for $p$ that lifts for $p_2$. Clearly, in both cases the result is a plot of $W_2^0\cup_{(\tilde{h}_0)^{-1}}W_1^0$, whence the conclusion.
The lemma just proven is a preliminary statement which will be needed to establish a link between the following two statements; all three put together will allows us to relate the gluing-dual commutativity to compatibility of (dual) pseudo-metrics.
#### The triple $W_1^0\cup_{\tilde{h}_0}W_2^0\cong(W_1\cup_{\tilde{h}}W_2)^0\cong(W_1\cup_{\tilde{h}}W_2)^*$
The facts that we prove here ensure a kind of total commutativity between $()^0$ (characteristic) and $()^*$ (dual); this phrase is of course very informal, what we really mean shall be clear from the two statements that follow.
\[dual:characteristic:gluing:commute:prop\] Let $\chi_1:W_1\to X_1$ and $\chi_2:W_2\to X_2$ be two locally trivial diffeological vector pseudo-bundles, let $W_1^0$ and $W_2^0$ be their characteristic sub-bundles, let $h:X_1\supseteq Y\to X_2$ be a smooth invertible map with smooth inverse, and let $\tilde{h}$ be its smooth fibrewise linear lift. Let $g_1$ and $g_2$ be pseudo-metrics on $W_1$ and $W_2$ respectively, compatible with the gluing along $(\tilde{h},h)$, and assume that the dual pseudo-metrics $g_2^*$ and $g_1^*$ are compatible with $(\tilde{h}^*,h^{-1})$. Let $\tilde{h}_0$ be the restriction of $\tilde{h}$ on $\mbox{Domain}(\tilde{h})\cap W_1^0$. Then:
1. $\tilde{h}_0$ is a diffeomorphism with values in $W_2^0$;
2. There is a pseudo-bundle diffeomorphism $$\Phi^0:W_1^0\cup_{\tilde{h}_0}W_2^0\to(W_1\cup_{\tilde{h}}W_2)^0$$ covering the identity map $X_1\cup_f X_2\to X_1\cup_f X_2$;
3. The natural pairing map $\Psi_{\tilde{g}}^0:(W_1\cup_{\tilde{h}}W_2)^0\to(W_1\cup_{\tilde{h}}W_2)^*$ associated to the pseudo-metric $\tilde{g}$ on $W_1\cup_{\tilde{h}}W_2$ is a diffeomorphism of the characteristic sub-bundle $(W_1\cup_{\tilde{h}}W_2)^0$ with the dual pseudo-bundle $(W_1\cup_{\tilde{h}}W_2)^*$.
1\. The compatibility of $g_1$ and $g_2$ means that their restrictions $g_1(y)$ and $g_2(h(y))$ on each fibre in the domain of definition and, respectively, the range of $\tilde{h}$ are compatible pseudo-metrics on the diffeological vector spaces $\chi_1^{-1}(y)$ and $\chi_2^{-1}(h(y))$. By Theorem \[criterio:exist:pseudometrics:vspaces:thm\] this means that the restriction $\tilde{h}_0$ of $\tilde{h}$ to $\mbox{Domain}(\tilde{h})\cap W_1^0$ is a smooth injection, whose range is contained in $\mbox{Range}(\tilde{h})\cap W_2^0$. Furthermore, by compatibility of $g_2^*$ and $g_1^*$ and Theorem \[crit:dual:pseudometrics:comp:vspaces:thm\], the dual spaces $(\chi_2^{-1}(h(y)))^*$ and $(\chi_1^{-1}(y))^*$ have the same dimension, which is equal to the dimension of the corresponding characteristic subspaces; therefore $\tilde{h}_0$ is also surjective. Finally, that its inverse is smooth, follows from local triviality and the fact that its restriction onto each fibre is a smooth linear map between finite-dimensional vector spaces.
2\. This is a direct consequence of the definition of a characteristic sub-bundle. The diffeomorphism $\Phi^0:W_1^0\cup_{\tilde{h}_0}W_2^0\to(W_1\cup_{\tilde{h}}W_2)^0$ is defined by $$\Phi^0=\left\{\begin{array}{ll} j_1^{W_1}\circ(j_1^{W_1^0})^{-1} & \mbox{over }i_1^{X_1}(X_1\setminus Y)\\
j_2^{W_2}\circ(j_2^{W_2^0})^{-1} & \mbox{over }i_2^{X_2}(X_2);
\end{array}\right.$$ it is essentially the natural inclusion map.
3\. The diffeomorphism in question is the pairing map $$\Psi_{\tilde{g}}^0:(W_1\cup_{\tilde{h}}W_2)^0\to(W_1\cup_{\tilde{h}}W_2)^*$$ restricted to the characteristic sub-bundle and defined in the usual way, *i.e.*, by $$\mbox{if }w\in(W_1\cup_{\tilde{h}}W_2)^0\Rightarrow\Psi_{\tilde{g}}^0(w)(\cdot)=\tilde{g}((\chi_1\cup_{(\tilde{h},h)}\chi_2)(w))(w,\cdot).$$ That it is smooth, follows from smoothness of $\tilde{g}$; that it is bijective, is easily obtained by examining its restriction on each fibre, where, since the fibres of characteristic sub-bundle have standard diffeology, it becomes the usual isomorphism-by-duality on some standard ${{\mathbb{R}}}^n$. Finally, the smoothness of its inverse follows from the assumption of local triviality.
Under the same assumptions as those of Proposition \[dual:characteristic:gluing:commute:prop\], we also have the following:
\[dual:characteristic:gluing:commute:second:part:prop\] There is a canonical diffeomorphism $$\Psi_{g_2}^0\cup_{(\tilde{h}_0^{-1},\tilde{h}^*)}\Psi_{g_1}^0:W_2^0\cup_{\tilde{h}_0^{-1}}W_1^0\to W_2^*\cup_{\tilde{h}^*}W_1^*,$$ whose restrictions onto the factors of gluing coincide with the natural pairing maps associated to a pair of compatible pseudo-metrics $g_2$ and $g_1$.
It follows from the assumptions, specifically, the assumption of compatibility of $g_2^*$ and $g_1^*$, that $\tilde{h}^*$ is a diffeomorphism; by Proposition \[dual:characteristic:gluing:commute:prop\] the $\tilde{h}_0^{-1}$ is also a diffeomorphism. Thus, the desired diffeomorphism of the two pseudo-bundles in the statement is the result $\Psi_{g_2}^0\cup_{(\tilde{h}_0^{-1},\tilde{h}^*)}\Psi_{g_1}^0$ of a gluing (see Section 1.3.1 for definitions) of the natural pairing maps $\Psi_{g_2}^0:W_2^0\to W_2^*$ and $\Psi_{g_1}^0:W_1^0\to W_1^*$. Since the gluing of two diffeomorphisms along a pair of diffeomorphisms yields again a diffeomorphism, we only need to check that $\Psi_{g_2}^0$ and $\Psi_{g_1}^0$ are $(\tilde{h}_0^{-1},\tilde{h}^0)$-compatible, that is, that for any $w_2\in\mbox{Domain}(\tilde{h}_0^{-1})$ we have $$\tilde{h}^*(\Psi_{g_2}^0(w_2))=\Psi_{g_1}^0(\tilde{h}_0^{-1}(w_2)).$$ Let us verify why this is true.
The left-hand side of the expression is by definition $$\tilde{h}^*(\Psi_{g_2}^0(w_2))(\cdot)=\tilde{h}^*(g_2(\chi_2(w_2))(w_2,\cdot))=g_2(\chi_2(w_2))(w_2,\tilde{h}(\cdot))=g_1(h^{-1}(\chi_2(w_2)))(\tilde{h}_0^{-1}(w_2),\cdot),$$ where the last equality follows from the compatibility of pseudo-metrics $g_1$ and $g_2$. The right-hand side of the expression is $$\Psi_{g_1}^0(\tilde{h}_0^{-1}(w_2))(\cdot)=g_1(\chi_1(\tilde{h}_0^{-1}(w_2)))(\tilde{h}_0^{-1}(w_2),\cdot),$$ and it remains to observe that $h^{-1}(\chi_2(w_2))=\chi_1(\tilde{h}_0^{-1}(w_2))$, simply because $\tilde{h}_0$ is a lift of $h$, and the statement is proven.
#### The map $\Psi_{g_2}^0\cup_{(\tilde{h}_0^{-1},\tilde{h}^*)}\Psi_{g_1}^0$
We conclude this section by giving the precise formula for the map $\Psi_{g_2}^0\cup_{(\tilde{h}_0^{-1},\tilde{h}^*)}\Psi_{g_1}^0$, which we will need in the next section. As follows from the general construction of gluing of two smooth maps, we have $$\left(\Psi_{g_2}^0\cup_{(\tilde{h}_0^{-1},\tilde{h}^*)}\Psi_{g_1}^0\right)(w^0)=\left\{\begin{array}{ll}
j_1^{W_2^*}\circ\Psi_{g_2}^0\circ(j_1^{W_2^0})^{-1} & \mbox{on }(\chi_2^0\cup_{(\tilde{h}_0^{-1},h^{-1})}\chi_1^0)^{-1}(i_1^{X_2}(X_2\setminus h(Y)))\\
j_2^{W_1^*}\circ\Psi_{g_1}^0\circ(j_2^{W_1^0})^{-1} & \mbox{on }(\chi_2^0\cup_{(\tilde{h}_0^{-1},h^{-1})}\chi_1^0)^{-1}(i_2^{X_1}(X_1)).
\end{array}\right.$$
### Proving the gluing-dual commutativity
We now give our final statement, which is a sufficient condition (and, together with Theorem \[gluing-dual:commute:dual:pseudometrics:compatible:thm\], a criterion) for the gluing-dual commutativity condition to be satisfied.
Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be diffeological vector pseudo-bundles, locally trivial and with finite-dimensional fibres, and let $(\tilde{f},f)$ be a gluing of $V_1$ to $V_2$, with an invertible $f$. Suppose that $V_1$ and $V_2$ admit pseudo-metrics compatible with this gluing, and let $g_1$ and $g_2$ be a fixed choice of such pseudo-metrics. Assume, finally, that the induced pseudo-metrics $g_2^*$ and $g_1^*$ on the dual pseudo-bundles $V_2^*$ and $V_1^*$ are compatible with the gluing along $(\tilde{f}^*,f^{-1})$. Then $V_1$, $V_2$, and $(\tilde{f},f)$ satisfy the gluing-dual commutativity condition.
Let us first show that $(V_1\cup_{\tilde{f}}V_2)^*$ and $V_2^*\cup_{\tilde{f}^*}V_1^*$ are diffeomorphic, and then explain why there is a canonical diffeomorphism between them. Applying Proposition \[dual:characteristic:gluing:commute:prop\](3) and then (2), we obtain $$(V_1\cup_{\tilde{f}}V_2)^*\cong(V_1\cup_{\tilde{f}}V_2)^0\cong V_1^0\cup_{\tilde{f}_0}V_2^0;$$ next, by Lemma \[switch:characteristic:is:diffeo:lem\] and Proposition \[dual:characteristic:gluing:commute:second:part:prop\] we obtain $$V_1^0\cup_{\tilde{f}_0}V_2^0\cong V_2^0\cup_{\tilde{f}_0^{-1}}V_1^0\cong V_2^*\cup_{\tilde{f}^*}V_1^*,$$ as wanted. It remains to see that the diffeomorphisms involved produce in the end the canonical commutativity diffeomorphism $\Phi_{\cup,*}$.
Let us now specify the final diffeomorphism that we obtain from the above sequence. Let $\tilde{g}$ be the pseudo-metric on $V_1\cup_{\tilde{f}}V_2$ induced by $g_1$ and $g_2$. Then the first diffeomorphism of the sequence is $(\Psi_{\tilde{g}}^0)^{-1}$; the second is $(\Phi^0)^{-1}$, the inverse of the diffeomorphism described in the proof of Proposition \[dual:characteristic:gluing:commute:prop\], the third is the switch-like map $\varphi_{V_1^0\leftrightarrow V_2^0}$, and the fourth is $\Psi_{g_2}^0\cup_{(\tilde{f}_0^{-1},\tilde{f}^*)}\Psi_{g_1}^0$. We need to see that the composition $$\Phi=\left(\Psi_{g_2}^0\cup_{(\tilde{f}_0^{-1},\tilde{f}^*)}\Psi_{g_1}^0\right)\circ\left(\varphi_{V_1^0\leftrightarrow V_2^0}\right)\circ(\Phi^0)^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}$$ coincides with the appropriate canonically defined map $\Phi_{\cup,*}$.
Indeed, after some obvious cancelations the pointwise description of the diffeomorphism $\Phi$ is as follows: $$\Phi(v)=\left\{\begin{array}{ll}
\left(j_2^{V_1^*}\circ\Psi_{g_1}^0\circ(j_1^{V_1})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(v) & \mbox{if }(\pi_1\cup_{(\tilde{f},f)}\pi_2)^*(v)\in i_1^{X_1}(X_1\setminus Y)\\
\left(j_2^{V_1^*}\circ\Psi_{g_1}^0\circ\tilde{f}_0^{-1}\circ(j_2^{V_2})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(v) & \mbox{if }(\pi_1\cup_{(\tilde{f},f)}\pi_2)^*(v)\in i_2^{X_2}(f(Y))\\
\left(j_1^{V_2^*}\circ\Psi_{g_2}^0\circ(j_2^{V_2})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(v) & \mbox{if }(\pi_1\cup_{(\tilde{f},f)}\pi_2)^*(v)\in i_2^{X_2}(X_2\setminus f(Y)).
\end{array}\right.$$ Let us consider the three cases indicated.
Let first $v$ be such that $(\pi_1\cup_{(\tilde{f},f)}\pi_2)^*(v)\in i_1^{X_1}(X_1\setminus Y)$; then we have
$\Phi(v)=\left(j_2^{V_1^*}\circ\Psi_{g_1}^0\circ(j_1^{V_1})^{-1}\right)\left((\Psi_{\tilde{g}}^0)^{-1}(v)\right),\mbox{ where } (\Psi_{\tilde{g}}^0)^{-1}(v)=v^0\in
j_1^{V_1}(\pi_1^{-1}(X_1\setminus Y))\cap(V_1\cup_{\tilde{f}}V_2)^0$
such that $v(\cdot)=g_1(x)((j_1^{V_1})^{-1}(v^0),(j_1^{V_1})^{-1}(\cdot))$ for $x=\pi_1((j_1^{V_1})^{-1}(v^0))$,
where $(\cdot)$ stands for the argument of $v\in(V_1\cup_{\tilde{f}}V_2)^*$, that is being taken in $j_1^{V_1}(V_1\setminus\pi_1^{-1}(Y))$. Hence we obtain $$\Phi(v)(\cdot)=j_2^{V_1^*}\left(g_1(x)((j_1^{V_1})^{-1}(v^0),\cdot)\right).$$ This time $(\cdot)$ stands for an element of $V_1^0$; it is related to the argument of $v(\cdot)$ by the map $j_1^{V_1^0}$, so we have in the end $$\Phi(v)(\cdot)=j_2^{V_1^*}\left(v(j_1^{V_1^0}(\cdot))\right)\Rightarrow \Phi(v)=\left(j_2^{V_1^*}\circ(j_1^{V_1})^*\right)(v)\Rightarrow\Phi=j_2^{V_1^*}\circ(j_1^{V_1})^*,$$ *i.e.*, the canonical form of the gluing-dual commutativity diffeomorphism.
The other two cases are rather similar. Let $v\in(V_1\cup_{\tilde{f}}V_2)^*$ be such that $(\pi_1\cup_{(\tilde{f},f)}\pi_2)^*(v)\in i_2^{X_2}(f(Y))$; we then have
$\Phi(v)=\left(j_2^{V_1^*}\circ\Psi_{g_1}^0\circ\tilde{f}_0^{-1}\circ(j_2^{V_2})^{-1}\right)\left((\Psi_{\tilde{g}}^0)^{-1}(v)\right)$, where $(\Psi_{\tilde{g}}^0)^{-1}(v)=v^0\in j_2^{V_2}\left(\pi_2^{-1}(f(Y))\right)\cap(V_1\cup_{\tilde{f}}V_2)^0$
such that $v(\cdot)=g_2(x)((j_2^{V_2})^{-1}(v^0),(j_2^{V_2})^{-1}(\cdot))$ for $x=\pi_2((j_2^{V_2})^{-1}(v^0))$.
From this, we obtain $$\Phi(v)=\left(j_2^{V_1^*}\circ\Psi_{g_1}^0\circ\tilde{f}_0^{-1}\circ(j_2^{V_2})^{-1}\right)(v^0)(\cdot)=j_2^{V_1^*}\left(g_1(f^{-1}(x))((\tilde{f}_0^{-1}\circ(j_2^{V_2})^{-1})(v^0),\cdot)\right).$$ Once again, we should relate this to the expression for $v(\cdot)$, this time keeping in mind the compatibility of the pseudo-metrics $g_1$ and $g_2$. It suffices to recall that the argument $(\cdot)$ in this case is being taken in $V_1^0$ and is related to that of $v(\cdot)$ by the map $j_2^{V_2}\circ\tilde{f}_0$. Therefore we can rewrite the expression for $\Phi(v)(\cdot)$ as follows: $$\Phi(v)(\cdot)=j_2^{V_1^*}\left(g_1(f^{-1}(x))((\tilde{f}_0^{-1}\circ(j_2^{V_2})^{-1})(v^0),\cdot)\right)=(j_2^{V_1^*}\circ\tilde{f}_0^*)\left(g_2(x)((j_2^{V_2})^{-1}(v^0),\tilde{f}_0(\cdot))\right),$$ which then allows us to conclude that $$\Phi(v)(\cdot)=\left(j_2^{V_1^*}\circ\tilde{f}_0^*\circ(j_2^{V_2})^*\right)(v)(\cdot)\Rightarrow\Phi=j_2^{V_1^*}\circ\tilde{f}_0^*\circ(j_2^{V_2})^*.$$
It remains to consider the third part of the definition of $\Phi$. Let $v\in(V_1\cup_{\tilde{f}}V_2)^*$ be such that $(\pi_1\cup_{(\tilde{f},f)}\pi_2)^*(v)\in i_2^{X_2}(X_2\setminus f(Y))$; we then have
$\Phi(v)=\left(j_1^{V_2^*}\circ\Psi_{g_2}^0\circ(j_2^{V_2})^{-1}\right)\left((\Psi_{\tilde{g}}^0)^{-1}(v)\right)$, where $(\Psi_{\tilde{g}}^0)^{-1}(v)=v^0\in
j_2^{V_2}(\pi_2^{-1}(X_2\setminus f(Y)))\cap(V_1\cup_{\tilde{f}}V_2)^0$
such that $v(\cdot)=g_2(x)((j_2^{V_2})^{-1}(v^0),(j_2^{V_2})^{-1}(\cdot))$ for $x=\pi_2((j_2^{V_2})^{-1}(v^0))$.
We therefore have $$\Phi(v)(\cdot)=\left(j_1^{V_2^*}\circ\Psi_{g_2}^0\circ(j_2^{V_2})^{-1}\right)(v^0)(\cdot)=j_1^{V_2^*}\left(g_2(x)((j_2^{V_2})^{-1}(v^0),\cdot)\right).$$ By the same considerations regarding the argument $(\cdot)$, which in this case is related to that of $v(\cdot)$ by the map $j_2^{V_2}$, we obtain $$\Phi(v)(\cdot)=j_1^{V_2^*}\left(v(j_2^{V_2^0}(\cdot))\right)\Rightarrow\Phi=j_1^{V_2^*}\circ(j_2^{V_2})^*,$$ therefore $\Phi$ has the canonical form also in the third case, whence the final claim.
Final remarks on the gluing-commutativity condition
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To conclude the discussion on the gluing-dual commutativity, we stress that the crucial point[^26] throughout was the dual map $\tilde{f}^*$ being a diffeomorphism (of its domain with its image). As follows from our proofs, it is this condition that is equivalent to both
- the *specific* map $\Phi_{\cup,*}$ being a diffeomorphism; and
- the dual pseudo-metrics $g_2^*$ and $g_1^*$ being compatible.
It remains to observe that this also justifies our choice to state right away the gluing-dual commutativity condition in terms of $\Phi_{\cup,*}$ being smooth, rather than just asking for the existence of *some* diffeomorphism between $(V_1\cup_{\tilde{f}}V_2)^*$ and $V_2^*\cup_{\tilde{f}^*}V_1^*$: the two are equivalent.
The pseudo-metrics $\tilde{g}^*$ and $\widetilde{g^*}$ on pseudo-bundles $(V_1\cup_{\tilde{f}}V_2)^*$ and $V_2^*\cup_{\tilde{f}^*}V_1^*$
========================================================================================================================================
Assuming that $V_1$, $V_2$, and $(\tilde{f},f)$ satisfy the gluing-dual commutativity condition implies, in particular, that there are two ways to construct a pseudo-metric on the pseudo-bundle $(V_1\cup_{\tilde{f}}V_2)^*\cong V_2^*\cup_{\tilde{f}^*}V_1^*$, which correspond, respectively, to the left-hand side and the right-hand side of this expression. Specifically, the (specific expression for the) pseudo-bundle on the left carries the pseudo-metric $\tilde{g}^*$ that is induced by, or dual to, the pseudo-metric $\tilde{g}$. The pseudo-bundle on the right is obtained from a given gluing of two pseudo-bundles carrying compatible pseudo-metrics; it therefore carries a pseudo-metric $\widetilde{g^*}$ that corresponds to this gluing (we mentioned this in Section 1; the details can be found in [@pseudometric-pseudobundle], and we recall what we need immediately below). We show that this is actually the same pseudo-metric.
The pseudo-metric $\tilde{g}^*$ on $(V_1\cup_{\tilde{f}}V_2)^*$
---------------------------------------------------------------
This is the pseudo-metric dual to[^27] the pseudo-metric $\tilde{g}$ on the pseudo-bundle $V_1\cup_{\tilde{f}}V_2$; the latter is defined as the composition $$\tilde{g}=\left(\Phi_{\cup,*}^{-1}\otimes\Phi_{\cup,*}^{-1}\right)\circ\Phi_{\otimes,\cup}\circ(g_2\cup_{(f^{-1},\tilde{f}^*\otimes\tilde{f}^*)}g_1)\circ\varphi_{X_1\leftrightarrow X_2},$$ where $\varphi_{X_1\leftrightarrow X_2}$ is the switch map, and $$\Phi_{\cup,*}:(V_1\cup_{\tilde{f}}V_2)^*\to V_2^*\cup_{\tilde{f}^*}V_1^*\,\,\mbox{ and }\,\, \Phi_{\otimes,\cup}:(V_2^*\otimes V_2^*)
\cup_{\tilde{f}^*\otimes\tilde{f}^*}(V_1^*\otimes V_1^*)\to(V_2^*\cup_{\tilde{f}^*}V_1^*)\otimes(V_2^*\cup_{\tilde{f}^*}V_1^*)$$ are the appropriate versions of the commutativity diffeomorphisms (see Section 4.2.1 for the explicit formula).
The pseudo-metric $\tilde{g}^*$ is then defined as the pseudo-metric dual to $\tilde{g}$, which by the usual definition means that, if $\Psi_{\tilde{g}}:V_1\cup_{\tilde{f}}V_2\to(V_1\cup_{\tilde{f}}V_2)^*$ is the (already-seen) pairing map relative to $\tilde{g}$, that is, $\Psi_{\tilde{g}}(v)=\tilde{g}((\pi_1\cup_{(\tilde{f},f)}\pi_2)(v))(v,\cdot)$, then for any $x\in X_1\cup_f X_2$ and any $v^*,w^*\in((\pi_1\cup_{(\tilde{f},f)}\pi_2)^*)^{-1}(x)$ we have by definition $$\tilde{g}^*(x)(v^*,w^*):=\tilde{g}(x)(v,w),$$ where $v,w\in(\pi_1\cup_{(\tilde{f},f)}\pi_2)^{-1}(x)$ are any two elements such that $\Psi_{\tilde{g}}(v)=v^*$ and $\Psi_{\tilde{g}}(w)=w^*$. We can avail ourselves of the already-mentioned restriction $\Psi_{\tilde{g}}^0$ of $\Psi_{\tilde{g}}$ to the characteristic sub-bundle $(V_1\cup_{\tilde{f}}V_2)^0$ and thus define $$\tilde{g}^*(x)(v^*,w^*):=\tilde{g}(x)\left((\Psi_{\tilde{g}}^0)^{-1}(v^*),(\Psi_{\tilde{g}}^0)^{-1}(w^*)\right).$$ Finally, an even more explicit formula for $\tilde{g}^*$ is $$\tilde{g}^*(x)(v^*,w^*)=\left\{\begin{array}{ll}
g_1((i_1^{X_1})^{-1}(x))(v_1,w_1), & \mbox{if }x\in\mbox{Range}(i_1^{X_1}),\\
g_2((i_2^{X_2})^{-1}(x))(v_2,w_2), & \mbox{if }x\in\mbox{Range}(i_2^{X_2}),
\end{array}\right.$$ where we have by definition $$\left\{\begin{array}{ll}
v_1=\left((j_1^{V_1})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(v^*), & w_1=\left((j_1^{V_1})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(w^*)\\
v_2=\left((j_2^{V_2})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(v^*), & w_2=\left((j_2^{V_2})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(w^*).
\end{array}\right.$$
The pseudo-metric $\widetilde{g^*}$ on $V_2^*\cup_{\tilde{f}^*}V_1^*$
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The pseudo-metric $\widetilde{g^*}$ is defined on the pseudo-bundle $V_2^*\cup_{\tilde{f}^*}V_1^*$ fibrewise, by imposing it to coincide with $g_2^*$ or $g_1^*$ on the appropriate subsets. Specifically, for $i=1,2$ let, as before, $\Psi_{g_i}:V_i\to V_i^*$ be the pairing map associated to $g_1$ and $g_2$ respectively; then for any given $x\in X_2\cup_{f^{-1}}X_1$ and any two $v^*,w^*\in(\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*)^{-1}(x)$ we define $$\widetilde{g^*}(x)(v^*,w^*)=\left\{\begin{array}{ll}
g_2^*((i_1^{X_2})^{-1}(x))((j_1^{V_2^*})^{-1}(v^*),(j_1^{V_2^*})^{-1}(w^*))=g_2((i_1^{X_2})^{-1}(x))(v_2,w_2) & \mbox{if }x\in\mbox{Range}(i_1^{X_2}), \\
g_1^*((i_2^{X_1})^{-1}(x))((j_2^{V_1^*})^{-1}(v^*),(j_2^{V_1^*})^{-1}(w^*))=g_1((i_2^{X_1})^{-1}(x))(v_1,w_1) & \mbox{if }x\in\mbox{Range}(i_2^{X_1}),
\end{array}\right.$$ where again we make reference to the characteristic sub-bundles and the corresponding invertible restrictions $\Psi_{g_2}^0$ of $\Psi_{g_2}$ and $\Psi_{g_1}^0$ of $\Psi_{g_1}$, since by construction $$\left\{\begin{array}{ll}
v_2=\left((\Psi_{g_2}^0)^{-1}\circ(j_1^{V_2^*})^{-1}\right)(v^*), & w_2=\left((\Psi_{g_2}^0)^{-1}\circ(j_1^{V_2^*})^{-1}\right)(w^*),\\
v_1=\left((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(v^*), & w_1=\left((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(w^*).
\end{array}\right.$$
Comparing $\tilde{g}^*$ and $\widetilde{g^*}$
---------------------------------------------
In the case we are considering, we have assumed[^28] that $V_1$, $V_2$, and $(\tilde{f},f)$ satisfy the gluing-dual commutativity condition. By the very definition of the latter, this means that the pseudo-bundles $(V_1\cup_{\tilde{f}}V_2)^*$ and $V_2^*\cup_{\tilde{f}^*}V_1^*$ are diffeomorphic, and in a canonical way. Since both of these pseudo-bundles also carry the canonical pseudo-metrics, described in the two sections above, it is natural to ask next whether their canonical identification, via the gluing-dual commutativity diffeomorphism $\Phi_{\cup,*}$, is an isometry relative to these pseudo-metrics. In this section we prove that it is.
More precisely, since $\tilde{g}^*$ and $\widetilde{g^*}$ are maps $$\tilde{g}^*:X_1\cup_f X_2\to(V_1\cup_{\tilde{f}}V_2)^{**}\otimes(V_1\cup_{\tilde{f}}V_2)^{**}\,\,\mbox{ and }\,\,\widetilde{g^*}:X_2\cup_{f^{-1}}X_1\to(V_2^*\cup_{\tilde{f}^*}V_1^*)^*\otimes
(V_2^*\cup_{\tilde{f}^*}V_1^*)^*,$$ we show that by adding the diffeomorphism $$(\Phi_{\cup,*}^{-1})^*\otimes(\Phi_{\cup,*}^{-1})^*:(V_1\cup_{\tilde{f}}V_2)^{**}\otimes(V_1\cup_{\tilde{f}}V_2)^{**}\to(V_2^*\cup_{\tilde{f}^*}V_1^*)^*\otimes(V_2^*\cup_{\tilde{f}^*}V_1^*)^*$$ and the switch map $\varphi_{X_1\leftrightarrow X_2}:X_1\cup_f X_2\to X_2\cup_{f^{-1}}X_2$, we obtain $$(\Phi_{\cup,*}^{-1})^*\otimes(\Phi_{\cup,*}^{-1})^*\circ\tilde{g}^*=\widetilde{g^*}\circ(\varphi_{X_1\leftrightarrow X_2}).$$ The full statement is as follows.
Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two finite-dimensional locally trivial diffeological vector pseudo-bundle, and let $(\tilde{f},f)$ be a gluing between them such that $f$ and $\tilde{f}^*$ are diffeomorphisms. Assume that there exist pseudo-metrics $g_1$ and $g_2$ on $V_1$ and $V_2$ respectively, that are compatible with the gluing along $(\tilde{f},f)$. Then the following is true: $$\left((\Phi_{\cup,*}^{-1})^*\otimes(\Phi_{\cup,*}^{-1})^*\right)\circ\tilde{g}^*=\widetilde{g^*}\circ(\varphi_{X_1\leftrightarrow X_2}).$$
Notice that the assumptions of the theorem provide for the existence of all the maps that appear in the claim, that is, for the existence of the smooth switch map $\varphi_{X_1\leftrightarrow X_2}$, that of the gluing-dual commutativity diffeomorphism $\Phi_{\cup,*}$, and the existence and compatibility of the dual pseudo-metrics $g_2^*$ and $g_1^*$, through which the pseudo-metrics $\tilde{g}^*$ and $\widetilde{g^*}$ are defined.
The actual meaning of the formula that we wish to prove is as follows: taken an arbitrary $x\in X_1\cup_f X_2$ and arbitrary $v^*,w^*\in(\pi_2^*\cup_{(\tilde{f}^*,f^{-1})}\pi_1^*)^{-1}(\varphi_{X_1\leftrightarrow
X_2}(x))\in V_2^*\cup_{\tilde{f}^*}V_1^*$, we should have $$\widetilde{g^*}(\varphi_{X_1\leftrightarrow X_2}(x))(v^*,w^*)=\tilde{g}^*(x)(\Phi_{\cup,*}^{-1}(v^*),\Phi_{\cup,*}^{-1}(w^*)).$$ Since this formula involves the switch map and a gluing-dual commutativity diffeomorphism, both of which are defined separately in three cases, we should check the desired equality in the same three cases as well. These cases are: $x\in i_1^{X_1}(X_1\setminus Y)$, $x\in i_2^{X_2}(f(Y))$, and $x\in i_2^{X_2}(X_2\setminus f(Y))$.
Let $x\in i_1^{X_1}(X_1\setminus Y)$; then $v^*,w^*\in\mbox{Range}(j_2^{V_1^*})$ and $\varphi_{X_1\leftrightarrow X_2}(x)=i_2^{X_1}((i_1^{X_1})^{-1}(x))$. Notice that the corresponding $v$ and $w$ that appear in the definition of the pseudo-metric $\widetilde{g^*}$ are then $$v=\left((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(v^*),\,\,\,w=\left((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(w^*),$$ therefore we have
$\widetilde{g^*}(\varphi_{X_1\leftrightarrow X_2}(x))(v^*,w^*)=\widetilde{g^*}(i_2^{X_1}((i_1^{X_1})^{-1}(x)))(v^*,w^*)=$
$=g_1((i_1^{X_1})^{-1}(x))\left(((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1})(v^*),((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1})(w^*)\right)$.
On the other hand, when $\tilde{g}^*$ is applied to two elements $v_1^*,w_1^*$ in the fibre of $(V_1\cup_{\tilde{f}}V_2)^*$ over a point in $i_1^{X_1}(X_1\setminus Y)$, its value is $g_1((i_1^{X_1})^{-1}(x))(v_1,w_1)$, where $$v_1=\left((j_1^{V_1})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(v_1^*),\,\,\,w_1=\left((j_1^{V_1})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(w_1^*);$$ in our case we will have $v_1^*:=\Phi_{\cup,*}^{-1}(v^*)$ and $w_1^*:=\Phi_{\cup,*}^{-1}(w^*)$. Observe now that $$\Phi_{\cup,*}^{-1}(v^*)=\left(((j_1^{V_1})^*)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(v^*)\,\,\,\mbox{ and }\,\,\,\Phi_{\cup,*}^{-1}(w^*)=\left(((j_1^{V_1})^*)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(w^*),$$ and that in the case we are considering the relation between $(\Psi_{\tilde{g}}^0)^{-1}$ and $(\Psi_{g_1}^0)^{-1}$ is as follows: $$(\Psi_{\tilde{g}}^0)^{-1}=j_1^{V_1}\circ(\Psi_{g_1}^0)^{-1}\circ(j_1^{V_1})^*.$$ Therefore we have $$v_1=\left((j_1^{V_1})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)\left(\Phi_{\cup,*}^{-1}(v^*)\right)=\left((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(v^*)$$ $$w_1=\left((j_1^{V_1})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)\left(\Phi_{\cup,*}^{-1}(w^*)\right)=\left((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(w^*),$$ hence
$\tilde{g}^*(x)(\Phi_{\cup,*}^{-1}(v^*),\Phi_{\cup,*}^{-1}(w^*))=$
$$=g_1\left((i_1^{X_1})^{-1}(x)\right)\left(((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1})(v^*),((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1})(w^*)\right)=$$
$=\widetilde{g^*}(\varphi_{X_1\leftrightarrow X_2}(x))(v^*,w^*)$,
as wanted.
Turning to the second case, let $x\in i_2^{X_2}(f(Y))$, so that $v^*,w^*\in\mbox{Range}(j_2^{V_1^*})$ and $\varphi_{X_1\leftrightarrow X_2}(x)=(i_2^{X_1}\circ f^{-1}\circ(i_2^{X_2})^{-1})(x)$. To calculate $\widetilde{g^*}(\varphi_{X_1\leftrightarrow X_2}(x))(v^*,w^*)$, write $$v=\left((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(v^*),\,\,\,w=\left((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1}\right)(w^*);$$ this implies that
$\widetilde{g^*}(\varphi_{X_1\leftrightarrow X_2}(x))(v^*,w^*)=$
$$=g_1\left((f^{-1}\circ(i_2^{X_2})^{-1})(x)\right)\left(((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1})(v^*),((\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1})(w^*)\right)=$$
$=g_2((i_2^{X_2})^{-1})(x))\left((\tilde{f}\circ(\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1})(v^*),(\tilde{f}\circ(\Psi_{g_1}^0)^{-1}\circ(j_2^{V_1^*})^{-1})(w^*)\right)$,
where we have used the compatibility of the pseudo-metrics $g_1$ and $g_2$ and the fact that $v$ and $w$ belong to the characteristic sub-bundle, on which $\tilde{f}$ is invertible by assumption.
To calculate now the second part of the identity to verify, recall that by definition $$\tilde{g}^*(x)(v_2^*,w_2^*)=g_2((i_2^{X_2})^{-1}(x))(v_2,w_2),$$ where $$v_2:=\left((j_2^{V_2})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(v_2^*)\,\,\,\mbox{ and }w_2:=\left((j_2^{V_2})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(w_2^*),$$ with $v_2^*$, $w_2^*$ denoting for brevity $$v_2^*:=\Phi_{\cup,*}^{-1}(v^*),\,\,\,w_2^*:=\Phi_{\cup,*}^{-1}(w^*).$$ We now recall that in the case we are considering, $$\Phi_{\cup,*}^{-1}(v^*)=\left(((j_1^{V_1})^*)^{-1}\circ\tilde{f}^*\circ(j_2^{V_2^*})^{-1}\right)(v^*)\,\,\,\mbox{ and }\,\,\,
\Phi_{\cup,*}^{-1}(w^*)=\left(((j_1^{V_1})^*)^{-1}\circ\tilde{f}^*\circ(j_2^{V_2^*})^{-1}\right)(w^*),$$ and there are the following relations between $\Psi_{\tilde{g}}^0$, $\Psi_{g_1}^0$, and $\Psi_{g_2}^0$ (which we state immediately for their inverses): $$(\Psi_{\tilde{g}}^0)^{-1}=j_2^{V_2}\circ(\Psi_{g_2}^0)^{-1}\circ(\tilde{f}^*)^{-1}\circ(j_1^{V_1})^*\,\,\,\mbox{ and }\,\,\,
(\Psi_{g_2}^0)^{-1}=\tilde{f}\circ(\Psi_{g_1}^0)^{-1}.$$ Thus, by direct calculation $$v_2=\left((\Psi_{g_2}^0)^{-1}\circ(j_2^{V_2^*})^{-1}\right)(v^*)\,\,\,\mbox{ and }\,\,\,w_2=\left((\Psi_{g_2}^0)^{-1}\circ(j_2^{V_2^*})^{-1}\right)(w^*),$$ which means that $$\tilde{g}^*(x)(v_2^*,w_2^*)=\widetilde{g^*}(\varphi_{X_1\leftrightarrow X_2}(x))(v^*,w^*),$$ again as wanted.
Finally, let us consider the third case, that of $x\in i_2^{X_2}(X_2\setminus f(Y))$; then $v^*,w^*\in\mbox{Range}(j_1^{V_2^*})$ and $\varphi_{X_1\leftrightarrow X_2}(x)=\left(i_1^{X_2}\circ(i_2^{X_2})^{-1}\right)(x)$. Furthermore, $$\widetilde{g^*}(\varphi_{X_1\leftrightarrow X_2}(x))(v^*,w^*)=g_2((i_2^{X_2})^{-1}(x))(v,w),$$ where $$v=\left((\Psi_{g_2}^0)^{-1}\circ(j_1^{V_2^*})^{-1}\right)(v^*)\,\,\,\mbox{ and }\,\,\,w=\left((\Psi_{g_2}^0)^{-1}\circ(j_1^{V_2^*})^{-1}\right)(w^*).$$
On the other side, we have $$\tilde{g}^*(x)(v_2^*,w_2^*)=g_2((i_2^{X_2})^{-1}(x))(v_2,w_2),$$ where $$v_2=\left((j_2^{V_2})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(v_2^*)\,\,\,\mbox{ and }\,\,\,w_2=\left((j_2^{V_2})^{-1}\circ(\Psi_{\tilde{g}}^0)^{-1}\right)(w_2^*),$$ and in turn $$v_2^*=(\Phi_{\cup,*})^{-1}(v^*)=\left(((j_2^{V_2})^*)^{-1}\circ(j_1^{V_2^*})^{-1}\right)(v^*),\,\,\,w_2^*=(\Phi_{\cup,*})^{-1}(v^*)=\left(((j_2^{V_2})^*)^{-1}\circ(j_1^{V_2^*})^{-1}\right)(w^*).$$ Finally, we observe that there is the following relation between $\Psi_{\tilde{g}}^0$ and $\Psi_{g_2}^0$ (stated again for their inverses): $$(\Psi_{\tilde{g}}^0)^{-1}=j_2^{V_2}\circ(\Psi_{g_2}^0)^{-1}\circ(j_2^{V_2})^*.$$ Thus, putting together consecutively the expressions for $v_2$, $v_2^*$, and $(\Psi_{\tilde{g}}^0)^{-1}$ (and likewise, for $w_2$, $w_2^*$, and $(\Psi_{\tilde{g}}^0)^{-1}$), we obtain $$v_2=\left((\Psi_{g_2}^0)^{-1}\circ(j_1^{V_2^*})^{-1}\right)(v^*)=v\,\,\,\mbox{ and }\,\,\,w_2=\left((\Psi_{g_2}^0)^{-1}\circ(j_1^{V_2^*})^{-1}\right)(w^*)=w,$$ which implies that $$\tilde{g}^*(x)(v_2^*,w_2^*)=\widetilde{g^*}(\varphi_{X_1\leftrightarrow X_2}(x))(v^*,w^*),$$ and therefore concludes our consideration of the third case. All cases having thus been exhausted, the proof is finished.
We can re-phrase our main conclusion by stating the following.
\[gluing:dual:commut:diffeo:is:isometry:cor\] The gluing-dual commutativity diffeomorphism $\Phi_{\cup,*}$ is a pseudo-bundle isometry between $\left((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*\right)$ and $\left(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*}\right)$.
The covariant Clifford algebras
===============================
We choose this umbrella term to refer to the pseudo-bundles of Clifford algebras that are associated to whatever pseudo-bundles we obtain by interposing the operations of gluing and taking the dual pseudo-bundle. These are, first of all, the pseudo-bundles of Clifford algebras associated to $(V_1\cup_{\tilde{f}}V_2)^*$ and $V_2^*\cup_{\tilde{f}^*}V_1^*$; as we have seen above, these pseudo-bundles are *a priori* different, but they are naturally identified under appropriate assumptions. Once these assumptions are imposed, we still have another *a priori* difference, that of the two natural pseudo-metrics $\tilde{g}^*$ and $\widetilde{g^*}$ on them being different, the possibility treated in the preceding section. Finally, there is a third option, that of the result of gluing of two pseudo-bundles of Clifford algebras, those associated to $V_2^*$ and $V_1^*$. In this section we complete our consideration of these three cases, and of how they are interrelated.
The diffeomorphism ${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)\cong{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$
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The diffeomorphism in question is easily obtained from the gluing-dual commutativity diffeomorphism $\Phi_{\cup,*}$, whose existence we assume and which in this case both guarantees that the pseudo-metrics $\tilde{g}^*$ and $\widetilde{g^*}$ exist and are well-defined, and, by Corollary \[gluing-dual:commute:dual:pseudometrics:compatible:thm\], is an isometry with respect to them. Extending $\Phi_{\cup,*}$ to a diffeomorphism between ${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)$ and ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ is then a standard procedure, whose result we denote by $$\Phi_{\cup,*}^{{C \kern -0.1em \ell}}:{C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*}).$$ To describe this diffeomorphism, it suffices to recall that for any equivalence class of form $$[v_1\otimes\ldots\otimes v_k]\in\left((\pi_1\cup_{(\tilde{f},f)}\pi_2)^{{C \kern -0.1em \ell}}\right)^{-1}(Y)\subset{C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*),$$ with a representative $v_1\otimes\ldots\otimes v_k$, its image is the equivalence class in ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ of $$\Phi_{\cup,*}(v_1')\otimes\ldots\otimes\Phi_{\cup,*}(v_k').$$ In other words, $\Phi_{\cup,*}^{{C \kern -0.1em \ell}}$ is the pushdown, by the quotient projections $$\pi^{T((V_1\cup_{\tilde{f}}V_2)^*)}:T((V_1\cup_{\tilde{f}}V_2)^*)\to{C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)\mbox{ and }
\pi^{T(V_2^*\cup_{\tilde{f}^*}V_1^*)}:T(V_2^*\cup_{\tilde{f}^*}V_1^*)\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*}),$$ of the map $\bigoplus_n\Phi_{\cup,*}^{\otimes n}$, so that we have $$\Phi_{\cup,*}^{{C \kern -0.1em \ell}}\circ\pi^{T((V_1\cup_{\tilde{f}}V_2)^*)}=\pi^{T(V_2^*\cup_{\tilde{f}^*}V_1^*)}\circ\left(\bigoplus_n\Phi_{\cup,*}^{\otimes n}\right).$$
The map $\Phi_{\cup,*}^{{C \kern -0.1em \ell}}:{C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ given by the identity $$\Phi_{\cup,*}^{{C \kern -0.1em \ell}}\circ\pi^{T((V_1\cup_{\tilde{f}}V_2)^*)}=\pi^{T(V_2^*\cup_{\tilde{f}^*}V_1^*)}\circ\left(\bigoplus_n\Phi_{\cup,*}^{\otimes n}\right)$$ is a well-defined diffeomorphism.
This is a consequence of Corollary \[gluing:dual:commut:diffeo:is:isometry:cor\].
The pseudo-bundle ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$
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There is a third pseudo-bundle, with all fibres Clifford algebras, that is naturally associated to our data, the pseudo-bundle ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$. It is obtained by gluing together the pseudo-bundles of Clifford algebras relative to, respectively, $(V_2^*,g_2^*)$ and $(V_1^*,g_1^*)$, along the map $(\tilde{F}^*)^{{C \kern -0.1em \ell}}$, induced by the map $\tilde{f}^*$; this is the construction described in Section 1.5.2. To indicate how this construction works specifically for this case, it suffices to say that, for any given equivalence class in $((\pi_2^*)^{{C \kern -0.1em \ell}})^{-1}(f(Y))\subset{C \kern -0.1em \ell}(V_2^*,g_2^*)$, with an arbitrary representative $v_1^*\otimes\ldots\otimes v_k^*$, its image under $(\tilde{F}^*)^{{C \kern -0.1em \ell}}$ is the equivalence class in ${C \kern -0.1em \ell}(V_1^*,g_1^*)$ of the element $$\tilde{f}^*(v_1^*)\otimes\ldots\otimes\tilde{f}^*(v_k^*).$$
The main point, however, is that the resulting pseudo-bundle ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ is naturally diffeomorphic to the pseudo-bundle ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ via the diffeomorphism $$\Phi^{{C \kern -0.1em \ell}(*)}:{C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$$ that is determined by the following formula: $$\Phi^{{C \kern -0.1em \ell}(*)}=\left\{\begin{array}{ll} (j_1^{V_2^*})^{{C \kern -0.1em \ell}}\circ(j_1^{{C \kern -0.1em \ell}(V_2^*)})^{-1} & \mbox{on }\left((\pi_2^*)^{{C \kern -0.1em \ell}}\cup_{((\tilde{F}^*)^{{C \kern -0.1em \ell}},f^{-1})}(\pi_1^*)^{{C \kern -0.1em \ell}}\right)^{-1}(i_1^{X_2}(X_2\setminus f(Y))) \\
(j_2^{V_1^*})^{{C \kern -0.1em \ell}}\circ(j_2^{{C \kern -0.1em \ell}(V_1^*)})^{-1} & \mbox{on }\left((\pi_2^*)^{{C \kern -0.1em \ell}}\cup_{((\tilde{F}^*)^{{C \kern -0.1em \ell}},f^{-1})}(\pi_1^*)^{{C \kern -0.1em \ell}}\right)^{-1}(i_2^{X_1}(X_1)),
\end{array}\right.$$ where $$(j_1^{V_2^*})^{{C \kern -0.1em \ell}}:{C \kern -0.1em \ell}(V_2^*,g_2^*)\setminus((\pi_2^*)^{Cl})^{-1}(f(Y))\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*}) \mbox{ and }
(j_2^{V_1^*})^{{C \kern -0.1em \ell}}:{C \kern -0.1em \ell}(V_1^*,g_1^*)\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$$ are the fibrewise extensions to ${C \kern -0.1em \ell}(V_2^*,g_2^*)\setminus((\pi_2^*)^{Cl})^{-1}(f(Y))$ and to ${C \kern -0.1em \ell}(V_1^*,g_1^*)$, respectively, of the two natural inclusions $$V_2^*\setminus(\pi_2^*)^{-1}(f(Y))\hookrightarrow{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})\mbox{ and } V_1^*\hookrightarrow{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*}).$$ The latter inclusions are in turn given by the compositions of either $j_1^{V_2^*}:V_2^*\setminus(\pi_2^*)^{-1}(f(Y))\to V_2^*\cup_{\tilde{f}^*}V_1^*$ or $j_2^{V_1^*}:V_1^*\to V_2^*\cup_{\tilde{f}^*}V_1^*$, with the standard inclusion $V_2^*\cup_{\tilde{f}^*}V_1^*\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$.
The map $\Phi^{{C \kern -0.1em \ell}(*)}:{C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ thus defined is a smooth diffeomorphism.
This is a consequence of Theorem 5.5 in [@clifford-alg], applied to $V_2^*$, $V_1^*$, and $\tilde{f}^*$; notice that $\widetilde{g^*}$ is exactly the counterpart of the pseudo-metric $\tilde{g}$ that appears in the statement of the theorem just cited, in that it is obtained from $g_2^*$ and $g_1^*$ in precisely the same way as $\tilde{g}$ is obtained from $g_1$ and $g_2$.
The three diffeomorphisms $\Phi_{\cup,*}^{{C \kern -0.1em \ell}}$, $\Phi^{{C \kern -0.1em \ell}(*)}$, and $\Phi_{\cup}^{{C \kern -0.1em \ell}(*)}$
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We now summarize the above by listing the three possible forms of the Clifford algebra pseudo-bundle, together with the assumptions that allow us to identify them to each other, and with the corresponding diffeomorphisms.
#### The assumptions
As (almost) everywhere throughout the paper, we consider two finite-dimensional diffeological vector pseudo-bundles $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$, and a gluing of the former to the latter along the pair $(\tilde{f},f)$. In order for the three diffeomorphisms to exist, we must also assume the following:
- the two pseudo-bundles are locally trivial;[^29]
- $f$ and $\tilde{f}^*$ are diffeomorphisms;
- $V_1$ and $V_2$ admit compatible pseudo-metrics $g_1$ and $g_2$ respectively.
#### The three shapes of the pseudo-bundle of covariant Clifford algebras, and their equivalences
Under the assumptions just listed, the following three pseudo-bundles are well-defined: $${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*),\,\,\,{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*}),\,\,\,{C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*).$$ Although *a priori* these could be three different pseudo-bundles, the same assumptions that guarantee that all three are well-defined at the same time, also guarantee that they are in fact equivalent, via the following diffeomorphisms, already described above:
- $\Phi_{\cup,*}^{{C \kern -0.1em \ell}}:{C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$;
- $\Phi^{{C \kern -0.1em \ell}(*)}:{C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)\to{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$; and
- $\left(\Phi^{{C \kern -0.1em \ell}(*)}\right)^{-1}\circ\Phi_{\cup,*}^{{C \kern -0.1em \ell}}=:\Phi_{\cup}^{{C \kern -0.1em \ell}(*)}:{C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)\to{C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$.
The pseudo-bundles of exterior algebras, and gluing
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We now turn to considering the pseudo-bundles of exterior algebras, first in the contravariant case, and then in the covariant case. We recall their construction, which is standard, and concentrate on the interactions with the operation of gluing.
The contravariant case
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We first consider the contravariant version of the exterior algebra, by which we mean the following. Let first $V$ be a diffeological vector space; for each tensor power of $V$ consider the alternating operator[^30] $$\mbox{Alt}:\underbrace{V\otimes\ldots\otimes V}_n\to\underbrace{V\otimes\ldots\otimes V}_n,$$ acting, as usual, by $$\mbox{Alt}(v_1\otimes\ldots\otimes v_n)=\frac{1}{n!}\sum_{\sigma}(-1)^{\mbox{sgn}(\sigma)}v_{\sigma(1)}\otimes\ldots\otimes v_{\sigma(n)}$$ and extended by linearity. In this section the $n$-th exterior power of $V$ is the image $$\bigwedge_n(V)=\mbox{Alt}(\underbrace{V\otimes\ldots\otimes V}_n);$$ the whole exterior algebra $\bigwedge_*(V)$ is the direct sum of all $\bigwedge_n(V)$. We obtain the pseudo-bundle $\bigwedge_*(V)$ of exterior algebras associated to a given pseudo-bundle $\pi:V\to X$ by employing the same operations in the pseudo-bundle version, and defining the alternating operator fibrewise.
### The induced gluing map $\tilde{f}^{\bigwedge_*}$
This map is provided by the universal factorization property for alternating maps. Specifically, let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two finite-dimensional diffeological vector pseudo-bundles, and let $(\tilde{f},f)$ be a gluing between them. Then the restriction $\tilde{f}|_{\pi_1^{-1}(y)}$ of $\tilde{f}$ to each fibre in its domain of definition is a smooth linear map between diffeological vector spaces $\pi_1^{-1}(y)$ and $\pi_2^{-1}(f(y))$, and the direct sum of all tensor degrees of $\tilde{f}|_{\pi_1^{-1}(y)}$ is again a smooth linear map between the tensor algebras of these spaces: $$\bigoplus_n\left(\tilde{f}|_{\pi_1^{-1}(y)}\right)^{\otimes n}:T(\pi_1^{-1}(y))\to T(\pi_2^{-1}(f(y))).$$ By construction, this map commutes with the two respective alternating operators, so its restriction, that we denote by $\left(\tilde{f}|_{\pi_1^{-1}(y)}\right)^{\bigwedge_*}$, to $\bigwedge_*(\pi_1^{-1}(y))$ is a smooth linear map between the exterior algebras of the two fibres: $$\left(\tilde{f}|_{\pi_1^{-1}(y)}\right)^{\bigwedge_*}:\bigwedge_*(\pi_1^{-1}(y))\to\bigwedge_*(\pi_2^{-1}(f(y))).$$ Finally, the collection $$\tilde{f}^{\bigwedge_*}:=\bigcup_{y\in Y}\left(\tilde{f}|_{\pi_1^{-1}(y)}\right)^{\bigwedge_*},$$ where $Y$ is the domain of definition of $f$, yields a smooth and fibrewise linear map $\tilde{f}^{\bigwedge_*}$ between the appropriate subsets of $\bigwedge_*(V_1)$ and $\bigwedge_*(V_2)$. Thus, it yields an induced gluing between the corresponding pseudo-bundles of contravariant exterior algebras.
### The pseudo-bundles $\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$ and $\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$
In a similar manner, for the pseudo-bundle $V_1\cup_{\tilde{f}}V_2$ there is its own alternating operator $\mbox{Alt}$, whose image is the pseudo-bundle $\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$. Since each fibre of $T(V_1\cup_{\tilde{f}}V_2)$ coincides with either a fibre of $T(V_1)$ or one of $T(V_2)$, and fibrewise each of the three alternating operators under consideration (those relative to $V_1$, $V_2$, and $V_1\cup_{\tilde{f}}V_2$) is the usual one of a diffeological vector space, it makes sense to expect the two pseudo-bundles of exterior algebras, $\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$ and $\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$, to be diffeomorphic. Indeed, they are, and it is not difficult to describe the natural diffeomorphism between them; it is based on the gluing-tensor product commutativity diffeomorphism $\Phi_{\cup,\otimes}$, as the next construction shows.
#### A preliminary remark
At this moment we explicitly impose the assumption that for each of our pseudo-bundles $V_1$ and $V_2$ (and accordingly, for all the results of their gluings, their duals, and any mixture of such), the set of the dimensions of their fibres has a finite upper limit. We denote it by $$\dim_V=\sup_{x_1\in X_1,x_2\in X_2}\{\dim(\pi_1^{-1}(x_1)),\dim(\pi_2^{-1}(x_2))\};$$ we do not go into any detail about how this correlates with any other assumptions of ours, just note that we will need for one of the diffeomorphisms that we define in the next paragraph (specifically, we use to ensure that $\Phi^{\Lambda_*}$ is indeed onto).
#### The diffeomorphism $\Phi_{\cup,\otimes}^{(\otimes n)}:\left(V_1\cup_{\tilde{f}}V_2\right)^{\otimes n}\to V_1^{\otimes n}\cup_{\tilde{f}^{\otimes n}}V_2^{\otimes n}$
We first describe the construction of this diffeomorphism, which is by induction on $n$. The base of the induction is $n=2$, in which case $\Phi_{\cup,\otimes}^{(\otimes n)}=\Phi_{\cup,\otimes}$, the already-mentioned gluing-tensor product commutativity diffeomorphism. Suppose that $\Phi_{\cup,\otimes}^{(\otimes(n-1))}$ has already been defined. Then $\Phi_{\cup,\otimes}^{(\otimes n)}$ is obtained as the composition $$\left(V_1\cup_{\tilde{f}}V_2\right)^{\otimes(n-1)}\otimes\left(V_1\cup_{\tilde{f}}V_2\right)\to\left(V_1^{\otimes(n-1)}\cup_{\tilde{f}^{\otimes n}}V_2^{\otimes
(n-1)}\right)\otimes\left(V_1\cup_{\tilde{f}}V_2\right)\to V_1^{\otimes n}\cup_{\tilde{f}^{\otimes n}}V_2^{\otimes n},$$ where the first arrow stands for $\Phi_{\cup,\otimes}^{(\otimes(n-1))}\otimes\mbox{Id}_{V_1\cup_{\tilde{f}}V_2}$, and the second one, for the version $\Phi_{\cup,\otimes}^{\tilde{f}^{\otimes(n-1)},\tilde{f}}$ of the gluing-tensor product commutativity diffeomorphism applied to the case of two factors, $V_1^{\otimes(n-1)}\cup_{\tilde{f}^{\otimes n}}V_2^{\otimes (n-1)}$ and $V_1\cup_{\tilde{f}}V_2$. We can summarize the whole construction as $$\Phi_{\cup,\otimes}^{(\otimes n)}=\Phi_{\cup,\otimes}^{\tilde{f}^{\otimes(n-1)},\tilde{f}}\circ\left(\Phi_{\cup,\otimes}^{(\otimes(n-1))}\otimes\mbox{Id}_{V_1\cup_{\tilde{f}}V_2}\right).$$ Note also that we will denote the inverse of $\Phi_{\cup,\otimes}^{(\otimes n)}$ by $\Phi_{\otimes,\cup}^{(\otimes n)}$. Finally, for all $k$ (limited in practice by $\dim_V$) we define a diffeomorphism $$\Phi_{\cup,\oplus}^{(k)}:\bigoplus_{n=0}^k\left(V_1^{\otimes n}\cup_{\tilde{f}^{\otimes n}}V_2^{\otimes n}\right)\to\left(\oplus_n V_1^{\otimes
n}\right)\cup_{\oplus_n\tilde{f}^{\otimes n}}\left(\oplus_n V_2^{\otimes n}\right);$$ the construction is exactly the same, just using the gluing-direct sum commutativity diffeomorphism $\Phi_{\cup,\oplus}$ in place of $\Phi_{\cup,\otimes}$.
#### The diffeomorphism $\Phi_{\cup,\otimes}^{(\otimes n)}$, and the alternating operators
We write $\mbox{Alt}^{(n)}$ for the restriction of the alternating operator $\mbox{Alt}$ on $V_1\cup_{\tilde{f}}V_2$ onto the $n$-th tensor degree; likewise, $\mbox{Alt}_i^{(n)}$ stands for the same restriction of the alternating operator on $V_i$, for $i=1,2$. It is then quite trivial to observe that we have $$\mbox{Alt}^{(n)}=\Phi_{\otimes,\cup}^{(\otimes n)}\circ\left(\mbox{Alt}_1^{(n)}\cup_{\left(\tilde{f}^{\otimes
n},\tilde{f}^{\otimes}\right)}\mbox{Alt}_2^{(n)}\right)\circ \Phi_{\cup,\otimes}^{(\otimes n)};$$ equivalently, $$\Phi_{\cup,\otimes}^{(\otimes n)}\circ\mbox{Alt}^{(n)}=\left(\mbox{Alt}_1^{(n)}\cup_{\left(\tilde{f}^{\otimes
n},\tilde{f}^{\otimes}\right)}\mbox{Alt}_2^{(n)}\right)\circ\Phi_{\cup,\otimes}^{(\otimes n)}.$$
#### The diffeomorphism $\Phi^{\bigwedge_*}:\bigwedge_*(V_1\cup_{\tilde{f}}V_2)\to \bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$
We can now define the desired diffeomorphism as $$\Phi^{\bigwedge_*}=\Phi_{\cup,\oplus}^{(\dim_V)}\circ\bigoplus_n\Phi_{\cup,\otimes}^{(\otimes n)}\mid_{\bigwedge_*(V_1\cup_{\tilde{f}}V_2)}.$$ It remains to observe that $\Phi^{\bigwedge_*}$ is indeed onto $\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$, as follows from its commutativity (in the sense explained in the previous paragraph) with the alternating operators.
The map $$\Phi^{\bigwedge_*}=\Phi_{\cup,\oplus}^{(\dim_V)}\circ\bigoplus_n\Phi_{\cup,\otimes}^{(\otimes n)}\mid_{\bigwedge_*(V_1\cup_{\tilde{f}}V_2)}$$ is a pseudo-bundle diffeomorphism $\bigwedge_*(V_1\cup_{\tilde{f}}V_2)\to\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$ covering the identity map on $X_1\cup_f X_2$.
The pseudo-bundles of covariant exterior algebras
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We now consider the covariant case. The basic definition is simple: the **covariant exterior algebra** $\bigwedge(V)$ of a pseudo-bundle $V$ is $\bigwedge_*(V^*)$, the contravariant exterior algebra of its dual pseudo-bundle. So the reason why we consider it separately is to study its behavior with respect to the gluing, which, as we know, is not always well-behaved with respect to duality.
### The induced map between $\bigwedge(V_2)$ and $\bigwedge(V_1)$
Indeed, let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two finite-dimensional diffeological vector pseudo-bundles, and let $(\tilde{f},f)$ be a gluing between them such that $f$ is invertible. The gluing map between $\bigwedge(V_2)$ and $\bigwedge(V_1)$ is defined exactly as $\tilde{f}^{\bigwedge_*}$, but it is based on the dual map $\tilde{f}^*$. This gluing map is denoted by $\tilde{f}^{\bigwedge}$ and is in fact $$\tilde{f}^{\bigwedge}:=(\tilde{f}^*)^{\bigwedge_*}.$$
### The pseudo-bundles $\bigwedge(V_1\cup_{\tilde{f}}V_2)$ and $\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$
As in the contravariant case, there are two natural pseudo-bundles of exterior algebras to consider, namely those mentioned in the title of this section. It is also natural to wonder whether they are diffeomorphic; we show that indeed they are, under the assumption that the gluing-dual commutativity condition is satisfied, by constructing a certain pseudo-bundle diffeomorphism $$\Phi^{\bigwedge}:\bigwedge(V_1\cup_{\tilde{f}}V_2)\to\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$$ covering the switch map.
#### The $n$-th degree component of $\Phi^{\bigwedge}$
Let $\Phi_{\cup,\otimes}^{(\otimes n)}:\left(V_2^*\cup_{\tilde{f}^*}V_1^*\right)^{\otimes
n}\to(V_2^*)^{\otimes n}\cup_{(\tilde{f}^*)^{\otimes n}}(V_1^*)^{\otimes n}$ be the diffeomorphism constructed in the previous section. The $n$-th tensor degree component of $\Phi^{\bigwedge}$ is the composition $$\Phi_{\cup,\otimes}^{(\otimes n)}\circ\Phi_{\cup,*}^{\otimes n}:\left((V_1\cup_{\tilde{f}}V_2)^*\right)^{\otimes n}\to
\left(V_2^*\cup_{\tilde{f}^*}V_1^*\right)^{\otimes n}\to(V_2^*)^{\otimes n}\cup_{(\tilde{f}^*)^{\otimes n}}(V_1^*)^{\otimes n}.$$ Notice that if $$\mbox{Alt}_{\cup,*}:\left((V_1\cup_{\tilde{f}}V_2)^*\right)^{\otimes n}\to\left((V_1\cup_{\tilde{f}}V_2)^*\right)^{\otimes n},\,\,\,
\mbox{Alt}_2:(V_2^*)^{\otimes n}\to(V_2^*)^{\otimes n}\mbox{ and }\mbox{Alt}_1:(V_1^*)^{\otimes n}\to(V_1^*)^{\otimes n}$$ are the $n$-th degree alternating operators on $(V_1\cup_{\tilde{f}}V_2)^*$, $V_2^*$, and $V_1^*$ respectively, then we have $$\left(\Phi_{\cup,\otimes}^{(\otimes n)}\circ\Phi_{\cup,*}^{\otimes n}\right)\circ\mbox{Alt}_{\cup,*}=
\left(\mbox{Alt}_2\cup_{\left((\tilde{f}^*)^{\otimes n},(\tilde{f}^*)^{\otimes n}\right)}\mbox{Alt}_1\right)\circ\left(\Phi_{\cup,\otimes}^{(\otimes n)}\circ\Phi_{\cup,*}^{\otimes n}\right).$$
#### The diffeomorphism $\Phi^{\bigwedge}$
We now employ also the gluing-direct sum commutativity diffeomorphism $\Phi_{\cup,\oplus}^{(dim_{V^*})}$, also from the previous section, to obtain $\Phi^{\bigwedge}$. Indeed, we define $$\Phi^{\bigwedge}=\Phi_{\cup,\oplus}^{(\dim_{V^*})}\circ\bigoplus_{n=0}^{(\dim_{V^*})}\left(\Phi_{\cup,\otimes}^{(\otimes n)}
\circ\Phi_{\cup,*}^{\otimes n}\right)\mid_{\bigwedge(V_1\cup_{\tilde{f}}V_2)}.$$ This is a well-defined injective, smooth and fibrewise linear map on $\bigwedge(V_1\cup_{\tilde{f}}V_2)$; that its image is $\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$, follows from the commutativity between each $\Phi_{\cup,\otimes}^{(\otimes n)}\circ\Phi_{\cup,*}^{\otimes n}$ and the appropriate alternating operators (see above).
The map $$\Phi^{\bigwedge}=\Phi_{\cup,\oplus}^{(\dim_{V^*})}\circ\bigoplus_{n=0}^{(\dim_{V^*})}\left(\Phi_{\cup,\otimes}^{(\otimes
n)}\circ\Phi_{\cup,*}^{\otimes n}\right)\mid_{\bigwedge(V_1\cup_{\tilde{f}}V_2)}$$ is a pseudo-bundle diffeomorphism $\bigwedge(V_1\cup_{\tilde{f}}V_2)\to\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$ covering the switch map $\varphi_{X_1\leftrightarrow X_2}$.
The Clifford actions
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In this section we consider all possible (shapes of) Clifford actions, first outlining what acts on what, and then establishing the various natural equivalences.
The outline
-----------
As we have seen in the preceding sections, there is a multitude of formally distinct, but (as we are about to see) equivalent with respect to the diffeomorphisms described in the previous two sections, Clifford actions relative to a given gluing of $(V_1,g_1)$ and $(V_2,g_2)$. In this section we give a list of these actions and their equivalences, with proofs and details appearing in the two sections immediately following. As before, we assume that $f$ and $\tilde{f}^*$ are diffeomorphisms, and $g_1$ and $g_2$ are compatible.
### The contravariant case
Recall that in this case we have two natural Clifford algebras,[^31] specifically $${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})\cong{C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$$ (recall that the diffeomorphism that we have between them is $\Phi^{{C \kern -0.1em \ell}}:{C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)\to{C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$) and two natural exterior algebras, $$\bigwedge_*(V_1\cup_{\tilde{f}}V_2)\cong\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2).$$
#### Summary of actions
The natural actions are, the standard action $c_*$ of ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ on $\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$, and the composite action $\tilde{c}_*:=c_1\cup_{(\tilde{F}^{{C \kern -0.1em \ell}},\tilde{f}^{\bigwedge_*})}c_2$ of ${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$ on $\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$, where $c_i$ is the standard action of ${C \kern -0.1em \ell}(V_i,g_i)$ on $\bigwedge_*(V_i)$. We will show that this is a partial case of the construction considered in [@clifford-alg].
#### The equivalence of the two actions
This is expressed by the formula: $$\Phi^{\bigwedge_*}(c_*(v)(e))=\tilde{c}_*\left((\Phi^{{C \kern -0.1em \ell}})^{-1}(v)\right)\left(\Phi^{\bigwedge_*}(e)\right)$$ for all $v\in{C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ and $e\in\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$ such that $\pi^{{C \kern -0.1em \ell}}(v)=\pi^{\bigwedge_*}(e)$. Below we will explain why this relation does hold.
### The covariant case
There are three Clifford algebras to consider: $${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)\cong{C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})\cong{C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*),$$ and essentially two exterior algebras: $$\bigwedge(V_1\cup_{\tilde{f}}V_2)\cong\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1),$$ to which we will also add the contravariant exterior algebra $\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$.
#### Summary of Clifford actions
We now outline which Clifford algebra (or the result of gluing of such) acts on which pseudo-bundle of exterior algebras:
- ${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)$ acts on $\bigwedge(V_1\cup_{\tilde{f}}V_2)$ via the standard Clifford action $c^*$;
- ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ acts on $\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$ via the action $c_{*,\cup}$ (see Proposition 6.8) induced by the standard Clifford actions $c_2^*$ and $c_1^*$ of ${C \kern -0.1em \ell}(V_2^*,g_2^*)$ and ${C \kern -0.1em \ell}(V_1^*,g_1^*)$ on $\bigwedge(V_2)$ and $\bigwedge(V_1)$ respectively;
- ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ has, again, the standard Clifford action, which we have not mentioned yet and which we now denote by $\tilde{c}_{*,\cup}$, on $\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$.
#### The equivalence of actions
As in the contravariant case, the actions $c^*$, $c_{*,\cup}$, and $\tilde{c}_{*,\cup}$ turn out to be equivalent, with the equivalence established via the diffeomorphisms $\Phi^{{C \kern -0.1em \ell}(*)}$, $\Phi_{\cup,*}^{{C \kern -0.1em \ell}}$, and $\Phi_{\cup}^{{C \kern -0.1em \ell}(*)}$, as well as $\Phi^{\bigwedge}$ and $\Phi_{\cup,*}^{\bigwedge}:\bigwedge(V_1\cup_{\tilde{f}}V_2)\to
\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$. Specifically, we have:
- the action $c^*$ of ${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)$ on $\bigwedge(V_1\cup_{\tilde{f}}V_2)$ is related to the action $\tilde{c}_{*,\cup}$ of ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ on $\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$, with respect to the diffeomorphisms $\Phi_{\cup,*}^{{C \kern -0.1em \ell}}$ and $\Phi_{\cup,*}^{\bigwedge}$, via $$\Phi_{\cup,*}^{\bigwedge}(c^*(v)(e))=\tilde{c}_{*,\cup}(\Phi_{\cup,*}^{Cl}(v))(\Phi_{\cup,*}^{\bigwedge}(e)),$$ that holds for all $v\in{C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)$ and $e\in\bigwedge(V_1\cup_{\tilde{f}}V_2)$ such that $\pi^{\bigwedge}(e)=\pi^{{C \kern -0.1em \ell}}(v)$;
- the action $c_{*,\cup}$ of ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ on $\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$ is equivalent to the action $\tilde{c}_{*,\cup}$ of ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ on $\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$, with respect to the diffeomorphisms $\Phi^{{C \kern -0.1em \ell}(*)}$ and $\Phi_{\cup}^{{C \kern -0.1em \ell}(*)}$, via $$\left(\Phi_{\cup,*}^{\bigwedge}\circ(\Phi^{\bigwedge})^{-1}\right)(c_{*,\cup}(v)(e))=\tilde{c}_{*,\cup}\left(\Phi^{cl(*)}(v)\right)\left((\Phi_{\cup,*}^{\bigwedge}\circ(\Phi^{\bigwedge})^{-1})(e)\right)$$ that is true for all $v\in{C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ and $e\in\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$ such that $(\pi_2^{\bigwedge}\cup_{(\tilde{f}^{\bigwedge},f^{-1})}\pi_1^{\bigwedge})(e)=(\pi_2^{{C \kern -0.1em \ell}}\cup_{((\tilde{F}^*)^{{C \kern -0.1em \ell}},f^{-1})}\pi_1^{{C \kern -0.1em \ell}})(v)$.
Notice that these equivalences imply also the equivalence of $c^*$ to $\tilde{c}_{*,\cup}$.
The standard Clifford action is smooth
--------------------------------------
The basis for several versions of the Clifford(-type) actions listed above is the usual action of the (contravariant) Clifford algebra on the corresponding (also contravariant) exterior algebra. This means the following.
Let $\pi:V\to X$ be any locally trivial finite-dimensional diffeological vector pseudo-bundle that admits a pseudo-metric; let $g$ be a fixed choice of a pseudo-metric on it. The **standard Clifford action** of ${C \kern -0.1em \ell}(V,g)$ on $\bigwedge_*(V)$ is the map $c:{C \kern -0.1em \ell}(V,g)\to\mathcal{L}(\bigwedge_*(V),\bigwedge_*(V))$ given by $$c(v)(v_1\wedge\ldots\wedge v_k)=v\wedge v_1\wedge\ldots\wedge v_k-\sum_{j=1}^k(-1)^{j+1}v_1\wedge\ldots\wedge g(\pi(v))(v,v_j)\wedge\ldots\wedge v_k.$$ On each fibre $\pi^{-1}(x)$ of $V$, this is the usual Clifford action of the Clifford algebra relative to the bilinear symmetric form $g(x)$ on the exterior algebra of $\pi^{-1}(x)$.
\[standard:clifford:action:is:smooth:lem\] The action $c$ is smooth as a map ${C \kern -0.1em \ell}(V,g)\to\mathcal{L}(\bigwedge_*(V),\bigwedge_*(V))$.
Notice first of all that the pseudo-bundle $\bigwedge_*(V)$ smoothly splits as the direct sum $\bigoplus_k\bigwedge^kV$. It then follows from the above presentation of the action $c$ and the definition of the diffeology of ${C \kern -0.1em \ell}(V,g)$, that it suffices to show that the following two maps $c_V,c_j:V\to\mathcal{L}(\bigwedge_*(V),\bigwedge_*(V))$ are smooth: $$c_V(v)(v_1\wedge\ldots\wedge v_k)=v\wedge v_1\wedge\ldots\wedge v_k\,\mbox{ and }\,c_j(v)(v_1\wedge\ldots\wedge v_k)=v_1\wedge\ldots\wedge g(\pi(v))(v,v_j)\wedge\ldots\wedge v_k.$$ Thus, $c_V$ acts as the exterior product, which is smooth by definition (recall that the diffeology on each exterior product degree can be described as the pushforward of the tensor product diffeology by the alternating operator, which makes it, and the exterior product as a consequence, automatically smooth).
The smoothness of the map $c_j$ follows from the smoothness of the pseudo-metric $g$. To be slightly more explicit, we note that on a small enough neighborhood $U$, we can write a plot of the $k$-th exterior degree as $(p_1,\ldots,p_k)$, where each $p_i$ is a plot of $V$, acting by $u\mapsto p_1(u)\wedge\ldots\wedge p_k(u)$. Therefore the evaluation map that determines the smoothness of $c_j$ is locally of form $$(u',u)\mapsto p_1(u)\wedge\ldots\wedge g(\pi(p(u')))(p(u'),p_j(u))\wedge\ldots\wedge p_k(u)$$ for some other plot $p:U'\to V$ of $V$. Since $(u',u)\mapsto g(\pi(p(u')))(p(u'),p_j(u))$ is a smooth function, and the diffeology of $\bigwedge_*(V)$ is a (vector) pseudo-bundle diffeology, we obtain a plot of $\bigwedge_*(V)$, whence the claim.
The compatibility of two standard Clifford actions
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Likewise, we can show that under certain assumptions, two standard Clifford actions are compatible with a given gluing; this happens precisely when the gluing itself is commutative. Here is the precise statement.
\[standard:clifford:actions:compatible:prop\] Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two locally trivial finite-dimensional diffeological vector pseudo-bundles, let $(\tilde{f},f)$ be a gluing between them such that $f$ and $\tilde{f}$ are diffeomorphisms, and let $g_1$ and $g_2$ be compatible pseudo-metrics on $V_1$ and $V_2$ respectively. Let $c_i$ for $i=1,2$ be the standard Clifford actions of ${C \kern -0.1em \ell}(V_i,g_i)$ on $\bigwedge_*(V_i)$. Then for all $v,v_1,\ldots,v_k\in V_1$ such that $\pi_1(v)=\pi_1(v_1)=\ldots=\pi_1(v_k)\in Y$ we have $$\tilde{f}^{\bigwedge_*}(c_1(v)(v_1\wedge\ldots\wedge v_k))=c_2(\tilde{F}^{{C \kern -0.1em \ell}}(v))(\tilde{f}^{\bigwedge_*}(v_1\wedge\ldots\wedge v_k)).$$
By the definition of $\tilde{F}^{{C \kern -0.1em \ell}}$ and that of $\tilde{f}^{\bigwedge_*}$, we have that $$c_2(\tilde{F}^{{C \kern -0.1em \ell}}(v))(\tilde{f}^{\bigwedge_*}(v_1\wedge\ldots\wedge v_k))=c_2(\tilde{f}(v))(\tilde{f}(v_1)\wedge\ldots\wedge\tilde{f}(v_k)).$$ The desired condition easily follows from this. Indeed,
$\tilde{f}^{\bigwedge_*}(c_1(v)(v_1\wedge\ldots\wedge v_k))=$
$=\tilde{f}^{\bigwedge_*}(v\wedge v_1\wedge\ldots\wedge v_k-\sum_{j=1}^k(-1)^{j+1}g_1(\pi_1(v))(v,v_j)v_1\wedge\ldots\wedge v_{j-1}\wedge v_{j+1}\wedge\ldots\wedge v_k)=$
$=\tilde{f}(v)\wedge\tilde{f}(v_1)\wedge\ldots\wedge\tilde{f}(v_k)-$
$-\sum_{j=1}^k(-1)^{j+1}g_1(\pi_1(v))(v,v_j)\tilde{f}(v_1)\wedge\ldots\wedge\tilde{f}(v_{j-1})\wedge\tilde{f}(v_{j+1})\wedge\ldots\wedge\tilde{f}(v_k)).$
Now, since
$c_2(\tilde{f}(v))(\tilde{f}(v_1)\wedge\ldots\wedge\tilde{f}(v_k))=\tilde{f}(v)\wedge\tilde{f}(v_1)\wedge\ldots\wedge\tilde{f}(v_k)-$
$-\sum_{j=1}^k(-1)^{j+1}g_2(\pi_2(\tilde{f}(v)))(\tilde{f}(v),\tilde{f}(v_j))\tilde{f}(v_1)\wedge\ldots\wedge\tilde{f}(v_{j-1})\wedge\tilde{f}(v_{j+1})\wedge\ldots\wedge\tilde{f}(v_k))$.
The pseudo-metrics $g_1$ and $g_2$ being compatible ensures that $g_2(\pi_2(\tilde{f}(v)))(\tilde{f}(v),\tilde{f}(v_j))=g_1(\pi_1(v))(v,v_j)$, whence the claim.
The contravariant case: the actions on $\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$ and $\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$
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We now describe the action $\tilde{c}_*$, and prove its equivalence (already stated above) to the action $c_*$. Notice that $c_*$ is an instance of the standard Clifford action, so it is smooth by Lemma \[standard:clifford:action:is:smooth:lem\].
#### The action $\tilde{c}_*=c_1\cup_{(\tilde{F}^{Cl},\tilde{f}^{\bigwedge_*})}c_2$
This is a partial case of a more general construction described in [@clifford-alg]. The construction bears some similarity to that of the gluing of smooth maps, although, as mentioned in the same source, it is not quite the same thing. To describe this action, let $v\in cl(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$; then $\tilde{c}_*(v)$ is an endomorphism of the fibre of $\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$ over the point $(\pi_1^{{C \kern -0.1em \ell}}\cup_{(\tilde{F}^{{C \kern -0.1em \ell}},f)}\pi_2^{{C \kern -0.1em \ell}})(v)\in X_1\cup_f X_2$. Then the action $\tilde{c}_*$ is defined as follows: $$\tilde{c}_*(v)(e)=\left\{\begin{array}{ll}
j_1^{\bigwedge_*(V_1)}\left(c_1((j_1^{{C \kern -0.1em \ell}(V_1,g_1)})^{-1}(v))((j_1^{\bigwedge_*(V_1)})^{-1}(e))\right) & \mbox{over }i_1^{X_1}(X_1\setminus Y)\\
j_2^{\bigwedge_*(V_2)}\left(c_2((j_2^{{C \kern -0.1em \ell}(V_2,g_2)})^{-1}(v))((j_2^{\bigwedge_*(V_2)})^{-1}(e))\right) & \mbox{over }i_2^{X_2}(X_2).
\end{array}\right.$$ In other words, we just pull back $v$ and $e$ to the respective factors of gluing, apply $c_1$ or $c_2$, as appropriate, and re-insert the result into the pseudo-bundle $\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$. It now suffices to note that by Proposition \[standard:clifford:actions:compatible:prop\] $c_1$ and $c_2$ are compatible as Clifford actions, so it follows from [@clifford-alg] that the action $\tilde{c}_{*,\cup}$ is smooth.
#### The equivalence of $c_*$ to $\tilde{c}_{*,\cup}$
We now prove the already-mentioned statement of equivalence for these actions.
\[gluing:compatible:actions:commutes:with:standard:thm\] Let $v\in{C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ and $e\in\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$ be such that $\pi^{{C \kern -0.1em \ell}}(v)=\pi^{\bigwedge_*}(e)$. Then $$\Phi^{\bigwedge_*}(c_*(v)(e))=\tilde{c}_*\left((\Phi^{{C \kern -0.1em \ell}})^{-1}(v)\right)\left(\Phi^{\bigwedge_*}(e)\right).$$
The proof is almost trivial if we adopt the following viewpoint: since both actions are fibrewise based on the standard Clifford action, it suffices to assume that $v$ is an element of the copy of $V_1\cup_{\tilde{f}}V_2$ naturally contained in ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$, and that $e$ belongs to the $k$-th exterior degree of $V_1\cup_{\tilde{f}}V_2$, that is, $e=v_1\wedge\ldots\wedge v_k$ for $v_1,\ldots,v_k\in V_1\cup_{\tilde{f}}V_2$. Formally, there are two cases to consider: that of $\pi^{{C \kern -0.1em \ell}}(v)\in i_1^{X_1}(X_1\setminus Y)$ and that of $\pi^{{C \kern -0.1em \ell}}(v)\in i_2^{X_2}(X_2)$.
Thus, suppose that $\pi^{{C \kern -0.1em \ell}}(v)\in i_1^{X_1}(X_1\setminus Y)$. Since $c_*$ is the standard action, we have $$c_*(v)(e)=v\wedge v_1\wedge\ldots\wedge v_k-\sum_{j=1}^k(-1)^{j+1}v_1\wedge\ldots\wedge\tilde{g}(\pi^{{C \kern -0.1em \ell}}(v))(v,v_j)\wedge \ldots\wedge v_k.$$ Therefore
$\Phi^{\bigwedge_*}(c_*(v)(e))=j_1^{\bigwedge_*(V_1)}((j_1^{V_1})^{-1}(v)\wedge(j_1^{V_1})^{-1}(v_1)\wedge\ldots\wedge(j_1^{V_1})^{-1}(v_k)- $
$-\sum_{j=1}^k(-1)^{j+1}(j_1^{V_1})^{-1}(v_1)\wedge\ldots\wedge g_1(\pi_1((j_1^{V_1})^{-1}(v)))((j_1^{V_1})^{-1}(v),(j_1^{V_1})^{-1}(v_j))\wedge\ldots\wedge(j_1^{V_1})^{-1}(v_k))$.
It therefore suffices to note that $j_1^{{C \kern -0.1em \ell}(V_1,g_1)}((j_1^{V_1})^{-1}(v))=(\Phi^{{C \kern -0.1em \ell}})^{-1}(v)$ and $j_1^{\bigwedge_*(V_1)}((j_1^{V_1})^{-1}(v_1)\wedge\ldots\wedge(j_1^{V_1})^{-1}(v_k))=\Phi^{\bigwedge_*}(v_1\wedge\ldots\wedge v_k)$, to draw the desired conclusion. Since the treatment of the case $\pi^{{C \kern -0.1em \ell}}(v)\in i_2^{X_2}(X_2)$ is exactly the same, the proof is finished.
The covariant case
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In this case we have three potential actions, corresponding to the three shapes of the Clifford algebra and those of the three exterior algebras (one of which is actually a contravariant algebra, trivially identified to a covariant one). After a detailed description of the actions, we prove their equivalences, already announced in Section 8.1.2.
### The action $c^*$ of ${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)$ on $\bigwedge(V_1\cup_{\tilde{f}}V_2)$
This is a case of a standard Clifford action, considered in Lemma \[standard:clifford:action:is:smooth:lem\]; this lemma, in particular, ensures, that $c^*$ is a smooth action. Recall indeed that $\bigwedge(V_1\cup_{\tilde{f}}V_2)=\bigwedge_*((V_1\cup_{\tilde{f}}V_2)^*)$ by its definition.
### The action $c_{*,\cup}$ of ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ on $\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$
Let $c_2^*:{C \kern -0.1em \ell}(V_2^*,g_2^*)\to\mathcal{L}(\bigwedge(V_2),\bigwedge(V_2))$ and $c_1^*:{C \kern -0.1em \ell}(V_1^*,g_1^*)\to\mathcal{L}(\bigwedge(V_1),\bigwedge(V_1))$ be the standard Clifford actions. By Proposition \[standard:clifford:actions:compatible:prop\], they are compatible with the gluings that yield respectively ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ and $\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$, with respect to the maps $\tilde{f}^{\bigwedge}$ and $(\tilde{F}^*)^{{C \kern -0.1em \ell}}$. Thus, the procedure described in [@clifford-alg] yields a smooth action $c_{*,\cup}=c_2^*\cup_{((\tilde{F}^*)^{{C \kern -0.1em \ell}},\tilde{f}^{\bigwedge})}c_1^*$. The formula that describes it is as follows.
Let $v\in{C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ and $e\in\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$ be such that $(\pi_2^{\bigwedge}\cup_{(\tilde{f}^{\bigwedge},f^{-1})}\pi_1^{\bigwedge})(e)=(\pi_2^{{C \kern -0.1em \ell}}\cup_{((\tilde{F}^*)^{{C \kern -0.1em \ell}},f^{-1})}\pi_1^{{C \kern -0.1em \ell}})(v)$. Then we will have $$c_{*,\cup}(v)(e)=\left\{\begin{array}{ll}
j_1^{\bigwedge(V_2)}\left(c_2^*((j_1^{{C \kern -0.1em \ell}(V_2^*,g_2^*)})^{-1}(v))((j_1^{\bigwedge(V_2)})^{-1}(e))\right) & \mbox{over }i_1^{X_2}(X_2\setminus f(Y))\\
j_2^{\bigwedge(V_1)}\left(c_1^*((j_2^{{C \kern -0.1em \ell}(V_1^*,g_1^*)})^{-1}(v))((j_2^{\bigwedge(V_1)})^{-1}(e))\right) & \mbox{over }i_2^{X_1}(X_1).
\end{array}\right.$$
### The action $\tilde{c}_{*,\cup}$ of ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ on $\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$
This is also a standard Clifford action; its smoothness follows, once again, from Lemma \[standard:clifford:action:is:smooth:lem\].
### The equivalence of $c^*$ to $\tilde{c}_{*,\cup}$, and that of $c_{*,\cup}$ to $\tilde{c}_{*,\cup}$
It now remains to prove the two equivalence formulae for the covariant actions $c^*$, $c_{*,\cup}$, and $\tilde{c}_{*,\cup}$.
#### The diffeomorphism $\Phi_{\cup,*}^{\bigwedge}:\bigwedge(V_1\cup_{\tilde{f}}V_2) \to\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$
This diffeomorphism is induced by the gluing-dual commutativity diffeomorphism $\Phi_{\cup,*}$ in a completely standard way and is, by definition, the map $$\Phi_{\cup,*}^{\bigwedge}=\bigoplus_{n=0}^{\dim_{V^*}}\Phi_{\cup,*}^{\otimes n}|_{\bigwedge(V_1\cup_{\tilde{f}}V_2)}.$$ That it has, in particular, the desired range follows from the commutativity of $\Phi_{\cup,*}^{\otimes n}$ with the relevant alternating operators.
#### The equivalence of $c^*$ to $\tilde{c}_{*,\cup}$
We now show that the action $c^*$ of ${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)$ on $\bigwedge(V_1\cup_{\tilde{f}}V_2)$ is equivalent to the action $\tilde{c}_{*,\cup}$ of ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ on $\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$, via the rule described in the following statement.
Let $v\in{C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)$ and $e\in\bigwedge(V_1\cup_{\tilde{f}}V_2)$ be such that $\pi^{\bigwedge}(e)=\pi^{{C \kern -0.1em \ell}}(v)$. Then $$\Phi_{\cup,*}^{\bigwedge}(c^*(v)(e))=\tilde{c}_{*,\cup}(\Phi_{\cup,*}^{{C \kern -0.1em \ell}}(v))(\Phi_{\cup,*}^{\bigwedge}(e)).$$
The proof uses the same kind of reasoning as that of Theorem \[gluing:compatible:actions:commutes:with:standard:thm\], in which it suffices observe that both diffeomorphisms $\Phi^{{C \kern -0.1em \ell}(*)}$ and $\Phi_{\cup,*}^{\bigwedge}$ are based on the same diffeomorphism $\Phi_{\cup,*}$, and that the latter essentially commutes with the exterior product.
#### The equivalence of $c_{*,\cup}$ to $\tilde{c}_{*,\cup}$
Let us now show that the action $c_{*,\cup}$ of ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ on $\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$ is equivalent to the action $\tilde{c}_{*,\cup}$ of ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$ on $\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$. Specifically, we have the following statement.
Let $v\in{C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ and $e\in\bigwedge(V_2)\cup_{\tilde{f}^{\bigwedge}}\bigwedge(V_1)$ be such that $(\pi_2^{\bigwedge}\cup_{(\tilde{f}^{\bigwedge},f^{-1})}\pi_1^{\bigwedge})(e)=(\pi_2^{{C \kern -0.1em \ell}}\cup_{((\tilde{F}^*)^{{C \kern -0.1em \ell}},f^{-1})}\pi_1^{{C \kern -0.1em \ell}})(v)$. Then $$\left(\Phi_{\cup,*}^{\bigwedge}\circ(\Phi^{\bigwedge})^{-1}\right)(c_{*,\cup}(v)(e))=\tilde{c}_{*,\cup}\left(\Phi^{{C \kern -0.1em \ell}(*)}(v)\right)\left((\Phi_{\cup,*}^{\bigwedge}\circ(\Phi^{\bigwedge})^{-1})(e)\right).$$
This is a direct consequence of Theorem \[gluing:compatible:actions:commutes:with:standard:thm\] applied to $(V_2^*,g_2^*)$, $(V_1^*,g_1^*)$, and $(\tilde{f}^*,f^{-1})$.
Examples
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The two examples that we describe in this section are chosen with the following considerations in mind. For one thing, even when we start with some usual smooth vector bundles (as in the first example below), the gluing of them may be defined on a non-open set, producing a result which is not a smooth manifold, yet is being treated as if it were one. In the second example we consider a non-diffeomorphic gluing of fibres.
The wedge of two lines
----------------------
We start with the two pseudo-bundles $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$. Let $V_1=V_2={{\mathbb{R}}}^2$ with the standard diffeology, let $X_1=X_2={{\mathbb{R}}}$, also standard, and let $\pi_1$ and $\pi_2$ be the two standard projections on the $x$-axis: $\pi_1:V_1\ni(x,y)\mapsto x\in{{\mathbb{R}}}=X_1$ and $\pi_2:V_2\ni(x,y)\mapsto x\in{{\mathbb{R}}}=X_2$. The pseudo-bundle structure is given by imposing on each fibre $(x,y_1)+(x,y_2)\mapsto(x,y_1+y_2)$ and $\lambda(x,y_1)\mapsto(x,\lambda y_1)$. The gluing of these two pseudo-bundles is given by $(\tilde{f},f)$, where $f:X_1\supset\{0\}\to\{0\}\subset X_2$ and $\tilde{f}$ is determined by a non-zero constant $a\in{{\mathbb{R}}}$ via the rule $$\tilde{f}(0,1)=(0,a)\in V_2={{\mathbb{R}}}^2.$$
Let $f_1,f_2:{{\mathbb{R}}}\to{{\mathbb{R}}}$ be two smooth everywhere positive functions; let $g_i$ be the pseudo-metric on $V_i$ given by $$g_i(x)(v,w)=f_i(x)\cdot e^2(v)\cdot e^2(w),$$ where $e^2$ is the second element of the usual dual basis of the canonical basis of ${{\mathbb{R}}}^2$ (relative to the notation used later on it can be written as $dy$). The compatibility condition is then $f_1(0)=a^2f_2(0)$.
#### The result of gluing
The pseudo-bundle $V_1\cup_{\tilde{f}}V_2$ can be described as the union $\{(x,0,z)\}\cup\{(0,y,z)\}$ of two planes in ${{\mathbb{R}}}^3$, and, accordingly, $X_1\cup_f X_2$ as the union $\{(x,0,0)\}\cup\{(0,y,0)\}$ of the two axes, with the projection $\pi_1\cup_{(\tilde{f},f)}\pi_2$ acting by $(x,0,z)\mapsto(x,0,0)$, $(0,y,z)\mapsto(0,y,0)$. The pseudo-metric $\tilde{g}$ is then $$\tilde{g}(x,y,0)=\left\{\begin{array}{ll} f_1(x)dz^2 & \mbox{if }y=0\mbox{ and }x\neq 0,\\ f_2(y)dz^2 & \mbox{if }x=0. \end{array}\right.$$
#### The pairing maps $\Psi_{g_1}$, $\Psi_{g_2}$, and $\Psi_{\tilde{g}}$
Since all fibres are standard, the characteristic sub-bundles coincide with the pseudo-bundles themselves, so all three maps are automatically invertible. They act by: $$\Psi_{g_1}(x,y)=f_1(x)ydy,\,\,\,\Psi_{g_2}(x,y)=f_2(x)ydy,$$ and then, using the just-mentioned presentation of $V_1\cup_{\tilde{f}}V_2$ as a subset of ${{\mathbb{R}}}^3$, we have $$\Psi_{\tilde{g}}(x,y,z)=\left\{\begin{array}{ll} f_1(x)zdz & \mbox{if }y=0,\\ f_2(y)zdz & \mbox{if }x=0 \end{array}\right.$$
#### The dual pseudo-metrics
The dual pseudo-metrics are therefore described in the same manner as $g_1$ and $g_2$, but the coefficients are inverted: $$g_2^*(x)(v^*,w^*)=\frac{1}{f_2(x)}\cdot v^*(e_2)\cdot w^*(e_2)\,\,\,\mbox{ and }\,\,\,g_1^*(x)(v^*,w^*)=\frac{1}{f_1(x)}\cdot v^*(e_2)\cdot w^*(e_2).$$ The compatibility condition for $g_2^*$ and $g_1^*$ is thus $\frac{1}{f_2(0)}=\frac{a^2}{f_1(0)}$, and so is equivalent to the one for $g_1$ and $g_2$. The pseudo-metric $\tilde{g}^*\equiv\widetilde{g^*}$ is therefore $$\tilde{g}^*(x',y',0)=\left\{\begin{array}{ll}
\frac{1}{f_1(x')}\frac{\partial}{\partial z}\otimes\frac{\partial}{\partial z} & \mbox{if }y'=0,\\
\frac{1}{f_2(x')}\frac{\partial}{\partial z}\otimes\frac{\partial}{\partial z} & \mbox{if }x'=0.
\end{array}\right.$$
#### The pseudo-bundles of Clifford algebras
All fibres in our case are $1$-dimensional, so each of ${C \kern -0.1em \ell}(V_1,g_1)$, ${C \kern -0.1em \ell}(V_2,g_2)$ is thus a trivial fibering of ${{\mathbb{R}}}^3$ over ${{\mathbb{R}}}$; the result of their gluing can be described as the subset in ${{\mathbb{R}}}^4$ given by the equation $xy=0$, so that $${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)=\{(x,0,z,w),\mbox{ where }x\neq 0\}\cup\{(0,y,z,w)\},$$ with the Clifford multiplication being defined by $$(x,0,z_1,w_1)\cdot_{{C \kern -0.1em \ell}}(x,0,z_2,w_2)=(x,0,z_1w_2+z_2w_1,-f_1(x)z_1z_2+w_1w_2),$$ $$(0,y,z_1,w_1)\cdot_{{C \kern -0.1em \ell}}(0,y,z_2,w_2)=(0,y,z_1w_2+z_2w_1,-f_2(y)z_1z_2+w_1w_2).$$
From this, it is also quite evident that the result trivially coincides with ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$, so much in fact, that we can only distinguish between the two by choosing two slightly different forms of designating the same subset in ${{\mathbb{R}}}^4$. Specifically, in the case of ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ we describe the set of its points as $$\{(x,y,z,w),\mbox{ where }xy=0\}.$$ Obviously, this is the same set as we described as the set of points of ${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$; the chosen presentation of the latter emphasizes its structure as the result of a gluing.
#### The pseudo-bundles of covariant Clifford algebras
Consider again the subset in ${{\mathbb{R}}}^4$ given by the equation $xy=0$. This is the subset that is identified with all three (shapes of) the Clifford algebra. For all three possibilities, we identify the copy of $V_1^*$ contained in either of them with the hyperplane $\{(x,0,z,0)\}$, and the copy of $V_2^*$, with the hyperplane $\{(0,y,z,0)\}$; the fourth coordinate $w$ corresponds to the scalar part of the Clifford algebra.
The distinction between the various shapes of the Clifford algebra is the following one. When this subset is viewed as ${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)$, we describe the Clifford multiplication as $$\left\{\begin{array}{l}
(x,0,z_1,w_1)\cdot_{{C \kern -0.1em \ell}}(x,0,z_2,w_2)=(x,0,z_1w_2+z_2w_1,-\frac{1}{f_1(x)}z_1z_2+w_1w_2)\mbox{ for }x\neq 0,\\
(0,y,z_1,w_1)\cdot_{{C \kern -0.1em \ell}}(0,y,z_2,w_2)=(0,y,z_1w_2+z_2w_1,-\frac{1}{f_2(2)}z_1z_2+w_1w_2)\mbox{ otherwise}.
\end{array}\right.$$ On the other hand, when we view the same subset as either ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$ or ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$, we describe the corresponding product by $$\left\{\begin{array}{l}
(x,0,z_1,w_1)\cdot_{{C \kern -0.1em \ell}}(x,0,z_2,w_2)=(x,0,z_1w_2+z_2w_1,-\frac{1}{f_1(x)}z_1z_2+w_1w_2)\mbox{ for all }x,\\
(0,y,z_1,w_1)\cdot_{{C \kern -0.1em \ell}}(0,y,z_2,w_2)=(0,y,z_1w_2+z_2w_1,-\frac{1}{f_2(2)}z_1z_2+w_1w_2)\mbox{ for }y\neq 0.
\end{array}\right.$$
#### The pseudo-bundles of exterior algebras
These can be presented in exactly the same way as those of Clifford algebras. In both the contravariant and the covariant case we have a unique presentation, again as a subset of ${{\mathbb{R}}}^4$ given by the equation $xy=0$, with the exterior product $$\left\{\begin{array}{l}
(x,0,z_1,w_1)\wedge(x,0,z_2,w_2)=(x,0,z_1w_2+z_2w_1,w_1w_2),\\
(0,y,z_1,w_1)\wedge(0,y,z_2,w_2)=(0,y,z_1w_2+z_2w_1,w_1w_2)
\end{array}\right.$$
#### The Clifford actions
In the contravariant case, we have two exterior algebras, $\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$ and $\bigwedge_*(V_1)\cup_{\tilde{f}_*^{\bigwedge}}\bigwedge_*(V_2)$, with the actions $c$ and $\tilde{c}$ of, respectively, ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ and ${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$. In the former case, we have
$c((0,0,z_2,w_2))(0,0,z,w)=(0,0,z_2,w_2)\wedge(0,0,z,w)-\left(0,0,0,\tilde{g}(0,0,0)((0,0,z_2),(0,0,z))\right)=$
$=(0,0,z_2w+w_2z,w_2w+f_2(0)z_2z)$;
in the latter case, the only thing that changes with respect to the formula just given, is that the term $\tilde{g}(0,0,0)((0,0,z_2),(0,0,z)$ is replaced by the term $g_2(0,0)((0,z_2),(0,z))$, whose value however is exactly the same.
The covariant case is analogous, although we have three exterior algebras, $\bigwedge(V_1\cup_{\tilde{f}}V_2)$, $\bigwedge_*(V_2^*\cup_{\tilde{f}^*}V_1^*)$, and $\bigwedge(V_2)\cup_{(\tilde{f}^*)^{\bigwedge}}\bigwedge(V_1)$, with the actions $c$, $c_{*,\cup}$, and $c_{\cup,*}$ of, respectively, ${C \kern -0.1em \ell}((V_1\cup_{\tilde{f}}V_2)^*,\tilde{g}^*)$, ${C \kern -0.1em \ell}(V_2^*\cup_{\tilde{f}^*}V_1^*,\widetilde{g^*})$, and ${C \kern -0.1em \ell}(V_2^*,g_2^*)\cup_{(\tilde{F}^*)^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_1^*,g_1^*)$. Once again, these actions have the same form everywhere except over the point of gluing (the origin), where we would formally write the formulae for $c((0,0,z_2,w_2))(0,0,z,w)$, $c_{*,\cup}((0,0,z_2,w_2))(0,0,z,w)$, and $c_{\cup,*}((0,0,z_2,w_2))(0,0,z,w)$ with respect to $\tilde{g}^*$, $\widetilde{g^*}$, or $g_1^*$, respectively:
$c((0,0,z_2,w_2))(0,0,z,w)=(0,0,z_2,w_2)\wedge(0,0,z,w)-\left(0,0,0,\tilde{g}^*(0,0,0)((0,0,z_2),(0,0,z))\right)=$
$=(0,0,z_2w+w_2z,w_2w+\frac{1}{f_2(0)}z_2z)$,
$c_{*,\cup}((0,0,z_2,w_2))(0,0,z,w)=(0,0,z_2,w_2)\wedge(0,0,z,w)-\left(0,0,0,\widetilde{g^*}(0,0,0)((0,0,z_2),(0,0,z))\right)=$
$=(0,0,z_2w+w_2z,w_2w-g_1^*(0,0)((0,z_2),(0,z)))=(0,0,z_2w+w_2z,w_2w+\frac{1}{f_1(0)}z_2z)$,
$c_{\cup,*}((0,0,z_2,w_2))(0,0,z,w)=(0,0,z_2,w_2)\wedge(0,0,z,w)-\left(0,0,0,g_1^*(0,0)((0,z_2),(0,z))\right)=$
$=(0,0,z_2w+w_2z,w_2w+\frac{1}{f_1(0)}z_2z)$.
A non-diffeomorphism $\tilde{f}$ and diffeomorphism $\tilde{f}^*$
-----------------------------------------------------------------
Let $\pi_2:V_2\to X_2$ be the same as in the previous section, *i.e.*, the standard projection ${{\mathbb{R}}}^2\to{{\mathbb{R}}}$; define $\pi_1:V_1\to X_1$ to be the projection of $V_1={{\mathbb{R}}}^3$ to $X_1={{\mathbb{R}}}$, where $X_1$ carries the standard diffeology, and $V_1={{\mathbb{R}}}\times{{\mathbb{R}}}\times{{\mathbb{R}}}$ carries the product diffeology relative to the standard diffeologies on the first two factors and the vector space diffeology generated by the plot ${{\mathbb{R}}}\ni x\mapsto|x|$ on the third factor.[^32] The projection $\pi_1$ is just the projection onto the first factor. The gluing map $f$ for the bases is the same, $\{0\}\to\{0\}$, and the one for the total spaces is almost the same, specifically, $\tilde{f}(0,y,z)=(0,ay)$ with $a\neq 0$ (again, notice that zeroing out the third coordinate is necessary for $\tilde{f}$ to be smooth). The pseudo-bundle $\pi_2:V_2\to X_2$ carries the same pseudo-metric $g_2$ as in the previous example, while the pseudo-metric $g_1$ on $\pi_1:V_1\to X_1$ extends the previous one in a trivial manner: $$g_1(x)((x,y_1,z_1),(x,y_2,z_2))=f_1(x)y_1y_2.$$ The compatibility condition remains the same.
The entire covariant case coincides with that of the example treated in the previous section. We only consider the pseudo-bundle $V_1\cup_{\tilde{f}}V_2$ and the corresponding contravariant constructions.
#### The pseudo-bundle $V_1\cup_{\tilde{f}}V_2$
We represent it as a subset in ${{\mathbb{R}}}^4$, specifically as the union of the plane given by the equations $x=0$ and $w=0$ (the part corresponding to $V_2$), and of the set $\{y=0\}\setminus\{x=0,y=0,w=0\}$; this is the part corresponding to $V_1$, where excising the line $\{x=0,y=0,w=0\}$ reflects how $V_1\cup_{\tilde{f}}V_2$ contains $V_1\setminus\pi_1^{-1}(Y)$, and not the entire $V_1$. Thus, the entire set can be described as $$\left\{\begin{array}{ll} (x,0,z,w) & \mbox{except the points }(0,0,z,0)\\ (0,y,z,0) & \mbox{for all }y,z. \end{array}\right.$$
#### The two Clifford algebras
The Clifford algebra of $V_2$ is the already seen one; relative to the presentation of $V_1\cup_{\tilde{f}}V_2$ given above, we could describe it as a subset of ${{\mathbb{R}}}^5$, adding the 5th coordinate $u_1$ for the scalar part of ${C \kern -0.1em \ell}(V_2,g_2)\cong{{\mathbb{R}}}\oplus V_2$. Thus, $${C \kern -0.1em \ell}(V_2,g_2)=\{(0,y,z,0,u_1)\},$$ with the Clifford multiplication given by $$(0,y,z',0,u_1')\cdot_{{C \kern -0.1em \ell}}(0,y,z'',0,u_1'')=(0,y,u_1''z'+u_1'z'',0,u_1'u_1''-f_2(y)z'z'').$$
The Clifford algebra ${C \kern -0.1em \ell}(V_1,g_1)$ is bigger; since the fibres of $V_1$ have dimension $2$, each fibre of $cl(V_1,g_1)$ has dimension $4$. Thus, we represent it as a subset in ${{\mathbb{R}}}^6$, by adding the coordinates $u_1,u_2$, where $u_1$ corresponds to the scalar part and $u_2$ corresponds to the degree $2$ vector part. Thus, $${C \kern -0.1em \ell}(V_1,g_1)=\{(x,0,z,w,u_1,u_2)\},$$ with the Clifford multiplication given by
$(x,0,z',w',u_1',u_2')\cdot_{Cl}(x,0,z'',w'',u_1'',u_2'')=$
$=(x,0,z'u_1''+z''u_1',w'u_1''+w''u_1'+f_1(x)w'',u_1'u_1''-f_1(x)z'z'',u_1'u_2''+u_1''u_2')$.
Finally, ${C \kern -0.1em \ell}(V_1\cup_{\tilde{f}}V_2,\tilde{g})$ can be described as the following subset in ${{\mathbb{R}}}^6$: $$\{(x,y,z,w,u_1,u_2)\mbox{ such that }xy=0,\,x=0\Rightarrow w=u_2=0\},$$ while ${C \kern -0.1em \ell}(V_1,g_1)\cup_{\tilde{F}^{{C \kern -0.1em \ell}}}{C \kern -0.1em \ell}(V_2,g_2)$ is presented as the subset in ${{\mathbb{R}}}^6$ of the following form: $$\{(x,0,z,w,u_1,u_2)\mbox{ such that }x\neq 0\}\cup\{(0,y,z,0,u_1,0)\mbox{ for all }y,z\}.$$ The fibrewise multiplication is described by uniting the two formulae just given.
#### The contravariant exterior algebras
Likewise, the exterior algebras $\bigwedge_*(V_1)$ and $\bigwedge_*(V_2)$ are given by the same sets. Both of these we immediately represent as subsets of ${{\mathbb{R}}}^6$, with the $5$-th coordinate being the scalar part and the $6$-th coordinate being the exterior product corresponding to the exterior product relative to the $3$-rd and the $4$-th coordinates; in the case of $V_2$, this part is obviously trivial. Thus, we have $$\bigwedge_*(V_1)=\{(x,0,z,w,u_1,u_2)\},\,\,\,\bigwedge_*(V_2)=\{(0,y,z,0,u_1,0)\},$$ with the exterior product given by
$(x,0,z',w',u_1',u_2')\wedge(x,0,z'',w'',u_1'',u_2'')=$
$=(x,0,u_1''z'+u_1'z'',u_1''w'+u_1'w'',u_1'u_1'',u_1''u_2'+u_1'u_2''+z'w''-z''w')$,
$(0,y,z',0,u_1',0)\wedge(0,y,z'',0,u_1'',0)=(0,y,u_1''z'+u_1'z'',0,u_1'u_1'',0)$.
The exterior algebras $\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$ and $\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$ are then represented respectively by the sets $$\bigwedge_*(V_1\cup_{\tilde{f}}V_2)=\{(x,y,z,w,u_1,u_2),\mbox{ where } xy=0,\,x=0\Rightarrow w=u_2=0\},$$ $$\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)=\{(x,0,z,w,u_1,u_2)\mbox{ such that }x\neq 0\}\cup\{(0,y,z,0,u_1,0)\}.$$ It is obvious that the two presentations determine the same set, with the second one possibly giving a better idea of the structure of the set, and the first one allowing for the uniform description of the exterior product, in the following way:
$(x,y,z',w',u_1',u_2')\wedge(x,y,z'',w'',u_1'',u_2'')=$
$=(x,y,u_1''z'+u_1'z'',u_1''w'+u_1'w'',u_1'u_1'',u_1''u_2'+u_1'u_2''+z'w''-z''w')$.
#### The Clifford actions
It remains to describe the corresponding Clifford actions. As is standard, in the case of ${C \kern -0.1em \ell}(V_1,g_1)$, it suffices to consider the action of elements of form $(x,0,z,0,0,0)$ and $(x,0,0,w,0,0)$ on elements of form $(x,0,z,0,0,0)$, $(x,0,0,w,0,0)$, $(x,0,0,0,u_1,0)$, and $(x,0,0,0,0,u_2)$.
For these elements the multiplication is determined as follows $$\left\{\begin{array}{l}
c_1(x,0,z,0,0,0)(x,0,z',0,0,0)=(x,0,0,0,-f_1(x)z^2,0)\\
c_1(x,0,z,0,0,0)(x,0,0,w,0,0)=(x,0,0,0,0,zw)\\
c_1(x,0,z,0,0,0)(x,0,0,0,u_1,0)=(x,0,u_1z,0,0,0)\\
c_1(x,0,z,0,0,0)(x,0,0,0,0,u_2)=(x,0,0,-u_2f_1(x)z,0,0)\\
c_1(x,0,0,w,0,0)(x,0,z,0,0,0)=(x,0,0,0,0,-zw)\\
c_1(x,0,0,w,0,0)(x,0,0,w',0,0)=(x,0,0,0,-f_1(x)ww',0)\\
c_1(x,0,0,w,0,0)(x,0,0,0,u_1,0)=(x,0,0,u_1w,0,0)\\
c_1(x,0,0,w,0,0)(x,0,0,0,0,u_2)=(x,0,0,0,0,0).
\end{array}\right.$$ In the case of ${C \kern -0.1em \ell}(V_2,g_2)$, it suffices to consider the action of $(0,y,z,0,0,0)$ on elements of form $(0,y,z,0,0,0)$ and $(0,y,0,0,u_1,0)$, and we have $$\left\{\begin{array}{l}
c_2(0,y,z,0,0,0)(0,y,z',0,0,0)=(0,y,0,0,-f_2(y)zz',0) \\
c_2(0,y,z,0,0,0)(0,y,0,0,u_1,0)=(0,y,u_1z,0,0,0)
\end{array}\right.$$ Finally, the Clifford action on both $\bigwedge_*(V_1\cup_{\tilde{f}}V_2)$ and $\bigwedge_*(V_1)\cup_{\tilde{f}^{\bigwedge_*}}\bigwedge_*(V_2)$ is obtained by concatenating the two lists; the difference between the two pseudo-bundles is not seen on the level of defining the action, but rather in how we determine the two sets of points (as already been indicated above), underlying the commutativity between the gluing and the exterior product.
[99]{} <span style="font-variant:small-caps;">K.T. Chen</span>, *Iterated path integrals*, Bull. Amer. Math. Soc. (5) **1977**, pp. 831-879. <span style="font-variant:small-caps;">J.D. Christensen – G. Sinnamon – E. Wu</span>, *The D-topology for diffeological spaces*, Pacific J. of Mathematics (1) **272** (2014), pp. 87-110. <span style="font-variant:small-caps;">J.D. Christensen – E. Wu</span>, *Tangent spaces and tangent bundles for diffeological spaces*, arXiv:1411.5425v1. <span style="font-variant:small-caps;">P. Iglesias-Zemmour</span>, *Fibrations difféologiques et homotopie*, Thèse de doctorat d’État, Université de Provence, Marseille, 1985. <span style="font-variant:small-caps;">P. Iglesias-Zemmour</span>, *Diffeology*, Mathematical Surveys and Monographs, 185, AMS, Providence, 2013. <span style="font-variant:small-caps;">Y. Karshon – J. Watts</span>, *Basic forms and orbit spaces: a diffeological approach*, SIGMA (2016). <span style="font-variant:small-caps;">E. Pervova</span>, *Multilinear algebra in the context of diffeology*, arXiv:1504.08186v2. <span style="font-variant:small-caps;">E. Pervova</span>, *Diffeological Clifford algebras and pseudo-bundles of Clifford modules*, arXiv:1505.06894v2. <span style="font-variant:small-caps;">E. Pervova</span>, *On the notion of scalar product for finite-dimensional diffeological vector spaces*, arXiv:1507.03787v1. <span style="font-variant:small-caps;">E. Pervova</span>, *Diffeological vector pseudo-bundles*, Topol. Appl. **202** (2016), pp. 269-300. <span style="font-variant:small-caps;">E. Pervova</span>, *Diffeologica gluing ofl vector pseudo-bundles and pseudo-metrics on them*, Topol. Appl. (2017), http://dx.doi.org/10.1016/j.topol.2017.02.002 <span style="font-variant:small-caps;">E. Pervova</span>, *Diffeological Dirac operators and diffeological gluing*, arXiv:1701.06785v1. <span style="font-variant:small-caps;">I. Satake</span>, *The Gauss-Bonnet theorem for V-manifolds*, J. Math. Soc. Japan (4) **9** (1957), pp. 464-492. <span style="font-variant:small-caps;">J.M. Souriau</span>, *Groups différentiels*, Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Mathematics, 836, Springer, (1980), pp. 91-128. <span style="font-variant:small-caps;">J.M. Souriau</span>, *Groups différentiels de physique mathématique*, South Rhone seminar on geometry, II (Lyon, 1984), Astérisque 1985, Numéro Hors Série, pp. 341-399. <span style="font-variant:small-caps;">A. Stacey</span>, *Comparative smootheology*, Theory Appl. Categ., 25(4) (2011), pp. 64-117. <span style="font-variant:small-caps;">M. Vincent</span>, *Diffeological differential geometry*, Master Thesis, University of Copenhagen, 2008. <span style="font-variant:small-caps;">J. Watts</span>, *Diffeologies, Differential Spaces, and Symplectic Geometry*, PhD Thesis, 2012, University of Toronto, Canada. <span style="font-variant:small-caps;">E. Wu</span>, *Homological algebra for diffeological vector spaces*, Homology, Homotopy $\&$ Applications (1) **17** (2015), pp. 339-376.
University of Pisa\
Department of Mathematics\
Via F. Buonarroti 1C\
56127 PISA – Italy\
\
ekaterina.pervova@unipi.it\
[^1]: In fact, each diffeology is just a set of maps, which may wholly contain another diffeology, or be contained in one; in fact, there is a complete lattice on them on any $X$.
[^2]: We stress that this can indeed be any quotient, with no restrictions on the equivalence relation $\sim$.
[^3]: There are analogous notions of a diffeological group, diffeological algebra, and so on.
[^4]: In contrast with non-vector space diffeologies, the finest of which is always the discrete one, consisting of constant maps only.
[^5]: *A priori*, this direct-sum-diffeology-from-subset-diffeologies is finer, so not all usual direct sum decompositions of diffeological vector spaces are smooth; see an example in [@pseudometric].
[^6]: The choice of the term *diffeological vector pseudo-bundle* is ours; the same object is called just a *diffeological fibre bundle* in [@iglFibre], a *regular vector bundle* in [@vincent], and a *diffeological vector space over $X$* in [@CWtangent]. The choice of the term pseudo-bundle is meant to distinguish these objects from the numerous other versions of bundles that have appeared so far.
[^7]: This obviously poses the question of the existence of such diffeology; this was considered in [@vincent]; see also [@CWtangent], Proposition 4.6 for a relevant methodology.
[^8]: This means the topology underlying the diffeological structure, the so-called D-topology, see [@iglFibre]. In most significant examples, however, the diffeology is put on a space already carrying a topological structure, and in way such that the D-topology coincides with it.
[^9]: Notice that in the case when the gluing map $f$ is invertible as map from its domain to its range, and only in this case, $X_1\cup_f X_2$ is a span of the spaces $X_1$ and $X_2$. However, *a priori* the gluing construction is more general.
[^10]: They are defined by composition of the obvious inclusions into $X_1\sqcup X_2$, with the quotient projection $\pi$.
[^11]: Which is a natural consequence of the construction and also its merit, as it allows to treat, for instance, conical singularities as the results of gluing to a one-point space. For this reason, although in most cases we deal with gluings along invertible maps, hence symmetric ones, we treat them as if they were not, to keep the discussion as general as possible.
[^12]: According to a personal preference, one could use the language of category theory and describe the gluing operation as the pushforward of the pair of maps $\mbox{Id}:X_1\to X_1$ and $f:X_1\supset Y\to X_2$ to the category of smooth maps between diffeologies, that enjoys the bifunctoriality property.
[^13]: Relative to the product diffeology on $C^{\infty}(X_1,Z)\times C^{\infty}(X_2,Z)$ which in turn comes from the functional diffeologies on $C^{\infty}(X_1,Z)$ and $C^{\infty}(X_2,Z)$.
[^14]: The rule-of-thumb is that $i_1^{smth}$ and $i_2^{smth}$ are used respectively for the left-hand and the right-hand factor in the base space of the pseudo-bundle under consideration, while $j_1^{smth}$ and $j_2^{smth}$ refer to the left-hand and the right-hand factor of the total space.
[^15]: For us this means, via a diffeomorphism covering the switch map.
[^16]: They may have different dimensions.
[^17]: That is, one determined by their usual smooth structure.
[^18]: If one wishes, these could be described as universal factorization properties in the category of diffeological pseudo-bundles
[^19]: The notion of a pseudo-metric is designed to be a generalization from a scalar product on a vector space, where the compatibility with a given $f$ is equivalent to $f$ being an isometry (of its domain with its range).
[^20]: Or of its appropriate vector subspace, see the remark above.
[^21]: Which at the moment does not appear to be particularly limiting, in the sense that we do not know of any non-locally-trivial pseudo-bundles that admit pseudo-metrics in the first place.
[^22]: Which means that the direct sum diffeology coincides with $V$’s or $W$’s own diffeology, or, alternatively, that the composition of each plot of $V$ (respectively $W$) with the projection on $V_0$ (respectively $W_0$) is a plot of the latter.
[^23]: The choices of $f$ however could be plenty; it suffices to take $V$ the standard ${{\mathbb{R}}}^n$ and $W$ any other diffeological vector space of dimension strictly smaller than $n$. Any linear map from $V$ to $W$ is then going to be smooth (see Section 3.9 in [@iglesiasBook]).
[^24]: As we have said already, $V_0$ is uniquely defined by $V$; this is not necessarily true of $V_1$, which is defined uniquely only when the pseudo-metric has been fixed already. However, there is always at least one choice of $V_1$; what we mean at the moment is that such a choice is fixed arbitrarily.
[^25]: It is easy prove that $\psi_g$ is smooth even if we do not assume $V$ to be locally trivial.
[^26]: Under the assumption that the pseudo-bundles involved are finite-dimensional and locally trivial.
[^27]: It would be more precise to say, induced by duality.
[^28]: As is, in fact, necessary for the two pseudo-metrics just described to be well-defined.
[^29]: This is sufficient but not necessary. What we really need is that the right inverse of the pairing map, that takes values in the characteristic sub-bundle be smooth, and so it suffices that this sub-bundle split off as a smooth direct summand.
[^30]: In other terms, the antisymmetrization operator.
[^31]: A note on slight change in terminology: in the remainder of the paper we will just say *Clifford algebra* instead of a *pseudo-bundle of Clifford algebras*, and *exterior algebra* instead of *pseudo-bundle of exterior algebras*; in the present context this is unlikely to cause confusion.
[^32]: In fact, any non-standard vector space diffeology would be sufficient for our purposes.
|
---
abstract: |
In this paper we study multi-dimensional reflected backward stochastic differential equations driven by Wiener-Poisson type processes. We prove existence and uniqueness of solutions, with reflection in the inward spatial normal direction, in the setting of certain time-dependent domains.\
2000 [*Mathematics Subject Classification.*]{}\
[*Keywords and phrases: backward stochastic differential equation, reflected backward stochastic differential equation, time-dependent domain, convex domain*]{}
address:
- |
Kaj Nyström\
Department of Mathematics, Uppsala University\
S-751 06 Uppsala, Sweden
- |
Marcus Olofsson\
Department of Mathematics, Uppsala University\
S-751 06 Uppsala, Sweden
author:
- 'K. Nystr[ö]{}m, M. Olofsson'
title: 'Reflected BSDE of Wiener-Poisson type in Time-dependent Domains'
---
Introduction
============
Backward stochastic differential equations, BSDEs for short, is by now an established field of research. The solution to a classical BSDE, driven by a Wiener process $W$, is a pair of processes $(Y,Z)$ such that $$\begin{aligned}
Y_t= \xi + \int_t ^T f(s,Y_s,Z_s) ds - \int_t ^T Z_s dW_s,\ 0\leq t\leq T,\end{aligned}$$ where $\xi$ is a random variable that becomes known, with certainty, only at time $T$. In this setting $Y_t\in\mathbb R^d$, $d\geq 1$, and in the following we refer to the case $d=1$ as the one-dimensional case and to the case $d>1$ as the multi-dimensional case. Classical BSDEs have turned out important in many areas of mathematics including mathematical finance, see [@EPQ] and the long list of references therein, stochastic control theory and stochastic game theory, see, e.g., [@CK] and [@HL], as well as in the connection to partial differential equations, see, e.g., [@BBP] and [@PP].
In [@EKPPQ] a notion of *reflected* BSDE was introduced. A solution to a one-dimensional reflected BSDE is a triple of processes $(Y,Z,\Lambda)$ satisfying $$\begin{aligned}
Y_t&=& \xi + \int_t ^T f(s,Y_s,Z_s) ds +\Lambda_T-\Lambda_t- \int_t ^T Z_s dW_s,\ 0\leq t\leq T,\notag\\
Y_t&\geq& S_t,\end{aligned}$$ where the barrier $S$ is a given (one-dimensional) stochastic process. $\Lambda$ is a continuous increasing process, with $\Lambda_0=0$, pushing the process $Y$ upwards in order to keep it above the barrier. This is done with minimal energy in the sense that $$\begin{aligned}
\int_0^T (S_t-Y_t)d\Lambda_t=0,\end{aligned}$$ and consequently $\Lambda$ increases only when $Y$ is at the boundary of the space-time domain $\{(t,s): s>S_t\}$. This type of reflected BSDE has important applications in the context of American options, optimal stopping and obstacle problem, see [@EKPPQ], as well as in the context of stochastic game problems, see [@CK].
In the multi-dimensional case there are at least two different types of reflected BSDEs studied in the literature.
The first type of multi-dimensional reflected BSDE was first studied in [@GP] where the authors considered reflected BSDEs of the form $$\begin{aligned}
\label{Kn4}
Y_t&=& \xi + \int_t ^T f(s,Y_s,Z_s) ds +\Lambda_T-\Lambda_t- \int_t ^T Z_s dW_s,\ 0\leq t\leq T,\notag\\
Y_t&\in & \Omega, \ 0\leq t\leq T,\end{aligned}$$ where $\Omega\subset\mathbb R^d$. In this case $\Lambda_0=0$ and $$\begin{aligned}
\label{Kn4+}
&&\Lambda_t=\int_{0}^{t}\gamma _{s}d\left\vert \Lambda \right\vert
_{s},\ \gamma _{s}\in N^{1}\left( Y_{s}\right),\notag\\
&& d\left\vert \Lambda \right\vert \left( \left\{ t\in \left[ 0,T\right]
:Y_{t} \in \Omega\right\} \right)=0,\end{aligned}$$ where $N^{1}\left( Y_{s}\right)$ is the unit inner normal to $\Omega$ at $Y_{s}$. In particular, the process $\Lambda_t$ is of bounded total variation $|\Lambda|$ and it increases only when $Y$ is at the boundary of $\Omega$. To be more precise, when $Y$ is at the boundary it is pushed into the domain along $\gamma\in N^1(Y)$. In [@GP] existence and uniqueness for this problem is established and we note that this problem, and its analysis, is inspired by and resemble the corresponding theory for reflected stochastic differential equations, see [@T], [@S], and [@LS]. Naturally one can attempt, as in the case of reflected SDEs, to study this problem with oblique reflection instead of reflection in the direction of the inner unit normal. However, to the best of our knowledge the case of oblique reflection is a less developed area of research in the context of BSDEs and we are only aware of the work in [@R], where the author studies an obliquely reflected BSDE in an orthant.
The second type of multi-dimensional reflected BSDEs occurs in the study of optimal switching problems and stochastic games, see, e.g., [@AF], [@AH], [@DHP], [@HT], [@HZ], and references therein. In the generic optimal switching problem a production facility is considered and it is assumed that the production can run in $d \geq 2$ different production modes. Furthermore, there is a stochastic process $X=(X_t)_{t\geq 0}$ which stands for the market price of the underlying commodities and other financial parameters that influence the production. When the facility is in mode $i$, the revenue per unit time is $f_i(t,X_t)$ and the cost of switching from mode $i$ to mode $j$, at time $t$, is $c_{ij}(t,X_t)$. Let $(Y_t^1,\dots,Y_t^d)$ be the value function associated with the optimal switching problem, on the time interval $[t,T]$, i.e., $Y_t^i$ stands for the optimal expected profit if, at time $t$, the production is in mode $i$. In this case, one can prove, under various assumptions, see [@AF], [@AH], and [@DHP], that $(Y_t^1,\dots,Y_t^d)$ solves the reflected BSDE $$\begin{aligned}
\label{eq6}
&Y^i_t=\xi_i+\int_t^Tf_i(s, X_s)ds-\int_t^TZ_s^idW_s+\Lambda_T^i-\Lambda_t^i,\notag\\
&Y^i_t\geq \max_{j\in A_i} \left (Y^j_t - c_{ij}(t,X_t)\right), \notag \\
&\int_0^T \left ( Y^i_t-\max_{j\in A_i} \left (Y^j_t -c_{ij}(t,X_t) \right ) \right )d\Lambda_t^i=0,\end{aligned}$$ where $i\in\{1,\dots,d\}$, $0\leq t\leq T,$ and $A_i=\{1,\dots,d\}\setminus\{i\}$. In this case the reflected BSDE evolves in the closure of the time-dependent domain $$\begin{aligned}
\label{eq6+}
D&=&\{(t,y)=(t, y_1,\dots,y_d)\in\mathbb R^{d+1}:\ 0\leq t\leq T,\notag\\
&&y_i \geq \max_{j\in A_i} \left ( y_j - c_{ij}(t,X_t) \right ), \mbox{ for all }i\in\{1,\dots,d\}\}.\end{aligned}$$ On the boundary of $D$ a reflection occurs and in [@HT] the authors refer to this as an oblique reflection. While this oblique reflection seems to have no clear relation to what is referred to as an oblique reflection in the context of , , it is still fair to refer to the problem in as an obliquely reflected BSDE. However, we emphasize that the problems in , and are significantly different.
In this paper we consider the problem in , in time-dependent domains and with underlying stochastic processes beyond Brownian motion. In light of , , and corresponding developments for reflected SDEs, see [@C], [@CGK], [@LS], [@NO], [@S], and [@T], it is natural to allow for time-dependent domains and in many cases this extra feature calls for additional arguments in comparison with the case of time-independent domains. In particular, we here consider , in the context of time-dependent domains, and along the lines of [@GP]. In addition, we allow the BSDE to be driven by a Wiener-Poisson type process and our main result is a generalization of [@GP] and [@O] to a time-dependent setting. In general, it seems difficult to generalize [@GP] and its proofs beyond the assumption of convexity of the time-slices of the domain. Indeed, the assumption on convexity is heavily explored in [@GP] and [@O]. Beyond ensuring the existence of projections, convexity establishes the positivity of certain terms appearing when applying the Ito formula. In this sense, one may say that the arguments are slightly rigid as the structural assumption of convexity seems crucial. In our analysis, it turns out that we are only able to pull the arguments of [@GP] through in the context of time-dependent domain having a similar rigidity in time. More precisely, in our case the time slices must be non-increasing and hence the domain must be non-expanding as a function of time. Under such a structural assumption though, we are able to generalize [@GP] and [@O] to a time-dependent setting. Finally, we note that it is an interesting open problem to understand if, in analogy with the connection between optimal switching problems and the problem in , the problem in , can be naturally associated to some stochastic optimization problem.
Statement of main result
========================
In this section we state our main result. To do this properly we first briefly discuss the geometry and processes of Wiener-Poisson type and define the reflected BSDE studied in this paper.
Geometry
--------
Given $d\geq 1$, we let $\left\langle \cdot ,\cdot
\right\rangle $ denote the standard inner product on $\mathbb{R}^{d}$ and $\left\vert z\right\vert =\left\langle z,z\right\rangle
^{1/2}$ be the Euclidean norm of $z\in\mathbb R^d.$ Whenever $z\in\mathbb{R}^{d}$ and $r>0$, we let $B_r(z)$ and $S_r(z)$ denote the ball and sphere of radius $r$, centered at $z$, respectively, i.e. $B_{r}\left( z\right) =\left\{ y\in\mathbb{R}^{d}:\left\vert z-y\right\vert <r\right\} $ and $S_{r}\left( z\right)=\left\{ y\in\mathbb{R}^{d}:\left\vert z-y\right\vert =r\right\}$. Moreover, given $F\subset\mathbb{R}^{d}$, $E\subset\mathbb{R}
^{d}$, we let $\bar{F}$, $\bar{E}$ be the closure of $F$ and $E$, respectively, and we let $d\left( y,E\right) $ denote the Euclidean distance from $y\in\mathbb{R}^{d}$ to $E$. Given $d\geq 1$, $T>0$ and an open, connected set $D^{\prime}\subset\mathbb{R}
^{d+1}$ we will refer to $$D=D^{\prime }\cap ([0,T]\times \mathbb{R}^{d}),$$as a time-dependent domain.
Given $D$ and $t\in \left[ 0,T\right] $, we define the time sections of $D$ as $$\label{eq:timeslice}
D_{t}=\left\{ z:\left( t,z\right) \in D\right\}.$$ We assume that $$\label{timedep+}
D_{t}\neq \emptyset, D_t\text{ is open, bounded and connected for every }t\in \left[ 0,T\right],$$and that $$D_{t}\text{ is convex
for every }t\in \left[ 0,T\right]. \label{timedep+1}$$Furthermore, following [CGK]{}, we let $$l\left( r\right) =\sup_{\substack{ s,t\in \lbrack 0,T] \\ \left\vert s-t\right\vert \leq r}}\,\sup_{z\in \overline{D_{s}}}d\left( z,D_{t}\right) ,$$ be the modulus of continuity of the variation of $D$ in time and we assume that $$\label{limitzero}
\lim_{r \to 0^{+}}l\left( r\right) =0.$$ We also assume that $$\label{timedep+2}
D_{t'}\subseteq D_t\text{ whenever $t'\geq t$, }t', t\in \left[ 0,T\right].$$Note that implies that $$l\left( r\right) =\sup_{\substack{ t\in \lbrack 0,T], [t-r,t+r]\in [0,T]}}\sup_{z\in \overline{D_{t-r}}}d\left( z,D_{t+r}\right).$$We let $\partial D$ and $\partial D_{t}$, for $t\in \left[ 0,T\right] $, denote the boundaries of $D$ and $D_{t}$, respectively, and we let $N_{t}\left(
z\right) $ denote the cone of inward normal vectors at $z\in \partial D_{t}$, $t\in \lbrack 0,T]$. Note that it follows from that $N_{t}\left(
z\right)\neq\emptyset $ for every $z\in \partial D_{t}$, $t\in \lbrack 0,T]$. In general, the cone $N_{t}\left( z\right) $ of inward normal vectors at $z\in \partial
D_{t}$, $t\in \lbrack 0,T]$, is defined as being equal to the set consisting of the union of the set $\left\{ 0\right\}$ and the set $$\left\{ v\in\mathbb{R}^{d}:v\neq 0,\exists \rho >0\text{ such that }B_{\rho }\left( z-\rho
v/\left\vert v\right\vert \right) \subset \mathbb{R}^{d}\setminus D_t\right\} .$$Note that this definition does not rule out the possibility of several unit inward normal vectors at the same boundary point. Given $N_{t}\left(
z\right) $, we let $N_{t}^{1}(z):=N_{t}(z)\cap S_{1}(0)$, so that $N_{t}^{1}(z)$ contains the set of vectors in $N_{t}(z)$ with unit length. In this paper we consider reflected BSDEs in the setting of time-dependent domains $D$ satisfying -. Furthermore, reflection at $z\in \partial D_{t}$, $t\in \lbrack 0,T]$, is considered in the direction of a unit spatial inward normal in the cone $N_{t}\left(
z\right)$.
Processes of Wiener-Poisson type
--------------------------------
Throughout the paper we let $$\left(\Omega ,\mathcal{F} ,\{ \mathcal{F}_t\}_{t \in [0,T]}, \mathbb{P}\right)$$ be a complete Wiener-Poisson space in $\mathbb R^n\times \mathbb R^m\setminus\{0\}$ with Levy measure $\lambda$. In particular, $\left(\Omega ,\mathcal{F} ,\mathbb P \right) $ is a complete probability space and $\{\mathcal{F}_t\},_{t\in[0,T]}$ is an increasing, right continuous family of complete sub $\sigma$-algebras of $\mathcal{F}$. We let $(W_t,\{F_t\})_{t \in [0,T]}$ be a standard Wiener process in $\R^n$ and $(\mu_t, \{\F_t\})_{ t\in [0,T]}$ be a martingale measure in $\R^m \setminus\{0\}$, which is assumed to be independent of $W$, and which corresponds to a standard Poisson random measure $p(t,A)$. Indeed, for any Borel measurable subset $A$ of $\R^m \setminus \{0\}$ such that the Levy measure $\lambda$ satisfies $\lambda(A) <+ \infty $, we have $$\mu_t(A) = p(t,A) - t \lambda(A)$$ where $p(t,A)$ satisfies $$E[ p(t,A) ] = t \lambda(A).$$ We let $U:=\R^m \setminus \{0\}$ and we let $\mathcal{U}$ be its Borel $\sigma$-algebra. We assume that $\{\F_t\}_{t\in [0,T]}$ is the filtration generated by $W_t$ and the jump process corresponding to the Poission random measure $p$, augmented with the $\mathbb P$-null sets of $\F$, i.e., $$\F_t=\sigma\left( \int \int _{A \times [0,s]} p(ds,dx): s\leq t, A\in \mathcal{U} \right ) \vee \sigma \left (W_s, s\leq t \right) \vee \F_0,$$ where $\F_0$ denotes the $\P$-null sets of $\F$ and $\sigma_1 \vee \sigma_2 $ denotes, given two $\sigma$-algebras $\sigma_1$ and $\sigma_2$, the ${\sigma}$-algebra generated by ${\sigma}_1 \cup {\sigma}_2$.
Reflected BSDEs
---------------
Given $T>0$, we let $\mathcal{D}\left( \left[ 0,T\right] ,
\mathbb{R}^{d}\right) $ denote the set of càdlàg functions $v(t)=v_{t}:\left[ 0,T\right] \rightarrow
\mathbb{R}^{d}$, i.e., functions which are right continuous and have left limits. We denote the set of functions $w(t)=w_{t}:\left[ 0,T\right]
\rightarrow \mathbb{R} ^{d}$ with bounded variation by $\mathcal{BV}\left( \left[ 0,T\right] , \mathbb{R}^{d}\right) $ and we let $\left\vert w \right\vert $ denote the total variation of $w \in \mathcal{BV}\left( \left[ 0,T\right] ,\mathbb{R}^{d}\right) $. Recall that the total variation process $\left\vert w \right\vert $ is defined as $$\left\vert w \right\vert_t=\sup\sum_{k=1}^{n}| w_{t_i}-w_{t_{i-1}}|,\ 0\leq t\leq T,$$ where the supremum is taken over all finite partitions $0=t_0<t_1<\dots<t_n=t$. Furthermore, we have that $$\begin{aligned}
\label{bff}
w_t=\int_0^t\nu_sd\left\vert w \right\vert_s\end{aligned}$$ where $\nu_s$ is a vector of length 1, i.e., $|\nu_s|=1$ for $\left\vert w \right\vert$-almost all $s$. Let $$\left(
\Omega ,\mathcal{F}, \{\mathcal{F}_t\}, \mathbb{P}, W_t,\mu_t, t\in[0,T] \right)$$ be the complete Wiener-Poisson space in $\mathbb R^n\times \mathbb R^m\setminus\{0\}$, with Levy measure $\lambda$, as outlined above. Let $L^2(\Omega, \mathcal{F}_T, \mathbb P)$ be the space of square integrable, $\mathcal{F}_T$-adapted random variables and let $L^2(U, \mathcal{U}, \lambda; \R^d)$ be the space of functions which are $\mathcal{U}$-measurable, maps values in $U$ to $\R^d$, and which are square integrable on $U$ with respect to the Levy-measure $\lambda$. In the following we let the norm $$\|z\|:=(\sum_{i,j}|z_{ij}|^2)^{1/2}$$ be defined on real-valued $(d\times n)$-dimensional matrices and we define the norm $$\| u(e) \| := \left (\int_V |u(e)|^2 \lambda(de) \right )^{1/2}$$ on $L^2(U, \mathcal{U}, \lambda; \R^d)$. Let $\xi=(\xi_1,\dots,\xi_d)$ be such that $$\begin{aligned}
\label{data}
\mbox{$\xi \in L^2(\Omega, \mathcal{F}_T, \mathbb P)$ and $\xi\in D_T$ a.s.}\end{aligned}$$ Let $f: \Omega \times [0,T] \times \R^d \times\R^{d\times n}\times L^2(U, \mathcal{U}, \lambda; \R^d) \to \R^d$ be a function such that $$\begin{aligned}
\label{data++}
(i)&&\mbox{$(\omega,t)\to f(\omega,t,y,z,u) $ is $\mathcal{F}_t$ progressively measurable whenever}\notag\\
&&\mbox{$(y,z,u)\in \R^d \times\R^{d\times n}\times L^2(U, \mathcal{U}, \lambda; \R^d)$},\notag \\
(ii)&&\mbox{$E \bigl [\int_0 ^T |f(\omega,t,0,0,0)|^2 dt\bigr ] < \infty $},\notag\\
(iii)&&\mbox{$|f(\omega, t, y, z, u)-f(\omega, t, y', z', u')|\leq c(|y-y'|+\|z-z'\|+\|u-u'\|)$}\notag\\
&&\mbox{for some constant $c$ whenever}\notag\\
&&\mbox{$(y,z,u), (y',z',u') \in \R^d \times \R^{d\times n} \times L^2(U, \mathcal{U}, \lambda; \R^d)$, $(\omega,t)\in \Omega \times [0,T]$}.\end{aligned}$$ In the context of BSDEs, $\xi$ and $f$ are usually referred to as terminal value and driver of the BSDE, respectively. We are now ready to formulate the notion of reflected BSDE considered in this paper.
\[rbsde\] Let $d\geq 1$ and $T>0$. Let $D\subset\mathbb{R}^{d+1}$ be a time-dependent domain satisfying . Given $(\xi,f)$ as in -, a quadruple $(Y_t, Z_t, U_t, \Lambda_t)$ of progressively measurable processes with values in $\R^d \times \R^{d \times m} \times L^2(U, \mathcal{U}, \lambda; \R ^d) \times \R^d$ is said to be a solution to a reflected BSDE, with reflection in the inward spatial normal direction, in $D$, and with data $(\xi,f),$ if the following holds. $Y\in \mathcal{D}\left( \left[ 0,T\right] ,
\mathbb{R}^{d}\right) $, $Z$ and $U$ are predictable processes, and $$\begin{aligned}
(i)&&E\left [\sup _{0 \leq t \leq T} |Y_t|^2\right ] <\infty, \\
(ii)&& E \left [\int_0 ^T \|Z_t\|^2 dt + \int _0^T \int _U|U_s(e)|^2 \lambda(de)ds \right ] < \infty,\notag\\
(iii)&&Y_t= \xi + \int_t ^T f(s,Y_s,Z_s,U_s) ds + \Lambda_T- \Lambda_t\notag\\
&&- \int_t ^T Z_s dW_s - \int _t^T \int _U U_s(e) \mu(de,ds) \quad \mbox{a.s.}, \\
(iv)&& Y_t \in\overline{D_t} \quad \mbox{a.s.},\end{aligned}$$ whenever $t \in [0, T]$. Furthermore, $\Lambda\in \mathcal{BV}\left( \left[ 0,T\right] ,\mathbb{R}
^{d}\right) $ and $$\begin{aligned}
(v)&&\Lambda_t=\int_{0}^{t^{}}\gamma _{s}d\left\vert \Lambda \right\vert
_{s},\ \gamma _{s}\in N_{s}^{1}\left( Y_{s}\right) \mbox{ whenever } Y_s \in \partial D_{s},\\
(vi)&& d\left\vert \Lambda \right\vert \left( \left\{ t\in \left[ 0,T\right]
:\left(t,Y_{t}\right) \in D\right\} \right)=0.\end{aligned}$$
Statement of the main result
----------------------------
Concerning reflected BSDEs we establish the following result.
\[maint1\] Let $d\geq 1$ and $T>0$. Let $D\subset \mathbb{R}^{d+1}$ be a time-dependent domain satisfying - and assume the terminal data $\xi$ and driver $f$ satisfy -. Then there exists a unique solution $(Y_t, Z_t, U_t, \Lambda_t)$ to the reflected BSDE, with reflection in the inward spatial normal direction, in $D$, and with data $(\xi,f),$ in the sense of Definition \[rbsde\].
Organization of the paper
-------------------------
The rest of the paper is organized as follows. Section \[sec:prel\] is of preliminary nature and we here focus on the geometry of the time-dependent domain as well as smooth approximations of it. We also recall the Ito formula in the context of Wiener-Poisson processes. In section \[sec:lemmas\] we introduce a sequence of non-reflected BSDEs, constructed by penalization techniques, and develop a number of technical lemmas for these. Finally, using the results of section \[sec:lemmas\], we prove the main result in section \[sec:proof\].
Preliminaries {#sec:prel}
=============
Let $d\geq 1$ and $T>0$. Let $D\subset\mathbb{R}^{d+1}$ be a time-dependent domain satisfying - . Let $N =N_{t}(z)=N (t,z)$ denote the cone of inward normal vectors given for all $z\in \partial D_{t}$, $t\in \lbrack 0,T]$. Note that by there exists, for any $y\in\mathbb{R}^{d}\setminus \overline{D}_{t}$, $t\in \lbrack 0,T]$, at least one projection of $y$ onto $\partial D_{t}$ along $N_{t}$, denoted $\pi\left( t,y\right) $, which satisfies$$\left\vert y-\pi\left(t, y\right) \right\vert=d\left( y,D_{t}\right).$$ To have $\pi(t,\cdot)$ well-defined for all $y\in\mathbb R^d$ we also let $\pi\left(t, y\right)=y$ whenever $y\in\overline{D_t}$. The following lemma summarizes a few standard results from convex analysis.
\[lemmaa\] Let $D\subset
\mathbb{R}^{d+1}$ be a time-dependent domain satisfying and assume and . Then the following holds whenever $t\in[0,T]$: $$\begin{aligned}
(i)&&\langle y^\prime-y,y-\pi\left(t, y\right)\rangle \leq 0,\mbox{ for } (y,y^\prime) \in \mathbb R^d\times\overline{D_t}, \mbox{ and }\notag \\
(ii)&&\langle y^\prime-y, y-\pi(t,y)\rangle \leq \langle y^\prime-\pi(t,y^\prime),y-\pi(t,y)\rangle,\notag \\
(iii)&& |\pi(t,y) - \pi(t,y')| \leq |y-y'|, $$ whenever $(y,y^\prime) \in \mathbb R^d\times\mathbb R^d$. Furthermore, there exists $P_T\in D_T$ and $\gamma$, $1\leq\gamma<\infty$, depending on $d(P_T, \partial D_T)$, such that $$\begin{aligned}
(iv)&& \langle y-P_T, y-\pi(t,y)\rangle \geq \gamma^{-1} |y-\pi(t,y)|, \mbox{ for any } y\in \mathbb R^d, t\in[0,T].\end{aligned}$$
Geometry of time-dependent domains - smooth approximations
----------------------------------------------------------
Note that the assumptions in , , and contain no particular smoothness assumption. Instead, we will in the following construct smooth approximations of $D$ to enable the use of Ito’s formula. In the following we let $h(t,y)=d(y,D_t)$ whenever $y\in\mathbb R^d$, $t\in [0,T]$, with the convention that $h(t,y)=0$ if $y\in\overline{D_t}$. Assuming that $D_T\neq\emptyset\neq D_0$ we let $h(t,y)=h(T,y)$ whenever $y\in\mathbb R^d$, $t>T$, and $h(t,y)=h(0,y)$ whenever $y\in\mathbb R^d$, $t<0$. Using this notation we see that $$\overline D=\{(t,x)\in [0,T]\times\mathbb R^d|\ h(t,x)=0\}.$$ Let $\phi=\phi(s,y)$ be a smooth mollifier in $\mathbb R^{d+1}$, i.e., $\phi\in C_0^\infty(\mathbb R^{d+1})$, $0\leq\phi\leq 1$, the support of $\phi$ is contained in the Euclidean unit ball in $\mathbb R^{d+1}$, centered at $0$, and $\int\phi dyds=1$. Let, for $\delta>0$ small, $\phi_\delta(s,y)=\delta^{n+1}\phi(\delta^{-1}s,\delta^{-1}y)$. Based on $\phi_\delta$ we let, whenever $(t,y)\in\mathbb R\times\mathbb R^d$, $$\begin{aligned}
h_{\delta}(t,y)=(\phi_{\delta}\ast h)(t,y)=\int_{\mathbb R}\int_{\mathbb R^n}\phi_{\delta}(t-s,y-x)h(s,x)dxds\end{aligned}$$ be a smooth mollification of $h$. Furthermore, we let $$h ( F, G ) = \max ( \sup \{ d ( y, F ) : y \in G \}, \sup \{ d ( y, G ) : y \in F \} )$$ denote the Hausdorff distance between the sets $ F, G \subset \mathbb R^d$. Based on $h_{\delta}$ we introduce a smooth approximation of $D$ as follows. Given $\eta$ fixed and $\delta>0$, we let $$D^{\eta}_{\delta} = \{(t,x)\in [0,T]\times\mathbb R^d|h_\delta(t,x)< \eta\}.$$ Then $D^{\eta}_{\delta}$ converges to $D^{\eta}:= \{(t,x)\in [0,T]\times\mathbb R^d|h(t,x)< \eta\}$ in the Hausdorff distance sense as $\delta\to 0$. Note that $D^\eta_\delta$ is a $C^\infty$-smooth domain. Hence, letting $\delta, \eta \to 0$ we have the following lemma.
\[lemmauu1-\] Let $D\subset
\mathbb{R}
^{d+1}$ be a time-dependent domain satisfying - . Then, for any $\epsilon>0$ there exists a time-dependent domain $D_\epsilon\subset \mathbb{R} ^{d+1}$ satisfying - such that $D_\epsilon$ is $C^\infty$-smooth and $$\begin{aligned}
h(D_{t},D_{\epsilon,t})<\epsilon\mbox{ for all $t\in[0,T]$},\end{aligned}$$ where $D_t$ is as defined in , and $D_{{\epsilon},t}=\{x: (x,t) \in D_{\epsilon}\}$.
Let, for all $t\in \lbrack 0,T]$, $N_\epsilon =N_{\epsilon,t}(z)=N_\epsilon(t,z)$ denote the cone of inward normal vectors at $z \in \partial D_{{\epsilon},t}$. Due to the smoothness of $\partial D_{\epsilon,t}$, $N_\epsilon(t,z)$ consists of a single vector. For any $y\in
\mathbb{R}
^{d}\setminus \overline{D}_{{\epsilon},t}$, $t\in \lbrack 0,T]$, we let $\pi_\epsilon\left( t,y\right) $ denote the projection of $y$ onto $\partial D_{\epsilon,t}$ along this unique direction. To have $\pi_\epsilon(t,\cdot)$ well-defined for all $y\in\mathbb R^d$ we also let $\pi_\epsilon\left(t, y\right)=y$ whenever $y\in\overline{D}_{\epsilon,t}$. In this setting, the following lemma can be proven as Lemma 2.2 in [@GP] as we are only considering fixed time slices $D_t$ of $D$.
\[lemmauu1\] Let $D\subset\mathbb{R}^{d+1}$ be a time-dependent domain satisfying - and let $D_{\epsilon}$ be constructed as above. There exists a constant $c$ such that, if $\epsilon\in (0,1)$, $y\in\mathbb R^d$ and $t\in[0,T]$, then $$\begin{aligned}
(i)&&|\pi(t,y)-\pi_\epsilon(t,y)| \leq c \sqrt{\epsilon^2+\epsilon d(y, D_{\epsilon,t})},\\
(ii)&&|\pi(t,y)-\pi_\epsilon(t,y)| \leq c \sqrt{\epsilon^2+\epsilon d(y, D_t)}.\end{aligned}$$
The following lemma is a corollary of Lemma \[lemmauu1\].
\[lemmauu2\] Let $D$ and $D_{\epsilon}$ be as in the statement of Lemma \[lemmauu1\]. There exists a constant $c$ such that, if $\epsilon\in (0,1)$ and $y\in\mathbb R^d$, $t\in[0,T]$, then $$\begin{aligned}
(i)&&|\pi(t,y)-\pi_\epsilon(t,y)| \leq c \sqrt{\epsilon}(1+d(y, D_{\epsilon,t})),\\
(ii)&&|\pi(t,y)-\pi_\epsilon(t,y)|\leq c \sqrt{\epsilon}\sqrt{d(y, D_{t,\epsilon})}\mbox{ whenever } d(y,D_{\epsilon,t})>\epsilon.\end{aligned}$$
We here also recall Ito’s formula in the context of Wiener-Poisson processes, see [@OS]. Here and in the following, we denote by $\C^{1,2}([0,T]\times \R^d, \R)$ the space of functions $\varphi(t,y) : [0,T]\times \R^d \to \R$ which are once continuosly differentiable with respect to $t\in [0,T]$ and twice continuosly differentiable with respect to $y \in \R^d$ and we let $A^\ast$ denote the transpose of the matrix $A$.
\[lemmauu2+\] Let $Y_t$ be a Levy process such that $$dY_t = f_t dt + \sigma _t dW_t + \int _U U_t(e) \mu (de,dt)$$ and let $\varphi(t,y) \in \C^{1,2}([0,T]\times \R^d, \R)$. Then $$\begin{aligned}
d\varphi(t, Y_t) &=& \partial_t\varphi(t, Y_{t}) dt+ (\nabla \varphi(t, Y_{t^-})) \left [ f_t dt + \sigma_t dW_t + \int _U U_s(e) \mu (de,ds) \right] \notag \\
&+& \sum_{i,j}\frac{1}{2} (\sigma_t\sigma_t^\ast)_{ij}\partial^2_{y_iy_j}\varphi_\epsilon(t, Y_{t}) dt \notag \\
&+& \int_U \left [\varphi(t,Y_{t^-} + U_t(e))-\varphi(t,Y_{t^-}) - \langle \nabla \varphi(t,Y_{t^-}), U_t(e) \rangle \right] p(de,dt).\end{aligned}$$
Based on the smooth domain $D_{\epsilon}$ we define the function $\varphi_\epsilon(t,y):= (d(y,D_{\epsilon,t}))^2 = |y-\pi_\epsilon(t,y)|^2$ to which Ito’s formula needs to be applied in the proof of Theorem \[maint1\],. Note that although $D_{\epsilon}$ is a smooth domain, the second (spatial) derivative of $\varphi_{\epsilon}$ is not continuous at the boundary of $D_{{\epsilon},t}$ and thus Lemma \[lemmauu2+\] is not directly applicable. To enable the use of Ito’s formula we therefore proceed along the lines of [@LS], see also [@GP], and extend our distance function $\varphi_{\epsilon}$ across the boundary and into the domain $D_{{\epsilon},t}$. Indeed, since $\partial D_{{\epsilon},t}$ is smooth there exists a neighbourhood $V_{{\epsilon},t}$ of $\partial D_{{\epsilon},t}$ such that, for $y \in D_{{\epsilon},t} \cap V_{{\epsilon},t}$, there exists a unique pair $(x, s) \in \partial D_{{\epsilon},t} \times \R^+$ such that $y=x+s\gamma$, where $\gamma \in N^1_{{\epsilon},t}(x)$. Recall that $N^1_{{\epsilon},t}(x)$, the cone of unit inward normal vectors to $D_{{\epsilon},t}$, at $x \in \partial D_{{\epsilon},t}$, contains only a single vector. By the convexity of $D_{\epsilon,t}$ we also have $$y=x+s \gamma, \ \mbox{ for } x=\pi_\epsilon(t,y),\ s=-d(y, D_{\epsilon,t}), \ \gamma \in N^1_{{\epsilon},t}(\pi_{\epsilon}(t,x)),$$ whenever $y\in\mathbb R^d\setminus\overline{D_{\epsilon,t}}$. Hence, for $t \in [0,T]$ fixed, we can define a smooth map $\phi_\epsilon : \R^d \to\R$ such that $$\begin{aligned}
\phi_\epsilon(t,y)&=&s\quad \mbox{ when } y\in (\mathbb R^d\setminus\overline{D_{\epsilon,t}})\cup (
V_{\epsilon,t}\cap \overline{D_{\epsilon,t}}),\notag\\
\phi_\epsilon(t,y)&>&0\quad \mbox{ otherwise. }$$ Using such a function $\phi_\epsilon$ we have $$\begin{aligned}
D_\epsilon&=&\{(t,y):\ t\in [0,T], y\in\mathbb R^d,\ \phi_\epsilon(t,y)>0\},\notag\\
\partial D_{\epsilon,t}&=&\{y\in\mathbb R^d,\ \phi_\epsilon(t,y)=0\},\mbox{ for }t\in [0,T],\notag\\
(\mathbb R^d\times [0,T])\setminus\overline{D_\epsilon}&=&\{(t,y):\ t\in [0,T], y\in\mathbb R^d,\ \phi_\epsilon(t,y)<0\}.\end{aligned}$$ Note that $\phi_{\epsilon}(t,y)$ is smooth also across the boundary of $D_{{\epsilon},t}$ and that $\varphi_{\epsilon}(t,y)=(\phi_{\epsilon}(t,y)^-)^2$. Following [@GP], we can now take an approximating sequence of smooth functions $\{g_n\}_{n\geq0}$, tending to $g(x)=(x^-)^2$ as $n \to \infty$, and construct a sequence of smooth functions $\{\varphi^n_{\epsilon}(t,y)=g_n(\phi_{\epsilon}(t,y))\}_{n\geq 0}$ such that Ito’s formula can be applied to $\varphi^n_{\epsilon}$ for every $n\geq 0$ and such that $\varphi^n_{\epsilon}(t,y)$, $\frac{\partial}{\partial t}\varphi^n_{\epsilon}(t,y)$, $\frac{\partial}{\partial y_i}\varphi^n_{\epsilon}(t,y)$, $\frac{\partial^2}{\partial y_iy_j}\varphi^n_{\epsilon}(t,y)$ tend to $\varphi_{\epsilon}(t,y)$, $\frac{\partial}{\partial t}\varphi_{\epsilon}(t,y)$, $\frac{\partial}{\partial y_i}\varphi_{\epsilon}(t,y)$, $\frac{\partial^2}{\partial y_iy_j}\varphi_{\epsilon}(t,y)$, respectively, as $n \to \infty$. Having such an approximation in mind, we will from here on in slightly abuse notation and apply the Ito formula directly to $\varphi_{\epsilon}(t,y)$.
Finally, the following lemma is the result of the geometric assumptions on $D$ that we will use in the context of Ito’s formula.
\[lemmauu2ny\] Let $D$ and $D_{\epsilon}$ be as in the statement of Lemma \[lemmauu1\] and let $\varphi_{\epsilon}(t,y)$ be defined as $$\varphi_\epsilon(t,y):= (d(y,D_{\epsilon,t}))^2 = |y-\pi_\epsilon(t,y)|^2,\ t\in [0,T],\ y\in\mathbb R^d.$$ Then $$\begin{aligned}
\label{lemmauu2+-}
&(i) &\partial_t\varphi_\epsilon(t,y)\geq 0, \notag \\
&(ii) &\partial^2_{y_iy_j}\varphi_\epsilon(t,y)\xi_i\xi_j\geq 0,\end{aligned}$$ whenever $t\in [0,T],\ y, \xi \in\mathbb R^d$, and $$\label{lemmauu2+-+}
\varphi_{{\epsilon}}(t,y+z) - \varphi_{{\epsilon}}(t,y) - \langle \nabla \varphi_{{\epsilon}}(t,y) ,z \rangle\geq 0$$ whenever $t\in [0,T],\ y,z\in\mathbb R^d$.
$(i)$ follows from and $(ii)$ follows from the convexity of $D_{{\epsilon},t}$. Finally, Taylors formula and $(ii)$ yields .
Estimates for approximating problems: technical lemmas {#sec:lemmas}
======================================================
To prove the existence part of Theorem \[maint1\] we use the method of penalization. Indeed, for each $n\geq 1$, we construct a quadruple $(Y_t^n, Z_t^n, U_t ^n, \Lambda_t^n)$ through penalization and we prove that $(Y_t^n, Z_t^n, U_t ^n, \Lambda_t^n)$ converges, as $n\to \infty$, to a solution $(Y_t, Z_t, U_ t, \Lambda_t)$ of the reflected backward stochastic differential equation, with reflection in the inward spatial normal direction, in $D$, and with data $(\xi,f),$ as defined in Definition \[rbsde\]. Uniqueness is then proved by an argument based on Ito’s formula. In the following we let $(\xi,f)$ be as in -, and we let $D\subset\mathbb{R}^{d+1}$ be a time-dependent domain satisfying - . Furthermore, we let $c$ denote a generic constant which may change value from line to line.
Construction of $(Y_t^n, Z_t^n, U_t^n, \Lambda_t^n)$
----------------------------------------------------
Let, for any $n\in\mathbb Z_+$, $$\begin{aligned}
\label{ffa1}
f_n(t,y,z,u):=f(t,y,z,u)-n(y-\pi(t,y)).\end{aligned}$$ Then, for $n$ fixed, $f_n$ satisfies since $\pi$ has the Lipschitz property in space, see Lemma \[lemmaa\] $(iii)$. Hence, using results concerning existence and uniqueness for (unconstrained) BSDEs driven by Wiener-Poisson type processes, see Lemma 2.4 in [@TL], we can conclude that there exist, for each $n\in\mathbb Z_+$, a unique triple $(Y_t^n, Z_t^n, U_t^n)$ and a constant $c_n$, $1\leq c_n<\infty$, such that
$$\begin{aligned}
\label{ffa}(i)&&E \left [\sup_{0 \leq t \leq T} |Y_t^n|^2 \right] \leq c_n,\notag\\
(ii)&& E \left [\int_0 ^T \|Z^n_t
\|^2 dt + \int _0^T \int _U|U^n_s(e)|^2 \lambda(de)ds \right] < \infty,
\notag\\
(iii)&&Y^n_t= \xi + \int_t ^T f_n(s,Y^n_s,Z^n_s,U^n_s) ds\notag\\
&&- \int_t ^T Z^n_s dW_s- \int_t^T \int _U U^n_s(e) \mu(de,ds).\end{aligned}$$
Note also that from [@TL] we have $Y^n \in \mathcal{D}\left( \left[ 0,T\right] , \R^{d}\right) $. Given $(Y_t^n, Z_t^n, U_t ^n)$ we define, for $n\in\mathbb Z_+$, the process $\Lambda_t^n$ through $$\label{ffa3}
\Lambda_t ^n =-n \int_0^t ( Y_s ^n-\pi(s,Y_s ^n)) ds.$$ Note that $$ \Lambda_t ^n =\int_0^t \frac{-( Y_s ^n-\pi(s,Y_s ^n))}{| Y_s ^n-\pi(s,Y_s ^n)|}d|\Lambda^n|_s$$ and that $-( Y_s ^n-\pi(s,Y_s ^n))/{| Y_s ^n-\pi(s,Y_s ^n)|}$ is an element in the inwards directed normal cone to $D_s$ at $\pi(s,Y_s ^n)\in\partial D_s$. Furthermore, using and we see that $(iii)$ can be rewritten as $$\begin{aligned}
Y_t ^n&=& \xi +\int_t ^T f(s, Y_s ^n, Z_s ^n, U_s ^n) ds +\Lambda^n_T-\Lambda_t^n\notag\\
&&-\int _t ^T Z_s^n dW_s - \int _t^T \int _U U_s^n(e) \mu(de,ds),\end{aligned}$$ for all $t\in [0,T]$. Recall that $Y^n, \Lambda^n, U^n$, $W_t$ and $Z^n$ are multi-dimensional processes. In particular, $Y_t^n=(Y_t ^{1,n}, \dots,Y_t ^{d,n})$, $\Lambda_t^n=(\Lambda_t ^{1,n},\dots,\Lambda_t ^{d,n})$, $U_t^n=(U_t ^{1,n},\dots, U_t ^{d,n})$, $W_t=(W_t^1,\dots,W_t^n)$ and $ Z_t^n$ is a $d\times n$-matrix with entries $Z_t^{i,j,n}$ and columns $Z_t^{j,n}$.
A priori estimates for $(Y_t^n, Z_t^n, U_t^n, \Lambda_t^n)$
-----------------------------------------------------------
\[ll1\] There exists a constant $c$, $1\leq c<\infty$, independent of $n$, such that $$\begin{aligned}
(i)&&E \left [\sup_{0 \leq t \leq T}|Y_t^n|^2 \right ] \leq c,\notag\\
(ii)&&E \left [\int _t ^T \|Z_s^n\|^2 + \int _t^T \int _U |U_s^n(e)|^2 \lambda(de)ds \right ]\leq c,\ t\in[0,T],\\
(iii)&& E \left [n\int _t ^T|Y_s^n-\pi(s,Y_s ^n)|ds \right ]\leq c, \ t\in[0,T].\end{aligned}$$
Let $P_T\in D_T$ be as in Lemma \[lemmaa\] $(iv)$. Applying Ito’s formula to the process $|Y_t ^n-P_T|^2$ we deduce, for $t\in [0,T]$, that $$\begin{aligned}
\label{jaj}
&&|Y_t ^n-P_T|^2 + \int _t ^T
\|Z_s^n\|^2 ds + \int _t ^T \int _U |U_s^n(e)|^2 p(de, ds) \notag \\
&=&|\xi-P_T|^2 + 2 \int _t ^T \langle Y_s ^n-P_T, f(s,Y_s ^n,Z_s ^n,U_s^n)\rangle ds - 2 \int _t ^T \langle Y_s ^n-P_T, Z_s^n dW_s \rangle\notag \\
&&- 2 \int _t ^T \int _U \langle Y_s ^n-P_T, U^n_s(e) \rangle \mu(de,ds) - 2 n \int _t ^T \langle Y_s ^n-P_T,Y_s^n-\pi(s,Y_s ^n)\rangle ds.\end{aligned}$$ Let $$\begin{aligned}
A_n:=|Y_t ^n-P_T|^2 + \int _t ^T \|Z_s^n\|^2 ds + \int _t ^T \int _U |U_s^n(e)|^2 p(de, ds).\end{aligned}$$ Rearranging , we find that $$\begin{aligned}
&&A_n + 2 n \int _t ^T \langle Y_s ^n-P_T,Y_s^n-\pi(s,Y_s ^n)\rangle ds \notag\\
&=& |\xi-P_T|^2 + 2 \int _t ^T \langle Y_s ^n-P_T, f(s,Y_s ^n,Z_s ^n,U_s^n)\rangle ds\notag\\
&&- 2 \int _t ^T \langle Y_s ^n-P_T, Z_s^n dW_s\rangle - 2 \int _t ^T \int _U \langle Y_s^n-P_T, U^n_s(e) \rangle \mu(de,ds).\end{aligned}$$ Furthermore, using Lemma \[lemmaa\] $(iv)$ we see that $$\begin{aligned}
\label{jaj++ny}
&&A_n + 2 \gamma^{-1} n \int _t ^T |Y_s^n-\pi(s,Y_s ^n)|ds \notag\\
&\leq& |\xi-P_T|^2 + 2 \int _t ^T \langle Y_s ^n-P_T, f(s,Y_s ^n,Z_s ^n,U_s^n)\rangle ds\notag\\
&&- 2 \int _t ^T \langle Y_s ^n-P_T, Z_s^n dW_s\rangle+ 2 \int _t ^T \int _U \langle Y_s^n-P_T, U^n_s(e) \rangle \mu(de,ds).\end{aligned}$$ Next, taking expectations in and using the fact that $\mu$ is a martingale measure, we can conclude that $$\begin{aligned}
\label{eq:inequal1}
&&E \left [A_n+2 \gamma^{-1} n \int _t ^T |Y_s^n-\pi(s,Y_s ^n)|ds \right ] \leq I_{t,T}\end{aligned}$$ where $$\begin{aligned}
\label{jaj++1}
I_{t,T}=E \left [ |\xi-P_T|^2 \right] + 2 E\left [\int _t ^T \langle Y_s ^n-P_T, f(s,Y_s ^n,Z_s ^n, U_s^n)\rangle ds\right].\end{aligned}$$ Using the Lipschitz character of $f$, $(iii)$, and the inequality $ab \leq \eta a^2 + \frac{b^2}{4\eta}$ it follows that we can estimate $I_{t,T}$ as, $$\begin{aligned}
\label{jaj++1+}
I_{t,T}&\leq& c\left( 1+ E \left[\int _t^T \left( |f(s,P_T,0,0)|^2+(1+2\eta)|Y^n_s-P_T|^2 ds \right ) \right] \right)\notag \\
&+& c E\left[ \int _t ^T \frac{1}{\eta} \left (\|Z_s^n\|^2 + \int _U |U_s^n(e)|^2 \lambda(de) \right) ds\right ]\end{aligned}$$ where $\eta>0$ is a degree of freedom and $c$, $1\leq c<\infty$, is a constant depending on $\xi$, $f$ and $\mbox{diam($D_T$)}$. If we let $\eta$ be such that $c/\eta \leq1/2$, it follows from and , after recalling the definition of $A_n$, that $$\begin{aligned}
\label{jaj++ha}
&&E[|Y_t ^n-P_T|^2]+ \frac 1 2 E \left [\int _t ^T \|Z_s^n\|^2 ds\right ] + \frac{1}{2}E \left [\int _t ^T \int _U |U_s^n(e)|^2 \lambda(de) ds \right ] \notag \\
&\leq& c \left( 1+ E \left [\int _t^T |Y^n_s-P_T|^2 ds \right ]\right).\end{aligned}$$ Using and Gronwall’s lemma we deduce that $$\label{ffa2--}
\sup_{0 \leq t \leq T}E[|Y_t^n-P_T|^2] \leq \tilde c\exp(\tilde c T)$$ where $\tilde c$ is independent of $n$. In particular, we can conclude that there exists $\hat c$, $1\leq\hat c<\infty$, independent of $n$, such that $$\begin{aligned}
\label{ffa2--+}
(i)&&\sup_{0 \leq t \leq T}E[|Y_t^n|^2] \leq \hat c, \quad \mbox {and}\notag\\
(ii)&&\ E \left [\int _0 ^T \|Z_s^n\|^2 ds + \int _0 ^T\int _U |U_s^n(e)|^2 \lambda(de) ds \right]\leq\hat c.\end{aligned}$$ We next prove that there exists $\check c$, $1\leq \check c<\infty$, independent of $n$, such that $$\begin{aligned}
\label{ffa2--++}
E[\sup_{0 \leq t \leq T}|Y_t^n|^2] \leq \check c.\end{aligned}$$ To do this we first note, using and the Lipschitz character of $f$, that, for a constant $c$, $1\leq c<\infty$, $$\begin{aligned}
\label{eq:BDGdoob}
&&c^{-1}|Y_t^n-P_T|^2\leq |\xi-P_T|^2+\biggl |\int _t ^T \langle Y_s ^n-P_T, f(s,P_T,0,0)\rangle ds\biggr |\notag\\
&&+\biggl |\int _t ^T |Y_s ^n-P_T|(|Y_s ^n-P_T|+\|Z_s ^n\|+\|U_s^n\|_2)ds\biggr |\notag\\
&&+\biggl |\int _t ^T \langle Y_s ^n-P_T, Z_s^n dW_s\rangle\biggr |+ \biggl | \int _t ^T \int _U \langle Y_s^n-P_T, U^n_s(e) \rangle \mu(de,ds)\biggr |.\end{aligned}$$ We treat the last two terms on the right hand side of using Hölders inequality and the Burkholder-Davis-Gundy inequality. Indeed, applying these yields $$\begin{aligned}
\label{jaj++nyhha}
&&E\left [\sup_{0 \leq t \leq T}\left |\int _t ^T \langle Y_s ^n-P_T, Z_s^n dW_s\rangle\right |\right]\notag\\
&&\leq \biggl (E \left [\sup_{0 \leq t \leq T}|Y_t ^n-P_T|^2\right ]\biggr )^{1/2}\biggl (E\biggl[\sup_{0 \leq t \leq T}\biggl |\int _t ^T Z_s^n dW_s \biggr |^2
\biggr ]\biggr )^{1/2}\notag\\
&&\leq c\biggl (E \left [\sup_{0 \leq t \leq T}|Y_t ^n-P_T|^2 \right ]\biggr )^{1/2}\biggl (E \left [\int _0 ^T \|Z_s^n\|^2 ds\right ]\biggr )^{1/2}.\end{aligned}$$ Similarly, $$\begin{aligned}
&&E\left [\sup_{0 \leq t \leq T}\biggl |\int _t ^T \int _U \langle Y_s^n-P_T, U^n_s(e) \rangle \mu(de,ds)\biggr |\right]\notag\\
&&\leq c\biggl (E \left [\sup_{0 \leq t \leq T}|Y_t ^n-P_T|^2 \right ]\biggr )^{1/2}\biggl (E \left [\int _0 ^T\int _U |U_s^n(e)|^2 \lambda(de) ds\right ]\biggr )^{1/2}.\end{aligned}$$ Using the above estimates as well as , and assumption we get after taking expectation in that $$\begin{aligned}
&&E \left [ \sup_{0\leq t \leq T}|Y_t^n-P_T|^2 \right ] \leq c \biggl (1+ \left ( E [\sup_{0 \leq t \leq T}|Y_t ^n-P_T|^2 ] \right )^{1/2} \biggr)\end{aligned}$$ from which we conclude that $E \left [ \sup_{0\leq t \leq T}|Y_t^n-P_T|^2 \right] \leq c$ for some constant $1 \leq c < \infty$ and, consequently, that holds. Finally, starting from and repeating the arguments above we also deduce that $$\begin{aligned}
E \left [n\int _t ^T|Y_s^n-\pi(s,Y_s ^n)|ds \right ]\leq\hat c\end{aligned}$$ for some constant $\hat c$ independent of $n$. This completes the proof of Lemma \[ll1\].
Uniform control of $d(Y_t^n,D_t)$
---------------------------------
We here prove the following lemma.
\[lemmbb\] Let $D\subset\mathbb{R}^{d+1}$ be a time-dependent domain satisfying - . Let, for $\epsilon>0$ small, $D_{\epsilon}$ be as in Lemma \[lemmauu1-\]. Then there exist $\epsilon_0>0$ and $c$, $1\leq c<\infty$, both independent of $n$, such that $$\begin{aligned}
(i)&&E \left [\sup{_{0\leq t\leq T}d(Y_t^n,D_{{\epsilon},t})^2} \right ] \leq c\biggl ( \frac 1 n +\epsilon+n\epsilon^2\biggr ),\notag\\
(ii)&&E \left [\int_0^T(d(Y_t^n,D_{{\epsilon},t}))^2dt \right ] \leq c\biggl ( \frac 1 {n^2} +\frac \epsilon n+\epsilon^2\biggr ),\end{aligned}$$ whenever $0<\epsilon<\epsilon_0$ and $n\geq 1$.
Let $\varphi_\epsilon(t,Y_t)=d(Y_t^n, D_{{\epsilon},t})^2=|Y_t^n-\pi_\epsilon(t,Y_t^n)|^2$. Then $\nabla_y\varphi_\epsilon(t,Y_t)=2(Y_t^n-\pi_\epsilon(t,Y_t^n))$. Using the Ito formula of Lemma \[lemmauu2+\] we see that $$\begin{aligned}
\label{aaa-li}
&&\varphi_{\epsilon}(t,Y_t^n) + \int_t^T (\partial_{s}\varphi_\epsilon)(s,Y_{s}^n)ds+ \frac{1}{2} \int_t^T \sum_{i,j} (Z_s^{n} Z_s^{n,\ast})_{ij} \partial^2_{y_iy_j}\varphi_\epsilon(s, Y_{s}) ds \notag\\
&&+ \int _t ^T \int _U [\varphi_{{\epsilon}}(s,Y_{s^-}^n+U^n_s(e)) - \varphi_{{\epsilon}}(Y^n_{s^-}) - \langle \nabla \varphi_{{\epsilon}}(s,Y^n_{s^-}) , U^n_s(e) \rangle ] p(de,ds) \notag \\
&&=\varphi_\epsilon(T,\xi)+I_1+I_2+I_3 + I_4,\end{aligned}$$ where $$\begin{aligned}
\label{aaa+}
I_1&=&2\int _t ^T \langle Y_s^n-\pi_\epsilon(s,Y_s^n),f(s,Y_s,Z_s^n, U_s^n)\rangle ds,\notag\\
I_2&=& - 2\int _t ^T \langle Y_s^n-\pi_\epsilon(s,Y_s^n),Z^n_s
dW_s \rangle,\notag\\
I_3&=&- 2n \int_t ^T\langle Y_s^n-\pi_\epsilon(s,Y_s^n),Y_s^n-\pi(s,Y_s^n)\rangle ds, \notag \\
I_4&=&- 2 \int_t ^T \int _U \langle Y_s^n-\pi_\epsilon(s,Y_s^n), U^n_s(e) \rangle \mu(de,ds).\end{aligned}$$ Using Lemma \[lemmauu2ny\] we see that $$\begin{aligned}
&&\int_t^T \partial_{s}\varphi_\epsilon(s,Y_s^n)ds+ \frac{1}{2} \int_t^T \sum_{i,j} (Z_s^{n} Z_s^{n,\ast})_{ij} \partial^2_{y_iy_j}\varphi_\epsilon(s, Y_s) ds \notag\\
&&+ \int _t ^T \int _U [\varphi_{{\epsilon}}(s,Y_{s^-}^n+U^n_s(e)) - \varphi_{{\epsilon}}(Y^n_{s^-} ) - \langle \nabla \varphi_{{\epsilon}}(s,Y^n_{s^-}) , U^n_s(e) \rangle ] p(de,ds)\geq 0,\end{aligned}$$ and hence $$\begin{aligned}
\label{aaa}
\varphi_{\epsilon}(t,Y_t^n)\leq \varphi_\epsilon(T,\xi)+I_1+I_2+I_3 + I_4.\end{aligned}$$ Since $\xi \in D_T$ a.s. we see, using Lemma \[lemmauu1\], that $\varphi _{\epsilon}(T,\xi)\leq c {\epsilon}^2$ a.s. To simplify the notation in what follows, we define $\chi_\epsilon (t,y) : [0,T]\times\mathbb R^d \to \{0,1\}$ as $$\chi_\epsilon(t,y) =\begin{cases}1& \mbox{ if } d(y,D_{\epsilon,t})>\epsilon \\
0 & \mbox{otherwise}
\end{cases}.$$ We first focus on the term $I_1$ in . Then, using the above introduced notation we see that $$\begin{aligned}
I_1&\leq &2\int _t ^T |\varphi_\epsilon(s,Y_s^n)|^{1/2}|f(s,Y_s^n,Z_s^n, U^n_s)|\chi_\epsilon(s,Y_s^n)ds\notag\\
&&+2\int _t ^T |\varphi_\epsilon(s,Y_s^n)|^{1/2}|f(s,Y_s^n,Z_s^n, U_s^n)|(1-\chi_\epsilon(s,Y_s^n))ds.\end{aligned}$$ Furthermore, by the inequality $ab \leq \eta a^2 + \frac{b^2}{4\eta}$ and $x \leq \max\{1,x^2\}$, $$\begin{aligned}
(i)&& 2|\varphi_\epsilon(s,Y_s^n)|^{1/2}|f(s,Y_s^n,Z_s^n, U^n_s)|\chi_\epsilon(s,Y_s^n)\notag\\
&\leq&
\frac{n}{4} \varphi_\epsilon(s,Y_s^n) \chi_\epsilon(s,Y_s^n) + \frac{4}{n} |f(s,Y_s^n,Z_s^n, U^n_s)|^2 \chi_\epsilon(s,Y_s^n),\notag\\
(ii)&& 2|\varphi_\epsilon(s,Y_s^n)|^{1/2}|f(s,Y_s^n,Z_s^n, U^n_s)|(1-\chi_\epsilon(s,Y_s^n))\notag\\
&\leq&2({{\epsilon}}+ {{\epsilon}} |f(s,Y_s^n,Z_s^n,U^n_s)|^2)(1-\chi_\epsilon(s,Y_s^n)).\end{aligned}$$
Next, focusing on the term $I_3$, we have by the bilinearity of $\langle \cdot, \cdot\rangle$ that $$\begin{aligned}
\label{aaa++}
I_3&=&-2n \int _t ^T|Y_s^n-\pi_\epsilon(s,Y_s^n)|^2\chi_\epsilon(s,Y_s^n) ds\notag\\
&&- 2n \int _t ^T\langle Y_s^n-\pi_\epsilon(s,Y_s^n),\pi_\epsilon(s,Y_s^n)-\pi(s,Y_s^n)\rangle\chi_\epsilon(s,Y_s^n) ds\notag\\
&&- 2n \int _t ^T\langle Y_s^n-\pi_\epsilon(s,Y_s^n),Y_s^n-\pi(s,Y_s^n)\rangle(1-\chi_\epsilon(s,Y_s^n)) ds\notag\\
&:=&I_{31}+I_{32}+I_{33}.\end{aligned}$$ By Lemma \[lemmauu1\] $(i)$ we immediately see that $|I_{33}|\leq c n\epsilon^2$. Furthermore, using Lemma \[lemmauu2\] $(ii)$ we see that $$\begin{aligned}
\label{aaa+++}
|I_{32}|\leq c\sqrt{\epsilon} n \int _t ^T(d(Y_s^n,D_{\epsilon,s}))^{3/2}\chi_\epsilon(s,Y_s^n) ds.\end{aligned}$$ Using the inequality $ ab \leq \frac{3a^{\frac{4}{3}}}{4}+\frac{b^4}{4} $ with $a= d(Y^n_s, D_{{\epsilon},s})^{\frac{3}{2}}$ and $b=c\sqrt{\epsilon}$ we deduce from that $$\begin{aligned}
\label{aaa+++jj}
|I_{32}|\leq cn{\epsilon^2}+n \int _t ^T(d(Y_s^n,D_{\epsilon,s}))^{2}\chi_\epsilon(s,Y_s^n) ds.\end{aligned}$$ Putting the estimate into together we can conclude that $$\begin{aligned}
I_3\leq cn{\epsilon}^2-n \int _t ^T|Y_s^n-\pi_\epsilon(s,Y_s^n)|^2\chi_\epsilon(s,Y_s^n) ds.\end{aligned}$$ Hence, putting the estimate for $I_1$ and $I_3$ together we can conclude that $$\begin{aligned}
\label{aaad}
I_1+I_3&\leq & c\epsilon+cn\epsilon^2-\frac 3 4n \int _t ^T\varphi_\epsilon(s,Y_s^n)\chi_\epsilon(s,Y_s^n) ds\notag\\
&&+c\int _t ^T\biggl (\frac 1 n+\epsilon\biggr ) |f(s,Y_s^n,Z_s^n, U^n_s)|^2 ds.\end{aligned}$$ Combining and we have proved that $$\begin{aligned}
\label{aaad+}
\varphi_{\epsilon}(t,Y_t^n)&\leq& c( {\epsilon}+n{\epsilon}^2) -\frac{3}{4} n \int _t ^T \varphi_{\epsilon}(s,Y^n_s) \chi_{\epsilon}(s,Y^n_s) ds\notag\\
&&+ c\int _t^T(\frac{1}{n} + {\epsilon}) |f(s,Y^n_s, Z^n_s, U^n_s)|^2 ds \notag \\
&&- 2\int _t ^T \langle Y_s^n-\pi_\epsilon(s,Y_s^n),Z^n_s
dW_s \rangle\notag\\
&&-2\int_t ^T \int _U \langle Y_s^n-\pi_\epsilon(s,Y_s^n), U^n_s(e) \rangle \mu(de,ds).\end{aligned}$$ In particular, $$\begin{aligned}
\label{aaad+ia}
&&\varphi_{\epsilon}(t,Y_t^n)+\frac{3}{4}n \int _t ^T \varphi_{\epsilon}(s,Y^n_s) \chi_{\epsilon}(s,Y^n_s) ds\notag\\
&\leq& c( {\epsilon}+n{\epsilon}^2)+ c\int _t^T(\frac{1}{n} + {\epsilon}) |f(s,Y^n_s, Z^n_s, U^n_s)|^2 ds \notag \\
&&-2\int _t ^T \langle Y_s^n-\pi_\epsilon(s,Y_s^n),Z^n_s
dW_s \rangle-2\int_t ^T \int _U \langle Y_s^n-\pi_\epsilon(s,Y_s^n), U^n_s(e) \rangle \mu(de,ds).\end{aligned}$$ where $c$ is independent of ${\epsilon}$ and $n$. The estimate in Lemma \[lemmbb\] $(ii)$ now follows from taking expectation in and using the Lipschitz property of $f$ and Lemma \[ll1\] $(i)$ and $(ii)$. Similarly, using we can conclude that $$\begin{aligned}
\sup_{0\leq t\leq T}E[\varphi_{\epsilon}(t,Y^n_t)] \leq c\bigl (\frac{1}{n} +{\epsilon}+ n{\epsilon}^2\bigr),\end{aligned}$$ for a constant $c$, independent of ${\epsilon}$ and $n$. Since $\varphi_{\epsilon}(t, Y_t) \geq 0$ for all $t \in [0,T]$ we can, repeating the arguments above, also conclude from that $$\begin{aligned}
\label{eq:allests}
&& E\left [ \int_t^T (\partial_{s}\varphi_\epsilon)(s,Y_{s}^n)ds \right ]+ E \left [\int_t^T \sum_{i,j} (Z_s^{n} Z_s^{n,\ast})_{ij} \partial^2_{y_iy_j}\varphi_\epsilon(s, Y_{s}) ds \right ] \notag\\
+&&E\left [ \int _t ^T \int _U [\varphi_{{\epsilon}}(s,Y_{s^-}^n+U^n_s(e)) - \varphi_{{\epsilon}}(Y^n_{s^-}) - \langle \nabla \varphi_{{\epsilon}}(s,Y^n_{s^-}) , U^n_s(e) \rangle ] p(de,ds) \right ] \notag \\
&& \leq c\bigl (\frac{1}{n} +{\epsilon}+ n{\epsilon}^2\bigr),\end{aligned}$$ for some constant $1\leq c < \infty$, for all $t \in [0,T]$. Once again using the Lipschitz property of $f$ and Lemma \[ll1\] in we see that to complete the proof of Lemma \[lemmbb\] $(i)$ it remains to control the terms $$\begin{aligned}
\label{eq:finests}
&(i)&E \left [\sup_{0\leq t \leq T} \left | \int _t ^T \langle Y_s^n-\pi_\epsilon(s,Y_s^n),Z^n_s dW_s \rangle \right | \right ] \notag \\
&(ii)&E \left [ \sup_{0\leq t \leq T}\left |\int_t ^T \int _U \langle Y_s^n-\pi_\epsilon(s,Y_s^n), U^n_s(e) \rangle \mu(de,ds) \right | \right ].$$
We first treat $(i)$. As in we use the Burkholder-Davis-Gundy inequality to see that $$\begin{aligned}
\label{eq:nuzest}
&E \left [\sup_{0\leq t \leq T} \left | \int _t ^T \langle Y_s^n-\pi_\epsilon(s,Y_s^n),Z^n_s dW_s \rangle \right | \right ] &\notag \\
\leq & E \left [ \left ( \int _0 ^T \left |(Y_s^n-\pi_\epsilon(s,Y_s^n))^\ast Z^n_s \right|^2 ds \right )^{1/2} \right ].\end{aligned}$$ By the convexity of $\varphi_{\epsilon}$ and equivalence of Euclidean norms (see [@GP] p. 115) we have that $$\frac {|(Y_s^n-\pi_{\epsilon}(s,Y_s^n))^\ast Z^n_s|^2}{|(Y_s^n-\pi_{\epsilon}(s,Y_s^n))|^2} \I_{\{Y^n_s \not \in D_{e,s}\}}\leq c \left ( \sum_{i,j} (Z_s^{n} Z_s^{n,\ast})_{ij} \partial^2_{y_iy_j}\varphi_\epsilon(s, Y_{s}) \right),$$ where $\I$ is the indicator function, i.e., $\I_{\{Y^n_s \not \in D_{{\epsilon},s}\}} = 1$ if $Y^n_s \not \in D_{{\epsilon},s}$ and $0$ otherwise. Hence, it follows from that for $t\in [0,T]$ $$\label{eq:nuzest1+}
\int_t ^T \frac{|(Y_s^n-\pi_{\epsilon}(s,Y_s^n))^\ast Z^n_s|^2}{|(Y_s^n-\pi_{\epsilon}(s,Y_s^n))|^2} \I_{\{Y^n_s \not \in D_{e,s}\}}ds\leq c\bigl (\frac{1}{n} +{\epsilon}+ n{\epsilon}^2\bigr).$$ Note also that $\left | Y_s^n-\pi_\epsilon(s,Y_s^n) \right |= \left |Y_s^n-\pi_\epsilon(s,Y_s^n) \right | \I_{\{Y^n_s \not \in D_{{\epsilon},s}\}}$ and that $$\begin{aligned}
\label{eq:nuzest2} \int _t ^T |(Y_s^n-\pi_{\epsilon}(s,Y_s^n))^\ast Z^n_s|^2 ds & \notag \\
\leq \sup_{t\leq s \leq T}& \varphi_{\epsilon}(s, Y^n_s) \int _t ^T \frac {| (Y_s^n-\pi_{\epsilon}(s,Y_s^n))^\ast Z^n_s|^2}{|(Y_s^n-\pi_{\epsilon}(s,Y_s^n))|^2} \I_{\{Y^n_s \not \in D_{e,s}\}} ds.\end{aligned}$$ We again use $ab \leq \eta a^2 + \frac{b^2}{\eta}$ to conclude from , and that $$\begin{aligned}
\label{eq:wrty}
&E \left [\sup_{0\leq t \leq T} \left | \int _t ^T \langle Y_s^n-\pi_\epsilon(s,Y_s^n),Z^n_s dW_s \rangle \right | \right ] \notag \\
\leq &\eta E \left [\sup_{0 \leq t \leq T} \varphi_{\epsilon}(t, Y^n_t) \right ] +c_\eta(\frac{1}{n} +{\epsilon}+ n{\epsilon}^2\bigr)\end{aligned}$$ for some constant $c_\eta <\infty$ depending on the degree of freedom $\eta >0$.
We now treat the term $(ii)$. Using Taylor’s theorem we see that $$\begin{aligned}
E\left [ \int _t ^T \int _U [\varphi_{{\epsilon}}(s,Y_{s^-}^n+U^n_s(e)) - \varphi_{{\epsilon}}(Y^n_{s^-}) - \langle \nabla \varphi_{{\epsilon}}(s,Y^n_{s^-}) , U^n_s(e) \rangle p(de,ds) \right ]& \notag \\
\geq \breve c E \left [ \int_t^T |U(e)|^2 \lambda(de) ds \right ],\end{aligned}$$ for some constant $\breve c\geq 0$. Furthermore, by the strong convexity of $\varphi_{\epsilon}$, and this is a consequence of , there exists a constant $\kappa >0$ such that $\breve c\geq \kappa >0$. Therefore we can, in a way similar to the above, conclude that $$\begin{aligned}
\label{eq:wrty2}
&E \left [ \sup_{0\leq t \leq T}\left |\int_t ^T \int _U \langle Y_s^n-\pi_\epsilon(s,Y_s^n), U^n_s(e) \rangle \mu(de,ds) \right | \right ] \notag \\
\leq & E \left [\left (\int_t ^T \int _U |Y_s^n-\pi_\epsilon(s,Y_s^n)|^2 |U^n_s(e)|^2 \lambda(de) ds \right) ^{1/2} \right ] \notag \\
\leq & E \left [ \left ( \sup_{t\leq s \leq T}\varphi_{\epsilon}(s,Y^n_s) \int_t ^T \int _U |U^n_s(e)|^2 \lambda(de) ds \right) ^{1/2}\right ] \notag \\
\leq &\eta E \left [\sup_{0 \leq t \leq T} \varphi_{\epsilon}(t, Y^n_t) \right ] +c_\eta(\frac{1}{n} +{\epsilon}+ n{\epsilon}^2\bigr).\end{aligned}$$ Once again, $c_\eta <\infty$ is a constant depending on the degree of freedom $\eta>0$.
The proof of Lemma \[lemmbb\] is now completed by choosing $\eta$ small enough (in analogy with ,) and combining with , .
\[lemmbbaa\] Let $D\subset \R^{d+1}$ be a time-dependent domain satisfying - . Then there exists $c$, $1\leq c<\infty$, independent of $n$ such that $$\begin{aligned}
(i)&&E \left [\sup_{0\leq t\leq T}(d(Y_t^n,D_{t}))^2 \right] \leq \frac{ c} n,\notag\\
(ii)&&E \left [\int_0^T(d(Y_t^n,D_{t}))^2dt \right] \leq \frac {c} {n^2},\end{aligned}$$ whenever $n\geq 1$.
Let, for $\epsilon>0$ small, $D_{\epsilon}$ be as in Lemma \[lemmauu1-\]. Then, using Lemma \[lemmauu1-\] we have $$\begin{aligned}
h(D_{t},D_{\epsilon,t})<\epsilon\mbox{ for all $t\in[0,T]$.}\end{aligned}$$ Hence, $$\begin{aligned}
d(Y_t^n,D_{t})\leq d(Y_t^n,D_{\epsilon,t})+\epsilon\mbox{ for all $t\in[0,T]$.}\end{aligned}$$ Applying Lemma \[lemmbb\] and letting ${\epsilon}\to 0$ completes the proof.
$(Y_t^n, Z_t^n, U_t^n)$ is a Cauchy sequence
---------------------------------------------
\[anotherlemma\] There exists a constant $c$ such that the following holds whenever $m, n \in \mathbb{Z}^+$: $$\begin{aligned}
E \left [\sup_{0 \leq t \leq T} |Y_t^n-Y_t^m|^2 + \int_0 ^t \|Z_t^n-Z_t^m\|^2 dt\right ] &\leq& c \left(\frac{1}{n} + \frac{1}{m}\right),\\
E\left [\int_0 ^T \int _U |U_s^n(e)-U^m_s(e)| ^2 \lambda(de) ds\right] &\leq& c \left(\frac{1}{n} + \frac{1}{m}\right).\end{aligned}$$
Applying Ito’s formula to $|Y^n_t-Y^m_t|^2$ we get that $$\begin{aligned}
\label{eq:itoexpmn}
&&|Y^n_t-Y^m_t|^2+\int _t ^T \|Z_s^n-Z_s^m\|^2 ds\notag\\
&& + \int _t ^T \int _U |U^n_s(e)- U^m_s(e)| ^2 p(de,ds) \notag\\
&=& 2 \int _t ^T \langle Y^n_s - Y^m_s , f(s,Y^n_s, Z^n_s, U^n_s) - f(s,Y^m_s, Z^m_s, U^m_s) \rangle ds \notag \\
&&- 2 \int _t ^T \langle Y^n_s - Y^m_s , (Z^n_s- Z^m_s) dW_s\rangle\notag\\
&&- 2 \int _t ^T \int _U \langle Y^n_s - Y^m_s , U^n_s(e)- U^m_s(e) \rangle \mu(de,ds) \notag \\
&&- 2n \int _t ^T \langle Y^n_s - Y^m_s , Y^n_s- \pi(s,Y^n_s) \rangle ds\notag\\
&& + 2m \int _t ^T \langle Y^n_s - Y^m_s , Y^m_s- \pi(s,Y^m_s) \rangle ds.\end{aligned}$$ Hence, taking expectation and using the Lipschitz character of $f$ we deduce that $$\begin{aligned}
\label{apa1}
&&E \left [|Y^n_t-Y^m_t|^2 \right]+E\left [\int _t ^T \|Z_s^n-Z_s^m\|^2 ds\right ]\notag\\
&& + E\left [\int _t ^T \int _U |U^n_s(e)- U^m_s(e)| ^2 \lambda(de)ds\right] \notag\\
&\leq& c E\left [ \int _t ^T (|Y^n_s - Y^m _s |^2 + |Y^n_s - Y^m _s |\|Z^n_s - Z^m _s \|)ds\right ]\notag \\
&&+c E\left [ \int _t ^T |Y^n_s - Y^m _s | \int _U |U^n_s(e) - U^m _s(e)| \lambda(de) ds\right ] \notag \\
&&- 2E\left [ \int _t ^T \langle Y^n_s - Y^m_s , n(Y^n_s - \pi(s,Y^n_s)) - m(Y^m_s - \pi(s,Y^m_s)) \rangle ds\right ].\end{aligned}$$ Note that $$\begin{aligned}
&&-\langle Y^n_s - Y^m_s , n(Y^n_s - \pi(s,Y^n_s)) - m(Y^m_s - \pi(s,Y^m_s)) \rangle \notag\\
&=&\langle Y^m_s - Y^n_s , n(Y^n_s - \pi(s,Y^n_s)) \rangle +\langle Y^n_s - Y^m_s , m(Y^m_s - \pi(s,Y^m_s)) \rangle.\end{aligned}$$ Using Lemma \[lemmaa\] $(ii)$ we have that $$\begin{aligned}
&&\langle Y^m_s - Y^n_s , n(Y^n_s - \pi(s,Y^n_s)) \rangle\leq n \langle Y^m_s - \pi(s,Y^m_s) , Y^n_s - \pi(s,Y^n_s)\rangle \notag\\
&&\langle Y^n_s - Y^m_s , m(Y^m_s - \pi(s,Y^m_s)) \rangle\leq m \langle Y^n_s - \pi(s,Y^n_s) , Y^m_s - \pi(s,Y^m_s)\rangle.\end{aligned}$$ Furthermore, $$\begin{aligned}
&&2E\left [ \int _t ^T \langle Y^m_s - Y^n_s , n(Y^n_s - \pi(s,Y^n_s)) \rangle ds\right ]\notag\\
&\leq& 2nE\left [ \int _t ^T |Y^m_s - \pi(s,Y^m_s)|| Y^n_s - \pi(s,Y^n_s)|ds\right ]\notag\\
&\leq & nE\left [ \int _t ^T \beta(d(Y^m_s,D_s))^2+\beta^{-1}(d(Y^n_s,D_s))^2ds\right ]\notag\\
&\leq &c(n\beta m^{-2}+\beta^{-1} n^{-1})\leq cm^{-1}\end{aligned}$$ where we have used Lemma \[lemmbbaa\] $(ii)$ and chosen the degree of freedom to equal $\beta=m/n$. This argument can be repeated with $n\langle Y^m_s - Y^n_s , n(Y^n_s - \pi(s,Y^n_s)) \rangle$ replaced by $m \langle Y^n_s - \pi(s,Y^n_s) , Y^m_s - \pi(s,Y^m_s)\rangle $ resulting in the bound $ cn^{-1}$. Put together we can conclude that $$\begin{aligned}
\label{apa}
&&- 2E\left [ \int _t ^T \langle Y^n_s - Y^m_s , n(Y^n_s - \pi(s,Y^n_s)) - m(Y^m_s - \pi(s,Y^m_s)) \rangle ds \right ]\notag\\
&\leq &c(n^{-1}+m^{-1}).\end{aligned}$$ Combining , and using Cauchy’s inequality as in , we can conclude that $$\begin{aligned}
\label{apa1+}
&&E \left [|Y^n_t-Y^m_t|^2 \right ]+ \frac{1}{2}E\left [\int _t ^T\|Z_s^n-Z_s^m\|^2 ds\right ]\notag\\
&& +\frac{1}{2} E\left [\int _t ^T \int _U |U^n_s(e)- U^m_s(e)| ^2 \lambda(de)ds\right ] \notag\\
&\leq& c E\left [ \int _t ^T |Y^n_s - Y^m _s |^2 ds\right ]+c(n^{-1}+m^{-1})\end{aligned}$$ where $c$ is independent of $n$ and $m$. By Gronwall’s inequality we then have, using , $$\begin{aligned}
\label{apa1++}
E \left [|Y^n_t-Y^m_t|^2 \right ]\leq c(n^{-1}+m^{-1}).\end{aligned}$$ Subsequently, $$\begin{aligned}
\label{apa1+a}
E\left [\int _t ^T\|Z_s^n-Z_s^m\|^2 ds\right ]&\leq& c(n^{-1}+m^{-1})\notag\\
E\left[\int _t ^T \int _U |U^n_s(e)- U^m_s(e)| ^2 \lambda(de)ds\right ]&\leq& c(n^{-1}+m^{-1}).\end{aligned}$$ Using , , Lemma \[lemmbbaa\] $(i)$, and now familiar arguments based on the Burkholder-Davis-Gundy inequality, we can, starting from , also deduce that $$E \left [ \sup_{0 \leq t \leq T} |Y^n_t -Y^m_t|^2 \right ] \leq c \left (\frac{1}{n}+\frac{1}{m}\right),$$ to complete the proof of Lemma \[anotherlemma\]. We omit further details.
The final argument: proof of Theorem \[maint1\] {#sec:proof}
===============================================
Using Lemma \[anotherlemma\] we can conclude that $(Y_t^n,Z_t^n,U_t^n)$ is a Cauchy sequence in the space of progressively measurable processes $(Y_t,Z_t,U_t)$ satisfying $$\begin{aligned}
E \left [\sup_{0 \leq t\leq T} |Y_t|^2 \right] +E \left [\int _0 ^T \|Z_t\|^2 ds \right] +E \left[\int _0 ^T \int _U |U_t(e)|^2 \lambda(de) ds \right]<\infty.\end{aligned}$$ Hence, taking a subsequence if necessary, we have a sequence $(Y_t^n,Z_t^n,U_t^n)_{n \geq 0}$ and a triple of processes $(Y_t,Z_t,U_t)$ such that $$Y_t=\lim _{n \to \infty} Y^n_t, \
Z_t=\lim _{n \to \infty} Z^n_t, \
U_t=\lim _{n \to \infty} U^n_t$$ in the sense that $$\begin{aligned}
\label{aadg}
(i)&&E \left [\sup_{0 \leq t\leq T} |Y_t^n-Y_t|^2 \right] \to 0,\notag\\
(ii)&&E\left [\int _0 ^T \|Z_t^n-Z_t\|^2 ds \right] \to 0, \notag \\
(iii)&& E \left [\int _0 ^T \int _U |U_t^n(e)-U_t(e)|^2 \lambda(de) ds \right] \to 0\end{aligned}$$ as $n \to \infty$. Furthermore, by Lemma \[ll1\] we have $$\begin{aligned}
\label{ffaagain}
(i)&&E \left[\sup_{0 \leq t \leq T} |Y_t|^2 \right] <\infty,\notag\\
(ii)&& E \left [\int_0 ^T \|Z_t\|^2 dt + \int _0^T \int _U|U_s(e)|^2 \lambda(de)ds \right ] < \infty.\end{aligned}$$ Note that, as a uniform limit of càdlàg functions $\{Y^n_t\}$, we immediately have that $Y_t\in \mathcal{D}\left( \left[ 0,T\right] ,\R^{d}\right)$ and, by Lemma \[lemmbbaa\] $(i)$, we have $Y_t \in \overline D$. Recall that $$\Lambda_t^n= -n \int _0 ^t (Y^n_s- \pi(s,Y^n_s))ds=\int_0^t -\frac{( Y_s ^n-\pi(s,Y_s ^n))}{| Y_s ^n-\pi(s,Y_s ^n)|}d|\Lambda^n|_s$$ and that $-( Y_s ^n-\pi(s,Y_s ^n))/{| Y_s ^n-\pi(s,Y_s ^n)|}$ is an element in the inward directed normal cone to $D_s$ at $\pi(s,Y_s ^n)\in\partial D_s$. Using $(iii)$, and it follows that there exists $\Lambda_t$ such that $$\Lambda_t=\lim _{n \to \infty}\Lambda^n= \lim _{n \to \infty} -n \int _0 ^t (Y^n_s- \pi(s,Y^n_s))ds$$ in the sense that $$E \left [\sup_{0 \leq t\leq T} |\Lambda_t^n-\Lambda_t|^2 \right ] \to 0.$$ Hence, as $(\Lambda^n_t(\omega))_{0\leq t \leq T}$ is continuous, $(\Lambda_t(\omega))_{0\leq t \leq T}$ is continuous in $t$ a.s.
Existence: $(Y_t, Z_t, U_t, \Lambda_t)$ is a solution
-----------------------------------------------------
We will now prove that the constructed quadruple $(Y_t,Z_t,U_t,\Lambda_t)$ is a solution to our original problem. We first note that, as a limit of $(Y^n_t, Z^n_t, U^n_t)$, $(Y_t,Z_t,U_t)$ are progressively measurable, $Y_t\in \mathcal{D}\left( \left[ 0,T\right] ,\R^{d}\right)$ and $Z$ and $U$ are predictable. Hence it remains to verify that $(Y_t, Z_t, U_t, \Lambda_t)$ satisfies $(i)$-$(vi)$ stated in Definition \[rbsde\] and that $\Lambda\in \mathcal{BV}\left( \left[ 0,T\right] ,\mathbb{R}
^{d}\right) $. That $(Y_t, Z_t, U_t, \Lambda_t)$ satisfies $(i)$-$(iii)$ was proved above and $(iv)$ is a consequence of Lemma \[lemmbbaa\] and $(i)$. Hence we in the following focus on properties $(v)$ and $(vi)$. As mentioned above, we have that $(\Lambda_t(\omega))_{0\leq t \leq T}$ is continuous in $t$ for almost all $\omega$ by uniform convergence. Furthermore, using that $$\int _t ^T d|\Lambda^n|_s=n \int _t ^T |Y_s^n - \pi(s, Y^n _s)|ds$$ we see from Lemma \[ll1\] $(iii)$ that $$E \left[\int _0 ^T d|\Lambda^n|_s \right ] = E \left [ n \int _0 ^T |Y^n_s-\pi(s,Y^n_s)| ds \right ]\leq c\mbox{ for all $n\in\mathbb Z_+$},$$ for some constant $c$ which is independent of $n$. It follows that $\Lambda_t(\omega)$ is of bounded total variation on $[0,T]$ for almost all $\omega$. Hence, it only remains to verify that $$\begin{aligned}
\label{defo}
(v)&&\Lambda_t=\int_{0}^{t}\gamma _{s}d\left\vert \Lambda \right\vert
_{s},\ \gamma _{s}\in N_{s}^{1}\left( Y_{s}\right) \ \mbox{ whenever } Y_s \in \partial D_{s},\notag\\
(vi)&& d\left\vert \Lambda \right\vert \left( \left\{ t\in \left[ 0,T\right]
:\left(t,Y_{t}\right) \in D\right\} \right)=0.\end{aligned}$$ To verify the statements in we will use the following lemma.
\[lemma:adgda\] Let $\{\Lambda^n\}_{n \in \mathbb{Z}_+}$ be a sequence of continuous functions, $\Lambda^n:[0,T]\to \R^d$, which converges uniformly to $\Lambda$ as $n\to \infty$. Assume $\Lambda^n\in \mathcal{BV}\left( \left[ 0,T\right] ,\mathbb{R}
^{d}\right) $ and that $|\Lambda^n|_T \leq c$, for some $c<\infty$, hold for all $n$. Let $\{f^n\}_{n \in \mathbb{Z}_+}$ be a sequence of càdlàg functions, $f^n:[0,T]\to \R^d$, converging uniformly to $f$ as $n\to \infty$. Then, $$\lim _{n \to \infty}\int _0 ^t \langle f^n_s, d\Lambda^n_s \rangle = \int _0 ^t \langle f_s, d\Lambda_s \rangle$$ for all $t \in [0, T]$.
This is essentially Lemma 5.8 in [@GP], see also [@S].
Using Lemma \[lemmaa\] $(i)$ we see that , $$\langle z_t- Y^n _t, Y^n_t-\pi(t,Y^n_t) \rangle \leq 0$$ for any càdlàg process $z_t$ taking values in $\overline{D_t}$. Hence, for any such process $z_t$ we have that $$\begin{aligned}
\int_0 ^t -n \langle z_s- Y^n _s, Y^n_s-\pi(s,Y^n_s) \rangle ds =\int_0^t \langle z_s-Y^n_s , d\Lambda^n_s \rangle \geq 0.
\end{aligned}$$ Passing to the limit we obtain, using Lemma \[lemma:adgda\], that $$\label{defo+}
\int_0 ^ t\langle Y_s- z_s, d\Lambda_s \rangle \leq 0$$ for all $z \in \mathcal{D}\left( \left[ 0,T\right], \R^{d}\right)$ taking values in $\overline D$, and for all $t\in [0,T]$. Next, let $\tau \in [0,T)$ be any time such that $Y_{\tau} \in D$ and let $\hat \gamma$ be a unit vector in $\R^d$. Since $Y_s$ is right-continuous, taking assumption into account, we see that there exists ${\epsilon}>0$ and $\delta > 0$ such that $Y_s+{\epsilon}\hat \gamma \in D$ and $Y_s-{\epsilon}\hat \gamma \in D$ whenever $s \in [\tau, \tau+\delta]$. However, this in combination with implies that $$0 \leq \int _{\tau} ^{\tau+\delta} \hat \gamma d\Lambda_t \leq 0,$$ which in turn implies $(vi)$ in . Hence $$\begin{aligned}
\Lambda_t=\int_{0}^{t}\gamma _{s}d\left\vert \Lambda \right\vert_{s},\end{aligned}$$ for some vector field $\gamma_s\in \R^d$, with support on $\partial D$, and such that $|\gamma_{s}|=1$ (see ). To conclude the existence part of Theorem \[maint1\] it remains to show $(v)$, i.e., that $\gamma_s\in N_{s}^{1}\left( Y_{s}\right)$ whenever $Y_s \in \partial D_{s}$. However, using the above and we see that to prove $\gamma_s\in N_{s}^{1}\left( Y_{s}\right)$ whenever $Y_s \in \partial D_{s}$, it is enough to prove that if $ Y_{s}\in\partial D_s$ and if $\langle Y_{s}-z_s,\gamma_s\rangle\leq 0$ for all $z_s \in D_s$, then ${\gamma_s}\in N_{s}\left( Y_{s}\right)$. To do this, take $\beta=Y_s-\gamma_s \in \mathbb R^d$. Then, $$|\beta-z_s|^2=|\beta-Y_{s}|^2+|Y_{s}-z_s|^2 + 2\langle \beta -Y_{s},Y_{s}-z_s\rangle$$ for all $z_s\in D_s$. Hence, if $\langle \beta -Y_{s},Y_{s}-z_s\rangle = -\langle Y_{s}-z_s,\gamma_s\rangle\geq 0$, then we have that $$|\beta-z_s|^2\geq|\beta-Y_{s}|^2$$ for all $z_s \in D_s$, which implies $\gamma_s \in N_s(Y_s)$. This proves $(v)$ and thus the proof of the existence part of Theorem \[maint1\] is complete. $\Box$
Uniqueness: $(Y_t, Z_t, U_t, \Lambda_t)$ is the only solution
-------------------------------------------------------------
We here prove the uniqueness part of Theorem \[maint1\] using Ito’s formula. Indeed, assume that $(Y^i,Z^i, U^i , \Lambda^i)$, $i=1,2$, are two solutions to the the reflected BSDE under consideration and define $$\{\Delta Y_t, \Delta Z_t, \Delta U_t, \Delta \Lambda_t\}= \{Y^1_t - Y^2_t, Z^1_t - Z^2_t, U^1_t - U^2_t, \Lambda^1_t - \Lambda^2_t \}.$$ Then, applying Ito’s formula to $|\Delta Y_t|^2$ and taking expectation we have that $$\begin{aligned}
& E \left [ |\Delta Y_t|^2+\int _t ^T |\Delta Z_s|^2 ds + \int _t ^T \int _ U |\Delta U_s(e)|^2 \lambda(de)ds \right ] \\
=& 2 E \left [ \int _ t ^T \langle \Delta Y_s, f(s,Y_s^1, Z_s^1, U_s^1)-f(s, Y_s^2,Z_s^2,U_s^2) \rangle ds \right ]\\
&+ 2 E \left [\int _t ^T \langle \Delta Y_s, d\Delta\Lambda_s \rangle \right].\end{aligned}$$ Using we see that the last term on the right hand side in the above display is $\leq0$. Next, using the Lipschitz character of $f$ and standard manipulations, see , , we can conclude that $$\begin{aligned}
& E \left [ |\Delta Y_t|^2+\int _t ^T |\Delta Z_s|^2 ds + \int _t ^T \int _ U |\Delta U_s(e)|^2 \lambda(de)ds \right ] \\
\leq& c E\left [ \int _ t ^T |\Delta Y_s|^2 ds + \frac{1}{2} \int _t ^T |Z_s| ^2 ds + \frac{1}{2} \int _t ^T |\Delta U_s(e)|^2 \lambda(de)ds \right ].\end{aligned}$$ Applying Gronwall’s lemma we see that $\{\Delta Y_t, \Delta Z_t, \Delta U_t\}$ must be identically zero a.s. By $(iii)$ of Definition 1, the same applies to $\Delta \Lambda_t$ and the proof is hence complete. $\Box$
[99]{}
B. El Asri and I. Fakhouri, [*Optimal multi-modes switching with the switching cost not necessarily positive*]{}, Arxiv preprint arXiv:1204.1683v1, 2012 - arxiv.org.
B. El-Asri and S. Hamadene, *The finite horizon optimal multi-modes switching problem: The viscosity solution approach*, Applied Mathematics & Optimization [**60**]{} (2009), 213-235.
G. Barles, R. Bukhdan and E. Pardoux, [*BSDE’s and integral-partial differential equations*]{}, Stochastics and Stochastics Report, [**60**]{} (1997), 57-83.
C. Costantini, [*The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations*]{}, Probability Theory and Related Fields, [**91**]{} (1992), 43-70.
C. Costantini, E. Gobet, and N. El Karoui. [*Boundary sensitivities for diffusion processes in time dependent domains*]{}, Applied Mathemathics & Optimization, [**54**]{} (2006), 159-187.
J. Cvitani´c and I. Karatzas, [*Backward stochastic differential equations with reflection and Dynkin games*]{}, The Annals of Probability, [**24**]{} (1996), 2024-2056.
B. Djehiche, S. Hamadene and A. Popier, *A finite horizon optimal multiple switching problem*, SIAM Journal on Control and Optimization, [**4**8]{} (2010), 2751-2770.
N. El-Karoui, C. Kampoudjian, E. Pardoux, S. Peng and M.C. Quenez *Reflected Solutions of backward SDE’s and related obstacle problems for PDE’s*, The Annals of Probability, [**25**]{} (1997) 702-737.
N. El Karoui, S. Peng and M.C. Quenez. [*Backward stochastic differential equations in finance*]{}, Mathematical finance, [**7**]{} (1997), 1-71.
A. Gegout-Petit and E. Pardoux, [*Equations differentielles stochastiques retrogardes reflechies dans un convexe*]{}, Stochastics and Stochastics Reports, [**57**]{} (1996), 111-128.
S. Hamadene and J. Lepeltier, [*Zero-zum stochastic differential games and backward equations*]{}, Systems & Control Letters, [**24**]{} (1995), 259-263.
Y. Hu and S. Tang *Multi-dimensional BSDE with oblique reflection and optimal switching*, Probability Theory and Related Fields, [**147**]{} (2010), 89-121.
S. Hamadene and J. Zhang *Switching problem and related system of reflected backward SDEs*, Stochastic Processes and their Applications, [**120**]{} (2010), 403-426.
P. L. Lions and A. S. Sznitman, [*Stochastic differential equations with reflecting boundary conditions*]{}, Communications on Pure and Applied Mathematics, [**37**]{} (1984), 511-537.
K. Nystr[ö]{}m and T. [Ö]{}nskog, [*The Skorohod Oblique Reflection Problem in Time-dependent Domains*]{}, Annals of Probability [**38**]{} (2010), 2170-2223.
Y. Ouknine, [*Reflected backward stochastic differential equations with jumps*]{}, Stochastics and Stochastics Reports, [**65**]{} (1998), 111-125.
B. Oksendal and A. Sulem, [*Applied Stochastic Control of Jump Diffusions*]{}, Springer-Verlag, Berlin, 2005.
E. Pardoux and S. Peng, [*Backward stochastic differential equations and quasiliner parabolic partial differential equations*]{}, Lecture Notes in CIS, [**176**]{} (1992), 200-217.
S. Ramasubramanian, [*Reflected backward stochastic differetial equations in an orthant*]{}, Proceedings of the Indian Academy of Science, [**112**]{} (2002), 347-360 Y. Saisho, [*Stochastic differential equations for multi-dimensional domain with reflecting boundary*]{}, Probability Theory and Related Fields, [**74**]{} (1987), 455-477.
H. Tanaka, [*Stochastic differential equations with reflecting boundary conditions in convex regions*]{}, Hiroshima Mathematical Journal, [**9**]{} (1979), 163-177. 1987.
S. Tang and X. Li, [*Necessary conditions for optimal control of stochastic systems with random jumps*]{}, SIAM Journal of Control and Optimization, [**32**]{} (1994), 1447-1475.
|
---
author:
- |
I. Ezeani, P. Rayson\
Lancaster University,\
Lancaster, UK\
I. Onyenwe, C. Uchechukwu\
Nnamdi Azikiwe University,\
Awka, Nigeria\
Mark Hepple\
Sheffield University,\
Sheffield, UK\
bibliography:
- 'ig\_en\_mt.bib'
title: |
Igbo-English Machine Translation:\
An Evaluation Benchmark
---
Introduction
============
Although researchers are pushing the boundaries and enhancing the capacities of NLP tools and methods, works on African languages are lagging behind. A lot of focus on well-resourced languages such as English, Japanese, German, French, Russian, Mandarin Chinese etc. Over 97% of the world’s 7000 languages, including African languages, are low-resourced for NLP i.e. they have little or no data, tools, and techniques for NLP research. For instance, only 5 out of 2965 (0.19%) authors of full-text papers in the ACL Anthology[^1] extracted from the 5 major conferences in 2018 (ACL, NAACL, EMNLP, COLING and CoNLL) are affiliated to African institutions[^2].
In this work, we discuss our effort toward building a standard evaluation benchmark dataset for Igbo-English machine translation tasks. Igbo[^3] is one of the 3 major Nigerian languages spoken by over 50 million people globally, 50% of whom are in southeastern Nigeria. Igbo is low-resourced despite some efforts toward developing IgboNLP such as part-of-speech tagging: [@onyenwe2014part], [@onyenwe2019toward]; and diacritic restoration: [@ezeani2016automatic], [@ezeani2018transferred].
Although there are exiting sources for collecting Igbo monolingual and parallel data, such as the *OPUS Project* ([@TIEDEMANN12.463]) or the [JW.ORG](JW.ORG), they have certain limitations. The OPUS Project is a good source training data but, given that there are no human validations, may not be good as an evaluation benchmark. [JW.ORG](JW.ORG) contents, on the other hand, are human generated and of good quality but the genre is often skewed to religious contexts and therefore may not be good for building a generalisable model.
This project focuses on creating and publicly releasing a standard evaluation benchmark dataset for Igbo-English machine translation research for the NLP research community. This project aims to build, maintain and publicly share a standard benchmark dataset for Igbo-English machine translation research . There are three key objectives:
1. Create a minimum of 10,000 English-Igbo human-level quality sentence pairs mostly from the news domain
2. To assemble and clean a minimum of 100,000 monolingual Igbo sentences, mostly from the news domain, as companion monolingual data for training MT models
3. To release the dataset to the research community as well as present it at a conference and publish a journal paper that details the processes involved.
Methods
=======
To achieve the objectives above, the task was broken down in the following phases:\
\
**Phase 1: Raw data collection and pre-processing:**\
This phase is to produce cleaned and pre-processed a minimum 10,000 sentences: 5,000 English and 5,000 Igbo. It involved the collection, cleaning and pre-processing (normalisation, diacritic restoration, spelling correction etc.) of Igbo and English sentences from freely available electronic texts (e.g. Wikipedia, CommonCrawl, local government materials, local TV/Radio stations etc).\
\
**Phase 2: Translation and correction**\
In this phase, the 10,000 sentence pairs are created manual translation and correction. The key tasks include:
1. Translating English sentences to Igbo (EN-IG)
2. Translating Igbo sentences to English (IG-EN)
3. Correcting the translations
5 Igbo speakers were engaged for the bidirectional of translations while 3 other Igbo speakers, including an Igbo linguist are assisting with the on-going corrections. Chunks ($\approx$ 250 each) of sentences are given to each translator in each direction (i.e. IG-EN and EN-IG). At the time of submission, we have $11,584$ sentence pairs as detailed in Table \[breakdown\] while the splits of the parallel data into *development*, *text* and *hidden test* sets is shown in Table \[splits\]
---------------- ---------- -------------------------------------------
*Igbo-English* 5,836 <https://www.bbc.com/igbo>
*English-Igbo* 5,748 Mostly from local newspapers (e.g. Punch)
*Total* $11,584$
---------------- ---------- -------------------------------------------
: Splits of the Benchmark Evaluation Parallel Data[]{data-label="splits"}
**Evaluation Splits** **IG-EN** **EN-IG**
----------------------- ----------- -----------
*Development Set* 5000
*Test set* 500
*Hidden Test* 336
: Splits of the Benchmark Evaluation Parallel Data[]{data-label="splits"}
**Phase 3: Manual checks and Inter-translator Agreement**\
This phase is currently on-going and it involves manually checking and correcting the 10,000 translated sentence pairs. This is to ensure that the translations conform with the contemporary communicative usage of the languages. Our approach so far is simplistic i.e. it seeks to establish absolute agreement between translators. We know it could overstate agreement ([@lommel2014assessing]), but we believe it will improve the quality of the translation. More work will be done in this area in future.\
**Phase 4: Monolingual Igbo sentence collection and pre-processing**\
The aim here is to collect and clean a minimum of 100,000 monolingual Igbo sentences. the cleaning process involves normalisation, diacritic restoration, spelling correction from freely available sources (news, government materials, Igbo literature, local TV/Radio stations etc).
A large chunk of the data is collected from the Jehova’s Witness Igbo[^4] contents. Though we included the Bible, more contemporary contents (books and magazine e.g. *Teta!* (*Awake!*), *Ulo Nche!* (*WatchTower*)) were the main focus. Also, we got contents from BBC-Igbo[^5] and Igbo-Radio (<https://www.bbc.com/igbo>) as well as Igbo literary works(*Eze Goes To School*[^6] and *Mmadu Ka A Na-Aria* by Chuma Okeke). This phase is still on-going but we have so far collected and cleaned $\approx$ 380k Igbo sentences as detailed in Table \[data\_sources\_counts\]. It is important to point out that we have also collected data in other formats (e.g. audio, non-electronic texts) from local media houses which we hope to also transcribe and include in our collection.
------------------------ ------------- --------------- ------------
eze-goes-to-school.txt 1272 25413 2616
mmadu-ka-a-na-aria.txt 2023 39731 3292
bbc-igbo.txt 34056 566804 28459
igbo-radio.txt 5131 191450 13391
jw-ot-igbo.txt 32251 712349 13417
jw-nt-igbo.txt 10334 253806 6731
jw-books.txt 142753 1879755 25617
jw-teta.txt 14097 196818 7689
jw-ulo-nche.txt 27760 392412 10868
jw-ulo-nche-naamu.txt 113772 1465663 17870
**383,449** **5,724,201** **69,091**
------------------------ ------------- --------------- ------------
: Data Sources and Counts[]{data-label="data_sources_counts"}
Access to data
==============
All data generated as described above are available under the Creative Commons license from this GitHub repository[^7] and will be regularly updated.
Conclusion
==========
This work presents an on-going project on building a benchmark evaluation dataset for Igbo–English machine translation project. The released dataset will hopefully be useful in fairly and more reliably comparing the performance of models built for IG-EN translations.
Our efforts in increasing the size of the sentence pairs as well as improving the quality of translations will continue in will be published as we progress. In addition to releasing the dataset to the research community, our plan for future works include building and comparing various machine translation models based on the current state-of-the-art methods. This will be followed by an in-depth analysis of their performances.
### Acknowledgments {#acknowledgments .unnumbered}
The authors wish to acknowledge and thank Facebook AI Research (Facebook AI) for funding this project. Our immense gratitude also goes to Marc’Aurelio Ranzato and Francisco Guzmán for initiating, facilitating the funding and providing us with a lot of technical ideas.
[^1]: <https://www.aclweb.org/anthology/>
[^2]: **Source:** <http://www.marekrei.com/blog/geographic-diversity-of-nlp-conferences/>
[^3]: **Igbo:** <https://en.wikipedia.org/wiki/Igbo_language>
[^4]: **Source:** <https://www.jw.org/ig/>
[^5]: <https://www.bbc.com/igbo/>
[^6]: [https:// bit.ly/2vdGvKN](https:// bit.ly/2vdGvKN)
[^7]: <https://github.com/IgnatiusEzeani/IGBONLP/tree/master/ig_en_mt>
|
---
abstract: 'The primary focus of autonomous driving research is to improve driving accuracy. While great progress has been made, state-of-the-art algorithms still fail at times. Such failures may have catastrophic consequences. It therefore is important that automated cars foresee problems ahead as early as possible. This is also of paramount importance if the driver will be asked to take over. We conjecture that failures do not occur randomly. For instance, driving models may fail more likely at places with heavy traffic, at complex intersections, and/or under adverse weather/illumination conditions. This work presents a method to learn to predict the occurrence of these failures, i.e. to assess how difficult a scene is to a given driving model and to possibly give the human driver an early headsup. A camera-based driving model is developed and trained over real driving datasets. The discrepancies between the model’s predictions and the human ‘ground-truth’ maneuvers were then recorded, to yield the ‘failure’ scores. Experimental results show that the failure score can indeed be learned and predicted. Thus, our prediction method is able to improve the overall safety of an automated driving model by alerting the human driver timely, leading to better human-vehicle collaborative driving.'
author:
- 'Simon Hecker$^{1}$, Dengxin Dai$^{1}$, and Luc Van Gool$^{1,2}$[^1][^2]'
bibliography:
- 'egbib.bib'
title: '**Failure Prediction for Autonomous Driving**'
---
INTRODUCTION {#sec:introduction}
============
Autonomous vehicles will have a substantial impact on people’s daily life, both personally and professionally. For instance, automated vehicles can largely increase human productivity by turning driving time into working time, provide personalized mobility to non-drivers, reduce traffic accidents, or free up parking space and generalize valet service [@autonomous:vehicle:guide:policymakers]. As such, developing automated vehicles is becoming the core interest of many, diverse industrial players. Recent years have witnessed great progress in autonomous driving [@deep:driving; @nvidia:driving:16; @end:driving:16; @perception:path:generation; @chen2017brain; @arxiv2018:hecker], resulting in announcements that autonomous vehicles have driven over many thousands of miles and that companies aspire to sell such vehicles in a few years. All this has fueled expectations that fully automated vehicles are coming soon.
Yet, significant technical obstacles must be overcome before assisted driving can be turned into full-fletched automated driving, a prerequisite for the above visions to materialize. To make matters worse, an automated car that from time to time will call on the driver to take over, will, by many drivers, be considered worse than having no automated driving at all. Indeed, in such a transition situation, the driver will be required to permanently pay attention to the road, as to not be out of context when s/he suddenly needs to act. And that does not go together well with the boredom coming with not having to intervene for a long time. The more successful the automation, the worse the issue. Add legal responsibilities to the picture, and the possibility that the human driver is called upon to take decisions, however rarely that is, may still be with us for a while.
{width="0.85\linewidth"}
With so much effort currently going into improving autonomous driving, such systems will certainly improve quickly. Yet, as said, during the coming years performance will probably not be strong enough such that occasional mistakes can be avoided altogether. Indeed, driving models may still fail due to congested traffic, bad weather, frontal illumination, road constructions, etc., or simply unexpectedly, due to the idiosyncrasies of the underlying algorithms. Failures of a vehicle can be catastrophic [@the:cost:of:accident], and it is therefore crucial to obtain an early warning for impending trouble. Despite this importance, the community has so far paid limited attention to the automated predictions of potential failures. We therefore decided to push for a capability where driving models can yield a warning such as *I am unable to make a reliable decision for the coming situation*, and can give the human driver an early warning about a possible need for human intervention.
We propose the concept of *Scene Drivability*, i.e. how easy a scene is for an automated car to navigate. A low *drivability* score means that the automated vehicle is likely to fail for the particular scene. Obviously, *scene drivability* is dependent on the autonomous driving system at hand. In order to quantify and learn this property, we therefore first need to pick a particular autonomous driving model. We developed one of our own, solely based on video observations. Videos from car-mounted cameras were used to train it. In keeping with modern machine learning, it automatically learned things like ‘when the vehicle is in the left-most lane, the only safe maneuvers are a right-lane change or keeping straight, unless the vehicle is approaching an intersection’. It is clear that such learning requires the system to be exposed to a representative sample of scenarios. We therefore trained the model on a large, real driving dataset, which contains video sequences and other time-stamped sensor measurements such as steering angles, speeds, and GPS coordinates [@arxiv2018:hecker]. The driving model achieves a performance similar to other recent approaches based on video observations [@end:driving:16; @driving:attention; @arxiv2018:hecker]. Discrepancies between the predictions by the trained driving model and the ground-truth maneuvers by human drivers are then used to assess the likelihood of failure, i.e. the *Scene drivability* score.
Due to the success of deep neural networks in supervised learning [@lecun2015deep] and especially in autonomous driving, we develop a Recurrent Convolutional Network (RCNet) with four CNNs [@vgg16] as visual encoders and three LSTMs [@lstm] to integrate the visual contents, temporal relationships, and the previous driving states (steering angle and speed) into one single prediction model. The model can be trained very efficiently in an end-to-end manner, and its architecture is shown in Figure \[fig:pipeline\]. This architecture is used for both tasks: car driving and its failure prediction. All layers, except for the task-specific fully-connected layers, are shared for computational efficiency.
Readers will notice that our model is quite simple. The emphasis of this paper is not on achieving the state-of-the-art driving performance. Rather, it is to provide a sensible driving model and infer failure prediction for it, as a contribution to let autonomous driving survive the risky market situation ahead. The choice of the model is also due to our access to sensors and data.
In this work, we quantize the *scene drivability* scores for particular driving scenes to two levels: *Safe* and *Hazardous*. They are intended to translate to the two driving modes ‘Full Automation’ and ‘Driver assisted’. Our experiments show that scene drivability can indeed be learned and predicted. Of course, the drivability will increase if the driving model is improved, especially when information from other sensors is added, such as from GPS, laser scanners, and radar. Our method is flexible enough to include those. We also do [*not*]{} claim to predict the drivability of scenes for any driving model out there, but rather propose a framework that can be trained to extract the drivability for other models as well.
Related Work {#sec:related}
============
Our work is relevant to both autonomous and assisted driving, and to vision system failure mode prediction.
Driving Models for Automated Cars
---------------------------------
Significant progress has been made in autonomous driving, especially due to the deployment of deep neural networks. Driving models can be clustered into two groups based on their working paradigms [@deep:driving]: mediated perception approaches and end-to-end mapping approaches. Mediated perception approaches require recognition of all driving-relevant objects, such as lanes, traffic signs, traffic lights, cars, pedestrians, etc. [@kitti:dataset; @Cityscapes; @3d:object:detection:AD]. Some of these recognition tasks could be tackled separately, and there are excellent works [@3d:traffic:scene] integrating the results. This group of methods represents the current state-of-the-art for autonomous driving, and most of the results are reported with diverse sensors used, such as laser scanners, GPS, radar and high-definition maps of the environment.
End-to-end mapping methods aim to construct a direct mapping from the sensory input to the maneuvers. The idea can be traced back to the 1980s, when a neural network was used to learn a direct mapping from images to steering angles [@network:autonomous:1980]. Another successful example of learning a direct mapping is [@LeCun:driving:05], which uses ConvNets to learn a human driver’s steering angles. The popularity of this idea is fueled by the success of end-to-end trained deep neural networks and the availability of large driving datasets [@nvidia:driving:16; @oxford:driving; @arxiv2018:hecker]. Recent advances have been shown in [@arxiv2018:hecker] by incorporating surround-view videos and route planning. The future end-to-end approaches may also need a mixture of sensors and modules for even better performance. Possible modules consist of traffic agent detection and tracking [@target:tracking:lidar:13; @multi-sensor:detection:tracking:14; @3d:object:detection:AD; @semantic:foggy:scene; @DomainAdaptiveFasterRCNN], future prediction of road agents’ location and behavior [@predicting:deeper; @sceneparse:opticalflow; @intent:aware:pedestrian:prediction], and driveability map generation [@driveability:maps:for:lane].
Assistive Features for Vehicles
-------------------------------
Over the last decades, more and more assistive technologies have been deployed, that help to increase driving safety. Technologies such as lane keeping, blind spot checking, forward collision avoidance, adaptive cruise control etc., alert drivers about potential dangers [@forecast:control; @control:driver:modeling; @kasper2012object]. In the same vein, drivers are monitored to avoid distraction and drowsiness [@look:at:driver; @gaze:driver], and maneuvers are anticipated [@car:knows:iccv15] to generate alerts in a timely manner. Readers are referred to [@looking:at:human] for an excellent overview of such work. Our work complements existing ADAS and driver monitoring techniques by equipping fully automated cars with an assistive feature to anticipate automation failures and yields a timely alert for the human driver to take over.
Failure Prediction
------------------
Performance-blind algorithms can be disastrous. As automated vision increasingly penetrates industrial applications, this issue is gaining attention [@dai:phd:thesis; @zhang2014predicting]. Notable examples in computer vision for learning model uncertainty or failure include: semantic image segmentation [@kendall2015bayesian], optical flow [@confidence:of:08; @flow:confidence], image completion [@completion:quality], stereo [@stereo:confidence], and image creation [@dai:synthesizability]. Our work adds autonomous driving to the list. In addition to creating warnings, performance-aware algorithms bring other benefits as well. For instance, they can speed up algorithms downstream, by adaptively allocating computing resources based on scene difficulty. For autonomous driving, this can also mean using sensors adaptively or selectively. Another paper relevant to ours is [@Anticipating:accidents:DB:18], which anticipates traffic accidents by learning from a large-scale incidents database.
Method {#sec:method}
======
In this section, we first present our end-to-end direct mapping method for autonomous driving, based on the recent success of recurrent neural networks. We then present how we use the same architecture to learn to predict the (un)certainty of the system, i.e. our drivability score.
$\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{./trainingtest.pdf} &
\includegraphics[width=0.38\linewidth]{./timeline.pdf} \\
\text{(a) training procedure} &
\text{(b) target space}
\end{tabular}$
Driving Model {#sec:drivingmodel}
-------------
In contrast to predicting the car’s ego-motion like previous work [@nvidia:driving:16; @end:driving:16], our model predicts the steering wheel (angle) and the speed of the cars directly. The goal of our driving model is to map directly from a frontal-view video to the steering wheel angle and speed of the car. Let us denote the video by $V$, and the vehicle’s steering wheel angle and speed by $A$ and $S$ respectively. We assume that the driving model works with discrete time and makes driving decisions every $1/f$ seconds. The inputs $V$, $A$ and $S$ are synchronized and sampled at a sampling rate $f$. In this work, $f=4$. Unless stated otherwise, our inputs and outputs all are represented in this discretized form.
Let us denote the current video frame by $V_t$, the current vehicle’s speed by $S_t$, and the current steering angle by $A_t$. The $k^{th}$ previous values can then be denoted by $V_{(t-k)}$, $S_{t-k}$ and $A_{t-k}$, resp., and all $k$ previous values can be denoted by $\mathbf{V}_{[t-k,t)} \equiv \langle V_{t-k}, ..., V_{t-1}\rangle$, $\mathbf{A}_{[t-k,t)} \equiv \langle A_{t-k}, ..., A_{t-1}\rangle$ and $\mathbf{S}_{[t-k,t)} \equiv \langle S_{t-k}, ..., S_{t-1}\rangle$, resp.. Our goal is to train a deep network that predicts driving actions from all inputs: $$F: (\mathcal{V}_{[t-k,t]}, \mathcal{A}_{[t-k,t-1)}, \mathcal{S}_{[t-k,t-1)}) \rightarrow \mathcal{A}_t \times \mathcal{S}_t
\label{eq:learning:fun1}$$ where $\mathcal{A}_t$ represents the steering angle space and $\mathcal{S}_t$ the speed space for the current time. $\mathcal{A}$ and $\mathcal{S}$ can be defined at several levels of granularity. We consider the continuous values directly recorded from the car’s CAN bus, where $\mathcal{S}=\{S | 0 \leq S \leq 180$ for speed and $\mathcal{A}=\{A | -720 \leq A \leq 720\}$ for steering angle. Here, kilometer per hour (km/h) is the unit of $S$, and degree ($^\circ$) the unit of $A$.
Given $N$ training samples collected during real drives, learning to predict the driving actions for the current situation is based on minimizing the following cost: $$\begin{split}
L(\theta) = \sum_{n=1}^N \Big( l({A}^{n}_t, F_{\text{a}}(\mathbf{V}^{n}_{[t-k, t]}, \mathbf{A}^{n}_{[t-k, t)}, \mathbf{S}^{n}_{[t-k, t)})) \\ + \lambda l({S}^{n}_t, F_{\text{s}} (\mathbf{V}_{[t-k, t]}^n, \mathbf{A}^{n}_{[t-k, t)}, \mathbf{S}^{n}_{[t-k, t)})) \Big),
\end{split}$$ where $\lambda$ is a parameter balancing the two losses, one for steering angle and the other for speed. We use $\lambda=1$ in this work due to prior CAN signal normalization. $F$ is the learned function for the driving model. For the continuous regression task, $l(.)$ is the $L2$ loss function. Our model learns from multiple previous frames in order to better understand traffic dynamics. We assume that the *current* video frame $V_t$ is already available for making the decision.
Failure Prediction
------------------
An automated car can fail due to many causes. Here we focus on *scene drivability* – a driving situation is too challenging for the driving model to make reliable decisions. We define failure scores based on the discrepancies between the predicted maneuvers (steering angles and speed) and the human driver’s maneuvers. In particular, we denote the predicted speed and steering angle by $\bar{S}_t$ and $\bar{A}_t$. Then, the failure for speed and steering angle estimation are signaled by: $$g^a_t = \text{sgn}(\|A_t-\bar{A_t}\|-T_a),$$ and $$g^s_t = \text{sgn}(\|S_t-\bar{S_t}\|-T_s),$$ where $$\text{sgn}(x) =
\begin{cases}
1 & \text{if } x \ge 0,\\
0 & \text{if } x < 0.
\end{cases},$$ and $T_a$ and $T_s$ are thresholds defining correct and incorrect predictions for steering angle and speed. Then, the failure occurrence for the current time is signaled by: $$g_t = g_t^a \vee g_t^s,
\label{eq:failure:1}$$ where $\vee$ is an OR operator: $x \vee y = 0$ if $x = y = 0$ and $x \vee y = 1$ otherwise. The definition by Equation \[eq:failure:1\] quantizes scene drivability into two levels: *Safe* and *Hazardous*. *Safe* scenes are defined as those with an absolute error of less than $T_{a}$ degrees for steering angle and with an absolute error of less than $T_{s}$ km/h for speed. *Hazardous* are those with a deviation in either category larger than the defined threshold. A *safe* scene allows for a driving mode of *High Automation* and a *hazardous* scene allows for a driving mode of *Partial/No Automation*. These thresholds can be set and tuned according to specific driving models and legal regulations.
Failure prediction is more useful, the earlier it can be done, i.e. the more time can be given to the human driver to take over. Therefore, the failure that our model is trained to predict is from current time $t$ to future time $t+m$: $$g_{\overrightarrow{[t,t+m]}} = g_t \vee g_{t+1} \vee g_{t+2} \vee ... \vee g_{t+m}.
\label{eq:failure:2}$$ By learning to predict $g_{\overrightarrow{[t,t+m]}}$, our model will alert the human driver if either the speed prediction and/or the steering angle prediction is going to fail at any of the time points in the time period $[t, t+m]$. The learning goal is then changed to training a deep network model to make a prediction for driving actions for current time $t$ and to make a prediction for the drivability score for the time period from $t$ to a future time point $t+m$. In particular, the learning target is changed from Equation \[eq:learning:fun1\] to: $$F: (\mathcal{V}_{[t-k,t]}, \mathcal{A}_{[t-k,t-1)}, \mathcal{S}_{[t-k,t-1)}) \rightarrow \mathcal{A}_t \times \mathcal{S}_t \times \mathcal{G}_{\overrightarrow{[t,t+m]}}
\label{eq:learning:fun2}$$ where $\mathcal{G}_{\overrightarrow{[t,t+m]}}=\{0,1\}$ denotes the space of our drivability score defined by Equation \[eq:failure:2\]. In this work, $m$ is set to $8$ to represent a period of $2$ seconds. A different length can be used if the application needs. Please see Figure \[fig:timeline\] for the illustrative flowchart of the training procedure and solution space of our driving model and the failure prediction model.
Implementations {#sec:implementation}
---------------
We adopt a deep neural network for our learning task. The model learns to predict three targets: the vehicle’s steering angle, its speed, and the failure score defined by Equation \[eq:failure:2\]. In particular, our model consists of four copies of convolutional neural networks (CNNs) with shared weights as visual encoders, combined with three Long Short Term Memory networks (LSTMs) to integrate the visual information, historical driving speed, and historical steering angles. The outputs of the three LSTMs are integrated by three fully connected networks (FCN) to make the final predictions for the vehicle’s steering angle, its speed, and the failure status of Equation \[eq:failure:2\]. As shown in Figure \[fig:pipeline\], all layers of the network are shared by the three tasks except for the top, task-specific layers.
{width="75.00000%"}
The image input sequence consists of four video frames, taken from a front-facing camera mounted on the roof of the vehicle. The sequence includes the current video frame along with three previous frames; this allows for images in the sequence to vary significantly and thus improves the predictive performance of the model. Input images are center cropped to $270\times270$ pixels from an initial resolution of $480\times270$, resized to $240\times240$ and finally randomly cropped to $224\times224$ during training and center cropped to the same dimensions during evaluation. Each image in the sequence is fed into a ResNet34 [@resnet] with shared convolutional layers. The convolutional layers are pre-trained on the classification task of ImageNet [@imagenet]. All layers of the ResNet are trainable with the final layers output being fed into a 2-layer FCN: $fc(1024)$ - Relu - $fc(1024)$ - Relu. The parameters of the FCN are randomly initialized. This results in a $4\times1024$ feature vector which describes the high level historical and current visual input into the system. We incorporate three parallel LSTMs [@lstm] with $128$, $16$ and $16$ hidden states and $4$, $2$ and $2$ layers, resp. The high level visual features of the ResNets, the historical speed information, and the steering angle information are fed into the three LSTMs, resp. Steering angle and speed information are sampled at the same sampling rate of $1/f=4$, the same as for the video. A temporal sequence of length $k=4$ is used for all three inputs. At this point, we have aggregated a total of three feature vectors, that describe visual information, historical steering angle and historical speed. These vectors are then concatenated. Our final prediction task varies depending on whether we train a driving agent or a failure prediction agent. Consequently the very top layers of our model architecture will vary depending on the task. In the driving agent case, the network is continuously trained to output the current steering angle and vehicle speed using two regression networks. The two regression networks consist of a 2-layer fully connected network of $fc(512)$ - Relu - $fc(1)$ each, and are tasked to output either steering angle or vehicle speed. For the failure prediction agent, we train our network on a two class *Safe* or *Hazardous* classification task as defined by Equation \[eq:failure:2\]. We predict to an interval of $m=8$ samples (i.e. $2$ seconds) into the future from the current time. This gives us the opportunity to notify the driver of a potential hazard ahead of time, allowing for an adequate response. Our classification task network architecture consists of a 2-layer fully connected network $fc(512)$ - Relu - $fc(2)$. This network is optimized via the cross entropy loss. We optimize our network with the Adam Optimizer [@adam] and a learning rate of $0.00001$. Our models train for 10 epochs with a mini-batch size of $32$ which results in around $15$ hours of training time each with a GeForce GTX TITAN X Graphics Card.
$\begin{tabular}{cccc}
\begin{turn}{90}\hspace{0mm} $\[T\_a, T\_s\] = \[10.0\^, 5.0\]$ \hspace{2mm} $\[T\_a, T\_s\] = \[7.0\^, 3.0\]$ \hspace{4mm} $\[T\_a, T\_s\] = \[5.0\^, 2.0\]$ \end{turn} &
\hspace{-1mm}
\includegraphics[width=0.95\linewidth, height=131mm]{./2x3_map_manual25.png}\\
& \hspace{15mm} (a) Driving Model \hspace{35mm} (b) Driving Model + Failure Prediction Model
\end{tabular}$
Experiments {#sec:experiment}
===========
Datasets and Training
---------------------
We train and evaluate our method on our autonomous vehicle dataset which consists of around $150,000$ unique sequences captured by a car mounted camera in Switzerland. Alongside the video data, time-stamped sensor measurements are provided by the dataset as well, such as the vehicle’s speed, steering wheel angle and GPS locations. Thus, this data is ideal for self-driving studies. The GPS coordinates allow for compelling visualizations of where the model fails. In order to properly train and evaluate our model, we split our dataset to three datasets of equal size: Dataset 1, Dataset 2, and Dataset 3. We train our driving model on Dataset 1, train the failure prediction model on Dataset 2, and evaluate both models on Dataset 3. The two models need to be trained on separate datasets, because the predictions of the driving model on its own training set are too optimistic to reflect the real failures. Please refer to Figure \[fig:timeline\] for the training procedure of our models.
$\begin{tabular}{cccc}
\hspace{-2mm}
\includegraphics[width=0.33\linewidth]{./methodCompare_a5_s2.png}
& \hspace{-4mm}
\includegraphics[width=0.33\linewidth]{./methodCompare_a7_s3.png}
& \hspace{-4mm}
\includegraphics[width=0.33\linewidth]{./methodCompare_a10_s5.png} \\
\text{(a) $[T_\theta, T_v] = [5.0^{\circ}, 2.0\text{ km/h}]$ } & \text{(b) $[T_\theta, T_v] = [7.0^{\circ}, 3.0\text{ km/h}]$} & \text{(c) $[T_\theta, T_v] = [10.0^{\circ}, 5.0\text{ km/h}]$ } \\
\end{tabular}$
**Model** **MAE speed** **MAE angle**
------------------------------- --------------- ---------------
CNN+LSTM [@driving:attention] N.A. 4.15
Our Model 0.15 3.66
: Control Performance. Comparison between mean absolute error (MAE) in m/s and degree. []{data-label="tab:stateoftheart"}
Driving Accuracy
----------------
We first compare the overall performance of our driving model to the state-of-the-art models based on video observations [@end:driving:16; @driving:attention], and find that our model yields results similar to these models. This is not only illustrated in Figure \[fig:performance:plot\], where our driving model is very close to predicting the human ground truth driving performance, but also in Table \[tab:stateoftheart\] that highlights the control performance in terms of mean absolute error.
$\begin{tabular}{cccc}
\begin{turn}{90}\hspace{1cm} \emph{Safe}\end{turn} &
\hspace{-4mm}
\includegraphics[width=0.31\linewidth]{./animated_6520.jpg}
& \hspace{-4mm}
\includegraphics[width=0.31\linewidth]{./animated_27697.jpg}
& \hspace{-4mm}
\includegraphics[width=0.31\linewidth]{./animated_26546.jpg} \\
\begin{turn}{90}\hspace{1cm} \emph{Hazardous}\end{turn} &
\hspace{-4mm}
\includegraphics[width=0.31\linewidth]{./animated_27583.jpg}
& \hspace{-4mm}
\includegraphics[width=0.31\linewidth]{./animated_23555.jpg}
& \hspace{-4mm}
\includegraphics[width=0.31\linewidth]{./animated_26705.jpg} \\
& \text{ High} & \text{ Medium} & \text{ Low } \\
\end{tabular}$
In addition to reporting the overall quantitative numbers, we visualize where the most common failures of our driving model occur. This is achieved by defining a failure of the driving model when either the predicted vehicle speed or steering wheel angle deviates more than a defined threshold, and plotting the performance as a function of color on the map. We show, in Figure \[fig:failures\], three different threshold settings: $[T_s, T_a] \leftarrow [2\text{ km/h},5^{\circ}]$, $[T_s, T_a)] \leftarrow [3\text{ km/h}, 7^{\circ}]$, and $[T_s, T_a] \leftarrow [5\text{ km/h}, 10^{\circ}]$, ranging from a maximum deviation of 5 degrees and 2 km/h for our tightest definition of failure up to 10 degrees and 5 km/h for our loosest definition of failure.
In particular, the model is more likely to fail at intersections, partially due to the unknown destination of the vehicle and thus the ambiguity of which route the driver will take. We acknowledge that this is largely due to the lack of route planning in our driving model. However, route planning has been used to improve driving models in our recent work [@arxiv2018:hecker] and a fusion of route planned aware driving models with failure prediction is considered as our next future work. In addition, we mainly observed failures during sharp corners, in congested traffic, and in urban environments when many pedestrians are involved.
Failure Prediction
------------------
In this section, we evaluate our failure prediction model. Accurate and timely requests for manual take-over do result in a safety gain. We now show that our model learns to accurately alert the driver of impeding driving agent failure, reducing the risk of collisions. This is also illustrated by the gain in safety for different budgets of manual driving time ranging from $1-100\%$ of total driving time.
For this work, we train our failure prediction model on three different thresholds for speed and angle. These range from a very tight definition of failure using a threshold of 5 degrees and 2 km/h, to 7 degrees and 3 km/h, and finally to a loose definition of 10 degrees and 5km/h. We use the same underlying driving agent for each of these three instances, and thus obtain different metrics for our failure prediction dataset. It is worth noticing that three failure prediction models are trained, one for each threshold setting. We evaluate the method in the setting of image retrieval. The scenes for which the driving model is mostly likely to fail are retrieved and handed over to the human driver to deal with. We then compute the reduction of failure as a function of manual driving time. We compare our method to a baseline method which requires the same amount of manual driving time but does not have a learned policy about when to ask for manual intervention; it rather notifies the driver at regular intervals to take control. We also compare our method to an uncertainty estimation method for neural networks [@dropout:bayesian], for which the human driver is asked to take over when the driving model is mostly uncertain about its output. The uncertainty is computed by the dropout technique [@dropout:bayesian].
The results in Figure \[fig:failurePrediction\] show that our method can effectively reduce the amount of driving-model induced failure by switching to manual driving timely and accurately. Our model performs significantly better than the baseline model and the uncertainty model [@dropout:bayesian], because our method is specifically trained for the purpose. This trend can also be observed in Table \[tab:gainSafety\], where we depict the percentage gained safety over the baseline model for all three threshold models. The experimental results show that failures of a driving model can be learned and predicted quite accurately, and the failure prediction can be used to improve driving safety in a human-vehicle collaborative driving system. Predicting failure is actually easier than predicting the correct driving decisions, and thus all the more worthwhile including.
While our method performs better in all three cases, we do notice a more pronounced improvement when our definition of failure is more lax. One possible reason is that for the case with a very strict definition of the failure, the noise in the recordings of low-level maneuvers is too influential.
---------------- --------------------------------- --------------------------------- ----------------------------------
Manual Driving $[5.0^{\circ}, 2 \text{ km/h}]$ $[7.0^{\circ}, 3 \text{ km/h}]$ $[10.0^{\circ}, 5 \text{ km/h}]$
10.0% +6.9% +23.1% +52.5%
15.0% +11.0% +35.5% +81.2%
20.0% +15.1% +51.2% +117.4%
25.0% +19.0% +64.7% +153.7%
30.0% +23.8% +78.7% +195.0%
35.0% +29.6% +96.4% +249.4%
40.0% +34.9% +116.2% +349.9%
---------------- --------------------------------- --------------------------------- ----------------------------------
: Safety gain of our failure prediction over a baseline method in a hybrid, human-vehicle collaborative driving system.
\[tab:gainSafety\]
Finally, we show in Figure \[fig:examples:pics\] several driving scenes with the predicted maneuvers (speed and steering angle) and the drivability scores. While the method is able to alert drivers that a driving scene is hazardous for the driving model, it is often hard to figure out the underlying reason. A brief explanation such as ‘too many road constructions’ or ‘road getting too narrow’ will significantly reduce the confusion caused to the driver. An investigation into such underlying reasons is future work.
Conclusion {#sec:conclusion}
==========
In this work, we have presented the concept of *Scene Drivability* for automated cars. It indicates how feasible a particular driving scene is for a particular automated driving method. In order to quantify it, we have developed a novel learning method based on recurrent neural networks. We treated the discrepancies between the predictions of the automated driving model and the human drivers’ maneuvers as the (un)drivability scores of the scenes. Experimental results show that such drivability scores can be learned and predicted, and the prediction can be used to improve the safety of automated cars. The learning framework is flexible and can be applied to other driving models with more sensors. To the best of our knowledge, this is the first attempt to predict the failures of automated driving models. Our future work includes 1) developing more sophisticated driving models (e.g. including recognition of traffic relevant objects, route planning, and 360 degree sensing); 2) extending our failure prediction model to the new driving models; and 3) adding diagnostics by making explicit the inferring reasons for the failures.
**Acknowledgment**: This work is supported by Toyota Motor Europe via the research project TRACE-Zurich.
[^1]: $^{1}$Simon Hecker, Dengxin Dai, and Luc Van Gool are with the Toyota TRACE-Zurich team at the Computer Vision Lab, ETH Zurich, 8092 Zurich, Switzerland [firstname.lastname@vision.ee.ethz.ch ]{}
[^2]: $^{2}$Luc Van Gool is also with the Toyota TRACE-Leuven team at the Dept of Electrical Engineering ESAT, KU Leuven 3001 Leuven, Belgium
|
---
abstract: 'We apply a functional renormalisation group to systems of four bosonic atoms close to the unitary limit. We work with a local effective action that includes a dynamical trimer field and we use this field to eliminate structures that do not correspond to the Faddeev-Yakubovsky equations. In the physical limit, we find three four-body bound states below the shallowest three-body state. The values of the scattering lengths at which two of these states become bound are in good agreement with exact solutions of the four-body equations and experimental observations. The third state is extremely shallow. During the evolution we find an infinite number of four-body states based on each three-body state which follow a double-exponential pattern in the running scale. None of the four-body states shows any evidence of dependence on a four-body parameter.'
author:
- 'Benjamín Jaramillo Ávila and Michael C. Birse'
title: 'Universal behaviour of four-boson systems from a functional renormalisation group'
---
\[section:introduction\]Introduction
====================================
Systems where two-body scattering lengths are much longer than ranges of the forces between the particles are important in various areas of physics. Their low-energy properties display universal scaling behaviour, controlled by the “unitary limit" in which the scattering length tends to infinity. In nuclear physics, the large scattering lengths are large enough that low-energy aspects of few-nucleon systems can be described in this framework [@bvk02; @hp10]. In atomic physics, the shallow dimer of $^4$He atoms leads to a scattering length that is about 100 times larger than the size of the atoms [@gsthks00]. Even better examples are provided by ultra-cold atoms in traps, where Feshbach resonances can be used to tune the scattering lengths to values very close to the unitary limit [@cgjt10].
In the unitary limit, three-boson systems display a remarkable effect, first predicted by Efimov in 1970 [@efim70; @efim71]. They possess an infinite tower of three-body bound states, with energies in a constant ratio of $\sim 515.0$. This breaks the expected scale invariance to a discrete symmetry, with one three-body parameter needed to fix the energies of all these states. In real systems, the sequence of deeply bound states is cut off by the range of the forces, and the shallowest ones by the finite scattering length. Three-fermion systems can also show Efimov behaviour, provided there are enough species to allow spatially symmetric states. Although there were suggestions that the $A=3$ nuclei $^3$H and $^3$He could be interpreted as Efimov states [@bhvk00], the first clear observation of such states was in an ultra-cold gas of caesium atoms [@k06]. Reviews of the field can be found in Refs. [@bh06; @fzbhng11; @hp11].
This behaviour in the three-body sector feeds through to four-body systems, where most numerical calculations find two bound states in each Efimov cycle [@hp07; @vsdg09; @delt10] whose energies are fixed ratios to the nearest three-body state. However, in contrast, Hadizadeh *et al.* find up to three four-body states per cycle, with energies that depend on an additional four-body parameter [@had11; @had12], supporting their earlier results of Ref. [@ytdf06]. Experimental evidence for two four-body states based on an Efimov three-body state has been seen in the recombination rates of trapped $^{133}$Cs atoms [@f09], with resonances that are consistent with the results of Refs. [@hp07; @vsdg09; @delt10].
Renormalisation-group methods have been applied to elucidate scaling behaviour in few-body systems [@bhvk99; @bhvk00; @bb05; @nish08; @hp11] and hence to determine their relevant parameters. Here we apply a functional renormalisation group (FRG) [@wett93; @btw02] to the four-boson problem. During the evolution we observe a double-exponential pattern of four-body states built on each three-body state, similar to the “super-Efimov" behaviour found by Nishida, Moroz and Son in a two-dimensional three-body system [@mns13]. These have energies that can be expressed in terms of a universal scaling function, similar to that in Refs. [@had11; @had12], but they show no evidence of dependence on an additional four-body parameter. The states in our “super-Efimov" pattern are not necessarily physical and, away from the unitary limit, we find that only three of them are present in the last Efimov cycle and so can appear as physical bound states. The two deepest of this states appear for scattering lengths that are in reasonable agreement with those found in studies of four-body equations [@vsdg09; @delt10] and experimental observations [@f09].
This paper is structured as follows. In Section \[section:FRG\_and\_running\_action\] we present the FRG and running action that we use to study four-atom systems. Previous results on the three-body sector are summarised in Sec. \[section:three\_body\], as they provide key input into our four-body equations. Those equations are presented in Sec. \[section:four\_body\] together with our results for the four-body sector. We summarise and conclude in Sec. \[section:conclusions\].
\[section:FRG\_and\_running\_action\]FRG and running action
===========================================================
The FRG we use is based on a running version of effective action that generates the one-particle irreducible Green’s functions [@wett93; @btw02]. A regulator is added to the theory to suppress fluctuations with momenta below some scale $k$. For large $k$, we start with a suitably parametrised “bare" action. The methd works by evolving from this bare action to the limit $k \to 0$, where all quantum fluctuations are included and the action becomes physical. Away from this limit, that is for $k > 0$, the running action is not physical because of the partial suppression of fluctuations. Even though it is fully nonperturbative, the driving term in the FRG equation for the action has the form of a one-loop integral. Instead of diagrammatic expansions, practical approximation schemes are obtained by truncating the effective action to a finite number of terms.
This FRG is being applied to systems of nonrelativistic particles, in order to study, in particular, dense matter [@bkmw05; @dgpw07a; @dfgpw10]. In that context, it provides an alternative to traditional many-body methods. As part of this programme, studies of few-body systems in the same framework are needed to fix the input parameters. These studies are also proving interesting in their own right [@fmsw09; @mfsw09; @sm10; @bkw11].
A key ingredient of our approach is a trimer field. Such fields have been introduced before, in Refs. [@fmsw09; @mfsw09; @sm10]. However in the previous application to the four-boson problem [@sm10], this field was used to explore the dependence of amplitudes on an external energy. In contrast, our approach emphasises its dynamical role. This allows us to describe the atom-trimer channel of the four-body system and hence to obtain equations with a structure like that of the Faddeev-Yakubovsky equations [@yak67].
In this work we study systems of up to four nonrelativistic bosonic “atoms". We represent the atoms by the field $\psi(x)$ and we also introduce dimer and trimer fields, $\phi(x)$ and $\chi(x)$, in order to include energy-dependent propagators for two- and three-body subsystems. The evolution equation for the effective action $\Gamma_k[\psi,\psi^*,\phi,\phi^*,\chi,\chi^*]$ takes the form [@btw02] $$\label{eq:flow:eq}
\partial_k\Gamma=-\frac{\mbox{i}}{2}\,\operatorname{Tr}\left[(\partial_k{\mathbf R})\,
\left(({\boldsymbol \Gamma}^{(2)}-{\mathbf R})^{-1}\right)\right]
+\frac{\delta\Gamma}{\delta\Phi}\cdot\partial_k\Phi,$$ where ${\boldsymbol \Gamma}^{(2)}$ denotes the matrix of second derivatives of the action with respect to the fields and ${\mathbf R}$ the regulator that is added to suppress low-momentum modes. The trace $\operatorname{Tr}$ and the scalar product in the final term include integrals over energy and three-momenta as well as sums over the different types of field. The final term in the equation appears when we include fields that depend explicitly on the scale $k$, as in Ref. [@gw02; @fmsw09; @sm10].
For our regulators, ${\mathbf R}$, we use the form suggested by Litim [@lit01], which is optimised for local interactions. This suppresses the contributions of modes with momenta $q < k$ by replacing the kinetic energy in the inverse propagator for each field with a constant. For the atom field it has the form $$\label{eq:regulator}
{R_{a}}(q,k)=\frac{k^2-q^2}{2m}\,\theta(k-q).$$ The dimer and trimer regulators have similar forms but also contain the wave-function renormalisation factors defined below.
The key ingredient in any practical application of the FRG is the choice of truncation for the running action. Here we work with only local interactions. This reduces the functional differential equation for the action to a set of coupled ordinary differential equations for renormalisation factors and coupling constants multiplying the terms in that appear in the action, as defined below. Large numbers of diagrams contribute to the driving terms, as in the versions without trimer fields studied in Refs. [@sm10; @bkw11].
The running action we use is
$$\begin{aligned}
\label{eq:running:action}
\Gamma_k[\psi,\psi^*,\phi,\phi^*,\chi,\chi^*]
&=&\int{\rm d}^4x\,\Biggl[
\psi^*\left({\rm i}\,\partial_0+\frac{\nabla^2}{2m}\right)\psi
+Z_d\,\phi^*\left({\rm i}\,\partial_0+\frac{\nabla^2}{4m}\right)\phi
+Z_t\,\chi^*\left({\rm i}\,\partial_0+\frac{\nabla^2}{6m}\right)\chi\cr
\noalign{\vspace{5pt}}
&&\qquad\qquad-u_d\phi^*\phi-u_t\chi^*\chi
-\frac{g}{2}\bigl(\phi^*\psi\psi+\psi^*\psi^*\phi\bigr)
-h\bigl(\chi^*\phi\psi+\phi^*\psi^*\chi\bigr)-\lambda\,\phi^*\psi^*\phi\psi\cr
\noalign{\vspace{5pt}}
&&\qquad\qquad-\frac{u_{dd}}{2}\bigl(\phi^*\phi\bigr)^2
-\frac{v_d}{4}\bigl(\phi^*\phi^*\phi\psi\psi+\phi^*\psi^*\psi^*\phi\phi\bigr)
-\frac{w}{4}\phi^*\psi^*\psi^*\phi\psi\psi\cr
&&\qquad\qquad\null -u_{tt}\, \chi^*\psi^*\chi\psi
-\frac{u_{dt}}{2}\bigl(\phi^*\phi^*\chi\psi+\chi^*\psi^*\phi\phi\bigr)
-\frac{v_t}{2}\bigl(\phi^*\psi^*\psi^*\chi\psi+\chi^*\psi^*\phi\psi\psi\bigr)
\Biggr].\end{aligned}$$
This contains kinetic terms for atom, dimer and trimer fields with wave-function renormalisation factors and interaction terms with up to four underlying atoms. This action was also used by Schmidt and Moroz [@sm10] (see in particular the Appendix to that paper) but they chose to eliminate the four-atom couplings with trimer fields ($u_{tt}$, $u_{dt}$ and $v_t$) so that channels with dynamic trimers are not needed. The analogous fermionic couplings without trimers were studied in Ref. [@bkw11].
The inverse propagators for the fields in Eq. (\[eq:running:action\]) are expanded up to first order in the energy, which impliess first-order time derivatives in the action. In each channel, the zero-energy point for this expansion is taken to be the threshold for breakup of an $n$-atom state into $n$ free atoms. Spatial derivatives appear at second order as required by Galilean invariance, which follows from our choice of regulator [@lit01].
The wave-function renormalisation factors $Z_{d,t}$, self-energies $u_{d,t}$ and couplings $h$, $\lambda$ etc. all run with the regulator scale $k$. In vacuum, there is no renormalisation factor for the atom field $\psi$ and the coupling $g$ remains constant during the evolution.
Even though atom-atom scattering near the unitary limit can be described by an atom-atom contact interaction, the running action in Eq. (\[eq:running:action\]) does not contain such a term. This is because it can be eliminated through a Hubbard-Stratonovich transformation at some large starting scale $K$. The atom-atom term is not regenerated by the evolution and so atom-atom scattering is mediated only by the coupling $g$ to dimers. At zero energy, the scattering is given by $g^2/u_d(k)$ where $u_d(k)$ evolves linearly with $k$. We choose its initial value $u_d(K)$ such that, in the physical limit, $u_d(0)$ gives the desired scattering length $a$ [@bkmw05; @dgpw07a].
In contrast, the atom-dimer interaction, $\lambda$, is regenerated even if we set it to zero initially. By introducing fields that depend explicitly on the scale $k$, as in Ref. [@gw02; @fmsw09; @sm10], we can cancel the evolution of this and some other couplings. If we set their initial values to zero at the starting scale $K$, then these couplings are effectively eliminated from the problem. Here we take the trimer to run as $$\label{eq:dk:chi}
\partial_k \chi = \zeta_1 \, \phi \psi
+ \zeta_2 \, \psi^{\dagger} \chi \psi
+ \zeta_3 \, \psi^{\dagger} \phi \phi
+ \zeta_4 \, \psi^{\dagger} \phi \psi \psi,$$ where the $\zeta_i(k)$ are
\[eq:zeta:all\] $$\begin{aligned}
\label{eq:zeta:a}
\zeta_1&=&-\, \frac{\partial_k\lambda}{2 \, h},\\
\noalign{\vspace{5pt}}
\label{eq:zeta:b}
\zeta_2&=& -\, \frac{ \partial_kv_t }{ h }
+\frac{ u_{tt} \, \partial_k\lambda }{ 2 \, h^2 }
- \frac{ u_t \, v_t \, \partial_k\lambda }{ 2 \, h^3 }
+ \frac{ u_t \, \partial_k w }{ 8 \, h^2 },\\
\noalign{\vspace{5pt}}
\label{eq:zeta:c}
\zeta_3&=&-\, \frac{ \partial_kv_d }{ 4 \, h }
+\frac{ u_{dt} \, \partial_k\lambda }{ 2 \, h^2 },
\\
\noalign{\vspace{5pt}}
\label{eq:zeta:d}
\zeta_4&=&-\, \frac{ \partial_kw }{ 8 \, h }
+\frac{ v_t \, \partial_k\lambda}{ 2 \, h^2 }.\end{aligned}$$
The first term in Eq. (\[eq:dk:chi\]) cancels the running of $\lambda$, and the others do the same for the four-atom couplings $v_d$, $w$ and $v_t$.
Once we have eliminated these couplings, the physical processes that give rise to their evolution are implicitly present in the flows of the remaining couplings through contributions to their flows from the final term in Eq. (\[eq:flow:eq\]). For example, if the contact interaction $\lambda$ is eliminated, atom-dimer scattering only occurs through coupling to the trimer. The effects responsible for the evolution of $\lambda$ are now codified in a term proportional to $u_t\zeta_1$ in the flow of $h(k)$, arising from the first term of Eq. (\[eq:dk:chi\]).
\[section:three\_body\]Three-body sector
========================================
The three-body sector, described by the couplings $h(k)$, $u_t(k)$ and $Z_t(k)$, has been studied using this action by Floerchinger *et al.* [@fmsw09]. We summarise its main features here to provide some “landmarks" for our four-body results. In the unitary limit, the flow equations for the three-atom couplings have the forms
\[eq:three:body:flows:all\] $$\begin{aligned}
\label{eq:three:body:flows:h}
\partial_k\bigl(h^2\bigr)
&=&-\,\frac{312}{125 \, k} \; h^2(k)
- \frac{256 }{125 \, k^3} \; u_t(k),\\
\noalign{\vspace{5pt}}
\label{eq:three:body:flows:uchi}
\partial_k u_t
&=&\frac{56 \, k}{125 } \, h^2(k),\\
\noalign{\vspace{5pt}}
\label{eq:three:body:flows:zchi}
\partial_k Z_t
&=&-\,\frac{448}{625 \, k} \; h^2(k),\end{aligned}$$
where, to simplify the expressions, $u_t$ and $Z_t$ have been redefined to absorb constant factors of $g^2 \, m$ and $1/g^2$ respectively.
These equations describe the flows for regulator scales $k \gg 1/a$, where $a$ is the two-body scattering length. In this limit, the equations are scale invariant and so we expect their solutions to scale as powers of $k$. Indeed this system of differential equations is satisfied if $h^2(k)$ and $Z_t(k)$ behave as $k^{ d }$ and $u_t(k)$ as $k^{ 2 + d }$ where $d$ has two possible values, $$\label{eq:three:body:exponent:in:h}
d_{\pm} = -281/125 \pm \mathfrak{i} \, \sqrt{535}/25.$$ Since these are a complex-conjugate pair, we can form real solutions and define rescaled quantities that oscillate periodically in $t = \ln(k/K)$:
\[eq:three:body:scalings:all\] $$\begin{aligned}
\label{eq:three:body:scalings:H}
\hat{H}(k)&=&k^{281/125} \; h^2(k),\\
\noalign{\vspace{5pt}}
\label{eq:three:body:scalings:uchi}
\hat{u}_t(k)&=&k^{31/125} \; u_t(k),\\
\noalign{\vspace{5pt}}
\label{eq:three:body:scalings:zchi}
\hat{Z}_t(k)&=&k^{281/125} \; Z_t(k).\end{aligned}$$
This periodic behaviour is a consequence of the Emimov effect [@efim70; @efim71] which breaks the scale invariance of theory to a discrete symmetry. It follows from the complex scaling exponents in Eq. (\[eq:three:body:exponent:in:h\]). For the truncated action and regulator used here, the scaling factor in momentum is $\sim 29.8$ [@fmsw09; @sm10], which yields longer cycles than the true value of $\sim 22.7$.
In this framework, atom-dimer scattering at zero energy is given by the combination $h(k)^2/u_t(k)$, which evolves in the same way as $\lambda(k)$ in the theory without the trimer [@fmsw09; @sm10]. It displays a sequence of poles that are equally spaced in $t$, reflecting the discrete scaling symmetry of the Efimov effect. Each of these poles corresponds to the passage of a three-body bound state through the three-atom threshold as $k$ is lowered. In the physical limit they build up the infinite tower of Efimov states.
Although the flow equations in the three-body sector require three initial conditions, only one of these defines a physical parameter. This fixes the initial phase of the periodic functions or, equivalently, the scale at which the first Efimov pole appears. Physical quantities are independent of the magnitudes of the couplings since they depend only on the ratios $h(k)^2/u_t(k)$ and $h(k)^2/Z_t(k)$.
\[section:four\_body\]Four-body sector
======================================
In the four-atom sector, we use the scale dependence of the trimer to eliminate the couplings $v_d$, $w$ and $v_t$ that include the dimer-atom-atom channel. This leaves only the ones involving the dimer-dimer and atom-trimer channels, $u_{dd}$, $u_{dt}$ and $u_{tt}$. The first of these, $u_{dd}$, describes dimer-dimer scattering at zero energy (the four-atom threshold). Similarly $u_{tt}$ describes atom-trimer scattering and $u_{dt}$ the coupling between the two channels. This choice reflects the structure of the Faddeev-Yakubovsky equations used in most direct calculations of four-body systems [@yak67]. In contrast, Schmidt and Moroz [@sm10] also introduced a trimer field to treat energy dependence but kept only the couplings $u_{dd}$, $v_d$ and $w$.
The evolution of the four-atom couplings, $u_{dd}$, $u_{dt}$ and $u_{tt}$, is governed by a system of three coupled nonlinear differential equations. We define regulated energies for atoms, dimers and trimers,
\[eq:regulated:energies:all\] $$\begin{aligned}
{E_{a}}(q,k) &=& \frac{q^2}{2 \, m} + {R_{a}}(q,k),
\label{eq:regulated:energy:atom}\\
\noalign{\vspace{5pt}}
{E_{d}}(q,k) &=& \frac{q^2}{4 \, m} + \frac{{R_{d}}(q,k)}{Z_d(k)} + \frac{u_d(k)}{Z_d(k)},
\label{eq:regulated:energy:dimer}\\
\noalign{\vspace{5pt}}
{E_{t}}(q,k) &=& \frac{q^2}{6 \, m} + \frac{{R_{t}}(q,k)}{Z_t(k)} + \frac{u_t(k)}{Z_t(k)},
\label{eq:regulated:energy:trimer}\end{aligned}$$
where the single-atom self-energy contains $$u_d(k,a) = \frac{M \, g^2}{\pi^2} \; \left( \frac{k}{6} - \frac{\pi}{4 \, a} \right).$$ From these we construct the quantities $$\begin{aligned}
{ T_{ \alpha, \beta, \gamma, \delta }^{ X } }
=\frac{\partial_k R_X \; \left( {Z_d}\right)^{-\beta-\gamma}
\left( {Z_t}\right)^{-\delta} }
{ \left( {E_{a}}\right)^\alpha \left( {E_{d}}\right)^\beta \left( {E_{a}}+ {E_{d}}\right)^\gamma
\left( {E_{a}}+ {E_{t}}\right)^\delta },\end{aligned}$$ for $X = a,d,t$. In terms of these, the system of equations can be written
\[eq:fbp:all\] $$\begin{aligned}
\partial_k u_{dd}&=&\int \! \frac{d^3q}{\left( 2 \, \pi \right)^3}
\bigg[\frac{3 \, g^4}{8} { T_{ 4, 0, 0, 0 }^{a} }
+ \frac{g^2 h^2}{2} \left( 2 \, { T_{ 3, 0, 0, 1 }^{a} } + { T_{ 2, 0, 0, 2 }^{a} } {Z_t}+ { T_{ 2, 0, 0, 2 }^{t} } \right)
+ \frac{\left( u_{dd} \right)^2}{2} { T_{ 0, 2, 0, 0 }^{d} } \left( {Z_d}\right)^{-1}
\nonumber \\
&&\qquad\qquad\quad-\, 2 \, g \, h \, u_{dt} \left( { T_{ 2, 0, 0, 1 }^{a} }
+ { T_{ 1, 0, 0, 2 }^{a} } {Z_t}+ { T_{ 1, 0, 0, 2 }^{t} } \right)
+ 2 \left( u_{dt} \right)^2 \left( { T_{ 0, 0, 0, 2 }^{a} } {Z_t}+ { T_{ 0, 0, 0, 2 }^{t} } \right)\bigg],\end{aligned}$$ $$\begin{aligned}
\partial_k u_{dt}&=&\int \! \frac{d^3q}{\left( 2 \, \pi \right)^3}
\bigg[- \frac{g^3 u_t \, u_{dd}}{4 \, h} \left( { T_{ 2, 1, 1, 0 }^{a} } + { T_{ 1, 1, 2, 0 }^{a} } {Z_d}+ { T_{ 1, 2, 1, 0 }^{d} } + { T_{ 1, 1, 2, 0 }^{d} } \right)
- \frac{g \, h \, u_{tt}}{2} \left( { T_{ 2, 0, 0, 1 }^{a} } + { T_{ 1, 0, 0, 2 }^{a} } {Z_t}+ { T_{ 1, 0, 0, 2 }^{t} } \right)\nonumber \\
&&\qquad\qquad\quad
-\, \frac{g \, h \, u_{dd}}{2} \left( { T_{ 0, 1, 2, 0 }^{a} } {Z_d}+ { T_{ 0, 2, 1, 0 }^{d} }
+ { T_{ 0, 1, 2, 0 }^{d} } \right)
+ \frac{g^4 u_t \, u_{dt}}{8 \, h^2} \left( 2 \, { T_{ 3, 0, 1, 0 }^{a} }
+ { T_{ 2, 0, 2, 0 }^{a} } {Z_d}+ { T_{ 2, 0, 2, 0 }^{d} } \right)\nonumber \\
&&\qquad\qquad\quad
-\, \frac{g^2 u_t \, u_{dt}}{2} \left( { T_{ 2, 0, 1, 1 }^{a} } + { T_{ 1, 0, 2, 1 }^{a} } {Z_d}+ { T_{ 1, 0, 1, 2 }^{a} } {Z_t}+ { T_{ 1, 0, 2, 1 }^{d} } + { T_{ 1, 0, 1, 2 }^{t} } \right)
+ \frac{u_{dd} \, u_{dt}}{2} { T_{ 0, 2, 0, 0 }^{d} } \left( {Z_d}\right)^{-1}
\nonumber \\
&&\qquad\qquad\quad
-\, h^2 u_{dt} \left( { T_{ 0, 0, 2, 1 }^{a} } {Z_d}+ { T_{ 0, 0, 1, 2 }^{a} } {Z_t}+ { T_{ 0, 0, 2, 1 }^{d} } + { T_{ 0, 0, 1, 2 }^{t} } \right)
+ u_{dt} \, u_{tt} \left( { T_{ 0, 0, 0, 2 }^{a} } {Z_t}+ { T_{ 0, 0, 0, 2 }^{t} } \right)
\nonumber \\
&&\qquad\qquad\quad
+\, \frac{g^3 h \, u_t}{4} \left( 2 \, { T_{ 3, 0, 1, 1 }^{a} } + { T_{ 2, 0, 2, 1 }^{a} } {Z_d}+ { T_{ 2, 0, 1, 2 }^{a} } {Z_t}+ { T_{ 2, 0, 2, 1 }^{d} } + { T_{ 2, 0, 1, 2 }^{t} } \right)
\nonumber \\
&&\qquad\qquad\quad
+\, \frac{g \, h^3}{2} \left( { T_{ 2, 0, 1, 1 }^{a} } + { T_{ 1, 0, 2, 1 }^{a} } {Z_d}+ { T_{ 1, 0, 1, 2 }^{a} } {Z_t}+ { T_{ 1, 0, 2, 1 }^{d} } + { T_{ 1, 0, 1, 2 }^{t} } \right)\bigg],\end{aligned}$$ $$\begin{aligned}
\partial_k u_{tt}&=&\int \! \frac{d^3q}{\left( 2 \, \pi \right)^3}
\bigg[ g^2 h^2 \left( { T_{ 2, 1, 1, 0 }^{a} } + { T_{ 1, 1, 2, 0 }^{a} } {Z_d}+ { T_{ 1, 2, 1, 0 }^{d} } + { T_{ 1, 1, 2, 0 }^{d} } \right)
+ \frac{g^4 u_t \, u_{tt}}{4 \, h^2} \left( 2 \, { T_{ 3, 0, 1, 0 }^{a} }
+ { T_{ 2, 0, 2, 0 }^{a} } {Z_d}+ { T_{ 2, 0, 2, 0 }^{d} } \right)
\nonumber \\
&&\qquad\qquad\quad
+\, g^4 \, u_t \left( 2 \, { T_{ 3, 1, 1, 0 }^{a} } + { T_{ 2, 1, 2, 0 }^{a} } {Z_d}+ { T_{ 2, 2, 1, 0 }^{d} } + { T_{ 2, 1, 2, 0 }^{d} } \right)
- 2 \, g \, h \, u_{dt} \left( { T_{ 0, 1, 2, 0 }^{a} } {Z_d}+ { T_{ 0, 2, 1, 0 }^{d} } + { T_{ 0, 1, 2, 0 }^{d} } \right)
\nonumber \\
&&\qquad\qquad\quad
+\, \frac{g^4 \left( u_t \right)^2}{4} \left( 2 \, { T_{ 3, 0, 2, 1 }^{a} }
+ 2 \, { T_{ 2, 0, 3, 1 }^{a} } {Z_d}+ { T_{ 2, 0, 2, 2 }^{a} } {Z_t}+ 2 \, { T_{ 2, 0, 3, 1 }^{d} } + { T_{ 2, 0, 2, 2 }^{t} } \right)
+ \left( u_{dt} \right)^2 { T_{ 0, 2, 0, 0 }^{d} } \left( {Z_d}\right)^{-1}
\nonumber \\
&&\qquad\qquad\quad
-\, 2 \, h^2 u_{tt} \left( { T_{ 0, 0, 2, 1 }^{a} } {Z_d}+ { T_{ 0, 0, 1, 2 }^{a} } {Z_t}+ { T_{ 0, 0, 2, 1 }^{d} } + { T_{ 0, 0, 1, 2 }^{t} } \right)
+ \left( u_{tt} \right)^2 \left( { T_{ 0, 0, 0, 2 }^{a} } {Z_t}+ { T_{ 0, 0, 0, 2 }^{t} } \right)
\nonumber \\
&&\qquad\qquad\quad
+\, g^2 h^2 u_t \left( { T_{ 2, 0, 2, 1 }^{a} } + 2 \, { T_{ 1, 0, 3, 1 }^{a} } {Z_d}+ { T_{ 1, 0, 2, 2 }^{a} } {Z_t}+ 2 \, { T_{ 1, 0, 3, 1 }^{d} }
+ { T_{ 1, 0, 2, 2 }^{t} } \right)
- g^2 u_t \, u_{tt} \left( { T_{ 2, 0, 1, 1 }^{a} } + { T_{ 1, 0, 2, 1 }^{a} } {Z_d}\right)
\nonumber \\
&&\qquad\qquad\quad
-\, g^2 u_t \, u_{tt} \left( { T_{ 1, 0, 1, 2 }^{a} } {Z_t}+ { T_{ 1, 0, 2, 1 }^{d} }
+ { T_{ 1, 0, 1, 2 }^{t} } \right)
+ \frac{g^6 \left( u_t \right)^2}{4 \, h^2} \left( 3 \, { T_{ 4, 1, 1, 0 }^{a} }
+ { T_{ 3, 1, 2, 0 }^{a} } {Z_d}+ { T_{ 3, 2, 1, 0 }^{d} } + { T_{ 3, 1, 2, 0 }^{d} } \right)
\nonumber \\
&&\qquad\qquad\quad
-\, \frac{g^3 u_t \, u_{dt}}{h} \left( { T_{ 2, 1, 1, 0 }^{a} } + { T_{ 1, 1, 2, 0 }^{a} } {Z_d}+ { T_{ 1, 2, 1, 0 }^{d} } + { T_{ 1, 1, 2, 0 }^{d} } \right)
\nonumber \\
&&\qquad\qquad\quad
+\, h^4 \left( 2 \, { T_{ 0, 0, 3, 1 }^{a} } {Z_d}+ { T_{ 0, 0, 2, 2 }^{a} } {Z_t}+ 2 \, { T_{ 0, 0, 3, 1 }^{d} } + { T_{ 0, 0, 2, 2 }^{t} } \right)\bigg].\end{aligned}$$
The appearance of $h^2(k)$, $u_t(k)$ and $Z_t(k)$ in the four-body flow equations, Eqs. (\[eq:fbp:all\]), means that they inherit the Efimov periodicity of the three-body sector. This also leads to two types of singularity in the equations. One arises from terms with denominators containing either one or two powers of the regulated energy of an atom plus a trimer, ${E_{a}}(k) + {E_{t}}(k)$. This passes through zero energy once in every Efimov cycle, at the point where a regulated atom-trimer threshold drops below the four-atom threshold as we lower $k$. At each crossing we expect additional contributions to the imaginary parts of the four-body couplings, as a channel with a new Efimov state becomes open.
The other type of divergent term has a factor of $1/\left(h(k)\right)^2$. These lead to unphysical singularities in the four-body couplings, which mark the start of a short region within each Efimov cycle where $h^2(k)$ and $Z_t(k)$ have opposite signs. In these regions, the trimer field has a ghost-like character, with a propagator $h^2(k)/(Z_t\,p_0-u_t(k))$ that has a negative residue at its pole. This is a warning that not all features of the effective action are physical for non-zero values of $k$. Fortunately these regions are well separated from the threshold regions where the phenomena of interest occur.
In the scaling regime the four-atom couplings display Efimov periodicity. This can be seen most clearly if they are multiplied by appropriate powers of $k$, analogously to the rescaling of the three-body sector in Eqs. (\[eq:three:body:scalings:all\]). Here we define the couplingss,
\[eq:fbp:scalings:all\] $$\begin{aligned}
\hat{u}_{dd}(k)&=&k^3 \;\, u_{dd}(k),
\label{eq:fbp:scalings:udd}\\
\hat{u}_{dt}(k)&=&k^{781/250} \;\, u_{dt}(k),
\label{eq:fbp:scalings:udt}\\
\hat{u}_{tt}(k)&=&k^{406/125} \;\, u_{tt}(k),
\label{eq:fbp:scalings:utt}\end{aligned}$$
where the powers of $k$ are determined from dimensional analysis of the running action and the scalings in the three-body sector, Eqs. (\[eq:three:body:flows:all\]).
The flow equations for these rescaled couplings can be written
\[eq:flow:all\] $$\begin{aligned}
\label{eq:flow:uu2}
\partial_t \hat{u}_{dd}&=& \frac{1}{\pi^2}+ 3 \, \hat{u}_{dd}
+ \frac{8\pi^2 \, \hat{u}_{dd}^2}{15}
+ \frac{\hat{H} \; \partial_t \hat{Z}_t}{45\pi^2 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{2 \, \hat{H} \; \hat{u}_t}{3\pi^2 \, \hat E_{at}^2 \hat{Z}_t^2}
+ \frac{1573 \, \hat{H}}{1875\pi^2 \, \hat E_{at}^2 \; \hat{Z}_t}
+ \frac{2 \, \hat{H}}{3\pi^2 \, \hat E_{at} \hat{Z}_t}
- \frac{2 \, \hat{U}_{dt} \; \partial_t \hat{Z}_t}{45\pi^2 \, \hat E_{at}^2
\; \hat{Z}_t^2}\nonumber \\
\noalign{\vspace{5pt}}
&&-\, \frac{2 \, \hat{u}_t \; \hat{U}_{dt}}{3\pi^2 \, \hat E_{at}^2 \; \hat{Z}_t^2}
- \frac{6938 \, \hat{U}_{dt}}{5625\pi^2 \, \hat E_{at}^2 \; \hat{Z}_t}
- \frac{2 \, \hat{U}_{dt}}{3\pi^2 \, \hat E_{at} \; \hat{Z}_t}
+ \frac{\hat{U}_{dt}^2 \; \partial_t \hat{Z}_t}{45\pi^2 \, \hat{H} \;
\hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{2219 \, \hat{U}_{dt}^2}{5625\pi^2 \, \hat{H} \; \hat E_{at}^2 \; \hat{Z}_t},\end{aligned}$$ $$\begin{aligned}
\label{eq:flow:h:times:udt}
\partial_t \hat{U}_{dt} &=&-\, \frac{496\pi^2 \, \hat{H} \; \hat{u}_{dd}}{375}
- \frac{1096\pi^2 \, \hat{u}_{dd} \; \hat{u}_t}{375}
+ \frac{2 \, \hat{H}^2 \; \partial_t \hat{Z}_t}{75 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{2 \, \hat{H} \; \hat{u}_t \; \partial_t \hat{Z}_t}{75 \, \hat E_{at}^2 \;
\hat{Z}_t^2}
+ \frac{18 \, \hat{H}^2 \; \hat{u}_t}{25 \, \hat E_{at}^2 \hat{Z}_t^2}
+ \frac{28 \, \hat{H} \; \hat{u}_t^2}{25 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{8938 \, \hat{H}^2}{9375 \, \hat E_{at}^2 \; \hat{Z}_t}
\nonumber \\
\noalign{\vspace{5pt}}
&&+\, \frac{11438 \, \hat{H} \; \hat{u}_t}{9375 \, \hat E_{at}^2 \; \hat{Z}_t}
+ \frac{66 \, \hat{H}^2}{125 \, \hat E_{at} \; \hat{Z}_t}
+ \frac{116 \, \hat{H} \; \hat{u}_t}{125 \, \hat E_{at} \; \hat{Z}_t}
- \frac{\hat{H} \; \hat{u}_{tt} \; \partial_t \hat{Z}_t}{90 \pi^2 \, \hat E_{at}^2 \; \hat{Z}_t^2}
- \frac{\hat{H} \; \hat{u}_t \; \hat{u}_{tt}}{6 \pi^2 \, \hat E_{at}^2 \; \hat{Z}_t^2}
- \frac{3469 \, \hat{H} \; \hat{u}_{tt}}{11250 \pi^2 \, \hat E_{at}^2 \; \hat{Z}_t}
\nonumber \\
\noalign{\vspace{5pt}}
&&-\, \frac{\hat{H} \; \hat{u}_{tt}}{6 \pi^2 \, \hat E_{at} \; \hat{Z}_t}
+ 3 \, \hat{U}_{dt}+ \frac{8\pi^2 \, \hat{u}_{dd} \; \hat{U}_{dt}}{15}
- \frac{2 \, \hat{H} \; \hat{U}_{dt} \; \partial_t \hat{Z}_t}{75 \, \hat E_{at}^2 \; \hat{Z}_t^2}
- \frac{2 \, \hat{u}_t \; \hat{U}_{dt} \; \partial_t \hat{Z}_t}{75 \, \hat E_{at}^2 \; \hat{Z}_t^2}
- \frac{8 \, \hat{H} \; \hat{u}_t \; \hat{U}_{dt}}{25 \, \hat E_{at}^2 \; \hat{Z}_t^2}
- \frac{18 \, \hat{u}_t^2 \; \hat{U}_{dt}}{25 \, \hat E_{at}^2 \; \hat{Z}_t^2}
\nonumber \\
\noalign{\vspace{5pt}}
&&-\, \frac{2146 \, \hat{H} \; \hat{U}_{dt}}{3125 \, \hat E_{at}^2 \; \hat{Z}_t}
- \frac{8938 \, \hat{u}_t \; \hat{U}_{dt}}{9375 \, \hat E_{at}^2 \; \hat{Z}_t}
- \frac{16 \, \hat{H} \; \hat{U}_{dt}}{125 \, \hat E_{at} \; \hat{Z}_t}
- \frac{66 \, \hat{u}_t \; \hat{U}_{dt}}{125 \, \hat E_{at} \; \hat{Z}_t}
+ \frac{\hat{u}_{tt} \; \hat{U}_{dt} \; \partial_t \hat{Z}_t}{90 \pi^2 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{2219 \, \hat{u}_{tt} \; \hat{U}_{dt}}{11250 \pi^2 \, \hat E_{at}^2 \; \hat{Z}_t},\end{aligned}$$ $$\begin{aligned}
\label{eq:flow:utt}
\partial_t \hat{u}_{tt}
&=& \frac{4384 \pi^2 \, \hat{H}}{375}
+ \frac{13568 \pi^2 \, \hat{u}_t}{375}
+ \frac{9184 \pi^2 \,\hat{u}_t^2}{375 \, \hat{H}}
+ \frac{8 \pi^2 \, \hat{H}^2 \; \partial_t \hat{Z}_t}{125 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{16 \pi^2 \, \hat{H} \; \hat{u}_t \; \partial_t \hat{Z}_t}{125 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{192 \pi^2 \, \hat{H}^2 \; \hat{u}_t}{125 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{8 \pi^2 \, \hat{u}_t^2 \; \partial_t \hat{Z}_t}{125 \, \hat E_{at}^2 \hat{Z}_t^2}
\nonumber \\
\noalign{\vspace{5pt}}
&&+\, \frac{624 \pi^2 \, \hat{H} \; \hat{u}_t^2}{125 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{432 \pi^2 \, \hat{u}_t^3}{125 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{33752 \pi^2 \, \hat{H}^2}{15625 \, \hat E_{at}^2 \, \hat{Z}_t}
+ \frac{87504 \pi^2 \, \hat{H} \; \hat{u}_t}{15625 \, \hat E_{at}^2 \; \hat{Z}_t}
+ \frac{53752 \pi^2 \; \hat{u}_t^2}{15625 \, \hat E_{at}^2 \; \hat{Z}_t}
+ \frac{384 \pi^2 \, \hat{H}^2}{625 \, \hat E_{at} \; \hat{Z}_t}
\nonumber \\
\noalign{\vspace{5pt}}
&&+\, \frac{1968 \pi^2 \, \hat{H} \; \hat{u}_t}{625 \, \hat E_{at} \; \hat{Z}_t}
+ \frac{1584 \pi^2 \, \hat{u}_t^2}{625 \, \hat E_{at} \; \hat{Z}_t}
+ \frac{406 \, \hat{u}_{tt}}{125}
+ \frac{256 \, \hat{u}_t \; \hat{u}_{tt}}{125 \, \hat{H}}
- \frac{4 \, \hat{H} \; \hat{u}_{tt} \; \partial_t \hat{Z}_t}{75 \, \hat E_{at}^2 \; \hat{Z}_t^2}
- \frac{4 \, \hat{u}_t \; \hat{u}_{tt} \; \partial_t \hat{Z}_t}{75 \, \hat E_{at}^2 \; \hat{Z}_t^2}
- \frac{16 \, \hat{H} \; \hat{u}_t \; \hat{u}_{tt}}{25 \, \hat E_{at}^2 \; \hat{Z}_t^2}
\nonumber \\
\noalign{\vspace{5pt}}
&&-\, \frac{36 \, \hat{u}_t^2 \; \hat{u}_{tt}}{25 \, \hat E_{at}^2 \; \hat{Z}_t^2}
- \frac{4292 \, \hat{H} \; \hat{u}_{tt}}{3125 \, \hat E_{at}^2 \; \hat{Z}_t}
- \frac{17876 \, \hat{u}_t \; \hat{u}_{tt}}{9375 \, \hat E_{at}^2 \; \hat{Z}_t}
- \frac{32 \, \hat{H} \; \hat{u}_{tt}}{125 \, \hat E_{at} \; \hat{Z}_t}
- \frac{132 \, \hat{u}_t \; \hat{u}_{tt}}{125 \, \hat E_{at} \; \hat{Z}_t}
+ \frac{\hat{u}_{tt}^2 \; \partial_t \hat{Z}_t}{90 \pi^2 \, \hat E_{at}^2 \; \hat{Z}_t^2}
+ \frac{2219 \, \hat{u}_{tt}^2}{11250 \pi^2 \hat E_{at}^2 \; \hat{Z}_t}
\nonumber \\
\noalign{\vspace{5pt}}
&&-\, \frac{1984 \pi^2 \, \hat{U}_{dt}}{375}
- \frac{4384 \pi^2 \, \hat{u}_t \; \hat{U}_{dt}}{375 \, \hat{H}}
+ \frac{16 \pi^2 \; \hat{U}_{dt}^2}{15 \, \hat{H}},\end{aligned}$$
where we have defined the rescaled atom-trimer energy $\hat E_{at} = 2/3 + \hat{u}_\chi/\hat{Z}_\chi$ and the modified coupling $\hat{U}_{dt} = \hat{H}^{1/2} \; \hat{u}_{dt}$. As in Eqs. (\[eq:three:body:flows:all\]) we have absorbed powers of the constants $g^2$ and $m$ into the couplings to try to simplify the expressions.
\[ht\] ![ \[fig:one\_cycle\] One Efimov cycle of the flow of the rescaled coupling $\hat{u}_{tt}(k)$ in the unitary limit, plotted against $t=\ln(k/K)$. The real part is shown by the solid curve and the imaginary part by the dashed one. The atom-trimer threshold corresponding to the vanishing of ${E_{a}}(k) + {E_{t}}(k)$ is marked by the grey vertical line at $t=t_3\simeq-4.85$.](01_image_4b_cycle.pdf "fig:"){width="\columnwidth"}
We have numerically integrated the coupled equations for $u_{dd}(k)$, $u_{dt}(k)$ and $u_{tt}(k)$ through several Efimov cycles, and we have checked that any transients caused by our choice of initial conditions die out within the first cycle. All three couplings show similar structures but they are most clearly visible in $u_{tt}(k)$ and so we present only results for its flow. One cycle of the rescaled coupling $\hat{u}_{tt}(k)$ in the unitary limit is shown in Fig. \[fig:one\_cycle\]. At the value of $t=\ln(k/K)$ where the atom-trimer threshold passes through zero energy, $t=t_3\simeq -4.85$, we see the expected discontinuity in the slope of the imaginary part signalling the opening of a new channel. The unphysical singularity arising from the zero of $h^2(k)$ is the structure that can be seen at $t\simeq-3.0$.
\[h\] ![\[fig:below\_threshold\] The imaginary part of $\hat{u}_{tt}(k) \, /\left( t - t_3 \right)$ just before the threshold $t_3\simeq-4.85$ shown in Fig. \[fig:one\_cycle\], plotted against $x = \ln \left( t - t_3 \right)$. Apart from the rightmost one, corresponding to the deepest four-body state, the poles are approximately equally spaced.](02_image_4b_threshold.pdf "fig:"){width="\columnwidth"}
Several simple poles can also be seen in Fig. \[fig:one\_cycle\], at $t\simeq-3.83$, $-4.67$, and just below the threshold. When we look more closely at the region close to an atom-trimer threshold, as in Fig. \[fig:below\_threshold\], we find an infinite sequence of these poles. These become equally spaced in the variable $x=\ln(t-t_3)$. These poles do not correspond to singularities in the equations but are generated by the evolution of the couplings. Like the singularities that appear in the three-body sector, we interpret them as bound states or, rather, narrow resonances since they have finite imaginary parts as a result of coupling to open channels with more deeply bound trimers. However, as we discuss below, not all of these poles may appear as physical states.
The introduction of the trimer field to describe energy dependence in the three-body sector is essential for generating these poles as they do not appear in the FRG equations for the couplings without trimer fields [@bkw13].[^1] The scales at which these poles appear follow a double-exponential, “super-Efimov" pattern, similar to that observed in the two-dimensional three-body system studied by Nishida *et al.* [@mns13].
Mathematically this structure arises from the forms of our differential equations which are analogous to that of the RG equation of Ref. [@mns13]. The key terms that lead to the “super-Efimov" behaviour are the ones that are singular at the atom-trimer threshold. These arise from diagrams that are similar to those in Fig. 2 of that paper. However we should stress these states appear for non-zero values of $k$, where the action is not physical. Moreover the four-body flow equations depend on a scale as a result of the breaking of scale invariance by the Efimov effect. These states can therefore move relative to the atom-trimer threshold during the evolution to the physical limit. In particular, they may pass through the nearby atom-trimer threshold to become virtual states. If so, only a finite number of bound states may persist in that limit. Furthermore, a theorem of Amado and Greenwood forbids an infinite number of four-body bound states based on a zero-energy trimer state [@ag73]. Nonetheless, the presence of these virtual states might be relevant to the rich structure of states being found in for body systems away from the unitary limit. For example, Deltuva [@delt12] has recently described a tower of four-body bound states lying just below the atom-atom-dimer threshold in systems with finite dimer binding energy.
The local form of the action, Eq. (\[eq:running:action\]), does not allow us to study the full energy dependence in the four-body channels and so we cannot directly determine the spectrum in the physical limit. Instead, we can examine where these states cross zero energy as we move away from the unitary limit by taking a non-zero atom-atom scattering length, $a<0$. Such zero-energy states are the ones observed in experiments on ultra-cold atoms in traps, as they lead to resonant enhancements of the loss of atoms at particular values of the scattering length [@fzbhng11; @f09].
\[t\] ![\[fig:last\_cycle\] The final cycle of the flow of the rescaled coupling $\hat{u}_{tt}(k)$ plotted against $t=\ln(k/K)$. The solid line corresponds to the real part and the imaginary to the dashed one. The atom-atom scattering length has been tuned so that the last three-body state appears at $k=0$ ($t=-\infty$).](03_image_4b_last_cycle.pdf "fig:"){width="\columnwidth"}
With a finite scattering length, the final Efimov cycle no longer has the same form as in the unitary limit. An example is shown in Fig. \[fig:last\_cycle\]. For $t\gtrsim -2.3$ the flow of the four-atom coupling matches Fig. \[fig:one\_cycle\], but beyond this point differences become increasingly visible. The example shown has the scattering length tuned so that the shallowest trimer state has exactly zero binding energy at $k=0$. In this case, we find three four-body states appearing in the final Efimov cycle (the poles close to $t=-4.1$, $-5.6$ and $-7.1$). There is thus no conflict with the theorem of Amado and Greenwood [@ag73] that there are only a finite number of these four-body states. We denote the corresponding scattering length by $a_3$. When we further decrease $a$, we find that the values $a_4^{(n)}$ at which these states cross the four-atom threshold are related to $a_3$ by $$a_4^{(0)}\!/a_3\simeq 0.438,\quad
a_4^{(1)}\!/a_3\simeq 0.877,\quad
a_4^{(2)}\!/a_3\simeq 0.9967.$$ For the two lowest states, these ratios are within 5% of the results of exact solutions to the four-body equations [@vsdg09; @delt10], and hence they are also in reasonable agreement with the experimental numbers [@f09]. The third state lies extremely close to the atom-trimer threshold. If it is real, then it will be a challenge to observe both numerically and experimentally. However this state may just be an artefact of our truncation since improvements to the action which shorten the Efimov cycle might make it unbound.
Returning to the double-exponential behaviour observed during the evolution, the scale $k_4^{(n)}$ at which the $n$-th excited four-body state appears can be written in the form $$k_4^{(n)}=k_3\,\exp\left[\alpha\,{\rm e}^{-\beta n}\right],
\label{eq:k4tok3}$$ where $\alpha\simeq 1.53$, $\beta\simeq 2.06$, and $k_3$ denotes the scale corresponding to the atom-trimer threshold for the next three-body Efimov state. This describes the energies of all states except the lowest ($n=0$) to a very good approximation. The ratios between scales for subsequent states can be expressed in the form of a universal scaling function, $$k_4^{(n+1)}/k_4^{(n)}=\left(k_3/k_4^{(n)}
\right)^{1-\exp(-\beta)}.$$ A similar scaling relation between the binding energies has also been found by Hadizadeh *et al.* [@had11; @had12], although its functional form is quite different and it predicts at most three four-body states in an Efimov cycle. More importantly, and in contrast to the results of those authors, the scales at which our states appear do not depend on any new four-body scale: the parameter $\alpha$ in Eq.(\[eq:k4tok3\]) has a fixed value which is independent of the initial conditions we impose on the four-body couplings. The independence of any four-body parameter also applies to the physical states discussed above.
\[section:conclusions\]Conclusions
==================================
In summary: we have used the FRG to study systems of four bosons close to the unitary limit. In contrast to previous approaches, we introduce a dynamical trimer field and use this to match the channel structure of the Faddeev-Yakubovsky equations. In the physical limit, where the cut-off scale tends to zero, we examine the points at which three- and four-body states pass through zero energy as we vary the atom-atom scattering length. We find three four-body states in the last Efimov cycle. The lowest two of these pass through zero for scattering lengths that are in good agreement with the results of exact solutions of the Faddeev-Yakubovsky equations [@vsdg09; @delt10] and with experimental observations [@f09]. The third state is extremely weakly bound and so may be an artefact of our truncated action.
In the unitary limit, the evolution generates an infinite number of four-body resonant states during each Efimov cycle, although it seems unlikely that all of these persist to the physical limit. These states lie just below each atom-trimer threshold and follow a double-logarithmic, or “super-Efimov" pattern [@mns13]. They obey a universal scaling relation analogous to that of Ref. [@had11]. However the scales at which they appear are independent of the initial conditions on the four-body couplings. This supports the conclusion of Refs. [@phm04; @hp07] that there is no additional relevant parameter in four-boson systems with contact interactions.
We are grateful to S. Floerchinger, B. Krippa, S. Moroz, R. Schmidt and N. Walet for helpful discussions. BJA acknowledges support from CONACyT.
[99]{} P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci. **52**, 339 (2002). H.-W. Hammer and L. Platter, Ann. Rev. Nucl. Part. Sci. **60**, 207 (2010). R. E. Grisenti, W. Schöllkopf, J. P. Toennies, G. C. Hegerfeldt, T. Köhler, and M. Stoll, Phys. Rev. Lett. **85**, 2284 (2000). C. Chin, R. Grimm, P. Julienne and E. Tiesinga, Rev. Mod. Phys. **82**, 1225 (2010). V. Efimov, Phys. Lett. B **33**, 563 (1970). V. N. Efimov, Sov. J. Nucl. Phys. **12**, 589 (1971). P. F. Bedaque, H.-W. Hammer and U. van Kolck, Nucl. Phys. A **676**, 357 (2000). T. Kraemer *et al.*, Nature **440**, 315 (2006). E. Braaten and H.-W. Hammer, Phys. Rep. **428**, 259 (2006). F. Ferlaino, A. Zenesini, M. Berninger, B. Huang, H.-C. Nägerl and R. Grimm, Few-Body Syst. **51**, 113 (2011). H.-W. Hammer and L. Platter, Phil. Trans. Roy. Soc. Lond. A **369**, 2679 (2011). H.-W. Hammer and L. Platter, Eur. Phys. J. A 32, 113 (2007). J. von Stecher, J. P. D’Incao, and C. H. Greene, Nature Phys. **5**, 417 (2009). A. Deltuva, Phys. Rev. A **82**, 040701 (2010). M. R. Hadizadeh, M. T. Yamashita, L. Tomio, A. Delfino and T. Frederico, Phys. Rev. Lett. **107**, 135304 (2011). M. R. Hadizadeh, M. T. Yamashita, L. Tomio, A. Delfino and T. Frederico, Phys. Rev. A **85**, 023610 (2012). M. T. Yamashita, L. Tomio, A. Delfino, and T. Frederico, Europhys. Lett. **75**, 555 (2006). F. Ferlaino *et al.*, Phys. Rev. Lett. **102**, 140401 (2009) P. F. Bedaque, H.-W. Hammer and U. van Kolck, Phys. Rev. Lett. **82**, 463 (1999). T. Barford and M. C. Birse, J. Phys. A **38**, 697 (2005). Y. Nishida, Phys. Rev. D **77**, 061703 (2008). C. Wetterich, Phys. Lett. B **301**, 90 (1993). J. Berges, N. Tetradis and C. Wetterich, Phys. Rep. **363**, 223 (2002). Y. Nishida, S. Moroz and D. T. Son, Phys. Rev. Lett. **110**, 235301 (2013). M. C. Birse, B. Krippa, J. A. McGovern and N. R. Walet, Phys. Lett. B **605**, 287 (2005). S. Diehl, H.Gies, J. M. Pawlowski and C. Wetterich, Phys.Rev. A **76**, 021602 (2007);. S. Diehl, S. Floerchinger, H. Gies, J. M. Pawlowski and C. Wetterich, Annalen Phys. **522**, 615 (2010). S. Floerchinger, S. Moroz, R. Schmidt, C. Wetterich, Phys. Rev. A**79**, 013603 (2009). S. Moroz, S. Floerchinger, R. Schmidt, C. Wetterich, Phys. Rev. A**79**, 042705 (2009). R. Schmidt and S. Moroz, Phys. Rev. A **81**, 052709 (2010). M. C. Birse, B. Krippa and N. R. Walet, Phys. Rev. A**83** 023621 (2011). O. A. Yakubovsky, Sov. J. Nucl. Phys. **5**, 937 (1967). H. Gies and C. Wetterich, Phys. Rev. D **65**, 065001 (2002). D. F. Litim, Phys. Rev. D **64**, 105007 (2001). M. C. Birse, B. Krippa and N. R. Walet, Phys. Rev. C**87**, 054001 (2013). S. Moroz and R. Schmidt, private communication. R. D. Amado and F. C. Greenwood, Phys. Rev. D **7**, 2517 (1973). A. Deltuva, Phys. Rev. A **85**, 012708 (2012). L. Platter, H.-W. Hammer, and U.-G. Meissner, Phys. Rev. **A** 70, 052101 (2004).
[^1]: The four-body states seen in Ref. [@sm10] have been found to be numerical artefacts [@bkw13; @smpc].
|
---
abstract: 'We have analysed an effect of the Bak-Sneppen predator-prey food-chain self-organization on nucleotide content of evolving species. In our model, genomes of the species under consideration have been represented by their nucleotide genomic fraction and we have applied two-parameter Kimura model of substitutions to include the changes of the fraction in time. The initial nucleotide fraction and substitution rates were decided with the help of random number generator. Deviation of the genomic nucleotide fraction from its equilibrium value was playing the role of the fitness parameter, $B$, in Bak-Sneppen model. Our finding is, that the higher is the value of the threshold fitness, during the evolution course, the more frequent are large fluctuations in number of species with strongly differentiated nucleotide content; and it is more often the case that the oldest species, which survive the food-chain competition, might have specific nucleotide fraction making possible generating long genes.'
author:
- 'Marta Dembska$^1$, Miros[ł]{}aw R. Dudek$^1$ and Dietrich Stauffer$^2$'
title: '**Food-chain competition influences gene’s size.**'
---
$^1$ [ *Institute of Physics, Zielona G[ó]{}ra University, 65-069 Zielona G[ó]{}ra, Poland*]{}\
$^2$ [*Institute of Theoretical Physics, Cologne University, D-50923 K[ö]{}ln, Euroland*]{} \
\
\
\
Keywords: DNA, Bak-Sneppen model, predator-prey, computer simulation.
PACS: 82.39.Pj, 87.15.Aa, 89.75.Fb
Model introduction
==================
To understand the way the higher organized species emerge during evolution we consider very simple model of evolving food chain consisting of $N$ species. In the model, each species is represented by nucleotide composition of their DNA sequence and the substitution rates between the nucleotides. There are four possible nucleotides, A, T, G, C, in a DNA sequence. In our model, the DNA sequence is represented, simply, by four reals, $F_A, F_T, F_G, F_C$, being the nucleotide fractions and
$$F_A+F_T+F_G+F_C=1.
\label{fractions}$$
The nucleotide fractions depend on time due to mutations and selection.
Our model is originating from the Bak-Sneppen model of co-evolution [@BS1] and Kimura’s neutral mutation hypothesis ([@Motoo_Kimura1],[@Wen-HsiungLi1]). According to Kimura’s hypothesis, neutral mutations are responsible for molecular diversity of species. In 1980, Kimura introduced two-parameter model [@Motoo_Kimura2], [@Wen-HsiungLi2], where the transitional substitution rate (substitutions $A \leftrightarrow G$ and $C \leftrightarrow T$) is different from the transversional rate (substitutions $A \leftrightarrow T$, $G \leftrightarrow T$, $A \leftrightarrow C$, $G \leftrightarrow C$) . If we use Markov chain notation, with discrete time $t$, then the transition matrix, ${\bf M}_{\rm nucl}$,
$$\begin{aligned}
{\bf M}_{\rm nucl} &=& \left(
\begin{array}{llll}
1-uW_{A} & u~W_{AT} & u~W_{AG} & u~W_{AC} \\
u~W_{TA} & 1-uW_{T} & u~W_{TG} & u~W_{TC}\\
u~W_{GA} & u~W_{GT} & 1-uW_{G} & u~W_{GC}\\
u~W_{CA} & u~W_{CT} & u~W_{CG} & 1-uW_{C}
\end{array}
\right)\\
&=&
\left(
\begin{array}{llll}
1-u(2v+s)& uv& us & uv \\
uv & 1-u(2v+s) & uv & us\\
us & uv & 1-u(2v+s) & uv\\
uv & us & uv & 1-u(2v+s) \\
\end{array}
\right),
\label{macierz1}
\nonumber\end{aligned}$$
representing rates of nucleotide substitutions in the two-parameter Kimura model fulfills the following equation
$${\overrightarrow F(t+1)} = {\bf M}_{\rm nucl} {\overrightarrow F(t)}
\label{evolution}$$
where [={$F_A(t),F_T(t),F_G(t),F_C(t)\}^T$]{} denotes nucleotide fractions at time $t$, $u$ represents substitution rate and the symbols $W_{ij}=s$ for transitions and $W_{ij}=v$ for transversions ($i,j=A,T,G,C$) represent relative substitution probability of nucleotide $j$ by nucleotide $i$. $W_{ij}$ satisfy the equation
$$\sum_{i,j=A,T,G,C} W_{ij}=1,$$
which in the case of the two-parameter Kimura model is converted into the following
$$4s+8v=1,
\label{suma}$$
and $W_j=\sum_{i\neq j} W_{ij}$.
Evolution described by Eq.(\[macierz1\]) has the property that starting from some initial value of $\overrightarrow F(t_0)$ at $t=t_0$ the solution tends to an equilibrium in which $F_A=F_T=F_G=F_C=0.25$. The example of this type of behavior has been presented in Fig.\[fig1\]. The two-parameter Kimura approximation is one of the simplest models of nucleotide substitutions. For example, in reconstructing the phylogenetic trees, one should use a more general form of the transition matrix in Eq.(\[macierz1\]) ([@Wen-HsiungLi2],[@Lobry1],[@Rzhetsky],[@Lobry2]). This is not necessary in our model, where we need only the property that the nucleotide frequencies are evolving to their equilibrium values.
![Dependence of nucleotide fractions on time in two-parameter Kimura model. Here, the initial fractions take the following values: $F_A=0.320964$, $F_T=0.246541$, $F_G=0.0252434$, $F_C=0.407252$. Besides, there has been plotted the maximum absolute deviation from the difference $\vert F_A-F_T \vert$ and $\vert F_G-F_C \vert$ (the dashed curve).[]{data-label="fig1"}](Figure1.eps)
More complicated prey-predator relations were simulated with a $5 \times 5$ Chowdhury lattice [@Stauffer2] with a fixed number of six food levels. Each lower (prey) level contains twice as many possible species as the upper (predator) level. Also this model does not contain an explicit bit-string as genome. We now introduced a composition vector as above, different for each different species, and let it evolve according to Eq.(3). Again, after many iterations all four fractions approached 0.25. This result, as we will show below, is qualitatively different from that in the model defined below, where we observe fluctuations of nucleotide frequency, instead.
Our model consists of $N$ species and for each species we define the set of random parameters, [$F_A$, $F_T$, $F_G$, $F_C$, $u$, $s$, $v$]{}, which satisfy only two equations, Eq.(\[fractions\]) and Eq.(\[suma\]), and we assume that $4s>8v$ to fulfill the condition that transitions ($s$) dominate transversions ($v$). The nucleotide fractions, representing each species, change in time according to Eq.(\[evolution\]).
The species are related according to food-chain. In the case of the nearest-neighbor relation the species $i+1$ preys on species $i$. The extension to further neighbors follows the same manner. The food-chain has the same dynamics as in Bak-Sneppen model (BS) [@BS1], i.e., every discrete time step $t$, we choose the species $i$ with minimum fitness $B_i$ and the chosen species is replaced by a new one together with the species linked to it with respect to food-chain relation. In the original BS model the nearest neighborhood of species $i$ is symmetrical, e.g. $\{B_{i-1},B_i,B_{i+1}\}$. The asymmetrical (directional) neighborhood applied for food-chain modeling has been discussed by Stauffer and Jan [@Stauffer1] and their results were qualitatively the same as in the BS model. The generalizations of food-chain onto more complex food-web structures are also available [@Ito], [@Stauffer2].
The new species, substituting the old ones, obtain new random values [$F_A$, $F_T$, $F_G$, $F_C$, $u$, $s$, $v$]{}. In our model the fitness $B_i$ of the species $i=1, 2, \ldots, N$ is represented by the parameter
$$B_i=1-D, \quad D=\max (\vert {F_A-F_T}\vert, \vert {F_G-F_C}\vert),
\label{selectionrule}$$
where $B_i \in [0,1]$ is a measure of the deviation from equilibrium of the nucleotide numbers $F_A-F_T$ and $F_G-F_C$. Thus, the species with the smallest value of $B_i$ (largest compositional deviation from equilibrium) are eliminated together with their right-hand neighbors with respect to food-chain. This elimination mechanism leads to self-organization. Namely, in the case of finite value of $N$ the statistically stationary distribution of the values of $B_i$ ($i=1, 2, \ldots, N$) is achieved after finite number of time steps with the property that the selected species with the minimum value $B_{min}$ is always below some threshold value $B_c$ or it is equal to the value. The typical snapshot, at transient time, of the distribution of the values of $B_i$ is presented in Fig.\[fig2\].
So, if Fig.\[fig2\] looks much the same as it had been resulted from the simulation of pure BS model, then what are the new results in our model? In the following, we will show that the higher value of the threshold fitness, during the evolution course, it is often the case that the winners of the food-chain competition become also species with specific nucleotide composition, which is generating long genes.
Discussion of results
=====================
We know, from Eq.(\[evolution\]) (see also Fig.\[fig1\]), that a single species tends to posses equilibrium nucleotide composition, which in this simple two-parameter Kimura model means asymptotically the same nucleotide composition $F_A=F_T=F_G=F_C=0.25$. The only distinction, which we could observe, if we had used a more general form of the substitution table, could be the resulting equilibrium nucleotide composition different from the uniform one. This would bring nothing new to the qualitative behavior of our model.
![Snapshot of the distribution of the species fitness at the transient time $t=5000$. In the example, $N=200$ and the substitution rate is a random real $u=0.01*rnd$, number of the nearest-neighbors $n=1$. The horizontal line is representing the value of threshold fitness.[]{data-label="fig2"}](Figure2.eps)
![Few examples of time dependence of the threshold fitness $B$ for different values of the upper bound of the applied substitution rate $u$.[]{data-label="fig3"}](Figure3.eps)
![ Distribution $P(k)$ of gene length $k$ in Chromosome IV of [*Saccharomyces cerevisiae*]{} genome and in the case of the approximate analytic formula (Eq.(\[sizeofgene\])), where the nucleotide fractions take the values as in Chr. IV, i.e., $F_A=0.31121$, $F_T=0.309727$, $F_G=0.190188$, $F_C=0.188875$. Parameter $k$ is representing number of codons (nucleotide triplets).[]{data-label="fig4"}](Figure4.eps)
Once, in the model under consideration, nucleotide composition of species is changing according to Eq.(\[macierz1\]), the species fitness $B_i$ depends on time. It is not the case in the BS model [@BS1], where the fitness of the evolving species is constant in time unless it is extincted. Although $B_i$ depends on time, the food-chain selection rule introduces mechanism, which forbids to achieve the equilibrium nucleotide composition ($B_i=1$). Instead, there appears a threshold value of $B_c$, below which the species become extinct. In our model the threshold value depends on substitution rate $u$. The examples of this dependence for transient time of $10^9$ generations have been plotted in Fig.\[fig3\].
Similarly, as in BS model, the SOC phenomenon disappears if the number of nearest neighbors $n=0$. Then, all species tend to the state with $B=1$.
We will discuss the influence of threshold fitness optimization on nucleotide composition of species and, in consequence, its influence on the possible maximum length of gene in species genome. To this aim, we assume that a gene has continuous structure (no introns) and it always starts from codon START (ATG) and ends with codon STOP (TGA, TAG or TAA). Then, the probability of generating any gene consisting of $k$ nucleotide triplets in a random genome with the fractions $F_A$, $F_T$, $F_G$, $F_C$ could be approximated by the following formulae (see also [@Cebrat]):
$$P(k)=\alpha F_A F_T F_G (2 F_A F_T F_G+F_A^2F_T ) (1-2 F_A F_T F_G-F_A^2F_T)^{k-1},
\label{sizeofgene}$$
where $\alpha$ is a normalization constant, which can be derived from the normalization condition
$$\sum_{k=1}^{k_{\rm cutoff}} P(k) = 1.
\label{normalization}$$
The value of $k_{\rm cutoff}$ in Eq.(\[normalization\]) could be associated with genome size.
In Fig.\[fig4\], there has been shown the relation between the empirical distribution of gene length $k$ in chromosome IV of [*Saccharomyces cerevisiae*]{} genome and the distribution $P(k)$ in Eq.(\[sizeofgene\]). Similar results we could obtain for other genomes. One can observe, that the approximation in Eq.(\[sizeofgene\]) is acceptable for small gene size, whereas it becomes wrong for large gene size. Generally, it is accepted that there is direct selective pressure on gene size for the effect. Examples of papers discussing the problem could be found [@WentianLi],[@Cebrat],[@proteomesize] together with analyses of rich experimental data.
The lowest frequency of gene size, $k$, in [*Saccharomyces cerevisiae*]{} genome is equal to $P_0 \approx 2.8 \times 10^{-5}$ (Fig.\[fig4\]). In many natural genomes $P_0$ takes value of the same order of magnitude, e.g., in the [*B.burgdorferi genome*]{} $P_0 \approx 5.7 \times 10^{-5}$. In our model, we have assumed that for all species holds $P_0=1 \times 10^{-6}$. We have also introduced maximum gene length, $k_{\max}$, which is the largest value of $k$ for which $P(k) \ge P_0$.
In the particular case of the same fractions of nucleotides in genome ($F_A=F_T=F_G=F_C=0.25$) the limiting value $k=k_{\max}$ for which $P(k) \ge P_0$ is equal to $k_{\max}=225$ nucleotide triplets ($675$ nucleotides). Thus, in our model, we could expect that for the oldest species the maximum gene length $k_{\max}$ should not exceed the value of $675$ nucleotides. The reason for that is that ageing species should approach equilibrium composition (Fig.\[fig1\]). However, surprisingly, we found that the self-organization phenomenon enforces a state, in which the oldest species may have much longer gene sizes than in genome with nucleotide composition corresponding to equilibrium composition. Actually, there start to appear fluctuations in the number of species with very short genes and very long ones.
![Time dependence of maximum gene size, $k_{\max}$, of the oldest species and the Guanine content in their genome in the evolving ecosystem when $N=500$, $u=0.1*rnd$, $n=1$. The data in the figure have been decimated.[]{data-label="fig5"}](Figure5.eps)
![The same parameters as in Fig.\[fig5\] but $n=0$.[]{data-label="fig6"}](Figure6.eps)
![Maximum gene length, $k_{\max}$, versus genomic fraction of nucleotide in the oldest species. The vertical lines correspond to equilibrium genome. In our model this means $F_A=F_T=F_G=F_C=0.25$. The same parameters as in Fig.\[fig5\] have been used.[]{data-label="fig7"}](Figure7.eps)
The selection towards the species with the smallest deviation from equilibrium nucleotide composition (the largest value of $B$) implicates that the species, which survive the selection, may have specific bias in nucleotide composition, which makes possible generating long genes. In our model, we have observed abundances of G+C content in the species with long genes. During simulation run, in each time step $t$, we have collected in a file data representing age of the oldest species, the corresponding gene size $k$ and nucleotide frequency. We have observed, that the closer the species fitness $B_i$ is to the threshold fitness the older might be the species and also the species might posses longer genes in its genome. There is no such effect in the case, when $n=0$. Even if there could appear, at some early time interval, a tendency to generate longer genes, this property would have disappeared after longer evolution time of the system of $N$ species. In Fig.\[fig5\], we have plotted time dependence of the recorded maximum length of gene in the oldest species and the corresponding Guanine fraction. One can compare this figure with Fig.\[fig6\], where there are no prey-predator relation in the ecosystem ($n=0$). In the latter case, the system is ageing in accordance with the Eq.(\[evolution\]) and $B_i \rightarrow 1$ ($i=1, 2, \ldots, N$) and self-organization has not been observed.
The observed property of the competing species has an analogy with the behavior of the model of evolution of evolving legal rules in Anglo-American court, introduced by Yee [@KentonKYee] (see Fig. 3 in his paper).
The relation between nucleotide fraction of genome and the possible maximum length of gene in such genome has been shown in the histogram in Fig.\[fig7\]. The presented data address, solely, the oldest species. Notice, $A \approx T$ and $G \approx C$ for genomes both with short genes and long ones, whereas $A \approx T \approx G \approx C \approx 0.25$ for genomes with nucleotide composition near equilibrium understood in terms of the two-parameter Kimura model. We should remember, that the substitution table for the two-parameter Kimura model (Eq.\[macierz1\]) is a symmetric matrix and the observed compositional asymmetry results directly from the predator-prey self-organization phenomenon. The right-hand wings, evident in the structure in Fig.\[fig7\], do vanish in the case when $n=0$ in spite of the same fitness parameter in Eq.\[selectionrule\] applied for selection.
We have not included strand structure in species genome, in our model, since it is represented only with the help of nucleotide fraction. Lobry and Sueoka, in their paper [@LobrySueoka], concluded that if there are no bias in mutation and selection between two DNA strands, then it is expected $A \approx T$ and $G \approx C$ within each strand, and that the observed variation in G+C content between species is an effect of another phenomenon than, simply, asymmetric mutation pressure. Here, we have shown, that such compositional fluctuations of genome could result from ecosystem optimization - no direct selection on genes length is present in our model.
The predator-prey rule, in the model under consideration, introduces large fluctuations in nucleotide frequency in the ageing ecosystem, if it is sufficiently old. However, we have not observed this frequency phenomenon in modeling speciation on the Chowdhury lattice [@Stauffer2], as we stated in the beginning. After we have introduced a small change of our model, in such a way, that new species arising in the speciation process were taken always from among the survived species, and we only slightly were modifying their nucleotide frequency by introducing $d\%$ of changes in their values, then the observed by us fluctuations ceased to exist in the limit $d \rightarrow 0$, as found in the Chowdhury model.
Conclusions
===========
The specific result of the food-chain self-organization of the competing species is that the oldest survivors of the competition might posses strong compositional bias in nucleotides, the abundance of G+C content. In our model, this resulting asymmetry makes possible generating long genes. There was no direct selection applied on the gene length, in the model. The fluctuation in number of species with long genes and short genes represents rather undirectional noise, the amplitude of which is increasing while the ecosystem is ageing. The effect ceases to exist if there is no species competition. The same is if we allow only $d\%$ changes of nucleotide frequency in the new formed species, in the limit $d \rightarrow 0$.
It could be, that the observed self-organization is an attribute of genes in genome evolution. Typically, many genes are coupled together in genome in a hierarchical dynamical structure, which resembles complex food-web structure. Some genes may be duplicated but also you can observe fusion of genes or even genomes.
\
We thank geneticist S. Cebrat for bringing us physicists together at a GIACS/COST meeting, September 2005. One of us, M.D., thanks A. Nowicka for useful discussion.
[92]{} P. Bak and K. Sneppen, Phys. Rev. Lett. [**74**]{}, 4083 (1993) M. Kimura, Nature [**217**]{} 624 (1968) Wen-Hsiung Li, Theor. Popul. Biol. [**49**]{}, 146 (1996) M. Kimura, J. Mol. Evol. [**16**]{}, 11 (1980) Wen-Hsiung Li, Xun Gu, Physica A [**221**]{}, 159 (1995) J.R. Lobry, J. Mol. Evol. [**40**]{}, 326 (1995) J.R. Lobry, Physica A [**273**]{}, 99 (1999) A. Rzhetsky and M. Nei, Mol. Biol. Evol. [**12**]{}, 131 (1995) D. Stauffer and N. Jan, Int. J. Mod. Phys. C 11, 147 (2000) H. C. Ito, T. Ikegami, J. Theor. Biol. [**238**]{},1 (2006) D. Chowdhury and D. Stauffer, Phys. Rev. E [**68**]{}, 041901 (2003) Wentian Li, Computer & Chemistry [**23**]{}, 283 (1999) A. Gierlik, P. Mackiewicz, M. Kowalczuk, S. Cebrat, M.R. Dudek, Int. J. Mod. Phys. C [**10**]{}, 1 (1999) P. M. Harrison, A. Kumar, N. Lang, Ml Snyder and M. Gerstein, Nucl. Acids Res. [**30**]{}, 1083 (2002) K. K. Yee, arXiv:nlin.AO/0106028 (2005) J. R. Lobry and N. Sueoka, Genome Biol. [**3(10)**]{}:research0058.1-0058.14. (2002) ([*The electronic version can be found online at http://genomebiology.com/2002/3/10/research/0058*]{}
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---
abstract: 'Site dilution of spin-gapped antiferromagnets leads to localized free moments, which can order antiferromagnetically in two and higher dimensions. Here we show how a weak magnetic field drives this order-by-disorder state into a novel *disordered-free-moment* phase, characterized by the formation of local singlets between neighboring moments and by localized moments aligned antiparallel to the field. This disordered phase is characterized by the absence of a gap, as it is the case in a Bose glass. The associated field-driven quantum phase transition is consistent with the universality of a superfluid-to-Bose-glass transition. The robustness of the disordered-free-moment phase and its prominent features, in particular a series of *pseudo*-plateaus in the magnetization curve, makes it accessible and relevant to experiments.'
author:
- Rong Yu
- Tommaso Roscilde
- Stephan Haas
title: 'The disordered-free-moment phase: a low-field disordered state in spin-gap antiferromagnets with site dilution'
---
Valence bond solids (VBS) in spin-gapped antiferromagnets represent some of the most fundamental examples of quantum-disordered states in condensed matter systems. The nature of such states is by now well understood theoretically and has been extensively verified experimentally. A variety of mechanisms, such as increased strength of certain bonds in the magnetic Hamiltonian [@SandvikS94; @Matsumotoetal01], magnetic frustration [@ShastryS81], and the Haldane mechanism [@Affleck89], can render classical Néel order unstable towards the formation of local singlets, which arrange themselves into a VBS. Evidence for such phases has been found in a large number of magnetic compounds, ranging from Haldane chains [@Regnaultetal94], to spin ladders [@DagottoR96], to weakly coupled dimer systems [@Cavadinietal00; @Sasagoetal97]. A central focus of theoretical and experimental investigations has been the effect of doping on such states. In particular, it was soon realized theoretically, [@ShenderK91; @SigristF96] and observed experimentally [@Azumaetal97; @Xuetal00], that doping a VBS with static, non-magnetic impurities leads to the intriguing phenomenon of *order-by-disorder* (OBD): free $S=1/2$ magnetic moments appear close to the impurity sites and interact effectively via a long-range network of unfrustrated (albeit random) couplings, which decay exponentially with the inter-moment distance. These interactions, although weak, are sufficient for the free moments (FMs) to order antiferromagnetically at experimentally relevant temperatures [@Azumaetal97].
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Given the random nature of the inter-moment couplings, the OBD state induced by doping is extremely inhomogeneous, as it contains a large variety of energy scales which depend exponentially on the spatial distribution of the impurities. In this paper we study the evolution of the OBD state upon application of a magnetic field, which represents a straightforward experimental probe of energy scales in magnetic systems. Precisely due to the large distribution of the effective couplings between the FMs, we find an amazingly rich response of the system to the applied field. The field scan reveals that the long-range order in the system is extremely tenuous for large spin gaps, and its field-driven destruction leaves behind local singlets (or spins oppositely polarized to the field) on even- (or odd-)numbered clusters of FMs, which are coupled at energies higher than the scale characteristic for Néel order (see Fig. \[f.DFM\]). The local polarization fields for these FM clusters cover a continuous range, so that the magnetization process with increasing field continues also after destruction of the OBD, and the resulting *disordered-free-moment* (DFM) phase [@Roscilde06] is *gapless*. The DFM phase persists up to the sizable field which polarizes all the FMs, and the magnetization curve within this phase shows prominent features of intermediate *pseudo*-plateaus, still retaining a finite albeit extremely small slope, related to the distribution of the strongly interacting clusters of FMs.
To quantitatively investigate the field response of a site-diluted spin-gapped antiferromagnet, we focus our attention on a two-dimensional $S=1/2$ model of weakly coupled dimers [@Matsumotoetal01; @Yasudaetal01], whose Hamiltonian reads
$$\begin{aligned}
H&=& J\sum_{i\in A} \epsilon_{i}\epsilon_{i+\hat x}
\bm{S}_{i}\cdot
\bm{S}_{i+\hat x}
+J'\sum_{i\in A}\epsilon_{i}\epsilon_{i+\hat y}
\bm{S}_{i}\cdot
\bm{S}_{i+\hat y} \nonumber\\
&+&J'\sum_{i\in B}\sum_{\hat{d} = \hat{x},\hat{y}}
\epsilon_{i}\epsilon_{i+\hat d}
\bm{S}_{i}\cdot \bm{S}_{i+\hat d}
-h \sum_{i}\epsilon_{i} \emph{S}^z_{i}.
\label{e.ham_dilute}\end{aligned}$$
Here $i$ runs over the two sublattices ($A$ and $B$) of a square lattice, $\hat x$ and $\hat y$ are the two lattice vectors, and $\epsilon_i$ is the random dilution variable taking values 0 and 1 with probability $p$ and $1-p$ respectively. $h = g\mu_B H$ is the applied field. The couplings $J>J'$ determine the subset of strong antiferromagnetic bonds: for $J'/J < 0.523..$ [@Matsumotoetal01] the bond anisotropy stabilizes a dimer-singlet ground state against the conventional Néel ordered state of the square-lattice antiferromagnet. All the results presented here refer to the field and doping effects deep within the dimer-singlet regime at $J'/J = 1/4$.
The presence of lattice vacancies induces FMs which are localized in the vicinity of the unpaired spins which have lost their $J$-neighbor. Perturbation theory [@SigristF96; @Mikeskaetal04; @Roscilde06] provides an effective coupling $J_{ij} \approx (-1)^{i-j-1} (J_1/r) \exp(-r/\xi_0)$ between these FMs, where $r=|i-j|$, $\xi_0$ is the correlation length of the undoped system. We choose $J_1 \approx J' \exp(1/\xi_0)$ in order for $J_{ij}$ to correctly reproduce the limit $J'$ for neighboring unpaired spins. For a deeper understanding of the Hamiltonian Eq. (\[e.ham\_dilute\]), it is illuminating [@Laflorencieetal04] to study an effective model for the network of FMs, consisting of randomly distributed $S=1/2$ spins with effective couplings $J_{ij}$ $$H_{\rm FM} = \frac{1}{2} \sum_{i,j} J_{ij} \bm{S}_{i}\cdot
\bm{S}_{j} - h \sum_{i} \emph{S}^z_{i}.
\label{e.ham_effective}$$
We investigate the original and the effective Hamiltonian, given by Eq. (\[e.ham\_dilute\]) and (\[e.ham\_effective\]), using Stochastic Series Expansion (SSE) Quantum Monte Carlo simulations based on the directed-loop algorithm [@SyljuasenS02]. For the original Hamiltonian Eq. (\[e.ham\_dilute\]) we study $L\times L$ lattices up to $L$=40 with dilution $p=1/8$, whereas for the effective model Eq. (\[e.ham\_effective\]) we randomly distribute spins on the same lattice sizes with a density $p$ equal to that of the vacancies in the original model. Disorder averaging is typically performed over $\approx 300$ realizations. The ground-state properties are systematically obtained using a $\beta$-doubling approach [@Sandvik02]. Inverse temperatures up to $\beta J = 2^{15}$ are necessary to observe the physical $T\rightarrow 0$ behavior. In the following, we focus our attention on the uniform magnetization per spin $m_u = 1/N_s \sum_i \langle S_i^z \rangle$, where $N_s$ is the total number of spins in the system considered; on the uniform susceptibility $\chi_u/J = \partial m_u / \partial h$; on the staggered magnetization $m_s = \sqrt{S^{\perp}(\pi,\pi)/L^2}$, where $S^{\perp}(\pi,\pi) = 1/(2L^2) \sum_{ij}
\langle S_i^x S_j^x + S_i^y S_j^y \rangle$ is the transverse static structure factor; on the correlation length $\xi$, extracted from the $q$-dependent structure factor; and on the superfluid density $\rho_s= 1/(2\beta J)\langle W_x^2+W_y^2 \rangle$, where $W_{x(y)}$ are the winding numbers of the SSE worldlines.
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In the absence of a magnetic field, the effective couplings $J_{ij}$ give rise to long-range magnetic order of the network of coupled free moments. In particular, for $J'/J$=1/4 and $p$=1/8 we extrapolate a staggered magnetization $m_s= 0.032(3)$ in the thermodynamic limit. Although the couplings $J_{ij}$ range between 0 and $J'$, the average coupling strength responsible for the long-range order turns out to be much smaller than $J'$ [@Roscilde06]. Fig. \[f.magn\] shows the evolution of the ordered moment under application of a field. It reveals that the antiferromagnetic order is already destroyed at a field $h\ll J'$, namely at $h=h_{\rm DFM}= 0.007(1)$. Yet, a striking feature of this disordered phase is that the destruction of long-range order is *not* accompanied by the full polarization of the FMs, as it would ordinarily happen in a homogeneous antiferromagnet. At the critical field $h_{\rm DFM}$ the uniform magnetization is found to be $m_u=0.0208(6)$, much less than the value $m_u=pS=1/16$ corresponding to fully polarized FMs, which is attained at a much larger field $h_{\rm plateau}/J \approx 0.5$. Consequently the DFM phase, appearing between $h_{\rm DFM}$ and $h_{\rm plateau}$, is highly unconventional, retaining a finite uniform susceptibility and a gapless spectrum. These unconventional properties have also been observed in the magnetic Bose-glass phase of triplet quasiparticles living on intact dimers [@Fisheretal89; @RoscildeH05; @Roscilde06; @Nohadanietal05]. Later we will argue that these two phases bear indeed strong analogies, although involving different degrees of freedom.
For $h>h_{\rm plateau}$ the saturation of the magnetization of FMs leads to the full restoration of a gapped disordered phase due to the field [@Mikeskaetal04]. Once the field reaches the value corresponding to the gap of the clean system, $h=\Delta=0.60(1)$, a further Bose-glass phase is established in which rare clean regions develop a local magnetization without the appearence of spontaneous order, corresponding to localized triplet quasiparticles [@RoscildeH05; @Roscilde06]. A delocalization transition of the triplet bosons into a superfluid condensate corresponds to a further onset of long-range transverse order ($m_s>0$) at even higher fields [@RoscildeH05; @Roscilde06].
As shown in Fig. \[f.magn\], the main features of the field dependence $m_u$ and $m_s$ for $h<h_{\rm plateau}$ in the doped coupled-dimer model of Eq. (\[e.ham\_dilute\]) are very well reproduced by the effective model Eq. (\[e.ham\_effective\]), for which we take $\xi_0= 1$ as found by simulations at $h=0$ and $p=0$. In particular, the fundamental appearence of a DFM phase with $m_s=0$ and $\chi_u>0$ is confirmed in the effective model. This reveals that the FMs are essentially the only degrees of freedom responding to a field $h<h_{\rm plateau}$ in the doped coupled-dimer system.
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The novel quantum phase transition (QPT) between the OBD and the DFM phase is studied using finite-size scaling analysis of the correlation length, $\xi = L F_{\xi}[L^{1/\nu}\delta h]$, of the superfluid density $\rho_s = L^{d-2+z} F_{\rho_s}[L^{1/\nu}\delta h]$, and of the staggered magnetization $m_s = L^{-\beta/\nu} F_{m_s}[L^{1/\nu}\delta h]$, where $\delta h = h-h_{\rm DFM}$, as shown in Fig. \[f.scaling\]. This allows us to extract the critical exponents $z$, $\nu$ and $\beta$. For the original Hamiltonian Eq. (\[e.ham\_dilute\]) we find $z=2.0(1)$, $\nu=1.0(1)$ and $\beta=0.9(1)$. These estimates are also confirmed in the effective FM model. The above exponents are fully consistent with those of the 2$d$ superfluid-to-Bose-glass (SF-BG) QPT previously studied in diluted bilayers [@Roscilde06], and with the exponents found at the BG-SF transition for higher fields for the model Eq. (\[e.ham\_dilute\]) [@Yuetal07]. In particular $z$ is in agreement with the general theoretical prediction $z=d$ [@Fisheretal89], and $\nu$ satisfies the fundamental Harris criterion $\nu\geq 2/d$. Altogether the present results and those of Refs. point towards a general SF-BG universality in $d=2$ for order-disorder transitions at which the uniform susceptibility $\chi_u$ remains finite, corresponding to the absence of a gap.
In the DFM phase, the magnetization curves of the original Hamiltonian Eq. (\[e.ham\_dilute\]) and the effective model Eq. (\[e.ham\_effective\]) show a dramatic feature: beside the large pleateau appearing at $h\geq h_{\rm plateau}$, one observes the presence of apparent intermediate plateaus at around 3/4 and $95\%$ of the saturation magnetization. A detailed study of the temperature-dependent susceptibility in this field region reveals that these features are actually *pseudo*-plateaus (PPs), which retain an extremely small slope (Fig. \[f.magn\]). For both $H$ and $H_{\rm FM}$ the first PP extends up to $h\approx 0.7J'$; a second PP markedly appears around $h\approx 1.2J'$ for $H$ (it is rounded off for $H_{\rm FM}$ [@Yuetal07]); the true saturation plateau is only attained at $h\approx 2J'$. These fundamental features can be understood within the picture of strongly interacting clusters of FMs in the DFM phase. As shown in Fig. \[f.DFM\], the zero-field OBD phase is essentially inhomogeneous due to the random nature of the couplings. A majority of FMs are spaced from each other by an average distance $\langle r \rangle = p^{-1/d}$, large in the small dilution limit, and interact via weak average couplings $\langle J_{\rm eff} \rangle \sim p J'$ [@Roscilde06]. However, fluctuations in the spatial distribution of the impurities also lead to small clusters of free moments located on neighboring sites, and thus interacting with much stronger couplings $J'$. If antiferromagnetically coupled in even-numbered clusters, the strongly interacting FMs participate only marginally in the OBD state of the system, and have a significant singlet component in their ground state wave function. This is directly revealed in a histogram of the bond energies (Fig. \[f.histo\](b)) $E_b = J_b \langle{\bm S}_{1,b} \cdot {\bm S}_{2,b} \rangle$ where $J_b = J, J'$ and $(1,b)$, $(2,b)$ are the two neighboring lattice sites participating in the bond $b$. Beside the peak at $E_b\approx -3J/4$, corresponding to singlets on intact dimers, a further peak at $E_b\approx -3J'/4$ is observed, corresponding to FM dimer singlets, as well as a peak at $E_b\approx -J'/2$ corresponding to FM trimers.
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Applying a magnetic field $h \gtrsim \langle J_{\rm eff} \rangle$ has clearly the effect of destroying the long-range order of the FMs, but at the same time the FM singlets are left intact while odd-numbered clusters are not fully polarized (Fig. \[f.DFM\]), with the fundamental consequence that the antiferromagnetic order disappears but the FMs are far from saturation. This is clearly seen in the histogram of the local magnetic moments $\langle S_i^z \rangle$ of the unpaired spins only (Fig. \[f.histo\](b)): for small fields, a double-peak structure appears with a peak at $\langle S_i^z \rangle = S$ corresponding to fully polarized FMs, and a strong quantum peak at $\langle S_i^z \rangle = 0$ corresponding to FM singlets. The large tails for $0 < \langle S_i^z \rangle < S$ and for $\langle S_i^z \rangle < 0$ come instead from FMs in odd-numbered clusters. In fact, we can resolve two more peaks at $\langle S_i^z \rangle =
1/3$ and $\langle S_i^z \rangle = -1/6$ at larger fields, which correspond to partially polarized spins in FM trimers. Local FM clusters have widely different local gaps to full polarization, both due to their geometric structure (dimers, trimers, quadrumers, etc.), and to the local field they experience from the other FMs. Yet the distribution of rare FM clusters clearly assigns dominant statistical weight to the dimers, and this simple geometric fact is the reason for the appearence of the first PP: the magnetization process nearly stops until the local gap of the dominant FM dimers is overcome at $h \lesssim J'$. Nonetheless, the slope remains finite because the FM dimers have a distribution of local gaps. The magnetization value and the field location of the PP can be quantitatively related to the statistics of FMs clustered in dimers [@Yuetal07]. Analogously, one can quantitatively associate the second plateau with the statistics of the FM trimers [@Yuetal07]. Higher-order plateaus associated with larger local polarization fields should be expected, but they cannot be resolved within the given numerical accuracy.
From the above data a clear picture of the DFM phase emerges: in this phase, a majority of the FMs are polarized, but antiparallel spins exist, corresponding to rare FM clusters. Upon a spin-to-hardcore-boson transformation, these antiparalell spins take the nature of bosonic spin-down quasiparticles ($\downarrow$-QPs) *localized* on rare regions of the lattice [@Yuetal07]. This aspect connects with the ordinary picture of a Bose glass [@Fisheretal89], and it further endows the OBD-to-DFM quantum phase transition with the nature of a localization transition: in the OBD phase the $\downarrow$-QPs form a superfluid condensate, which is progressively depleted by the applied field (acting as a negative chemical potential), up to the point where the $\downarrow$-QPs undergo localization into a Bose-glass state, losing superfluidity but retaining compressibility, which corresponds to a finite uniform susceptibility.
It is evident from the above results that the DFM phase is relevant for experiments on site-diluted spin-gapped antiferromagnets which display an OBD phase in zero field. The fundamental condition for the observation of the DFM phase is that the spin gap of the pure system be much larger than the maximum energy scale of the FM interaction (average inter-dimer coupling in weakly coupled dimer systems, inter-chain coupling in Haldane chains). This condition is necessary to ensure that the physics of the field response of the FMs is well separated in energy from that of the field-induced ordered state (for a detailed discussion, see Ref. ). Joint magnetometry and neutron scattering measurements at relatively low fields should be sufficient to fully pinpoint this phase by demonstrating the absence of spontaneous order and the finite susceptibility down to zero temperature. Furthermore, NMR measurements can show the rich structure of the distribution of local magnetic moments, similar to the histogram of Fig. \[f.histo\]. The low-field location of the DFM phase and its strong physical signatures in the magnetic observables make it the most accessible novel disordered phase in quantum magnets with lattice randomness.
Useful discussions with T. Giamarchi, W. Li, O. Nohadani, P. Sengupta, M. Sigrist are gratefully acknowledged. It is a pleasure to thank M. Vojta for a remark which has sparked the present investigation. T.R. is supported by the E.U. through the SCALA integrated project. R.Y. and S.H. are supported by DOE under grant No. DE-FG02-05ER46240.
[99]{}
A. W. Sandvik *et al.*, Phys. Rev. Lett. [**72**]{}, 2777 (1994). M. Matsumoto *et al.*, Phys. Rev. B [**65**]{}, 014407 (2001). B. S. Shastry *et al.*, Physica [**108B**]{}, 1069 (1981). I. Affleck, J. Phys.: Condens. Matter [**1**]{}, 3047 (1989). L. P. Regnault *et al.*, Phys. Rev. B [**50**]{}, 9174 (1994). E. Dagotto and T. M. Rice, Science [**235**]{}, 1196 (1987). N. Cavadini *et al.*, J. Phys.: Condens. Matter [**12**]{}, 5463 (2000). Y. Sasago *et al.*, Phys. Rev. B [**55**]{}, 8357 (1997). E. F. Shender *et al.*, Phys. Rev. Lett. [**66**]{}, 2384 (1991). M. Sigrist *et al.*, J. Phys. Soc. Jpn. [**65**]{}, 2385 (1996). M. Azuma *et al.*, Phys. Rev. B [**55**]{}, R8658 (1997). G. Xu *et al.*, Science [**21**]{}, 419 (2000). T. Roscilde, Phys. Rev. B [**74**]{}, 144418 (2006). M. P. A. Fisher *et al.*, Phys. Rev. B [**40**]{}, 546 (1989). T. Roscilde *et al.*, Phys. Rev. Lett. [**95**]{}, 207206 (2005). C. Yasuda *et al.*, Phys. Rev. B [**64**]{}, 092405 (2001). H.-J. Mikeska *et al.* Phys. Rev. Lett. [**93**]{}, 217204 (2004). N. Laflorencie *et al.*, Phys. Rev. B [**69**]{}, 212412 (2004). O.F. Syljuåsen *et al.*, Phys. Rev. E [**66**]{}, 046701 (2002). A. W. Sandvik, Phys. Rev. B [**66**]{}, 024418 (2002). O. Nohadani *et al.*, Phys. Rev. Lett. [**95**]{}, 227201 (2006). R. Yu *et al.*, in preparation.
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---
abstract: 'We report the results from some of the deepest Keck/Multi-Object Spectrometer For Infra-Red Exploration data yet obtained for candidate $z \gtrsim 7$ galaxies. Our data show one significant line detection with 6.5$\sigma$ significance in our combined 10 hr of integration which is independently detected on more than one night, thus ruling out the possibility that the detection is spurious. The asymmetric line profile and non-detection in the optical bands strongly imply that the detected line is Ly$\alpha$ emission from a galaxy at $z$(Ly$\alpha)=7.6637 \pm 0.0011$, making it the fourth spectroscopically confirmed galaxy via Ly$\alpha$ at $z>7.5$. This galaxy is bright in the rest-frame ultraviolet (UV; $M_{\rm UV} \sim -21.2$) with a moderately blue UV slope ($\beta=-2.2^{+0.3}_{-0.2}$), and exhibits a rest-frame Ly$\alpha$ equivalent width of EW(Ly$\alpha$) $\sim 15.6^{+5.9}_{-3.6}$ Å. The non-detection of the 11 other $z \sim$ 7–8 galaxies in our long 10 hr integration, reaching a median 5$\sigma$ sensitivity of 28 Å in the rest-frame EW(Ly$\alpha$), implies a 1.3$\sigma$ deviation from the null hypothesis of a non-evolving distribution in the rest-frame EW(Ly$\alpha$) between $3<z<6$ and $z=$ 7–8. Our results are consistent with previous studies finding a decline in Ly$\alpha$ emission at $z>6.5$, which may signal the evolving neutral fraction in the intergalactic medium at the end of the reionization epoch, although our weak evidence suggests the need for a larger statistical sample to allow for a more robust conclusion.'
author:
- 'Mimi Song, Steven L. Finkelstein, Rachael C. Livermore, Peter L. Capak, Mark Dickinson, and Adriano Fontana'
title: 'Keck/MOSFIRE Spectroscopy of $\lowercase{z}=$ 7–8 Galaxies: L$\alpha$ Emission from a Galaxy at $\lowercase{z} = 7.66$'
---
Introduction
============
The Ly$\alpha$ emission line is a unique tool as the line properties encode information about the scattering medium through which the photons have passed. During the past few years, in the present absence of a sensitive 21 cm signal from reionization, investigating the redshift evolution of the “Ly$\alpha$ fraction”, the fraction of Lyman-break galaxies (LBGs) which exhibit strong Ly$\alpha$ emission, has served as a valuable and feasible means of providing constraints on the ionization state of the intergalactic medium (IGM). Spectrosopic follow-up of LBGs has revealed that the Ly$\alpha$ fraction (typically defined as LBGs with rest-frame Ly$\alpha$ EW $> 25$ Å) steadily increases from $z=3$ to $z=6$, reaching $\sim$50% for faint galaxies ($M_{\rm UV} > -20.25$) at $z \sim 6$ [@stark10; @stark11]. At higher redshifts of $z \sim 7$, however, initial expectations and attempts based on an extrapolation of the trend of the increasing Ly$\alpha$ fraction seen at lower redshifts found a reverse of the trend, showing only 20%–30% of faint galaxies with Ly$\alpha$ emission [e.g., @fontana10; @pentericci11; @ono12; @schenker12; @finkelstein13; @pentericci14]. This steep decrease beyond $z \sim 6$ is in line with measurements of Gunn–Peterson troughs [@gunn65] in the spectra of distant quasars [@fan06], which signal the (near) completion of reionization by $z \sim 6$.
Several attempts have been made to interpret the observed drop in the Ly$\alpha$ fraction in connection with the neutral fraction of the IGM or different models of reionization. Earlier works suggested, assuming that the observed drop in the Ly$\alpha$ fraction from $z \sim 6$ to $z \sim 7$ is entirely driven by the change in the IGM transmission, that it requires a steep increase in the volume-averaged neutral fraction of $\Delta x_{\rm H\,{\sc I}}>$ 0.4–0.5 over $\Delta z=1$ [@dijkstra11; @pentericci11]. Alternatives have subsequently been proposed that account for the possibility of other sources of Ly$\alpha$ attenuation which alleviate the amount of the required increase in the neutral fraction. For example, @dijkstra14 suggested that the change in the intrinsic physical properties of galaxies such as an increase in the escape fraction of ionizing photons can explain the observed drop with a mild increase in the neutral fraction of $\Delta x_{\rm H\,{\sc I}} =$ 0.1–0.2, and @bolton13 argued that the rise of the neutral fraction of only $\Delta x_{\rm H\,{\sc I}} =$ 0.1 by $z=7$ is sufficient when accounting for self-shielding absorption systems (Lyman limit systems; LLSs) in the IGM, which are expected to be abundant near the end of reionization (though see @mesinger15). At $z \sim 7$, a sufficient sample has been assembled to start discerning between ‘patchy’ and ‘smooth’ models of Ly$\alpha$ attenuation. @pentericci14 found from a compilation of observations at $z \sim 7$ that the ‘patchy’ model of Ly$\alpha$ attenuation (which does not necessarily literally mean a patchy reionization process but may instead signal the abundant LLSs; @mesinger15), is favored over the ‘smooth’ attenuation model. Although the interpretation is not straightforward, these studies all highlight the potential of studying the Ly$\alpha$ fraction as a valuable probe of reionization.
[lccccccccc]{} z8\_GSD\_17938 & 3:32:49.94 & $-$27:48:18.1 & 25.7 & 25.7 & $-$21.6 & 8.07 & $[7.87, 8.37]$ & 0.70 & $<$ 12\
z7\_GSD\_10175 & 3:32:50.48 & $-$27:46:56.0 & 25.7 & 25.6 & $-$21.2 & 6.93 & $[6.14, 7.22]$ & 0.37 & $<$ 15\
z7\_GSD\_12816 & 3:32:44.89 & $-$27:47:21.8 & 26.9 & 27.2 & $-$20.2 & 6.81 & $[6.02, 7.20]$ & 0.32 & $<$ 45\
z7\_MAIN\_2852 & 3:32:42.56 & $-$27:46:56.6 & 26.0 & 26.0 & $-$20.9 & 6.85 & $[6.75, 6.93]$ & 0.08 & $<$ 25\
z7\_MAIN\_4005 & 3:32:39.55 & $-$27:47:17.5 & 26.5 & 26.5 & $-$20.7 & 7.55 & $[6.30, 7.55]$ & 0.53 & $<$ 27\
z7\_MAIN\_3474 & 3:32:38.80 & $-$27:47:07.2 & 27.0 & 27.0 & $-$20.0 & 7.41 & $[7.08, 7.54]$ & 0.92 & $<$ 55\
z8\_GSD\_2135 & 3:32:42.88 & $-$27:45:04.3 & 26.9 & 26.8 & $-$20.2 & 7.76 & $[1.84, 8.05]$ & 0.49 & $<$ 39\
z7\_GSD\_568 & 3:32:40.69 & $-$27:44:16.7 & 26.9 & 26.8 & $-$20.1 & 7.20 & $[6.62, 7.45]$ & 0.62 & $<$ 35\
z7\_GSD\_431 & 3:32:40.26 & $-$27:44:09.9 & 26.6 & 26.7 & $-$20.4 & 7.37 & $[6.66, 7.71]$ & 0.70 & $<$ 28\
z7\_GSD\_1273 & 3:32:36.00 & $-$27:44:41.7 & 26.5 & 26.5 & $-$20.4 & 6.86 & $[6.66, 7.05]$ & 0.30 & $<$ 31\
z7\_GSD\_3811 & 3:32:32.03 & $-$27:45:37.1 & 25.8 & 25.9 & $-$21.2 & 7.42 & $[6.71, 7.62]$ & 0.73 & $<$ 15\
z7\_ERS\_12098 & 3:32:35.44 & $-$27:42:55.1 & 26.3 & 26.3 & $-$20.7 & 7.17 & $[6.23, 7.25]$ & 0.49 & $<$ 23
Because Ly$\alpha$ is redshifted into the near-infrared, pushing the study of Ly$\alpha$ emission to a higher redshift of $z > 7$ had been relatively slow. However, the advent of a new generation of ground-based near-infrared spectrographs with multiplexing capability and increased sensitivity has been changing the game by enabling more systematic searches for Ly$\alpha$ emission in $z \gtrsim 7$ galaxies. However, the current sample at $z>7$ lacks the statistical power to discern between the two models of Ly$\alpha$ attenuation [e.g., @tilvi14], as the required sample size is predicted to be at least several tens [@treu12].
As expected, previous attempts in the search for Ly$\alpha$ emission at $z>7$ have revealed that spectroscopically confirming galaxies at $z>7$ via Ly$\alpha$ is challenging, yielding, in addition to two galaxies confirmed via Ly$\alpha$-break and/or dust continuum [@watson15; @oesch16], only 10 spectroscopically confirmed galaxies via Ly$\alpha$ so far (@vanzella11 [@ono12; @schenker12; @schenker14; @shibuya12; @finkelstein13; @oesch15; @roberts-borsani15; @zitrin15]; see review in @finkelstein15a), and only four at $z>7.5$, possibly due to an increased neutral fraction in the IGM. Despite these challenges, spectroscopic follow-up of galaxy candidates at these high redshifts, either yielding detections or non-detections, is valuable toward building up a statistical sample that is large enough to constrain the reionization process as well as studying in detail the physical properties of galaxies via further follow-up observations, and is thus being actively pursued.
This paper extends such previous and on-going attempts. In this study, we report Ly$\alpha$ emission from a galaxy at $z=7.66$ in the Great Observatories Origins Deep Survey South [GOODS-S; @giavalisco04] field. This is from a very deep spectroscopic follow-up campaign of $z \sim$ 7–8 galaxy candidates with the Multi-Object Spectrometer For Infra-Red Exploration [MOSFIRE; @mclean12] on the Keck I 10 m telescope, where we push the median 5$\sigma$ limiting sensitivity in line flux down to $\sim 5 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$ between sky lines. Although limited by the small number of observed galaxies, we discuss the implications of our results in the context of the evolution of the Ly$\alpha$ visibility.
This paper is organized as follows. Section \[sec:data\] describes our target selection, deep spectroscopic observations with MOSFIRE, and data reduction. Section \[sec:results\] and \[sec:sedfit\] present the results from our spectroscopy and our stellar population modeling, respectively. The implication of our observations on the Ly$\alpha$ visibility is presented in Section \[sec:simul\]. The discussion and summary follow in Section \[sec:discussion\]. Throughout the paper, we adopt a concordance $\Lambda$CDM cosmology with $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_M = 0.3$, and $\Omega_{\Lambda} = 0.7$. We use the AB magnitude system [@oke83] and a @salpeter55 initial mass function (IMF) between 0.1 [$M_{\odot}$]{} and 100 [$M_{\odot}$]{}. We refer to the [*Hubble Space Telescope*]{} ([*HST*]{}) bands F435W, F606W, F775W, F814W, F850LP, F098M, F105W, F125W, F140W, and F160W as *B$_{435}$, V$_{606}$, i$_{775}$, I$_{814}$, z$_{850}$, Y$_{098}$, Y$_{105}$, J$_{125}$, JH$_{140}$, [and]{} H$_{160}$*, respectively. All quoted uncertainties are at 68% confidence intervals.
Data {#sec:data}
====
*HST* Data and Sample Selection
-------------------------------
The targets were selected in the GOODS-S field from the parent sample from @finkelstein15b. The parent sample was selected via photometric redshifts, which were estimated with [EAZY]{} [@brammer08], using the [*HST*]{} data set from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey [CANDELS; @grogin11; @koekemoer11] which incorporates all earlier imaging data over the field as described by @koekemoer11 [@koekemoer13]. The MOSFIRE slit design was prepared using the MAGMA configurable slit unit (CSU) design tool. This tool takes as an input a list of objects, along with relative priorities. Our priority scheme was based on two quantities: the $J_{125}$-band magnitude of the source and the fraction of the source’s redshift probability distribution function (PDF; $p(z)$) of which Ly$\alpha$ would be encompassed by the MOSFIRE $Y$ band (7.0 $\lesssim z \lesssim$ 8.2). We first assigned an initial priority based on the continuum magnitude, and then prioritized galaxies within that continuum magnitude bin by the normalized redshift integral. In this way, for two galaxies with similar redshift PDFs, the higher priority would go to the brighter one, while a faint galaxy with $z_{\rm phot} \sim$ 7.5 would be prioritized over a bright galaxy with $z_{\rm phot} \sim$ 6.0. In sum, we targeted 12 (8) galaxy candidates with $z_{\rm phot}=$ 6.8–8.2 (7.0–8.2). Of these, six galaxies have more than half of their redshift PDF placing Ly$\alpha$ in the MOSFIRE $Y$ band. The rest of slits in the mask were assigned to 18 galaxy candidates at lower redshifts of $z_{\rm phot}=$ 4–6 and one relatively bright star to monitor transparency and pointing accuracy. The median rest-frame absolute UV magnitude ($M_{\rm UV}$) of our targets (assuming they are at their photometric redshifts) is $-20.4$ for the $z=$ 7–8 sample, ranging from $-21.6$ to $-20.0$. The median $H_{160}$-band magnitude is 26.5, ranging \[25.6–27.2\]. The full list of our $z=$ 7–8 sample is tabulated in Table \[tab:target\].
MOSFIRE Y-band Observation {#sec:obs}
--------------------------
Observations were taken with MOSFIRE on the Keck I telescope over 4 nights during January 11 and January 13–15, 2015. We used the $Y$-band filter, to search for Ly$\alpha$ emission at $7.0 < z < 8.2$, with a 07 slit width correpsonding to a spectral resolution of $\sim$3 Å ($R=3500$). Most of the data were taken with 180 s exposures per frame, except that for the data taken on one night (January 15; for a total of 0.9 hr integration time) 60 s exposures per frame were used. We adopted an ABBA dither pattern with an $\pm$ 125 offset along the slit for sky subtraction. The seeing measured from the star placed on a slit was in the range of 06–09, with a median/mean of 07. In total, we obtained a total on-source integration time of $\sim$10 hr (from 2.8 hr (January 11) + 3.2 hr (January 13) + 3.2 hr (January 14) + 0.9 hr (January 15)), among which $\sim$7.3 hr was obtained in good conditions. These observations are among the deepest observations ever taken for $z \gtrsim 7$ galaxies.
Data Reduction
--------------
Data reduction was performed with the public MOSFIRE data reduction pipeline (DRP; version 2015A), in which flat fielding, wavelength calibration, sky subtraction, and rectification were performed to create two-dimensional (2D) spectra with a spectral resolution of 1.09 Å pixel$^{-1}$ and a spatial resolution of 018 pixel$^{-1}$. Upon monitoring the centroid of the slit star in each raw frame, we identified a $\sim$1 pixel hr$^{-1}$ drift along the slit, which was also noted by several other studies [e.g., @kriek15; @oesch15]. We thus split the data on each night into $\sim$1 hr chunks and reduced them seperately, to prevent loss of signal due to this drift.
Following this, analysis was done using our custom software. From the 2D spectrum created by the pipeline, we combined the data from the four nights by generating final inverse-variance-weighted stacks for each object, following @gawiser06. Spatial offsets between data chunks due to the drift were accounted for when combining data based on the centroids of the slit star. We extracted one-dimensional (1D) spectra at the expected position of each source with a width of 13 (about a 1.8$\times$ the median Gaussian FWHM), using an optimal extraction algorithm described in @horne86. This extraction scheme is similar to inverse-variance weighting, but additional weight is given for each spatial pixel based on the expected spatial profile for each source (which is a Gaussian for our unresolved sources), reducing statistical noise in the extracted spectra compared to a simple boxcar extraction scheme.
[ll]{}
$F_{\rm Ly\alpha}$ (10$^{-18}$ erg s$^{-1}$ cm$^{-2}$) & $5.5 \pm 0.9$ ($\pm 1.7$)\
Signal-to-noise Ratio & 6.5\
EW$_{\rm Ly\alpha}$ (Å) & 15.6$^{+5.9}_{-3.6}$ ($\pm 4.7$)\
$z_{\rm Ly\alpha}$ & $7.6637 \pm 0.0011$\
$\sigma_{\rm blue}$ (Å) & $0.33^{+5.51}_{-0.32}$\
$\sigma_{\rm red}$ (Å) & $6.49^{+0.32}_{-4.76}$\
FWHM$_{\rm red}$ (Å) & $15.0 \pm 2.7$\
$\sigma_{\rm red}$ (km s$^{-1}$) & $180 \pm 30$\
\
log $M_*$ ([$M_{\odot}$]{}) & $9.3^{+0.5}_{-0.4}$\
UV slope $\beta$ & $-2.2^{+0.3}_{-0.2}$\
$M_{\rm UV}$ & $-21.22^{+0.06}_{-0.10}$\
$E(B-V)$ & $0.06^{+0.10}_{-0.04}$\
SFR$_{\rm UV,obs}$ ([$M_{\odot}$]{}yr$^{-1}$) & $19^{+2}_{-1}$\
SFR$_{\rm UV,corr}$ ([$M_{\odot}$]{}yr$^{-1}$) & $33^{+56}_{-9}$
Correcting for telluric absorption was done using the @kurucz93 model spectrum of the spectral type of the slit star (G5I). Absolute flux calibration was performed by comparing and scaling the spectrum to the WFC3 $Y_{105}$-band magnitude of the slit star. This procedures accounts for the slit loss, assuming our targets are point sources unresolved under the seeing FWHM of our observations, which is a good approximation given the small size of high-redshift galaxies.
To check our flux calibration, we compared our calibration array with the total MOSFIRE $Y$-band throughput curve.[^1] We also utilized two bright continuum sources which were serendipitously included in our mask, to further verify our absolute flux calibration. Taking a similar approach to that of @kriek15, we first convolved [*HST*]{}/[*Y$_{105}$*]{} images of the two sources and the slit star with a Gaussian kernel with width ${\rm FWHM}_{\rm kernel}^2={\rm FWHM}_{\rm seeing}^2-{\rm FWHM}_{\rm {\it H}_{160}}^2$, to generate the $Y$-band image under the seeing of our spectroscopic observations. Then, we calculated the fraction of light of the two sources that are within our MOSFIRE slit layout. Comparing them to the fraction of light of the star within the slit (on which our absolute flux calibration is based), we calculated the expected flux ratio between our spectroscopic data and the broadband flux (i.e., [*HST*]{}/[*Y$_{105}$*]{}) for the two sources due to the difference in the slit loss. This comparison shows that our absolute calibration (which affects our measurements of line flux and equivalent width, but not the significance of the detection) is accurate within 20%–25%. We thus conservatively add a 30% systematic uncertainty in calibration in our error budget. The systematic uncertainties are indicated in Table \[tab:3811\], while the quoted uncertainties in the rest of the paper refer to random uncertainties.
Finally, to make sure that the error spectrum initially obtained from the pipeline does not underestimate the noise level, we scaled the error spectrum such that the standard deviation of the signal-to-noise ratio (S/N) in the sky dominated region is unity. The typical scale factor was $3.0 \pm 0.1$.
Results {#sec:results}
=======
Line Detection
--------------
We visually searched for emission lines in the extracted 1D spectra as well as 2D spectra at the expected positions of our targets. We take a conservative appoach of presenting objects for which an emission line is independently detected on more than one night, minimizing the possibility of a spurious detection. In other words, we regarded it as a spurious detection if the emission was detected on only one night out of four nights. This criterion yielded only one line detection among the 30 objects originally targeted, at $\lambda_{\rm obs}= 10532.2 \pm 1.3 $ Å, and with $6.5 \sigma$ significance. The rest remained undetected ($< 3\sigma$). Figure \[fig:spec\] shows the 1D and 2D spectra of the object with emission, z7\_GSD\_3811. The emission is detected on more than one night at the same spatial and spectral location, with two negative peaks at the expected position from the adopted dithering pattern, ensuring that the line is real and not spurious.
Normally, we expect an asymmetric line profile with a sharp blue edge and gradually declining red tail for Ly$\alpha$ emission at high redshift due to absorption by neutral hydrogen in the interstellar and intergalactic medium. However, most of the proposed Ly$\alpha$ detections in other $z \gtrsim 7$ candidates have not shown highly significant evidence for asymmetry, possibly due to the low S/N for most of the detections. We find that our detected line displays an asymmetric line profile, making this object one of the first notable detections of asymmetry for a $z>7$ Ly$\alpha$ line candidate. However, the significance is not strong due to the low S/N: the Gaussian line width on the blue and red side of the line is $0.33^{+5.51}_{-0.32}$ Å and $6.49^{+0.32}_{-4.76}$ Å, respectively. Due to the vicinity of a sky line located blueward of the line, the uncertainty in the line width on the blue side of the line ($\sigma_{\rm blue}$) is large, yielding a weak constraint on the ratio between the line width on the red and blue side ($\sigma_{\rm red}/\sigma_{\rm blue}= 19.5^{+0.2}_{-19.3}$).
Assuming the line is Ly$\alpha$, the implied redshift (based on the line centroid defined as the wavelength of the peak of the Ly$\alpha$ emission) is $z({\rm Ly\alpha})= 7.6637 \pm 0.0011$,[^2] placing it as presently the third most distant spectroscopically confirmed galaxy via Ly$\alpha$ and the only galaxy at $z>7$ in the GOODS-S field with a significant Ly$\alpha$ detection. The photometric redshift, estimated with [EAZY]{} [@brammer08], is $z_{\rm phot}=7.42^{+0.20}_{-0.71}$, in good agreement with the spectrosopic redshift, as shown in the inset of Figure \[fig:sed\].
The line-of-sight velocity dispersion, derived from the Ly$\alpha$ line width on the red side of the line and corrected for instrumental resolution, is 180 $\pm$ 30 km s$^{-1}$, similar to previously spectroscopically confirmed galaxies at similar redshifts [@oesch15; @zitrin15]
We fit an asymmetric Gaussian to the line to estimate the line flux of (5.5 $\pm$ 0.9) $\times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$. We estimated the rest-frame EW of Ly$\alpha$ emission from the observed Ly$\alpha$ flux and continuum flux density of the best-fit stellar population synthesis (SPS) model in a rest-frame 100 Å box redward of the Ly$\alpha$ line (see Section \[sec:sedfit\]). The inferred rest-frame EW of Ly$\alpha$ emission is modest with $15.6^{+5.9}_{-3.6}$ Å, thus this object would not be classified as an Ly$\alpha$ emitter according to the traditional criterion of EW(Ly$\alpha$) $>$ 20 Å. This value is also below the cutoff of EW(Ly$\alpha$) $>$ 25 Å [@stark11] often adopted in the study of the evolution of the Ly$\alpha$ fraction at high redshift.
Low-z interpretations {#sec:lowz}
---------------------
We examined the possibility that the object is a foreground [\[O[ii]{}\]]{}$\lambda\lambda$3726, 3729, H$\beta$, [\[O[iii]{}\]]{}$\lambda\lambda$4959, 5007, or H$\alpha$ emitter. First, if the detected line is H$\beta$ or one of the [\[O[iii]{}\]]{} doublet, the other two lines would have been detected within our spectral coverage in regions free from sky lines. We did not find any signal at the expected wavelengths of these lines.
Practically, the strong break observed between the [*z$_{850}$*]{} and [*Y$_{105}$*]{} bands (see Figure \[fig:sed\]) rules out the possibility that the detected emission is H$\beta$, [\[O[iii]{}\]]{}, or H$\alpha$, and leaves the only alternative possibility of the detected line being the [\[O[ii]{}\]]{} doublet. If the detected emission line is an [\[O[ii]{}\]]{} doublet at $z=1.83$, the spectral resolution of MOSFIRE $Y$-band grating ($\sim$3 Å) is sufficient to resolve the doublet. The possibility of the detected line being one of the two peaks, however, cannot be entirely ruled out. If the emission is the first peak of the [\[O[ii]{}\]]{} doublet at $\lambda_{\rm rest}=3726$Å, we would have detected the second peak (at $\lambda_{\rm rest}=3729$Å) at 2–10$\sigma$ significance at wavelengths clear of sky lines. On the other hand, if the emission is the second peak of the doublet, the centroid of the first peak would be behind the sky line located blueward of the detected line. To examine these possibilities, we performed simulations in which we inserted mock lines representing either the first or second peak of the [\[O[ii]{}\]]{} doublet at the expected positions in the 2D spectrum. The spatial and spectral line profile of the mock line was assumed to be the same as that of the observed emission, and the flux was assigned based on the most unfavorable flux ratio that is physically allowed (i.e., the weakest line possible; 0.35 $<$ $f$([\[O[ii]{}\]]{}$\lambda$3729)/$f$([\[O[ii]{}\]]{}$\lambda$3726) $<$ 1.5; @pradhan06). Our simulation results indicate that due to its low flux and broad line profile, we would not be able to completely rule out the existence of the other line of the doublet based solely on our 2D spectrum. If the detected emission is indeed one of the [\[O[ii]{}\]]{} doublet, the broad line width of the detected emission (FWHM $\sim$ 400 km s$^{-1}$) is atypical for its mass ($\log(M_*/M_{\odot})= 9.1^{+0.05}_{-0.09}$), exhibiting a factor of 3 deviation from the Tully–Fisher relation [@miller12]. The line width, together with the red spectral energy distribution (SED) and lack of detection in X-rays, indicates either that *if* this line is [\[O[ii]{}\]]{} then this galaxy likely hosts an type-2 active galactic nucleus (AGN) or that the galaxy has strong outflows. The direct constraint on the abundance of such population is not feasible currently at this redshift and in low-mass regime.
As discussed above, it is unlikely that the detected line is an *unresolved* [\[O[ii]{}\]]{} doublet given the spectral resolution. However, since the detected line has a moderate S/N of 6.5$\sigma$, we conservatively leave this possibility open but further suggest evidence against it in Section \[sec:serendi\] and \[sec:sedfit\].
Serendipitious Line Detections at $z \sim$ 1–2 {#sec:serendi}
----------------------------------------------
In addition to the detected emission in z7\_GSD\_3811 from our targets, we identified two other emission lines in objects which serendipitiously fell in slits.
The first object (R.A. = 3:32:43.22, decl. = $-$27:47:12.9 (J2000)) shows an emission line with two peaks. Assuming that the detected line is an [\[O[ii]{}\]]{} doublet, we derived its redshift to be $z=1.94$. Its photometric redshift, $z_{\rm phot}=1.87^{+0.07}_{-0.08}$ [@dahlen13], is in excellent agreement with the inferred [\[O[ii]{}\]]{} redshift, thus we conclude that the detected line is the [\[O[ii]{}\]]{} doublet. This [\[O[ii]{}\]]{} doublet strengthens the possibility that the detected emission in z7\_GSD\_3811 is Ly$\alpha$ and not an *unresolved* [\[O[ii]{}\]]{} doublet. The left panel of Figure \[fig:serendi\] shows that the doublet in this object is spectrally well-resolved both in the 1D and 2D spectra, yet the observed wavelength and S/N are similar to those of z7\_GSD\_3811.
The second object (R.A. = 3:32:50.48, decl. = $-$27:46:56.0 (J2000)) shows a prominent emission at $\lambda_{\rm obs}=$ 10398 Å, which we identified as an [\[O[iii]{}\]]{}$\lambda$5007 line (right panel of Figure \[fig:serendi\]). The other line of the doublet ([\[O[iii]{}\]]{}$\lambda$4959) is behind a sky line but still visible, and H$\beta$ is detected at 5.6$\sigma$. Upon close inspection, we noted that the emission has an offset of 4–5 pixels along the spatial axis from our original target, which corresponds to 07–09. We identified a galaxy in proximity of our original target at this distance, thus we concluded that the emission is not from our target but from a foreground galaxy at $z=1.08$.
Stellar Population Modeling and Stacking Analysis {#sec:sedfit}
=================================================
We performed a SED fitting analysis to the observed [*HST*]{}/Advanced Camera for Surveys (ACS; *B$_{435}$, V$_{606}$, i$_{775}$, I$_{814}$, z$_{850}$*), [*HST*]{}/WFC3 (*Y$_{105}$, J$_{125}$, H$_{160}$*), and VLT/Hawk-I $K$-band photometry of z7\_GSD\_3811, using the @bruzual03 SPS models. Details on our modeling are described in @song15. In addition to the [*HST*]{} bands originally included in the SED fitting in @song15, in this work we included the $K$-band photometry from the Hawk-I UDS and GOODS Survey [@fontana14] in the official CANDELS GOODS-S catalog (version 1.1). The [*Spitzer*]{}/IRAC photometry was excluded from the modeling, because z7\_GSD\_3811 is unfortunately heavily contaminated by a nearby bright source in IRAC. Thus, we do not have constraints on whether this galaxy exhibits the 4.5 $\mu$m color excess due to the strong [\[O[iii]{}\]]{} line falling in the 4.5 $\mu$m band that some other studies have reported for spectroscopically confirmed $z \sim $ 7–8 galaxies [@finkelstein13; @oesch15; @roberts-borsani15; @zitrin15].
As discussed in Section \[sec:lowz\], the only alternative interpretation of the detected emission in z7\_GSD\_3811 is the [\[O[ii]{}\]]{} doublet. Thus, we performed the SED fitting two times with a fixed redshift, first assuming the emission is Ly$\alpha$, and then, assuming the emission is an [\[O[ii]{}\]]{} doublet at $z$([\[O[ii]{}\]]{}) = 1.83.
Figure \[fig:sed\] shows the model fit and stamp images. The results of our SED fitting analysis show that the high-$z$ solution is preferred over the low-$z$ solution, albeit mildly. For the high-z interpretation, because we did not fit bands shortwards of the Ly$\alpha$ line due to the large uncertainty in modeling the IGM attenuation, and because the source is highly contaminated by a nearby bright source in IRAC channels, only four bands (*Y$_{105}$, J$_{125}$, H$_{160}$*, and $K$) were used to constrain the fit, which can be perfectly matched by SPS models with a certain combination of free parameters and nebular emission strengths, yielding $\chi^2_r \sim 0$.
For the low-$z$ interpretation, the non-detection in the deep optical bands[^3] and the strong break between *z$_{850}$* and *Y$_{105}$* of $\sim$1.8 magnitude yield the only possible solution to be a dusty low-mass ($\log(M_*/M_{\odot})= 9.1 \pm 0.1$) starburst galaxy with specific star formation rate (sSFR) of log(sSFR yr$^{-1}$)$= -7.1 \pm 0.2$. While the reduced chi-square of $\chi^2_r \sim 1.6$ for the low-$z$ solution indicates that it is still regarded a “good” fit, this low-$z$ solution is disfavored by non-detections in deep optical bands: as another measure of goodness-of-fit, we compared the distribution of normalized residuals to the standard normal distribution with ($\mu, \sigma$) = (0, 1). The comparison quantified using a Kolmogorov–Smirnov test [@kolmogorov33; @smirnov48] indicates that the likelihood that the normalized residuals come from the normal distribution is less than 20%, implying that the low-$z$ solution is not a preferred model for this galaxy.
To further probe the existence of any low level flux below the detection threshold of individual optical bands, we created a stack of [*V$_{606}$*]{}-, [*i$_{775}$*]{}-, [*I$_{814}$*]{}-, and [*z$_{850}$*]{}-band images. Prior to the stacking, the spatial resolution of the images were matched to that of the [*H$_{160}$*]{} band and the units were converted to a physical unit. Then, the stack and stack rms map were generated by inverse-variance weighting, on which the stack flux and flux error were measured within a 04 diameter aperture using the Source Extractor package [@bertin96] and aperture-corrected using the ratio between the flux within a 04 aperture and total flux measured in the [*H$_{160}$*]{} band. We quantified the background noise as the Gaussian width of the flux distribution measured from 10$^4$ randomly placed apertures of the same size used in our original photometry in source-free regions of the stacked image. We checked that the flux error (3.8 nJy) measured from Source Extractor is slightly larger than the background noise (2.6 nJy), thus conservatively took the larger one. The stamp image and 1$\sigma$ upper limit for the flux of the stack are shown in Figure \[fig:sed\]. The stacking yielded no identifiable emission at the position of the source. The measured stack flux is $6 \sigma$ lower than the prediction from the low-$z$ solution, further indicating the preference for the high-$z$ interpretation of the source.
We conclude that the detected line is Ly$\alpha$. z7\_GSD\_3811 is a galaxy bright in the UV with the rest-frame UV absolute magnitude of $M_{\rm UV} \sim -21.2$, about two times brighter in luminosity than the characteristic UV magnitude of the rest-frame UV luminosity function at $z=8$ of $M_{{\rm UV}, z=8}^*=-20.48$ [@finkelstein15a]. Other physical properties inferred from our SED fitting analysis indicate that z7\_GSD\_3811 is a typical galaxy at $z=$ 7–8 [*for its UV magnitude*]{}, with a moderately blue UV slope ($\beta= -2.2^{+0.3}_{-0.2}$), dust-corrected UV-based star formation rate (SFR) of $33^{+56}_{-9}$ [$M_{\odot}$]{} yr$^{-1}$, and stellar mass of $\log(M_*$/[$M_{\odot}$]{})$=9.3^{+0.5}_{-0.4}$. This galaxy was noted as a promising $z\gtrsim7$ candidate by several other previous [*HST*]{} imaging studies as well [@bouwens10; @mclure13]. Table \[tab:3811\] summarizes the physical properties of z7\_GSD\_3811.
L$\alpha$ Visibility {#sec:simul}
====================
Even with our deep integration of 10 hr, we detected only one Ly$\alpha$ emission line with a moderate rest-frame Ly$\alpha$ EW of 16 Å. To put this result in context, we computed the number of detections expected from our observations, with the aim of placing constraints on the evolution of the Ly$\alpha$ visibility with redshift.
First, we quantified the limiting sensitivity of our observations by simulating Ly$\alpha$ lines in our MOSFIRE spectra. We modeled the Ly$\alpha$ line as an asymmetric Gaussian, similar to the detected line in z7\_GSD\_3811. Then, we inserted the lines with varying line fluxes into each of the actual 1D spectra in our mask at varying positions between the MOSFIRE $Y$-band wavelength coverage (9800–11200 Å), to find the line flux as a function of wavelength that ensures an $X$-$\sigma$ detection ($X \ge 3$). The upper left panel of Figure \[fig:simul\] presents the results, showing that our deep spectroscopy reaches a median 5$\sigma$ limiting sensitivity in line flux of $\sim$5 $\times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$ between sky lines. Scaling our limiting sensitivity by $\sqrt{t}$, where $t$ is the integration time, we find it consistent with the quoted limits of other MOSFIRE $Y$-band observations reported by @wirth15.
For each object in our mask, we computed the $X$-$\sigma$ limit in the rest-frame EW(Ly$\alpha$) as a function of redshift (i.e., observed wavelength) for our observations. This was done via 1000 Monte Carlo realizations of the photometry for each object, for which we performed SED fitting. In each realization, the redshift was randomly drawn from the $p(z)$ distribution (thus the contaminant fraction, which is given by our $p(z)$, is accounted for in our results), and the corresponding continuum flux density redward of the Ly$\alpha$ was calculated from the best-fit SPS model. The ratio of the limiting sensitivity, for which we take the median value at each wavelength as all the targets were observed in the same conditions in one MOSFIRE mask, to the continuum flux density in each realization gives the rest-frame $X$-$\sigma$ EW limit as a function of redshift (bottom left panel of Figure \[fig:simul\]).
By assuming an intrinsic rest-frame EW distribution for Ly$\alpha$ before being processed by the neutral gas in the IGM, we can compute how many sources are expected to be detected above our $X$-$\sigma$ EW limit. For the intrinsic rest-frame EW distribution for Ly$\alpha$, $p$(EW$_{\rm intrinsic})$, we adopted a log-normal form given by @schenker14, which is based on the compilation of observations at $3<z<6$ when the universe is ionized. Then, $p$(EW$_{\rm intrinsic})$ and our $X$-$\sigma$ EW limit inferred from our fake source simulation at the corresponding wavelength is compared, to estimate the probability that the line is detected. Here, we assumed that the $p$(EW$_{\rm intrinsic})$ does not evolve as a function of redshift from $3<z<6$ to $z=$ 7–8. Our analysis takes into account the effect of a sensitive wavelength dependancy due to sky lines and the incomplete spectral coverage of the redshift probability distribution ($p(z)$), and is properly weighted by $p(z)$. The resulting probability distribution of the expected number of detections from our observations is shown in the right panel of Figure \[fig:simul\]. Depending on the detection threshold adopted, our results show a 1–2$\sigma$ deviation from the null hypothesis of no evolution. For example, based on the Ly$\alpha$ EW distribution at lower redshift of $z \sim$ 3–6 (assuming no evolution with redshift), we expect to detect $1.7^{+0.6}_{-0.5}$ ($2.4^{+0.8}_{-0.8}$) objects with $> 5\sigma$ ($3\sigma$) significance, for which our observations weakly reject at the 1.3$\sigma$ (2$\sigma$) confidence level. Our results are conservative in the sense that had we assumed a zero low-$z$ interloper fraction or used an extrapolation of the EW(Ly$\alpha$) distribution from lower redshifts to $z\sim$ 7–8, the inferred deviation from the expectation (and thus the implied decline in the Ly$\alpha$ fraction) would be higher.[^4]
Discussion and Summary {#sec:discussion}
======================
We have presented results from deep near-infrared $Y$-band spectroscopy targeting 12 galaxy candidates with $z_{\rm phot}=$ 7–8 in the GOODS-S field. Our long integration of $\sim$10 hr with Keck/MOSFIRE enabled us to probe the Ly$\alpha$ emission down to a median 5$\sigma$ rest-frame EW(Ly$\alpha$) limit of 28 Å (ranging \[12–55\] Å; listed in Table \[tab:target\]). Despite our deep spectroscopy, out of our 30 targets, we identified only one emission line at 6.5$\sigma$ significance.
We claim that the detection is real, given that $i$) it was detected independently on more than one night, $ii$) at the expected spatial location, and $iii$) with two negative peaks at the positions expected from our dithering pattern.
This line is likely Ly$\alpha$ emission from a galaxy at $z=7.6637$, based on $i$) its asymmetric line profile characteristic of Ly$\alpha$ at high redshift, $ii$) the non-detection in the optical bands as well as an optical stack ([*V$_{606}$*]{}+[*i$_{775}$*]{}+[*I$_{814}$*]{}+[*z$_{850}$*]{} bands), and $iii$) the inferred redshift in good agreement with its photometric redshift. While we cannot completely rule out the possibility that the detected line is an unresolved [\[O[ii]{}\]]{} doublet from a galaxy at $z=1.83$, we find that it is unlikely, as a serendipitious [\[O[ii]{}\]]{} emitter at $z \sim 1.9$ that falls in one of the slits, with the redshift difference of only $\Delta z \sim 0.1$ and with a similar S/N to that of z7\_GSD\_3811, shows clearly resolved double peaks both in our final stack and on individual nights. However, although rare, it is still feasible that the detected line is one of the two peaks of a broad [\[O[ii]{}\]]{} doublet indicative of an AGN or strong outflows.
The detected Ly$\alpha$ line has a modest rest-frame EW of 16 Å and a line flux of $(5.5\pm0.9) \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$. This galaxy is bright in the UV ($M_{\rm UV}= -21.2$; $\sim 2L^*_{z=8}$), and is a typical for its UV brightness in terms of UV slope ($\beta=-2.2$) and stellar mass ($\log(M_*/M_{\odot})=9.3$).
Identifying its nature via follow-up observations would be challenging but not impossible. Assuming this galaxy is an [\[O[ii]{}\]]{} emitter at $z=1.83$, other strong rest-frame optical emission lines (H$\beta$, [\[O[iii]{}\]]{}, and H$\alpha$) all fall in between ground-based near-infrared bands, thus deep spaced-based grism may be the only possibility to detect those lines before the advent of the [*James Webb Space Telescope*]{}. If this galaxy is indeed at $z=7.6637$ (with a normal stellar population), other emission features (e.g., [C[iii]{}\]]{}$\lambda\lambda$1907, 1909) would be too weak to be detected in currently available data sets (e.g., [*HST*]{} grism) based on the typical flux ratio, unless Ly$\alpha$ is attenuated more than a factor of 15 by the IGM. However, this is unlikely given the Ly$\alpha$ EW distribution found by @stark11 and @schenker14 for its UV luminosity in galaxies at $3<z<6$. Additional integration in $Y$-band (for Ly$\alpha$) or deep $H$-band observations (for [C[iii]{}\]]{}) can help verifying its identity. Alternatively, the Atacama Large Millimeter Array (ALMA) provides an opportunity to detect the [\[C[ii]{}\]]{} line at 158 $\mu$m with less than an hour of integration, assuming that the empirical relation between SFR and [\[C[ii]{}\]]{} 158 $\mu$m luminosity found for normal star-forming galaxies at high redshift [@capak15] holds.
The rest of the targeted galaxies remain undetected, showing a $1.3\sigma$ ($2\sigma$) deviation from the expected number of detections (with $>$5$\sigma$ ($>$3$\sigma$) significance) when assuming no evolution in the Ly$\alpha$ EW distribution from lower redshifts of $3<z<6$ to $z=$ 7–8. Our observations thus support the decline in the EW of Ly$\alpha$ at $z>6.5$ of earlier studies [e.g., @schenker14; @tilvi14; @pentericci14], which may be due to the increase of neutral gas in the IGM. However, the evidence from our observations alone is not conclusive due to the large statistical uncertainties. The addition of our sample to the compilation of previous data would be only incremental, thus we defer a detailed analysis on the evolution of the IGM neutrality to future studies with a larger statistical sample.
However, our results from very deep spectroscopy have implications for future observations. Recently, several studies [@roberts-borsani15; @zitrin15] have claimed a high Ly$\alpha$ visibility in bright galaxies at $z>7.5$ in the EGS field, which were selected based on red IRAC \[3.6\]$-$\[4.5\] colors indicative of strong [\[O[iii]{}\]]{} emission. Combined with the recent discovery of Lyman continuum leakers among strong [\[O[iii]{}\]]{} emitters at low redshifts [@izotov16; @vanzella16] and the lack of significant Ly$\alpha$ detections in the GOODS-S field at comparable redshifts (before this study), this may signal the inhomogeneity of the reionization process on large scales. Indeed, while LAEs at lower redshifts of $3<z<6$ show that faint LAEs on average have a larger Ly$\alpha$ EW than bright ones [@stark11], most spectrosopic campaigns at higher redshift targeting $z>7$ galaxy candidates have only succeeded in detecting Ly$\alpha$ emission in bright galaxies [@finkelstein13; @oesch15; @roberts-borsani15; @zitrin15]. Our results are in line with these studies, yielding one Ly$\alpha$ detection from a bright ($L \sim 2L^*$) galaxy. However, it is noteworthy that our sole detection in z7\_GSD\_3811 is among those with the lowest EW limit (cyan circles in the bottom left panel of Figure \[fig:simul\]). This indicates that current spectroscopic campaigns at $z>7$ are only reaching a sufficient depth for the brightest galaxies, leaving the possibility of detecting several galaxies in Ly$\alpha$ emission with modest Ly$\alpha$ EW in fainter galaxies open with deeper spectroscopy. Extremely deep spectroscopy (either by performing long integrations on blank fields or by utilizing magnification due to gravitational lensing) to better quantify the Ly$\alpha$ EW distribution, along with quantifying large-scale spatial fluctuation in the reionization process from spatial clustering of Ly$\alpha$ emission from wide area surveys, will remain as a valuable probe of reionization in the near future.
We would like to thank the anonymous referee for valuable suggestions which improved this paper, and N. Scoville, L. Murchikova, S. Manohar, and B. Darvish for their work during observing runs. M.S. acknowledges support from the NSF AAG award AST-1518183. M.S. and S.L.F. acknowledge support from the NASA Astrophysics and Data Analysis Program award \#NNX15AM02G issued by JPL/Caltech, as well as a NASA Keck PI Data Award, administered by the NASA Exoplanet Science Institute. This research is based on observations made with the Keck Telescope. The Observatory was made possible by the financial support of the W. M. Keck Foundation. We recognize and acknowledge the cultural role and reverence that the summit of Mauna Kea has within the indigenous Hawaiian community.
[*Facilities:*]{} , ,
Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393 Bolton, J. S., & Haehnelt, M. G. 2013, , 429, 1695 Bouwens, R. J., Illingworth, G. D., Gonz[á]{}lez, V., et al. 2010, , 725, 1587 Brammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, , 686, 1503 Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000 Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682 Calzetti, D., Kinney, A. L., & Storchi-Bergmann, T. 1994, , 429, 582 Capak, P. L., Carilli, C., Jones, G., et al. 2015, , 522, 455 Dahlen, T., Mobasher, B., Faber, S. M., et al. 2013, , 775, 93 Dijkstra, M., Wyithe, S., Haiman, Z., Mesinger, A., & Pentericci, L. 2014, , 440, 3309 Dijkstra, M., Mesinger, A., & Wyithe, J. S. B. 2011, , 414, 2139 Fan, X., Strauss, M. A., Becker, R. H., et al. 2006, , 132, 117 Finkelstein, S. L. 2015, arXiv:1511.05558 Finkelstein, S. L., Papovich, C., Dickinson, M., et al. 2013, , 502, 524 Finkelstein, S. L., Papovich, C., Salmon, B., et al. 2012, , 756, 164 Finkelstein, S. L., Ryan, R. E., Jr., Papovich, C., et al. 2015, , 810, 71 Fontana, A., Dunlop, J. S., Paris, D., et al. 2014, , 570, A11 Fontana, A., Vanzella, E., Pentericci, L., et al. 2010, ApJL, 725, L205 Gawiser, E., van Dokkum, P. G., Herrera, D., et al. 2006, , 162, 1 Giavalisco, M., Ferguson, H. C., Koekemoer, A. M., et al. 2004, ApJL, 600, L93 Grogin, N. A., Kocevski, D. D., Faber, S. M., et al. 2011, ApJS, 197, 35 Gunn, J. E., & Peterson, B. A. 1965, , 142, 1633 Horne, K. 1986, , 98, 609 Izotov, Y. I., Orlitov[á]{}, I., Schaerer, D., et al. 2016, , 529, 178 Kennicutt, Jr., R. C. 1998, ARA&A, 36, 189 Koekemoer, A. M., Anton, M., Faber, S. M., et al. 2011, ApJS, 197, 36 Koekemoer, A. M., Ellis, R. S., McLure, R. J., et al. 2013, , 209, 3 Kolmogorov, A. 1933, G. Inst. Ital. Attuari, 4, 83 Kriek, M., Shapley, A. E., Reddy, N. A., et al. 2015, , 218, 15 Kurucz, R. 1993, ATLAS9 Stellar Atmosphere Programs and 2 km/s Grid. Kurucz CD-ROM No. 13. (Cambridge, MA: Smithsonian Astrophysical Observatory), 13 McLean, I. S., Steidel, C. C., Epps, H. W., et al. 2012, , 8446, 84460J McLure, R. J., Dunlop, J. S., Bowler, R. A. A., et al. 2013, , 432, 2696 Mesinger, A., Aykutalp, A., Vanzella, E., et al. 2015, , 446, 566 Miller, S. H., Ellis, R. S., Sullivan, M., et al. 2012, , 753, 74 Oesch, P. A., Brammer, G., van Dokkum, P. G., et al. 2016, , 819, 129 Oesch, P. A., van Dokkum, P. G., Illingworth, G. D., et al. 2015, ApJL, 804, L30 Oke, J. B., & Gunn, J. E. 1983, ApJ, 266, 713 Ono, Y., Ouchi, M., Mobasher, B., et al. 2012, , 744, 83 Pentericci, L., Fontana, A., Vanzella, E., et al. 2011, , 743, 132 Pentericci, L., Vanzella, E., Fontana, A., et al. 2014, , 793, 113 Pradhan, A. K., Montenegro, M., Nahar, S. N., & Eissner, W. 2006, , 366, L6 Roberts-Borsani, G. W., Bouwens, R. J., Oesch, P. A., et al. 2015, arXiv:1506.00854 Salpeter, E. E. 1955, ApJ, 121, 161 Schenker, M. A., Ellis, R. S., Konidaris, N. P., & Stark, D. P. 2014, , 795, 20 Schenker, M. A., Stark, D. P., Ellis, R. S., et al. 2012, , 744, 179 Shibuya, T., Kashikawa, N., Ota, K., et al. 2012, , 752, 114 Smirnov, N. 1948, The Annals of Mathematical Statistics, 19, 279 Song, M., Finkelstein, S. L., Ashby, M. L. N., et al. 2016, , in press (arXiv:1507.05636) Song, M., Finkelstein, S. L., Gebhardt, K., et al. 2014, , 791, 3 Stark, D. P., Ellis, R. S., Chiu, K., Ouchi, M., & Bunker, A. 2010, , 408, 1628 Stark, D. P., Ellis, R. S., & Ouchi, M. 2011, ApJL, 728, L2 Steidel, C. C., Erb, D. K., Shapley, A. E., et al. 2010, , 717, 289 Tilvi, V., Papovich, C., Finkelstein, S. L., et al. 2014, , 794, 5 Treu, T., Trenti, M., Stiavelli, M., Auger, M. W., & Bradley, L. D. 2012, , 747, 27 Vanzella, E., de Barros, S., Vasei, K., et al. 2016, arXiv:1602.00688 Vanzella, E., Pentericci, L., Fontana, A., et al. 2011, ApJL, 730, L35 Watson, D., Christensen, L., Knudsen, K. K., et al. 2015, , 519, 327 Wirth, G. D., Trump, J. R., Barro, G., et al. 2015, , 150, 153 Zitrin, A., Labb[é]{}, I., Belli, S., et al. 2015, ApJL, 810, L12
[^1]: http://www2.keck.hawaii.edu/inst/mosfire/throughput.html
[^2]: Due to the IGM absorption and Ly$\alpha$ kinematics, the systemic redshift is likely to be slightly lower than the inferred redshift from the Ly$\alpha$ line. The systemic redshift (not corrected for IGM absorption) would be $\sim$0.01 lower than the inferred redshift for the average velocity offsets of 200–400 km s$^{-1}$ found in Ly$\alpha$ emitters and LBGs at lower redshift of $z \sim$ 2–3 [e.g., @song14; @steidel10].
[^3]: Formally, our elliptical aperture photometry yields a 2.3$\sigma$ detection in $I_{\rm 814}$ band. However, the $I_{814}$-band stamp image shows that all identifiable emissions are off-center and do not line up with near-infrared emission, indicating that they are likely background noise or from another unresolved faint source. Using a smaller, circular 04 diameter aperture centered on the near-infrared emission, we find no detection ($< 1\sigma$) in any optical band.
[^4]: For reference, in a more traditional framework developed by @stark10 of “Ly$\alpha$ fraction”, z7\_GSD\_3811 is not regarded as a Ly$\alpha$-emitting galaxy, as the rest-frame EW is below the cutoff of 25 or 55 Å. Thus, the inferred Ly$\alpha$ fraction from our observation at $z \sim 7.5$ with EW $>$ 25 or 55 Å is $X_{\rm Ly\alpha}
< 0.37$ for the UV-bright galaxies ($-21.75 < M_{\rm UV} < -20.25$) and $X_{\rm Ly\alpha} < 0.61$ for UV-faint galaxies ($-20.25 < M_{\rm UV} < -18.75$; $1\sigma$).
|
---
abstract: 'The effect of translationese has been studied in the field of machine translation (MT), mostly with respect to training data. We study in depth the effect of translationese on test data, using the test sets from the last three editions of WMT’s news shared task, containing 17 translation directions. We show evidence that (i) the use of translationese in test sets results in inflated human evaluation scores for MT systems; (ii) in some cases system rankings do change and (iii) the impact translationese has on a translation direction is inversely correlated to the translation quality attainable by state-of-the-art MT systems for that direction.'
author:
- |
Mike Zhang\
Information Science Programme\
University of Groningen\
The Netherlands\
`j.j.zhang.1@student.rug.nl`\
Antonio Toral\
Center for Language and Cognition\
University of Groningen\
The Netherlands\
`a.toral.ruiz@rug.nl`
bibliography:
- 'acl2019.bib'
title: The Effect of Translationese in Machine Translation Test Sets
---
Introduction
============
Translated texts in a human language exhibit unique characteristics that set them apart from texts originally written in that language. It is common then to refer to translated texts with the term [*translationese*]{}. The characteristics of translationese can be grouped along the so-called universal features of translation or translation universals [@baker1993corpus], namely simplification, normalisation and explicitation. In addition to these three, interference is recognised as a fundamental law of translation [@toury2012descriptive]: “phenomena pertaining to the make-up of the source text tend to be transferred to the target text". In a nutshell, compared to original texts, translations tend to be simpler, more standardised, and more explicit and they retain some characteristics that pertain to the source language.
The effect of translationese has been studied in machine translation (MT), mainly with respect to the training data, during the last decade. Previous work has found that an MT system performs better when trained on parallel data whose source side is original and whose target side is translationese, rather than the opposite [@kurokawa2009automatic; @lembersky2013effect].
A recent paper has studied the effect of translationese on test sets [@toral2018attaining], in the context of assessing the claim of human parity made on Chinese-to-English WMT’s 2017 test set [@hassan2018achieving]. The source side of this test set, as it is common in WMT [@bojar2016findings; @bojar2017findings; @bojar2018findings], was half original and half translationese. It was found out that the translationese part was artificially easier to translate, which resulted in inflated scores for MT systems.
Noting that this finding was based on one test set for a single translation direction, we explore this topic in more depth, studying the effect of translationese in all the language pairs of the news shared task of WMT 2016 to 2018. Our research questions (RQs) are the following:
- RQ1. Does the use of translationese in the source side of MT test sets unfairly favour MT systems in general or is this just an artifact of the Chinese-to-English test set from WMT 2017?
- RQ2. If the answer to RQ1 is yes, does this effect of translationese have an impact on WMT’s system rankings? In other words, would removing the part of the test set whose source side is translationese result in any change in the rankings?
- RQ3. If the answer to RQ1 is yes, would some language pairs be more affected than others? E.g. based on the level of the relatedness between the two languages involved.
The remainder of the paper will be organized as follows. provides an overview of previous work about the effect of translationese in MT. Next, describes the data sets used in our research. This is followed by , and , where we conduct the experiments for RQ1, RQ2 and RQ3, respectively. Finally, outlines our conclusions and lines of future work.
Related Work {#s:related_work}
============
There is previous research in the field of MT that has looked at the impact of translationese, mostly on training data, but there are works that have focused also on tuning and testing data sets.
The pioneering work on this topic by @kurokawa2009automatic showed that French-to-English statistical MT systems trained on human translations from French to English (original source and translationese target, henceforth referred to as O$\rightarrow$T) outperformed systems trained on human translations in the opposite direction (i.e. translationese source and original target, henceforth referred to as T$\rightarrow$O). These findings were corroborated by @lembersky2013effect, who also adapted phrase tables to translationese, which resulted in further improvements. @lembersky2012language focused on the monolingual data used to train the language model of a statistical MT system and found that using translated texts led to better translation quality than relying on original texts.
@stymne2017effect investigated the effect of translationese on tuning for statistical MT, using data from the WMT 2008–2013 [@bojar-EtAl:2013:WMT] for three language pairs. The results using O$\rightarrow$T and T$\rightarrow$O tuning texts were compared; the former led to a better length ratio and a better translation, in terms of automatic evaluation metrics. Finally, @toral2018attaining investigated the effect of translationese on the Chinese$\rightarrow$English (ZH$\rightarrow$EN) test set from WMT’s 2017 news shared task. They hypothesized that the sentences originally written in EN are easier to translate than those originally written in ZH, due to the simplification principle of translationese, namely that translated sentences tend to be simpler than their original counterparts [@laviosa1998universals]. Two additional universal principles of translation, explicitation and normalisation, would also indicate that a ZH text originally written in EN would be easier to translate. In fact, they looked at a human translation and the translation by an MT system [@hassan2018achieving] and observed that the human translation outperforms the MT system when the input text is written in the original language (ZH), but the difference between the two is not significant when the original language is translationese (ZH input originally written EN). Therefore, they concluded that the use of translationese as the source language in test sets distorts the results in favour of MT systems.
Data Sets {#s:data}
=========
We use the test data from WMT16, WMT17, and WMT18 news translation tasks (*newstest2016, newstest2017, and newstest2018*) exclusively, because they provide results using the *direct assessment* (DA) score [@graham2013continuous; @graham2014machine; @graham2017can], which is the metric we will use in our experiments. DA is a crowd-sourced human evaluation metric to determine MT quality. To elaborate, after participants submit their translations produced by their MT systems, a human evaluation campaign is run. This is to assess the translation quality of the systems, and to rank them accordingly. Human evaluation scores are provided via crowdsourcing and/or by participants, using Appraise [@federmann2012appraise]. Human assessors are asked to rate a given candidate translation by how adequately it expresses the meaning of the corresponding reference translation, thus avoiding the use of the source texts and therefore not requiring bilingual speakers. The rating is done on an analogue scale, which corresponds to an absolute 0-100 scale.
To prevent differences in scoring strategies of distinct human assessors, the human assessment scores for translations are standardized according to each individual human assessor’s overall mean and standard deviation score, which is indicated as the $z$-score in WMT finding papers. Average standardized scores for individual segments belonging to a given system are then computed, before the final overall DA score for that system is computed as the average of its standardized segment scores.
Finally, systems are ranked to produce the shared task results. There is of course the possibility that some systems score similarly in the shared task. If that is the case, those systems are clustered together. Specifically, clusters are determined by grouping systems together, and comparing the scores they obtained. According to the Wilcoxon rank-sum test, if systems do not significantly outperform others, they are in the same cluster, the opposite is the case if they do outperform each other [@bojar2016findings; @bojar2017findings; @bojar2018findings]. provides an overview of the number of systems, segments, and assessments in the previously mentioned editions of WMT for all available language directions. These are the datasets that we use in this work.
Effect of Translationese on Direct Assessment Scores {#s:rq1}
====================================================
The test sets used by @bojar2016findings [@bojar2017findings; @bojar2018findings] are bilingual, thus having two sides: source text and reference translation. The source is written in the language that is to be translated from (original language), while the reference is written in the language into which the source text is to be translated (target language). In all the test sets used in our experiments English is one of the two languages involved, being either the source or the target.
Taking as an example of WMT test set the one for Chinese-to-English from 2017, this contains 2,001 sentence pairs. Out of these, 1,000 sentences were originally written in Chinese and translated by a human translator into English, hence the target text is translationese. The other half consists of 1,001 sentences that were originally written in English and translated by a human translator into Chinese, hence the source text is translationese in this subset. A graphical depiction of this can be found in Figure \[fig:dist\]. The advantage of this procedure is that the same test set can be used for the English-to-Chinese direction, thus reducing the costs involved in creating test sets in half.
Language Direction
------------------------------ ------ ---------- ---------- ------ ----------- ----------- ------ ---------- ----------
WMT ORG TRS WMT ORG TRS WMT ORG TRS
Chinese$\rightarrow$English 73.2 [-1.5]{} [+3.9]{} 78.8 [-1.3]{} [+2.0]{}
English$\rightarrow$Chinese 73.2 [-4.1]{} [+5.0]{} 80.7 [-4.0]{} [+2.3]{}
Czech$\rightarrow$English 75.4 [-5.8]{} [+5.7]{} 74.6 [-4.3]{} [+4.2]{} 71.8 [-1.6]{} [+1.6]{}
English$\rightarrow$Czech 62.0 [-5.8]{} [+7.4]{} 67.2 [-6.6]{} [+7.2]{}
Estonian$\rightarrow$English 73.3 [-4.0]{} [+4.0]{}
English$\rightarrow$Estonian 64.9 [-4.1]{} [+3.9]{}
Finnish$\rightarrow$English 66.9 [-3.2]{} [+3.0]{} 73.8 [-2.1]{} [+2.2]{} 75.2 [-2.4]{} [+2.3]{}
English$\rightarrow$Finnish 59.6 [-5.1]{} [+5.6]{} 64.7 [-7.7]{} [+8.0]{}
German$\rightarrow$English 75.8 [-4.1]{} [+4.1]{} 78.2 [-2.4]{} [+2.2]{} 79.9 [-3.8]{} [+4.3]{}
English$\rightarrow$German 72.9 [-5.1]{} [+4.4]{} 85.5 [-1.9]{} [+1.9]{}
Latvian$\rightarrow$English 76.2 [-0.4]{} [+0.6]{}
English$\rightarrow$Latvian 54.4 [-11.2]{} [+11.7]{}
Romanian$\rightarrow$English 73.9 [-0.4]{} [+0.5]{}
Russian$\rightarrow$English 74.2 [-1.2]{} [+1.8]{} 82.0 [-0.7]{} [+0.6]{} 81.0 [-0.1]{} 0.0
English$\rightarrow$Russian 75.4 [-5.8]{} [+5.8]{} 72.0 [-7.4]{} [+7.4]{}
Turkish$\rightarrow$English 57.1 [-1.6]{} [+1.6]{} 68.8 [-3.8]{} [+3.9]{} 74.3 [-3.2]{} [+3.9]{}
English$\rightarrow$Turkish 53.4 [-13.4]{} [+11.8]{} 66.3 [-4.1]{} [+5.5]{}
Source and reference files contain documents, each of which is provided with a label indicating in which language it was originally written. In our experiments we compute the DA scores for each test set (i) on the whole test set, which corresponds to the results reported in WMT, (ii) on the subset for which the source text was originally written in the source language (referred to as ORG in our experiments) and (iii) on the remaining subset, for which the source text was originally written in the target language, and is thus translationese (referred to as TRS in our experiments).
shows the absolute difference in DA score for the ORG and TRS subsets, taking the whole test set (WMT) as starting point for the comparison. We observe a clear and common trend: using original input results in a lower DA score, while using translationese input increases the DA score. This trend is consistent for all the 17 translation directions considered and for all the 3 years of WMT studied, thus providing enough evidence to answer RQ1: the use of translationese as input of test sets results in higher DA scores for MT systems.
Effect of Translationese on Rankings {#s:rq2}
====================================
We compute Kendall’s $\tau$ to give an overview of to what degree rankings change for each translation direction. The $\tau$ coefficient is obtained by comparing WMT rankings to the resulting rankings if only the ORG subset is used as input. Since systems can share the same cluster, and thus the same ranking, we compute Kendall’s $\tau$ both with and without ties. With ties, all systems in the same cluster are considered to occupy the same rank, hence the correlation with ties is sensitive only to changes that go beyond clusters. E.g. if a system moves from the second cluster to the first one. In contrast, without ties all the ranking changes are considered, even if a system changes position but remains within the same cluster. shows the Kendall’s $\tau$ correlations for all translation directions between the rankings on the whole test set (WMT) and on the ORG subset. We do see that some of the translation directions have a $\tau$ coefficient of 1, which means that the agreement between the two rankings is perfect, i.e. the rankings in WMT and ORG are exactly the same. However, we observe that there were few systems submitted to such translation directions (e.g. $\tau=1$ for Romanian$\rightarrow$English in 2017, for which 7 systems were submitted, see ). Apart from those, other language directions show that there are at least slight rank changes between the WMT rankings and ORG rankings. Looking at the low ranked translation directions, we observe that some are close to a $\tau$ coefficient of 0, especially in correlations without ties, such as German$\rightarrow$English in WMT 2017 ($\tau=0.345$). This means that some rankings have only a weak correlation. Probably related to the differences in DA scores between WMT and ORG (RQ1), we also find that systems’ rankings change for most language pairs when comparing WMT and ORG rankings. We see that there is no perfect correlation between rankings, apart from a few language directions for which only a few systems were submitted. This indicates that the rankings do change to a certain degree. Computing Kendall’s $\tau$ with ties results in higher correlation coefficients than without ties, implying that systems do shift, but tend to stay in the same cluster they occupied in the WMT ranking. In some editions of WMT, the rankings for certain language pairs change considerably. The biggest change in terms of ranking takes place for PROMT’s rule-based system RU$\rightarrow$EN for WMT16. This system advances four positions in the ranking when only original source text is considered, going from rank 5 to rank 1 (although tied with several other systems). It is worth noting that while the DA score for the majority of systems decreases when using original source text, the opposite happens for PROMT’s system.
Thus far we have looked at a single result per translation direction and year, based on the best system in , and on the correlation between systems in . Now we zoom in on a translation direction: Chinese$\rightarrow$English. shows how DA scores change between the whole test set (WMT) and the subsets ORG and TRS, both in terms of raw and standarized scores. In addition, the table depicts how many positions a system goes up or down in the ranking. In the table we observe consistently that the DA score for ORG input is lower than that for WMT, while that for TRS is higher than that for WMT. It is also worth noting that most top scoring systems change in rankings, and that system clusters shift. Due to limited space we provide equivalent tables to for the remaining 16 translation directions as an appendix.
Effect of Translationese on Different Language Pairs {#s:rq3}
====================================================
We aim to find out not only whether translationese has an effect on test sets (RQ1 and RQ2), but also to study whether some language pairs are more affected than others (RQ3). Two hypotheses in this regard are as follows: (i) the degree of translationese’s impact has to do with the translation quality attainable for a translation direction, as represented by the DA score of the best MT system submitted; (ii) the degree of translationese’s impact has to do with how related are the two languages involved.
In order to test the second hypothesis, the degree of similarity between languages has to be quantified. We make use of the lang2vec tool [@littell2017uriel] using the URIEL Typological Database [@littell2016uriel] to compute the similarity between pairs of languages. Similar to the approach of @berzak2017predicting, all the 103 available morphosyntactic features in URIEL are obtained; these are derived from the World Atlas of Language Structures (WALS) [@dryer2013wals], Syntactic Structures of the World’s Languages (SSWL) [@collins2009syntactic] and Ethnologue [@lewis2009ethnologue]. Missing feature values are filled with a prediction from a $k$-nearest neighbors classifier. We also extract URIEL’s 3,718 language family features derived from Glottolog [@hammarstrom2019glottolog]. Each of these features represents membership in a branch of Glottolog’s world language tree. Truncating features with the same value for all the languages present in our study, 87 features remain, consisting of 60 syntactic features and 27 family tree features. We then measure the level of relatedness between two languages using the linguistic similarity (LS) by @berzak2017predicting (), i.e. the cosine similarity between the URIEL feature vectors for two languages $v_y$ and $v_y^{\prime}$. $$L S_{y, y^{\prime}}=\frac{v_{y} \cdot v_{y^{\prime}}}{\left\|v_{y}\right\|\left\|v_{y^{\prime}}\right\|}
\label{eq:ls}$$
Together with the LS for a language direction, we take the best system of the most recent year in our data set, WMT18, for that language direction. The motivation behind is that a top performing system from the most recent campaign should be representative of the current state-of-the-art in machine translation for the translation direction it was submitted to. To look into the effect of translationese across different language pairs, we present two approaches, following the hypotheses put forward at the beginning of this section: (i) compare the DA score of the best system for each translation direction on subset ORG to the relative or absolute difference in DA score for that system between subset ORG and the whole set (WMT); (ii) compare the LS of the two languages in each translation direction to the relative or absolute difference in DA scores for the best system between subset ORG and the whole set (WMT);
{width=".5\linewidth"} {width=".5\linewidth"}
{width=".5\linewidth"} {width=".5\linewidth"}
shows the Pearson correlation and 95% confidence region of the DA score of the best scoring system for each language direction on subset ORG against the absolute and relative difference of the DA scores of those systems between WMT input and ORG input. We observe an interesting trend; higher scoring systems tend to have lower differences in score, which indicates that translationese has less effect. Considering either relative or absolute differences, the correlations are in both cases significant and strong ($p<0.001$, $|R|>0.75$).
shows the Pearson correlation and 95% confidence region of the LS of a language pair (English compared to another language in our data sets) against the absolute and relative difference of the DA scores of the best system for each translation direction between WMT input and ORG input. Here, we see a less obvious trend, and in fact both correlations are very weak and non-significant. However, just as in the previous figure we can see that most of the out-of-English systems tend to have a higher relative and absolute difference than systems that translate into English.
On a side note, we created different feature combinations from the earlier mentioned features for LS. Apart from syntactic and family tree features, phonological features are also present in URIEL. However, other combinations did not seem to alter the LS difference score, compared to using the mentioned features in the experimental setup.
Conclusion and Future Work {#s:conclusions}
==========================
This paper has looked in depth at the effect of translationese in bidirectional test sets, commonly used in machine translation shared tasks, by conducting a series of experiments on data sets for 17 translation directions in the three last editions of the news shared task from WMT. Specifically, we have recomputed the direct assessment (DA) scores separately for the whole test set (WMT), and for the subsets whose source side contains original language (ORG) and translationese (TRS). Results show that using original language input lowers the DA scores, and translationese input increases the scores (RQ1), and perhaps more importantly, system rankings do change (RQ2). We have also investigated the degree to which these rankings change, by measuring the correlation between the rankings with a non-parametric correlation metric that supports ties (Kendall’s $\tau$). Results show that systems do change in absolute ranking, but tend to stay more in the same cluster as they were before.
Last, we looked at whether the effect of translationese correlates with certain characteristics of translation directions. We did not find a correlation between the effect of translationese and the level of relatedness of the two languages involved but we did find a correlation between the effect of translationese and the translation quality attainable for translation directions (RQ3). In other words, human evaluation for better performing systems would seem to be less affected by translationese. Related, we observe that translation directions that contain an under-resourced language tend to obtain low DA scores. Hence, we could say that the effect of translationese tends to be high specially when an under-resourced language is present, which could distort (inflate) the expectations in terms of translation quality for these languages.
As for future work, we plan to focus on studying what the characteristics of translationese are. I.e. what are the traits that set apart the language used in original test sets from translationese test sets. All the code and data used in our experiments are available on GitHub[^1].
Supplemental Material {#sec:supplemental}
=====================
These are the supplementary tables for the paper “The Effect of Translationese in Machine Translation Test Sets". Provided are the remaining 16 tables of each language direction. These tables are of the same structure as Table 4 in the paper.
[^1]: https://github.com/jjzha/translationese
|
---
author:
- 'N. D. R. Bhat'
- 'Y. Gupta'
- 'M. Kramer'
- 'A. Karastergiou'
- 'A. G. Lyne'
- 'S. Johnston'
date: Accepted 19 October 2006
title: 'Simultaneous Single-Pulse Observations of Radio Pulsars: V. On the Broadband Nature of The Pulse Nulling Phenomenon in PSR B1133+16'
---
Introduction {#s:intro}
============
Radio emission from pulsars is known to vary on a wide range of time scales, from as short as one pulse period to many hours or days. Many pulsars are known to exhibit the phenomenon of “nulling," where the emission appears to cease, or is greatly diminished, for a certain number of pulse periods. Typical time scales of nulling are of the order of a few pulse periods, however it may last for up to many hours in certain pulsars; for example, PSR B0826–34 which is active for only $\sim$20% of the time [@durdin1979].
Ever since its discovery [@backer1970] and early investigations [@ritchings1976], the phenomenon of nulling has remained as a vital clue for understanding the elusive pulsar emission mechanism. Several authors have investigated this phenomenon in detail (e.g., @rankin1986 [@biggs1992]), and it is fairly well established that (i) the phenomenon is intrinsic to the pulsar, (ii) it is possibly broadband, and (iii) the null fraction (NF) is strongly correlated to the pulse period [@ritchings1976; @biggs1992]. It is also known that the null fraction (NF) depends on the pulsar class, and that single-core pulsars have rather low values of NF [@rankin1986]. However, the suggested correlation between the null fraction and age is not strongly supported within a given pulsar class. As is the case with several other phenomena of pulsar signals, a satisfactory explanation for nulling, particularly the physics that governs it, remains largely elusive.
Studies of individual pulses from simultaneous multifrequency observations provide valuable insights into the pulsar emission mechanism. In particular, such studies help address the broadband nature and frequency dependence of the intrinsic phenomena such as drifting and nulling. Such studies however require coordinated multi-station observations which are hard to realise in practice. Consequently, few such studies have been reported in the past, and most of these were made in the early days of pulsar research [@robinson1968; @backer1974; @davies1984]. Moreover, most of these experiments involved observations at only 2 to 3 frequencies, and focussed primarily on lower frequencies ($\la$ 2.6 GHz). These observations led to a general understanding that phenomena such as drifting, nulling and moding are generally of broadband nature at low observing frequencies.
This work is fifth in a series of papers describing simultaneous multifrequency observations of radio pulsars. Observations were made in January 2000 as part of a joint collaborative effort between the European Pulsar Network (EPN; @lorimer1998) and the Giant Metrewave Radio Telescope (GMRT) in India. These observations led to long stretches of high quality data for five pulsars over a wide frequency range, from 0.24 to 4.85 GHz [@bhat2001]. Data at 1.4 and 4.85 GHz (from the Lovell and Effelsberg telescopes respectively) were recorded in full polarisation, while data at lower frequencies (from the GMRT) were recorded in total power. The science goals included investigation of a variety of phenomena related to pulsar emission. The earlier papers in this series focussed on polarisation and spectra of individual pulses ([@karas2001; @karas2002; @karas2003; @kramer2003]; hereafter Paper IV). In this paper we present an in-depth analysis of the pulse nulling phenomenon using data at 4 frequencies for PSR B1133+16, with particular emphasis on investigating the broadband nature of the phenomenon.
The remainder of the paper is organised as follows. In §2 we describe the details of observations and data reduction. Pulse energy time series are presented in §3, and we discuss the statistics of nulling in §4. In later sections we focus our attention on some key results from our analysis, most notable of which is evidence for a “selective nulling" phenomenon that is characterised by mostly single-peaked, narrow emission features at the highest frequency, 4850 MHz, and nulls at lower frequencies. Our conclusions are presented in § \[s:conc\].
Simultaneous Multifrequency Observations {#s:simobs}
========================================
The data presented in this paper was obtained from simultaneous observations at four different frequencies, viz. 325, 610, 1400 and 4850 MHz. Data at the higher two frequencies were recorded over bandwidths of 500 MHz (Effelsberg) and 32 MHz (Lovell), while the data-taking system at the GMRT employed a bandwidth of 16 MHz. The observations using Effelsberg and Lovell have been described by @karas2002. Observations at the GMRT capitalised on the instrument’s unique capability whereby observations at more than one frequency can be done simultaneously, by configuring the telescope in a multiple sub-array mode [@gupta2002]. In this mode, data at the two frequencies were recorded by the same data logging system, where the signals from both the frequencies were added together after square-law detection, and the pulsar dispersion delay was used to separate the two data streams in the off-line analysis. Data from all telescopes were stored for off-line processing.
At Effelsberg and Lovell, data were recorded as a single, uninterrupted session, whereas data-taking at the GMRT was split into two sessions (of durations 1932 and 2441 pulse periods, i.e., in total 4373 pulse periods) due to maximum file size limitations for a single scan. For the convenience of analysis, as well as to allow useful cross-checks of any statistical analyses that we perform, we treat data from the two sessions as two separate data sets, and hereafter refer to them as and , respectively. Data from all three telescopes were converted to a common EPN format and time-aligned following the procedures detailed in @karas2001. Further, data for all four frequencies were smoothed to an effective time resolution of 1159.8 $\mu$s (off-line dispersion was performed on data from the GMRT, while data at Lovell were recorded after correcting for dispersion on-line)[^1]. A small fraction of the data (roughly 9%), particularly at the GMRT frequencies, was affected by radio frequency interference (RFI), and is hence excluded from subsequent analyses.
As described in Paper IV, flux density calibration was performed at all four frequencies. At Effelsberg and Lovell, the pulsar data was compared to that of a pre-calibrated noise diode, which itself was compared to the strength of known flux calibrators observed during pointing observations before, after and during the observations. The estimated uncertainty resulting from this calibration scheme is about 10%. At the GMRT, known flux calibrators such as 3C147, 3C295 and 3C286 were observed before and after every pulsar observation. Unlike the pulsar data, the calibration data were gathered separately at two frequencies, i.e., one frequency at a time, using the summed signal from only those antennas that were selected for the frequency. From these data, an effective calibration scale was computed for each of the two frequencies. Using the nearest available calibration source, this scale was then applied to the “on–off” deflection of the pulsar, separately for each frequency of observation, in order to flux calibrate the pulsar signal at both frequencies. As demonstrated in Paper IV, the average flux densities estimated from our calibration scheme are consistent with what is known from literature at the frequencies of observation.
Pulse Energy Time Series {#s:epul}
========================
Detection of null and on states
-------------------------------
Conventionally, a null state signifies our inability to detect pulsar emission over one or more pulse periods. This [*inability*]{} depends on the sensitivity of the observing system, and consequently, sensitive single-pulse observations are ideal for accurate characterisation of pulsar nulling properties. For PSR B1133+16, typical flux densities at our observing frequencies are $\sim$20 to $\sim$100 times larger than the minimum detectable flux densities obtained from our observing parameters. Such high quality data allow an in-depth study of the nulling behaviour of this pulsar.
Traditionally, the occurrence of nulling in a pulsar is characterised by the pulse nulling statistics that are derived from pulse energy distributions (e.g. Biggs 1992; Ritchings 1976). ON and OFF pulse energy histograms for our four frequency observations are shown in Fig. \[fig:nullhist\]. At lower frequencies, we can see a classical separation of the ON pulse histograms into two components, with a zero energy excess that signifies the presence of nulling. At the higher two frequencies, the distributions tend to merge with each other, as the signal-to-noise ratio (S/N) for the pulses is, on the whole, poorer.
Using these distributions and a method similar to that described by Ritchings (1976), we have estimated the null fraction (NF) values for the two data sets (Table \[tab:alpha\]), at the two lower frequencies. The method does not yield meaningful estimates at the higher frequencies 1400 and 4850 MHz, where the ON and OFF distributions tend to merge. The lower and upper limits obtained in this manner are denoted as and respectively. Values for the NF at a given frequency are quite consistent between the two data sets. Furthermore, they are consistent with values reported (at 408 MHz) in the literature for this pulsar (Biggs 1992; Ritchings 1976).
Whereas pulse energy distributions provide a good tool for estimating statistical values related to the nulling phenomenon, we need a different approach to identify individual null pulses at each frequency of observation. We do this by comparing the ON pulse energy estimate with a threshold based on the system noise level. The uncertainty in the pulse energy estimate is given by $\sqrt {\non} \sigoff$, where is the number of ON pulse bins and is the rms of the OFF pulse region. Using this as a threshold, we define pulses with ON pulse energy smaller than $ 3 \times \sigep $ as null pulses.
Results from this classification are illustrated in Fig. \[fig:epul\] which shows the time series of pulse energies (on a logarithmic scale for clarity). The blue band of filled circles denotes the 3-$\sigma$ threshold. As can be seen, the single pulse energies vary by large amounts. Some of the slower time scale variations are likely due to interstellar scintillations (ISS), and this effect is discussed in detail in Appendix I. At each frequency, a significant number of individual null pulses can easily be detected.
[{width="10cm"}]{} [{width="10cm"}]{}
Binary Time Series of Pulse States {#s:bin}
----------------------------------
From the above classification scheme for null pulses, we derive a “binary time series" of pulse states where the “on" and “off" (null) states of pulsar emission are assigned values of “1" and “0", respectively. An example of such a time series of pulse states is shown in Fig. \[fig:bin\] where, for clarity, the length of the time series is restricted to $\approx$200 pulse periods. This is enough to illustrate the general trends, and can be summarised as follows:
1. To first order, we find that the nulling phenomenon is broadband – in that most pulses are in null state simultaneously at all four frequencies – over a wide range of 0.3 to 4.9 GHz, and spanning a frequency ratio of $\sim$15, i.e., 4 octaves.
2. However, there is a significant number of pulses where nulling does [*not always*]{} occur simultaneously at all frequencies.
3. Furthermore, there appears to be a tendency for what we term as “selective nulling,” wherein the null state is restricted to two or three frequencies of observation (more often the lower frequencies), and sometimes just to a single frequency.
To study these aspects more quantitatively, we compute different statistics from the four frequency binary time series of pulse energies, and investigate their dependencies on frequency and frequency separation.
Statistics of Null Pulses {#s:stat}
=========================
The simplest useful statistical quantity is the null count, which is simply the total number of pulses that are in a null state in the binary time series. This null count, when normalised by the total number of pulses in the data set, can be thought of as an estimate of the null fraction. The usefulness of our data set is that such null counts can be estimated for each frequency, as well as for different combinations of frequencies. For the latter, only pulses that null simultaneously for the desired combination of frequencies are included in the null count.
Table \[tab:nf\] shows a summary of these results, where numbers are tabulated separately for data sets I and II. Column (1) is a listing of the specific class (i.e., frequency or the combination of frequencies) for which the null count is computed. Columns (2) and (4) list the number of null pulses (and their percentages) for a given class.
The first thing to note about the results in this table is that our single frequency null counts are fairly consistent with our estimates of null fraction in Table \[tab:alpha\], with the exception of the 610 MHz estimate for which is biased by the effect of ISS (see Appendix A). Furthermore, the null counts generally appear to reduce systematically as one goes from the single frequency counts to those for two or more frequency combinations, suggesting that the nulling phenomenon is somewhat frequency dependent. Thus, for example, the single frequency data show a null fraction of typically $\sim$16%, while the estimate is approximately half this value for the combination of all observing frequencies. Furthermore, a scrutiny of the numbers in columns (2) and (4) also reveals a tendency for the null fraction to decrease with the frequency coverage of the data. This is better illustrated in Fig. \[fig:nullmean\] which shows a plot of “mean null counts” against the number of frequencies in the null class type under consideration (column 1 of Table \[tab:nf\]). Hence, our observations strongly suggest that the nulling phenomenon is [*not always*]{} broadband. The most significant exceptions to the above trends involve data from 610 and 1400 MHz observations, especially for data set II. We believe that the large fluctuations and deep fading of pulse intensities at these frequencies, which lead to increased null counts, is due to the phenomenon of ISS. This is addressed in detail in Appendix A.
Nulls and Bursts: Evidence for a Frequency Dependence?
------------------------------------------------------
We digress briefly to address the question of whether the null and burst durations have any frequency dependence, as this is something that has never been reported before in the literature. Fig. \[fig:bin\] hints at such a possibility, with a tendency for shorter null durations at higher frequencies.
In order to quantify this, we have computed individual null and burst durations from our binary time series of pulse states and the results are summarised in Fig. \[fig:nullburst\] in the form of histograms of null and burst lengths. The following conclusions can be drawn from these results:
1. This pulsar shows predominantly short duration nulls (up to four pulse periods). A vast majority of the observed nulls have durations of just one pulse period, with a smaller fraction extending to durations of two or more pulse periods.
2. The number of longer duration nulls (i.e. two or more pulse periods) seems to decrease with an increase in frequency. This trend is very clear for data set I.
3. The burst duration varies over a wide range from 1 to 30 pulse periods and seems to follow a roughly exponential-type distribution. Majority of these bursts have durations of 1 to 10 pulse periods.
4. Although there does not seem to be any systematic trend for burst durations with frequency, short duration bursts are comparatively larger in number at our highest observing frequency of 4850 MHz.
To further explore the frequency dependence of null durations, we study the ratio of the number of nulls of single pulse duration, to the number of nulls of two, three and four pulse durations. Figure \[fig:nullratio\] shows these ratios for each of the four frequencies. It is clear from this that the relative number of longer duration nulls does indeed decrease significantly with increasing frequency. Thus this pulsar appears to null more often for relatively longer durations at lower frequencies.
Selective Nulling: “Exclusive Null” Pulses {#s:exnulls}
------------------------------------------
In order to characterise the selective nulling phenomenon pointed out in section §\[s:bin\], we compute the number of “exclusive nulls" for every frequency combination – these are the counts of pulses that null [*exclusively*]{} for that frequency combination. For instance, for the 325+610+1400 MHz combination[^2], it is the difference between the null count for this combination and that for all four frequencies. For two-frequency combinations, we further exclude selective null pulses for the relevant 3-frequency combinations (e.g., 325+610+1400 MHz and 325+610+4850 MHz for the case of 325+610 MHz). Columns (3) and (5) in Table \[tab:nf\] tabulate such exclusive null counts (and their percentages).
As per Table \[tab:nf\], the most striking case among the various classes of selective nulling is a significantly large number of exclusive null pulses for the frequency combination 325+610+1400 MHz (3.5% and 7.5% respectively for and ). The combination 325+610 MHz also shows a significant number of exclusive nulls (2.7% and 1.5% for and respectively). In most other cases (barring the peculiar cases of 610+1400 MHz for , which we address in Appendix A), the number of such exclusive nulls is relatively small ($\la$ 0.5%), and a careful examination of the relevant pulses do not show evidence for any general trend. Therefore, in the bulk of the remainder of this paper, we focus our attention on the most interesting case of simultaneous exclusive nulls at 325+610+1400 MHz. In order to distinguish this special class of nulls from those that are broadband over the full range of observing frequencies, we refer to them as “low frequency nulls” (LF nulls).
Selective Nulling: Low Frequency Nulls at 325, 610 and 1400 MHz {#s:lfbn}
===============================================================
We have carefully examined the individual pulses that are grouped as “low frequency nulls," and find, quite interestingly, that the emission at the highest frequency (4850 MHz) is often marked by a fairly strong, narrow pulse. Examples of this kind of pulses are shown in Fig. \[fig:eg\]. In this section, we characterise their properties, and compare and contrast them with those of normal emission seen at this frequency.
Characterisation of Emission at 4850 MHz {#s:char}
----------------------------------------
To characterise the high frequency pulses occurring during the LF nulls, we use basic properties such as their width, location and strength. This is done in a two-step process. First, we perform a box car analysis in order to obtain some first order estimates of these quantities. For this, we trial a large number of box car widths (within a range from 0.1 to 0.9 times the main pulse width), deriving the amplitude, location and peak signal-to-noise ratio (S/N) in each case. The case that yields maximum S/N is then taken as the closest representation of the pulse. Using these values as starting points ensures a quick and easy convergence of the finer grid search that is performed subsequently. Here, the amplitude, width and location of the best fit Gaussian is determined using standard chi-square minimisation techniques. These three parameters, along with the peak S/N, are presented in Fig. \[fig:gaus\]. From this, we can infer the following about the nature of these high frequency emissions:
1. The emission features are often quite strong, with peak S/N typically $\sim$50, while stronger pulses are seen with S/N as large as $\sim$300.
2. Their occurrence is largely confined to a narrow longitude range of $\la$ 10 ms, starting roughly 0.25$w_p$ from the leading edge of the pulse profile, where $w_p$ is the ON pulse window.
3. There is some evidence for a second preferred location (roughly at 0.75$w_p$ from the leading edge), albeit this is mainly seen in .
4. These pulses typically have widths of a few milliseconds, and are usually narrower than the typical subpulses seen during the normal emission.
Histograms of the above-mentioned parameters are plotted in Fig. \[fig:full\] (shaded regions). These are for , however similar trends are also seen for . The S/N and amplitude seem to follow somewhat exponential-like distributions on a linear scale, but we have chosen to plot them on a logarithmic scale in order to better compare and contrast them with the corresponding distributions for the normal population of pulses (see next section). The width distribution is somewhat skewed and is slightly double-peaked in nature. The longitude distributions are nearly symmetric.
Comparison with “Normal" Emission at 4850 MHz {#s:comp}
---------------------------------------------
In order to compare the properties of these high frequency (HF) pulses with those of “normal” pulses, we compute similar parameters for the sample of normal pulses at 4850 MHz. Normal pulses are those which do not belong to either the LF null class or the all-frequency null class. Since this pulsar shows a two-component profile, and given that the high frequency emission during LF nulls tends to occur preferentially in the longitude range of the leading component, we find that it is more appropriate to compare with the properties derived from the leading component only. The results of this exercise are summarised in the histograms shown in Fig. \[fig:full\] (for ), where quantities for the normal population of pulses are shown as unshaded bars. For the reasons described earlier, the S/N and amplitude histograms are plotted on a logarithmic scale. Further, the two distributions are normalised for equal peaks to allow an easy comparison.
From these figures we see that the S/N distribution is narrower and less skewed for the the HF emission during LF nulls and it appears to peak at a comparatively higher S/N. The amplitude distributions are also narrower and more symmetric than those for normal pulses. The distributions for widths and locations also show significant differences between the two classes. As inferred qualitatively earlier, the width distribution for the HF pulses is narrower than that for the normal pulses. There is significant spread in the longitudes of occurrence (for both classes), and the spread is more for the normal pulses, thus reinforcing the conclusion that the HF pulses are confined to a narrower range of longitude. Furthermore, the longitude distributions for the LF null class peak at slightly earlier longitudes than those for the normal pulses, indicating that this emission arrives at an earlier phase than that of the normal pulses. Some of these results are clearly seen in a comparison of the average profiles obtained for the two classes of pulses (Fig. \[fig:profs\]). These reinforce the conclusions that the HF pulses during LF nulls occur predominantly under the leading component and arrive at an earlier longitude than the normal pulses.
Comparison with “Giant Pulses” at 4850 MHz {#s:giant}
------------------------------------------
A different analysis of this data set in Paper IV revealed an interesting sub-class of “giant” pulses for this pulsar at 4850 MHz. These are some of the strongest pulses seen among the non-nulling detections, with typical flux densities more than $ 10 \times \avS $ (where is the mean flux density of the entire data), which is the commonly used threshold for detection of giant pulses (e.g. @johnston2002). The emission is often narrow in width and tends to occur preferentially at the [*trailing*]{} edge of the leading pulse component (see Fig. 17 of Paper IV). Based on their large flux densities and from the hint of an emerging power-law component in the cumulative distribution of their energies (Fig. 16 of Paper IV), we referred to them as possible giant pulses. However, unlike the classical giant pulses as have been observed for the Crab and a few other pulsars [@cordes2004; @johnston2002], which are characterised by a power-law energy distribution and a spectral index that is comparable to or steeper than that of the normal pulses, these appear to be of a significantly flatter spectrum. This conjecture stems largely from (i) their apparent absence at our lower frequencies and (ii) from observations at a higher frequency of 8450 MHz (Maron & Löhmer, private communication).
It is important to note that in Paper IV we studied the same data set but excluded all pulses with nulls at any frequency in order to compute the spectra. Hence, when studying the nulling pulses and their frequency behaviour as we do in this paper, it is the study of a different subset of pulses. Given this, it is interesting to compare and contrast the population of the afore-mentioned giant pulses with the HF pulses, as seen during the LF nulls. Both classes are generally strong and narrow, with a mostly single-peaked emission, though the flux densities of the latter are well below the $ 10 \times \avS $ threshold of the former. Further, they both tend to occur preferentially over some restricted longitude ranges of the leading component: giants near the trailing edge of the leading component (i.e. arriving at a slightly later phase than the normal pulses), and the HF pulses near the leading edge (arriving at a slightly earlier phase than normal pulses). Both show low-number statistics (40 giants and 195 HF pulses) which prevent us from a detailed study of their energy distributions. Their rates of occurrence – typically 1 in 100 for giants and 1 in 20 for HF pulses – are comparable or even larger than the rates of occurrence of giant pulses that are seen for the Crab pulsar [e.g. @cordes2004]. Thus there are some similarities between these two special classes of pulses, and it is possible that they may have related origins.
Selective Nulling at 325 and 610 MHz {#s:exnullgmrt}
------------------------------------
The next most interesting sample in our data set is a subset of pulses that null [*exclusively*]{} at 325 and 610 MHz. Some examples are shown in Fig. \[fig:exnulllow\]. Although an analysis similar to that described in § \[s:char\] and § \[s:comp\] was attempted for this class of nulls, the poor statistics of the sample did not allow us to obtain any meaningful characterisation of the emission at higher frequencies for this class of LF nulls. Qualitative examination shows that a majority ($\sim$80–90%) of these pulses are characterised by single-peaked emission at 1400 and 4850 MHz, although in most cases the pulse widths do not seem to be significantly narrower than typical sub-pulse widths. As in the case of the first class of HF pulses, the emission here (at 4850 MHz) also tends to occur preferentially at the longitude of the leading component, with only a minority of these pulses appearing at the longitude of the trailing component. Only occasionally, the emission at 1400 and/or 4850 MHz is seen as a double-peaked pulse that is characteristic of the normal emission from this pulsar.
The average profiles (at 1400 and 4850 MHz) of the pulses in this null class are shown in Fig. \[fig:2freqprofs\] (green curves), along with those of the full sample of normal pulses (red curves). As was the case for the first class of HF pulses, these pulses tend to arrive at a slightly earlier phase than the normal emission, at 4850 MHz. However, no such offset is readily visible for the emission at 1400 MHz.
Discussion {#s:res}
==========
In this paper, we focus on the analysis of the pulse nulling phenomenon in PSR B1133+16 using high-quality single-pulse data from simultaneous multifrequency observations. The most important finding from our analysis is that nulling [*does not always occur simultaneously*]{} at all frequencies. Our observations uncover a significant number of pulses ($\approx$6%) that null at the three lower frequencies of our observation, however are marked by quite narrow and strong pulses at the highest frequency of 4850 MHz. Their properties seem to be quite different from those of the normal pulses seen at this frequency, but interestingly, show some striking similarities with the subset of the strongest pulses in such a sample. Our analysis shows some evidence for the number of simultaneous null pulses to decrease with the frequency coverage of the data, and also suggests that longer-duration nulls are relatively more common at the lower observing frequencies. In the remainder of the paper, we review what has been learnt so far about the nulling behaviour of this pulsar from past observations and discuss possible implications for pulsar emission.
Comparison with Previous Nulling Studies {#s:prev}
----------------------------------------
PSR B1133+16 is among the well-studied pulsars since the early days of pulsar observations [@backer1972; @ferguson1978; @kardashev1982; @boriakoff1983; @smirnova1994]. Its nulling behaviour is not extreme as in the case of PSR B0031$-$07 and PSR B1944+17 [@hugu1970; @deich1986], which are known for their long durations of nulling (null fractions of the order $\sim$50%). In fact, most null durations of this pulsar are within the range of one to a few pulse periods, with a moderately large null fraction ($\sim$10 to 20%) over typical observing durations. It is also important to recognise that most nulling studies to date have been based on data taken at a single observing frequency. Interestingly, our estimates of null fraction at single frequencies (with the exception of 4850 MHz for and barring the special cases of 610 and 1400 MHz for ; see Appendix A), are comparable to the published estimate of 15$\pm$3% from earlier studies (Biggs 1992; Ritchings 1976). However, the null fraction of real “broadband” nulls (i.e., nulls that simultaneously occur at all four frequencies of observation) is only $\approx$7.5%. Thus, in general, most published estimates for null fraction are likely to be overestimates if broadbandness is adopted as an additional criterion to define a truly null state.
There have been several single-pulse studies of PSR B1133+16 in the past based on data from simultaneous observations at multiple frequencies [@boriakoff1983; @kardashev1982; @boriakoff-ferguson1981; @bartel-sieber1978; @backer1974], and it is fairly well established that the pulse energy and structure of this pulsar are well correlated at single-pulse, sub-pulse and micro-pulse levels. Specifically, the works of @boriakoff1983 and @kardashev1982 addressed the broadbandness of micropulse and subpulse structure, while @bartel-sieber1978 focussed on the microstructure of single pulses in this pulsar. However, none of these studies were able to address the simultaneity in pulse nulling at different observing frequencies. In fact, there have been few studies that have even peripherally addressed this interesting phenomenon. The only relevant work that we are aware of in this context is by @davies1984 who studied simultaneous data of PSR B0809+74 taken at 102 and 406 MHz. In their short stretch of data (348 pulse periods), they recorded 9 nulls at 102 MHz, and only 3 at 406 MHz. Interestingly nulls at 406 MHz always correspond to nulls at 102 MHz, but not vice versa. While this may not qualify as a strong supporting evidence for selective nulling, it does suggest that the pulsar nulling characteristics vary with observing frequency.
Implications for Pulsar Emission Models {#s:impli}
---------------------------------------
In sections §\[s:char\] and §\[s:comp\], we compared and contrasted the characteristics of the high frequency emission during low frequency nulls with the “normal” emission at 4850 MHz. We also compared them with those of “giant” pulses as reported in our earlier paper (§\[s:giant\]). Based on our analysis, we would like to conjecture that (a) the high frequency emission pulses are different from the normal pulses and are likely to be related to the “giant” pulses; (b) the spectrum of these “giants” is probably flatter, and consequently they are more easily seen at higher frequencies (Paper IV); and (c) we see them when the normal radio emission is off or too weak to be detectable (i.e. nulling) at high frequencies (this paper).
One of the most interesting results from our analysis is the observation of narrow high frequency pulses at times of low frequency nulls. The simplest possible interpretation could be the presence of an [*additional*]{} process of emission that does [*not*]{} turn off when the pulsar switches to a “null state” at low frequencies. It is also quite possible that such a process is more like a gradual effect, becoming predominant above a cutoff frequency. Unfortunately, our data are insufficient to address such a hypothesis.
Our data also show several instances of significant emission at both 1400 and 4850 MHz, when there occurs a null at the lower frequencies. However, the results are not conclusive enough to suggest a frequency dependence for such an additional process. A likely scenario is that the emission tends to be narrower and peaks at an earlier phase with an increase in frequency.
It is possible that such an additional emission process is also present during the normal emission. However, given that the pulsar’s emission is typically double-peaked and has a strong leading pulse component, its presence might be hard to discern in non-nulling pulses. Assuming this may be the case, we would expect a significant increase in relative strength of leading and trailing pulse components with an increase in frequency, which is indeed seen for this pulsar (Fig. \[fig:allprofs\]).
An alternate, and perhaps more likely, possibility involves this emission being related to the population of “giant” pulses (as discussed in § \[s:giant\]). In particular, given their preferred longitudes of occurrence near the leading pulse component, a likely flat-spectrum nature, and above all, a tendency to occur even when the pulsar nulls at low frequencies, may also suggest that they probably originate in a different part of the magnetosphere which does not participate in nulling. If this is the case, it may support an outer gap origin as opposed to a polar cap region (where nulling is probably more effective). Such a scenario might also suggest a possible link to some high-energy emission processes that occur in the outer parts of the magnetosphere. Future simultaneous observations of this pulsar, preferably with a denser sampling of frequency in the $\sim$1–5 GHz range, should enable us to address this in more detail.
Conclusions {#s:conc}
===========
We have studied high-quality single-pulse data of PSR B1133+16 from simultaneous multifrequency observations conducted using the GMRT, Lovell and Effelsberg to perform an in-depth analysis of the pulse nulling phenomenon. Our observations provided long data stretches over a wide frequency range (from 0.3 to 4.85 GHz), which allow, for the first time, investigation of nulling as a function of observing frequency, separation in frequency as well as combination of frequencies. The data shows that the pulsar spends approximately 15% of the time in a null state at our frequencies of observation. However, only roughly half of these nulls occur simultaneously at all four frequencies. In other words, much in contrary to the traditional notion of being a broadband phenomenon, nulling does [*not always*]{} occur simultaneously at all four frequencies.
Our most interesting finding of this “selective nulling” phenomenon is a significantly large number of pulses ($\approx$5%) that show an emission at the highest frequency of our observation, 4850 MHz, while there occurs a null at all three lower frequencies (325, 610 and 1400 MHz). This emission is often seen as fairly strong and narrow pulses, and is different from the broad, double-peaked emission normally seen from this pulsar. We have characterised these high frequency pulses in terms of their amplitudes, widths and locations, and compared their statistical properties with those of the leading component of the normal pulses. Our analysis reveals significant differences in the properties between the two classes. Specifically, the population of HF pulses shows an amplitude distribution that is more skewed, a comparatively narrower width distribution, and a tendency to arrive at a slightly earlier phase than the leading pulse component. Additionally, we note some interesting similarities between these narrow pulses and the “giant” pulses as identified for this pulsar in our earlier paper.
We do not have any clear and convincing explanation to offer at this point, apart from the simplest (and rather speculative) interpretation involving the presence of an additional emission process that does not turn off even when the pulsar nulls at lower frequencies. Alternatively, such an additional process may be a gradual effect that becomes more predominant at higher frequencies. While our data are insufficient to address such hypotheses, such scenarios could potentially result in observable effects such as an increase in relative strength between the leading and trailing pulse components at higher observing frequencies, which is indeed seen for this pulsar.
Given the significant similarities between this emission and “giant” pulses as seen at 4850 MHz, it is quite possible that they share a common, or at least related, origin. In particular, their tendency to occur at specific pulse longitudes, along with the pulse shapes and spectral characteristics that are different from the normal emission, add support to such a conjecture and are suggestive of a likely origin in the outer parts of the magnetosphere. We hope future simultaneous observations will help to shed more light on such possibilities.
[*Acknowledgements:*]{} We are grateful to all the people working at the participating telescopes who made these observations possible. In particular we thank S. Kudale, M. Jangam and C. Lange for assistance with the observations, and R. Wielebinski for the encouragement and support towards this project. We also thank an anonymous referee for a critical review and several insightful comments which helped improve the contents and presentation of the paper. The Giant Metrewave Radio Telescope (GMRT) is operated by the National Centre for Radio Astrophysics of Tata Institute of Fundamental Research.
Backer, D. C. 1970, Nature, 228, 42 Backer, D. C. 1972, , 174, L157 Backer, D. C. & Fisher, J. R. 1974, , 189, 137 Bartel, N. & Sieber, W. 1978, , 70, 307 Bhat, N. D. R., Rao, A. P., & Gupta, Y. 1999, , 121, 483 Bhat, N. D. R., Karastergiou, A., Gupta, Y., Kramer, M., & Lyne, A. G. 2001, BAAS, 34, 569 Biggs, J. D. 1992, , 394, 574 Boriakoff, V. 1983, , 272, 687 Boriakoff, V. & Ferguson, D. C. 1981, IAU Symp. 95: Pulsars: 13 Years of Research on Neutron Stars, 95, 191 Cordes, J. M., Bhat, N. D. R., Hankins, T. H., McLaughlin, M. A., & Kern, J. 2004, , 612, 375 Cordes, J. M., Pidwerbetsky, A., & Lovelace, R. V. E. 1986, , 310, 737 Davies, J. G., Lyne, A. G., Smith, F. G., Izvekova, V. A., Kuzmin, A. D., & Shitov, I. P. 1984, , 211, 57 Deich, W. T. S., Cordes, J. M., Hankins, T. H., & Rankin, J. M. 1986, , 300, 540 Durdin, J. M., Large, M. I., Little, A. G., Manchester, R. N., Lyne, A. G., & Taylor, J. H. 1979, , 186, 39 Ferguson, D. C., & Seiradakis, J. H. 1978, , 64, 27 Gupta, Y., Gothoskar, P., & Bhat, N. D. R. 2002, IAU Symposium, 199, 369 Huguenin, G. R., Taylor, J. H., & Troland, T. H. 1970, , 162, 727 Johnston, S. & Romani, R. W. 2002, , 332, 109 Karastergiou, A., von Hoensbroech, A., Kramer, M., Lorimer, D. R., Lyne, A. G., Doroshenko, O., Jessner, A., Jordan, C. & Wielebinski, R. 2001, , 379, 270 Karastergiou, A., Kramer, M., Johnston, S., Lyne, A. G., Bhat, N. D. R., & Gupta, Y. 2002, , 391, 247 Karastergiou, A., Johnston, S., & Kramer, M. 2003, , 404, 325 Kardashev, N. S., Nikolaev, N. I., Novikov, A. I., Popov, M. V., Soglasnov, V. A., Kuzmin, A. D., Smirnova, T. V., Bartel, N., Sieber, W. & Wielebinski, R. 1982, , 109, 340 Kramer, M., Karastergiou, A., Gupta, Y., Johnston, S., Bhat, N. D. R., & Lyne, A. G. 2003, , 407, 655 Lorimer, D. R., Jessner, A., Seiradakis, J. H., Lyne, A. G., D’Amico, N., Athanasopoulos, A., Xilouris, K. M., Kramer, M. & Wielebinski, R. 1998, , 128, 541 Rankin, J. M. 1986, , 301, 901 Rickett, B. J. 1990, , 28, 561 Ritchings, R. T. 1976, , 176, 249 Robinson, B. J., Cooper, B. F. C., Gardner, F. F., et al. 1968, Nature, 218, 1143 Romani, R. W., Narayan, R., & Blandford, R. 1986, , 220, 19 Smirnova, T. V., Tul’bashev, S. A., & Boriakoff, V. 1994, , 286, 807
[**Appendix A: Scintillation Fading and Nulling Statistics**]{}\[s:scnt\]
Here we address the peculiar case of a large number of nulls seen at 610 and 1400 MHz for . On a careful examination, we find that a majority of these nulls are largely clustered into two groups, the first in the range $\sim$1200 to $\sim$1500 in pulse number, and the second in the range $\sim$2200 to $\sim$2500. Interestingly, these regions of occurrence correlate well with the regions of flux fading seen at these frequencies. We therefore propose that the apparent excess in null counts and simultaneous nulls exclusive to these frequencies is essentially a manifestation of scintillation-induced flux fading. Below we elaborate on our arguments.
As is well known, pulsar signals are subject to interstellar scintillation (ISS), observable effects of which are strongly frequency dependent. At low observing frequencies, most pulsars are in the strong scintillation regime, where the observable effects can be grouped into two distinct classes, viz., diffractive and refractive scintillation [e.g. @rickett1990; @cordes1986; @romani1986], with consequences such as flux density variations exhibiting two characteristic time scales. At frequencies higher than a few GHz, most nearby pulsars transition to the weak scintillation regime, which is marked by a single characteristic time scale [@rickett1990]. PSR B1133+16 is expected to be in the strong scattering regime at 325, 610 and 1400 MHz, and in weak scintillation at our highest observing frequency of 4850 MHz (see Paper IV for details on relevant scintillation parameters).
For observations at low observing frequencies ($\la$ 1 GHz), the dominant ISS effect is random intensity modulations due to diffractive scintillation [e.g. @bhat1999]. The pulse intensity decorrelates over narrow ranges in time and frequency, and the decorrelation widths ($\nud$ and $\tiss$ respectively in frequency and time) are strongly frequency dependent ($\nud \sim \nu^{4.4}$, $\tiss \sim \nu^{1.2}$, where $\nu$ is the frequency of observation). Thus, intensity modulations as deep as 100% can be expected over time and frequency scales of $\tiss$ and $\nud$. For observations made over time durations () much larger than $\tiss$, three regimes may be identified for apparent intensity modulations, depending on the ratio of the scintillation bandwidth ($\nud$) to the total bandwidth of observation, $\delB$: (i) $\delB \gg \nud$: here the effective flux modulations will be quenched to some extent depending on the ratio $\delB/\nud$; (ii) $\delB \sim \nud$, i.e. when the observing bandwidth is comparable to the scintillation bandwidth: flux fading as large as $\sim$100% is likely to occur over durations of $\tiss$; (iii) $\delB \ll \nud$, such deep flux fading could still occur over $\tiss$, albeit with somewhat less likelihood.
Consider the scenario where simultaneous observations made at three different frequencies correspond to the three different cases described above. There could be rare instances of simultaneous flux fading at the two frequencies that correspond to cases (ii) and (iii). Such apparent nulls may lead to consequences such as an increase in null counts as well as an increase in the number of simultaneous nulls exclusive to those frequencies. A comparison of the scintillation and observing parameters for PSR B1133+16 at 610 and 1400 MHz ($\nud$ $\sim$11 MHz, $\tiss$ $\sim$ 6 min at 610 MHz; $\nud$ $\sim$ 270 MHz, $\tiss$ $\sim$ 16 min at 1400 MHz) supports such a conjecture.
Some simple checks could help reaffirm this. Since increases with the observing frequency, we would expect the time span of such apparent nulls to be limited to at the lower frequency, i.e. 610 MHz. Interestingly, the observed duration of 300 pulses ($\approx$6 min) is in excellent agreement with the expected value for at 610 MHz. Further, at 1400 MHz is expected to be roughly 3 times larger than that at 610 MHz. This means we can expect null counts at these two frequencies to increase by roughly 4% and 12% in comparison to the measured values for the . Though the observed excess is somewhat larger than this, it does conform to the expectation that larger excess should be seen at 1400 MHz. In short, the increase in null counts and a larger number of exclusive nulls at 610 and 1400 MHz for can be attributed to the combination of ISS and our observing parameters.
[^1]: The dispersion effect is negligible at the highest frequency 4850 MHz, and hence no correction was applied for data from Effelsberg.
[^2]: To denote a specific frequency combination, we insert the “+" symbol in between the relevant frequencies, e.g. the notation “325+610 MHz" refers to the null pulses that are exclusive to 325 and 610 MHz.
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abstract: 'A hypothesis about the average phase-space distribution of resonance eigenfunctions in chaotic systems with escape through an opening is proposed. Eigenfunctions with decay rate $\gamma$ are described by a classical measure that ${\ensuremath{(i)}}$ is conditionally invariant with classical decay rate $\gamma$ and ${\ensuremath{(ii)}}$ is uniformly distributed on sets with the same temporal distance to the quantum resolved chaotic saddle. This explains the localization of fast-decaying resonance eigenfunctions classically. It is found to occur in the phase-space region having the largest distance to the chaotic saddle. We discuss the dependence on the decay rate $\gamma$ and the semiclassical limit. The hypothesis is numerically demonstrated for the standard map.'
author:
- Konstantin Clauß
- 'Martin J. Körber'
- Arnd Bäcker
- Roland Ketzmerick
title: Resonance eigenfunction hypothesis for chaotic systems
---
*Introduction.*—Eigenvalue spectra and the structure of eigenfunctions are the key to understanding any quantum system. Universal properties are usually expected for quantum systems with chaotic classical dynamics. For closed systems the *statistics of eigenvalues* follows random matrix theory [@BohGiaSch1984; @Ber1985; @SieRic2001; @MueHeuBraHaaAlt2004], and the *structure of eigenfunctions* is described by the [semiclassical eigenfunction hypothesis]{} [@Vor1979; @Ber1977b; @Ber1983]. It states that eigenfunctions are concentrated on those regions explored by typical classical orbits. If the dynamics is ergodic, this is proven by the quantum ergodicity theorem [@Shn1974; @CdV1985; @Zel1987; @ZelZwo1996; @Bie2001; @NonVor1998], showing that almost all eigenfunctions converge to the uniform distribution on the energy shell in phase space [@BaeSchSti1998]. These fundamental results for single particle quantum chaos recently had strong impact in many-body systems, e.g. for thermalization [@Sre1994; @AleKafPolRig2016]. Experimentally one often deals with chaotic scattering systems [@Gas2014b], which appear in many fields of physics, such as nuclear reactions [@MitRicWei2010], microwave resonators [@Sto1999], acoustics [@TanSoe2007], quantum dots [@Jal2016], and optical microcavities [@CaoWie2015]. Thus the counterparts of the fundamental results of closed systems are desired for scattering systems. This has been achieved for the *statistics of resonances* [@HaaIzrLehSahSom1992; @FyoSom1997; @Alh2000; @JacSchBee2003; @FyoSav2012; @KumDieGuhRic2017; @Sjo1990; @Lin2002; @LuSriZwo2003; @SchTwo2004; @NonZwo2005; @She2008; @RamPraBorFar2009; @KopSch2010; @KoeMicBaeKet2013]. In particular the fractal Weyl law [@Sjo1990; @Lin2002; @LuSriZwo2003; @SchTwo2004; @NonZwo2005; @She2008; @RamPraBorFar2009; @KopSch2010; @KoeMicBaeKet2013] relates the growth rate of the number of long-lived resonances to the fractal dimension of the chaotic saddle of the classical dynamics. For the *structure of resonance eigenfunctions* some aspects have been studied, e.g. for open billiards [@IshSaiSadBer2001; @KimBarStoBir2002; @BaeManHucKet2002; @WeiRotBur2005; @WeiBarKuhPolSch2014], optical microcavities [@GmaCapNarNoeStoFaiSivCho1998; @LeeRimRyuKwoChoKim2004; @WieHen2008; @ShiHarFukHenSasNar2010; @ShiWieCao2011; @HarShi2015], potential systems [@RamPraBorFar2009], and maps [@CasMasShe1999b; @SchTwo2004; @KeaNovPraSie2006; @NonRub2007; @ErmCarSar2009; @LipRyuLeeKim2012; @CarWisErmBenBor2013; @SchAlt2015; @KoeBaeKet2015]. However, there exists no analogue to the semiclassical eigenfunction hypothesis for scattering systems. This fundamental open problem of the structure of resonance eigenfunctions is addressed in this paper.
![Average Husimi phase-space distribution of resonance eigenfunctions (top) compared to constructed classical measures ${\ensuremath{\mu_{\gamma}^h}}$ (bottom) with decay rates $\gamma = 0.6$ (left) and $\gamma = 2$ (right) for $h = 1/1000$. Chaotic standard map with $\kappa = 10$ on phase space $\Gamma = [0,1)\times[0,1)$ with opening $\Omega = [0.2, 0.4]\times[0, 1)$ (blue dashed line). Colormap with fixed maximum for each $\gamma$. Prominent localization for $\gamma = 2$ and overall quantum-classical agreement. []{data-label="FIG:title"}](fig1.pdf)
For simplicity of the presentation we focus in the rest of the paper on time-discrete maps with chaotic dynamics and escape through an opening. The resulting discussion is straightforwardly generalized to autonomous systems like the paradigmatic three-disk scattering system or the Hénon-Heiles potential. Maps arise naturally, e.g., from a Poincaré section in autonomous systems or from a stroboscopic Poincaré section in time-periodically driven systems. Quantizing such a map yields a subunitary propagator whose non-orthogonal, right eigenfunctions have varying decay rates $\gamma$. Note that these eigenfunctions extend into the opening for the chosen ordering of escape before the mapping, see Fig. \[FIG:title\]. Surprisingly, there occurs a localization of their average phase-space distribution within the opening. This localization is more prominent for resonance eigenfunctions with large decay rates $\gamma$, as visualized for the standard map (introduced below) in Fig. \[FIG:title\] (top). Thus, the following questions arise: What is the origin of this localization? What distinguishes the phase-space region of localization? More generally, is this effect caused by quantum interference (like dynamical localization [@Fis2010] or scarring due to periodic orbits [@Hel1984]) or by properties of the classical dynamics? Before answering these questions, let us briefly introduce the classical and quantum mechanical background. Classically, for chaotic dynamics of a map with escape almost all points on phase-space $\Gamma$ will be mapped into the opening $\Omega$ eventually and thus escape [@LaiTel2011]. Only a set of measure zero does not leave the system under forward and backward iteration. This invariant set usually is a fractal and is called the *chaotic saddle* ${\ensuremath{\Gamma_{\text{s}}}}$, see Fig. \[FIG:TIMES\](a). Its unstable manifold consists of points approaching ${\ensuremath{\Gamma_{\text{s}}}}$ under the inverse map and is therefore called the *backward-trapped set* ${\ensuremath{\Gamma_{\text{b}}}}$, see Fig. \[FIG:TIMES\](b). Generic initial phase-space distributions asymptotically converge to the uniform distribution on ${\ensuremath{\Gamma_{\text{b}}}}$, the so-called *natural measure* ${\ensuremath{\mu_{\text{nat}}}}$, with corresponding decay rate ${\ensuremath{\gamma_{\text{nat}}}}$ [@PiaYor1979; @KanGra1985; @Tel1987; @LopMar1996; @DemYou2006; @AltPorTel2013]. Quantum mechanically, the support of resonance eigenfunctions is given by the backward trapped set ${\ensuremath{\Gamma_{\text{b}}}}$ [@CasMasShe1999b; @KeaNovPraSie2006]. Furthermore, long-lived eigenfunctions with decay rates $\gamma\approx{\ensuremath{\gamma_{\text{nat}}}}$ are distributed as the natural measure ${\ensuremath{\mu_{\text{nat}}}}$ on phase space [@CasMasShe1999b], which corresponds to the steady-state distribution in the context of optical microcavities [@LeeRimRyuKwoChoKim2004]. There are a few supersharp resonances with $\gamma$ significantly smaller than ${\ensuremath{\gamma_{\text{nat}}}}$ [@Nov2012]. Instead, we focus on the large number of shorter-lived eigenfunctions ($\gamma > {\ensuremath{\gamma_{\text{nat}}}}$). For their integrated weight on $\Omega$ and on each of its preimages the dependence on the decay rate $\gamma$ was derived in reference [@KeaNovPraSie2006]. This concept is generalized by so-called conditionally invariant measures [@PiaYor1979; @LopMar1996; @DemYou2006; @NonRub2007]. Recently, we suggested a specific conditionally invariant measure proportional to ${\ensuremath{\mu_{\text{nat}}}}$ on the opening $\Omega$, describing classically the weight of eigenfunctions on either side of a partial barrier [@KoeBaeKet2015]. None of these results, however, explains the observed localization phenomenon.
In this paper we propose a hypothesis for resonance eigenfunctions in chaotic systems predicting their average phase-space distribution. The hypothesis defines a conditionally invariant measure of the classical system for given decay rate $\gamma$ and effective Planck’s constant $h$. It gives a classical explanation for the localization of resonance eigenfunctions in those phase-space regions having the largest distance to the chaotic saddle. This is demonstrated in Fig. \[FIG:title\] for the chaotic standard map. We discuss the dependence on $\gamma$ and $h$, and briefly speculate about the semiclassical limit. *Resonance eigenfunction hypothesis.*—We postulate that in chaotic systems with escape through an opening the average phase-space distribution of resonance eigenfunctions with decay rate $\gamma$ for effective Planck’s constant $h$ is described by a measure that ${\ensuremath{(i)}}$ is conditionally invariant with decay rate $\gamma$ and ${\ensuremath{(ii)}}$ is uniformly distributed on sets with the same temporal distance to the $h$-resolved chaotic saddle.
Combining both properties yields a measure $$\begin{aligned}
{\ensuremath{\mu_{\gamma}^h}}(A) = \frac{1}{\mathcal{N}} \int_A
{\ensuremath{\text{e}}}^{{\ensuremath{t_h(x)}}\cdot(\gamma - {\ensuremath{\gamma_{\text{nat}}}})}\,{\ensuremath{\text{d}}}{\ensuremath{\mu_{\text{nat}}}}(x),
\label{EQ:mugh0}\end{aligned}$$ for all $A\subset \Gamma$ with normalization constant $\mathcal{N}$. Here the temporal *saddle distance* ${\ensuremath{t_h(x)}}\in\mathbb{R}$ fulfills $$\begin{aligned}
{\ensuremath{t_h({\ensuremath{M}}^{-1}(x))}} = {\ensuremath{t_h(x)}}- 1 \label{EQ:dist}\end{aligned}$$ for almost all $x\in{\ensuremath{\Gamma_{\text{b}}}}$, i.e. each backward iteration of the map ${\ensuremath{M}}$ on ${\ensuremath{\Gamma_{\text{b}}}}$ reduces the saddle distance by one. An important implication of Eq. is that ${\ensuremath{\mu_{\gamma}^h}}$ is enhanced with increasing $\gamma > {\ensuremath{\gamma_{\text{nat}}}}$ in those regions of ${\ensuremath{\Gamma_{\text{b}}}}$ having the largest saddle distance, due to the exponential factor. These regions must be in the opening $\Omega$, which is easily shown by contradiction. Thus the hypothesis leads to a classical prediction for the localization of resonance eigenfunctions in chaotic systems.
![Classical sets for the considered standard map. (a) Chaotic saddle ${\ensuremath{\Gamma_{\text{s}}}}$, (b) backward trapped set ${\ensuremath{\Gamma_{\text{b}}}}$, (c) opening $\Omega$ and preimages ${\ensuremath{M}}^{-1}(\Omega), {\ensuremath{M}}^{-2}(\Omega)$ (from dark to light), (d, e) partition of the backward trapped set ${\ensuremath{\Gamma_{\text{b}}}}$ with colored sets ${\ensuremath{\mathcal{E}^h_{n}}}$ with integer saddle distance $n \leq {\ensuremath{m_h}}$ for (d) $h = 1/1000$ with ${\ensuremath{m_h}}= 3$ and (e) $h = 1/16000$ with ${\ensuremath{m_h}}= 4$. Regions with $n \leq 0$ are within the $h$-resolved saddle ${\ensuremath{\Gamma^h_{\text{s}}}}$. []{data-label="FIG:TIMES"}](fig2.pdf)
We will now discuss properties ${\ensuremath{(i)}}$ and ${\ensuremath{(ii)}}$ in more detail. A measure $\mu$ is called conditionally invariant with decay rate $\gamma$ under a map ${\ensuremath{M}}$ with escape through an opening, if it is invariant under time evolution up to an overall decay, $$\begin{aligned}
\mu({\ensuremath{M}}^{-1}(A)) = {\ensuremath{\text{e}}}^{-\gamma}\,\mu(A), \label{EQ:ci}\end{aligned}$$ for all $A\subset \Gamma$ [@PiaYor1979; @LopMar1996; @DemYou2006]. Equation states that the set $M^{-1}(A)$, which consists of points that are mapped onto $A$, has a measure that is smaller by a factor ${\ensuremath{\text{e}}}^{-\gamma}$ than the measure of $A$. The support of conditionally invariant measures is the backward trapped set ${\ensuremath{\Gamma_{\text{b}}}}$. The most important of these measures is the natural measure ${\ensuremath{\mu_{\text{nat}}}}$ with decay rate ${\ensuremath{\gamma_{\text{nat}}}}$ [@LopMar1996; @DemYou2006]. This measure is uniformly distributed on the backward trapped set ${\ensuremath{\Gamma_{\text{b}}}}$. We stress that for any decay rate $\gamma$ there are infinitely many different conditionally invariant measures [@DemYou2006; @NonRub2007]. So far it is unknown, if any of these classical measures corresponds to resonance eigenfunctions with arbitrary decay rates. Property ${\ensuremath{(ii)}}$ selects a specific class of measures which are uniformly distributed on subsets of ${\ensuremath{\Gamma_{\text{b}}}}$. Uniform distribution with respect to ${\ensuremath{\Gamma_{\text{b}}}}$ (the support of conditionally invariant measures) is equivalent to proportionality to the natural measure, explaining the appearance of ${\ensuremath{\mu_{\text{nat}}}}$ in Eq. . In analogy to quantum ergodicity for closed systems it is reasonable to consider for resonance eigenfunctions a uniform distribution on ${\ensuremath{\Gamma_{\text{s}}}}$, as classically this is an invariant set with chaotic dynamics. The quantum mechanical uncertainty relation, however, implies a finite phase-space resolution $h$ replacing ${\ensuremath{\Gamma_{\text{s}}}}$ by a quantum resolved saddle ${\ensuremath{\Gamma^h_{\text{s}}}}$. It is desirable to combine the assumption of uniformity on the saddle, the finite quantum resolution, and conditional invariance. This is achieved by introducing a temporal distance ${\ensuremath{t_h(x)}}$ to the quantum resolved saddle ${\ensuremath{\Gamma^h_{\text{s}}}}$ for all $x\in{\ensuremath{\Gamma_{\text{b}}}}$ and assuming uniformity on all sets with the same temporal distance. The resulting measures ${\ensuremath{\mu_{\gamma}^h}}$, Eq. , are conditionally invariant according to Eq. as can be shown using Eq. .
For the saddle distance ${\ensuremath{t_h(x)}}$ we now provide a conceptually and numerically simple implementation. For this we consider as a convenient definition of ${\ensuremath{\Gamma^h_{\text{s}}}}$ a symmetric surrounding of ${\ensuremath{\Gamma_{\text{s}}}}$,${\ensuremath{\Gamma^h_{\text{s}}}}= \{x\in\Gamma : d(x, {\ensuremath{\Gamma_{\text{s}}}}) \leq \sqrt{\hbar/2} \}$, with Euclidean distance $d$ smaller than the width of coherent states. We define an *integer saddle distance* $n\in\mathbb{Z}$ for $x\in{\ensuremath{\Gamma_{\text{b}}}}$ as the number of backward steps to enter the $h$-resolved saddle, $$\begin{aligned}
{\ensuremath{t_h(x)}}= n \ \, \Leftrightarrow \ \,
{\ensuremath{M}}^{-n}(x) \in {\ensuremath{\Gamma^h_{\text{s}}}},\end{aligned}$$ with ${\ensuremath{M}}^{-i}(x) \notin {\ensuremath{\Gamma^h_{\text{s}}}}$ for all $i < n$. For points inside of ${\ensuremath{\Gamma^h_{\text{s}}}}$ this leads to $n\leq 0$. Defining ${\ensuremath{\mathcal{E}^h_{n}}}:= \{x\in{\ensuremath{\Gamma_{\text{b}}}}: {\ensuremath{t_h(x)}}= n\}$ as the sets with integer saddle distance $n$ we obtain a partition of ${\ensuremath{\Gamma_{\text{b}}}}$ with ${\ensuremath{\mathcal{E}^h_{n}}}= {\ensuremath{M}}^n{\ensuremath{\mathcal{E}^h_{0}}}$. There is a maximal saddle distance ${\ensuremath{m_h}}$, and consequently the regions ${\ensuremath{\mathcal{E}^h_{n}}}$ with $n > {\ensuremath{m_h}}$ are empty sets. With this Eq. simplifies to $$\begin{aligned}
{\ensuremath{\mu_{\gamma}^h}}(A) &= \frac{1}{\mathcal{N}}\sum_{n = -\infty}^{{\ensuremath{m_h}}}
{\ensuremath{\text{e}}}^{n(\gamma -
{\ensuremath{\gamma_{\text{nat}}}})}
{\ensuremath{\mu_{\text{nat}}}}(A\cap {\ensuremath{\mathcal{E}^h_{n}}}), \label{EQ:mugh}\end{aligned}$$ for all $A\subset \Gamma$, which will be applied in the following.
*Example system.*—Throughout this paper we use the paradigmatic example of the standard map [@Chi1979] in its symmetric form $(q, p) \mapsto (q + p^\ast, p^\ast + v(q + p^\ast))$ with $p^\ast = p + v(q)$ and $v(q) = (\kappa/4\pi) \sin(2\pi q)$, considered on the torus $q\in[0,1)$, $p\in[0,1)$ with periodic boundary conditions. We consider a kicking strength $\kappa = 10$ to ensure a fully chaotic phase space. The opening is chosen as a vertical strip $\Omega$, such that $ q\in [0.2, 0.4]$ and $p\in [0, 1)$, see Fig. \[FIG:title\]. Position and size of $\Omega$ determine the classical decay rate ${\ensuremath{\gamma_{\text{nat}}}}\approx 0.21$ of the natural measure ${\ensuremath{\mu_{\text{nat}}}}$. We consider the Floquet quantization ${\ensuremath{U}_\text{cl}}$ [@BerBalTabVor1979; @ChaShi1986] of the closed map on a Hilbert space of dimension $1/h$ with effective Planck’s constant $h$. The quantum map is opened as ${\ensuremath{U}}= {\ensuremath{U}_\text{cl}}\cdot (\mathbbm{1} - P_{\Omega})$ with projector $P_\Omega$ on the opening [@BorGuaShe1991]. The eigenvalue problem of this subunitary propagator, $U\psi = \lambda\,\psi$, leads to eigenvalues with modulus less than unity, $|\lambda|^2 \equiv {\ensuremath{\text{e}}}^{-\gamma} < 1$. The decay rate $\gamma$ characterizes the time evolution of the corresponding resonance eigenfunction $\psi$. There is a broad distribution of decay rates $\gamma$ [@BorGuaShe1991; @SchTwo2004]. We compute the Husimi phase-space distribution $\mathcal{H}(q, p) = 1/h\ |\langle q, p | \psi\rangle|^2$ for each eigenfunction $\psi$ by taking the overlap with symmetric coherent states $|q, p\rangle$ centered at $(q, p)\in\Gamma$.
While Husimi distributions $\mathcal{H}$ of individual resonance eigenfunctions show strong quantum fluctuations, we want to explain their average behavior. Therefore we calculate the average Husimi distribution ${
\ensuremath{\langle \mathcal{H} \rangle_{{\gamma}}}}$, where the average is taken over eigenfunctions from the interval $[{\gamma}\cdot c, {\gamma}/c]$ around some $\gamma$-value of interest with constant $c = 0.95$. We improve this averaging by increasing the number of contributing Husimi distributions in two ways: First, we vary the Bloch phase $\theta_p \in \{0.04, 0.08, \dots, 0.96\}$ of the quantization $U$. Secondly, the inverse Planck’s constant is varied in $\{0.94, 0.96, 0.98, 1, 1.02, 1.04, 1.06\}\cdot h^{-1}$ for $h=1/1000$. Classical measures ${\ensuremath{\mu_{\gamma}^h}}$ are obtained as follows. Using the sprinkler method [@LaiTel2011] we approximate the chaotic saddle ${\ensuremath{\Gamma_{\text{s}}}}$ as a point set with more than $10^7$ points not leaving the system under ten forward and backward time steps, see Fig. \[FIG:TIMES\](a). Tenfold forward iteration of this set gives an approximation of ${\ensuremath{\Gamma_{\text{b}}}}$, see Fig. \[FIG:TIMES\](b). The uniform distribution on this point set approximates ${\ensuremath{\mu_{\text{nat}}}}$ which is used in Eq. . We partition ${\ensuremath{\Gamma_{\text{b}}}}$ into sets ${\ensuremath{\mathcal{E}^h_{n}}}$ by determining the integer saddle distance $n$ for each $x\in {\ensuremath{\Gamma_{\text{b}}}}$, such that $d({\ensuremath{M}}^{-n}(x), {\ensuremath{\Gamma_{\text{s}}}}) \leq \sqrt{\hbar/2}$ and $d({\ensuremath{M}}^{-n + 1}(x), {\ensuremath{\Gamma_{\text{s}}}}) > \sqrt{\hbar/2}$, shown in Figs. \[FIG:TIMES\](d) and (e) for two values of $h$. Note that the region with maximal saddle distance ${\ensuremath{m_h}}$ is similar for both considered $h$. The saddle distance $n$ varies for points on ${\ensuremath{\Gamma_{\text{b}}}}$ and in particular on the opening $\Omega$ for two reasons: the geometric distance along the manifold to the quantum resolved saddle ${\ensuremath{\Gamma^h_{\text{s}}}}$ and the variation of the local stretching, i.e. finite time Lyapunov exponents. In order to construct ${\ensuremath{\mu_{\gamma}^h}}$, we assign to each $x\in{\ensuremath{\Gamma_{\text{b}}}}$ a weight ${\ensuremath{\text{e}}}^{n(\gamma - {\ensuremath{\gamma_{\text{nat}}}})}$ according to the factor in Eq. . Integrating these weights over grid cells with chosen resolution $800 \times 800$ and normalizing we obtain a phase-space density numerically approximating ${\ensuremath{\mu_{\gamma}^h}}$.
*Comparison.*—In Fig. \[FIG:title\] we show the average phase-space distributions ${
\ensuremath{\langle \mathcal{H} \rangle_{\gamma}}}$ for $\gamma = 0.6$ and $\gamma = 2$ for $h=1/1000$. Because $\mathcal{H}(q, p)$ is the expectation value of the projector on a coherent state $|q, p\rangle$, we compute the classical analogue. This is obtained by a convolution of the constructed measures ${\ensuremath{\mu_{\gamma}^h}}$ with a Gaussian of the same width as the coherent state, i.e. with standard deviation $\sqrt{\hbar/2}$. This allows for quantum-classical comparison on the phase space. Overall we observe very good agreement concerning the support of the distributions, their weight on the opening $\Omega$, and their localization within $\Omega$. The Husimi distributions show the following features: First, they are supported by the smoothed backward trapped set. Secondly, one observes that their density on the opening $\Omega$ is larger than on its surrounding. The other stripes with larger density (than their surrounding) fall on the preimages ${\ensuremath{M}}^{-1}(\Omega)$ and ${\ensuremath{M}}^{-2}(\Omega)$, shown in Fig. \[FIG:TIMES\](c). Thirdly and most importantly, the Husimi distributions within $\Omega$ are not uniform on ${\ensuremath{\Gamma_{\text{b}}}}$, but show localization, which is stronger for larger $\gamma$. The same three observations hold for the constructed measures ${\ensuremath{\mu_{\gamma}^h}}$, where they directly follow from properties ${\ensuremath{(i)}}$ and ${\ensuremath{(ii)}}$. The first two observations are implied by conditional invariance. Note that the integrated weight on $\Omega$ increases with $\gamma$ as ${\ensuremath{\mu_{\gamma}^h}}(\Omega) = 1 - {\ensuremath{\text{e}}}^{-\gamma}$, which follows from Eq. . It also implies for the $k$-th preimage of the opening ${\ensuremath{\mu_{\gamma}^h}}(M^{-k}(\Omega)) = {\ensuremath{\text{e}}}^{-k\gamma}{\ensuremath{\mu_{\gamma}^h}}(\Omega)$, which agrees with the quantum mechanical analysis [@KeaNovPraSie2006]. For the third observation we explicitly need the saddle distance in our classical construction, which follows from property ${\ensuremath{(ii)}}$. Those parts of $\Omega$ with maximal saddle distance ${\ensuremath{m_h}}$, see Fig. \[FIG:TIMES\](d), show the largest enhancement due to the exponential factor in Eq. . Consequently, regions with smaller saddle distance are less enhanced. In conclusion we have found a classical explanation for the localization of resonance eigenfunctions. In particular, this shows that it is not an interference effect. Note that our previously proposed measures [@KoeBaeKet2015], which do not depend on $h$, only resemble the first two observations, but not the localization effect within the opening. Thus those measures fail to describe resonance eigenfunctions on a detailed level.
![Average Husimi distribution of resonance eigenfunctions (top) compared to constructed classical measures ${\ensuremath{\mu_{\gamma}^h}}$ (bottom) with $\gamma \in \{ {\ensuremath{\gamma_{\text{nat}}}}, 0.6, 1, 2 \}$ for (a) $h=1/1000$ and (b) $h=1/16000$ on phase-space region $[0.15, 0.45]\times[0.15, 0.45]$. Colormap as in Fig. \[FIG:title\], with fixed maximum for each $\gamma$ in (a) and in (b).[]{data-label="FIG:Zoom"}](fig3.pdf "fig:")\
*Dependence on $\gamma$.*—In Fig. \[FIG:Zoom\](a) we illustrate quantum (top) and classical (bottom) phase-space distributions zoomed into the phase-space region $(q,p) \in [0.15, 0.45]\times[0.15,0.45]$ for increasing decay rates $\gamma$ starting with ${\ensuremath{\gamma_{\text{nat}}}}$ for $h = 1/1000$. This region is chosen to contain the significant peaks in $\Omega$. As expected, at the natural decay rate ${\ensuremath{\gamma_{\text{nat}}}}$ the Husimi distribution is almost perfectly resembled by the (smoothed) natural measure ${\ensuremath{\mu_{\text{nat}}}}$. Eigenfunctions with larger $\gamma$ show an increasingly prominent localization. Classically, this is reproduced using the measures . Note that at $\gamma = 2$ also differences between classical and quantum densities can be seen. The main peak is sharper and stronger localized quantum mechanically than for the classical construction. We attribute this to the chosen simplification using an integer saddle distance. *Dependence on $h$.*—Figure \[FIG:Zoom\](b) shows the corresponding sequence of plots for much smaller effective Planck’s constant $h=1/16000$. The eigenfunctions resolve finer structures of the backward trapped set. Again, similarly good agreement between quantum and classical densities is found. In particular one observes stronger density variations on ${\ensuremath{\Gamma_{\text{b}}}}$ in form of arcs, e.g. for $\gamma = 1$. Classically their origin is the increased maximum saddle distance ${\ensuremath{m_h}}= 4$ and the finer partition of ${\ensuremath{\Gamma_{\text{b}}}}$ seen in Fig. \[FIG:TIMES\](e) especially in the opening. Furthermore, the sets of maximal saddle distance ${\ensuremath{m_h}}$ are similar, see Figs. \[FIG:TIMES\](d) and (e), such that the localization occurs in a similar region in Figs. \[FIG:Zoom\](a) and (b). Again, at $\gamma = 2$ sharper and stronger peaks occur in the quantum distribution than classically.
While numerically it is not possible to go to much smaller values of the effective Planck’s constant $h$, we briefly speculate about the semiclassical limit. Decreasing $h$ gives a smaller surrounding of ${\ensuremath{\Gamma_{\text{s}}}}$, such that the saddle distance ${\ensuremath{t_h(x)}}$ increases for all $x\in{\ensuremath{\Gamma_{\text{b}}}}$, including the maximum ${\ensuremath{m_h}}$. If for decreasing $h$ the difference ${\ensuremath{m_h}}- {\ensuremath{t_h(x)}}$ converges, one can show that the measures ${\ensuremath{\mu_{\gamma}^h}}$ converge towards a family of $\gamma$-dependent measures ${\ensuremath{\mu_{\gamma}}}$. In this case according to the hypothesis a semiclassical convergence of the eigenfunctions is expected. If such limit measures ${\ensuremath{\mu_{\gamma}}}$ exist, it is a challenging question, whether and how they can be calculated directly. Moreover one would have to test, whether the structure of resonance eigenfunctions for finite $h$ is well enough explained by $\mu_\gamma$. *Discussion.*—We have shown that the proposed resonance eigenfunction hypothesis for chaotic systems reproduces the average phase-space distribution of resonance eigenfunctions down to scales of order $h$. In particular the resulting measures ${\ensuremath{\mu_{\gamma}^h}}$ give a classical explanation of the quantum mechanically observed localization. Small deviations might be improved by more elaborate definitions of ${\ensuremath{\Gamma^h_{\text{s}}}}$ and the saddle distance ${\ensuremath{t_h(x)}}$, e.g. by considering in the definition of ${\ensuremath{\Gamma^h_{\text{s}}}}$ the distance along the unstable manifold or by considering continuous saddle distances from a smooth quantum resolved saddle. An application of the hypothesis to time-continuous systems, like open billiards and potential systems, is straightforward. A future challenge is the application to optical microcavities, which requires a generalization to partial transmission and reflection.
We are grateful to E. G. Altmann, L. Bunimovich, T. Harayama, E. J. Heller, S. Nonnenmacher, and H. Schomerus for helpful comments and stimulating discussions, and acknowledge financial support through the Deutsche Forschungsgemeinschaft under Grant No. KE 537/5-1.
[54]{} natexlab\#1[\#1]{}bibnamefont\#1[\#1]{}bibfnamefont\#1[\#1]{}citenamefont\#1[\#1]{}url\#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
O. Bohigas, M. J. Giannoni, and C. Schmit[: [ *Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.52.1)[**52**]{} (1984), 1–4.
M. V. Berry: [*Semiclassical theory of spectral rigidity*]{}, Proc. R. Soc. Lon. A [**400**]{} (1985), 229–251.
M. Sieber and K. Richter[: [ *Correlations between periodic orbits and their rôle in spectral statistics*]{}, Phys. Scripta ](http://dx.doi.org/10.1238/Physica.Topical.090a00128)[**2001**]{} (2001), 128–133.
S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland[: [ *Semiclassical Foundation of Universality in Quantum Chaos*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.93.014103)[**93**]{} (2004), 014103.
A. Voros[: [*Semi-classical ergodicity of quantum eigenstates in the [W]{}igner representation*]{}, in: [ *Stochastic Behavior in Classical and Quantum Hamiltonian Systems*]{} (Eds. G. Casati and J. Ford), vol. 93 of [*Lecture Notes in Physics*]{}, 326–333, (Springer Berlin Heidelberg, Berlin), ](http://dx.doi.org/10.1007/BFb0021732) (1979).
M. V. Berry[: [*Regular and irregular semiclassical wavefunctions*]{}, J. Phys. A ](http://dx.doi.org/10.1088/0305-4470/10/12/016)[**10**]{} (1977), 2083–2091.
M. V. Berry: [*Semiclassical mechanics of regular and irregular motion*]{}, in: [*Comportement Chaotique des Syst[è]{}mes D[é]{}terministes — Chaotic Behaviour of Deterministic Systems*]{} (Eds. G. Iooss, R. H. G. Helleman and R. Stora), 171–271, ([N]{}orth-[H]{}olland, [A]{}msterdam), (1983).
A. I. Shnirelman[: [*Ergodic properties of eigenfunctions (in [Russian]{})*]{}, Usp. Math. Nauk ](http://mi.mathnet.ru/eng/umn4463)[**29**]{} (1974), 181–182.
Y. [Colin de Verdière]{}[: [*Ergodicité et fonctions propres du laplacien (in [French]{})*]{}, Commun. Math. Phys. ](http://projecteuclid.org/euclid.cmp/1104114465)[**102**]{} (1985), 497–502.
S. Zelditch[: [ *Uniform distribution of eigenfunctions on compact hyperbolic surfaces*]{}, Duke. Math. J. ](http://dx.doi.org/10.1215/S0012-7094-87-05546-3)[**55**]{} (1987), 919–941.
S. Zelditch and M. Zworski[: [ *Ergodicity of eigenfunctions for ergodic billiards*]{}, Commun. Math. Phys. ](http://projecteuclid.org/euclid.cmp/1104276097)[**175**]{} (1996), 673–682.
S. De Bi[è]{}vre[: [*Quantum chaos: a brief first visit*]{}, in: [*[Second [[Summer School]{}]{} in [[Analysis]{}]{} and [[Mathematical Physics]{}]{} ([[Cuernavaca]{}]{}, 2000)]{}*]{} (Eds. S. P[é]{}rez-Esteva and C. Villegas-Blas), Contemp. Math. [**289**]{}, 161–218. [Amer. Math. Soc., Providence, RI]{}, ](http://dx.doi.org/10.1090/conm/289/04878) (2001).
S. Nonnenmacher and A. Voros[: [*Chaotic Eigenfunctions in Phase Space*]{}, J. Stat. Phys. ](http://dx.doi.org/10.1023/A:1023080303171)[**92**]{} (1998), 431–518.
A. Bäcker, R. Schubert, and P. Stifter[: [*Rate of quantum ergodicity in Euclidean billiards*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.57.5425)[**57**]{} (1998), 5425–5447, ; erratum ibid. [**58**]{} (1998) 5192.
M. Srednicki[: [*Chaos and quantum thermalization*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.50.888)[**50**]{} (1994), 888–901.
L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol[: [*From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics*]{}, Adv. Phys. ](http://dx.doi.org/10.1080/00018732.2016.1198134)[**65**]{} (2016), 239–362.
P. Gaspard[: [ *Quantum Chaotic Scattering*]{}, Scholarpedia](http://dx.doi.org/10.4249/scholarpedia.9806) [ **9**]{}(6): 9806, (2014).
G. E. Mitchell, A. Richter, and H. A. Weidenm[ü]{}ller[: [ *Random Matrices and Chaos in Nuclear Physics: [N]{}uclear Reactions*]{}, Rev. Mod. Phys. ](http://dx.doi.org/10.1103/RevModPhys.82.2845)[**82**]{} (2010), 2845–2901.
H.-J. Stöckmann[: [ *Quantum Chaos: [A]{}n [I]{}ntroduction*]{}, (Cambridge University Press, Cambridge), ](http://dx.doi.org/10.1017/CBO9780511524622) (1999).
G. Tanner and N. S[ø]{}ndergaard[: [ *Wave Chaos in Acoustics and Elasticity*]{}, J. Phys. A ](http://dx.doi.org/10.1088/1751-8113/40/50/R01)[**40**]{} (2007), R443–R509.
R. A. Jalabert[: [ *Mesoscopic Transport and Quantum Chaos*]{}, Scholarpedia](http://dx.doi.org/10.4249/scholarpedia.30946) [**11**]{}(1): 30946, (2016).
H. Cao and J. Wiersig[: [ *Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics*]{}, Rev. Mod. Phys. ](http://dx.doi.org/10.1103/RevModPhys.87.61)[**87**]{} (2015), 61–111.
F. Haake, F. Izrailev, N. Lehmann, D. Saher, and H.-J. Sommers[: [*Statistics of complex levels of random matrices for decaying systems*]{}, Z. Phys. B ](http://dx.doi.org/10.1007/BF01470925)[ **88**]{} (1992), 359–370.
Y. V. Fyodorov and H.-J. Sommers[: [*Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance*]{}, J. Math. Phys. ](http://dx.doi.org/10.1063/1.531919)[**38**]{} (1997), 1918–1981.
Y. Alhassid: [*The statistical theory of quantum dots*]{}, Rev. Mod. Phys. [**72**]{} (2000), 895–968.
P. Jacquod, H. Schomerus, and C. W. J. Beenakker[: [ *Quantum [A]{}ndreev Map: [A]{} Paradigm of Quantum Chaos in Superconductivity*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.90.207004)[**90**]{} (2003), 207004.
Y. V. Fyodorov and D. V. Savin[: [ *Statistics of Resonance Width Shifts as a Signature of Eigenfunction Nonorthogonality*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.108.184101)[**108**]{} (2012), 184101.
S. Kumar, B. Dietz, T. Guhr, and A. Richter[: [ *Distribution of Off-Diagonal Cross Sections in Quantum Chaotic Scattering: Exact Results and Data Comparison*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.119.244102)[**119**]{} (2017), 244102.
J. Sj[ö]{}strand[: [ *Geometric Bounds on the Density of Resonances for Semiclassical Problems*]{}, Duke Math. J. ](http://dx.doi.org/10.1215/S0012-7094-90-06001-6)[**60**]{} (1990), 1–57.
K. K. Lin[: [*Numerical Study of Quantum Resonances in Chaotic Scattering*]{}, J. Comput. Phys. ](http://dx.doi.org/10.1006/jcph.2001.6986)[ **176**]{} (2002), 295–329.
W. T. Lu, S. Sridhar, and M. Zworski[: [ *Fractal Weyl Laws for Chaotic Open Systems*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.91.154101)[ **91**]{} (2003), 154101.
H. Schomerus and J. Tworzyd[ł]{}o[: [ *Quantum-to-Classical Crossover of Quasibound States in Open Quantum Systems*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.93.154102)[**93**]{} (2004), 154102.
S. Nonnenmacher and M. Zworski[: [*Fractal Weyl laws in discrete models of chaotic scattering*]{}, J. Phys. A ](http://dx.doi.org/10.1088/0305-4470/38/49/014)[ **38**]{} (2005), 10683.
D. L. Shepelyansky[: [ *Fractal Weyl law for quantum fractal eigenstates*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.77.015202)[ **77**]{} (2008), 015202.
J. A. Ramilowski, S. D. Prado, F. Borondo, and D. Farrelly[: [*Fractal Weyl law behavior in an open Hamiltonian system*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.80.055201)[ **80**]{} (2009), 055201.
M. Kopp and H. Schomerus[: [*Fractal [Weyl]{} laws for quantum decay in dynamical systems with a mixed phase space*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.81.026208)[**81**]{} (2010), 026208.
M. J. Körber, M. Michler, A. Bäcker, and R. Ketzmerick[: [ *Hierarchical Fractal [W]{}eyl Laws for Chaotic Resonance States in Open Mixed Systems*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.111.114102)[**111**]{} (2013), 114102.
H. Ishio, A. I. Saichev, A. F. Sadreev, and K.-F. Berggren[: [*Wave Function Statistics for Ballistic Quantum Transport through Chaotic Open Billiards: [S]{}tatistical Crossover and Coexistence of Regular and Chaotic Waves*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.64.056208)[**64**]{} (2001), 056208.
Y.-H. Kim, M. Barth, H.-J. St[ö]{}ckmann, and J. P. Bird[: [*Wave Function Scarring in Open Quantum Dots: [A]{} Microwave-Billiard Analog Study*]{}, Phys. Rev. B ](http://dx.doi.org/10.1103/PhysRevB.65.165317)[**65**]{} (2002), 165317.
A. Bäcker, A. Manze, B. Huckestein, and R. Ketzmerick[: [ *Isolated resonances in conductance fluctuations and hierarchical states*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.66.016211)[**66**]{} (2002), 016211.
B. Weingartner, S. Rotter, and J. Burgd[ö]{}rfer[: [ *Simulation of Electron Transport through a Quantum Dot with Soft Walls*]{}, Phys. Rev. B ](http://dx.doi.org/10.1103/PhysRevB.72.115342)[**72**]{} (2005), 115342.
T. Weich, S. Barkhofen, U. Kuhl, C. Poli, and H. Schomerus[: [ *Formation and interaction of resonance chains in the open three-disk system*]{}, New J. Phys. ](http://dx.doi.org/10.1088/1367-2630/16/3/033029)[**16**]{} (2014), 033029.
C. Gmachl, F. Capasso, E. E. Narimanov, J. U. N[ö]{}ckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho[: [*High-Power Directional Emission from Microlasers with Chaotic Resonators*]{}, Science ](http://dx.doi.org/10.1126/science.280.5369.1556)[**280**]{} (1998), 1556–1564.
S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim[: [*Quasiscarred Resonances in a Spiral-Shaped Microcavity*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.93.164102)[**93**]{} (2004), 164102.
J. Wiersig and M. Hentschel[: [ *Combining Directional Light Output and Ultralow Loss in Deformed Microdisks*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.100.033901)[**100**]{} (2008), 033901.
S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, and E. E. Narimanov[: [ *Chaos-Assisted Directional Light Emission from Microcavity Lasers*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.104.163902)[**104**]{} (2010), 163902.
J.-B. Shim, J. Wiersig, and H. Cao[: [*Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.84.035202)[**84**]{} (2011), 035202.
T. Harayama and S. Shinohara[: [ *Ray-Wave Correspondence in Chaotic Dielectric Billiards*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.92.042916)[**92**]{} (2015), 042916.
G. Casati, G. Maspero, and D. L. Shepelyansky[: [ *Quantum fractal eigenstates*]{}, Physica D ](http://dx.doi.org/10.1016/S0167-2789(98)00265-6)[**131**]{} (1999), 311–316.
J. P. Keating, M. Novaes, S. D. Prado, and M. Sieber[: [ *Semiclassical Structure of Chaotic Resonance Eigenfunctions*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.97.150406)[**97**]{} (2006), 150406.
S. Nonnenmacher and M. Rubin[: [*Resonant eigenstates for a quantized chaotic system*]{}, Nonlinearity ](http://dx.doi.org/10.1088/0951-7715/20/6/004)[**20**]{} (2007), 1387.
L. Ermann, G. G. Carlo, and M. Saraceno[: [ *Localization of Resonance Eigenfunctions on Quantum Repellers*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.103.054102)[**103**]{} (2009), 054102.
D. Lippolis, J.-W. Ryu, S.-Y. Lee, and S. W. Kim[: [*On-manifold localization in open quantum maps*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.86.066213)[**86**]{} (2012), 066213.
G. G. Carlo, D. A. Wisniacki, L. Ermann, R. M. Benito, and F. Borondo[: [ *Classical transients and the support of open quantum maps*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.87.012909)[**87**]{} (2013), 012909.
M. Schönwetter and E. G. Altmann[: [*Quantum signatures of classical multifractal measures*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.91.012919)[**91**]{} (2015), 012919.
M. J. Körber, A. Bäcker, and R. Ketzmerick[: [ *Localization of Chaotic Resonance States due to a Partial Transport Barrier*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.115.254101)[**115**]{} (2015), 254101.
S. Fishman[: [ *[Anderson]{} Localization and Quantum Chaos Maps*]{}, Scholarpedia](http://dx.doi.org/10.4249/scholarpedia.9816) [**5**]{}(8): 9816, (2010).
E. J. Heller[: [ *Bound-State Eigenfunctions of Classically Chaotic [H]{}amiltonian Systems: [S]{}cars of Periodic Orbits*]{}, Phys. Rev. Lett. ](http://dx.doi.org/10.1103/PhysRevLett.53.1515)[**53**]{} (1984), 1515–1518.
Y.-C. Lai and T. Tél: [*Transient Chaos: [C]{}omplex Dynamics on Finite Time Scales*]{}, no. 173 in Applied Mathematical Sciences, (Springer Verlag, New York), 1st edn., (2011).
G. Pianigiani and J. A. Yorke[: [ *Expanding Maps on Sets Which Are Almost Invariant: [D]{}ecay and Chaos*]{}, Trans. Amer. Math. Soc. ](http://dx.doi.org/10.2307/1998093)[**252**]{} (1979), 351–366.
H. Kantz and P. Grassberger[: [ *Repellers, semi-attractors, and long-lived chaotic transients*]{}, Physica D ](http://dx.doi.org/10.1016/0167-2789(85)90135-6)[**17**]{} (1985), 75–86.
T. Tél[: [*Escape rate from strange sets as an eigenvalue*]{}, Phys. Rev. A ](http://dx.doi.org/10.1103/PhysRevA.36.1502)[**36**]{} (1987), 1502–1505.
A. Lopes and R. Markarian[: [*Open Billiards: [I]{}nvariant and Conditionally Invariant Probabilities on [C]{}antor Sets*]{}, [SIAM]{} J. Appl. Math. ](http://dx.doi.org/10.1137/S0036139995279433)[**56**]{} (1996), 651–680.
M. F. Demers and L.-S. Young[: [*Escape rates and conditionally invariant measures*]{}, Nonlinearity ](http://dx.doi.org/10.1088/0951-7715/19/2/008)[**19**]{} (2006), 377–397.
E. G. Altmann, J. S. E. Portela, and T. Tél[: [*Leaking chaotic systems*]{}, Rev. Mod. Phys. ](http://dx.doi.org/10.1103/RevModPhys.85.869)[**85**]{} (2013), 869–918.
M. Novaes[: [*Supersharp resonances in chaotic wave scattering*]{}, Phys. Rev. E ](http://dx.doi.org/10.1103/PhysRevE.85.036202)[**85**]{} (2012), 036202.
B. V. [Chirikov]{}[: [*[A universal instability of many-dimensional oscillator systems]{}*]{}, Phys. Rep. ](http://dx.doi.org/10.1016/0370-1573(79)90023-1)[**52**]{} (1979), 263–379.
M. V. Berry, N. L. Balazs, M. Tabor, and A. Voros[: [*Quantum maps*]{}, Ann. Phys. (N.Y.) ](http://dx.doi.org/10.1016/0003-4916(79)90296-3)[**122**]{} (1979), 26–63.
S.-J. Chang and K.-J. Shi[: [ *Evolution and exact eigenstates of a resonant quantum system*]{}, Phys. Rev. A ](http://dx.doi.org/10.1103/PhysRevA.34.7)[**34**]{} (1986), 7–22.
F. Borgonovi, I. Guarneri, and D. L. Shepelyansky[: [ *Statistics of Quantum Lifetimes in a Classically Chaotic System*]{}, Phys. Rev. A ](http://dx.doi.org/10.1103/PhysRevA.43.4517)[**43**]{} (1991), 4517–4520.
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---
abstract: 'In this paper we present a computation of the mean first-passage times both for a random walk in a discrete bounded lattice, between a starting site and a target site, and for a Brownian motion in a bounded domain, where the target is a sphere. In both cases, we also discuss the case of two targets, including splitting probabilities, and conditional mean first-passage times. In addition, we study the higher-order moments and the full distribution of the first-passage time. These results significantly extend our earlier contribution \[Phys. Rev. Lett. [**95**]{}, 260601\].'
author:
- 'S. Condamin'
- 'O. Bénichou'
- 'M. Moreau'
title: 'Random walks and Brownian motion: a method of computation for first-passage times and related quantities in confined geometries'
---
Introduction
============
The time it takes to a random walker to go from a starting site to a target site, the so called first-passage time (FPT), is an especially important quantity that underlies a wide range of physical processes[@VanKampen; @Redner]. Indeed, numerous real situations, such as diffusion limited reactions [@Rice] or animals searching for food [@nous], can be rephrased as first-passage problems. In all these situations, the FPT is a limiting factor. As a consequence, it is crucial to determine how this quantity depends on the parameters of the problem.
Among these parameters, geometrical factors turn out to be determining. For example, the mean first-passage time (MFPT) between a starting site and a target site for a 2D random walker is infinite if the walk is not bounded. On the contrary, it becomes finite as soon as the walk is confined. But how does the MFPT depend on the confining surface? In fact, the answer to this general question appears as a difficult task, because explicit determinations of FPT are most of the time limited to very artificial geometries, such as 1D and spherically symmetric problems [@Redner].
However, in most of the real situations, the searcher performs a random walk in more general confining geometries. This is for example the case in biology, where biomolecules often follow a complicated series of transformations, which are located at precise parts of the cell. Determining the influence of the shape of the cell on the FPT actually appears as a first step in the understanding of the global kinetics of the process.
This question of determining first passage properties in general confined geometries has raised a growing attention (see for example [@henry1; @henry2; @henry3; @ferraro1; @ferraro2; @blanco; @mazzolo; @benichou05; @condamin; @LevitzJPCM05; @LevitzPRL06]). Two important results have notably been obtained. First, in the case of discrete random walks, an expression for the mean first passage time (MFPT) between two nodes of a general network has been found [@Noh]. However, no quantitative estimation of the MFPT was derived in this paper. Second, the leading behavior of MFPT of a continuous Brownian motion at a small absorbing window of a general reflecting bounded domain has been given [@Holcman05; @bere]. These studies have even been extended to a situation with a deep potential well, leading to a generalization of the Kramers formula [@Langer]. In the case when this window is a small sphere within the domain, the behavior of MFPT has also been derived [@Pinsky]. This result is rigorous, but does not give access to the dependence of the MFPT with the starting site.
Very recently [@CondaminPRL], we have proposed a novel approach which allowed us to propose accurate estimations of first passage times of [*discrete*]{} random walks in confined geometry. Preliminary results concerning a continuous Brownian motion have also been announced. The main purpose of this paper is to provide a detailed analysis of this [*continuous*]{} case, relevant to many real physical situations. In addition, we extend our previous work in several directions, for both discrete and continuous cases: the complete distribution of FPTs is obtained; extra quantities, as conditional MFPT in the case of several targets or mean exit times by a small aperture of a general reflecting bounded domain, are derived.
The paper is structured as follows. In Sec. \[discret\], we first present the computation method of FPTs in the case of random walks on discrete lattices. This study includes the obtention of the MFPT, a comprehensive derivation of the expression of the higher order moments as well as the complete distribution of the FPT, whose physical meaning is extensively analysed. The situation with two competitive targets is also studied, and we compute MFPT, splitting probabilities, and conditional MFPT.
In Sec. \[continu\], we extend all these results to the case of a continuous Brownian motion, and detail the specific difficulties encountered in this case.
The explicit results obtained in Secs. \[discret\] and \[continu\] involve pseudo-Green functions of a Laplace type operator, with given boundary conditions. The Appendix \[approximations\] is devoted to the evaluation of these pseudo-Green functions. For several domain shapes, an exact formula can be obtained, which gives, for the quantities computed in the article, exact explicit expressions in the discrete case, or accurate approximations in the continuous case. For other domain shapes, basic approximations are proposed.
These results are briefly summarized in Sec. \[discussion\], with a discussion of the important parameters to take into account and of the qualitative behavior of the MFPT
Random walks on discrete lattices\[discret\]
============================================
Mean first-passage time
-----------------------
Let us consider a point performing a random walk on an arbitrary bounded lattice with reflecting boundaries. We want to compute the MFPT ${\langle\mathbf{T}\rangle}$ of the random walker at target site $T$, starting from a site $S$ at time $0$. We summarized this computation in a previous article [@CondaminPRL].
However, since it is the basis of all the developments explained in this article, we found useful to give it here in full detail, with the addition of several necessary precisions.
Our method is based on a formula given by Kac [@Aldous], concerning irreducible graphs, such that any point can be reached from any other point. An irreducible graph admits a unique stationary probability $\pi({\bf r})$ to be at site ${\bf r}$ (physically, this is the probability for a particle which has been in the domain for a long time to be at site ${\bf r}$. If the transition probabilities are symmetric this stationary probability is uniform.). We consider random walks starting from an arbitrary point of a subset $\Sigma$ of the lattice, chosen with probability $\pi({\bf r})/\pi(\Sigma)$, where $\pi(\Sigma) = \sum_{{\bf r} \in \Sigma} \pi({\bf r})$. Then, Kac’s formula asserts that the mean number of steps needed to return to any point of $\Sigma$, i.e. the mean first-return time (MFRT) to $\Sigma$ is $1/\pi(\Sigma)$. A simple proof of this result and of its extension to higher-order moments, which will be used later on, is given in Appendix \[AnnKac\].
Kac’s formula can be used to derive the MFPT ${\langle\mathbf{T}\rangle}$ by slightly modifying the original lattice (see Fig.\[astuce\]): we suppress all the original links starting from the target site $T$, and add a new one-way link from $T$ to the starting point $S$, whereas all other links are unchanged. In this new lattice, any trajectory starting from $T$ goes to $S$ at its first step, so that the MFRT to $T$ is just the MFPT from $S$ to $T$ in the former lattice, plus one.
An exact, formal expression for the MFPT can thus be derived for the most general finite graph. Consider $N$ points at positions ${\bf r}_1, \ldots, {\bf r}_N$ in an arbitrary space. The transition rates from point $j$ to point $i$ are denoted $w_{ij}$. If we assume that one transition takes place during each time unit we have: $$\sum_{i} w_{ij} = 1$$ Let ${\bf r}_T$ be the position of the target site, ${\bf r}_S$ be the position of the starting site, and $\pi({\bf r})$ be the stationary probability of the modified lattice. We write $\pi({\bf r}_T) = J$. According to Kac’s formula, the MFRT to $T$ on the modified graph is $1/J$, so that the MFPT from $S$ to $T$ in the original graph is $ \langle \mathbf{T}\rangle
= 1/J - 1$. All we need to find is the stationary probability $\pi$. It satisfies the following equation: $$\pi({\bf r}_i) = \sum_j w_{ij}\pi({\bf r}_j) + J\delta_{iS} -
J w_{iT}$$ where $\delta$ is the Kronecker symbol. To solve this equation, we define the auxiliary function $\pi'$, such that $\pi'({\bf r}_i)=\pi({\bf r}_i) - J\delta_{iT}$. It satisfies: $$\pi'({\bf r}_i) = \sum_j w_{ij}\pi'({\bf r}_j) + J\delta_{iS} -
J \delta_{iT}
\label{eqpip}$$ so that $\pi'$ has the following expression: $$\pi'({\bf r}_i) = (1-J)\pi_0({\bf r}_i) + J H({\bf r}_i|{\bf r}_S) -
J H({\bf r}_i|{\bf r}_T),
\label{valpip}$$ where $\pi_0$ is the stationary probability of the original lattice, and $H$ is the discrete pseudo-Green function [@Barton], which satisfies the two following equations: $$H({\bf r}_i|{\bf r}_j) = \sum_k w_{ik} H({\bf r}_k|{\bf r}_j)+
\delta_{ij} - \frac1N
\label{pseudogreen}$$ $$\sum_i H({\bf r}_i|{\bf r}_j) \equiv \bar{H},
\label{hbar}$$ where $\bar{H}$ is independent of $j$. Moreover, if $w_{ij}$ is symmetric, which will be the case in all the practical cases considered, $H$ will also be symmetric in its arguments. The pseudo-Green function can be seen as a generalization of the usual infinite-space Green function to a bounded domain. Indeed, Eq. (\[pseudogreen\]) without the $-1/N$ term corresponds to the definition of the infinite-space Green function, which would not have any solution for a finite domain with reflecting boundary conditions: it is necessary in this case to compensate the source term $\delta_{ij}$, and the simplest way to do so is to add the $-1/N$ term. The properties of this function are further discussed in Appendix \[AnnGreen\]. We can thus see that the solution (\[valpip\]) satisfies equation (\[eqpip\]), and ensures that $\pi$ is normalized.
The condition $\pi'({\bf r}_T)=0$ allows us to compute $J$ and to deduce the following exact expression: $$\langle \mathbf{T}\rangle = \frac{1}{\pi_0({\bf r}_T)}
[ H({\bf r}_T|{\bf r}_T) - H({\bf r}_T|{\bf r}_S) ]$$ If $w_{ij}$ is symmetric, and we will consider that this is the case in the rest of the paper, we simply have $\pi_0 = 1/N$, and we get the simpler formula: $$\langle \mathbf{T}\rangle = N
[ H({\bf r}_T|{\bf r}_T) - H({\bf r}_T|{\bf r}_S) ]
\label{randwalk}$$
This result may be obtained by an alternative and complementary approach. We consider that in the domain there is a constant flux $J$ of particles per time unit entering the domain at the source point $S$. The particles are absorbed when they reach the target, and, since all particles are eventually absorbed, we have an outcoming flux $J$ at the target. The average number of particles in the domain satisfies: $\mathcal{N}=J{\langle\mathbf{T}\rangle}$, which will allow the determination of ${\langle\mathbf{T}\rangle}$. Indeed, the average density of particles $\rho({\bf r})$ satisfies the following equation: $$\rho({\bf r}_i) = \sum_j w_{ij}\rho({\bf r}_j) + J\delta_{iS} - J\delta_{iT}.
\label{alternative}$$ The three terms of the equation correspond respectively to the diffusion of particles, the incoming flux in $S$, and the outgoing flux in $T$. This is exactly the same equation as Eq.(\[eqpip\]), with the same condition $\rho({\bf r}_T) = 0$, and thus admits a similar solution, with the difference that the total number of particles in the domain is not fixed a priori. The solution is thus: $$\rho({\bf r}_i) = \rho_0 + J H({\bf r}_i|{\bf r}_S) - J H({\bf
r}_i|{\bf r}_T)
\label{rhodiscret}$$ which gives, with the condition $\rho({\bf r}_T) = 0$ and the relation $J{\langle\mathbf{T}\rangle}= \mathcal{N} = N\rho_0$, the same result as before for the mean first-passage time, namely Eq.(\[randwalk\]). This formula is equivalent to the one given in [@Noh], but is expressed in terms of pseudo-Green functions. One advantage of the present method is that it may be easily extended to more complex situations, as it will be shown. Another advantage is that, although the pseudo-Green function $H$ is not known in general, it is well suited for approximations when the graph is a bounded regular lattice. The simplest one in this case is to approximate the pseudo-Green function by its infinite-space limit, the “usual” Green function: $H({\bf r}|{\bf r}')\simeq G_0({\bf r}-{\bf r}')$, which satisfies: $$G_0({\bf r}) = \frac{1}{\sigma}\sum_{{\bf r}' \in N({\bf r})}
G_0({\bf r}')+ \delta_{0{\bf r}}.
\label{green}$$ where $N({\bf r})$ is the ensemble of neighbours of ${\bf r}$, and $\sigma$ the coordination number of the lattice. The value of $G_0(0)$ and the asymptotic behaviour of $G_0$ are well-known [@Hughes]. For instance, for the 3D cubic lattice, we have: $G_0(0)
= 1.516386...$ and $G_0({\bf r}) \simeq 3/(2\pi r)$ for $r$ large. For the 2D square lattice, we have $G_0(0)-G_0({\bf r}) \simeq
(2/\pi) \ln(r) + (3/\pi) \ln2 + 2\gamma/\pi$, where $\gamma$ is the Euler gamma constant, and $(3/\pi) \ln2 + 2\gamma/\pi = 1.029374...$ . These estimations of $G_0$ are used for all the practical applications in the following. In some cases (especially in three dimensions when the target is far from any boundary), approximating $H$ by $G_0$ will give accurate results (see Fig. \[distst\]). The small correction is due to boundary effects, which are further discussed in Section \[discussion\]. In other cases it will only give an order of magnitude. In the case of a rectangular or parallepipedic domain an exact expression of $H$ is known [@CondaminJCP], and the FPT from any point to any other point in the domain can be computed exactly. This exact result and simple approximations, which can be used in other cases, are given in Appendix \[approximations\].
Application: absorbing opening in a reflecting boundary
-------------------------------------------------------
Another situation that may arise and can easily be dealt with is the case of an absorbing opening in a (locally) flat reflecting boundary of a bounded domain: we are interested in the mean time a particle takes to exit from the domain, if it may only exit by this opening. (see Fig. \[ouverture\]). We only consider regular lattices of dimension $d=2$ or $3$. We can define a target site, just behind the flat boundary. The problem here is that the pseudo-Green function for the domain plus the target site is difficult to compute, whereas the pseudo-Green function near a flat boundary can be easily evaluated, and is even known exactly if the domain is rectangular or parallepipedic. To solve this problem, we will call the site next to the target the approach site $A$. We indeed have to go through this approach site in order to reach the target site. We will call $\langle\mathbf{T}\rangle_{ST}$ the average time to reach the target site, starting from the source; $\langle\mathbf{T}\rangle_{SA}$ the average time to reach the approach site, still starting from the source; $\langle\mathbf{T}\rangle_{AA}$ the average time to return to the approach site, assuming the random walk does not go to the target site after exiting the approach site; $\langle\mathbf{T}\rangle_{AT}$ the average time to reach the target site, starting from the approach site. We have the following equations: $$\langle\mathbf{T}\rangle_{ST} = \langle\mathbf{T}\rangle_{SA} +
\langle\mathbf{T}\rangle_{AT}$$ since the random walk has to go through the approach site, and $$\langle\mathbf{T}\rangle_{AT} = \frac{2d-1}{2d}\left(\left< \mathbf{T}
\right>_{AA} +
\left<\mathbf{T}\right>_{AT}\right)+\frac1{2d}
\label{eqat}$$ Once the random walker is at the approach site, it may either go directly to the target site (probability $\frac1{2d}$, where $d$ is the dimension of the lattice) or go another way, in which case it has to go back to the approach site before finding the target site. Thus, $$\left<\mathbf{T}\right>_{AT} = (2d-1)\left<\mathbf{T}\right>_{AA} + 1$$ To compute $\langle \mathbf{T}\rangle_{AA}$, we have to remember that, if the boundary was fully reflecting, we would have the average return time: it is given by Kac’s formula, and is $N$. We then have, with arguments similar to Eq. \[eqat\]: $$N = \frac{2d-1}{2d}\langle \mathbf{T}\rangle_{AA}+ \frac1{2d}$$ We then have: $$(2d-1) \langle \mathbf{T}\rangle_{AA}= 2dN - 1$$ and thus: $${\langle\mathbf{T}\rangle}_{AT}=2dN$$ As for the average time needed to reach the approach site, starting from the starting site, it is exactly the same as in the case where the boundary is totally reflecting: $$\langle \mathbf{T}\rangle_{SA} = N [ H({\bf r}_A|{\bf r}_A) - H({\bf
r}_A|{\bf r}_S) ],$$ and finally: $$\langle \mathbf{T}\rangle_{ST} = N [2d+ H({\bf r}_A|{\bf r}_A) - H({\bf
r}_A|{\bf r}_S)],$$
To evaluate $H$, we have to take into account the effect of the boundary. Since the boundary is flat, the simplest way to check the boundary condition is to write $H({\bf r}|{\bf r}') \simeq G_0({\bf r}-{\bf r}')+
G_0({\bf r}-s({\bf r}'))$, where $s({\bf r})$ is the point symmetrical to ${\bf r}$ with respect to the boundary. We will use this approximation in the following, (cf. Appendix \[approximations\] for a discussion of this approximation) We note $G_0(1)=G_0(0)-1$ the Green function for the sites surrounding the origin, and notice that $T$ is symmetrical to $A$ with respect to the boundary. The mean exit time is then: $$\langle\mathbf{T}\rangle_{ST} \simeq N[2d+G_0(0)+G_0(1)-
G_0({\bf r}_S-{\bf r}_A)-G_0({\bf r}_S-{\bf r}_T)]$$
Higher-order moments
--------------------
Moreover, we are able to evaluate the higher-order moments and distribution of the FPT in the 3D case, provided the domain is not too elongated, i.e. the typical distance between a point and a boundary is $N^{1/3}$. The computation of the moments is detailed in Appendix \[computationdiscret\]. However, we cannot compute the higher-order moments and distribution of the FPT in two dimensions, or with a too elongated 3D domain. The computational reasons behind this are explained in Appendix \[computationdiscret\], but we will also explain it later from a physical point of view. We obtain the following result for higher-order moments: $$\left<\mathbf{T}^n\right>_i = n!N^n\left[ \left(
H({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_i)\right)
\left( H({\bf r}_T|{\bf r}_T) - \bar{H} \right)^{n-1}
+\mathcal{O}(nN^{-2/3}) \right],
\label{eqmoments}$$ where $\bar{H}$ is defined by Eq. (\[hbar\])
To check these results, we computed the moments with a numerical simulation (cf. Appendix \[simulations\] for the simulation method), and found (see Fig.\[moments\]) a good agreement with the theoretical estimation (\[eqmoments\]), where $H$ is approximated by $G_0$, and $\bar{H}$ is approximated by is its value for a spherical domain, computed in the continuous limit, $\bar{H} = (18/5)(3/(4\pi))^{2/3}N^{-1/3}$ (cf. Eq. (\[valuehbar\]) for the computation).
The study of the distribution in the limit of large $N$ will enable us to go even further. Indeed, if we neglect the corrections in $nN^{-2/3}$ in Eq. (\[eqmoments\]), the moments of $T/N$ are those of the following probability density $p$: $$p(t) = \left(\frac{H({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_S)}
{(H({\bf r}_T|{\bf r}_T) - \bar{H})^2}\right)
\exp\left(-\frac{t}{H({\bf r}_T|{\bf r}_T) - \bar{H}}\right)
+ \frac{H({\bf r}_T|{\bf r}_S)-\bar{H}}{H({\bf r}_T|{\bf r}_T) - \bar{H}}
\, \delta(t)
\label{probdist}$$ The large-$N$ limit of this probability density is rigorous (since the corrections to the moments vanish). In this limit, $H({\bf r}_T|{\bf r}_T)$ tends to $G_0(0)$; $\bar{H}$ tends to $0$. Thus, the probability density of $\mathbf{T}/N$ tends to the following probability density, the relative position of $i$ and $T$ being fixed: $$p(t) = \left(\frac{G_0(0)-G_0({\bf r}_T-{\bf r}_S)}{G_0^2(0)}\right)
\exp\left(-\frac{t}{G_0(0)}\right)
+ \frac{G_0({\bf r}_T-{\bf r}_i)}{G_0(0)} \delta(t)
\label{problimit}$$
These results have been confronted to numerical simulations (Fig.\[distrib2\]). We computed the exact distribution for several domain sizes, and may notice that the curve divides in two at short times. This is due to the fact that, at short times, the parity of the step is important: as long as the walk does not touch the boundary, the distance between the starting point and the walker has the same parity as the time elapsed. The time needed for the two curves to collapse into one shows very well the time needed to erase the memory of the starting position. The curves before this time correspond to the Dirac part of the probability density (\[probdist\]); however, we can see that, once the two curves have collapsed, the resulting curves fit very well the theoretical prediction (\[probdist\]), which is indeed more accurate than the limit probability density (\[problimit\]).
To analyse the physical meaning of this result, we may first notice that, if the probability density (\[probdist\]) is averaged on the starting point, the Dirac part of the density vanishes, and we simply have an exponential distribution of the first-passage time. This property sheds a new light on the quasi-chemical approximation [@bere], which assumes that if a particle starts randomly in a volume, and may only exit through a small hole, it has a constant probability to exit at each time step. This approximation leads to an exponential distribution of the exit times. If we consider that the target site is the exit point for a particle, then the exit time is exactly the FPT. Thus, we have an evaluation of the accuracy of the quasi-chemical approximation (or at least of its moments) in this case.
The interpretation of the probability density (\[probdist\]) is the following: the first part of the density, which decays exponentially, corresponds to the decay of the probability distribution of the FPT if the particle starts randomly in the set of points. The second part corresponds to a particle reaching the target in a time negligible with respect to $N$. Here we must remember that a free 3D walk is transient: the particle may never reach the target in infinite space. Thus, one can interpret the Dirac term as the probability to reach the target *without touching the boundaries*. For $N$ large enough it is equivalent to the probability to reach the target at all in infinite space. And, for this kind of trajectory, the probability distribution of the FPT does not depend of $N$, and, thus, the probability density of $\mathbf{T}/N$ will tend to $\delta(0)$ for large $N$. On the other hand, if the particle *does* reach the boundary (it happens after a typical time $N^{2/3}$, since the boundaries are at a typical distance $N^{1/3}$, and the typical time needed to cross a distance $r$ is $r^2$), its position will become random in a time negligible with respect to $N$, and, thus, the probability density of $\mathbf{T}/N$ will be the same as if the particle started in a random position in this latter case. This argument fails for an elongated domain, which can be seen as a physical reason why we are not able to compute the FPT distribution in this case. We can check that the probability to reach the origin for a random walk in infinite space is indeed $\frac{G_0({\bf r})}{G_0(0)}$ [@Hughes; @Spitzer].
Here we can see an important physical difference between the 2D and 3D cases: in two dimensions the random walk is recurrent. We can thus conclude that the large-$N$ limit probability density of $\mathbf{T}/N$ will be a simple delta function, since, in the limit of infinite space, the particle almost certainly reaches the target in a finite time, even if the MFPT is infinite! However, the probability distribution for $\emph{finite}$ $N$ will be much more difficult to compute: Indeed, there will also be two regimes, of low $\mathbf{T}/N$, when the particles have not touched the boundary and thus the distribution is the same as in free space; and the regime of high $\mathbf{T}/N$, where the distribution decays exponentially (since the system has lost the memory of its starting point). The transition between the two regimes happens at a finite $\mathbf{T}/N$ (since the time needed to reach the boundaries is of order $N$). Thus, the low $\mathbf{T}/N$ regime will have a much stronger influence on the values of the moments than on the $3D$ case, which may explain why the computation of the moments and distribution is much more delicate in this case. We can see in Fig. \[distrib2D\] typical probability distributions for different domain sizes. One can very well see that the transition between the two regimes takes place at a finite $\mathbf{T}/N$ no matter the size of the domain, and that the long-time regime indeed corresponds to an exponential decay.
Case of two targets
-------------------
We can now assume that the lattice contains not one but two target points $T_1$ and $T_2$. The problems that may arise in this case are the mean time needed to reach one of the two targets, which we will call *mean absorption time* and note $\langle\mathbf{T}\rangle$, and the *splitting probabilities*, i.e. the probabilities $P_1$ to reach $T_1$ before $T_2$ and $P_2$ to reach $T_2$ before $T_1$. This model corresponds to the case of a diffusing particle which may be absorbed either by the target $T_1$ or the target $T_2$. We can also, even if it will be less straightforward, study the *conditional* mean absorption time, i.e. the mean absorption time $\langle \mathbf{T}_1 \rangle$ (resp. $\langle \mathbf{T}_2 \rangle$), for particles which are absorbed by the target $T_1$ (resp. $T_2$). This is relevant in many chemical applications [@Rice], and may be useful in biology to determine to which extent cellular variability may be controlled by diffusion [@Zon].
To compute these quantities, it is more convenient to use the alternative approach presented on page : we consider a constant incoming flux of particles $J$, and we have an average outcoming flux of particles $J_1$ in $T_1$, and $J_2$ in $T_2$. Since all particles are eventually absorbed, $J_1+J_2=J$. The probability to reach the target $i$ is then $P_i = J_i/J$. The total number of particles $\mathcal{N}$ in the domain satisfies $\mathcal{N} = J{\langle\mathbf{T}\rangle}$, and the mean density of particles satisfies the following equation: $$\rho({\bf r}_i) = \sum_j w_{ij}\rho({\bf r}_j) + J\delta_{iS} - J_1\delta_{iT_1}
- J_2\delta_{iT_2}$$ We then get: $$\rho({\bf r}_i) = \rho_0 + J H({\bf r}_i|{\bf r}_S) -
J_1 H({\bf r}_i|{\bf r}_{T_1}) - J_2 H({\bf r}_i|{\bf r}_{T_2}),
\label{2trho}$$ then, writing $\rho({\bf r}_{T_1})=\rho({\bf r}_{T_2})=0$, we get the following set of equations: $$\left\{ \begin{array}{rcl}
\rho_0 + JH_{1s} - JP_1H_{01} - JP_2H_{12}
& = & 0 \\
\rho_0 + JH_{2s} - JP_2H_{02} - JP_1H_{12}
& = & 0 \\
P_1+P_2 & = & 1 \\
\end{array} \right.$$ where $H_{12} = H({\bf r}_{T_1}|{\bf r}_{T_2})$ and, for $i = 1$ or $2$, $H_{is} = H({\bf r}_{T_i}|{\bf r}_S)$, $H_{0i} = H({\bf r}_{T_i}|{\bf r}_{T_i})$. From this set of equation we can deduce $P_1$, $P_2$ and $\rho_0 = J{\langle\mathbf{T}\rangle}/N$. We thus get exact expressions for the mean absorption time and the splitting probabilities, respectively: $$\langle \mathbf{T}\rangle = N
\frac{(H_{01}-H_{1s})(H_{02}-H_{2s}) - (H_{12}-H_{2s})(H_{12}-H_{1s})}
{H_{01}+H_{02}-2H_{12}}$$ $$\left\{
\begin{array}{l}
P_1 = \frac{H_{1s}+H_{02}-H_{2s}-H_{12}}{H_{01}+H_{02}-2H_{12}} \\
P_2 = \frac{H_{2s}+H_{01}-H_{1s}-H_{12}}{H_{01}+H_{02}-2H_{12}} \\
\end{array}
\right.$$
This result can be extended if necessary to more than two targets; if there are $n$ targets, we have $n+1$ unknown variables ($\rho_0$ and the $n$ probabilities $P_k$), with $n+1$ equations, namely $\sum P_k = 1$ and the $n$ equations $\rho({\bf r}_{T_k})=0$, which is enough to determine all the unknown variables. However, this may quickly become computationally expensive for a large number of targets.
We compared the two-target results to simulations (Fig. \[2t\]). Note that if we use the exact value for $H$, which we can compute for a cube (cf. Appendix \[approximations\]), it is indeed impossible to see a difference between the theoretical predictions and the simulations.
It is interesting to underline an important qualitative difference between the 2D and 3D cases. In 3D, the furthest target always has a significant probability to be reached first, since the most important terms in the probabilities $P_i$ are $H_{01}$ and $H_{02}$. In 2D, if a target is much closer from the source than the other, it will almost certainly be reached first, since $H({\bf r}_i|{\bf r}_j)$ scales as $\ln|{\bf r}_i-{\bf r}_j|$. Actually, the probability for the furthest target to be reached first decreases logarithmically. These properties are related to the transient character of the infinite 3D walk, and the recurrent character of the 2D walk: indeed, an infinite 2D walk explores all the sites of the lattice, whereas an infinite 3D walk does not; we may thus consider that the 2D walk will explore most of the sites surrounding the source before going much further, whereas the 3D walk will not, which qualitatively explains the difference of behaviour.
We can also determine the conditional absorption times ${\langle\mathbf{T}_{1}\rangle}$ and ${\langle\mathbf{T}_{2}\rangle}$. For this, we will compute $\mathcal{N}_k$, the average number of particles in the domain which will eventually be absorbed by $T_k$. We have $\mathcal{N}_k = J_k{\langle\mathbf{T}_{k}\rangle}$, which will allow us to compute ${\langle\mathbf{T}_{k}\rangle}$. To compute $\mathcal{N}_k$, we can simply notice that the density of particles that will eventually be absorbed by $T_k$ at the point $i$ is simply $\rho({\bf r}_i)P_k({\bf r}_i)$, where $P_k({\bf r}_i)$ is the probability to be absorbed by $T_k$ if the walk starts from $i$. We thus have: $$\mathcal{N}_k = \sum_{i}\rho({\bf r}_i)P_k({\bf r}_i)
\label{Ncondtimes}$$ This equation is exact but may prove quite difficult to compute, especially in two dimensions if $H$ is not known exactly. However, in three dimensions, we may use the same kind of approximations as for the computation of the high-order moments of the FPT (with the same limitations, i.e. the 3D domain should not be too elongated) to estimate the conditional probabilities. If we note $H_{iS} = H({\bf r}_i|{\bf r}_s)$ and $H_{ik}=H({\bf r}_i|{\bf r}_{T_k})$, we have: $$\mathcal{N}_1 = \sum_i\frac{(H_{i1}-H_{i2}+H_{02}-H_{12})
(\rho_0+JH_{iS}-J_1H_{i1}-J_2H_{i2})}{H_{01}+H_{02}-2H_{12}}$$ We use the properties $\sum_i H({\bf r}_i|{\bf r}_j) = N\bar{H}$ (cf. Eq. (\[hbar\])) and $\sum_i H({\bf r}_i|{\bf r}_j)H({\bf r}_i|{\bf r}_k) = \mathcal{O}(N^{1/3})$ (cf. Eq. \[scaling\]) to write: $$\mathcal{N}_1 = N\frac{(H_{02}-H_{12})\rho_0 + \mathcal{O}(N^{-2/3})}{H_{01}+
H_{02}-2H_{12}}$$ And we can conclude: $${\langle\mathbf{T}_{1}\rangle} = \frac{1}{P_1}\frac{H_{02}-H_{12}+\mathcal{O}(N^{-2/3})}{H_{01}+
H_{02}-2H_{12}}{\langle\mathbf{T}\rangle}\label{condtimes}$$ The expression for ${\langle\mathbf{T}_{2}\rangle}$ is of course equivalent. This expression is not exact, but is very accurate: the relative difference between the numerical simulations (see Fig.\[time2t\]) and the expression (\[condtimes\]) is of about $0.01 \%$, for a domain of size $N = 51^3$.
Finally, we have a wide range of quantities which can be computed exactly, or with a very good accuracy, provided we know the pseudo-Green function $H$. Unfortunately, there are only a few cases in which it can be computed exactly. Otherwise, we will have to use approximations, which, of course, give less accurate results. Both exact results and approximations are detailed in Appendix \[approximations\].
Brownian motion on continuous media\[continu\]
==============================================
We may consider a similar problem in a continuous medium (see Fig. \[schemabrown\]): if we have a Brownian motion whose diffusion coefficient is $D$, how much time does it take to reach a target? A difference with the discrete case is that the target has a finite size $a$ which is an important parameter of the problem. We will consider a spherical target $T$, of radius $a$, centered in ${\bf r}_T$. The Brownian motion starts from the starting point $S$ (its position is denoted by ${\bf r}_S$). It is restricted to a domain $\mathcal{D}$ of volume $V$ (for $2D$ domains we will call the area $A$) , and we note $\mathcal{D}^*$ the domain deprived of the target. We will derive the same quantities as in the discrete case, but the results are this time only approximate; we can thus add some refinements to the method, in order to increase the accuracy. These refinements are given in Appendix \[refinements\], and are used in practical computations of the MFPT in Appendix \[approximations\] when the target is close to a boundary. It should be emphasised that, in the cases where the pseudo-Green function is known, such as the case of a spherical domain, the method gives accurate explicit expressions for all the MFPT and the other quantities studied here.
Mean first passage time
-----------------------
The mean first passage time (MFPT) ${\langle\mathbf{T}({\bf r}_{s})\rangle}$ at the target satisfies the following equations [@Risken]: $$D \Delta {\langle\mathbf{T}({\bf r}_{s})\rangle}=-1\;{\rm if}\; {\bf r_s}\in{\mathcal D}^*$$ $${\langle\mathbf{T}({\bf r}_{s})\rangle}=0 \;{\rm if}\; {\bf r_s}\in \Sigma_{\rm abs}$$ $$\partial_n{\langle\mathbf{T}({\bf r}_{s})\rangle}=0 \;{\rm if}\; {\bf r_s}\in \Sigma_{\rm refl}$$ where $\Sigma_{\rm abs}$ (resp. $\Sigma_{\rm refl}$) stands for the surface of the absorbing target sphere (resp. the reflecting confining surface) and $\partial_n$ denotes the normal derivative. The boundaries have to be regular enough (twice continuously differentiable is sufficient, but not necessary) for these definitions to make sense.
To solve this problem, we introduce the following Green function $G({\bf r}|{\bf r'})$ defined by $$\label{G1}
-\Delta G({\bf r}|{\bf r'}) =\delta({\bf r}-{\bf r'}) \;{\rm if}\; {\bf r}\in{\mathcal D}^*$$ $$\label{G2}
G({\bf r}|{\bf r'})=0 \;{\rm if}\; {\bf r}\in \Sigma_{\rm abs}$$ $$\label{G3}
\partial_n G({\bf r}|{\bf r'})=0 \;{\rm if}\; {\bf r}\in \Sigma_{\rm refl}$$ Note that this Green function may also be seen as the stationary density of particles if there is an unit incoming flux of particles in ${\bf r}'$, and the diffusion coefficient is set to $1$. It should not be confused with the free Green function $G_0$, and is rather the continuous equivalent of the average density of particles $\rho$ defined in Eq. (\[alternative\]) with $J=1$. It depends implicitly on the target position through Eq. (\[G2\]).
Using Green’s formula, $$\int_{{\mathcal D}^*}\left(\left< \mathbf{T(r)}\right>\Delta G({\bf r}|{\bf r'})
-G({\bf r}|{\bf r'})\Delta\left< \mathbf{T(r)}\right>\right)d^d{\bf r}=
\int_{\Sigma_{\rm abs}+\Sigma_{\rm refl}} \left(\left< \mathbf{T(r)}\right>
\partial_n G({\bf r}|{\bf r'})-G({\bf r}|{\bf r'})
\partial_n\left< \mathbf{T(r)}\right>\right)d^{d-1}{\bf r},$$ we easily find that the MFPT is given by $$\label{Texpl}
{\langle\mathbf{T}({\bf r}_{S})\rangle}=\frac{1}{D}\int_{\mathcal{D}^*} G({\bf r}|{\bf r}_S)
d^d{\bf r}$$
To approximate $G({\bf r}|{\bf r}_S)$ we can use a direct transposition to the continuous case of Eq. (\[rhodiscret\]) : $$\label{Gapprox}
G({\bf r}|{\bf r}_S)\simeq \rho_0({\bf r}_S)+H({\bf r}|{\bf r}_S)-
H({\bf r}|{\bf r}_T)$$ where $\rho_0$ is defined by $G({\bf r}|{\bf
r}_S)\simeq0$ if ${\bf r}\in \Sigma_{\rm abs}$ and $H({\bf r}|{\bf
r}')$ is the pseudo-Green function [@Barton], which satisfies: $$\label{1}
- \Delta H({\bf r}|{\bf r'}) =\delta({\bf
r}-{\bf r'}) -\frac{1}{V}\;{\rm if}\; {\bf r}\in{\cal D}$$ $$\label{2}
\partial_n H({\bf r}|{\bf r'})=0 \;{\rm if}\; {\bf r}\in \Sigma_{\rm
refl}$$ $$\label{3}
H({\bf r}|{\bf r'})=H({\bf r'}|{\bf r})$$ $$\label{4}
\int_{\mathcal{D}}H({\bf r}'|{\bf r})d^d{\bf
r}' \equiv V\bar{H},$$ $\bar{H}$ being independent of ${\bf r}$. This latter equation can be easily deduced from the three previous ones.
The choice (\[Gapprox\]) of $G({\bf r}|{\bf r'})$ is the simplest one which satisfies formally Eqs (\[G1\]) and (\[G3\]). However, (\[G2\]) can only be approximately satisfied. To take into account this latter equation, we will approximate, on the target sphere, $H({\bf r}|{\bf r}_S)$ by $H({\bf r}_T|{\bf r}_S)$ and $H({\bf r}|{\bf r}_T)$ by $G_0({\bf r}-{\bf r}_T)+H^*({\bf r}_T|{\bf r}_T)$, where $G_0$ is the well-known free Green function( $(2\pi)^{-1} \ln(r)$ in 2D, $1/(4\pi r)$ in 3D), and $H^*$ is defined by: $$H^*({\bf r}|{\bf r}') \equiv H({\bf r}|{\bf r}') - G_0({\bf r}-{\bf r}').$$ Note that $H^*({\bf r}|{\bf r}_T)$ has no singularity in ${\bf r}_T$. Thus on the surface of the target sphere we have: $$\rho_0({\bf r}_S)+H({\bf r}_T|{\bf r}_S)-G_0(a)-H^*({\bf r}_T|{\bf r}_T) = 0 ,$$ where $G_0(a)$ is the value of $G_0({\bf r})$ when $|{\bf r}| = a$. We can now compute $${\langle\mathbf{T}({\bf r}_{S})\rangle} = \frac1D \int_{\mathcal{D}^*} \left( \rho_0({\bf r}_S) +
H({\bf r}|{\bf r}_S) - H({\bf r}|{\bf r}_T) \right) d^d{\bf r}$$ Since the target is small compared to the domain, the integral over $\mathcal{D}^*$ is almost equal to the integral over $\mathcal{D}$, the relative order of magnitude of the correction being $a^3/V$ in 3D and $a^2/A$ in 2D. Using the property (\[4\]), we can then compute the integral, and find the result: $${\langle\mathbf{T}({\bf r}_{S})\rangle} = \frac{V\rho_0({\bf r}_S)}{D} = \frac{V}{D}\left(G_0(a)+
H^*({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_S)\right)+ \mathcal{O}
\left(\frac{a^dG_0(a)}{D}\right)
\label{simplebrownian}$$ This equation is very close to (\[randwalk\]), with the correspondence $H({\bf r}|{\bf r}) \rightarrow G_0(a)+H^*({\bf r}|{\bf r})$, but one should pay attention to the fact that this is only an approximation! One may expect deviations from this expression when the variations of $H({\bf r}|{\bf r}_S)$ or $H^*({\bf r}|{\bf r}_T)$ will not be negligible over the target sphere; it corresponds to the cases when the target is either near the source or near a boundary. However, if we compare the expression obtained with simulations (see Fig.\[ctpos2D\]) when the target is near the source, we see no such deviation; this is justified in Appendix \[refinements\]. On the other hand, there is indeed a deviation near the boundaries. This deviation scales as $a/d$ in two dimensions, or $a/d^2$ in three dimensions, where $d$ is the distance between the target and the boundary. It is possible to compute a correction, which is explicited in Appendix \[refinements\], and used in practical applications in Appendix \[approximations\].
The exact value of $H$ is known analytically for disks and spheres [@Barton], we will detail this in Appendix \[approximations\]. This is why we will test the expressions we obtain in such geometries. If no exact expression is known, the simplest approximation of $H$ is simply $H = G_0$. More accurate approximations are also discussed in Appendix \[approximations\]. We give the estimations of ${\langle\mathbf{T}({\bf r}_{S})\rangle}$ with the basic approximation, to show the order of magnitude: $${\langle\mathbf{T}({\bf r}_{S})\rangle} = \frac{V}{4\pi D}\left(\frac1a-\frac1R\right) \;\;{\rm (3D)}
\label{base3D}$$ $${\langle\mathbf{T}({\bf r}_{S})\rangle} = \frac{A}{2\pi D}\ln\frac{R}{a} \;\;{\rm (2D)}
\label{base2D}$$ $R$ being the source-target distance. This already improves the (exact) asymptotic results of Pinsky [@Pinsky], which only give the leading term in $a$.
Higher-order moments
--------------------
The higher-order moments and density of the FPT in the three-dimensional case can also be computed. The computation is detailed in Appendix \[computationcontinu\]; the results are quite similar to the results obtained in the discrete case, and the physical interpretation is essentially the same. The results obtained are the following:
$${\langle\mathbf{T}^{n}({\bf r}_{S})\rangle} = \frac{n!V^n}{D^n}\left[
\left(G_0(a) + H^*({\bf r}_T|{\bf r}_T) - H({\bf r}_T|{\bf r}_S)\right)
\left(G_0(a) + H^*({\bf r}_T|{\bf r}_T) - \bar{H}\right)^{n-1}
+\mathcal{O}\left(nV^{-2/3}a^{2-n} \right) \right]$$
We may also deduce from this information about the probability density of the absorption time $p(t)$: If we drop the term $\mathcal{O}\left(nV^{-2/3}a^{2-n}\right)$, we have: $$\begin{aligned}
p(t) & = & \frac{D}{V}
\frac{G_0(a)+H^*({\bf r}_T|{\bf r}_T )-H({\bf r}_T|{\bf r}_S)}{\left(G_0(a)+H^*({\bf r}_T|{\bf r}_T )-\bar{H}\right)^2}
\exp\left(\frac{-Dt}{V\left(G_0(a)+H^*({\bf r}_T|{\bf r}_T )-\bar{H}\right)}\right) \\
&&
+ \frac{H({\bf r}_T|{\bf r}_S)-\bar{H}}{G_0(a)+H^*({\bf r}_T|{\bf r}_T )-\bar{H}} \delta(t) \nonumber\end{aligned}$$ In the limit $a \rightarrow 0$, with the position of ${\bf r}_S$ fixed, the $H$ terms are constant since they only depend on the shape of the domain, and $G_0(a)$ tends towards infinity. The probability density then simply becomes exponential: $$p(t) = \frac{4\pi aD}{V}\exp\left(-\frac{4\pi aDt}{V}\right)$$ In the limit $a \rightarrow 0$, with the quantity $R/a$ fixed, $H({\bf r}_S|{\bf r}_T) \sim G_0(R)$, and the probability density becomes: $$p(t) =
\frac{4\pi Da}{V}\left(1 -\frac{a}R\right)
\exp\left(-\frac{4\pi aDt}{V}\right)
+ \frac{a}{R} \delta(t)$$
We did not test these results with a numerical simulation, since the continuous simulation method (see Appendix \[simulations\]) is not adapted to the computation of the FPT density, and would require a large computation time to give accurate results. Furthermore, the approximations made (cf. Appendix \[computation\]) are the same as on the discrete case, and the discrete results have been successfully compared to an exact numerical simulation (cf. Fig. \[distrib2\]).
Case of two targets
-------------------
For the case of two targets, we will compute the same quantities as in the discrete case; however, we may notice that the radius $a_1$ and $a_2$ of the two targets may differ, which adds another parameter to the problem. With two targets, we will use the same Green function as before, but $\Sigma_{\rm abs}=\Sigma_1+\Sigma_2$ will be the reunion of the surfaces of the two absorbing target spheres. The mean absorption time ${\langle\mathbf{T}({\bf r}_{S})\rangle}$ satisfies the equation (\[Texpl\]); the splitting probability $P_1({\bf r}_S)$ satisfies the following equations [@VanKampen]: $$\Delta P_1({\bf r}) = 0$$ $$P_1({\bf r}) = 1 \; {\rm if} \; {\bf r} \in \Sigma_1$$ $$P_1({\bf r}) = 0 \; {\rm if} \; {\bf r} \in \Sigma_2$$ $$\partial_n P_1({\bf r}) = 0 \; {\rm if} \; {\bf r} \in \Sigma_{\rm refl}$$ Using Green’s formula, we get: $$P_1({\bf r}_S) = - \int_{\Sigma_1} \partial_n G({\bf r}|{\bf r}_S) d{\bf r}
\label{expP1},$$ The expression for $P_2$ is of course similar. Note that the normal derivative is oriented towards the inside of the target. A simple approximation of $G$, equivalent to the discrete Eq. (\[2trho\]) is: $$G({\bf r}|{\bf r}_S) = \rho_0({\bf r}_S) + H({\bf r}|{\bf r}_S) -
P_1({\bf r}_S)H({\bf r}|{\bf r}_{T_1}) - P_2({\bf r}_S) H({\bf r}|{\bf r}_{T_2}).$$ This expression satisfies Eq.(\[G1\]),(\[G3\]) and (\[expP1\]), and $\rho_0$, $P_1$ and $P_2$ are set in order to satisfy Eq.(\[G2\]) approximately. We use the same approximations as in the one-target case, which gives the following set of equations: $$\left\{ \begin{array}{rcl}
\rho_0({\bf r}_S) + H_{1s} - P_1H_{01} - P_2H_{12}
& = & 0 \\
\rho_0({\bf r}_S) + H_{2s} - P_2H_{02} - P_1H_{12}
& = & 0 \\
P_1+P_2 & = & 1 \\
\end{array} \right.
\label{2targetarray}$$ where $H_{12} = H({\bf r}_{T_1}|{\bf r}_{T_2})$ and, for $i = 1$ or $2$, $H_{is} = H({\bf r}_{T_i}|{\bf r}_S)$, $H_{0i} = G_0(a_i) + H^*({\bf r}_{T_i}|{\bf r}_{T_i})$. These equations are exactly identical to the discrete equations, only the meaning of the $H_{0i}$ changes. We thus can deduce, using the same relation between $\rho_0$ and ${\langle\mathbf{T}\rangle}$ as in Eq. (\[simplebrownian\]): $${\langle\mathbf{T}({\bf r}_{S})\rangle} = \frac{V}{D}
\frac{(H_{01}-H_{1s})(H_{02}-H_{2s}) - (H_{12}-H_{2s})(H_{12}-H_{1s})}
{H_{01}+H_{02}-2H_{12}}
\label{tcontinu}$$ $$\left\{
\begin{array}{l}
P_1 = \frac{H_{1s}+H_{02}-H_{2s}-H_{12}}{H_{01}+H_{02}-2H_{12}} \\
P_2 = \frac{H_{2s}+H_{01}-H_{1s}-H_{12}}{H_{01}+H_{02}-2H_{12}} \\
\end{array}
\right.
\label{pcontinu}$$
We show in Fig. \[c2t2D\] and \[c2t3D\] the results of the numerical simulations. We can see that they are accurate, with a small correction (the relative correction scales as $(a/d)\ln(d/a)$ in 2D or $a^2/d^2$ in 3D, $d$ being the distance between the two targets) when the two targets are close (an explicit correction is given in Appendix \[refinements\]), and when one target is near a boundary (exactly as in the one-target case).
The curves themselves deserve a few qualitative remarks. Unsurprisingly, the splitting probability $P_2$ is maximal when $T_2$ is the closest to the source. When the two targets have different sizes, an interesting phenomenon appears (Fig. \[c2t3D\]): the probability to hit the largest target ($T_2$) has a second maximum when it is close to the other target. One can understand this by a scaling argument. If the two targets are far away, $P_1$ will be about $a_1/(a_1+a_2)$. If the two targets touch one another, and $a_1 \ll a_2$, then the target $T_1$ covers a surface $\pi a_1^2$ of the target $T_2$. It can thus be expected that the probability $P_1$ will scale as $a_1^2/a_2^2$, and thus be much lower than if the two targets were far away. These arguments are for the 3D case, but the qualitative behavior would be the same in the 2D case. However, the behavior of the splitting probabilities when one target is much further than the other from the source will be different in 2D and 3D, for the same reasons as in the discrete case. In the figures the domain is not large enough to make the difference obvious. The mean absorption time has a similar qualitative behavior in both cases: an unsurprising minimum when the moving target is close to the source, maxima when the moving target is near a boundary, due to boundary effects, and a maximum when the two targets are close, which deserves a few more comments. This could indeed be predicted directly from Eq.(\[tcontinu\]), but, physically, this comes from the fact that, if the two targets are close, a particle undergoing a Brownian motion, which reaches one target, often would have reached the other shortly afterwards in a single target situation. Thus, the mean time gained, compared to the single target situation, will be much lower when the two targets are close. To analyse the values themselves, one should keep in mind that the times are normalized by $V/D$; the order of magnitude of the normalized times will then be $G_0(a)-G_0(R)$, which explains the values around $0.05$ obtained in the 3D case.
As for the conditional FPTs ${\langle\mathbf{T}_{1}({\bf r}_{S})\rangle}$ and ${\langle\mathbf{T}_{2}({\bf r}_{S})\rangle}$, we have the following relations [@VanKampen]: $$D \Delta (P_1({\bf r}){\langle\mathbf{T}_{1}({\bf r}_{})\rangle})= -P_1({\bf r}) \;{\rm if}\; {\bf r}
\in{\mathcal D}^*$$ $$P_1({\bf r}){\langle\mathbf{T}_{1}({\bf r}_{})\rangle} = 0 \;{\rm if}\; {\bf r}\in \Sigma_{\rm abs}$$ $$\partial_n(P_1({\bf r}){\langle\mathbf{T}_{1}({\bf r}_{})\rangle})=0 \;{\rm if}\; {\bf r}\in \Sigma_{\rm refl},$$ and of course the equivalent relations for ${\langle\mathbf{T}_{2}({\bf r}_{})\rangle}$. We use as usual Green’s formula, and obtain: $$P_1({\bf r}_S){\langle\mathbf{T}_{1}({\bf r}_{S})\rangle} = \int_{\mathcal{D}^*} G({\bf r}|{\bf r}_S)P_1({\bf r})
d{\bf r}$$ This equation is very similar to the discrete Eq. (\[Ncondtimes\]) and the following calculations for the 3D case are exactly identical, and give: $${\langle\mathbf{T}_{1}({\bf r}_{S})\rangle} = \frac{1}{P_1({\bf r}_S)}
\frac{H_{02}-H_{12}+\mathcal{O}(aV^{-2/3})}{H_{01}+H_{02}-2H_{12}}{\langle\mathbf{T}({\bf r}_{S})\rangle}
\label{ccondprob}$$
We show in Fig.\[ccondtime\] the result of numerical simulations. The noise is more important than in other simulations, especially for ${\langle\mathbf{T}_{1}\rangle}$. This is due to the fact that the probability $P_1$ is often small, which reduces the number of processes on which the time is averaged, and thus increases the noise.
We thus are able to compute first-passage times, splitting probabilities and absorption times with a good accuracy (especially with the improvements given in Appendix \[refinements\]), provided we know the pseudo-Green function $H$. The computation of $H$ is discussed extensively in Appendix \[approximations\] and more briefly in the following.
Discussion \[discussion\]
=========================
The computation of the pseudo-Green function can be a difficult problem. Indeed, there are a few cases when it can be computed exactly (see Appendix \[exact\]), namely in the discrete case for a rectangular/parallepipedic domain or for periodic boundary conditions, and in the continuous case when the domain is a disk, a sphere, or the surface of a sphere. Otherwise, we have to use an approximation, the simplest ones being presented and discussed in Appendix \[approximations\].
In the following we present a synthetic and qualitative description of the important parameters which have to be taken into account when it comes to computing the mean-first passage time.
The first and most important parameter is the size of the domain. Indeed, the MFPT is proportional to the size of the domain, both in two and three dimensions. The second essential parameter is the size of the target for the continuous case: once we have these two parameters we already have a rough order of magnitude of the MFPT. The third important parameter is the distance between the source and the target. In three dimensions this parameter is important as long as it is of the same order of magnitude as the target size; its influence is inversely proportional to the source-target distance. In two dimensions this parameter will be important at any distance, since the MFPT depends logarithmically of this distance. Once these parameter have been taken into account we have a good approximation of the MFPT (Eq. (\[base2D\]) and (\[base3D\])) if both source and target are far from any boundary. To see what *far* means in this case, a good criterion is that any correction involving a boundary (see below) is negligible. Otherwise we have an order of magnitude, and to proceed further we will have to take into account the precise position of the boundaries.
The qualitative effect of the boundary is to increase the MFPT when the target is near a boundary, and to decrease it when the source is near a boundary (it can be seen in the following equations). The first effect is much more important than the second: in three dimensions, with a flat boundary, a basic approximation gives: $${\langle\mathbf{T}\rangle}= \frac{1}{4\pi}\left(\frac1a - \frac1{|{\bf r}_S - {\bf r}_T|}
+ \frac{1}{|{\bf r}_T - {\bf s}({\bf r}_T)|} -
\frac1{|{\bf s}({\bf r}_S) - {\bf r}_T|}\right),$$ where ${\bf s}({\bf r})$ denotes the point symmetrical to ${\bf r}$ with respect to the boundary. One can see that the influence of the boundary is inversely proportional to the distance between the target and the boundary. This is also true if the source is near a boundary, which is why the most important parameter is indeed the position of the target. One may note, however, that if the target or the source lies in a corner, these effects are amplified.
In two dimensions the influence of the position of the boundary is more important, and the position of the source is a relevant parameter: a basic approximation with a flat boundary gives: $${\langle\mathbf{T}\rangle}= \frac{1}{2\pi}\left(\ln\frac{|{\bf r}_S-{\bf r}_T|}{a} +
\ln\frac{|{\bf s}({\bf r}_S)-{\bf r}_T|}{|{\bf s}({\bf r}_T)-{\bf r}_T|}\right)
\label{2Dboundary}$$ If the target is much closer to the boundary than the source the effect can be to double the MFPT; on the other hand, if the source only is near a boundary, the related correction is bounded. The corners also have an amplifying effect in two dimensions.
The quantitative estimates thus obtained are generally more accurate in three dimensions than in two dimensions, due to the fact that the effect of the boundaries on the pseudo-Green function is essentially local in three dimensions. In two dimensions there is still room for improvement, but an extensive discussion would be beyond the scope of this article.
Conclusion {#conclusion .unnumbered}
==========
In this article we managed to compute the mean first-passage times, the splitting probability and the full probability density of the first-passage time (in three dimensions) with a good accuracy for spherical or rectangular domains. For other shapes (with a regular enough boundary), we gave the basic tools to approximately estimate these quantities. These results are especially important in the analysis of diffusion-limited reactions: The first-passage time corresponds to the reaction time if one of the reactants is static, and the reaction rate is infinite. Two promising extensions of our work would be to take into account *finite* reaction rates, which would increase the relevance of our work to reaction-diffusion processes; and to study the same problem with anomalous diffusion, which is relevant in many physical situations.
Acknowledgements {#acknowledgements .unnumbered}
================
We gratefully thank Jean-Marc Victor for useful discussions and comments, and Sidney Redner for suggesting us the discrete simulation method.
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\[approximations\] Evaluation of the pseudo-Green function
==========================================================
Exact formulas\[exact\]
-----------------------
### Periodic boundary condition and rectangular domains for a discrete pseudo-Green function
There are two specific cases where the discrete pseudo-Green function $H$ may be computed exactly: when the domain is rectangular (parallepipedic in three dimensions) or when the boundary conditions are periodic [@CondaminJCP]. These results are interesting in themselves, but, moreover, for a domain which is almost rectangular/parallepipedic, they will give a good approximation for $H$. For periodic boundary conditions. if we consider a domain with $X$ sites in the $x$ direction, $Y$ sites in the $y$ direction, and $Z$ sites in the $z$ direction, a straightforward Fourier analysis gives: $$H({\bf r}|{\bf r}') = \frac1N \sum_{m=0}^{X-1}\sum_{n=0}^{Y-1}
\sum_{p=\delta_{(m,n)(0,0)}}^{Z-1}\frac{\exp\left(\frac{2im\pi(x-x')}{X}+
\frac{2in\pi(y-y')}{Y}+\frac{2ip\pi(z-z')}{Z}\right)}{1-\frac13\left(
\cos\frac{2m\pi}{X}+\cos\frac{2n\pi}{Y}+\cos\frac{2p\pi}{Z}\right)}$$ In two dimensions, we have a similar formula for $H$: $$H({\bf r}|{\bf r}') = \frac1N \sum_{m=0}^{X-1}\sum_{m=\delta_{n0}}^{Y-1}
\frac{\exp\left(\frac{2im\pi(x-x')}{X}+
\frac{2in\pi(y-y')}{Y}\right)}{1-\frac12\left(
\cos\frac{2m\pi}{X}+\cos\frac{2n\pi}{Y}\right)}$$
For a parallepipedic domain we get a slightly more complicated expression, and we have to use semi-integer coordinates for the points: $x$(resp. $y$ and $z$) varies between $1/2$ and $X$(resp. $Y$ and $Z$) $-1/2$. The result is the following: $$\begin{aligned}
H({\bf r}|{\bf r}') &=& \frac8N\sum_{m=1}^{X-1}\sum_{n=1}^{Y-1}\sum_{p=1}^{Z-1}
\frac{\cos\frac{m\pi x'}{X}\cos\frac{n\pi y'}{Y}
\cos\frac{p\pi z'}{Z}\cos\frac{m\pi x}{X}\cos\frac{n\pi y}{Y}
\cos\frac{p\pi z}{Z}}{1-\frac13\left(\cos\frac{m\pi}{X}+\cos\frac{n\pi}{Y}+
\cos\frac{p\pi}{Z}\right)} \label{exactcubic}\\
&& + \frac6N\sum_{m=1}^{X-1}\sum_{n=1}^{Y-1}
\frac{\cos\frac{m\pi x'}{X}\cos\frac{n\pi y'}{Y}
\cos\frac{m\pi x}{X}\cos\frac{n\pi y}{Y}
}{1-\frac12\left(\cos\frac{m\pi}{X}+\cos\frac{n\pi}{Y}
\right)} +
\frac6N\sum_{p=1}^{Z-1}
\frac{\cos\frac{p\pi z'}{Z}
\cos\frac{p\pi z}{Z}
}{1-\cos\frac{p\pi}{Z}} \nonumber \\
&& + \frac6N\sum_{m=1}^{X-1}\sum_{p=1}^{Z-1}
\frac{\cos\frac{m\pi x'}{X}\cos\frac{p\pi z'}{Z}
\cos\frac{m\pi x}{X}\cos\frac{p\pi z}{Z}
}{1-\frac12\left(\cos\frac{m\pi}{X}+\cos\frac{p\pi}{Z}
\right)} +
\frac6N\sum_{n=1}^{Y-1}
\frac{\cos\frac{n\pi y'}{Y}
\cos\frac{n\pi y}{Y}
}{1-\cos\frac{n\pi}{Y}} \nonumber\\
&& + \frac6N\sum_{n=1}^{Y-1}\sum_{p=1}^{Z-1}
\frac{\cos\frac{n\pi y'}{Y}\cos\frac{p\pi z'}{Z}
\cos\frac{n\pi y}{Y}\cos\frac{p\pi z}{Z}
}{1-\frac12\left(\cos\frac{n\pi}{Y}
+\cos\frac{p\pi}{Z}\right)}
+ \frac6N\sum_{m=1}^{X-1}
\frac{\cos\frac{m\pi x'}{X}
\cos\frac{m\pi x}{X}}
{1-\cos\frac{p\pi}{Z}} \nonumber \end{aligned}$$ In two dimensions the expression is slightly less imposing: $$\begin{aligned}
H({\bf r}|{\bf r}')&=& \frac4N \sum_{m=1}^{X-1} \sum_{n=1}^{Y-1}
\frac{\cos\frac{m\pi x'}{X}\cos\frac{n\pi y'}{Y}
\cos\frac{m\pi x}{X}\cos\frac{n\pi y}{Y}}{
1-\frac12\left(\cos\frac{m\pi}{X}+\cos\frac{n\pi}{Y}\right)}\nonumber \\
&& + \frac4N \sum_{m=1}^{X-1}\frac{\cos\frac{m\pi x'}{X}\cos\frac{m\pi x}{X}}{
1 - \cos\frac{m\pi}{X}} +
\frac4N \sum_{n=1}^{Y-1}\frac{\cos\frac{n\pi y'}{Y}\cos\frac{n\pi y}{Y}}{
1 - \cos\frac{n\pi}{Y}} \end{aligned}$$
These formulae have the advantage of being exact, which enables us to compute exactly all the quantities studied in this article for such geometries. However, the computation of $H$ may be computationally expensive for large domains. In the continuous case, the same method can be applied, but $H$ can only be expressed as an infinite series[@Barton]. We give the result for a $2D$ rectangle $X \times Y$: $$\begin{aligned}
H({\bf r}|{\bf r}') & = & \frac4{XY} \sum_{m=1}^{\infty} \sum_{n=1}^{\infty}
\frac{\cos\frac{m\pi x'}{X}\cos\frac{n\pi y'}{Y}
\cos\frac{m\pi x}{X}\cos\frac{n\pi y}{Y}}{
\left(\frac{m \pi}{X}\right)^2+\left(\frac{n \pi}{Y}\right)^2}\nonumber \\
&& + \frac2{XY} \sum_{m=1}^{\infty}\frac{\cos\frac{m\pi x'}{X}\cos
\frac{m\pi x}{X}}{
\left(\frac{m\pi}{X}\right)^2} +
\frac2{XY} \sum_{n=1}^{\infty}\frac{\cos\frac{n\pi y'}{Y}\cos\frac{n\pi y}{Y}}{
\left(\frac{n\pi}{Y}\right)^2} \end{aligned}$$
### Disks and spheres for the continuous pseudo-Green functions \[Disks\]
In the continuous case there is however a case where the pseudo-Green function is known exactly: if the domain is a disk or sphere of radius $b$. We will simply give the results; the detailed computation can be found in [@Barton].
In both formulas, we use the image of ${\bf r}'$, that we note $\tilde{\bf r}'$, which is aligned with ${\bf r}$ and the center of the disk/sphere $O$, and at a distance $\tilde{r}' = b^2/r'$. We note $R = |{\bf r}-{\bf r}'|$, $\tilde{R} = |{\bf r}-\tilde{\bf r}'|$, and $\mu = \cos\gamma$, $\gamma$ being the angle between ${\bf r}$ and ${\bf r}'$. In two dimensions, the result is the following: $$H({\bf r}|{\bf r}') = \frac1{2\pi}\left(\ln\frac{b}{R}+\ln\frac{b}{\tilde{R}}
+ \ln\frac{b}{r'}+\frac{r^2+r'^2}{2 b^2}\right)
\label{exactdisk}$$ The first term corresponds to $G_0$, the second to the image of ${\bf r}'$, the third term is needed to ensure the symmetry of $H$, and the last term corresponds to the $-1/V$ term in the definition of the pseudo-Green function. The three-dimensional result is a bit more complicated, with a logarithmic term whose physical signification is unclear: $$H({\bf r}|{\bf r}') = \frac{1}{4\pi}\left(\frac1R+\frac{b}{r'\tilde{R}}
-\frac1b\ln\left(\frac{r'\tilde{R}}{b^2}+1-\frac{rr'\mu}{b^2}\right)
+ \frac{r^2+r'^2}{2b^3}\right)
\label{exactsphere}$$ These results are very useful by themselves, but they will also be useful to approximate $H$ near a curved boundary, as we will see in the following. The result for a sphere can also be used to estimate $\bar{H}$ when one uses the approximation $H=G_0$ in non-elongated 3D domains. Indeed the exact result enables one to take into account the corrections to $G_0$, which are negligible when the source and the target are close, but give a substantial correction to the value of $\bar{H}$. To compute $\bar{H}$, one can use Eq. (\[4\]), and choose for ${\bf r}$ the centre of the sphere. We have in this case: $$H({\bf r}'|{\bf r}=0) = \frac{1}{4\pi}\left(\frac1R+\frac{R^2}{2b^3}\right)$$ A constant $(1 -\ln(2))/(4\pi b)$ has been suppressed, in order to have a final result relevant for the approximation $H=G_0$. From this expression of $H$ it is straightforward to get an expression for $\bar{H}$: $$\bar{H} = \frac{3}{5}\left(\frac{3}{4\pi}\right)^{2/3}V^{-1/3}$$ If one wants to use this result in the discrete case, it should be noted that the continuous limit of the discrete model corresponds to $D=1/2d$ and not $D = 1$. This diffusion coefficient is included in the discrete pseudo-Green function, and the discrete estimation of $\bar{H}$ is thus: $$\bar{H} = \frac{18}{5}\left(\frac{3}{4\pi}\right)^{2/3}N^{-1/3}
\label{valuehbar}$$
### Surface of spheres
Another case where we can compute exactly $H$ is the case of the surface of a sphere. Indeed in this case we have exactly: $$H({\bf r}|{\bf r}') = -\frac1{2\pi}\ln|{\bf r}-{\bf r}'|$$ Since $H$ is isotropic in this case it simplifies things: $G_0(a) +
H^*({\bf r}_T|{\bf r}_T)$ can be replaced by $H(a)$ in Eq.(\[simplebrownian\]) This gives back the result obtained by a straightforward computation of the FPTs in a sphere [@Desbois]. Moreover this will give good approximations of all the two-target quantities, which was, to our knowledge, not known until now. This result is not used elsewhere in the paper, but is however important due to the physical relevance of the diffusion on the surface of a sphere.
Use of the approximations\[use\]
--------------------------------
The next step is to study cases where no exact formula for $H$ is known. The simplest approximation to $H$ is the infinite-space Green function $G_0$, but this approximation in often unsatisfying. We thus present a few ways to improve it. Before we present them, it must be emphasised that, in general, all the $H$ terms should be derived with the same approximation: $H$ is defined up to a constant, and this constant depends of the approximation used! However, for complicated expression involving $H$, this constraint can be relaxed: if the expression can be decomposed into terms of the form $(H({\bf r}_1|{\bf r}_2) - H({\bf r}_3|{\bf r}_4))$, these terms may be computed with different approximations, since they do not depend on the constant up to which $H$ is defined. For example, in the two-target problem, we have $P_1 = \frac{H_{1s}+H_{02}-H_{2s}-H_{12}}{H_{01}+H_{02}-2H_{12}} $. We can use if necessary two approximations, one accurate around $T_1$, which we note $H^{(1)}$, and another accurate around $T_2$, which we note $H^{(2)}$. Then, to compute $P_1$, we use them the following way: $$P_1 = \frac{H^{(1)}_{1s}+H^{(2)}_{02}-H^{(2)}_{2s}-H^{(1)}_{12}}{
H^{(1)}_{01}+H^{(2)}_{02}-H^{(1)}_{12}-H^{(2)}_{12}}$$ This trick can be especially useful if one has to deal with two targets near two different boundaries.
Approximations\[boundary\]
--------------------------
The most basic approximation is already known: it is the approximation $H = G_0$. Its physical meaning is to ignore the presence of the boundaries, as far as the pseudo-Green function is concerned. To improve this approximation, there are essentially two ways: the first is to take the boundaries into account locally, and to satisfy the boundary conditions at the nearest boundary, we will see how in the following. The second one is to take the boundaries into account globally, by taking into account the terms $-1/N$ or $-1/V$ in the definition of $H$. The order of magnitude of the related correction will be of about $({\bf r}-{\bf r}')^2/N$ in the discrete case, or $({\bf r}-{\bf r}')^2/4A$ in the 2D continuous case, $({\bf r}-{\bf r}')^2/6V$ in the 3D continuous case. It is thus much weaker in 3D (the maximal relative correction scales as $N^{-1/3}$ or $a/V^{1/3}$) than in 2D (where the maximal relative correction scales as $1/\ln(N)$ or $1/\ln(V/a^3)$). A more detailed discussion of this kind of corrections would be technical and beyond the scope of this article, but the above order of magnitude can be a good evaluation of the accuracy of the following boundary approximation.
This approximation takes explicitly into account a planar boundary, and ignores all the others. It can be used both in the continuous and in the discrete case. If we note ${\bf s}({\bf r})$ the point symmetrical to ${\bf r}$ with respect to the boundary, then the local approximation: $$H({\bf r}|{\bf r}') = G_0({\bf r}-{\bf r}') + G_0({\bf r}-{\bf s}({\bf r}'))
\label{Hlocal}$$ satisfies the boundary conditions on the flat boundary, and is symmetric. It thus can be a good approximation for the pseudo-Green function.
Figs. \[distsw2D\] and \[quarter3D1\] show the efficiency of this approximation in two different cases: in a 2D discrete domain, and in a 3D continuous domain. In both cases the approximation improves the basic approximation $H=G_0$, but when the source is near another boundary, a systematic deviation appears, due to the influence of the other boundary.
The curvature of the boundary may be taken into account by approximating the pseudo-Green function by the pseudo-Green function inside a circle (\[exactdisk\]) or a sphere (\[exactsphere\]), or outside a circle or a sphere (it can be found in [@Barton]).
Computation of the higher-order moments\[computation\]
======================================================
Discrete case\[computationdiscret\]
-----------------------------------
In this part we will compute the higher-order moments of the FPT. To do this, we start from an extension of Kac’s formula (see Appendix \[AnnKac\]), which is the relation between the Laplace transforms of the FRT to the subset $\Sigma$, averaged on $\Sigma$, and of the FPT to this subset, the starting point being averaged over the complementary subset $\bar\Sigma$.
$$\pi(\Sigma)\left( \left\langle e^{-s\mathbf{T}}\right\rangle_\Sigma -
e^{-s} \right) = (1 -\pi(\Sigma))\left(e^{-s} - 1\right) \left\langle
e^{-s\mathbf{T}} \right\rangle_{\bar{\Sigma}}
\label{HighKac}$$
Both averages are weighted by the stationary probability $\pi$, in the following sense: $$<\varphi(\mathbf{T})>_\Sigma = \frac1{\pi(\Sigma)} \sum_{i \in
\Sigma}\pi({\bf r}_i) \sum_{t=1}^{\infty} p_i(\mathbf{T}=t) \varphi(t)$$ $$<\varphi(\mathbf{T})>_{\bar\Sigma} = \frac1{1-\pi(\Sigma)} \sum_{i
\notin \Sigma}\pi({\bf r}_i) \sum_{t=1}^{\infty} p_i(\mathbf{T}=t) \varphi(t),$$ where $p_i(\mathbf{T}=t)$ is the probability for the FRT (or the FPT, according to whether the point $i$ belongs to $\Sigma$ or not) to be $t$, if the random walk starts from the point $i$. To apply the equation (\[HighKac\]) to the determination of the FPTs, we may notice that the FPT from any point of the graph (except target) is the same on the original graph and on the modified graph: indeed, the behaviour of a random walk is exactly the same on both lattices as long as they do not reach $T$, and what happens afterwards does not matter. Moreover, the FRT to $T$ is still the FPT from $S$ to $T$, plus one. Thus if we apply the formula (\[HighKac\]) to the modified graph, $\Sigma$ being reduced to $T$, we get the following relation between the Laplace transform of the FPT from $S$ and the FPT averaged over the whole set of points (without $T$): $$J\left( \left<e^{-s\mathbf{T}}\right>_S - 1 \right) =
(1-J)\left(1 - e^s\right) \left< e^{-s\mathbf{T}} \right>_{\bar\Sigma}$$ $J$ is still $\pi({\bf r}_T)$. We have to pay attention to one thing: the average over $\bar\Sigma$ is weighted by the weights for the stationary distribution of the modified lattice. To go further we will have to consider *all* the modified lattices with $T$ as target point, the starting point being any point of the set. We will denote $\pi_i$ the stationary distribution associated with the modified graph whose starting point is $i$, and $J_i=\pi_i({\bf r}_T) $. Thus, we may note: $$J_i\left( \left<e^{-s\mathbf{T}}\right>_i - 1 \right) =
\left(1 - e^s\right) \sum_{j \neq T} \pi_i({\bf r}_j)
\left< e^{-s\mathbf{T}} \right>_j$$ From this, we may deduce the recurrence equation for the moments: $$\left<\mathbf{T}^n\right>_i = \frac1{J_i}\sum_{m=1}^n
\sum_{j \neq T} (-1)^{m+1} {n \choose m} \pi_{i}({\bf r}_j)
\left<\mathbf{T}^{n-m}\right>_j$$ We may thus compute explicitly the second moment. $$\left<\mathbf{T}^2\right>_i = \frac1{J_i}
\sum_{j \neq T} \pi_i({\bf r}_j)
\left( 2 \left<\mathbf{T}\right>_j - 1 \right)$$
If we replace $\pi$ and $<\mathbf{T}>$ by their values, we get $$\left<\mathbf{T}^2\right>_i = \frac{2N}{J_i} \sum_{j \neq T}
\left(\frac{1-J_i}N + J_i H({\bf r}_j|{\bf r}_i) - J_i H({\bf r}_j|{\bf r}_T)
\right)\left( H({\bf r}_T|{\bf r}_T)- H({\bf r}_T|{\bf r}_j)\right)
- \frac{1-J_i}{J_i}$$ We then may use the value of $\frac{1-J}J$, which we know: $$\begin{aligned}
\left<\mathbf{T}^2\right>_i & = & 2N \sum_{j \neq T}
\left(H({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_i) + H({\bf r}_j|{\bf r}_i)
- H({\bf r}_j|{\bf r}_T) \label{exactt2}
\right)\left( H({\bf r}_T|{\bf r}_T)- H({\bf r}_T|{\bf r}_j)\right)\nonumber
\\ && - N (H({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_i))\end{aligned}$$ This equation is exact, but it is difficult to evaluate properly in the general case. We will thus use approximations to evaluate this expression in the case of a 3-D regular lattice, with $N$ large and the boundaries far from the target, at a typical distance $N^{\frac13}$. We can thus neglect the term $N (H({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_i))$ in the R.H.S. of equation (\[exactt2\]). If we develop the rest of the formula, we get: $$\begin{aligned}
\left<\mathbf{T}^2\right>_i & = & 2N \left(NH^2({\bf r}_T|{\bf r}_T) - N
H({\bf r}_T|{\bf r}_T)H({\bf r}_T|{\bf r}_i)+ H({\bf r}_T|{\bf r}_T)
\sum_{j \neq T}(H({\bf r}_j|{\bf r}_i)-2H({\bf r}_T|{\bf r}_j))\right.
\nonumber \\ && \left. +
H({\bf r}_i|{\bf r}_T)\sum_{j \neq T}H({\bf r}_T|{\bf r}_j)
- \sum_{j \neq T}H({\bf r}_T|{\bf r}_j)H({\bf r}_j|{\bf r}_i)
+ \sum_{j \neq T}H^2({\bf r}_T|{\bf r}_j) \right)\end{aligned}$$ We can now drop the least important terms in this formula by evaluating the order of magnitude of the various sums over $j$. We have (cf. Eq. (\[hbar\])): $$\frac1N \sum_{j} H({\bf r}_i|{\bf r}_j) = \bar{H}
\label{barh}$$
Since $G_0({\bf r}) \sim 1/r$ in 3D, and the corrections are, on the worst case,of the same order of magnitude, we can see that $\bar{H}$ scales as $N^{-1/3}$. If we consider the sums $\sum_j H^2({\bf r}_T|{\bf r}_j)$ and $ \sum_j H({\bf r}_T|{\bf r}_j)H({\bf r}_j|{\bf r}_i)$, we may first notice that: $$\sum_j H({\bf r}_T|{\bf r}_j)H({\bf r}_j|{\bf r}_i) \leq
\left(\sum_j H^2({\bf r}_T|{\bf r}_j)
\sum_j H^2({\bf r}_i|{\bf r}_j) \right)^{1/2}
\label{scaling}$$ We thus only need to consider the case of $(1/N) \sum_j H^2({\bf r}_i|{\bf r}_j)$. And, for the same reasons as above, we can see that it scales as $N^{-2/3}$. Putting all this together, we have: $$\left<\mathbf{T}^2\right>_i = 2N^2\left[ \left(
H({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_i)\right)
\left( H({\bf r}_T|{\bf r}_T) - \bar{H} \right)
+\mathcal{O}(N^{-2/3}) \right]$$ It is possible to generalize this expression to higher-order moments; we will obtain the following result, for a given $n$: $$\left<\mathbf{T}^n\right>_i = n!N^n\left[ \left(
H({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_i)\right)
\left( H({\bf r}_T|{\bf r}_T) - \bar{H} \right)^{n-1}
+\mathcal{O}(N^{-2/3}) \right]$$ We can prove this by recurrence: if this is true for $m < n$, then: $$\left<\mathbf{T}^n\right>_i = \frac{n}{J_i}
\sum_{j \neq T} \pi_i({\bf r}_j)
\left<\mathbf{T}^{n-1}\right>_j$$ The others terms are negligible (their relative order of magnitude is at most $1/N$), and we will thus ignore them. We replace everything by its value, which gives: $$\left<\mathbf{T}^n\right>_i = n!N^{n-1}\sum_{j \neq T}
\left(\begin{array}{l}
H({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_i)\\
+H({\bf r}_j|{\bf r}_i)-H({\bf r}_j|{\bf r}_T)\\
\end{array}
\right)\left(\begin{array}{l}
H({\bf r}_T|{\bf r}_T)\\
-H({\bf r}_T|{\bf r}_j)\\
\end{array}\right)
\left(\begin{array}{l}
\left(H({\bf r}_T|{\bf r}_T)-\bar{H}\right)^{n-2}\\
+\mathcal{O}(N^{-\frac23}) \\
\end{array}
\right)$$ Using exactly the same approximations as above (the computation is identical), we get: $$\left<\mathbf{T}^n\right>_i = n!N^n\left[ \left(
H({\bf r}_T|{\bf r}_T)-H({\bf r}_T|{\bf r}_i)\right)
\left( H({\bf r}_T|{\bf r}_T) - \bar{H} \right)^{n-1}
+\mathcal{O}(N^{-2/3}) \right]$$ As for the dependence with $n$ of the correction, since we perform exactly the same operation at each step $n \rightarrow n+1$, the correction will be proportional to $n$, which may help estimate the validity of the approximation.
This computation fails for elongated domains: two main hypotheses are not satisfied in this case, namely that the boundaries are at a typical distance $N^{1/3}$, and that the corrections to $G_0$ have the same order of magnitude. The method can not either be applied to the 2D case, since the terms $1/N \sum_j H^2({\bf r}_i|{\bf r}_j)$ are no longer negligible.
Continuous case\[computationcontinu\]
-------------------------------------
In the continuous case we can perform a similar computation. The higher-order moments of the FPT at the target satisfy the following equations[@Risken]: $$D \Delta {\langle\mathbf{T}^{n}({\bf r}_{})\rangle}
=-n{\langle\mathbf{T}^{n-1}({\bf r}_{})\rangle}\;{\rm if}\;
{\bf r}\in{\mathcal D}^*$$ $${\langle\mathbf{T}^{n}({\bf r}_{})\rangle} = 0 \;{\rm if}\;
{\bf r} \in \Sigma_{\rm abs}$$ $$\partial_n{\langle\mathbf{T}^{n}({\bf r}_{})\rangle} = 0 \;{\rm if}\; {\bf r} \in \Sigma_{\rm refl}$$ Using a new time the Green function defined by Eqs. (\[G1\],\[G2\],\[G3\]) and the Green formula, we have $${\langle\mathbf{T}^{n}({\bf r}_{S})\rangle}
=\frac{n}{D}\int_{\mathcal{D}^*} G({\bf r}|{\bf r}_S)
{\langle\mathbf{T}^{n-1}({\bf r}_{})\rangle} \rangle d^d{\bf r}$$
With the knowledge of $G({\bf r}|{\bf r}')$ for all starting points ${\bf r}$, it is possible to compute the full distribution. In three dimensions, it is possible to find an expression for $\left<\mathbf{T}^n\right>$ similar to the one found in the discrete case. We will start from Eq.(\[simplebrownian\]). We can now compute the second moment, using the values for $\langle\mathbf{T}\rangle$ and $\rho_0$:
$$\begin{aligned}
{\langle\mathbf{T}^{2}({\bf r}_{S})\rangle} & = & \frac{2V}{D^2}
\int_{\mathcal{D}^*}
\left[G_0(a)+ H^*({\bf r}_T|{\bf r}_T)-
H({\bf r}_T|{\bf r}_S) +H({\bf r}|{\bf r}_S) - H({\bf r}|{\bf r}_T)\right]\\
&&
\left[G_0(a)+H^*({\bf r}_T|{\bf r}_T)-H({\bf r}|{\bf r}_T)\right]
d^d{\bf r}
\nonumber\end{aligned}$$
To compute this, we will use the two following equations, equivalent to Eq. (\[barh\]) and (\[scaling\]) for discrete random walks: $$\int_{\mathcal{D}^*}H({\bf r}_0|{\bf r})d^d{\bf r} = V\bar{H}
+ \mathcal{O}\left(a^2\right)$$ $$\int_{\mathcal{D}^*}H({\bf r}_1|{\bf r})H({\bf r}_2|{\bf r})
d^d{\bf r} = \mathcal{O}\left(V^{1/3}\right)$$
This gives: $${\langle\mathbf{T}^{2}({\bf r}_{S})\rangle} = \frac{2V^2}{D^2}\left[
\left(G_0(a) + H^*({\bf r}_T|{\bf r}_T) - H({\bf r}_T|{\bf r}_S)\right)
\left(G_0(a) + H^*({\bf r}_T|{\bf r}_T) - \bar{H}\right)
+\mathcal{O}\left(V^{-2/3}\right)\right]$$ This result may be extended by recurrence to higher-order moments, in exactly the same way that in the discrete case, which gives: $${\langle\mathbf{T}^{n}({\bf r}_{S})\rangle} = \frac{n!V^n}{D^n}\left[
\left(G_0(a) + H^*({\bf r}_T|{\bf r}_T) - H({\bf r}_T|{\bf r}_S)\right)
\left(G_0(a) + H^*({\bf r}_T|{\bf r}_T) - \bar{H}\right)^{n-1}
+\mathcal{O}\left(nV^{-2/3}a^{2-n} \right) \right]$$
\[refinements\]Refinements of the continuous theory
===================================================
In this Appendix we will see how to improve the results of Section \[continu\], provided we know the pseudo-Green function $H$. The results we obtained in Section \[continu\] are not perfectly satisfying for three reasons:
- [When the source and the target are close, the approximation works better than one could naively expect, given that it does not satisfy Eq. (\[G2\]) very accurately. It would be interesting to understand why.]{}
- [The approximation lacks accuracy when the target is near a boundary.]{}
- [In the two-target case the accuracy is not very good when the two targets are close.]{}
We will treat the first point in detail, and give the corrections, and the method used to compute them, for the second and third point.
A better evaluation of $G$
--------------------------
To understand this, we will first notice that the Green function we use could also be used in an electrostatic problem: the source is equivalent to a point charge, and the absorbing spheres are equivalent to conducting spheres set at a null potential. We can thus apply the well-known method of images [@Jackson] to our problem. If we have an image charge $q$ $$q({\bf r}_S) = \left\{
\begin{array}{ll}
-\frac{a}{|{\bf r}_S-{\bf r}_T|}\equiv -\frac{a}{R}&\;{\rm in}\;{\rm 3D} \\
-1 & \;{\rm in}\;{\rm 2D}
\end{array}\right.$$ placed on ${\bf i}({\bf r}_S)$, located on the line between the center of the sphere and the source, at a distance $R'=a^2/R$ of the target, where $R$ is the source-target distance, then the solution: $$G({\bf r}|{\bf r}_S) = \rho_0({\bf r}_S) + G_0({\bf r}|{\bf r}_S) -
G_0({\bf r}|{\bf r}_T) +
q({\bf r}_S)(G_0({\bf r}|{\bf i}({\bf r}_S))-G_0({\bf r}|{\bf r}_T))$$ satisfies exactly the boundary condition (\[G2\]) on the target sphere: we have, for ${\bf r} \in \Sigma_{\rm abs}$ $$G_0({\bf r}|{\bf r}_S) - G_0({\bf r}|{\bf r}_T)
+ q({\bf r}_S)(G_0({\bf r}|{\bf i}({\bf r}_S))-G_0({\bf r}|{\bf r}_T)) =
G_0({\bf r}_S|{\bf r}_T) - G_0(a)
\label{spherebc}$$ However, this solution does not satisfy the reflecting boundary conditions, and we will rather use the solution: $$G({\bf r}|{\bf r}_S) = \rho_0({\bf r}_S) + H({\bf r}|{\bf r}_S) - H({\bf r}|{\bf r}_T)
+ {q_{}({\bf r}_S)(H({\bf r}|{\bf i}_{}({\bf r}_S))-
H({\bf r}|{\bf r}_{T_{}}))}$$ which approximately satisfies (\[G2\]), provided that we neglect the variations of $H^*({\bf r}|{\bf r}_S)$ and $H^*({\bf r}|{\bf r}_T)$ on the target sphere. With this approximation we get: $$\rho_0({\bf r}_S) = G_0(a) - H({\bf r}|{\bf r}_S) + H^*({\bf r}_T|{\bf r}_T)
+{q_{}({\bf r}_S)(H^*({\bf i}_{}({\bf r}_S)|{\bf r}_{T_{}})-
H^*({\bf r}_{T_{}}|{\bf r}_{T_{}}))}$$ Note that the last term ${q_{}({\bf r}_S)(H^*({\bf i}_{}({\bf r}_S)|{\bf r}_{T_{}})-
H^*({\bf r}_{T_{}}|{\bf r}_{T_{}}))}$ can be neglected, since the variations of $H^*$ over the target sphere are neglected. Finally, to find the Eq. (\[simplebrownian\]), the only condition is to neglect the variations of $H^*$ over the target sphere, which will be a good approximation as soon as the target is far from any boundary. If this condition is satisfied, the approximation for the MFPT is accurate, even if the source is near the target.
Influence of a boundary
-----------------------
If the target is near a boundary, however, $H^*$ can no longer be considered as constant over the target sphere. To have a good approximation of $H$, one has to decompose the function one step further: $$H({\bf r}|{\bf r}') = G_0({\bf r}|{\bf r}') + G_0({\bf r}|{\bf s}({\bf r}'))
+ H^{**}({\bf r}|{\bf r}'),$$ where ${\bf s}({\bf r})$ is the point symmetrical to ${\bf r}$ with respect to the boundary. This simply explicits the image charges due to the boundary, which themselves have images on the target sphere. The real and image charges are depicted in Fig. \[symetrie\].
If we take into account all these charges, it is possible to obtain the following expression for the MFPT, valid as long as the target sphere does not touch the boundary: :
$${\langle\mathbf{T}({\bf r}_{S})\rangle} = \frac{V}{D}\left(
G_0(a) - H({\bf r}_T|{\bf r}_S) + H^*({\bf r}_T|{\bf r}_T) - K({\bf r}_S)
-K({\bf s}({\bf r}_S)) + K({\bf s}({\bf r}_T)) \right),
\label{boundarybrownian}$$
where $K({\bf r}) = {q_{}({\bf r})(H^*({\bf i}_{}({\bf r})|{\bf r}_{T_{}})-
H^*({\bf r}_{T_{}}|{\bf r}_{T_{}}))}$.
Two close targets
-----------------
The two-target case can be treated likewise: by considering the images of $T_1$ and $T_2$ on the other sphere, it is possible to compute corrections to the terms $H_{01}$, $H_{02}$, $H_{1s}$ and $H_{12}$ used in Eqs. (\[tcontinu\]) and (\[pcontinu\]). These correction are: $$H_{1s} = H({\bf r}_{T_1}|{\bf r}_S) + {q_{2}({\bf r}_S)(H({\bf r}_{T_1}|{\bf i}_{2}({\bf r}_S))-
H({\bf r}_{T_1}|{\bf r}_{T_{2}}))},
\label{c1}$$ $$H_{01} = G_0(a_1) + H^*({\bf r}_{T_1}|{\bf r}_{T_1})
+ {q_{2}({\bf r}_{T_1})(H({\bf r}_{T_1}|{\bf i}_{2}({\bf r}_{T_1}))-
H({\bf r}_{T_1}|{\bf r}_{T_{2}}))},
\label{c2}$$ $$H_{12} = H({\bf r}_{T_1}|{\bf r}_{T_2}),
\label{c3}$$ and similar corrections for $H_{02}$ and $H_{2S}$. $q_k({\bf r})$ and ${\bf i}_k({\bf r})$ denote the value and the position of the image charge of ${\bf r}$ inside $T_k$.
Proof of Kac’s formula and of its extension {#AnnKac}
===========================================
The model
---------
We use the notations of Section \[discret\]: $R$ is an arbitrary finite set of points $1,2, \hdots, N$, with positions ${\bf r}_1, {\bf r}_2, \hdots, {\bf r}_N$. $w_{ij}$ is the transition probability from $j$ to $i$, and we assume that any couple of points $i$ and $j$ in $R$ can be joined by at least one succession of links with non-zero transition probabilities.
Among the points of $R$, we now arbitrarily define a subset $\Sigma$, and note the complementary subset $\bar{\Sigma}$. Practically, the following properties will mostly be interesting if the number of points in $\Sigma$ is much smaller than the total number $N$ of points, but it is not necessary for the definitions.
With the definitions, the Perron-Frobenius theorem [@VanKampen] assures that there exists a stationary probability $\pi({\bf r}_i)$, which satisfies: $$\label{closed}
\pi({\bf r}_i) = \sum_{j\in R}w_{ij}\pi({\bf r}_j)$$ From now on, we will consider that $\Sigma$ is absorbing, which means that the particle is absorbed as soon as it goes to the subset. However, it may start from it and go away on the following step without being absorbed. Thus, we state that, on any state ${\bf r}_i$, the particle has a probability $p_d({\bf r}_i)$ to be absorbed on its next step equal to: $$p_d({\bf r}_i) = \sum_{j \in \Sigma}w_{ji}$$
Obtention of the formula
------------------------
Now, the probability $p({\bf r}_i,t)$ that the conditional particle is adsorbed exactly at time $t$, starting from state $i$ at time $0$, obeys the backward equation:
$$p({\bf r}_i,t) = \sum_{j \in \bar{\Sigma}}
p({\bf r}_j,t-1)
w_{ji}$$
if $t \geq 2$, and $$p({\bf r}_i,1) = p_d({\bf r}_i)$$
As a result, the Laplace transform $\hat{p}$ of $p({\bf r}_i,t)$ satisfies:
$$\hat{p}({\bf r}_i,s) - e^{-s}\sum_{j \in \Sigma}
w_{ji} = e^{-s} \sum_{j\in\bar{\Sigma}}
\hat{p}({\bf r}_j,s) w_{ji},$$
where $p({\bf r}_i,1)$ has been replaced by its value. We multiply this equation by the stationary probability $\pi({\bf r}_i)$ and sum up over all $i \in R$. We notice that, from (\[closed\]) $$\sum_{i\in\Sigma} w_{ji}\pi({\bf r}_i) = \pi({\bf r}_j)$$
We thus obtain:
$$\label{prefin}
\sum_{i\in R} \hat{p}({\bf r}_i,s)
\pi({\bf r}_i) - e^{-s}\sum_{j\in\Sigma} \pi({\bf r}_j) =
e^{-s} \sum_{j\in\bar{\Sigma}}
\hat{p}({\bf r}_j,s) \pi({\bf r}_j)$$
We now define two kinds of average for a quantity $\varphi(t)$:
([*i*]{}) the *volume average* $$\left< \varphi(\mathbf{T}) \right>_{\bar{\Sigma}} =
\frac1{\pi(\bar{\Sigma})} \sum_{i\in\bar{\Sigma}} \pi({\bf r}_i)
\sum_{t=1}^{\infty} \varphi(t) p({\bf r}_i,t)$$
([*ii*]{}) the *surface average* $$\left< \varphi(\mathbf{T}) \right>_{\Sigma} =
\frac1{\pi(\Sigma)} \sum_{i\in\Sigma} \pi({\bf r}_i)
\sum_{t=1}^{\infty} \varphi(t) p({\bf r}_i,t)$$
where $\pi(\bar{\Sigma})$ and $\pi(\Sigma)$ are the respective stationary probabilities of the volume and the surface: $$\pi(\bar{\Sigma}) = \sum_{i\in\bar{\Sigma}} \pi({\bf r}_i)$$ $$\pi(\Sigma) = \sum_{i\in\Sigma} \pi({\bf r}_i),$$
and $\mathbf{T}$ denotes the absorption time, which corresponds to the FPT to $\Sigma$, or the FRT to $\Sigma$, depending on whether the starting point is on $\bar{\Sigma}$ or $\Sigma$. We thus simply get from (\[prefin\]) the following equation: $$\pi(\Sigma){\left< e^{-s\mathbf{T}} \right>_{\Sigma}} + \pi(\bar\Sigma){\left< e^{-s\mathbf{T}} \right>_{\bar{\Sigma}}} - \pi(\Sigma)=
e^{-s}\pi(\bar\Sigma){\left< e^{-s\mathbf{T}} \right>_{\bar{\Sigma}}}$$ or $$\pi(\Sigma) \left({\left< e^{-s\mathbf{T}} \right>_{\Sigma}} - e^{-s} \right) = \pi(\bar\Sigma ) \left(
e^{-s} - 1 \right) {\left< e^{-s\mathbf{T}} \right>_{\bar\Sigma }},$$ which is the extended Kac’s formula, relating the Laplace transforms of the FRTs and the FPTs. Thus, for the first moment of $\mathbf{T}$ we obtain the very simple and general result: $$\left<\mathbf{T}\right>_{\Sigma} = \frac{1}{\pi(\Sigma)}$$ (Kac’s formula [@Aldous])
Simulation methods {#simulations}
==================
Random walks
------------
For random walks we use a method based on the exact enumeration method [@Majid]. The exact enumeration method allows one to compute the exact distribution probability up to a given time: at each time step ($t >
0$), we compute the full probability distribution of the random walker, using the master equation: $$p({\bf r},t) = \frac1\sigma \sum_{{\bf r}' \in N({\bf r})} p({\bf r}',t-1)$$ $p$ here is the probability of the random walker to be at position ${\bf r}$ at time $t$ and to never have reached the target site before. $N({\bf r})$ is the ensemble of neighbours of ${\bf r}$, which includes ${\bf r}$ itself if ${\bf r}$ is a boundary site. The initial condition is of course $p({\bf r}, 0)=\delta({\bf r},{\bf r}_S)$. Note that if we set $T=S$ the algorithm will compute the distribution of the FRT. After this first step, we have the probability distribution $p(t)$ of the FPT: $$p(t) = p({\bf r}_T, t)$$ The last step of the algorithm is to set $p({\bf r}_T,t)$ to 0, and we can then proceed to the computation for the time $t+1$. This enables us to compute the exact probability distribution, but of course the algorithm has to stop at a certain time. To go further, we can notice that the tail of the probability distribution is exponential (this corresponds to the highest eigenvalue of the transition matrix, the transition probabilities to and out of the target being set to 0 to take the absorption into account). If $p \sim e^{-\alpha t}$ for high enough $t$, then we can compute the distribution up to a time $t_0$, then estimate: $${\langle\mathbf{T}\rangle}= \sum_{t=0}^{t_0-1}p(t)+\frac{p(t_0)t_0}{1-e^{-\alpha}}+
\frac{p(t_0)e^{-\alpha}}{\left(1-e^{-\alpha}\right)^2}$$ The two latter terms correspond to $\sum_{t=t_0}^{\infty}p(t_0)e^{-\alpha(t-t_0)}$. Since $\alpha$ is small, its order of magnitude being $1/N$, they are approximated by $p(t_0)t_0/\alpha + p(t_0)/\alpha^2$. To estimate $\alpha$, we take $$\alpha = \frac{1}{10}\ln\frac{p(t_0-10)}{p(t_0)}$$ (we took $10$ steps and not one in order to avoid parity effects). To select $t_0$, we run a few trial simulations, with a large maximum time, and we determine the minimal $t_0$ which gives a result differing by at most $0.1 \% $ from the result obtained with a larger $t_0$. We add a small security margin, and then run the simulation. We use similar methods for all the other quantities studied. The error on the simulation results is thus guaranteed to be less than $0.1 \% $!
Brownian motion
---------------
Unfortunately, for the Brownian motion, we do not have such an accurate algorithm, and we thus used a Brownian-dynamics-based algorithm [@Bere97]: we average the time needed to reach the target on $n=10^5$ Brownian processes. To simulate the Brownian motion, we use the following algorithm:
1. [Find the distance between the particle and the nearest obstacle (target, non-flat boundary).]{}
2. [Multiply that distance by a constant $\alpha$ (we used $\alpha = 0.2$) to get a trial typical step length.]{}
3. [If we are very close to a boundary, or very close to the target, this trial step length would be too small. We thus add a lower cutoff to this trial step length (we took $0.01$ near the target, of typical size radius $1$, and $0.2$ near the curved boundaries, whose radius of curvature was typically $25$), and get the typical step length $r_{step}$.]{}
4. [We use this step length to determine our time step $t_{step} = r_{step}^2$. (we have $D=1$).]{}
5. [For each direction $x,y,z$, we add to the position a Gaussian random variable, of variance $2t_{step}$. To get such a variable, we use two random variables $\nu$ and $\mu$ uniformly distributed between 0 and 1, and then the random variable $
r_{step} \sqrt{-2\ln(\nu)}\cos(2\pi\mu)$ is indeed a Gaussian with the required variance.]{}
6. [If we are outside the domain, we move the particle inside the domain, to a position symmetrical with respect to the boundary.]{}
7. [If we are inside the target, we end the process, otherwise we take another step.]{}
This algorithm is less accurate than the one we used in the discrete case, and is computationally more expensive. Moreover, the study of the probability density of the FPT is delicate with this algorithm.
Properties of the pseudo-Green function $H$ {#AnnGreen}
===========================================
The properties of the *continuous* Pseudo-Green function are well described in [@Barton], and we will just describe the properties of the discrete one. We consider the case of symmetric transition probabilities. We define the discrete Laplacian operator: $$(-\Delta)_{ij} = \delta_{ij} - w_{ij}$$ This operator is hermitian, which will be useful. We define $\Phi_p$ and $\lambda_p$ the eigenvectors and (real) eigenvalues of the operator $-\Delta$, ordered from $0$ to $N-1$ in increasing order. We have $\lambda_0 = 0$, and $\Phi_0 = 1/\sqrt{N}$, with the usual normalization. Since the operator is hermitian, we can take $\Phi_p^* = \Phi_p$. We define: $$H({\bf r}_i|{\bf r}_j) = \sum_{p = 1}^{N-1}
\Phi_p^*({\bf r}_j)\Phi_p({\bf r}_i)/\lambda_p$$ This solution satisfies: $$-\Delta H({\bf r}_i|{\bf r}_j) = \delta_{ij} - \frac1N,$$ which corresponds to the definition we used for $H$, and we thus found the solution (up to a constant) to the equation (\[pseudogreen\]) we used to define $H$. This shows that $H$ is symmetric in its arguments if $W=\{ w_{ij} \}$ is symmetric. To prove that the sum $\bar{H}_j = \frac1N \sum_{i=1}^N
H({\bf r}_i|{\bf r}_j)$ is independent of $j$, we will simply sum up the equation (\[pseudogreen\]) over all $i$, and use the fact that $H$ is symmetric. This gives: $$-\Delta \bar{H}_j = 0$$ and $\bar{H}$ is proportional to $\Phi_0$, and thus is a constant.
|
---
abstract: 'Approximate heavy-quark spin and flavor symmetry and chiral symmetry play an important role in our understanding of the nonperturbative regime of strong interactions. In this work, utilizing the unitarized chiral perturbation theory, we explore the consequences of these symmetries in the description of the interactions between the ground-state singly charmed (bottom) baryons and the pseudo-Nambu-Goldstone bosons. In particular, at leading order in the chiral expansion, by fixing the only parameter in the theory to reproduce the $\Lambda_b(5912)$ \[$\Lambda_b^*(5920)$\] or the $\Lambda_c(2595)$ \[$\Lambda_c^*(2625)$\], we predict a number of dynamically generated states, which are contrasted with those of other approaches and available experimental data. In anticipation of future lattice QCD simulations, we calculate the corresponding scattering lengths and compare them to the existing predictions from a $\mathcal{O}(p^3)$ chiral perturbation theory study. In addition, we estimate the effects of the next-to-leading-order potentials by adopting heavy-meson Lagrangians and fixing the relevant low-energy constants using either symmetry or naturalness arguments. It is shown that higher-order potentials play a relatively important role in many channels, indicating that further studies are needed once more experimental or lattice QCD data become available.'
author:
- 'Jun-Xu Lu'
- Yu Zhou
- 'Hua-Xing Chen'
- 'Ju-Jun Xie'
- 'Li-Sheng Geng'
title: 'Dynamically generated $J^P=1/2^-(3/2^-)$ singly charmed and bottom heavy baryons'
---
Introduction
============
In recent years, heavy-flavor hadron physics has yielded many surprising results and attracted a lot of attention due to intensive worldwide experimental activities, such as *BABAR* [@Bernard:2013zwa], Belle [@SANTEL:2013jua; @Kato:2014nga], CLEO [@Seth:2011qd], BES [@Collaboration):2014uxa], LHCb [@Simone:2014yla], and CDF [@Palni:2012zxa]. The discoveries and confirmations of the many $XYZ$ particles have established the existence of exotic mesons made of four quarks, such as the $Z_c(3900)$ [@Ablikim:2013mio; @Liu:2013dau] and the $Z(4430)$ [@Aaij:2014jqa; @Choi:2007wga], and aroused great interest in the theoretical and lattice QCD community to understand their nature, though no consensus has been reached yet (see, e.g., Ref. [@Brambilla:2010cs]).
Different from the case of heavy-meson states, no similar exotic states have been firmly established in the heavy-flavor baryon sector, partly due to the fact that their production is more difficult. Up to now, there have only been a few experimental observations of excited charmed and bottom baryons (see Ref. [@Crede:2013kia] for a recent and comprehensive review). In the bottom baryon sector, the LHCb Collaboration has reported two excited $\Lambda_b$ states, the $\Lambda_b(5912)$ and the $\Lambda_b(5920)$ [@Aaij:2012da], with the latter being recently confirmed by the CDF Collaboration [@Aaltonen:2013tta]. In the charmed baryon sector, a number of excited states have been confirmed by various experiments, including the $\Lambda_c(2595)$, the $\Xi_c(2790)$, the $\Lambda_c(2625)$, and the $\Xi_c(2815)$ [@Agashe:2014kda]. The spin parities of the first two states and the last two states are assumed to be $1/2^-$ and $3/2^-$, respectively, according to quark model predictions.
The conventional picture is that these states are the orbital excitations of the corresponding ground states. There are, however, different interpretations; namely, they are dynamically generated states from the interactions between the ground-state charmed (bottom) baryons with the pseudo-Nambu-Goldstone bosons (and other coupled channels) [@Lutz:2003jw; @GarciaRecio:2008dp; @GarciaRecio:2012db; @Liang:2014eba; @Liang:2014kra] . The idea of dynamically generated states is an old one but has recently received a lot of attention. It has been quite successful in solving some long-standing difficulties encountered in hadron spectroscopy, e.g., the nature of the $\Lambda(1405)$ or the lowest-lying scalar nonet (see, e.g., Ref. [@Hyodo:2011ur] for a recent review). In the charmed and bottom baryon sector, [^1] various approaches have been adopted to study final-state interactions and resulting dynamically generated states, including the so-called unitarized chiral perturbation theory (UChPT) [@Lutz:2003jw], hidden-gauge symmetry inspired approaches [@Liang:2014eba; @Liang:2014kra; @Wu:2010jy; @Wu:2010vk; @Wu:2010rv; @Xiao:2013yca; @Xiao:2013jla], and heavy-quark symmetry inspired approaches [@GarciaRecio:2008dp; @GarciaRecio:2012db; @Flynn:2011gf; @Romanets:2012hm; @Romanets:2012ce; @Garcia-Recio:2013gaa; @Guo:2013xga] .
In the present work, we choose the UChPT to study the interactions between the ground-state charmed (bottom) baryons and the pseudo-Nambu-Goldstone bosons using the leading order (LO) chiral Lagrangians. In the charmed baryon case, our study differs from that of Ref. [@Lutz:2003jw] in the following respects. First, we adopt different regularization schemes to regularize the loop function in the UChPT. Second, to identify dynamically generated states, we search for poles on the complex plane instead of examining speed plots. Furthermore, we extend the UChPT to study the bottom baryons and study the effects of next-to-leading-order (NLO) potentials.
The paper is organized as follows. In Sec. II we briefly recall the UChPT in the description of the interactions between the pseudo Nambu-Goldstone mesons and the ground-state singly charmed (bottom) baryons at LO. Our main results are presented in Sec. III. In Sec. IV, we perform an exploratory NLO study, followed by a short summary and outlook in Sec. IV.
Theoretical framework
=====================
In this section, we briefly recall the essential ingredients of the UChPT. There are two building blocks in the UChPT: a kernel provided by chiral Lagrangians up to a certain order and a unitarization procedure. The kernel is standard except in the sector where baryons or heavy hadrons are involved, where nonrelativistic chiral Lagrangians are frequently used. Common unitarization procedures include the Bethe–Salpeter equation method [@Kaiser:1995eg; @Oller:1997ti; @Oset:1997it; @Krippa:1998us; @Nieves:1999bx; @Meissner:1999vr; @Lutz:2001yb; @GarciaRecio:2002td; @Hyodo:2002pk; @Borasoy:2006sr], the numerator/denominator (N/D) method [@Oller:1998zr], and the inverse amplitude method [@Truong:1988zp; @Dobado:1989qm; @Dobado:1992ha; @Dobado:1996ps; @Guerrero:1998ei; @GomezNicola:2001as; @Pelaez:2004xp]. In the present work, we choose to work with relativistic chiral Lagrangians and in the Bethe–Salpeter equation framework.
The Bethe–Salpeter equation can be written schematically as $$T=V+VGT,$$ where $T$ is the unitarized amplitude, $V$ is the potential, and $G$ is the one-loop two-point scalar function. In the context of the UChPT, the integral Bethe–Salpeter equation is often simplified and approximated to be an algebraic equation with the use of the on-shell approximation [@Oller:1997ti; @Oset:1997it]. This approximations works very well. See Ref. [@Altenbuchinger:2013gaa] for a recent study of off-shell effects in the UChPT and early references on this subject.
The leading-order interaction between a singly charmed baryon of the ground-state sextet and antitriplet and a pseudoscalar meson of the pion octet is provided by the chiral Lagrangian [@Lutz:2003jw; @Liu:2012uw] $$\begin{aligned}
\label{Eq:LOLag}
\begin{split}
\mathcal{L}
& =\frac{i}{16f_0^{2}}\mathrm{Tr}(\bar{H}_{[\bar{3}]}(x)\gamma^{\mu}[H_{[\bar{3}]}(x),[\phi(x),(\partial_{\mu}\phi(x))]_{-}]_{+})\\
& +\frac{i}{16f_0^{2}}\mathrm{Tr}(\bar{H}_{[6]}(x)\gamma^{\mu}[H_{[6]}(x),[\phi(x),(\partial_{\mu}\phi(x))]_{-}]_{+}),
\end{split}\end{aligned}$$ where $f_0$ is the pseudoscalar decay constant in the chiral limit, $\phi$ collects the pseudoscalar octet, and $H_{[\overline{3}]}$ and $H_{[6]}$ collect the charmed (bottom) baryons, respectively, $$\phi=\sqrt{2}
\left(
\begin{array}{ccc}
\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}} & \pi^{+} & K^{+} \\
\pi^{-} & -\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}} & K^{0} \\
K^{-} & \overline{K^{0}} & -\frac{2}{\sqrt{6}}\eta
\end{array}
\right),$$ $$H_{[\overline{3}]}=
\left(
\begin{array}{ccc}
0 & \Lambda_{c}^{+} & \Xi_{c}^{+} \\
-\Lambda_{c}^{+} & 0 & \Xi_{c}^{0} \\
-\Xi_{c}^{+} & -\Xi_{c}^{0} & 0
\end{array}
\right),$$ $$H_{[6]}=
\left(
\begin{array}{ccc}
\Sigma_{c}^{++} & \frac{\Sigma_{c}^{+}}{\sqrt{2}} & \frac{\Xi_{c}^{'+}}{\sqrt{2}} \\
\frac{\Sigma_{c}^{+}}{\sqrt{2}} & \Sigma_{c}^{0} & \frac{\Xi_{c}^{'0}}{\sqrt{2}} \\
\frac{\Xi_{c}^{'+}}{\sqrt{2}} & \frac{\Xi_{c}^{'0}}{\sqrt{2}} & \Omega_{c}^{0}
\end{array}
\right).$$ The $H$’s for the corresponding ground-state bottom baryons can be obtained straightforwardly by replacing the charm quark content by its bottom counterpart.
Expanding the Lagrangian of Eq. (\[Eq:LOLag\]) up to two pseudoscalar fields, one obtains the interaction kernel needed to describe the $\phi (p_2) B (p_1) \rightarrow \phi (p_4) B (p_3)$ process, where $p_i$’s are the 4-momenta of the respective particles, $$\begin{aligned}
\label{Eq:LOKernel}
V=\frac{C_{ij}^{(I,S)}}{4f_0^{2}}\gamma^{\mu}(p_{2}^{\mu}+p_{4}^{\mu})\approx\frac{C_{ij}^{(I,S)}}{4f_0^{2}}(E_2+E_4),\end{aligned}$$ where $C_{ij}^{(I,S)}$ are the Clebsch–Gordan coefficients given in the Appendix. In deriving the final form of $V$, we have assumed that the 3-momentum of a baryon is small compared to its mass. This is a valid assumption since in the present study we are only interested in the energy region close to the threshold of the respective coupled channels.
The loop function $G$ in the Bethe–Salpeter equation has the following simple form in four dimensions: $$G=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{2M_B}{[(P-q)^{2}-m_\phi^{2}+i\epsilon][q^{2}-M_B^{2}+i\epsilon]}.$$ This loop function is divergent and needs to be properly regularized. In principle, one can either adopt the dimensional regularization scheme or the cutoff regularization scheme. In Ref. [@Altenbuchinger:2013vwa], a so-called heavy-quark symmetry (HQS) inspired regularization scheme has been suggested, which manifestly satisfies both the chiral power counting and the heavy-quark spin and flavor symmetry up to $1/M_H$, where $M_H$ is a generic heavy-hadron mass. In the present work, we adopt the HQS regularization scheme, which reads
$$G_{HQS}=G_{\overline{MS}}-\frac{2\mathring{M}}{16\pi^{2}}\left(\log\left(\frac{\mathring{M}^{2}}{\mu^{2}}\right)-2\right)+\frac{2 m_\mathrm{sub}}{16\pi^{2}}\left(\log\left(\frac{\mathring{M}^{2}}{\mu^{2}}\right)+a\right),$$
$$\begin{aligned}
\label{Eq:MSbar}
\begin{split}
G_{\overline{MS}}(s,M^{2},m^{2}) & =\frac{2 M}{16\pi^{2}}\left[\frac{m^{2}-M^{2}+s}{2s}\log\left(\frac{m^{2}}{M^{2}}\right)\right.\\
& -\frac{q}{\sqrt{s}}(\log[2q\sqrt{s}+m^{2}-M^{2}-s]+\log[2q\sqrt{s}-m^{2}+M^{2}-s]\\
& -\log[2q\sqrt{s}+m^{2}-M^{2}+s]-\log[2q\sqrt{s}-m^{2}+M^{2}+s])\\
& \left.+\left(\log\left(\frac{M^{2}}{\mu^{2}}\right)\underline{-2}\right)\right].
\end{split}\end{aligned}$$
In the above equations, $m_\mathrm{sub}$ is a generic pseudoscalar meson mass, which can take the value of $m_\pi$ in the $u$, $d$ flavor case or an average of the pion, the kaon, and the eta masses in the $u$, $d$, and $s$ three-flavor case. $\mathring{M}$ is the chiral limit value of the charmed or bottom baryon masses. In the present study, we use the averaged antitriplet and sextet charmed or bottom baryon masses given in Table \[Table:masses\], instead. The difference is of higher chiral order. Clearly, the HQS inspired regularization method is a straightforward extension of the minimal subtraction scheme, which, in spirit, is very similar to the extended-on-mass-shell scheme [@Fuchs:2003qc]. In our present work, for the sake of comparison, we also present results obtained with the cutoff regularization scheme, where $$G_\mathrm{cut}=\int\limits_0^\Lambda \frac{q^2\,dq}{2\pi^2}\frac{E_M+E_m}{2 E_M E_m}\frac{2 M}{s-(E_M+E_m)^2+i\epsilon},$$ with $E_M=\sqrt{q^2+M^2}$ and $E_m=\sqrt{q^2+m^2}$. In the UChPT framework, one usually replaces the underlined $-2$ of Eq. (\[Eq:MSbar\]) by a subtraction constant to approximate unknown short-range or higher-order interactions. In the following, we refer to this regularization scheme as the $\overline{\mathrm{MS}}$ scheme.
![ Loop function $G(M)$ as a function of the heavy-hadron mass $M$ in different regularization schemes: HQS, $\overline{\mathrm{MS}}$ , the cutoff regularization scheme (CUT ), and the exact heavy-quark limit (HH). The subtraction constants or cutoff values have been fixed by reproducing the $\Lambda_b(5912)$ (left panel) or the $\Lambda_c(2595)$ (right panel). In calculating the loop function $G$, the pseudoscalar meson mass is fixed at that of the pion $m=138$ MeV, and the renormalization scale in the dimensional regularization methods is fixed at $\mu=1$ GeV.[]{data-label="Fig:loop"}](Fig1.eps "fig:"){width="45.00000%"} ![ Loop function $G(M)$ as a function of the heavy-hadron mass $M$ in different regularization schemes: HQS, $\overline{\mathrm{MS}}$ , the cutoff regularization scheme (CUT ), and the exact heavy-quark limit (HH). The subtraction constants or cutoff values have been fixed by reproducing the $\Lambda_b(5912)$ (left panel) or the $\Lambda_c(2595)$ (right panel). In calculating the loop function $G$, the pseudoscalar meson mass is fixed at that of the pion $m=138$ MeV, and the renormalization scale in the dimensional regularization methods is fixed at $\mu=1$ GeV.[]{data-label="Fig:loop"}](Fig2.eps "fig:"){width="45.00000%"}
In Fig. \[Fig:loop\], the loop functions $G$ calculated in different regularization schemes are compared with each other. The subtraction constants or cutoff values have been fixed by reproducing the $\Lambda_b(5912)$ (left panel) or the $\Lambda_c(2595)$ (right panel). In calculating the loop function $G$, the pseudoscalar meson mass is fixed at that of the pion $m=138$ MeV, and the renormalization scale in the dimensional regularization methods is fixed at $\mu=1$ GeV. The loop function in the exact heavy-quark limit is obtained by replacing $\mathring{M}$ with $M$ and expanding $G_{HQS}$ in inverse powers of $M$ up to $\mathcal{O}(1/M)$ [@Altenbuchinger:2013vwa]. It is clear that the loop functions of both the HQS scheme and the cutoff scheme seem to satisfy the heavy-quark symmetry to a few percent, while the naive $\overline{\mathrm{MS}}$ scheme strongly breaks the symmetry, consistent with the finding of Ref. [@Altenbuchinger:2013vwa]. To be conservative, in the following study of dynamically generated charmed (bottom) baryons, unless otherwise mentioned, we shall present the results obtained in both regularization schemes.
------------------------------ ----------------- -------------------- ------------------------------ -------------------- ----------------- -------------------- --
$\mathring{M}_c^{[\bar{3}]}$ $M_{\Lambda_c}$ $M_{\Xi_c}$ $\mathring{M}_c^{[6]} $ $M_{\Sigma_c}$ $M_{\Xi'_c}$ $M_{\Omega_c}$
$2408.5$ 2286.5 2469.5 2534.9 2453.5 2576.8 2695.2
$\mathring{M}_b^{[\bar{3}]}$ $M_{\Lambda_b}$ $M_{\Xi_b}$ $\mathring{M}_b^{[6]}$ $M_{\Sigma_b}$ $M_{\Xi'_b}$ $M_{\Omega_b}$
$5732.8$ 5619.4 5789.5 5890.0 5813.4 5926 6048
$M_{\Sigma^{*}_c}$ $M_{\Xi^{*}_c}$ $M_{\Omega^{*}_c}$ $\mathring{M}_{c^{*}}^{[6]}$ $M_{\Sigma^{*}_b}$ $M_{\Xi^{*}_b}$ $M_{\Omega^{*}_b}$
2517.9 2645.9 2765.9 2601.9 5833.5 5949.3 6069
$\mathring{M}_{b^{*}}^{[6]}$ $m_\pi$ $m_K$ $m_\eta$ $m_{sub}$ $f_\pi$ $f_0=1.17f_\pi$
5911.35 138.0 495.6 547.9 368.1 92.21 107.8
------------------------------ ----------------- -------------------- ------------------------------ -------------------- ----------------- -------------------- --
Results and discussions
=======================
At leading order, the only unknown parameter in the UChPT is related to the regularization of the loop function $G$, i.e., the subtraction constant $a$ in the dimensional regularization scheme or the cutoff value $\Lambda$ in the cutoff regularization scheme. Conventionally, in the latter method one often chooses a cutoff of the order of 1 GeV (the chiral symmetry breaking scale). Requiring the $G$ function evaluated at threshold to be equal in both methods, one can fix a “natural” value for the subtraction constant. In most cases, the above-mentioned prescription allows one to assign some of the dynamically generated states to their experimental counterparts. Once the identification is done, one can slightly fine-tune $\Lambda$ or $a$ so that the dynamically generated state coincides with its experimental counterpart and then use the so-obtained $\Lambda$ or $a$ to make predictions. We follow the same line of argument in the present work. As in previous works, we can identify the $\Lambda_c(2595)$ and the $\Lambda_b(5912)$ as dynamically generated states in their respective coupled channels.
Approximate heavy-quark spin symmetry implies that the interactions between a ground-state spin-1/2 (3/2) baryon and a pseudoscalar meson are the same in the limit of infinite heavy-quark masses. Therefore, one can extend the LO study of the $1/2^-$ sector to the $3/2^-$ sector. As a first approximation, we only need to replace the masses of the $1/2^+$ baryons by their $3/2^+$ counterparts (see Table \[Table:masses\]). Searching for poles on the complex plane, we find two states in the charmed and bottom sector with the same quantum numbers as those of the $\Lambda^{*}_{c}(2625)$ and $\Lambda^{*}_{b}(5920)$. Namely, they can be identified as the heavy-quark spin partners of the $\Lambda_c(2595)$ and $\Lambda_b(5912)$.
It should be noted that, unlike the heavy meson sector, the present lattice QCD simulations of charmed [@Briceno:2012wt; @Alexandrou:2014sha; @Liu:2009jc; @Namekawa:2013vu; @Brown:2014ena] or bottom [@Liu:2009jc; @Lewis:2008fu; @Brown:2014ena] baryons still focus on the ground states with the exception of Refs. [@Padmanath:2013zfa] and [@Meinel:2012qz], where excited triply charmed and bottom states were studied, respectively. Future lattice QCD simulations of the excited singly charmed and bottom baryons will be extremely valuable to test the predictions of the present work and those of other studies.[^2]
Pole position $(S,I)^{M}$ Main channels (threshold) Exp. [@Agashe:2014kda]
--------------- ------------------------------------- --------------------------------------------------------- ------------------------ --
$(6081,-i57)$ $(0,1)^{[\overline{3}]}$ $\Xi_{b}K$(6285.1)
$(5921,0)$ $(0,0)^{[\overline{3}]}$ $\Xi_{b}K$(6285.1)
$(5868,0)$ $(-1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{b}\overline{K}$(6115.0)
$(6118,-i50)$ $(-1,\frac{1}{2})^{[\overline{3}]}$ $\Xi_{b}\eta$(6337.4),$\Xi_{b}\pi$(5927.5)
$(6201,0)$ $(-2,0)^{[\overline{3}]}$ $\Xi_{b}\overline{K}$(6285.1)
$(6201,0)$ $(1,\frac{1}{2})^{[6]}$ $\Sigma_{b}K$(6308.6)
$(5967,-i9)$ $(0,1)^{[6]}$ $\Sigma_{b}\pi$(5951.0)
$(6223,-i14)$ $(0,1)^{[6]}$ $\Sigma_{b}\eta$(6360.9)
$(5912,0)$ $(0,0)^{[6]}$ $\Sigma_{b}\pi$(5951.0) $\Lambda_b(5912)$
$(6307,-i12)$ $(0,0)^{[6]}$ $\Xi'_{b}K$(6421.6)
$(6213,-i25)$ $( -1,\frac{3}{2})^{[6]}$ $\Sigma_{b}\overline{K}$(6308.6)
$(5955,0)$ $( -1,\frac{1}{2})^{[6]}$ $\Sigma_{b}\overline{K}$(6308.6)
$(6101,-i15)$ $( -1,\frac{1}{2})^{[6]}$ $\Xi'_{b}\pi$(6064.0)
$(6364,-i27)$ $(-1,\frac{1}{2})^{[6]}$ $\Omega_{b}K$(6543.6)
$(6361,-i59)$ $(-2,1)^{[6]}$ $\Omega_{b}\pi$(6186.0)
$(6169,0)$ $(-2,0)^{[6]}$ $\Xi'_{b}\overline{K}$(6421.6),$\Omega_{b}\eta$(6595.6)
: Dynamically generated bottom baryons of $J^P=1/2^-$. The subtraction constant is fixed in a way such that the $\Lambda_{b}(5912)$ mass is produced to be 5912 MeV with $a=-14.15$. All energies are in units of MeV and $(S,I)^M$ denotes (strangeness, isospin)$^\mathrm{SU(3) multiplet}$.[]{data-label="Table:bsub"}
Pole positions $(S,I)^{M}$ Main channels Ref. [@Agashe:2014kda]
---------------- ------------------------------------- -------------------------------------------- ------------------------ --
$(6083,-i72)$ $(0,1)^{[\overline{3}]}$ $\Xi_{b}K$(6285.1)
$(5908,0)$ $(0,0)^{[\overline{3}]}$ $\Xi_{b}K$(6285.1)
$(5867,0)$ $(-1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{b}\overline{K}$(6115.0)
$(6116,-i55)$ $(-1,\frac{1}{2})^{[\overline{3}]}$ $\Xi_{b}\eta$(6337.4),$\Xi_{b}\pi$(5927.5)
$(6198,0)$ $(-2,0)^{[\overline{3}]}$ $\Xi_{b}\overline{K}$(6285.1)
$(6221,0)$ $(1,\frac{1}{2})^{[6]}$ $\Sigma_{b}K$(6308.6)
$(5966,-i9)$ $(0,1)^{[6]}$ $\Sigma_{b}\pi$(5951.0)
$(6234,-i20)$ $(0,1)^{[6]}$ $\Sigma_{b}\eta$(6360.9)
$(5912,0)$ $(0,0)^{[6]}$ $\Sigma_{b}\pi$(5951) $\Lambda_b(5912)$
$(6305,-i17)$ $(0,0)^{[6]}$ $\Xi'_{b}K$(6421.6)
$(6226,-i27)$ $( -1,\frac{3}{2})^{[6]}$ $\Sigma_{b}\overline{K}$(6308.6)
$(5951,0)$ $( -1,\frac{1}{2})^{[6]}$ $\Sigma_{b}\overline{K}$(6308.6)
$(6089,-i10)$ $( -1,\frac{1}{2})^{[6]}$ $\Xi'_{b}\pi$(6064.0)
$(6343,-i35)$ $(-1,\frac{1}{2})^{[6]}$ $\Omega_{b}K$(6543.6)
$(6347,-i55)$ $(-2,1)^{[6]}$ $\Omega_{b}\pi$(6186.0)
$(6139,0)$ $(-2,0)^{[6]}$ $\Xi'_{b}\overline{K}$(6421.6)
: The same as Table \[Table:bsub\], but obtained in the cutoff regularization scheme with $\Lambda=2.17$ GeV.[]{data-label="Table:bcut"}
Pole position $(S,I)^{M}$ Main channels (threshold) Ref. [@Agashe:2014kda]
--------------- --------------------------- -------------------------------------- ------------------------ --
$(6181,0)$ $(1,\frac{1}{2})^{[6]}$ $\Sigma^{*}_{b}K$(6329.1)
$(5971,0)$ $(0,1)^{[6]}$ $\Sigma^{*}_{b}\pi$(5971.5)
$(6202,-i12)$ $(0,1)^{[6]}$ $\Sigma^{*}_{b}\eta$(6381.4)
$(5920,0)$ $(0,0)^{[6]}$ $\Sigma^{*}_{b}\pi$(5971.5) $\Lambda^{*}_b(5920)$
$(6289,-i11)$ $(0,0)^{[6]}$ $\Xi^{*}_{b}K$(6444.9)
$(6197,-i19)$ $( -1,\frac{3}{2})^{[6]}$ $\Sigma^{*}_{b}\overline{K}$(6329.1)
$(5950,0)$ $( -1,\frac{1}{2})^{[6]}$ $\Sigma^{*}_{b}\overline{K}$(6329.1)
$(6102,-i7)$ $( -1,\frac{1}{2})^{[6]}$ $\Xi^{*}_{b}\pi$(6087.3)
$(6344,-i23)$ $(-1,\frac{1}{2})^{[6]}$ $\Omega^{*}_{b}K$(6564.6)
$(6349,-i45)$ $(-2,1)^{[6]}$ $\Omega^{*}_{b}\pi$(6207.0)
$(6152,0)$ $(-2,0)^{[6]}$ $\Xi^{*}_{b}\overline{K}$(6444.9)
: Dynamically generated bottom baryons of $J^P=3/2^-$. The subtraction constant is fixed in a way such that the $\Lambda^*_{b}(5920)$ mass is produced to be 5920 MeV with $a=-16.27$. All energies are in units of MeV and $(S,I)^M$ denotes (strangeness, isospin)$^\mathrm{SU(3) multiplet}$.[]{data-label="Table:bsub2"}
Pole position $(S,I)^{M}$ Main channels (threshold) Ref. [@Agashe:2014kda]
--------------- --------------------------- -------------------------------------- ------------------------ --
$(6193,0)$ $(1,\frac{1}{2})^{[6]}$ $\Sigma^{*}_{b}K$(6329.1)
$(5971,0)$ $(0,1)^{[6]}$ $\Sigma^{*}_{b}\pi$(5971.5)
$(6205,-i15)$ $(0,1)^{[6]}$ $\Sigma^{*}_{b}\eta$(6381.4)
$(5920,0)$ $(0,0)^{[6]}$ $\Sigma^{*}_{b}\pi$(5971.5) $\Lambda^{*}_b(5920)$
$(6281,-i13)$ $(0,0)^{[6]}$ $\Xi^{*}_{b}K$(6444.9)
$(6202,-i19)$ $( -1,\frac{3}{2})^{[6]}$ $\Sigma^{*}_{b}\overline{K}$(6329.1)
$(5946,0)$ $( -1,\frac{1}{2})^{[6]}$ $\Sigma^{*}_{b}\overline{K}$(6329.1)
$(6091,-i3)$ $( -1,\frac{1}{2})^{[6]}$ $\Xi^{*}_{b}\pi$(6087.3)
$(6321,-i24)$ $(-1,\frac{1}{2})^{[6]}$ $\Omega^{*}_{b}K$(6564.6)
$(6331,-i39)$ $(-2,1)^{[6]}$ $\Omega^{*}_{b}\pi$(6207.0)
$(6128,0)$ $(-2,0)^{[6]}$ $\Xi^{*}_{b}\overline{K}$(6444.9)
: The same as Table \[Table:bsub\], but obtained in the cutoff regularization scheme with $\Lambda=2.60$ GeV.[]{data-label="Table:bcut2"}
Dynamically generated bottom baryons
------------------------------------
In Refs. [@GarciaRecio:2012db; @Liang:2014eba], the $\Lambda_b(5912)$ is found to be dynamically generated. In Ref. [@GarciaRecio:2012db], the dominant coupled channel is identified as $\bar{B}N$, while in Ref. [@Liang:2014eba] it is identified as $\bar{B}^* N$. In our approach, this state appears naturally as a $\Sigma_b \pi$ state. It is useful to point out the major differences among the approaches of Ref. [@GarciaRecio:2012db], Ref. [@Liang:2014eba], and the present work. The kernel looks similar in all the three cases. However, the three approaches differ in the number of coupled channels included and how the transition amplitudes between different channels are obtained. In Ref. [@GarciaRecio:2012db], the transition amplitudes between different coupled channels are obtained by invoking the SU(6) symmetry and heavy-quark spin symmetry, while in Ref. [@Liang:2014eba], they are obtained through the vector meson exchange or pion exchange. The number of coupled channels considered is the largest in Ref. [@GarciaRecio:2012db], while it is the smallest in our approach. In other words, we only consider the minimum number of channels needed to construct the LO chiral Lagrangians. In addition, up to the order at which we are working, the \[$\bar{3}$\] multiplet and the \[$6$\] multiplet do not mix. Therefore, one needs to be careful when comparing our predictions with those of Refs. [@GarciaRecio:2012db; @Liang:2014eba].
To study the interaction between the ground-state bottom baryons and the pseudoscalar mesons, we fix the cutoff value or the subtraction constant in such a way that the mass of the $\Lambda_b(5912)$ is produced to be 5912 MeV. Broken SU(3) chiral symmetry then predicts a number of additional resonances or bound states as shown in Tables \[Table:bsub\] and \[Table:bcut\]. It can be seen that in addition to the $\Lambda_b(5912)$, both regularization schemes generate a number of other states, the experimental counterparts of which cannot be identified . Future experiments are strongly encouraged to search for these states.
In Ref. [@GarciaRecio:2012db], only $(S,I)=(0,0)$ and $(-1,1/2)$ sectors are studied. For $J=1/2$, three $\Lambda_b$ states and three $\Xi_b$ states are identified. From the couplings of those dynamically generated states to the corresponding coupled channels (Tables III and IV of Ref. [@GarciaRecio:2012db]), it is clear that none of those states couples dominantly to the coupled channels considered in the present study. For instance, their $\Lambda_b(5912)$ and the bound state with $M=6009.3$ MeV couple only moderately to $\Sigma_b\pi$ and $\Lambda_b\eta$, respectively. The same is true for the two $\Xi_b$ states with $M=6035.4$ MeV and $M=6072.8$ MeV.
In Ref. [@Liang:2014eba], ten states are found in the $J^P=1/2^-$ sector. Among them, one $I=0$ state with $M=5969.5$ and one $I=1$ state with $M=6002.8$ MeV couple strongly to $\Sigma_b\pi$. Because of the different coupled channels considered in both works and the fact that the \[$\bar{3}$\] and \[6\] multiplets do not mix at leading order chiral perturbation theory, we must refuse the temptation to associate them to our dynamically generated states. One needs to keep in mind that in our present work only the smallest number of coupled channels that are dictated by approximate SU(3) chiral symmetry is taken into account. The introduction of additional coupled channels inevitably requires further less justified assumptions.
In principle, one can use the same subtraction constants or cutoff values for the $3/2^-$ sector. However, to make the prediction more precise, we slightly fine-tune them to reproduce the experimental mass of the $\Lambda_b^*(5920)$. The relative change of the subtraction constant or cutoff value reflects the effect of the heavy-quark symmetry breaking (at least in our framework). The results are shown in Tables \[Table:bsub2\] and \[Table:bcut2\], For the reason we mentioned above, the results are quite similar to those in the $J^P=1/2^-$ sector. In total, as in the $1/2^-$ sector, 11 states are found, the $J^P=1/2^-$ partners of which can be easily identified.
Pole position $(S,I)^{M}$ Main channels (threshold) Ref. [@Agashe:2014kda]
--------------------------- ------------------------------------- --------------------------------------------------------- ------------------------ --
$(2721,0)$ $(0,0)^{[\overline{3}]}$ $\Xi_{c}K$(2965.1) $\Lambda_c(2765)?$
$(2623,-i12)$ $(-1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{c}\overline{K}$(2782.1)
$(2965,0)$ $(-2,0)^{[\overline{3}]}$ $\Xi_{c}\overline{K}$(2965.1)
$(2948,0)$ $(1,\frac{1}{2})^{[6]}$ $\Sigma_{c}K$(2949.1)
$(2674,-i51)$ $(0,1)^{[6]}$ $\Sigma_{c}\pi$(2591.5)
$(2999,-i16)$ $(0,1)^{[6]}$ $\Sigma_{c}\eta$(3001.4),$\Xi'_{c}K$(3072.4)
$(2591,0)$ $(0,0)^{[6]}$ $\Sigma_{c}\pi$(2591.5) $\Lambda_c(2595)$
$(3069,-i12)$ $(0,0)^{[6]}$ $\Xi'_{c}K$(3072.4) $\Lambda_c(2940)?$
$\underline{(2947,-i34)}$ $( -1,\frac{3}{2})^{[6]}$ $\Sigma_{c}\overline{K}$(2949.1)
$(2695,0)$ $(-1,\frac{1}{2})^{[6]}$ $\Sigma_{c}\overline{K}$(2949.1)
$(2827,-i55)$ $(-1,\frac{1}{2})^{[6]}$ $\Xi'_{c}\pi$(2714.7) $\Xi_c(2790)$?
$\underline{(3123,-i44)}$ $(-1,\frac{1}{2})^{[6]}$ $\Omega_{c}K$(3190.8) $\Xi_c(3123)?$
$(2946,0)$ $(-2,0)^{[6]}$ $\Xi'_{c}\overline{K}$(3072.4),$\Omega_{c}\eta$(3243.1)
: Dynamically generated charmed baryons of $J^P=1/2^-$. The subtraction constant is fixed in a way such that the $\Lambda_{c}(2595)$ mass is produced to be 2591 MeV with $a=-8.27$. All energies are in units of MeV and $(S,I)^M$ denotes (strangeness, isospin)$^\mathrm{SU(3) multiplet}$.[]{data-label="Table:csub"}
Pole positions $(S,I)^{M}$ Main channels (threshold) Ref. [@Agashe:2014kda]
--------------------------- ------------------------------------- ----------------------------------- ------------------------ --
$(2707,0)$ $(0,0)^{[\overline{3}]}$ $\Xi_{c}K$(2965.1) $\Lambda_c(2765)?$
$(2622,-i12)$ $(-1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{c}\overline{K}$(2782.1)
$\underline{(2965,0)}$ $(-2,0)^{[\overline{3}]}$ $\Xi_{c}\overline{K}$(2965.1)
$\underline{(2949,0)}$ $(1,\frac{1}{2})^{[6]}$ $\Sigma_{c}K$(2949.1)
$(2672,-i53)$ $(0,1)^{[6]}$ $\Sigma_{c}\pi$(2591.5)
$\underline{(2996,-i21)}$ $( 0,1)^{[6]}$ $\Sigma_{c}\eta$(3001.4)
$(2591,0)$ $(0,0)^{[6]}$ $\Sigma_{c}\pi$(2591.5) $\Lambda_c(2595)$
$(3072,-i15)$ $(0,0)^{[6]}$ $\Xi'_{c}K$(3072.4) $\Lambda_c(2940)?$
$\underline{(2946,-i35)}$ $( -1,\frac{3}{2})^{[6]}$ $\Sigma_{c}\overline{K}$(2949.1)
$(2683,0)$ $(-1,\frac{1}{2})^{[6]}$ $\Sigma_c\overline{K}$(2949.1)
$(2813,-i44)$ $(-1,\frac{1}{2})^{[6]}$ $\Xi'_{c}\pi$(2714.7) $\Xi_c(2790)$?
$(3121,-i61)$ $(-1,\frac{1}{2})^{[6]}$ $\Omega_cK$(3190.8) $\Xi_c(3123)?$
$(2909,0)$ $(-2,0)^{[6]}$ $\Xi'_{c}K$(3072.4)
: The same as Table \[Table:csub\], but obtained in the cutoff regularization scheme with $\Lambda=1.35$ GeV.[]{data-label="Table:ccut"}
Pole position $(S,I)^{M}$ Main channels (threshold) Exp. [@Agashe:2014kda]
--------------- --------------------------- -------------------------------------- ------------------------ --
$(2952,0)$ $(1,\frac{1}{2})^{[6]}$ $\Sigma^{*}_{c}K$(3013.5)
$(2685,-i15)$ $(0,1)^{[6]}$ $\Sigma^{*}_{c}\pi$(2655.9)
$(2977,-i23)$ $(0,1)^{[6]}$ $\Sigma^{*}_{c}\eta$(3065.8)
$(2625,0)$ $(0,0)^{[6]}$ $\Sigma^{*}_{c}\pi$(2655.9) $\Lambda^{*}_c(2625)$
$(3066,-i19)$ $(0,0)^{[6]}$ $\Xi^{*}_{c}K$(3141.5)
$(2968,-i33)$ $( -1,\frac{3}{2})^{[6]}$ $\Sigma^{*}_{c}\overline{K}$(3013.5)
$(2656,0)$ $(-1,\frac{1}{2})^{[6]}$ $\Sigma^{*}_{c}\overline{K}$(3013.5)
$(2827,-i17)$ $(-1,\frac{1}{2})^{[6]}$ $\Xi^{*}_{c}\pi$(2783.9) $\Xi_c^*(2815)?$
$(3113,-i45)$ $(-1,\frac{1}{2})^{[6]}$ $\Omega^{*}_{c}K$(3261.5)
$(3118,-i80)$ $(-2,1)^{[6]}$ $\Omega^{*}_{c}\pi$(2903.9)
$(2885,0)$ $(-2,0)^{[6]}$ $\Xi^{*}_{c}\overline{K}$(3141.5)
: Dynamically generated charmed baryons of $J^P=3/2^-$. The subtraction constant is fixed in a way such that the $\Lambda^*_{c}(2625)$ mass is produced to be 2625 MeV with $a=-12.0$. All energies are in units of MeV and $(S,I)^M$ denotes (strangeness, isospin)$^\mathrm{SU(3) multiplet}$.[]{data-label="Table:csub2"}
Pole position $(S,I)^{M}$ Main channels (threshold) Exp. [@Agashe:2014kda]
--------------- --------------------------- -------------------------------------- ------------------------ --
$(2962,0)$ $(1,\frac{1}{2})^{[6]}$ $\Sigma^{*}_{c}K$(3013.5)
$(2684,-i15)$ $(0,1)^{[6]}$ $\Sigma^{*}_{c}\pi$(2655.9)
$(2980,-i28)$ $(0,1)^{[6]}$ $\Sigma^{*}_{c}\eta$(3065.8)
$(2625,0)$ $(0,0)^{[6]}$ $\Sigma^{*}_{c}\pi$(2655.9) $\Lambda^{*}_c(2625)$
$(3059,-i22)$ $(0,0)^{[6]}$ $\Xi^{*}_{c}K$(3141.5)
$(2974,-i33)$ $( -1,\frac{3}{2})^{[6]}$ $\Sigma^{*}_{c}\overline{K}$(3013.5)
$(2653,0)$ $(-1,\frac{1}{2})^{[6]}$ $\Sigma^{*}_{c}\overline{K}$(3013.5)
$(2816,-i13)$ $(-1,\frac{1}{2})^{[6]}$ $\Xi^{*}_{c}\pi$(2783.9) $\Xi_c^*(2815)?$
$(3093,-i51)$ $(-1,\frac{1}{2})^{[6]}$ $\Omega^{*}_{c}K$(3261.5)
$(3103,-i74)$ $(-2,1)^{[6]}$ $\Omega^{*}_{c}\pi$(2903.9)
$(2858,0)$ $(-2,0)^{[6]}$ $\Xi^{*}_{c}\overline{K}$(3141.5)
: The same as Table \[Table:csub\], but obtained in the cutoff regularization scheme with $\Lambda=2.13$ GeV.[]{data-label="Table:ccut2"}
Dynamically generated charmed baryons
-------------------------------------
Once the subtraction constant is fixed in the HQS approach, one can use the same constant to predict the counterparts of the dynamically generated bottom baryons. We have performed such a calculation and found that the $\Lambda_c(2595)$ can indeed be identified as a $\Sigma_c \pi$ state, as first pointed out in Refs. [@Lutz:2003jw; @Hofmann:2005sw]. To account for moderate heavy-quark flavor symmetry breaking corrections, we slightly fine-tune the subtraction constant in the dimensional regularization scheme or the cutoff value in the cutoff regularization scheme so that the mass of the $\Lambda_c(2595)$ is reproduced to be 2591 MeV.[^3] The predictions are then tabulated in Tables \[Table:csub\] and \[Table:ccut\].
A comparison with the predictions of Refs. [@GarciaRecio:2008dp; @Liang:2014kra] is again complicated by the same factors as mentioned previously. For instance, four $(S=0,I=0)$ states and five $(S=0,I=1)$ states are predicted in Ref. [@Liang:2014kra]. The number of dynamically generated states in Ref. [@GarciaRecio:2008dp] is even larger. Somehow, it seems that the number of states generated is proportional to the number of coupled channels considered.
In addition, our $\Lambda_c(2595)$ is predominantly a $\Sigma_c \pi$ state, where it is more of a $DN$ state in Ref. [@Liang:2014kra] and a $D^* N$ state in Ref. [@GarciaRecio:2008dp]. Despite of the different dominant components, it is clear that coupled channel effects or multiquark components may not be negligible in the wave function of the $\Lambda_c(2595)$. The same can be said about the $\Lambda_b(5912$).
In Tables \[Table:csub\], we have temporarily identified the states appearing at $\sqrt{s}=(2721,0)$ MeV, $\sqrt{s}=(3069-i12)$ MeV, $\sqrt{s}=(2827-i55)$ MeV, and $\sqrt{s}=(3123,-i44)$ MeV as the $\Lambda_c(2765)$, $\Lambda_c(2940)$, $\Xi_c(2790)$, and $\Xi_c(3123)$. These identifications are mainly based on the masses of these states [@Agashe:2014kda]. Since the spin parities of these states are not yet known, the associations of our states with their experimental counterparts should be taken with care. A second complication comes from the fact that coupled channels other than those considered here may not be negligible as can be seen from Fig. \[Fig:threshold\].
In Ref. [@He:2006is], the $\Lambda_c(2940)$ was suggested to be a molecular state with spin parity $J^P=1/2^-$ or $3/2^-$ because of its proximity to the $D^{*0} p$ threshold. In Ref. [@GarciaRecio:2008dp], none of the dynamically generated states with $J^P=1/2^-$ or $3/2^-$ can be associated to the $\Lambda(2940)$. In Ref. [@Liang:2014kra], a state at $2959$ MeV with a small width could be associated to the $\Lambda_c(2940)$, which, however, couples mostly $\rho \Sigma_c$. In our present study, since the $DN$ ($D^*N$) channels are not taken into account explicitly, we have found only two states located about 3050 MeV (see Tables \[Table:csub\] and \[Table:csub2\]), one of which we tentatively associate with the $\Lambda_c^{(*)}(2940)$. However, one definitely needs to take into account the missing $D^{(*)} N$ channels to be more conclusive. It should be noted that in the molecular picture Dong *et al*. have studied the strong two-body decays of the $\Lambda_c(2940)$ and shown that the $J^P=1/2^+$ assignment is favored [@Dong:2009tg]. Assuming this particular quantum number, they later studied the radiative [@Dong:2010xv] and strong three-body [@Dong:2011ys] decays of the $\Lambda_c(2940)$. The molecular nature of the $\Lambda_c(2940)$ has recently been studied in the framework of QCD sum rules [@Zhang:2012jk], the constituent quark model [@Ortega:2012cx], and the effective Lagrangians method [@He:2010zq], as well.
In Tables \[Table:csub2\] and \[Table:ccut2\], we tabulate the dynamically generated states in the $3/2^-$ sector. It should be noted that, compared to the $1/2^-$ sector, an extra pole is produced in the $(S,I)=(-2,1)$ channel. On the other hand, its counterpart is found in both the $3/2^-$ and $1/2^-$ bottom sectors. This seems to indicate that the breaking of the heavy-quark flavor symmetry is larger than that of the heavy-quark spin symmetry, as naively expected.
It should be noted that to confirm the identification of the dynamically generated states with their experimental counterparts, one needs to study their decay branching ratios, since many approaches used the masses of these states to fix (some of) their parameters. Strong and radiative decays are both very important in this respect since they may probe different regions of their wave functions. In the past few years, many such studies of the decays of charmed baryons have been performed, see, e.g., Refs. [@Dong:2009tg; @Dong:2010xv; @Dong:2011ys; @Cheng:2006dk; @Chen:2007xf; @Zhong:2007gp; @Liu:2012sj; @Yasui:2014cwa; @Gamermann:2010ga].[^4]
Further discussions
-------------------
Superficially, exact heavy-quark flavor symmetry would dictate that the number of dynamically generated states in the bottom sector and that in the charm sector is the same. A comparison of Tables \[Table:bsub\] and \[Table:csub\] (or Tables \[Table:bcut\] and \[Table:ccut\]) shows that this is almost the case, but not exactly. For instance, some counterparts of the dynamically generated bottom baryons in the charm sector are missing, such as the counterparts of the \[$\bar{3}$\] states at $\sqrt{s}=(6081-i57)$ MeV and $\sqrt{s}=(6118-i50)$ MeV. A closer look at these channels reveals that they simply become too broad and develop a width of $200\sim300$ MeV. It should be noted that we have not considered any states broader than 200 MeV in our study,
The broadening of these states can be traced back partially to the weakening of the corresponding potentials and partially to the calibration of our framework to reproduce the $\Lambda_b(5912)$ in the bottom sector and to reproduce the $\Lambda_c(2595)$ in the charmed sector. Since the $\Lambda_c(2595)$ is much closer to the threshold of its main coupled channel than the $\Lambda_b(5912)$, the calibration implies a weaker potential in the charm sector than in the bottom sector. Because of this weakening, the dynamical generation of some charmed baryons requires a slight readjustment of the potential by changing either $f_0$ or $a$ slightly within a few percent. Otherwise, they will show up as cusps. The pole positions of these states have been underlined to denote such a fine-tuning.
One should note that we have used an averaged pseudoscalar decay constant, $f_0=1.17 f_\pi$, in our calculations. Using the pion decay constant, $f_0=f_\pi$, will not change qualitatively our results and conclusions but can shift the predicted baryon masses by a few tens of MeV depending on the particular channel. We have not given explicitly such uncertainties in Tables \[Table:bsub\], \[Table:bcut\], \[Table:bsub2\], \[Table:bcut2\], \[Table:csub\], \[Table:ccut\], \[Table:csub2\], and \[Table:ccut2\], but one should keep in mind the existence of such uncertainties (or freedom) in our approach. In addition, the differences between the results obtained in the dimensional regularization scheme and those obtained in the cutoff regularization scheme also indicate inherent theoretical uncertainties of the UChPT method, which can be as large as 30–40 MeV depending on the channel. It should be mentioned that, although formally the dimensional regularization scheme might be preferred to the cutoff regularization scheme, they yield quite similar results in our present work, both in terms of heavy-quark symmetry conservation and in terms of the prediction of dynamically generated states once the relevant parameters are fixed in such a way that the $\Lambda_c(2595)$ and the $\Lambda_b(5912)$ are produced.
As mentioned previously, compared to the studies of Refs. [@GarciaRecio:2008dp; @GarciaRecio:2012db; @Liang:2014eba; @Liang:2014kra], we have only considered the minimum number of coupled channels dictated by chiral symmetry and its breaking. Such an approach is only appropriate if close to the dynamically generated states no other coupled channels with the same quantum numbers exist. Otherwise, one may need to take into account those channels involving either vector mesons (light or heavy) or noncharmed (bottom) baryons. As can be seen from Fig. \[Fig:threshold\], it is clear that for the dynamical generation of the $\Lambda_c(2595)$ and the $\Lambda_b(5912)$, our minimum coupled channel space indeed includes the most relevant channels, i.e., the $\Lambda_c \pi$ and the $\Sigma_b \pi$, while the next-closest coupled channels excluded in our space, the $N D$ and $N\bar{B}$, are roughly 200 MeV above. On the other hand, the $\Lambda_c(2940)$ state is close not only to the $\Xi'_c K$ channel taken into account in our framework but also to $\Lambda_c\omega$ and $\Lambda D_s$. As a result, our model space may be too restricted, and the result should be taken with care. This might be the reason why our prediction is about 100 MeV off the experimental mass of this resonance.
It has long been an important and challenging work to differentiate hadronic states of different nature, e.g., whether being composite states of other hadrons or being “genuine” (multi)quark states. Half a century ago, Weinberg proposed the so-called compositeness condition, which allowed him to tell that the deuteron is a weakly bound state of a proton and a neutron, instead of a genuine six-quark state [@Weinberg:1965zz]. With renewed interests in hadron spectroscopy, there have been some recent works on this issue [@Hanhart:2010wh; @Baru:2003qq; @Cleven:2011gp]. Extensions to larger binding energies in the $s$ wave for bound states [@Gamermann:2009uq] and resonances [@YamagataSekihara:2010pj] and to higher partial waves for mesonic states [@Aceti:2012dd; @Xiao:2012vv] and baryonic states [@Aceti:2014ala; @Aceti:2014wka] have been performed.[^5]
Following Ref. [@Aceti:2014ala], one can define the weight of a hadron-hadron component in a composite particle as $$X_i=-\mathrm{Re}\left[g_i^2\left[\frac{\partial G_i^{II}(s)}{\partial \sqrt{s}}\right]_{\sqrt{s}=\sqrt{s_0}}\right]£¬$$ where $\sqrt{s_0}$ is the pole position, $G_i^{II}$ is the loop function evaluated on the second Riemann sheet, and $g_i$ is the couplings of the respective resonance or bound state to channel $i$ calculated as $$g_i^2=\mathop{\mathrm{lim}}_{\sqrt{s}\rightarrow\sqrt{s_0}}(\sqrt{s}-\sqrt{s_0})T_{ii}^{II},$$ where $T_{ii}^{II}$ is the $ii$ element of the $T$ amplitude on the second Riemann sheet.
The deviation of the sum of $X_i$ from unity is related to the energy dependence of the $s$-wave potential, $$\sum_i X_i=1-Z,$$ where $$Z=-\sum_{ij}\left[g_i G_i^{II}(\sqrt{s})\frac{\partial V_{ij}(\sqrt{s})}{\partial \sqrt{s}}G_j^{II}(\sqrt{s})g_j\right]_{\sqrt{s}=\sqrt{s}_0}.$$ Although in certain cases $Z$ can be attributed to the weight of the missing channels (see, e.g., Ref. [@Aceti:2014ala]), it is not clear how to interpret $Z$ obtained from the smooth energy dependence of the chiral potential $V$ [@Aceti:2014wka]. In addition, in a coupled-channel scenario, we noticed that different treatments of the regularization schemes can reshuffle the contributions between $\sum_i X_i$ and $Z$, thus complicating the interpretation of the so-called compositeness [@Geng:2015]. To complicate things more, for processes involving short distances, it is the wave function at the origin that matters ($g_i G_i$ for the $s$ wave) [@Gamermann:2009fv]. For an extensive discussion on this issue, see Ref. [@Aceti:2014wka], which concluded that to judge the relevance of each channel one has to study different physical processes. In the present context, we may similarly conclude that the relevance of the channels neglected in the present work compared with those of Refs. [@GarciaRecio:2008dp; @GarciaRecio:2012db; @Liang:2014eba; @Liang:2014kra] can only be reliably evaluated in specific physical processes, which will be left for future studies.[^6]
![Thresholds of the coupled channels considered in different works for the singly charmed and bottom baryon sector with $J^P=1/2^-$ and $(S=0,I=0)$: Liang *et al.* [@Liang:2014eba; @Liang:2014kra], Hofmann *et al.* [@Hofmann:2005sw],Garcia-Recio *et al.* [@GarciaRecio:2008dp; @GarciaRecio:2012db],and the experiment in Ref. [@Agashe:2014kda]. In the left figure, two model spaces denoted by dot-dot-dashed lines (pseudoscalar-baryon) and dashed lines (vector-baryon), respectively, were studied in Ref. [@Liang:2014kra]. []{data-label="Fig:threshold"}](lambdac2593.eps "fig:"){width="48.00000%"} ![Thresholds of the coupled channels considered in different works for the singly charmed and bottom baryon sector with $J^P=1/2^-$ and $(S=0,I=0)$: Liang *et al.* [@Liang:2014eba; @Liang:2014kra], Hofmann *et al.* [@Hofmann:2005sw],Garcia-Recio *et al.* [@GarciaRecio:2008dp; @GarciaRecio:2012db],and the experiment in Ref. [@Agashe:2014kda]. In the left figure, two model spaces denoted by dot-dot-dashed lines (pseudoscalar-baryon) and dashed lines (vector-baryon), respectively, were studied in Ref. [@Liang:2014kra]. []{data-label="Fig:threshold"}](lambdab5912.eps "fig:"){width="48.00000%"}
$(S,I)^{M}$ Channel $a$ $(S,I)^{M}$ Channel $a$
------------------------------------- ----------------------------------- ----------------- -------------------------- ---------------------------------- ----------------- -- --
$(1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{b}K$(6115.0) $-0.111$ $(1,\frac{3}{2})^{[6]}$ $\Sigma_{b}K$(6308.6) $-0.138$
$(0,1)^{[\overline{3}]}$ $\Xi_{b}K$(6285.1) $-0.239-i0.040$ $(1,\frac{1}{2})^{[6]}$ $\Sigma_{b}K$(6308.6) $-0.419$
$(0,1)^{[\overline{3}]}$ $\Lambda_{b}\pi$(5757.4) $0.003$ $(0,2)^{[6]}$ $\Sigma_{b}\pi$(5951.0) $-0.102$
$(0,0)^{[\overline{3}]}$ $\Xi_{b}K$(6285.1) $-0.204-i0.003$ $(0,1)^{[6]}$ $\Xi'_{b}K$(6421.6) $-0.211-i0.007$
$(0,0)^{[\overline{3}]}$ $\Lambda_{b}\eta$(6167.3) $-0.150$ $(0,1)^{[6]}$ $\Sigma_{b}\eta$(6360.9) $-0.273-i0.014$
$(-1,\frac{3}{2})^{[\overline{3}]}$ $\Xi_{b}\pi$(5927.5) $-0.067$ $(0,1)^{[6]}$ $\Sigma_{b}\pi$(5951.0) $1.162$
$(-1,\frac{1}{2})^{[\overline{3}]}$ $\Xi_{b}\pi$(5927.5) $-0.245$ $(0,0)^{[6]}$ $\Xi'_{b}K$(6421.6) $-0.398-i0.019$
$(-1,\frac{1}{2})^{[\overline{3}]}$ $\Xi_{b}\eta$(6337.4) $-0.208-i0.028$ $(0,0)^{[6]}$ $\Sigma_{b}\pi$(5951.0) $-0.598$
$(-1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{b}\overline{K}$(6115.0) $-0.181-i0.206$ $(-1,\frac{3}{2})^{[6]}$ $\Xi'_{b}\pi$(6064.0) $0.012$
$(-2,1)^{[\overline{3}]}$ $\Xi_{b}\overline{K}$(6285.1) $-0.118$ $(-1,\frac{3}{2})^{[6]}$ $\Sigma_{b}\overline{K}$(6308.6) $-0.350-i0.061$
$(-2,0)^{[\overline{3}]}$ $\Xi_{b}\overline{K}$(6285.1) $-0.507$ $(-1,\frac{1}{2})^{[6]}$ $\Xi'_{b}\pi$(6064.0) $0.497$
$(-1,\frac{1}{2})^{[6]}$ $\Xi'_{b}\eta$(6473.9) $-0.222-i0.020$ $(-2,1)^{[6]}$ $\Omega_{b}\pi$(6186.0) $0.086$
$(-1,\frac{1}{2})^{[6]}$ $\Omega_{b}K$(6543.6) $-0.279-i0.014$ $(-2,0)^{[6]}$ $\Xi'_{b}\overline{K}$(6421.6) $-0.214$
$(-1,\frac{1}{2})^{[6]}$ $\Sigma_{b}\overline{K}$(6308.6) $-0.185-i0.008$ $(-2,0)^{[6]}$ $\Omega_{b}\eta$(6595.9) $-0.187-i0.003$
$(-2,1)^{[6]}$ $\Xi'_{b}\overline{K}$(6421.6) $-0.245-i0.112$ $(-3,\frac{1}{2})^{[6]}$ $\Omega_{b}\overline{K}$(6543.6) $-0.153$
: $\phi B$ scattering lengths $a$ (in units of fm) in the bottom sector with $J^P=1/2^-$.[]{data-label="Table:scat3b"}
$(S,I)^{M}$ Channel $a$ $(S,I)^{M}$ Channel $a$
------------------------------ -------------------------------------- ----------------- ------------------------------ -------------------------------------- ----------------- -- --
$(1,\frac{3}{2})^{[6^{*}]}$ $\Sigma^{*}_{b}K$(6329.1) $-0.126$ $(-1,\frac{1}{2})^{[6^{*}]}$ $\Xi^{*}_{b}\pi$(6087.3) $0.971$
$(1,\frac{1}{2})^{[6^{*}]}$ $\Sigma^{*}_{b}K$(6329.1) $-0.325$ $(-1,\frac{1}{2})^{[6^{*}]}$ $\Xi^{*}_{b}\eta$(6497.2) $-0.186-0.009i$
$(0,2)^{[6^{*}]}$ $\Sigma^{*}_{b}\pi$(5971.5) $-0.096$ $(-1,\frac{1}{2})^{[6^{*}]}$ $\Omega^{*}_{b}K$(6564.6) $-0.233-0.008i$
$(0,1)^{[6^{*}]}$ $\Xi^{*}_{b}K$(6444.9) $-0.183-0.004i$ $(-1,\frac{1}{2})^{[6^{*}]}$ $\Sigma^{*}_{b}\overline{K}$(6329.1) $-0.151-0.024i$
$(0,1)^{[6^{*}]}$ $\Sigma^{*}_{b}\eta$(6381.3) $-0.223-0.007i$ $(-2,1)^{[6^{*}]}$ $\Xi^{*}_{b}\overline{K}$(6444.9) $-0.222-0.056i$
$(0,1)^{[6^{*}]}$ $\Sigma^{*}_{b}\pi$(5971.5) $8.412$ $(-2,1)^{[6^{*}]}$ $\Omega^{*}_{b}\pi$(6207.0) $0.109$
$(0,0)^{[6^{*}]}$ $\Xi^{*}_{b}K$(6444.9) $-0.312-0.009i$ $(-2,0)^{[6^{*}]}$ $\Xi^{*}_{b}\overline{K}$(6444.9) $-0.184$
$(0,0)^{[6^{*}]}$ $\Sigma^{*}_{b}\pi$(5971.5) $-0.425$ $(-2,0)^{[6^{*}]}$ $\Omega^{*}_{b}\eta$(6616.9) $-0.165-0.002i$
$(-1,\frac{3}{2})^{[6^{*}]}$ $\Xi^{*}_{b}\pi$(6087.3) $0.042$ $(-3,\frac{1}{2})^{[6^{*}]}$ $\Omega^{*}_{b}\overline{K}$(6564.6) $-0.139$
$(-1,\frac{3}{2})^{[6^{*}]}$ $\Sigma^{*}_{b}\overline{K}$(6329.1) $-0.276-0.029i$
: $\phi B$ scattering lengths $a$ (in units of fm) in the bottom sector with $J^P=3/2^-$.[]{data-label="Table:scat3b2"}
------------------------------------- --------------------------- ----------------- ------------------------- -------------------------- -------------------------- ----------------- -----------------------
$(S,I)^{M}$ Channel $(S,I)^{M}$ Channel
Present work Ref. [@Liu:2012uw] Present work Ref. [@Liu:2012uw]
$(1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{c}K$ $-0.135$ $-0.032\pm0.038$ $(1,\frac{3}{2})^{[6]}$ $\Sigma_{c}K$ $-0.181$ $-0.44$$\mp$0.23
$(0,1)^{[\overline{3}]}$ $\Xi_{c}K$ $-0.281-i0.308$ $0.77+i0.18$ $(1,\frac{1}{2})^{[6]}$ $\Sigma_{c}K$ $-8.114$ 0.62$\pm$0.12
$(0,1)^{[\overline{3}]}$ $\Lambda_{c}\pi$ 0.002 0.006 $(0,2)^{[6]}$ $\Sigma_{c}\pi$ $-0.119$ $-0.25$$\mp$0.031
$(0,0)^{[\overline{3}]}$ $\Xi_{c}K$ $-0.338-i0.020$ 0.99$\pm$0.076 $(0,1)^{[6]}$ $\Xi'_{c}K$ $-0.365-i0.097$ $(0.18+i0.37)\mp$0.12
$(0,0)^{[\overline{3}]}$ $\Lambda_{c}\eta$ $-0.281$ $(0.35+i0.19)\pm0.044$ $(0,1)^{[6]}$ $\Sigma_{c}\eta$ $-0.787-i0.942$ $(0.18+i0.2)\mp$0.034
$(-1,\frac{3}{2})^{[\overline{3}]}$ $\Xi_{c}\pi$ $-0.072$ $-0.11$$\pm$0.0052 $(0,1)^{[6]}$ $\Sigma_{c}\pi$ 0.376 0.28
$(-1,\frac{1}{2})^{[\overline{3}]}$ $\Xi_{c}\pi$ 1.600 0.32$\pm$0.0052 $(0,0)^{[6]}$ $\Xi'_{c}K$ $-1.361-i1.174$ $(1.4+i0.56)\mp$0.12
$(-1,\frac{1}{2})^{[\overline{3}]}$ $\Xi_{c}\eta$ $-0.266-i0.197$ $(0.54+i0.098)\pm0.011$ $(0,0)^{[6]}$ $\Sigma_{c}\pi$ $-28.204$ 0.65$\mp$0.078
$(-1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{c}\overline{K}$ $-0.237-i0.148$ $(0.79+i0.27)\pm0.038$ $(-1,\frac{3}{2})^{[6]}$ $\Xi'_{c}\pi$ $-0.025$ $-0.19$$\mp$0.016
$(-2,1)^{[\overline{3}]}$ $\Xi_{c}\overline{K}$ $-0.141$ $-0.028$$\pm$0.038 $(-1,\frac{3}{2})^{[6]}$ $\Sigma_{c}\overline{K}$ $0.141-i0.770$ $0.12+i0.37$
$(-2,0)^{[\overline{3}]}$ $\Xi_{c}\overline{K}$ 12.014 1.7$\mp$0.038 $(-1,\frac{1}{2})^{[6]}$ $\Xi'_{c}\pi$ $-0.022$ 0.23$\mp$0.016
$(-1,\frac{1}{2})^{[6]}$ $\Xi'_{c}\eta$ $-0.196-i0.269$ $(0.55+i0.49)\mp0.24$ $(-2,1)^{[6]}$ $\Omega_{c}\pi$ 0.046 $-0.062$
$(-1,\frac{1}{2})^{[6]}$ $\Omega_{c}K$ $-0.508-i0.160$ $(1.4+i0.56)\mp0.23$ $(-2,0)^{[6]}$ $\Xi'_{c}\overline{K}$ $-0.413$ 0.61$\mp$0.12
$(-1,\frac{1}{2})^{[6]}$ $\Sigma_{c}\overline{K}$ $-0.345-i0.013$ $(2.0+i0.092)\mp0.35$ $(-2,0)^{[6]}$ $\Omega_{c}\eta$ $-0.277-i0.015$ $(0.68+i0.4)\mp$0.14
$(-2,1)^{[6]}$ $\Xi'_{c}\overline{K}$ $-0.088-i0.168$ $(-0.11+i0.37)\mp 0.12$ $(-3,\frac{1}{2})^{[6]}$ $\Omega_{c}\overline{K}$ $-0.197$ $-0.33$$\mp$0.23
------------------------------------- --------------------------- ----------------- ------------------------- -------------------------- -------------------------- ----------------- -----------------------
: $\phi B$ scattering lengths $a$ (in units of fm) in the charmed sector with $J^P=1/2^-$.[]{data-label="Table:scat3c"}
------------------------------ -------------------------------------- ----------------- --------------------------- ------------------------------ -------------------------------------- ----------------- -------------------------
$(S,I)^{M}$ Channel $(S,I)^{M}$ Channel
Present work Ref. [@Liu:2012uw] Present work Ref. [@Liu:2012uw]
$(1,\frac{3}{2})^{[6^{*}]}$ $\Sigma^{*}_{c}K$(3013.5) $-0.147$ $-0.45\mp0.23$ $(-1,\frac{1}{2})^{[6^{*}]}$ $\Xi^{*}_{c}\pi$(2783.9) $ 0.459$ $(0.23-0.027i)\mp0.016$
$(1,\frac{1}{2})^{[6^{*}]}$ $\Sigma^{*}_{c}K$(3013.5) $-0.683$ $0.63\mp0.12$ $(-1,\frac{1}{2})^{[6^{*}]}$ $\Xi^{*}_{c}\eta$(3193.8) $-0.263-0.060i$ $(0.57+0.5i)\mp0.24$
$(0,2)^{[6^{*}]}$ $\Sigma^{*}_{c}\pi$(2655.9) $-0.104$ $-0.25\mp0.031$ $(-1,\frac{1}{2})^{[6^{*}]}$ $\Omega^{*}_{c}K$(3261.5) $-0.324-0.036i$ $(1.4+0.56i)\mp0.23$
$(0,1)^{[6^{*}]}$ $\Xi^{*}_{c}K$(3141.5) $-0.246-0.020i$ $(0.13+0.37i)\mp0.12$ $(-1,\frac{1}{2})^{[6^{*}]}$ $\Sigma^{*}_{c}\overline{K}$(3013.5) $-0.222-0.009i$ $(2.0+0.092i)\mp0.35$
$(0,1)^{[6^{*}]}$ $\Sigma^{*}_{c}\eta$(3065.8) $-0.398-0.059i$ $(0.16+0.2i)\mp0.034$ $(-2,1)^{[6^{*}]}$ $\Xi^{*}_{c}\overline{K}$(3141.5) $-0.199-0.211i$ $(-0.12+0.37i)\mp0.12$
$(0,1)^{[6^{*}]}$ $\Sigma^{*}_{c}\pi$(2655.9) $0.820$ $0.27-0.021i$ $(-2,1)^{[6^{*}]}$ $\Omega^{*}_{c}\pi$(2903.9) $0.085$ $-0.062$
$(0,0)^{[6^{*}]}$ $\Xi^{*}_{c}K$(3141.5) $-0.572-0.072i$ $(1.5+0.56i)\mp0.12$ $(-2,0)^{[6^{*}]}$ $\Xi^{*}_{c}\overline{K}$(3141.5) $-0.243$ $(0.52-0.0032i)\mp0.12$
$(0,0)^{[6^{*}]}$ $\Sigma^{*}_{c}\pi$(2655.9) $-0.761$ $(0.67+0.032i)\mp0.078$ $(-2,0)^{[6^{*}]}$ $\Omega^{*}_{c}\eta$(3313.8) $-0.201-0.005i$ $(0.64+0.4i)\mp0.14$
$(-1,\frac{3}{2})^{[6^{*}]}$ $\Xi^{*}_{c}\pi$(2783.9) $ 0.022$ $(-0.19-0.0027i)\mp0.016$ $(-3,\frac{1}{2})^{[6^{*}]}$ $\Omega^{*}_{c}\overline{K}$(3261.5) $-0.157$ $-0.34\mp0.23$
$(-1,\frac{3}{2})^{[6^{*}]}$ $\Sigma^{*}_{c}\overline{K}$(3013.5) $-0.539-0.242i$ $0.13+0.37i$
------------------------------ -------------------------------------- ----------------- --------------------------- ------------------------------ -------------------------------------- ----------------- -------------------------
: $\phi B$ scattering lengths $a$ (in units of fm) in the charmed sector with $J^P=3/2^-$.[]{data-label="Table:scat3c2"}
Scattering lengths
------------------
Scattering lengths provide vital information on the strong interactions. Although direct experimental measurements of the scattering lengths between a charmed (bottom) baryon and a pseudoscalar meson cannot be foreseen in the near future, rapid developments in lattice QCD may soon fill the gap. In Tables \[Table:scat3b\], \[Table:scat3b2\], \[Table:scat3c\], and \[Table:scat3c2\], we tabulate the scattering lengths calculated in the dimensional regularization scheme, defined as $$a_{jj}=-\frac{M_j}{4\pi(M_j+m_j)}T^{(S,I)}_{jj},$$ for channel $j$ with strangeness $S$ and isospin $I$, where $M_j$ and $m_j$ are the respective baryon and meson masses of that channel. For the sake of comparison, we list the chiral perturbation theory results of Ref. [@Liu:2012uw]. One should note, however, that Ref. [@Liu:2012uw] calculated the scattering lengths up to $\mathcal{O}(p^3)$ . While in our study, only the leading-order ($\mathcal{O}(p)$) chiral perturbation theory kernel is used, and in addition we work with the UChPT.
Examining the scattering lengths in the charmed sector, we notice that because of the existence of a bound state just below their respective thresholds, the scattering lengths for the $\Sigma_c K$ channel with $(S,I)^M=(1,1/2)^{[6]}$ and for the $\Sigma_c\pi$ channel with $(S,I)^M=(0,0)^{[6]}$ are quite large and negative, i.e., $a_{\Sigma_c K}=-8.114$ and $a_{\Sigma_c\pi}=-28.204$. Therefore, a future lattice QCD study of these two channels may be able to test to what extent the scenario of these states being dynamically generated is true.
Exploratory NLO study of the $1/2^-$ sector
===========================================
In this section, we study the effects of the NLO potentials. In principle, higher-order effects in the UChPT can be taken into account systematically if relevant low-energy constants (LECs) can be fixed reliably. However, this is not the case in the present study. Therefore, we will turn to some phenomenological means to fix some of the LECs and vary others within their natural range to study the effects of the NLO potentials. As an exploratory study, we limit ourselves to the $1/2^-$ sector.
To reduce the number of unknown LECs, we use the following NLO Lagrangians in the heavy-meson formulation [@Liu:2012uw]: $$\begin{aligned}
\label{Eq:LOLag2}
\begin{split}
\mathcal{L}_{H\Phi}^{(2)}
& = \bar{c}_{0}Tr[\bar{H}_{\bar{3}}H_{\bar{3}}]Tr[\chi_{+}] + \bar{c}_{1}Tr[\bar{H}_{\bar{3}}\tilde{\chi}_{+}H_{\bar{3}}] + (\bar{c}_{2}-\frac{2g_{6}^{2}+g_{2}^{2}}{4M_{0}})Tr[\bar{H}_{\bar{3}}v \cdot uv \cdot uH_{\bar{3}}]\\
& +(\bar{c}_{3}-\frac{2g_{6}^{2}-g_{2}^{2}}{4M_{0}})\bar{H}_{\bar{3}}^{ab}v \cdot u_{a}^{c}v \cdot u_{b}^{d}H_{\bar{3},cd} + c_{0}Tr[\bar{H}_{6}H_{6}]Tr[\chi_{+}] + c_{1}Tr[\bar{H}_{6}\tilde{\chi}_{+}H_{6}]\\
& +(c_{2}-\frac{2g_{2}^{2}+g_{1}^{2}}{4M_{0}})Tr[\bar{H}_{6}v \cdot uv \cdot uH_{6}] + (c_{3}+\frac{2g_{2}^{2}-g_{1}^{2}}{4M_{0}})\bar{H}_{6}^{ab}v \cdot u_{a}^{c}v \cdot u_{b}^{d}H_{6,cd}\\
& +c_{4}Tr[\bar{H}_{6}H_{6}]Tr[v \cdot uv \cdot u] ,
\end{split}\end{aligned}$$ where $\chi_{+}$ and $\tilde{\chi}_{+}$ are defined as $$\begin{aligned}
\label{chi}
\begin{split}
\chi_{\pm} & = \xi^{+}\chi\xi^{+} \pm \xi\chi\xi \\
\chi & = \mathrm{diag}(m_{\pi}^{2},m_{\pi}^{2},2m_{K}^{2}-m_{\pi}^{2})\\
\tilde{\chi}_{\pm} & = \chi_{\pm}-\frac{1}{3}Tr[\chi_{\pm}],
\end{split}\end{aligned}$$ with $\xi=\exp(i\frac{\phi}{2f})$.
The LECs $g_2$ and $g_4$ can be fixed by reproducing the $\Sigma_c$ and $\Sigma_c^*$ widths, while the other $g_{i}$’s can be related to them using either quark model symmetries or the heavy-quark spin symmetry. The LECs $\bar{c}_{i}$ and $c_{i}$ are fixed using the (broken) SU(4) symmetry in Ref. [@Liu:2012uw]. In the present work, we follow Ref. [@Liu:2012uw] and use the values determined there and reproduced in Tables \[Table:constantg\], \[Table:constant1\], and \[Table:constant2\]. [^7] The LEC $\alpha'$ cannot be determined, and we will estimate its contribution below assuming a natural value within the range of $-1\sim 1$ as in Ref. [@Liu:2012uw].
$|g_{1}|$ $|g_{2}|$ $|g_{3}|$ $|g_{4}|$ $|g_{5}|$ $|g_{6}|$
----------- ----------- ----------- ----------- ----------- -----------
0.98 0.60 0.85 1.0 1.5 0
: Constants in Eq. (\[Eq:LOLag2\]) for the antitriplet.[]{data-label="Table:constantg"}
$\bar{c}_{0}$ $\bar{c}_{1}$ $\bar{c}_{2}$ $\bar{c}_{3}$
--------------- --------------- ------------------------------------------- -------------------------------------------
$-0.32$ $-0.52$ $-1.78+\frac{1}{3}\frac{\alpha'}{4\pi f}$ $-0.03-\frac{1}{3}\frac{\alpha'}{4\pi f}$
: Constants in Eq. (\[Eq:LOLag2\]) for the antitriplet (in units of $GeV^{-1}$).[]{data-label="Table:constant1"}
$c_{0}$ $c_{1}$ $c_{2}$ $c_{3}$ $c_{4}$
--------- --------- --------------------------------- --------- --------------------------
$-0.61$ $-0.98$ $-2.07-2\frac{\alpha'}{4\pi f}$ $-0.84$ $\frac{\alpha'}{4\pi f}$
: Constants in Eq. (\[Eq:LOLag2\]) for the sextet (in units of $GeV^{-1}$).[]{data-label="Table:constant2"}
In the NLO study, we fix the subtraction constant in the same way as in the LO case and search for poles on the complex plane. The results are tabulated in Tables \[Table:csub1\] and \[Table:bsub1\].
Compared to the LO case, we find some substantial changes when the NLO potentials are taken into account. For instance, in the charmed sector, one dynamically generated state in the antitriplet sector disappears while a new one appears with $\alpha'=-1.0$. In the bottom sector, one dynamically generated state in the antitriplet sector disappears as well. This implies that the NLO chiral potential has a huge impact on the predicted states in the antitriplet sector.
In the sextet sector, on the other hand, the changes are more moderate. Most states move a few tens of MeV compared to their LO counterparts with a few exceptions. However, the unknown LEC $\alpha'$ seems to affect the predictions a lot. In particular, when $\alpha'=-1$, many states disappear. Clearly, from the comparison with the LO results, we may conclude that $\alpha'=-1$ is not preferred.
One of the possible reasons why the results in the antitriplet sector change more dramatically than those in the sextet sector is the following. In the sextet sector, we have refitted the subtraction constant to produce the states at $(2591,0)$ and $(5912,0)$ MeV, while no such readjustments have been made for the antitriplet sectors. Nevertheless, one should note that even at NLO the $\Lambda_c(2595)$ and $\Lambda_b(5912)$ appears naturally as dynamically generated states without the need for an unnatural subtraction constant.
In fact, due to the lack of enough experimental information to have good control on the NLO LECs, none of the above observations is surprising. In Ref. [@Liu:2012uw], Liu and Zhu already found that in many cases the NLO potentials are larger than the LO ones (see Tables I, II, and III of their paper). Our studies confirmed their findings and showed that some of the LO predictions are subject to substantial modifications while some others may remain relatively stable. More experimental or lattice QCD inputs are clearly needed to check the results and clarify the situation. On one hand, one needs to be cautious about those results where higher-order potentials are shown to be particularly relevant. On the other hand, one should note that the NLO contributions depend critically on the way the relevant LECs are estimated. If we had put them equal to zero, the contributions would vanish. Clearly, the LECs should be determined in a more reliable way in order to study the effects of higher-order potentials.
--------------------------- ----------------- ------------------- ---------------- ------------------------------------- --------------------------------------------------------- --------------------
\*[LO]{} \*[$(S,I)^{M}$]{} Main channels \*[Ref. [@Agashe:2014kda]]{}
$\alpha'=0$ $\alpha'=1.0$ $\alpha'=-1.0$ (threshold)
- - - $(2936,-i15)$ $(0,1)^{[\overline{3}]}$ $\Xi_{c}K$(2965.1)
$(2721,0)$ $(2807,0)$ $(2794,0)$ $(2820,0)$ $(0,0)^{[\overline{3}]}$ $\Xi_{c}K$(2965.1)
$(2623,-i12)$ - - - $(-1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{c}\overline{K}$(2782.1),$\Xi_{c}\pi$(2607.5)
$(2965,0)$ $(2736,0)$ $(2741,0)$ $(2732,0)$ $(-2,0)^{[\overline{3}]}$ $\Xi_{c}\overline{K}$(2965.1)
$(2948,0)$ $(2918,0)$ $(2848,0)$ - $(1,\frac{1}{2})^{[6]}$ $\Sigma_{c}K$(2949.1)
$(2674,-i51)$ $(2699,-i107)$ $(2702,-i102)$ $(2699,-i105)$ $(0,1)^{[6]}$ $\Sigma_{c}\pi$(2591.5)
$(2999,-i16)$ $(2985,-i0.01)$ $(2984,-i33)$ - $(0,1)^{[6]}$ $\Sigma_{c}\eta$(3001.4),$\Xi'_{c}\overline{K}$(3072.4)
$(2591,0)$ $(2591,0)$ $(2591,0)$ $(2591,0)$ $(0,0)^{[6]}$ $\Sigma_{c}\pi$(2591.5) $\Lambda_c(2595)$
$(3069,-i12)$ $(3025,-i19)$ $(2954,-i11)$ - $(0,0)^{[6]}$ $\Xi'_{c}K$(3072.4) $\Lambda_c(2940)?$
$\underline{(2947,-i34)}$ $(2925,-i13)$ $(2857,-i11)$ - $( -1,\frac{3}{2})^{[6]}$ $\Sigma_{c}\overline{K}$(2949.1)
$(2695,0)$ $(2567,0)$ $(2568,0)$ $(2565,0)$ $(-1,\frac{1}{2})^{[6]}$ $\Sigma_{c}\overline{K}$(2949.1)
$(2827,-i55)$ $(2824,-i74)$ $(2813,-i70)$ $(2836,-i75)$ $(-1,\frac{1}{2})^{[6]}$ $\Xi'_{c}\pi$(2714.7) $\Xi_c(2790)$?
$\underline{(3123,-i44)}$ $(3084,-i26)$ $(3038,-i8)$ - $(-1,\frac{1}{2})^{[6]}$ $\Omega_{c}K$(3190.8)
- - $(3005,-i38)$ - $(-2,1)^{[6]}$ $\Xi'_{c}\overline{K}$(3072.4)
$(2946,0)$ $(2815,0)$ $(2821,0)$ $(2809,0)$ $(-2,0)^{[6]}$ $\Xi'_{c}\overline{K}$(3072.4),$\Omega_{c}\eta$(3243.1)
--------------------------- ----------------- ------------------- ---------------- ------------------------------------- --------------------------------------------------------- --------------------
: Dynamically generated charmed baryons of $J^P=1/2^-$ in the NLO UChPT in comparison with those in the LO. At NLO, three values for the LEC $\alpha'$ have been taken. The subtraction constant is fixed in the same way as in the LO case. All energies are in units of MeV and $(S,I)^M$ denotes (strangeness, isospin)$^\mathrm{SU(3) multiplet}$.[]{data-label="Table:csub1"}
--------------- --------------- ------------------- --------------- ------------------------------------- -------------------------------------------------------- -------------------
\*[LO]{} \*[$(S,I)^{M}$]{} Main channels \*[Ref. [@Agashe:2014kda]]{}
alpha=0 alpha=1.0 alpha=-1.0 (threshold)
$(6082,-i57)$ $(5888,-i13)$ $(5890,-i13)$ $(5886,-i13)$ $(0,1)^{[\overline{3}]}$ $\Xi_{b}K$(6285.1)
$(5922,0)$ $(6042,0)$ $(6032,0)$ $(6050,0)$ $(0,0)^{[\overline{3}]}$ $\Xi_{b}K$(6285.1)
$(5869,0)$ - - - $(-1,\frac{1}{2})^{[\overline{3}]}$ $\Lambda_{b}\overline{K}$(6115.0)
$(6118,-i51)$ $(6026,-i61)$ $(6023,-i56)$ $(6028,-i68)$ $(-1,\frac{1}{2})^{[\overline{3}]}$ $\Xi_{b}\eta$(6337.4)
$(6202,0)$ $(5848,0)$ $(5848,0)$ $(5848,0)$ $(-2,0)^{[\overline{3}]}$ $\Xi_{b}\overline{K}$(6285.1)
$(6201,0)$ $(6193,0)$ $(6125,0)$ - $(1,\frac{1}{2})^{[6]}$ $\Sigma_{b}K$(6308.6)
$(5967,-i9)$ $(5928,0)$ $(5926,0)$ $(5930,0)$ $(0,1)^{[6]}$ $\Xi'_{b}K$(6421.6),$\Sigma_{b}\pi$(5951.0)
$(6223,-i14)$ $(6215,-i3)$ $(6160,-i1)$ - $(0,1)^{[6]}$ $\Sigma_{b}\eta$(6360.9),$\Xi'_{b}K$(6421.6)
$(5912,0)$ $(5912,0)$ $(5912,0)$ $(5912,0)$ $(0,0)^{[6]}$ $\Sigma_{b}\pi$(5951.0) $\Xi'_{b}K$(6421.6) $\Lambda_b(5912)$
$(6307,-i12)$ $(6298,-i16)$ $(6228,-i10)$ - $(0,0)^{[6]}$ $\Xi'_{b}K$(6421.6)
$(6213,-i25)$ $(6189,-i6)$ $(6123,-i2)$ - $( -1,\frac{3}{2})^{[6]}$ $\Sigma_{b}\overline{K}$(6308.6)
$(5955,0)$ - - - $( -1,\frac{1}{2})^{[6]}$ $\Sigma_{b}\overline{K}$(6308.6)
$(6101,-i15)$ $(6089,-i15)$ $(6084,-i13)$ $(6094,-i16)$ $( -1,\frac{1}{2})^{[6]}$ $\Xi'_{b}\pi$(6064.0),$\Omega_{b}K$(6543.6)
$(6364,-i27)$ $(6345,-i18)$ $(6295,-i5)$ - $(-1,\frac{1}{2})^{[6]}$ $\Omega_{b}K$(6543.6)
$(6361,-i59)$ $(6340,-i28)$ $(6278,-i14)$ - $(-2,1)^{[6]}$ $\Omega_{b}\pi$(6186.0),$\Xi'_{b}\overline{K}$(6421.6)
$(6169,0)$ $(5984,0)$ $(5985,0)$ $(5984,0)$ $(-2,0)^{[6]}$ $\Xi'_{b}\overline{K}$(6421.6)
--------------- --------------- ------------------- --------------- ------------------------------------- -------------------------------------------------------- -------------------
: The same as Table \[Table:csub1\] but for the bottom baryons.[]{data-label="Table:bsub1"}
Summary and outlook
===================
We have studied the interaction between a singly charmed (bottom) baryon and a pseudoscalar meson in the unitarized chiral perturbation theory using leading-order chiral Lagrangians. It is shown that the interactions are strong enough to generate a number of dynamically generated states. Some of them can be naturally assigned to their experimental counterparts, such as the $\Lambda_b(5912)$ \[$\Lambda_b^*(5920)$\] and the $\Lambda_c(2595)$ \[$\Lambda_c^*(2625)$\]. By slightly fine-tuning the subtraction constant in the dimensional regularization scheme or the cutoff value in the cutoff regularization scheme so that the masses of these states are produced, we predict a number of additional states, the experimental counterparts of which remain unknown. We strongly encourage future experiments to search for these states.
In anticipation of future lattice QCD simulations of scattering lengths, as already happened in the light-baryon sector or the heavy-meson sector, we have calculated the scattering lengths between the charmed (bottom) baryons and the pseudoscalar mesons. A comparison between our results and those of the $\mathcal{O}(p^3)$ chiral perturbation theory study confirmed that there is indeed strong attraction in some of the coupled channels, which hints at the possible existence of dynamically generated states.
In future, the effects of higher-order potentials in the unitarized chiral perturbation theory need to be studied more carefully once relevant experimental or lattice QCD data become available. It should be noted, however, that the $\Lambda_c(2595)$ and $\Lambda_b(5912)$ and their $J^P=3/2^-$ counterparts seem to qualify as dynamically generated states even at next-to-leading order in the unitarized chiral perturbation theory.
Acknowledgements
================
This work is supported in part by the National Natural Science Foundation of China under Grants No. 11375024, No. 11205011, and No. 11105126; the New Century Excellent Talents in University Program of Ministry of Education of China under Grant No. NCET-10-0029; and the Fundamental Research Funds for the Central Universities.
Appendix: Clebsch–Gordan coefficients
=====================================
In this section, we tabulate the Clebsch–Gordan coefficients appearing in Eq. (\[Eq:LOKernel\]) for the antitriplet (Tables \[Table:coeff3f\] to \[Table:coeff3l\]) and sextet (Tables \[Table:coeff6f\] to \[Table:coeff6l\]) ground-state charmed or bottom baryons interacting with the pseudoscalar mesons.
$\Lambda_{c}$K
----------------- ----------------
$\Lambda_{c}$K 1
: $(S=1,I=1/2)$\[Table:coeff3f\]
$\Xi_{c}$K $\Lambda_{c}\pi$
------------------ ------------ ------------------
$\Xi_{c}$K 0 $1$
$\Lambda_{c}\pi$ $1$ 0
: $(S=0,I=1)$
$\Xi_{c}$K $\Lambda_{c}\eta$
------------------- ------------- -------------------
$\Xi_{c}$K $-2$ $-\sqrt{3}$
$\Lambda_{c}\eta$ $-\sqrt{3}$ 0
: $(S=0,I=0)$
$\Xi_{c}\pi$
-------------- --------------
$\Xi_{c}\pi$ 1
: $(S=-1,I=3/2)$
$\Xi_{c}\pi$ $\Xi_{c}\eta$ $\Lambda_{c}\overline{K}$
--------------------------- -------------- --------------- ---------------------------
$\Xi_{c}\pi$ $-2$ 0 $\sqrt{3/2}$
$\Xi_{c}\eta$ 0 0 $-\sqrt{3/2}$
$\Lambda_{c}\overline{K}$ $\sqrt{3/2}$ $-\sqrt{3/2}$ $-1$
: $(S=-1,I=1/2)$
$\Xi_{c}\overline{K}$
----------------------- -----------------------
$\Xi_{c}\overline{K}$ 1
: $(S=-2,I=1)$
$\Xi_{c}\overline{K}$
----------------------- -----------------------
$\Xi_{c}\overline{K}$ $-1$
: $(S=-2,I=0)$\[Table:coeff3l\]
$\Sigma_{c}$K
--------------- ---------------
$\Sigma_{c}$K $2$
: $(S=1,I=3/2)$\[Table:coeff6f\]
$\Sigma_{c}$K
--------------- ---------------
$\Sigma_{c}$K $-1$
: $(S=1,I=1/2)$
$\Sigma_{c}\pi$
----------------- -----------------
$\Sigma_{c}\pi$ $2$
: $(S=0,I=2)$
$\Xi^{'}_{c}K$ $\Sigma_{c}\eta$ $\Sigma_{c}\pi$
------------------ ---------------- ------------------ -----------------
$\Xi^{'}_{c}K$ 0 $-\sqrt{3}$ $-\sqrt{2}$
$\Sigma_{c}\eta$ $-\sqrt{3}$ 0 0
$\Sigma_{c}\pi$ $-\sqrt{2}$ 0 $-2$
: $(S=0,I=1)$
$\Xi^{'}_{c}$K $\Sigma_{c}\pi$
----------------- ---------------- -----------------
$\Xi^{'}_{c}$K $-2$ $-\sqrt{3}$
$\Sigma_{c}\pi$ $-\sqrt{3}$ $-4$
: $(S=0,I=0)$
$\Xi^{'}_{c}\pi$ $\Sigma_{c}\overline{K}$
-------------------------- ------------------ --------------------------
$\Xi^{'}_{c}\pi$ $1$ $\sqrt{2}$
$\Sigma_{c}\overline{K}$ $\sqrt{2}$ 0
: $(S=-1,I=3/2)$
$\Xi^{'}_{c}\pi$ $\Xi^{'}_{c}\eta$ $\Omega_{c}K$ $\Sigma_{c}\overline{K}$
-------------------------- ---------------------- ---------------------- --------------- --------------------------
$\Xi^{'}_{c}\pi$ $-2$ 0 $-\sqrt{3}$ $\frac{1}{\sqrt{2}}$
$\Xi^{'}_{c}\eta$ 0 0 $-\sqrt{3}$ $\frac{3}{\sqrt{2}}$
$\Omega_{c}K$ $-\sqrt{3}$ $-\sqrt{3}$ $-2$ 0
$\Sigma_{c}\overline{K}$ $\frac{1}{\sqrt{2}}$ $\frac{3}{\sqrt{2}}$ 0 $-3$
: $(S=-1,I=1/2)$
$\Xi^{'}_{c}\overline{K}$ $\Omega_{c}\pi$
--------------------------- --------------------------- -----------------
$\Xi^{'}_{c}\overline{K}$ $1$ $\sqrt{2}$
$\Omega_{c}\pi$ $\sqrt{2}$ 0
: $(S=-2,I=1)$
$\Xi^{'}_{c}\overline{K}$ $\Omega_{c}\eta$
--------------------------- --------------------------- ------------------
$\Xi^{'}_{c}\overline{K}$ $-1$ $\sqrt{6}$
$\Omega_{c}\eta$ $\sqrt{6}$ 0
: $(S=-2,I=0)$
$\Omega_{c}\overline{K}$
-------------------------- --------------------------
$\Omega_{c}\overline{K}$ $2$
: $(S=-3,I=1/2)$\[Table:coeff6l\]
[99]{} D. Bernard \[BaBar Collaboration\], PoS DIS [**2013**]{}, 179 (2013) \[arXiv:1311.0968 \[hep-ex\]\].
D. Santel \[Belle Collaboration\], PoS DIS [**2013**]{}, 162 (2013).
Y. Kato \[Belle Collaboration\], PoS Hadron [**2013**]{}, 053 (2014).
K. K. Seth, arXiv:1107.2641 \[hep-ex\].
H. -B. Li \[BESIII Collaboration\], EPJ Web Conf. [**72**]{}, 00011 (2014). P. d. Simone \[LHCb Collaboration\], EPJ Web Conf. [**73**]{}, 03007 (2014).
P. Palni \[CDF Collaboration\], Nucl. Phys. Proc. Suppl. [**233**]{}, 151 (2012).
M. Ablikim [*et al.*]{} \[BESIII Collaboration\], Phys. Rev. Lett. [**110**]{}, 252001 (2013) \[arXiv:1303.5949 \[hep-ex\]\].
Z. Q. Liu [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Lett. [**110**]{}, 252002 (2013) \[arXiv:1304.0121 \[hep-ex\]\].
R. Aaij [*et al.*]{} \[LHCb Collaboration\], Phys. Rev. Lett. [**112**]{}, 222002 (2014) \[arXiv:1404.1903 \[hep-ex\]\].
S. K. Choi [*et al.*]{} \[BELLE Collaboration\], Phys. Rev. Lett. [**100**]{}, 142001 (2008) \[arXiv:0708.1790 \[hep-ex\]\].
N. Brambilla, S. Eidelman, B. K. Heltsley, R. Vogt, G. T. Bodwin, E. Eichten, A. D. Frawley and A. B. Meyer [*et al.*]{}, Eur. Phys. J. C [**71**]{}, 1534 (2011) \[arXiv:1010.5827 \[hep-ph\]\]. V. Crede and W. Roberts, Rept. Prog. Phys. [**76**]{}, 076301 (2013) \[arXiv:1302.7299 \[nucl-ex\]\].
RAaij [*et al.*]{} \[LHCb Collaboration\], Phys. Rev. Lett. [**109**]{}, 172003 (2012) \[arXiv:1205.3452 \[hep-ex\]\].
T. A. Aaltonen [*et al.*]{} \[CDF Collaboration\], Phys. Rev. D [**88**]{}, 071101 (2013) \[arXiv:1308.1760 \[hep-ex\]\].
K. A. Olive [*et al.*]{} \[Particle Data Group Collaboration\], Chin. Phys. C [**38**]{}, 090001 (2014).
M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A [**730**]{}, 110 (2004) \[hep-ph/0307233\].
C. Garcia-Recio, V. K. Magas, T. Mizutani, J. Nieves, A. Ramos, L. L. Salcedo and L. Tolos, Phys. Rev. D [**79**]{}, 054004 (2009) \[arXiv:0807.2969 \[hep-ph\]\].
C. Garcia-Recio, J. Nieves, O. Romanets, L. L. Salcedo and L. Tolos, Phys. Rev. D [**87**]{}, 034032 (2013) \[arXiv:1210.4755 \[hep-ph\]\].
W. H. Liang, C. W. Xiao and E. Oset, Phys. Rev. D [**89**]{}, 054023 (2014) \[arXiv:1401.1441 \[hep-ph\]\]. W. H. Liang, T. Uchino, C. W. Xiao and E. Oset, Eur. Phys. J. A [**51**]{}, 16 (2015) \[arXiv:1402.5293 \[hep-ph\]\].
T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. [**67**]{}, 55 (2012) \[arXiv:1104.4474 \[nucl-th\]\].
E. E. Kolomeitsev and M. F. M. Lutz, Phys. Lett. B [**582**]{}, 39 (2004) \[hep-ph/0307133\].
F. K. Guo, P. N. Shen, H. C. Chiang, R. G. Ping and B. S. Zou, Phys. Lett. B [**641**]{}, 278 (2006) \[hep-ph/0603072\]. D. Gamermann, E. Oset, D. Strottman and M. J. Vicente Vacas, Phys. Rev. D [**76**]{}, 074016 (2007) \[hep-ph/0612179\].
A. Faessler, T. Gutsche, V. E. Lyubovitskij and Y. L. Ma, Phys. Rev. D [**76**]{}, 014005 (2007) \[arXiv:0705.0254 \[hep-ph\]\]. G. J. Ding, J. F. Liu and M. L. Yan, Phys. Rev. D [**79**]{}, 054005 (2009) \[arXiv:0901.0426 \[hep-ph\]\]. Y. Dong, A. Faessler, T. Gutsche, S. Kovalenko and V. E. Lyubovitskij, Phys. Rev. D [**79**]{}, 094013 (2009) \[arXiv:0903.5416 \[hep-ph\]\]. A. E. Bondar, A. Garmash, A. I. Milstein, R. Mizuk and M. B. Voloshin, Phys. Rev. D [**84**]{}, 054010 (2011) \[arXiv:1105.4473 \[hep-ph\]\]. C. Hidalgo-Duque, J. Nieves and M. P. Valderrama, Phys. Rev. D [**87**]{}, 076006 (2013) \[arXiv:1210.5431 \[hep-ph\]\].
N. Li, Z. F. Sun, X. Liu and S. L. Zhu, Phys. Rev. D [**88**]{}, 114008 (2013) \[arXiv:1211.5007 \[hep-ph\]\]. M. T. Li, W. L. Wang, Y. B. Dong and Z. Y. Zhang, Int. J. Mod. Phys. A [**27**]{}, 1250161 (2012) \[arXiv:1206.0523 \[nucl-th\]\]. M. Cleven, Q. Wang, F. K. Guo, C. Hanhart, U. G. Meissner and Q. Zhao, Phys. Rev. D [**87**]{}, 074006 (2013) \[arXiv:1301.6461 \[hep-ph\]\]. M. Altenbuchinger, L. -S. Geng and W. Weise, Phys. Rev. D [**89**]{}, 014026 (2014) \[arXiv:1309.4743 \[hep-ph\]\].
J. -J. Wu, R. Molina, E. Oset and B. S. Zou, Phys. Rev. Lett. [**105**]{}, 232001 (2010) \[arXiv:1007.0573 \[nucl-th\]\]. J. -J. Wu, R. Molina, E. Oset and B. S. Zou, Phys. Rev. C [**84**]{}, 015202 (2011) \[arXiv:1011.2399 \[nucl-th\]\]. J. -J. Wu and B. S. Zou, Phys. Lett. B [**709**]{}, 70 (2012) \[arXiv:1011.5743 \[hep-ph\]\].
C. W. Xiao, J. Nieves and E. Oset, Phys. Rev. D [**88**]{}, 056012 (2013) \[arXiv:1304.5368 \[hep-ph\]\]. C. W. Xiao and E. Oset, Eur. Phys. J. A [**49**]{}, 139 (2013) \[arXiv:1305.0786 \[hep-ph\]\].
J. M. Flynn, E. Hernandez and J. Nieves, Phys. Rev. D [**85**]{}, 014012 (2012) \[arXiv:1110.2962 \[hep-ph\]\].
O. Romanets, L. Tolos, C. Garcia-Recio, J. Nieves, L. L. Salcedo and R. G. E. Timmermans, Phys. Rev. D [**85**]{}, 114032 (2012) \[arXiv:1202.2239 \[hep-ph\]\].
O. Romanets, L. Tolos, C. GarcÂa-Recio, J. Nieves, L. L. Salcedo and R. Timmermans, Nucl. Phys. A [**914**]{}, 488 (2013) \[arXiv:1212.3943 \[hep-ph\]\]. C. Garcia-Recio, J. Nieves, O. Romanets, L. L. Salcedo and L. Tolos, Phys. Rev. D [**87**]{}, 074034 (2013) \[arXiv:1302.6938 \[hep-ph\]\]. F. -K. Guo, C. Hidalgo-Duque, J. Nieves and M. P. Valderrama, Phys. Rev. D [**88**]{}, 054014 (2013) \[arXiv:1305.4052 \[hep-ph\]\].
N. Kaiser, P. B. Siegel and W. Weise, Nucl. Phys. A [**594**]{}, 325 (1995) \[nucl-th/9505043\].
J. A. Oller and E. Oset, Nucl. Phys. A [**620**]{}, 438 (1997) \[Erratum-ibid. A [**652**]{}, 407 (1999)\] \[hep-ph/9702314\]. E. Oset and A. Ramos, Nucl. Phys. A [**635**]{}, 99 (1998) \[nucl-th/9711022\].
B. Krippa, Phys. Rev. C [**58**]{}, 1333 (1998) \[hep-ph/9803332\]. J. Nieves and E. Ruiz Arriola, Nucl. Phys. A [**679**]{}, 57 (2000) \[hep-ph/9907469\]. U. G. Meissner and J. A. Oller, Nucl. Phys. A [**673**]{}, 311 (2000) \[nucl-th/9912026\]. M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A [**700**]{}, 193 (2002) \[nucl-th/0105042\]. C. Garcia-Recio, J. Nieves, E. Ruiz Arriola and M. J. Vicente Vacas, Phys. Rev. D [**67**]{}, 076009 (2003) \[hep-ph/0210311\]. T. Hyodo, S. I. Nam, D. Jido and A. Hosaka, Phys. Rev. C [**68**]{}, 018201 (2003) \[nucl-th/0212026\].
B. Borasoy, U.-G. Meissner and R. Nissler, Phys. Rev. C [**74**]{}, 055201 (2006) \[hep-ph/0606108\].
J. A. Oller and E. Oset, Phys. Rev. D [**60**]{}, 074023 (1999) \[hep-ph/9809337\].
T. N. Truong, Phys. Rev. Lett. [**61**]{}, 2526 (1988). A. Dobado, M. J. Herrero and T. N. Truong, Phys. Lett. B [**235**]{}, 134 (1990). A. Dobado and J. R. Pelaez, Phys. Rev. D [**47**]{}, 4883 (1993) \[hep-ph/9301276\]. A. Dobado and J. R. Pelaez, Phys. Rev. D [**56**]{}, 3057 (1997) \[hep-ph/9604416\].
F. Guerrero and J. A. Oller, Nucl. Phys. B [**537**]{}, 459 (1999) \[Erratum-ibid. B [**602**]{}, 641 (2001)\] \[hep-ph/9805334\]. A. Gomez Nicola and J. R. Pelaez, Phys. Rev. D [**65**]{}, 054009 (2002) \[hep-ph/0109056\]. J. R. Pelaez, Mod. Phys. Lett. A [**19**]{}, 2879 (2004) \[hep-ph/0411107\].
M. Altenbuchinger and L. -S. Geng, Phys. Rev. D [**89**]{}, 054008 (2014) \[arXiv:1310.5224 \[hep-ph\]\].
Z. -W. Liu and S. -L. Zhu, Phys. Rev. D [**86**]{}, 034009 (2012) \[arXiv:1205.0467 \[hep-ph\]\].
T. Fuchs, J. Gegelia, G. Japaridze and S. Scherer, Phys. Rev. D [**68**]{}, 056005 (2003) \[hep-ph/0302117\].
A. Ali Khan, T. Bhattacharya, S. Collins, C. T. H. Davies, R. Gupta, C. Morningstar, J. Shigemitsu and J. H. Sloan, Phys. Rev. D [**62**]{}, 054505 (2000) \[hep-lat/9912034\].
R. A. Briceno, H. -W. Lin and D. R. Bolton, Phys. Rev. D [**86**]{}, 094504 (2012) \[arXiv:1207.3536 \[hep-lat\]\].
C. Alexandrou, V. Drach, K. Jansen, C. Kallidonis and G. Koutsou, Phys. Rev. D [**90**]{}, 074501 (2014) \[arXiv:1406.4310 \[hep-lat\]\].
L. Liu, H. -W. Lin, K. Orginos and A. Walker-Loud, Phys. Rev. D [**81**]{}, 094505 (2010) \[arXiv:0909.3294 \[hep-lat\]\].
Y. Namekawa [*et al.*]{} \[PACS-CS Collaboration\], Phys. Rev. D [**87**]{}, 094512 (2013) \[arXiv:1301.4743 \[hep-lat\]\]. Z. S. Brown, W. Detmold, S. Meinel and K. Orginos, Phys. Rev. D [**90**]{}, 094507 (2014) \[arXiv:1409.0497 \[hep-lat\]\]. R. Lewis and R. M. Woloshyn, Phys. Rev. D [**79**]{}, 014502 (2009) \[arXiv:0806.4783 \[hep-lat\]\]. M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, Phys. Rev. D [**90**]{}, 074504 (2014) \[arXiv:1307.7022 \[hep-lat\]\]. S. Meinel, Phys. Rev. D [**85**]{}, 114510 (2012) \[arXiv:1202.1312 \[hep-lat\]\].
M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, arXiv:1311.4806 \[hep-lat\]. M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, arXiv:1311.4354 \[hep-lat\].
J. Hofmann and M. F. M. Lutz, Nucl. Phys. A [**763**]{}, 90 (2005) \[hep-ph/0507071\].
X. G. He, X. Q. Li, X. Liu and X. Q. Zeng, Eur. Phys. J. C [**51**]{}, 883 (2007) \[hep-ph/0606015\]. Y. Dong, A. Faessler, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D [**81**]{}, 014006 (2010) \[arXiv:0910.1204 \[hep-ph\]\].
Y. Dong, A. Faessler, T. Gutsche, S. Kumano and V. E. Lyubovitskij, Phys. Rev. D [**82**]{}, 034035 (2010) \[arXiv:1006.4018 \[hep-ph\]\]. Y. Dong, A. Faessler, T. Gutsche, S. Kumano and V. E. Lyubovitskij, Phys. Rev. D [**83**]{}, 094005 (2011) \[arXiv:1103.4762 \[hep-ph\]\].
J. R. Zhang, Phys. Rev. D [**89**]{}, 096006 (2014) \[arXiv:1212.5325 \[hep-ph\]\]. P. G. Ortega, D. R. Entem and F. Fernandez, Phys. Lett. B [**718**]{}, 1381 (2013) \[arXiv:1210.2633 \[hep-ph\]\]. J. He, Y. T. Ye, Z. F. Sun and X. Liu, Phys. Rev. D [**82**]{}, 114029 (2010) \[arXiv:1008.1500 \[hep-ph\]\].
H. Y. Cheng and C. K. Chua, Phys. Rev. D [**75**]{}, 014006 (2007) \[hep-ph/0610283\]. C. Chen, X. L. Chen, X. Liu, W. Z. Deng and S. L. Zhu, Phys. Rev. D [**75**]{}, 094017 (2007) \[arXiv:0704.0075 \[hep-ph\]\]. X. H. Zhong and Q. Zhao, Phys. Rev. D [**77**]{}, 074008 (2008) \[arXiv:0711.4645 \[hep-ph\]\]. L. H. Liu, L. Y. Xiao and X. H. Zhong, Phys. Rev. D [**86**]{}, 034024 (2012) \[arXiv:1205.2943 \[hep-ph\]\]. S. Yasui, Phys. Rev. D [**91**]{}, 014031 (2015) \[arXiv:1408.3703 \[hep-ph\]\]. D. Gamermann, C. E. Jimenez-Tejero and A. Ramos, Phys. Rev. D [**83**]{}, 074018 (2011) \[arXiv:1011.5381 \[hep-ph\]\].
D. Gamermann, L. R. Dai and E. Oset, Phys. Rev. C [**76**]{}, 055205 (2007) \[arXiv:0709.2339 \[hep-ph\]\]. M. F. M. Lutz and M. Soyeur, Nucl. Phys. A [**813**]{}, 14 (2008) \[arXiv:0710.1545 \[hep-ph\]\]. Y. b. Dong, A. Faessler, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D [**77**]{}, 094013 (2008) \[arXiv:0802.3610 \[hep-ph\]\]. X. Liu, Chin. Phys. C [**33**]{}, 473 (2009). Y. L. Ma, Phys. Rev. D [**82**]{}, 015013 (2010) \[arXiv:1006.1276 \[hep-ph\]\]. M. Nielsen and C. M. Zanetti, Phys. Rev. D [**82**]{}, 116002 (2010) \[arXiv:1006.0467 \[hep-ph\]\]. F. Aceti, R. Molina and E. Oset, Phys. Rev. D [**86**]{}, 113007 (2012) \[arXiv:1207.2832 \[hep-ph\]\]. Y. Dong, A. Faessler, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D [**88**]{}, 014030 (2013) \[arXiv:1306.0824 \[hep-ph\]\].
S. Weinberg, Phys. Rev. [**137**]{}, B672 (1965). C. Hanhart, Y. S. Kalashnikova and A. V. Nefediev, Phys. Rev. D [**81**]{}, 094028 (2010) \[arXiv:1002.4097 \[hep-ph\]\]. V. Baru, J. Haidenbauer, C. Hanhart, Y. Kalashnikova and A. E. Kudryavtsev, Phys. Lett. B [**586**]{}, 53 (2004) \[hep-ph/0308129\]. M. Cleven, F. K. Guo, C. Hanhart and U. G. Meissner, Eur. Phys. J. A [**47**]{}, 120 (2011) \[arXiv:1107.0254 \[hep-ph\]\]. D. Gamermann, J. Nieves, E. Oset and E. Ruiz Arriola, Phys. Rev. D [**81**]{}, 014029 (2010) \[arXiv:0911.4407 \[hep-ph\]\].
J. Yamagata-Sekihara, J. Nieves and E. Oset, Phys. Rev. D [**83**]{}, 014003 (2011) \[arXiv:1007.3923 \[hep-ph\]\]. F. Aceti and E. Oset, Phys. Rev. D [**86**]{}, 014012 (2012) \[arXiv:1202.4607 \[hep-ph\]\]. C. W. Xiao, F. Aceti and M. Bayar, Eur. Phys. J. A [**49**]{}, 22 (2013) \[arXiv:1210.7176 \[hep-ph\]\]. F. Aceti, L. R. Dai, L. S. Geng, E. Oset and Y. Zhang, Eur. Phys. J. A [**50**]{}, 57 (2014) \[arXiv:1301.2554 \[hep-ph\]\]. F. Aceti, E. Oset and L. Roca, Phys. Rev. C [**90**]{}, 025208 (2014) \[arXiv:1404.6128 \[hep-ph\]\].
T. Sekihara, T. Hyodo and D. Jido, PTEP [**2015**]{}, 063D04 \[arXiv:1411.2308 \[hep-ph\]\].
Xun-Ju Lu and Li-Sheng Geng, in preparation. D. Gamermann and E. Oset, Phys. Rev. D [**80**]{}, 014003 (2009) \[arXiv:0905.0402 \[hep-ph\]\]. C. Garcia-Recio, C. Hidalgo-Duque, J. Nieves, L. L. Salcedo and L. Tolos, arXiv:1506.04235 \[hep-ph\].
[^1]: It is to be noted that in the charmed mesonic sector, many newly observed resonances have been claimed to of composite nature based on phenomenological Lagrangians or effective field theories (see, e.g., Refs. [@Kolomeitsev:2003ac; @Guo:2006fu; @Gamermann:2006nm; @Faessler:2007gv; @Ding:2009vj; @Dong:2009yp; @Bondar:2011ev; @HidalgoDuque:2012pq; @Li:2012ss; @Li:2012mqa; @Cleven:2013sq; @Altenbuchinger:2013vwa]).
[^2]: We note that preliminary results on the excited-state spectroscopy of singly and doubly charmed baryons have recently been presented at conferences [@Padmanath:2013bla; @Padmanath:2013laa].
[^3]: Experimentally, the $\Lambda_c(2595)$ is found at $2592.25\pm0.28$ MeV with a width of $2.6\pm0.6$ MeV [@Agashe:2014kda] . We need to slightly increase $f_0$ to put the $\Lambda_c(2595)$ exactly at this position because of the closeness of the $\Sigma_c\pi$ threshold.
[^4]: For similar studies in the heavy-flavor mesonic sector; see, e.g., Refs. [@Gamermann:2007bm; @Lutz:2007sk; @Dong:2008gb; @Liu:2009zg; @Ma:2010xx; @Nielsen:2010ij; @Aceti:2012cb; @Dong:2013iqa].
[^5]: See also Ref. [@Sekihara:2014kya] and references cited therein.
[^6]: In a recent work, the compositeness of the strange, charmed, and beauty $\Lambda$ states have been studied in the extended Weinberg–Tomozawa framework supplemented by the SU(6) and the heavy-quark symmetries [@Garcia-Recio:2015jsa].
[^7]: The values are slightly different from those of Ref. [@Liu:2012uw] because some relations among the LECs are stated incorrectly there.
|
---
abstract: 'Recent experimental data and progress in nuclear structure modeling have lead to improved descriptions of astrophysically important weak-interaction processes. The review discusses these advances and their applications to hydrostatic solar and stellar burning, to the slow and rapid neutron-capture processes, to neutrino nucleosynthesis, and to explosive hydrogen burning. Special emphasis is given to the weak-interaction processes associated with core-collapse supernovae. Despite some significant progress, important improvements are still warranted. Such improvements are expected to come from future radioactive ion-beam facilities.'
author:
- 'K. Langanke'
- 'G. Martínez-Pinedo'
title: 'Nuclear weak-interaction processes in stars'
---
Introduction {#sec:introduction}
============
The weak interaction is one the four fundamental forces in nature. Like the other three – strong, electromagnetic and gravitation – it plays a keyrole in many astrophysical processes. This can be nicely illustrated by the observation that new insights into the nature of the weak interaction usually were closely followed by the recognition of their importance in some astrophysical context. Shortly after Pauli postulated the existence of the neutrino and Fermi developed the first theory of weak interaction [@Fermi:1934], Gamow and Schoenberg speculated about the possible role of neutrinos in stellar evolution and proposed their production in the star as an important source for stellar energy losses [@Gamow.Schoenberg:1940; @Gamow:1941; @Gamow.Schoenberg:1941]. The development of the universal $V-A$ theory [@Feynman.Gell-Mann:1958] led Pontecorvo to realize that the bremsstrahlung radiation of neutrino pairs by electrons would be a very effective stellar energy loss mechanism [@Pontecorvo:1959]. Just after the discovery of neutral weak current @Freedman:1974, @Mazurek:1975, and @Sato:1975 recognized that this interaction would result in a sizable elastic scattering cross section between neutrinos and nucleons, leading to neutrino trapping during the core collapse of a massive star in a type II supernova.
The unified model of electroweak interaction [@Weinberg:1967; @Salam:1968; @Glashow.Iliopoulos.Maiani:1970] allows derivation of accurate cross sections for weak processes among elementary particles (i.e. electrons, neutrinos, quarks), but also for neutron and protons if proper formfactors are taken into account which describe the composite nature of the nucleons. However, the situation is different for weak interaction processes involving nuclei. Clearly, the smallness of the weak interaction coupling parameter allows treatment of these processes in perturbation theory, reducing the calculation basically to a nuclear structure problem. However, it has been the inability to adequately treat the nuclear many-body problem, which has – and in many cases still does - introduced a substantial uncertainty into some of the key weak interaction rates used in astrophysical simulations. However, the recent few years have witnessed a tremendous progress in nuclear many-body theory, made possible by new approaches and novel computer realizations of established models, but also by the availability of large computational capabilities. This progress allowed calculation of the rates for many of the stellar weak-interaction processes involving nuclei with significantly improved accuracy or for the first time. To actually know that the calculations are more reliable, implies the availability of experimental data which test, constrain and guide the theoretical models. Thus, the advances in modelling nuclear weak-interaction processes in stars also reflects the progress made by experimentalists in recent years which have succeeded to measure data which are relevant for the astrophysical applications discussed in this review either directly, e.g. half-lives for some short-lived nuclei on the r-process path [@Pfeiffer.Kratz.ea:2001], or indirectly like the Gamow-Teller distributions for nuclei in the iron mass range [@Osterfeld:1992] which decisively constrain the nuclear models. Another recent experimental first has been the measurement of charged- and neutral-current neutrino-nucleus cross sections.
This review will report about progress in modelling nuclear weak-interaction processes and their possible implications for stellar evolution and nucleosynthesis. We will restrict ourselves to advances achieved by improved nuclear models, treating the weak interaction within the standard model. Of course, it has long been recognized [e.g. @Sato.Sato:1975] that stars can be used as *laboratories for fundamental physics* [@Raffelt:1996] searching for new weakly interacting particles or constraining exotic components of the weak interaction outside the standard model. This field is rapidly growing .
Our review is structured as follows. Following a very brief discussion of the required ingredients of the weak interaction we introduce the nuclear many-body models which have been used in the studies of the weak-interaction processes (section \[sec:theor-descr\]). The remaining sections are devoted to the results of these calculations and their applications to astrophysics which include the solar nuclear reaction network and neutrino problem, the core collapse of massive stars, s- and r-process nucleosynthesis, neutrino nucleosynthesis, explosive hydrogen burning and type Ia supernovae.
Although generally quite important, weak-interaction processes constitute only a part of the many nuclear reactions occuring in stars. For recent reviews about other stellar nuclear reactions networks and nucleosynthesis the reader is referred to the comprehensive and competent work by [@Wallerstein.Iben.ea:1997; @Arnould.Takahashi:1999; @Boyd:2000; @Smith.Rehm:2001].
Theoretical description {#sec:theor-descr}
=======================
Weak interactions in nuclei {#sec:weak-inter-nucl}
---------------------------
![Semileptonic weak processes that occur during the evolution of stars. For each process the hadronic current is on the left and the leptonic current to the right. The dashed circle indicates a bound electron in the initial or final state. The four-momentum transfer $q_\lambda = (-\omega,\bm{q})$ for each process is given in terms of the charged lepton four-momentum $k_\lambda =
(\epsilon,\bm{k})$ and the neutrino four-momentum $\nu_\lambda =
(\nu,\bm{\nu})$. $\omega$, $\epsilon$, and $\nu$ represent the energy transfer, lepton energy and neutrino energy, respectively. In the case of antiparticles the directions of the momenta are show as an arrow close to the four-momentum label. The first row shows the usual decay modes in the laboratory. The second and third rows show processes that can occur under stellar conditions.\[fig:weakprocess\]](weakp.eps){width="0.9\linewidth"}
Processes mediated by the weak interaction in stars can be classified as leptonic (all interacting particles are leptons) and semileptonic (leptons interact with hadrons via the weak interaction). Leptonic processes can be straightforwardly computed using the standard electroweak model [@Grotz.Klapdor:1990]. The calculation of semileptonic processes (i.e. neutrino-nucleus reactions, charged-lepton capture, and $\beta$-decay) is more complicated due to the description of the nuclear states involved. Fortunately the momenta of the particles turn out to be small compared with the masses of the $Z,W$ bosons. Thus it is sufficient to consider the semileptonic processes of interest in the lowest-order approximation in the weak interaction. Then the interaction can be described by a current-current Hamiltonian density:
$$\label{eq:1}
\mathcal{H}(\bm{x})= -\frac{G}{\sqrt{2}}
\mathcal{J}_\mu(\bm{x})j_\mu(\bm{x}),$$
where $G=G_F V_{ud}$ for charge-current processes and $G=G_F$ for neutral current processes, with $G_F$ the Fermi coupling constant and $V_{ud}$ the up-down entry of the Cabibbo-Kobayashi-Maskawa matrix [@PDBook]. $j_\mu(\bm{x})$ and $\mathcal{J}_\mu(\bm{x})$ are the weak leptonic and hadronic density operators [@Walecka:1975; @Walecka:1995; @Donnelly.Peccei:1979]. The structure of the leptonic current $j_\mu(\bm{x})$ for a particular process is given by the standard electroweak model [@Weinberg:1967; @Salam:1968; @Glashow.Iliopoulos.Maiani:1970], and contains both, vector and axial-vector components. The standard model describes the hadronic current in terms of quark degrees of freedom. Since we are only interested in the matrix elements of $\mathcal{J}_\mu(\bm{x})$ in nuclei we need only to retain the pieces which involve $u$ and $d$ quarks. (The contribution from strange quarks is normally neglected, but see the discussion in section \[sec:delay-supern-mech\].) As in nuclear physics the nucleons are treated as elementary spin-1/2 fermions, the Standard Model current is not immediately applicable. Moreover, nucleons in nuclei interact also via the strong interaction. It is then convenient to define an effective hadronic current using arguments of Lorentz covariance and isospin invariance of the strong interaction. The effective hadronic current can be decomposed into strong isoscalar ($T=0$) and isovector ($T=1$) components and contains both vector ($V$) and axial-vector ($A$) pieces. The weak charge-changing current is isovector with $M_T = \pm 1$ and can be written in a general form as:
$$\label{eq:2}
\mathcal{J}_\mu = V_\mu^{1M_T} + A_\mu^{1M_T}.$$
This current governs processes (see figure \[fig:weakprocess\]) such as $\beta^\pm$-decay, $e$-capture, neutrino $(\nu_l,l^-)$ and anti-neutrino $(\bar{\nu}_l,l^+)$ reactions ($l=e, \mu $ or $\tau$). Under the conserved vector current (CVC) hypothesis [@Feynman.Gell-Mann:1958] the current $V_\mu^{1M_T}$ has a structure identical to the isovector part of the electromagnetic current. As a consequence of this hypothesis the weak charge-changing vector current is a conserved quantity. For the weak neutral current one has $M_T =0$ and, in general, both $T=0$ and $T=1$ pieces can occur. The general form of this current is
$$\label{eq:3}
\mathcal{J}_\mu = \beta_V^{(0)} V_\mu^{00} + \beta_A^{(0)}
A_\mu^{00} + \beta_V^{(1)} V_\mu^{10} + \beta_A^{(1)} A_\mu^{10}.$$
Assuming that the coupling constants are given by the Standard Model we have: $\beta_V^{(0)}=-2\sin^2 \theta_W$, $\beta_A^{(0)}= 0$, $\beta_V^{(1)}= 1-2\sin^2 \theta_W$, $\beta_A^{(1)}= 1$ [@Donnelly.Peccei:1979 p. 25]. $\theta_W$ is the weak mixing angle. The neutral current describes weak interactions such as neutrino $(\nu,\nu^\prime)$ and anti-neutrino $(\bar{\nu},\bar{\nu}^\prime)$ scattering.
The nuclear transitions that are induced by such weak currents (operators) involve initial and final states that are usually assumed to be eigenstates of angular momentum, parity, as well as isospin. It is then convenient to do a multipole expansion of the current operators. In that way one obtains the Coulomb, longitudinal, transverse electric and transverse magnetic multipoles defined in [@Walecka:1975 p. 136]. The expressions necessary for the calculation of the processes shown in figure \[fig:weakprocess\] can be obtained from references [@Walecka:1975; @Donnelly.Peccei:1979] in terms of the multipole operators. In general the multipole operators are $A$-body nuclear operators (with $A$ the nucleon number). In practice, at the energy scales we are interested in, weak interactions perturb the nucleus only slightly, so that to a good approximation one-body components dominate most of the transitions. Two-body meson exchange currents and other many body effects are neglected [see @Marcucci.Schiavilla.ea:2001; @Schiavilla.Wiringa:2002 for a description of the nuclear current including two-body operators]. It is further assumed that a nucleon in a nucleus undergoing a weak interaction can be treated as a free nucleon, which for the purpose of constructing interaction operators satisfies the Dirac equation. This latter approximation is known as the impulse approximation. For a single free nucleon, we have, using Lorentz covariance, conservation of parity, time-reversal invariance, and isospin invariance, the following general form for the vector and axial vector currents
\[eq:4\] $$\begin{aligned}
\langle \bm{k}' \lambda'; 1/2 m_{t'} | V_\mu^{TM_T} | \bm{k}
\lambda; 1/2 m_t \rangle = i \bar{u}(\bm{k}' \lambda') \left[
F_1^{(T)} \gamma_\mu + F_2^{(T)} \sigma_{\mu\nu}q_\nu \right]
u(\bm{k}\lambda) \langle 1/2 m_{t'} | I_T^{M_T} | 1/2 m_t
\rangle, \label{eq:4a}\\
\langle \bm{k}' \lambda'; 1/2 m_{t'} | A_\mu^{TM_T} | \bm{k}
\lambda; 1/2 m_t \rangle = i \bar{u}(\bm{k}' \lambda') \left[
F_A^{(T)} \gamma_5 \gamma_\mu - i F_P^{(T)} \gamma_5 q_\mu \right]
u(\bm{k}\lambda) \langle 1/2 m_{t'} | I_T^{M_T} | 1/2 m_t
\rangle. \label{eq:4b}
\end{aligned}$$
Here, the plane-wave single nucleon states are labelled with the three-momenta $\bm{k}$ ($\bm{k}'$), helicities $\lambda$ ($\lambda'$), isospin $1/2$ and isospin projections $m_t$ ($m_{t'}$). The momentum transfer, $q_\mu^2 = q^2 - \omega^2$, with $q=|\bm{q}|$, is defined in figure \[fig:weakprocess\]. Bold letters denote the three-momentum. The single-nucleon form factors $F_X^{(T)}=F_X^{(T)}(q_\mu^2)$, $T=0,1$, $X=1,2,A,P$ (vector Dirac, vector Pauli, axial, and pseudoscalar) are all functions of $q_\mu^2$ . Second class currents are not included in equation . The isospin dependence in equations is contained in
$$\label{eq:5}
I_T^{M_T} \equiv \frac{1}{2} \times \left\{
\begin{array}{ll}
1 & T=0, M_T =0 \\
\tau_0 & T=1, M_T = 0 \\
\tau_{\pm 1} = \mp \frac{1}{\sqrt{2}} (\tau_1 \pm i \tau_2) &
T=1, M_T = \pm 1
\end{array} \right.$$
To evaluate weak-interaction processes in nuclei, one needs matrix elements of the multipole operators between nuclear many-body states, labeled $|J_i M_{J_i}; T_i M_{T_i}\rangle$ which are complicated nuclear configurations of protons and neutrons. Using the Wigner-Eckart theorem we can write the matrix element of an arbitrary multipole operator $\hat{T}_{JM_J;TM_T}$ as [@Edmonds:1960]
$$\begin{aligned}
\label{eq:6}
\langle J_1 M_{J_1}; T_1 M_{T_1}|\hat{T}_{JM_J;TM_T} (q)|J_2 M_{J_2}; T_2
M_{T_2} \rangle & = & (-1)^{J_1-M_{J_1}} \left(
\begin{array}{ccc}
J_1 & J & J_2 \\
-M_{J_1} & M_J & M_{J_2}
\end{array} \right) \nonumber \\
& \times & (-1)^{T_1 - M_{T_1}}
\left(\begin{array}{ccc}
T_1 & T & T_2 \\
-M_{T_1} & M_T & M_{T_2}
\end{array}\right) \langle J_1; T_1|||\hat{T}_{J;T}(q)|||J_2; T_2
\rangle \end{aligned}$$
where the symbol $|||$ denotes that the matrix element is reduced in both angular momentum and isospin. If we assume that the multipole operators are one-body operators, we can write [@Heyde:1994]
$$\label{eq:7}
\langle J_1; T_1|||\hat{T}_{J;T}(q)|||J_2; T_2
\rangle= \sum_{\alpha \alpha'} \frac{\langle J_1;
T_1|||[a^\dagger_{\alpha'} \otimes \tilde{a}_\alpha]_{J;T}|||J_2; T_2
\rangle}{\sqrt{(2 J+1)(2 T+1)}} \langle \alpha' ||| T_{J;T}(q)
|||\alpha\rangle$$
with the sums extending over complete sets of single-particle wavefunctions $\alpha={n,l,j}$. The tensor product involves the single-particle creation operator $a^\dagger_\alpha \equiv
a^\dagger_{\alpha; m_{j_\alpha} m_{t_\alpha}}$ and $\tilde{a}_\alpha
\equiv (-1)^{j_\alpha - m_\alpha} (-1)^{1/2 - m_{t_\alpha}} a_{\alpha;
-m_{j_\alpha} -m_{t_\alpha}}$, with $a$ the destruction operator. The phase factor is introduced so that the operator $\tilde{a}$ transforms as a spherical tensor [@Edmonds:1960].
In practice the infinite sums in equation are approximated to include a finite number of (hopefully) dominant terms. The number of terms to include depends both of the computed observable and the model used (Shell-Model, Random phase approximation, …). Typical nuclear models are non-relativistic, requiring a non-relativistic reduction of the single-particle operators; the respective expressions are given for example by @Walecka:1975 and @Donnelly.Peccei:1979. @Donnelly.Haxton:1979 give the expressions for the single-particle matrix elements of these operators with harmonic oscillator wave functions. @Donnelly.Haxton:1980 provide expressions for general wave functions.
The above discussion presents the general theory of semileptonic processes. However, in many applications the momentum transfers involved are small compared with the typical nuclear momentum $Q
\approx R^{-1}$, with R the nuclear radius. In that case, the above formulas can be expanded in powers of $(qR)$ (long-wavelength limit) and one obtains the standard approximations to allowed (Gamow-Teller and Fermi) and forbidden transitions . In these limits the effect of the electromagnetic interaction on the initial or final charged lepton, that has been neglected in the above expressions, can be included [@Schopper:1966].
Nuclear models {#sec:nuclear-models}
--------------
As discussed in the previous section one of the basic ingredients for the evaluation of weak-interaction processes involving nuclei is the description of the nuclear many-body states. Moreover, the calculation of weak processes in stars have to account for the peculiarities of the medium (high temperatures and densities) and the presence of an electron plasma. When the temperatures and densities are small (for example during the r- and s-processes) weak transitions could be determined using the experimentally measured half-lives (in some cases one has to account for the presence of low lying isomeric states). However, as many of the very neutron-rich nuclei that participate in the r-process, are not currently accessible experimentally [see @Pfeiffer.Kratz.ea:2001 for recent experimental advances in the study of r-process nuclei], the necessary nuclear properties have to be extracted from theoretical models.
{width="0.9\linewidth"}
As the degrees of freedom increase drastically with the number of nucleons, models of different sophistication have to be chosen for the various regions in the nuclear charts. Exact calculations using realistic nucleon-nucleon interactions, e.g. by Green’s Function Monte Carlo techniques, are restricted to light nuclei with mass $A \le
10$ [@Carlson.Schiavilla:1998; @Wiringa.Pieper.ea:2000; @Pieper:2002]. As an alternative, methods based on effective field theory [@Kolck:1999; @Beane.Bedaque.ea:2001] have recently been developed for very light nuclei ($A \le 3$) . For heavier nuclei different approximations are required. In particular, restricted model spaces are used so that effective interactions and operators are necessary [@Hjorth-Jensen.Kuo.Osnes:1995]. For medium-mass nuclei ($A \le
70$) the shell model is the method of choice [@Talmi:1993]. This model explicitly treats all two-body correlations among a set of valance particles by a residual interaction. By diagonalizing the respective Hamiltonian matrix in the model space spanned by the independent particle states of the valence particles a quite satisfactory description of the ground state, the spectrum at moderate excitation energies and the electromagnetic and weak transitions among these states are obtained [@Caurier.Zuker.ea:1994; @Martinez-Pinedo.Poves.ea:1997]. In recent times due to progress both in computer technology and programming techniques shell-model calculations are now possible in model spaces which seemed impossible only a few years ago; i.e. the diagonalization codes <span style="font-variant:small-caps;">antoine</span> or <span style="font-variant:small-caps;">nathan</span> developed by Etienne Caurier allow for complete calculations in the $pf$-shell where the maximum dimension currently attained is $2.3 \times 10^9$ $M=0$ slater determinants for a complete diagonalization of $^{60}$Zn [@Caurier:2002; @Mazzocchi.Janas.ea:2001]. To treat even larger model spaces in the diagonalization shell model, different schemes are required. One method is to expand the nuclear many-body wave functions in terms of a few symmetry-projected Hartree-Fock-Bogoliubov (HFB) type quasiparticle determinants [@Schmid.Grummer.Faessler:1987; @Schmid.Zheng.ea:1989; @Schmid:2001]. A novel approach, introduced by @Honma.Mizusaki.Otsuka:1995 [see also @Otsuka.Honma.ea:2001], employs stochastical methods to determine the most important Slater determinants in the chosen model space. As an alternative to the diagonalization method, the Shell Model Monte Carlo (SMMC) [@Johnson.Koonin.ea:1992; @Koonin.Dean.Langanke:1997] allows calculation of nuclear properties as thermal averages, employing the Hubbard-Stratonovich transformation to rewrite the two-body parts of the residual interaction by integrals over fluctuating auxiliary fields. The integrations are performed by Monte Carlo techniques, making the SMMC method available for basically unrestricted model spaces. While the strength of the SMMC method is the study of nuclear properties at finite temperature, it does not allow for detailed nuclear spectroscopy.
The evaluation of nuclear matrix elements for the Fermi operator is straightforward. The Gamow-Teller operator connects Slater determinants within a model space spanned by a single harmonic oscillator shell ($0 \hbar \omega$ space). The shell model is then the method of choice to calculate the nuclear states involved in weak-interaction processes dominated by allowed transitions as complete or sufficiently converged truncated calculations are nowadays possible for such $0 \hbar \omega$ model spaces. The practical calculation of the Gamow-Teller distribution is achieved by adopting the Lanczos method [@Wilkinson:1965] as proposed by @Whitehead:1980; [see also @Poves.Nowacki:2001; @Langanke.Poves:2000].
The calculation of forbidden transitions, however, involves nuclear transitions between different harmonic oscillator shells and thus requires multi-$\hbar \omega$ model spaces. These are currently only feasible for light nuclei where *ab initio* shell model calculations are possible [@Navratil.Vary.Barrett:2000; @Caurier.Navratil.ea:2001]. Such multi-$\hbar\omega$ calculations have been used for the calculation of neutrino scattering from $^{12}$C [@Hayes.Towner:2000; @Volpe.Auerbach.ea:2000]. However, for heavier nuclei one has to rely on more strongly truncated nuclear models. As the kinematics of stellar weak-interaction processes are often such that forbidden transitions are dominated by the collective response of the nucleus the Random Phase Appoximation [@Rowe:1968] is usually the method of choice (figure \[fig:rpasm\]). Another advantage of this method is that, in contrast to the shell model, it allows for global calculations of these processes for the many nuclei often involved in nuclear networks. An illustrative example is the evaluation of nuclear half-lives based on the calculation of the GT strength function within the Quasiparticle RPA model [@Krumlinde.Moeller:1984; @Moeller.Randrup:1990]. The RPA method considers the residual correlations among nucleons via one particle one hole (1p-1h) excitations in large multi-$\hbar \omega$ model spaces. Compared to the shell model, the neglect of higher-order correlations renders the RPA method inferior for matrix elements between individual, non-collective states. A prominent example is the GT transition from the $^{12}$C ground state to the $T=1$ triad in the $A=12$ nuclei [e.g. @Engel.Kolbe.ea:1996]. While the shell model is able to reproduce the GT matrix element between these states [@Cohen.Kurath:1965; @Warburton.Brown:1992], RPA calculations miss an important part of the nucleon correlations and overestimate these matrix elements by about a factor of 2 [@Kolbe.Langanke.Vogel:1994; @Engel.Kolbe.ea:1996]. Recent developments have extended the RPA method to include the complete set of 2p-2h excitations in a given model space [@Drozdz.Nishizaki.ea:1990]. Such 2p-2h RPA models have, however, not yet been applied to semileptonic weak processes in stars. Moreover, the RPA allows for the proper treatment of the momentum-dependence in the different multipole operators, as it can be important in certain stellar neutrino-nucleus processes (see below), and for the inclusion of the continuum [@Buballa.Drozdz.ea:1991]. Detailed studies indicate that standard and continuum RPA calculations yield nearly the same results for total semileptonic cross sections [@Kolbe.Langanke.Vogel:2000]. This is related to the fact that both RPA versions obey the same sumrules. The RPA has also been extended to deal with partial occupation of the orbits so that configuration mixing in the same shell is included schematically [@Rowe:1968; @Kolbe.Langanke.Vogel:1999].
Hydrogen burning and solar neutrinos
====================================
The tale of the solar neutrinos and their ‘famous’ problem took an exciting twist from its original goal of measuring the central temperature of the Sun to providing convincing evidence for neutrino oscillations, thus opening the door to physics beyond the standard model of the weak interaction. In 1946, Pontecorvo suggested [@Pontecorvo:1946; @Pontecorvo:1991] [later independently proposed by @Alvarez:1949] that chlorine would be a good detector material for neutrinos and subsequently in the 1950’s Davis built a radiochemical neutrino detector which observed reactor neutrinos via the $^{37}$Cl($\nu_e,e^-)^{37}$Ar reaction [@Davis:1955]. After the $^{3}$He($\alpha,\gamma){}^7$Be cross section at low energies had been found to be significantly larger than expected [@Holmgren.Johnston:1958] and, slightly later, the $^7$Be$(p,\gamma){}^8$B cross section at low energies had been measured [@Kavanagh:1960], it became clear that the Sun should also operate by what are now known as the ppII and ppIII chains and in that way generate neutrinos with energies high enough to be detectable by a chlorine detector [@Fowler:1958; @Cameron:1958]. This idea was then seriously pursued by Davis, in close collaboration with Bahcall. The observed solar neutrino flux turned out to be lower than predicted by the solar models (the original solar neutrino problem) [@Davis.Harmer.Hoffman:1968; @Bahcall.Bahcall.Shaviv:1968], triggering the development of further solar neutrino detectors, initiating the field of neutrino oscillation experiments and, after precision helioseismology data [@Dalsgaard:2002] boosted the confidence in the solar models, finally culminating in the conclusive evidence for neutrino oscillations in the solar flux. A detailed recent review of the solar hydrogen burning and neutrino problem is given in [@Kirsten:1999].
![The energy spectrum of neutrinos predicted by the standard solar model [@Bahcall.Pinsonneault.Basu:2001]. The neutrino fluxes from continuum sources (like pp and $^8$B) are given in the units of number per cm$^2$ per second per MeV, while the line fluxes (e.g. $^7$Be) are given in number per cm$^2$ per second. The ranges of neutrino energies observable in the various detectors are indicated by arrows. The uncertainties in the various fluxes are given in percent (courtesy of J. N. Bahcall). \[fig:snuspect\]](snuspectrum.ps){width="0.9\linewidth"}
------------------------------------------------------------------ --------- ----------------------------------------------------
Term $\nu$ Energy
(%) (MeV)
$p + p \rightarrow {}^2\text{H} + e^+ + \nu_e$ 99.96 $\leq
0.423$
$p + e^- + p \rightarrow {}^2\text{H} + \nu_e$ 0.44 1.445
\[1.5\] $^2\text{H} + p \rightarrow {}^3\text{H} + \gamma$ 100
$^3\text{He} + {}^3\text{He} \rightarrow \alpha + 2p$ 85
$^3\text{He} + {}^4\text{He} \rightarrow {}^7\text{Be} + \gamma$ 15
$\qquad {}^7\text{Be} + e^- \rightarrow {}^7\text{Li} + \nu_e$ 15 $\left\{\begin{array}{r} 0.863\ 90\%\\ 0.385\ 10\%
\end{array}\right.$
$\qquad {}^7\text{Li} + p \rightarrow 2\alpha$
$^7\text{Be} + p \rightarrow {}^8\text{B} + \gamma $ 0.02
$\qquad {}^8\text{B} \rightarrow {}^8\text{Be}^* + e^+ + \nu_e$ $< 15$
$\qquad {}^8\text{Be}^* \rightarrow 2\alpha$
$^3\text{He} + p \rightarrow {}^4\text{He} + e^+ + \nu_e$ 0.00003 $<18.8$
------------------------------------------------------------------ --------- ----------------------------------------------------
: The solar pp chains. The neutrino terminations are from the BP2000 solar model [@Bahcall.Pinsonneault.Basu:2001]. The neutrino energies include the solar corrections [@Bahcall:1997]. \[tab:ppchain\]
The Sun generates its energy from nuclear fusion reactions in the pp chain (see table \[tab:ppchain\]), with a small contribution by the CNO cycle. Several of these reactions are mediated by the weak interaction, and hence create (electron) neutrinos which in the standard solar model can leave the Sun unhindered. The predicted flux of solar neutrinos on the surface of the earth is shown in figure \[fig:snuspect\]. These predictions depend on the knowledge of the relevant nuclear cross sections at solar energies (a few keV) which, with the notable exception of the $^3$He($^3$He,2p)$^4$He reaction [@Bonetti.Broggini.ea:1999] which has been measured directly in the underground laboratory in the Gran Sasso, relies on the extrapolation of data taken at higher energies. As these reactions are all non-resonant, the extrapolations are quite mild and appear to be under control [@Adelberger.Austin.ea:1998]. The cross section for the initial $p+p$ fusion reaction is so low that no data exist and the respective solar reaction rate relies completely on theoretical modelling. Nevertheless the underlying theory is thought to be under control and the uncertainty in this important rate is estimated to be about $1\%$ [@Kamionkowski.Bahcall:1994], based on potential model calculations, and it is probably even smaller if effective field theory is applied [@Park.Marcucci.ea:2001a; @Park.Kubodera.ea:1998; @Kong.Ravndal:2001]. In the solar plasma the reaction rates are slightly enhanced due to screening effects [@Dzitko.Turck-Chieze.ea:1995; @Gruzinov.Bahcall:1998]. The most significant plasma modification is found for the lifetime of $^7$Be with respect to electron capture where capture of continuum and bound electrons with the relevant screening corrections have to be accounted for [@Johnson.Kolbe.ea:1992; @Gruzinov.Bahcall:1998]. It is generally believed that the $^7$Be$(p,\gamma){}^8$B reaction is the least known nuclear input in nuclear models. Although this reaction occurs in the weak ppIII chain, the decay of $^8$B is the source of the high-energy neutrinos observed by the solar neutrino detectors. Recent direct and indirect experimental methods have improved the knowledge of the $^7$Be$(p,\gamma)^8$B rate considerably (@Davids.Anthony.ea:2001 and references therein). While these data point to an astrophysical S-factor in the range of 18–20 eV b, a very recent direct measurement with special emphasis on the control and determination of the potential errors yielded a slightly larger value [@Junghans.Mohrmann.ea:2001; @Junghans.Snover:2002]. The neutrino energy distribution arising from the subsequent $^8$B decay has been measured precisely by @Ortiz.Garcia.ea:2000. In principle, high-energy neutrinos are also produced in the $^3\text{He}+p$ fusion reaction which, however, occurs only in a weak branch in the solar pp cycles. Although the calculation of this cross section represents a severe theoretical challenge, it appears to be determined now with the required accuracy using state-of-the-art few-body methods .
The solar nuclear cross sections have been reviewed by @Adelberger.Austin.ea:1998, including also the reactions occuring in the CNO cycle. Except for some discrepancies in the $^{14}$N(p,$\gamma)^{15}$O cross section at low energies [@Adelberger.Austin.ea:1998; @Angulo.Descouvemont:2001], all relevant solar rates are sufficiently well known.
There are currently 5 solar neutrino detectors operating. Three of them, the homestake chlorine detector [@Bahcall:1989 p. 487], GALLEX[^1] [@Anselmann.Hampel.ea:1992], and SAGE [@Abdurashitov.Faizov.ea:1994]) can only observe charge-current (electron) neutrino reactions, while the two water Cerenkov detectors (Super-Kamiokande [@Fukuda.Hayakawa.ea:1998a], SNO [@Boger.Hahn.ea:2000]) also observe neutral-current events, which can be triggered by all neutrino flavors. All neutrino detectors have characteristic energy thresholds for neutrino detection, dictated by the various observation schemes; i.e. the detectors are blind for neutrinos with energies less than the threshold energy $E_{th}$. The pioneering chlorine experiment of Davis uses the $^{37}$Cl($\nu_e,e^-)^{37}$Ar reaction as detector, with $E_{th}=814 $ keV. Gallex and Sage detect neutrinos via $^{71}$Ga($\nu_e,e^-)^{71}$Ge with the threshold energy $E_{th}$=233.2 keV. In Super-Kamiokande (SK) solar neutrinos are identified by the observation of relativistic electrons produced from inelastic $\nu+e^-$ scattering. Due to high background at low energies, the observational threshold is set to $\sim 7$ MeV. SNO has an inner vessel of heavy water, surrounded by normal water. Like SK, this detector can also observe neutrinos via inelastic scattering off electrons. Additionally, and more importantly, SNO can also detect neutrinos by the dissociation of the deuteron in heavy water, with the threshold energy of order 6 MeV.
The threshold energies and the predicted solar neutrino fluxes are shown in figure \[fig:snuspect\]. One notes that SK and SNO are only sensitive to $^8$B neutrinos (neglecting the weak $hep$ flux), the chlorine experiment detects mainly $^8$B (76$\%$ of the predicted flux by @Bahcall.Pinsonneault.Basu:2001) and $^7$Be neutrinos, while Gallex and Sage can also observe neutrinos generated in the main solar energy source, the $p+p$ fusion reaction (54$\%$ of the predicted flux). It is important to note that the solar neutrino detectors have been calibrated, using known neutrino sources.
{width="0.9\linewidth"}
The original solar neutrino problem constitutes the fact that the earthbound detectors observe less neutrinos than predicted by the solar model. The current comparison is depicted in figure \[fig:snudetect\]. Importantly, Sage and Gallex, in close agreement to each other, observe at least a neutrino flux which is consistent with the fact that the current solar luminosity is powered by the $p+p$ fusion reaction [@Hampel.Handt.ea:1999; @Abdurashitov.Gavrin.ea:1999]. With improved input (nuclear reaction rates, opacities, etc.) the solar models evolved and, as a milestone, passed the stringent test of detailed comparison to the soundspeed distribution derived from helioseismology [@Christensen-Dalsgaard.Dappen.ea:1996]. It became clear that the solution to the solar neutrino problem pointed to weak-interaction physics beyond the standard model. This line of reasoning was supported by the observation [@Heeger.Robertson:1996] that any solar model assuming standard weak-interaction physics leads to contradictions between the observed fluxes in the various detectors.
It has been speculated already for a long time [@Pontecorvo:1968] that the solution to the deficient observed neutrino flux lies in the possibility that neutrinos change their flavor on their way from the center of the Sun to the earthbound detectors. Neutrino oscillations can occur if the flavor eigenstates (the physical $\nu_e, \nu_\mu,
\nu_\tau$ neutrinos) are not identical with the mass eigenstates $(\nu_1,\nu_2,\nu_3$) of the weak Hamiltonian, but rather are given by a unitary transformation of these states defined by a set of mixing angles. Importantly, oscillations between two flavor states can only occur if at least one of these states does not propagate with the speed of light implying that this neutrino has a mass different from zero; more precisely $\Delta m^2 = m_1^2 - m_2^2 \ne 0$, where $m_{1,2}$ are the masses of the oscillating neutrinos. As all neutrino masses are assumed to be zero in the Weinberg-Salam model, the observation of neutrino oscillations opens the door to new physics beyond the standard model of weak interaction. Neutrino oscillations can occur for free-propagating neutrinos (vacuum oscillations). However, their occurence can also be influenced by the environment. In particular, it has been pointed out that the high-energy ($\nu_e$) solar neutrinos can, for a certain range of mixing angles and mass differences, transform resonantly into other flavors, mediated by the interaction of the $\nu_e$ neutrinos with the electrons in the solar plasma, resulting in matter-enhanced oscillations [the so-called MSW effect @Wolfenstein:1978; @Mikheyev.Smirnov:1986].
![Flux of $\nu_{\mu,\tau}$ neutrinos vs. electron neutrinos as deduced from the SNO and SK $^8$B neutrino data. The diagonal bands show the total $^8$B flux as predicted by the standard solar model (SSM, dashed lines) [@Bahcall.Pinsonneault.Basu:2001] and that derived from the SNO and SK measurements (solid lines). The intercepts with the axis represent $1\sigma$ errors [from @Ahmad.Allen.ea:2001]. \[fig:SNO\]](SNOSK.eps){width="0.9\linewidth"}
First clear evidence for neutrino oscillations was reported by the SK collaboration which observed a deficit of $\nu_\mu$-induced events from atmospheric neutrinos and could link this deficit to $\nu_\mu
\rightleftarrows \nu_\tau$ oscillations [@Fukuda.Hayakawa.ea:1998b; @Fukuda.Hayakawa.ea:1999b]. \[Further evidence for neutrino oscillations has been given by the LSND collaboration . This result, however, was, for most of the allowed parameter space not confirmed by the Karmen experiment [@Armbruster.Blair.ea:1998a; @Armbruster.Blair.ea:1998b]. The complete LSND result will be tested by the MiniBoone[^2] experiment which is currently under construction.\] A clear link between neutrino oscillations and the solar neutrino problem has been presented recently by a combined analysis [@Ahmad.Allen.ea:2001] of the first SNO data with the precise SK data [@Fukuda.Fukuda.ea:2001b]. SNO measured the integrated event rate above the kinetic energy threshold $T_{\text{eff}}=6.75 $ MeV (the electron energy threshold then is $E_{\text{th}}=T_{\text{eff}}+0.511$ MeV) for charged-current (CC) reactions on the deuteron and inelastic electron scattering (ES). As no evidence for a deviation of the spectral shape from the predicted shape under the no-oscillation hypothesis has been observed, the integrated rate could be converted into the measured $^8$B neutrino flux, resulting in $\Phi^{\text{CC}}_{\text{SNO}} (\nu_e) =
1.75 \pm 0.07 (\text{stat}) \pm 0.12 (\text{syst}) \pm 0.05
(\text{theor}) \times 10^6$ cm$^{-2}$ s$^{-1}$, $\Phi^{\text{ES}}_{\text{SNO}} (\nu) = 2.39 \pm 0.34 (\text{stat}) \pm
0.15 (\text{syst}) \times 10^6$ cm$^{-2}$ s$^{-1}$. The SNO electron scattering flux result agrees with the more precise measurement from SK which yields $\Phi^{\text{ES}}_{\text{SK}} (\nu) = 2.32 \pm 0.03
(\text{stat}) \pm 0.08 (\text{syst}) \times 10^6$ cm$^{-2}$ s$^{-1}$. We note that the charged-current reaction can only be triggered by $\nu_e$ neutrinos at the energies of the solar neutrinos. Thus, from the measurement of the $\nu_e+D$ event rate, SNO has determined the solar $\nu_e$-flux arriving on earth stemming from the decay of $^8$B. On the other hand, neutrino-electron scattering can occur for all neutrino types, whereby the $\nu_e+e^-$ cross section is about seven times larger than the $\nu_{\mu,\tau} + e^-$ cross section. If no oscillations involving solar $\nu_e$ neutrinos occur, the SNO charged-current flux $\Phi^{\text{CC}}_{\text{SNO}}$ and the SK inelastic electron scattering flux $\Phi^{\text{ES}}_{\text{SK}}$ should be the same; that is excluded by 3.3 $\sigma$. \[The exclusion is even slightly more severe if the recent revision of the $\nu_e+D$ cross section including radiative corrections is considered [@Kurylov.Ramsey-Musolf.Vogel:2002].\] If $\nu_e \rightleftarrows
\nu_{\mu,\tau}$ oscillations occur, $\Phi^{\text{ES}}_{\text{SK}}$ should be larger than $\Phi^{\text{CC}}_{\text{SNO}}$ as it then contains additional neutral-current contributions from $\nu_{\mu,\tau}$ neutrinos. From a best fit to the SNO and SK data (see figure \[fig:SNO\]), this contribution has been determined as (with 1$\sigma$ uncertainty) $\Phi_{\mu,\tau} = 3.69 \pm 1.13 \times
10^6$ cm$^{-2}$ s$^{-1}$ [@Ahmad.Allen.ea:2001], implying that the total solar flux is $\Phi_{\text{SNO}}^{\text{CC}} (\nu_e) +\Phi
(\nu_{\mu,\tau}) =5.44 \pm 0.99 \times 10^6$ cm$^{-2}$ s$^{-1}$. This result agrees very nicely with the $^8$B neutrino flux predicted by the solar model [@Bahcall.Pinsonneault.Basu:2001] ($5.05 \times
10^6$ cm$^{-2}$ s$^{-1}$).
For years the measurement of the neutral-current $\nu$+D reaction at SNO has been anticipated as the ‘smoking gun’ for solar neutrino oscillations. After finishing this review, the first results of this milestone experiment have been published [@Ahmad.Allen.ea:2002a]. They lead to the same conclusions as the earlier SNO results [@Ahmad.Allen.ea:2001] showing a clear excess of neutral-current over charged-current events, as expected if neutrino oscillations are the origin of the solar neutrino problem. Furthermore, the observed neutral-current event rate is again consistent with the prediction of the solar model [@Bahcall.Pinsonneault.Basu:2001].
A global analysis of the latest solar neutrino data including the SNO charged-current rate favors matter-enhanced neutrino oscillations with large mixing angles [@Krastev.Smirnov:2001]. Considering the recent constraints on the $^7$Be$(p,\gamma){}^8$B cross section and the respectively predicted $^8$B solar neutrino flux, vacuum oscillations are essentially excluded. A similar result is obtained by @Bahcall.Gonzalez-Garcia.Pena-Garay:2002 including the recent day-night asymmetry measured at SNO [@Ahmad.Allen.ea:2002b]
For more than 30 years the solar neutrino problem has been a demanding challenge for experimentalists and theorists, for nuclear, particle and astrophysicists alike. The challenge appears to be mastered, leading to new physics and without the need of the many desperate solution attempts put forward over the years.
Late-stage stellar evolution
============================
General remarks
---------------
Weak interactions play an essential role already during hydrostatic burning. Its importance lies in the fact that the neutrinos generated by these processes can leave the star unhindered, thus carrying away energy and hence cooling the star. While the consideration of energy losses by neutrinos is already required during hydrogen burning (see above), the heat flux in the early stages of stellar burning is predominantly by radiation. This changes, following helium burning, when the stellar temperatures reach $\sim 5 \times 10^8$ K and neutrino-antineutrino pair production and emission becomes the leading energy loss mechanism. The respective cooling rate is a local property of the star depending on density $\rho$ and, very sensitively, on temperature $T$; i.e. the energy loss rate for $\nu{\bar{\nu}}$ emission scales approximately like $T^{11}$, implying that the hot inner regions of the star cool most effectively. However, the dominant nuclear reactions, occuring after helium burning, have even stronger temperature dependences. For example, the heat production $\epsilon$ in the $^{12}$C+$^{12}$C or $^{16}$O+$^{16}$O fusion reactions, which dominate hydrostatic carbon and oxygen burning, scales like $\epsilon
\approx T^{22}$ and $\approx T^{35}$ around $T= 10^9$ K. As a consequence of the temperature gradient in the stellar interior and the vast difference in the temperature sensitivity, nuclear reaction heating overcomes the neutrino energy loss in the center. However, in the cooler mantle region surrounding the core neutrino cooling dominates. The resulting entropy difference leads to convective instabilities [see @Arnett:1996]. First attempts of modelling late-stage stellar burning and nucleosynthesis including a two-dimensional treatment of convection is reported in [@Baleisis.Arnett:2001].
{width="0.90\linewidth"}
The importance of convection has, of course, already been noticed before and is accounted for in one-dimensional models within the so-called mixing-length theory [@Clayton:1968]. It has been found that this convective transport is far more efficient at carrying energy and mixing the matter composition than radiation transport. For example, convection dominates the envelope region in massive stars during helium shell burning, as can be seen in figure \[fig:Kippenhahn\] which shows the energy history of a 22 $M_\odot$ star [@Heger.Woosley.pvt]. The figure also identifies the various subsequent energy reservoirs of the star: hydrogen, helium, carbon, neon, oxygen, and silicon core and shell burning. However, the figure also demonstrates the importance of neutrino losses which, following oxygen core burning, can overcome the nuclear energy generation, except at the high temperatures in the very inner core. Obviously weak-interaction processes are crucial in this late epoch of massive stars. This is not only true for the star’s energy budget, but these processes can also alter the matter composition and entropy which, in turn, can affect the location and extension of convective shells, e.g. during oxygen and silicon burning, with subsequent changes in the stellar structure. Such effects have recently been observed after the improved shell-model weak-interaction rates (subsection III.2) have been incorporated into stellar models. (figure \[fig:Kippenhahn\] is already based on these rates.)
Shell-model electron capture and $\beta$ decay rates
----------------------------------------------------
The late evolution stages of massive stars are strongly influenced by weak interactions which act to determine the core entropy and electron to baryon ratio, $Y_e$, of the presupernova star, hence its Chandrasekhar mass which is proportional to $Y_e^2$. Electron capture reduces the number of electrons available for pressure support, while beta-decay acts in the opposite direction. Both processes generate neutrinos which, for densities $\rho\lesssim 10^{11}$ g cm$^{-3}$, escape the star carrying away energy and entropy from the core.
![The figure shows schematically the electron capture and beta decay processes in the stellar environment. Electron capture proceeds by Gamow-Teller transitions to the GT$_+$ resonance. In the case of beta decay both the Fermi and Gamow-Teller resonances are typically outside of the $Q_\beta$ window, and hence are not populated in laboratory decays. Due to the finite temperature in stars excited states in the decaying nucleus can be thermally populated. Some of these states have strong GT transitions to low-lying states in the daughter nucleus. These states in the decaying nucleus are called “backresonances”.\[fig:ecbeta\]](betamenos-new.eps){width="0.90\linewidth"}
Electron capture and beta decay during the final evolution of a massive star are dominated by Fermi and Gamow-Teller (GT) transitions. While the treatment of Fermi transitions (important only in beta decays) is straightforward, a correct description of the GT transitions is a difficult problem in nuclear structure. In the astrophysical environment nuclei are fully ionized so one has continuum electron capture from the degenerate electron plasma. The energies of the electrons are high enough to induce transitions to the Gamow-Teller resonance. Shortly after the discovery of this collective excitation @Bethe.Brown.ea:1979 recognized its importance for stellar electron capture. The presence of the degenerate electron gas blocks the phase space for the produced electron in beta decay. Then the decay rate of a given nuclear state is greatly reduced or even completely blocked at high densities. However, due to the finite temperature excited states in the decaying nucleus can be thermally populated. Some of these states are connected by strong GT transitions to low-lying states in the daughter nucleus that with increased phase space can significantly contribute to the stellar beta decay rates. The importance of these states in the parent nucleus for beta-decay was first recognized by Fuller, Fowler and Newman (commonly abbreviated as FFN) who coined the term “backresonances” (see figure \[fig:ecbeta\]).
Over the years, many calculations of weak interaction rates for astrophysical applications have become available . For approximately 15 years though, the standard in the field has been the tabulations of . These authors calculated rates for electron capture, positron capture, beta-decay, and positron emission plus the associated neutrino losses for all the astrophysically relevant nuclei ranging in mass number from 21 to 60. Their calculations were based upon an examination of all available experimental information in the mid 1980s for individual transitions between ground states and low-lying excited states in the nuclei of interest. Recognizing that this only saturated a small part of the Gamow-Teller distribution, they added the collective strength via a single-state representation. Both, energy position and strength collected in this single state were determined using an independent particle model (IPM).
![Comparison of shell model GT$_+$ distributions with experimental data for selected nuclei. The shell model results (discrete lines) have been folded with the experimental resolution (histograms). The arrows indicate the positions where @Fuller.Fowler.Newman:1982a placed the GT resonance in their calculations of the stellar weak-interaction rates [adapted from @Caurier.Langanke.ea:1999]. \[fig:GTstrength\]](GT+.eps){width="0.90\linewidth"}
Recent experimental data on GT distributions in iron group nuclei measured in charge exchange reactions [@Goodman.Goulding.ea:1980; @Osterfeld:1992], show that the GT strength is strongly quenched, compared with the independent particle model value, and fragmented over many states in the daughter nucleus. Both effects are caused by the residual interaction among the valence nucleons and an accurate description of these correlations is essential for a reliable evaluation of the stellar weak-interaction rates due to the strong phase space energy dependence, particularly of the stellar electron-capture rates. The shell model is the only known tool to reliably describe GT distributions in nuclei [@Brown.Wildenthal:1988]. Indeed, @Caurier.Langanke.ea:1999 demonstrated that the shell model reproduces all measured GT$_+$ distributions (in this direction a proton is changed into a neutron, like in electron capture) for nuclei in the iron mass range very well and gives a very reasonable account of the experimentally known GT$_-$ distributions (in this direction, a neutron is changed into a proton, like in $\beta$ decay). Further, the lifetimes of the $pf$-shell nuclei and their spectroscopy at low energies are simultaneously also described well. Figure \[fig:GTstrength\] compares the shell model GT$_+$ distributions to the pioneering measurement performed at TRIUMF. These measurements had a typical energy resolution of $\sim 1$ MeV. Recently developed techniques, involving for example ($^3\text{He},t$) [@Fujita.Akimune.ea:1996] and ($d,{}^2$He) [@Woertche:2001] charge-exchange reactions at intermediate energies, demonstrated in pilot experiments an improvement in the energy resolution by an order of magnitude or more. Again, the shell model calculations agree quite favorably with the improved data.
Several years ago, @Aufderheide:1991 and pointed out that the interacting shell model is the method of choice for the calculation of stellar weak-interaction rates. Following the work by @Brown.Wildenthal:1988, @Oda.Hino.ea:1994 calculated shell-model rates for all the relevant weak processes for $sd$-shell nuclei ($A=17$–39). This work was then extended to heavier nuclei ($A=45$–65) by @Langanke.Martinez-Pinedo:2001 based on shell-model calculations in the complete $pf$ shell. Following the spirit of FFN, the shell model results have been replaced by experimental data (energy positions, transition strengths) wherever available.
Weak interaction rates have also been computed using the proton-neutron quasiparticle RPA model [@Nabi.Klapdor-Kleingrothaus:1999a; @Nabi.Klapdor-Kleingrothaus:1999b] and the spectral distribution theory [@Kar.Ray.Sarkar:1994; @Sutaria.Ray:1995]
![Shell model electron-capture rates as a function of temperature ($T_9$ measures the temperature in $10^9$ K) and for selected densities ($\rho_7$ defines the density in $10^7$ g cm$^{-3}$) and nuclei. For comparison, the FFN rates are given by the full points.\[fig:ecrates\]](ecapture.ps){width="0.9\linewidth"}
![Shell model beta-decay rates as a function of temperature ($T_9$ measures the temperature in $10^9$ K) and for selected densities ($\rho_7$ defines the density in $10^7$ g cm$^{-3}$) and nuclei. For comparison, the FFN rates are given by the full points [from @Martinez-Pinedo.Langanke.Dean:2000]. \[fig:brates\]](beta.ps){width="0.9\linewidth"}
After oxygen burning, the important weak processes are electron captures and beta decays on nuclei in the iron mass range ($A \sim
45$–65). Conventional stellar models described these weak processes using the rates estimated by @Fuller.Fowler.Newman:1982a. These rates are compared to the shell model electron capture rates in figure \[fig:ecrates\] at relevant temperatures and densities. Importantly the shell model rates are nearly always lower than the FFN rates. Thus this difference represents a systematic trend, which is not expected to be washed out if the many nuclei in the stellar composition are considered. The difference is caused, for example, by the reduction of the Gamow-Teller strength (quenching) compared to the IPM value and a systematic misplacement of the Gamow-Teller centroid in nuclei with certain pairing structure [@Langanke.Martinez-Pinedo:2000]. In some cases, experimental data, which were not available to FFN, but could be used by @Langanke.Martinez-Pinedo:2001, led to significant changes. The FFN and shell-model beta decay rates are compared in figure \[fig:brates\], @Martinez-Pinedo.Langanke.Dean:2000 discuss the differences between the two rate sets.
Consequences of the shell model rates in stellar models
-------------------------------------------------------
@Heger.Langanke.ea:2001 [@Heger.Woosley.ea:2001] have investigated the influence of the shell model rates on the late-stage evolution of massive stars by repeating the calculations of @Woosley.Weaver:1995 keeping the stellar physics, except for the weak rates, as close to the original studies as possible. The new calculations have incorporated the shell-model weak interaction rates for nuclei with mass numbers $A=45$–65, supplemented by rates from @Oda.Hino.ea:1994 for lighter nuclei. The earlier calculations of Weaver and Woosley (WW) used the FFN rates for electron capture and an older set of beta decay rates [@Mazurek:1973; @Mazurek.Truran.Cameron:1974]. As a side-remark we note that late-stage evolution of massive stars is quite sensitive to the still not sufficiently well known $^{12}$C($\alpha,\gamma)^{16}$O rate. The value adopted in the standard WW and in the @Heger.Langanke.ea:2001 models \[$S(E=300\ \text{keV}) =
170$ keV b\] agrees, however, rather nicely with the recent data analysis \[$S(300)= 165\pm50$ keV b [@Kunz.Jaeger.ea:2001]\] and the value derived from nucleosynthesis arguments by @Weaver.Woosley:1993, \[$S(300) = 170\pm20$ keV b\].
{width="0.3\linewidth"} {width="0.3\linewidth"} {width="0.3\linewidth"}
Figure \[fig:presn\] illustrates the consequences of the shell model weak interaction rates for presupernova models in terms of the three decisive quantities: the central electron-to-baryon ratio $Y_e$, the entropy, and the iron core mass. The central values of $Y_e$ at the onset of core collapse increased by 0.01-0.015 for the new rates. This is a significant effect. For example, a change from $Y_e=0.43$ in the WW model for a 20 $M_\odot$ star to $Y_e=0.445$ in the new models increases the respective Chandrasekhar mass by about 0.075 $M_\odot$. We note that the new models also result in lower core entropies for stars with $M \leq 20\ M_\odot$, while for $M
\geq 20\ M_\odot$, the new models actually have a slightly larger entropy. The iron core masses are generally smaller in the new models where the effect is larger for more massive stars ($M \ge 20\
M_\odot$), while for the most common supernovae ($M \le 20\ M_\odot$) the reduction is by about 0.05 $M_\odot$. \[We define the iron core as the mass interior to the point where the composition becomes at least $50 \%$ of iron group elements $(A \geq 48)$\]. This reduction of the iron core mass appears to be counterintuitive at first glance with respect to the slower electron capture rates in the new models. It is, however, related to changes in the entropy profile during silicon shell burning which reduces the growth of the iron core just prior to collapse [@Heger.Langanke.ea:2001].
It is intriguing to speculate what effects these changes might have for the subsequent core collapse and supernova explosion. At first we note that in the current supernova picture gravitation overcomes the resisting electron degeneracy pressure in the core, leading to increasing densities and temperatures. Shortly after neutrino trapping at densities of a few $10^{11}$ g cm$^{-3}$, an homologous core, which stays in sonic communication, forms in the center. Once the core reaches densities somewhat in excess of nuclear matter density (a few $10^{14}$ g cm$^{-3}$) the nuclear equation of state stiffens and a spring-like bounce is created triggering the formation of a shock wave at the surface of the homologous core [@Bethe:1990]. This shock wave tries to traverse the rest of the infalling matter in the iron core. However, the shock loses its energy by dissociation of the infalling matter and by neutrino emission, and it is generally believed now that supernovae do not explode promptly due to the bounce shock. Probably, this happens in the ‘delayed mechanism’ [@Wilson:1985] where the shock is revived by energy deposition from the neutrinos generated by the cooling of the proto-neutron star, the remnant in the center of the explosion.
With the larger $Y_e$ values, obtained in the calculations with the improved weak rates, the core contains more electrons whose pressure acts against the collapse. It is also expected that the size of the homologous core, which scales like $\sim Y_e^2$ with the $Y_e$ value at neutrino trapping, should be larger. This, combined with the smaller iron cores, yields less material which the shock has to traverse. Furthermore, the change in entropy will affect the mass fraction of free protons, which in the later stage of the collapse contribute significantly to the electron capture. For presupernova models with masses $M<20\ M_\odot$, however, the number fraction of protons is very low [$\lesssim
10^{-6}$, @Heger.Langanke.ea:2001] so that for these stars electron capture should still be dominated by nuclei, even at densities in access of $10^{10}$ g cm$^{-3}$. We will return to this problem below.
![Evolution of the $Y_e$ value in the center of a 15 $M_\odot$ (upper part) and 25 $M_\odot$ (lower part) as a function of time until bounce. The most important $Y_e$-changing nuclei for the calculations adopting the shell model rates are indicated at different times, where the upper nucleus refers to electron capture and the lower to $\beta$-decay.\[fig:Ye\]](yemost.eps){width="0.9\linewidth"}
![Comparison of the change of the $Y_e$ value with time, $|dY_e/dt|$ due to electron capture and beta decay in a 15 $M_\odot$ star. In general, the $Y_e$ value decreases with time during the collapse, caused by electron captures. The loops indicate that during this period beta-decay, which increases $Y_e$, dominates over electron capture.\[fig:dyeye\]](dyeye-LM-s15.eps){width="0.9\linewidth"}
To understand the origin of these differences it is illustrative to investigate the role of the weak-interaction rates in greater details. The evolution of $Y_e$ during the collapse phase is plotted in figure \[fig:Ye\]. Weak processes become particularly important in reducing $Y_e$ below 0.5 after oxygen depletion ($\sim 10^7$ s and $10^6$ s before core collapse for the 15 $M_\odot$ and 25 $M_\odot$ stars, respectively) and $Y_e$ begins a decline which becomes precipitous during silicon burning. Initially electron capture occurs much more rapidly than beta decay. As the shell model rates are generally smaller than the FFN electron capture rates, the initial reduction of $Y_e$ is smaller in the new models; the temperature in these models is correspondingly larger as less energy is radiated away by neutrino emission.
An important feature of the new models is demonstrated in figure \[fig:dyeye\]. Beta decay becomes temporarily competitive with electron capture after silicon depletion in the core and during silicon shell burning [this had been foreseen in @Aufderheide.Fushiki.ea:1994b]. The presence of an important beta decay contribution has two effects. Obviously it counteracts the reduction of $Y_e$ in the core, but equally important, beta decays are an additional neutrino source and thus they add to the cooling of the core and a reduction in entropy. This cooling can be quite efficient as often the average neutrino energy in the involved beta decays is larger than for the competing electron captures. As a consequence the new models have significantly lower core temperatures than the WW models after silicon burning. At later stages of the collapse, beta decay becomes unimportant again as an increased electron chemical potential drastically reduces the phase space.
We note that the shell model weak interaction rates predict the presupernova evolution to proceed along a temperature-density-$Y_e$ trajectory where the weak processes are dominated by nuclei rather close to stability. Thus it will be possible, after next generation radioactive ion-beam facilities become operational, to further constrain the shell model calculations by measuring relevant GT distributions for unstable nuclei by charge-exchange reaction, where we like to point out again that the GT$_+$ distribution is also crucial for stellar $\beta$-decays. Figure \[fig:Ye\] identifies those nuclei which dominate (defined by the product of abundance times rate) the electron capture and beta decay during various stages of the final evolution of 15 $M_\odot$ and 25 $M_\odot$ stars. @Heger.Woosley.ea:2001 give an exhaustive list of the most important nuclei for both electron capture and beta decay during the final stages of stellar evolution for stars of different masses.
In total, the weak interaction processes shift the matter composition to smaller $Y_e$ values (see Fig. \[fig:Ye\]) and hence more neutron-rich nuclei, subsequently affecting the nucleosynthesis. Its importance for the elemental abundance distribution, however, strongly depends on the location of the mass cut in the supernova explosion. It is currently assumed that the remnant will have a baryonic mass between the iron core and oxygen shell masses [@Woosley.Heger.Weaver:2002]. As the reduction of $Y_e$ occurs mainly during silicon burning, it is essential to determine how much of this material will be ejected. Another important issue is the possible long-term mixing of material during the explosion [e.g. @Kifonidis.Plewa.ea:2000]. Changes of the elemental abundances due to the improved weak-interaction rates are rather small as the differences, compared to FFN, occur in regions of the star which are probably not ejected (however, for type Ia supernovae, see below). The weak interaction also determines the decay of the newly synthesized nuclei in supernova explosions. Some of them are proton-rich nuclei that decay by orbital electron capture, leaving atomic K-shell electron vacancies. The X-rays emitted can escape the supernova ejecta for sufficiently long-lived isotopes and can possibly be detected by the current generation of X-ray telescopes [@Leising:2001].
In dense stellar environment the electron capture rates have to be corrected for screening effects caused by the relativistically degenerate electron liquid. Such studies have been recently performed within the linear response theory [@Itoh.Tomizawa.ea:2002] who find typical screening corrections of order a few percent.
Collapse and post-bounce stage
==============================
The models, as we have described them above, constitute the so-called presupernova models. They follow the late-stage stellar evolution until core densities just below $10^{10}$ g cm$^{-3}$ and temperatures between 5 and 10 GK. (More precisely, @Woosley.Weaver:1995 define the final presupernova models as the time when the collapse velocity near the edge of the iron core first reached 1000 km s$^{-1}$.) As we have stressed above, stellar evolution until this time requires the consideration of an extensive nuclear network, but is simplified by the fact that neutrinos need only be treated as a source for energy losses. This is no longer valid at later stages of the collapse. As neutrinos will eventually be trapped in the collapsing core and their interaction with the surrounding matter is believed to be crucial for the supernova explosion, computer simulations of the collapse, bounce and explosion necessitate a detailed time- and space-dependent bookkeeping of the various neutrino ($\nu_e, \nu_\mu, \nu_\tau$ neutrinos and their antiparticles) distributions in the core. Built on the pioneering work by @Bruenn:1985, this is done by multi-group (neutrinos of different flavor and energy) Boltzmann neutrino transport . Advantageously, the temperature during the collapse and explosion are high enough that the matter composition is given by nuclear statistical equilibrium without the need of reaction networks for the strong and electromagnetic interaction. The transition from a rather complex global nuclear network, involving many neutron, proton and $\alpha$ fusion reactions and their inverse, to a quasi-statistical equilibrium, in which reactions within mini-cycles are fast enough to bring constrained regions of the nuclear chart into equilibrium, to finally global nuclear statistical equilibrium is extensively discussed by [@Woosley:1986].
Presupernova models are the input for collapse and explosion simulations. Currently, one-dimensional models with sophisticated neutrino transport do not explode , including first attempts with the presupernova models derived with the improved weak-interaction rates discussed above [@Messer.Others:2002]. Explosions can, however, be achieved if the shock revival in the delayed mechanism is modelled by two-dimensional hydrodynamics allowing for more efficient neutrino energy transfer [@Herant.Benz.ea:1994; @Burrows.Hayes.Fryxell:1995; @Janka.Mueller:1996]. Thus the intriguing question arises: Are supernova explosions three-dimensional phenomena requiring convective motion and perhaps rotation and magnetic fields? Or do one-dimensional models fail due to incorrect or insufficient nuclear physics input? Although first steps have been taken in modelling the multi-dimensional effects [for reviews and references see @Janka.Kifonidis.Rampp:2001; @Woosley.Heger.Weaver:2002], these require extremely demanding and computationally challenging simulations. In the following we will briefly discuss some nuclear physics ingredients in the collapse models and their possible improvements.
The crucial weak processes during the collapse are [@Bruenn:1985; @Rampp.Janka:2002; @Burrows:2001]:
\[eq:weakp\] $$\begin{aligned}
p + e^- & \rightleftarrows & n + \nu_e \label{eq:epnnu}\\
n + e^+ & \rightleftarrows & p + \bar{\nu}_e \label{eq:enpnu}\\
(A,Z) + e^- & \rightleftarrows & (A,Z-1) + \nu_e \label{eq:eAnuA}\\
(A,Z) + e^+ & \rightleftarrows & (A,Z+1) + \bar{\nu}_e
\label{eq:posA}\\
\nu + N & \rightleftarrows & \nu + N \label{eq:nuNnuN} \\
N + N & \rightleftarrows & N + N + \nu + \bar{\nu}
\label{eq:NNNNnunu}\\
\nu + (A,Z) & \rightleftarrows & \nu + (A,Z) \label{eq:nuAnuA}\\
\nu + e^\pm & \rightleftarrows & \nu + e^\pm \label{eq:nuenue}\\
\nu + (A,Z) & \rightleftarrows & \nu + (A,Z)^* \label{eq:nuAnuAin}\\
e^+ + e^- & \rightleftarrows & \nu + \bar{\nu} \label{eq:eenunu}\\
(A,Z)^* & \rightleftarrows & (A,Z) + \nu + \bar{\nu}
\label{eq:AAnunu} \end{aligned}$$
Here, a nucleus is symbolized by its mass number $A$ and charge $Z$, $N$ denotes either a neutron or a proton and $\nu$ represents any neutrino or antineutrino. In the early collapse stage, before trapping, these reactions proceed dominantly to the right. We note that, due to the generally accepted collapse picture [e.g. @Bethe:1990], elastic scattering of neutrinos on nuclei is mainly responsible for the trapping, as it determines the diffusion time scale of the outwards streaming neutrinos. Shortly after trapping, the neutrinos are thermalized by the energy downscattering, experienced mainly in inelastic scattering off electrons . The relevant cross sections for these processes are readily derived [@Bruenn:1985]. For elastic neutrino-nucleus scattering one usually makes the simplifying assumption that the nucleus has a $J=0^+$ spin/parity assignment, as appropriate for the ground state of even-even nuclei. The scattering process is then restricted to the Fermi part of the neutral current (pure vector coupling) [@Freedman:1974; @Tubbs.Schramm:1975] and gives rise to coherent scattering; i.e. the cross section scales with $A^2$, except from a correction $\sim (N-Z)/A$ arising from the neutron excess. This assumption is, in principle, not correct for the ground states of odd-$A$ and odd-odd nuclei and for all nuclei at finite temperature, as then $J \ge 0$ and the cross section will also have an axial-vector Gamow-Teller contribution. However, the relevant GT$_0$ strength is not concentrated in one state, but rather fragmented over many nuclear levels. Thus, one can expect that the GT contributions to the elastic neutrino-nucleus cross sections are in general small enough to be neglected.
Reactions and are equally important, as they control the neutronization of the matter and, in a large portion, also the star’s energy losses. Due to their strong phase space sensitivity ($\sim E_e^5$), the electron capture cross sections increase rapidly during the collapse as the density (the electron chemical potential scales like $\sim \rho^{1/3}$) and the temperature increase. We already observed above that beta-decay is rather unimportant during the collapse due to Pauli-blocking of the electron phase space in the final state. We also noted how sensitively the electron capture rate on nuclei depends on a proper description of nuclear structure. As we will discuss now, this is also expected for this stage of the collapse, although the relevant nuclear structure issues are somewhat different.
Electron capture on nuclei
--------------------------
The new presupernova models indicate that electron capture on nuclei will still be important, at least in the early stage of the collapse. Although capture on free protons, compared to nuclei, is favored by the significantly lower Q-value, the number fraction of free protons $Y_p$, i.e. the number of free protons divided by the total number of nucleons, is quite low [$Y_p \sim 10^{-6}$ in the 15 $M_\odot$ presupernova model of @Heger.Woosley.ea:2001]. This tendency had already been observed before, but has been strengthened in the new presupernova models, where the $Y_e$ values are significantly larger and thus the nuclei present in the matter composition are less neutron-rich, implying lower Q-values for electron capture. Furthermore, the entropy is smaller in stars with $\lesssim 20\
M_\odot$, yielding a smaller fraction of free protons.
As the entropy is rather low [@Bethe.Brown.ea:1979], most of the collapsing matter survives in heavy nuclei. However, $Y_e$ decreases during the collapse making the matter composition more neutron-rich, hence energetically favoring increasingly heavy nuclei. In computer studies of the collapse, the ensemble of heavy nuclei is described by one representative which is generally chosen to be the most abundant in the nuclear statistical equilibrium composition. Due to a simulation of the infall phase , such representative nuclei are $^{70}$Zn and $^{88}$Kr at different stages of the collapse [@Mezzacappa.pvt].
In current collapse simulations the treatment of electron capture on nuclei is schematic and rather simplistic. The nuclear structure required to derive the capture rate is then described solely on the basis of an independent particle model for iron range nuclei, i.e., considering only Gamow-Teller transitions from $f_{7/2}$ protons to $f_{5/2}$ neutrons . In particular, this model predicts that electron capture vanishes for nuclei with charge number $Z<40$ and neutron number $N\ge40$, arguing that Gamow-Teller transitions are blocked by the Pauli principle, as all possible final neutron orbitals are already occupied in nuclei with $N\ge40$ (closed $pf$ shell) [@Fuller:1982]. Such a situation would, for example, occur for the two nuclei $^{70}$Zn and $^{88}$Kr with $(Z=30,N=40)$ and $(Z=36,N=52)$, respectively. It has been pointed out [@Cooperstein.Wambach:1984] that this picture is too simple and that the blocking of the GT transitions will be overcome by thermal excitation which either moves protons into $g_{9/2}$ orbitals or removes neutrons from the $pf$ shell, in both ways reallowing GT transitions. In fact, due to this ‘thermal unblocking’, GT transitions again dominate the electron capture on nuclei for temperatures of order 1.5 MeV [@Cooperstein.Wambach:1984]. An even more important unblocking effect, which is already relevant at lower temperatures is, however, expected by the residual interaction which will mix the $g_{9/2}$ (and higher) orbitals with those in the $pf$ shell.
\
\
A consistent calculation of the electron capture rates for nuclei with neutron numbers $N>40$ and proton numbers $20<Z<40$, including configuration mixing and finite temperature, is not yet feasible by direct shell model diagonalization due to the large model spaces and many states involved. It can, however, be performed in a reasonable way adopting a hybrid model: The capture rates are calculated within the RPA approach with partial occupation formalism, including allowed and forbidden transitions. The partial occupation numbers represent an ‘average’ state of the parent nucleus and depend on temperature. They are calculated within the Shell Model Monte Carlo approach at finite temperature [@Koonin.Dean.Langanke:1997] and include an appropriate residual interaction. Exploratory studies, performed for a chain of germanium isotopes ($Z=32$), confirm that the GT transition is not blocked for $N\ge40$ and still dominates the electron capture process for such nuclei at stellar conditions [@Langanke.Kolbe.Dean:2001]. This is demonstrated in figure \[fig:gecapt\], which compares electron capture rates for $^{78}$Ge calculated within the hybrid model with the results in the independent particle model (IPM). For this nucleus ($N=46$) the rate in the IPM is given solely by forbidden transitions (mainly induced by $1^-$ and $2^-$ multipoles). However, correlations and finite temperature unblock the GT transitions in the hybrid model which increases the rate significantly. The differences are particularly important at lower densities (a few $10^{10}$ g cm$^{-3}$) where the electron chemical potential does not suffice to induce forbidden transitions.
We note again that many nuclei are present with similar mass abundances during the supernova collapse phase and that their relative abundances are approximately described by nuclear statistical equilibrium. Figure \[fig:gecapt\] shows the capture rates for several representative nuclei during the collapse phase, identified by the average charge and mass number of the matter composition following the time evolution of a certain ($M=0.6\ M_\odot$) mass trajectory [@Liebendoerfer.Mezzacappa.ea:2001]. (For the conditions shown in figure \[fig:gecapt\] $^{93}$Kr and $^{72}$Zn are examples of representative nuclei at $\rho_{11}=10.063$ and 0.601, respectively). The general trend of the rates reflects the competition of the two main energy scales of the capture process: the electron chemical potential $\mu_e$, which grows like $\rho^{1/3}$ during infall, and the reaction Q-value. As the Q-value is smaller for free protons ($Q=1.29$ MeV) than for neutronrich nuclei ($Q \sim$ few MeV), the capture rate on free protons is larger than for the heavy nuclei. However, this difference diminishes with increasing density. This is expected because the electron energies involved (for example, the electron chemical potential is $\mu_e \sim 18$ MeV at $\rho=10^{11}$ g cm$^{-3}$) are then significantly higher than the $Q$-values for the capture reactions on the abundant nuclei, (i.e. $^{93}$Kr has a $Q$-value of about 11 MeV). As also nuclear structure effects at the relatively high temperatures involved are rather unimportant, the capture rates on the abundant nuclei at the later stage of the collapse are rather similar. However, the capture rate is quite sensitive to the reaction $Q$-value for the lower electron chemical potentials. To quantify this argument we take the point of the stellar trajectory from figure \[fig:gecapt\] with the lowest electron chemical potential ($\mu_e \sim 8$ MeV) as an example. Under these conditions the capture rates on $^{70}$Cu and $^{76}$Ga (both nuclei have $Q$-values around 4 MeV) are noticeably larger than for $^{78}$Ge and $^{72}$Zn with $Q$-values around 8 MeV. However, in nuclear statistical equilibrium the relative mass fraction of $^{72}$Zn (about $1.2 \times 10^{-2}$) is larger than for $^{70}$Cu ($4.0 \times 10^{-3}$) or $^{76}$Ga ($1.8 \times 10^{-3}$). The most abundant nucleus, $^{66}$Ni, has a mass fraction of $4.3 \times
10^{-2}$ and a capture rate comparable to $^{72}$Zn. $^{93}$Kr is too neutron-rich to have a significant abundance at this stage of the collapse. This discussion indicates that the most abundant nuclei are not necessarily the nuclei which dominate the electron capture in the infall phase. Thus, a single-nucleus approximation can be quite inaccurate and should be replaced by an ensemble average.
What matters for the competition of capture on nuclei compared to that on free protons is the product of number abundance times capture rate. Figure \[fig:gecapt\] shows the time evolution of the number abundances for free neutrons, protons, $\alpha$-particles and heavy nuclei, calculated for the same stellar trajectory [obtained from @Liebendoerfer.pvt] for which the capture rates have been evaluated under the assumption of nuclear statistical equilibrium (NSE). (We note that the commonly adopted equations of states [@Lattimer.Swesty:1991; @Shen.Toki.ea:1998a] yield somewhat larger $Y_p$ fractions than obtained in NSE.) Importantly the number abundance of heavy nuclei is significantly larger than that of free protons (by more than two orders of magnitude at the example point discussed above) to compensate for the smaller capture rates on heavy nuclei. It appears thus that electron capture on nuclei cannot be neglected during the collapse. We note that the average energies of the neutrinos produced by capture on nuclei are significantly smaller than for capture on free protons making this process a potentially important source for low-energy neutrinos.
The neutron number $N=40$ is not magic in nuclear structure, nor for stellar electron capture rates. Thus the anticipated strong reduction of the capture rate on nuclei will not occur and we expect capture on nuclei to be an important neutronization process probably until neutrino trapping. The magic neutron number $N=50$ is also no barrier as for nuclei like $^{93}$Kr ($N=57$), the neutron $pf$-shell is nearly completely occupied, but due to correlations protons occupy, for example, the $g_{9/2}$ orbital, and thus unblock GT transitions by allowing transformations into $g_{9/2},g_{7/2}$ neutrons. The description of electron capture on nuclei in the collapse simulations needs to be improved.
In the current simulations the inverse reaction rates of the weak processes listed above are derived by detailed balance. Thus an improved description of electron capture will then also affect the neutrino absorption on nuclei, although this process is strongly suppressed by Pauli blocking in the final state.
Neutrino rates
--------------
![Normalized neutrino spectra for stellar electron capture on the six most important ‘electron-capturing nuclei’ in the presupernova model of a 15 $M_\odot$ star, as identified in [@Heger.Langanke.ea:2001]. The stellar parameters are $T=7.2
\times 10^9$ K, $\rho = 9.1 \times 10^9$ g cm$^{-3}$ and $Y_e=0.43$. The solid lines represent the spectra derived from the shell model electron capture rates. The dashed and dashed-dotted lines correspond to parametrizations recommended by @Langanke.Martinez-Pinedo.Sampaio:2001 and @Bruenn:1985, respectively.\[fig:nusp\]](nuspectrum.eps){width="0.9\linewidth"}
![Differential $^{56}$Fe$(\nu_e,e^-){}^{56}$Co cross section for the KARMEN neutrino spectrum, coming from the decay at rest of the muon, as a function of the excitation energy in $^{56}$Co. The figure shows the allowed contributions, while the inset gives the contribution of the 1$^-$ and 2$^-$ multipolarities.\[fig:fe56nue\]](fe56nue.eps){width="0.9\linewidth"}
In the capture process on nuclei, the electron has to overcome the Q-values of the nuclei and the internal excitation energy of the GT states in the daughter, so the final neutrino energies are noticeably smaller than for capture on free protons. Typical neutrino spectra for a presupernova model are shown in figure \[fig:nusp\]. In this stage of the collapse the neutrino energies are sufficiently small that they only excite allowed transitions. Consequently neutrino-nucleus cross sections for $pf$-shell nuclei can be determined on the basis of GT distributions determined in the shell model. In the later stage of the collapse, the increased density results also in higher energy electrons ($E_e \sim \rho^{1/3}$) which in turn, if captured by protons or nuclei, produce neutrinos with energies larger than 15–20 MeV. For such neutrinos forbidden (mainly $\lambda=1$ dipole and spin-dipole) transitions can significantly contribute to the neutrino-nucleus cross section. Such a situation is shown in figure \[fig:fe56nue\] which shows the differential cross section for the process $^{56}$Fe$(\nu_e,e^-){}^{56}$Co computed using the Shell-Model for the Gamow-Teller contribution and the CRPA for the forbidden contributions [@Kolbe.Langanke.Martinez-Pinedo:1999]. The calculation adopts a neutrino spectrum corresponding to a muon decaying at rest. The average energy, $\bar{E}_\nu\approx 30$ MeV, and momentum transfer, $q \approx 50$ MeV, represent the maximum values for $\nu_e$ neutrinos expected during the supernova collapse phase, i.e., the maximum contribution expected from forbidden transitions to the total neutrino-nucleus cross sections. In this particular case the $1^+$ multipole (Gamow-Teller at the $q=0$ limit) represents 50% of the cross section. The $^{56}$Fe$(\nu_e,e^-){}^{56}$Co cross section for neutrinos from muon-decay at rest has been measured by the Karmen collaboration. The measured cross section ($2.56\pm1.08({\rm
stat})\pm 0.43({\rm syst}) \times 10^{-40}$ cm$^2$) [@Zeitnitz:1994] agrees with the result calculated in the shell model (allowed transitions) plus CRPA (forbidden transitions) approach ($2.38 \times 10^{-40}$ cm$^2$) [@Kolbe.Langanke.Martinez-Pinedo:1999].
Although the most important neutrino reactions during collapse are coherent elastic scattering on nuclei and inelastic scattering off electrons, it has been noted [@Haxton:1988; @Bruenn.Haxton:1991] that neutrino-induced reactions on nuclei can happen as well. Using $^{56}$Fe as a representative nucleus, Bruenn and Haxton concluded that charged-current $(\nu_e,e^-)$ reactions do not have an appreciable effect on the evolution of the core during infall, due to the high-threshold for neutrino absorption. Based on shell model calculations of the GT strength distributions, @Sampaio.Langanke.Martinez-Pinedo:2001 confirmed this finding for other, more relevant nuclei in the core composition. The same authors showed that finite-temperature effects can increase the $(\nu_e,e^-)$ cross sections for low neutrino energies drastically [@Sampaio.Langanke.Martinez-Pinedo:2001]. But this increase is found to be significantly smaller than the reduction of the cross section caused by Pauli blocking of the final phase space, i.e. due to the increasing electron chemical potential. This environmental effect ensures that neutrino absorption on nuclei is unimportant during the collapse compared with inelastic neutrino-electron scattering.
![Inelastic neutrino cross sections for $^{56}$Fe (left) and $^{59}$Co (right) as function of initial neutrino energy and for selected temperatures (upper part). Only allowed Gamow-Teller transitions have been considered. Temperatures are in MeV. For $T=0$, the cross section is calculated for the ground state only. At $T>0$, the cross sections have been evaluated for a thermal ensemble of initial states. The corresponding neutrino energy distribution in the final state is shown in the lower part, assuming an initial neutrino energy of $E_\nu =7.5$ MeV. Due to threshold effects a significant portion of the neutrinos are upscattered in energy for even-even nuclei.\[fig:neut\]](neutral.eps){width="0.9\linewidth"}
@Bruenn.Haxton:1991 observed that inelastic neutrino scattering off nuclei plays the same important role of equilibrating electron neutrinos with matter during infall as neutrino-electron scattering. The influence of finite temperature on inelastic neutrino-nucleus scattering was studied in [@Fuller.Meyer:1991], using an independent particle model. While the study in [@Bruenn.Haxton:1991] was restricted to $^{56}$Fe, additional cross sections have been calculated for inelastic scattering of neutrinos on other nuclei based on modern shell-model GT strength distributions [@Sampaio.Langanke.ea:2002]. Again, for low neutrino energies the cross sections are enhanced at finite temperatures (figure \[fig:neut\]). This is caused by the possibility that, at finite temperatures, the initial nucleus can reside in excited states which can be connected with the ground state by sizable GT matrix elements. These states can then be deexcited in inelastic neutrino scattering. Note that in this case the final neutrino energy is larger than the initial (see figure \[fig:neut\]) so that the deexcitation occurs additionally with larger phase space. Until neutrino trapping there is little phase space blocking in inelastic neutrino-nucleus scattering. @Toivanen.Kolbe.ea:2001 presented the charged- and neutral-current cross sections for neutrino-induced reactions on the iron isotopes $^{52-60}$Fe, using a combination of shell model and RPA approach. Other possible neutrino processes, e.g. nuclear deexcitation by neutrino pair production , have been discussed in [@Fuller.Meyer:1991], but the estimated rates are probably too small for these processes to be important during the collapse.
Finally we remark that coherent elastic scattering on nuclei scales like $\sim E_\nu^2$ so that neutrinos with low energies are the last to be trapped. In order to fill this important sink for entropy and energy, processes which affect the production of neutrinos with low energies can be quite relevant for the collapse. Inelastic neutrino scattering on nuclei, including finite temperature effects, is one such process [@Bruenn.Haxton:1991]. The significantly lower energies of the neutrinos generated by electron capture on nuclei than the ones generated by capture on free protons is another reason to implement these processes with appropriate care in collapse simulations.
Delayed supernova mechanism {#sec:delay-supern-mech}
---------------------------
![Sketch of the stellar core during the shock revival phase. $R_\nu$ is the neutrinosphere radius, from which neutrinos are expected to stream out freely, $R_{ns}$ is the radius of the protoneutron star, $R_g$ the gain radius (see text) and $R_s$ the radius at which the shock is stalled. The shock expansion is impeded by mass infall at a rate ${\dot M}$, but supported by convective energy transport from the region of strongest neutrino heating to the stalled shock. Convection inside the protoneutron star (PNS) as well as correlations in the dense nuclear medium increase the neutrino luminosity [adapted from @Janka.Kifonidis.Rampp:2001]. \[fig:Janka1\]](sncore.eps){width="0.9\linewidth"}
In the delayed supernova mechanism the fate of the explosion is determined by several distinct neutrino processes. When the shock reaches the $\nu_e$ neutrinosphere, from which $\nu_e$ are expected to stream out freely, electron capture on the shock-heated and shock-dissociated matter increases the $\nu_e$ production rate significantly. Additionally neutrinos are produced by the transformation of electron-positron pairs into $\nu \bar{\nu}$ pairs (equation \[eq:eenunu\]). This process is strongly temperature dependent [e.g. @Brown.Soyeur:1979] and occurs most effectively in the shock-heated regions of the proto-neutron star. Electron-positron pair annihilation and nucleon-nucleon bremsstrahlung (equation \[eq:NNNNnunu\]) generate pairs of all three neutrino flavors with the same probability and thus are the main mechanisms for the production of $\nu_\mu$, $\nu_\tau$ neutrinos and antineutrinos . The emitted $\nu_e$ and $\bar{\nu}_e$ neutrinos, however, can be absorbed again by the free nucleons behind the shock. Due to the temperature and density dependences of the neutrino processes involved, neutrino emission wins over neutrino absorption in a region inside a certain radius (the *gain radius*), while outside the gain radius matter is heated by neutrino interactions that are dominated by absorption of electron neutrinos and antineutrinos on free nucleons which have been previously liberated by dissociation due to the shock (see figure \[fig:Janka1\]). As a net effect, neutrinos transport energy across the gain radius to the layers behind the shock. Due to the smaller abundances, neutrino-induced reactions on finite nuclei are expected to contribute only modestly to the shock revival. It has been also suggested that the shock revival is supported by ‘preheating’ [@Haxton:1988]. In this scenario the electron neutrinos, which have been trapped during the final collapse phase and are liberated in a very short burst (with luminosities of a few $10^{53}$ erg s$^{-1}$ lasting for about 10 ms), can partly dissociate the matter (e.g. iron and silicon isotopes) prior to the shock arrival. As reliable neutrino-induced cross sections on nuclei have not been available until recently, the neutrino-nucleus reactions have not been included in collapse and post-bounce simulations.
To describe the important neutrino-nucleon processes, most core collapse simulations use the same lowest order cross section for both neutrinos and antineutrinos [@Bruenn:1985; @Horowitz:2002], i.e., they neglect terms of order $E_\nu/M$, where $E_\nu$ is the neutrino energy and $M$ the nucleon mass. The most important corrections to the cross section at this order are the nucleon recoil and the weak magnetism related to the form factor $F_2$ in Eq. [@Horowitz:2002]. The recoil correction is the same for neutrinos and antineutrinos and decreases the cross sections. However, the weak magnetism corrects the cross sections via its parity-violating interference with the dominant axial-vector component. As the interference is constructive for neutrinos and destructive for antineutrinos, inclusion of the weak magnetism correction increases the neutrino cross section, while it decreases the $\bar{\nu}$-nucleon cross sections. It is then expected that corrections up to order $E_\nu/M$ decrease the antineutrino cross section noticeably (by about $25\%$ for 40 MeV antineutrinos), while the $\nu$-nucleon cross sections are only affected by a few percents for $E_\nu \le 100$ MeV [@Horowitz:2002].
![Shock trajectories of a 20 $M_\odot$ star, calculated with (dashed) and without (solid) an isoscalar strange axialform factor in the neutrino-nucleon elastic cross sections (courtesy of M. Liebendörfer). \[fig:strange\]](NH20.ps){width="0.9\linewidth"}
Neutral-current processes are sensitive to possible strange quark contributions in the nucleon which would give rise to an isoscalar piece $g_A^s$ in the axial-vector form factor besides the standard isovector form factor $g_A \bm{\tau}$ [@Jaffe.Manohar:1990; @Beise.McKeown:1991]. The current knowledge on $g_A^s$ comes from a $\nu p$ elastic scattering experiment performed at Brookhaven yielding $g_A^s=-0.15\pm0.08$ [@Ahrens.Aronson.ea:1987], but is considered rather uncertain [@Garvey.Louis.White:1993]. With $g_A=1.26$ and assuming axial-vector dominance, i.e. the cross section scales like $\sigma
\sim |g_A \bm{\tau} - g_A^s |^2$, a non-vanishing strange axial-vector form factor would reduce the elastic scattering cross section on neutrons and increase the $\nu p$ elastic cross section [@Garvey.Krewald.ea:1992; @Garvey.Kolbe.ea:1993; @Horowitz:2002]. As the matter behind the shock is neutron-rich, the net effect will be a reduction of the neutrino-nucleon elastic cross section. This increases the energy transfer to the stalled shock, however, a simulation has shown that this increase is not strong enough for a successful shock revival [@Liebendoerfer.Messer.ea:2002 see figure \[fig:strange\]].
The physics involved in the attempt to revive the shock by neutrino heating is exhaustively reviewed by @Janka.Kifonidis.Rampp:2001 [see also @Burrows.Goshy:1993]. These authors show that the fate of the stalled shock does not only depend on the neutrino heating above the gain radius, but is also influenced by the energy loss in the cooling region below the gain radius [see also @Bethe.Wilson:1985]. @Janka:2001 also demonstrates the existence of a critical value for the neutrino luminosity from the neutron star needed to revive the shock. This critical luminosity depends on the neutron star mass and radius and on the mass infall to the shock. One expects that the shock expansion is eased for high mass infall rates, which increase the matter pile-up on the neutron star and push the shock outwards, and for high $\nu_e$ and $\bar{\nu}_e$ luminosities from the neutron star, which lead to an enhancement of neutrino absorption relative to neutrino emission in the gain region.
The explosion depends crucially on the effectiveness by which energy is transported by neutrinos to the region where the shock has stalled. As stressed before, one-dimensional models including sophisticated neutrino transport [e.g. @Rampp.Janka:2000; @Liebendoerfer.Mezzacappa.ea:2001] fail to explode. However, the neutrino energy transport is very sensitive to: i) the effect of nucleon-nucleon correlations on the neutrino opacities in dense matter and ii) convection both in the neutrino-heated region and in the proto-neutron star [@Janka.Kifonidis.Rampp:2001].
In his pioneering work, @Sawyer:1989 calculated the neutrino mean free path, or equivalently the neutrino opacity, in uniform nuclear matter and showed that effects due to strong interaction between nucleons are important. The same conclusion has been reached by @Raffelt.Seckel.Sigl:1996 who demonstrated that the average neutrino-nucleon cross section in the medium is reduced due to spin fluctuations induced by the spin-dependent interaction among nucleons. [For earlier calculations of the neutrino mean free path in uniform nuclear matter, see @Sawyer:1975; @Friman.Maxwell:1979; @Iwamoto.Pethick:1982]. @Sawyer:1989 exploited the relation between the equation of state (EOS) of the matter and the long-wavelength excitations of the system to calculate the weak interaction rates. However, consistency between the EOS and the neutrino opacities are more difficult to achieve for large energy ($q_0$) and momentum ($q$) transfer of the neutrinos. Here, particle-hole and particle-particle interactions are examples of effects which might influence the EOS and the neutrino opacities. For the following discussion it is quite illuminating to realize the similarity of the neutrino-induced excitations of nuclear matter with the physics of multipole giant resonances in finite nuclei.
For muon and tau neutrinos, neutral current reactions are the only source of opacities. Here, the energy and momentum transfer is limited by the matter temperature alone. For electron neutrinos the mean free path is dominated by charged-current reactions, for which the energy transfer is typically of the order of the difference between neutron and proton chemical potentials. During the early deleptonization epoch of the proto-neutron star the typical neutron momenta are large ($\sim 100$–200 MeV) and the mismatch of proton, neutron and electron Fermi momenta can be overcome by the neutrino momenta. This is not longer possible in later stages when the neutrino energies are of order $k_b T$; then momentum conservation restricts the available phase space for the absorption reaction. Pauli blocking of the lepton in the final state increases the mean free path for charged-current and neutral-current reactions.
We note an important and quite general consequence of the fact that muon and tau neutrinos react with the proto-neutron star matter only by neutral-current reactions: The four neutrino types have similar spectra. Due to universality, $\nu_\mu$, $\nu_\tau$ and $\bar{\nu}_\mu$, $\bar{\nu}_\tau$ have identical spectra. It is usually even assumed that neutrinos and antineutrinos have the same spectra (one therefore refers to the 4 neutrino types unifyingly as $\nu_x$ neutrinos) exploiting axial-vector dominance in the neutrino cross sections. However, the interference of the axial-vector and the weak magnetism components makes the $\bar{\nu}_x$ spectra slightly hotter than the $\nu_x$ spectra. The $\nu_x$ neutrinos decouple deepest in the star, i.e., at a higher temperature, than electron neutrinos and antineutrinos, and hence have higher energies. As the matter in the proto-neutron star is neutron-rich, electron neutrinos, which are absorbed by neutrons, decouple at a larger radius than their antiparticles, which interact with protons by charged-current reactions. As a consequence decoupled electron neutrinos have, on average, smaller energies than electron antineutrinos. Calculated supernova neutrino spectra can be found in [@Janka.Hillebrandt:1989; @Yamada.Janka.ea:1999], which yield the average energies of the various supernova neutrinos approximately as: $\langle E_{\nu_e} \rangle =11$ MeV, $\langle E_{\bar{\nu}_e} \rangle
=16$ MeV, and $\langle E_{\nu_x} \rangle =25$ MeV. @Burrows.Young.ea:2000 find the same hierarchy, but somewhat smaller average neutrino energies.
For a much deeper and detailed description of the neutrino mean free paths in dense matter the reader is refered to [@Reddy.Prakash.ea:1999; @Prakash.Lattimer.ea:2001] and the earlier work [@Reddy.Prakash.Lattimer:1998; @Burrows.Sawyer:1998; @Burrows.Sawyer:1999]. We will here only briefly summarize the essence of the work presented in these references.
![Ratio of neutrino mean free paths in neutron matter calculated in RPA and Hartree-Fock approaches at various temperatures [@Margueron.Navarro.ea:2002]. The interaction is the Gogny force D1P. The neutrino energy is taken as $E_\nu=3T$.\[fig:nufree\]](vangiai2.eps){width="0.9\linewidth"}
Collapse simulations describe neutrino opacities typically on the mean-field level or even by a nucleon gas. Then an analytical expression can be derived for the vector and axial-vector response of the medium which in turn determines the charged- and neutral-current cross sections. Effects due to the strong interaction between nucleons are considered by a medium-dependent effective mass in the dispersion relation. Like in finite nuclei, collective excitations in nuclear matter arise due to nucleon-nucleon correlations beyond the mean-field approximation. As it is believed that single-pair excitations dominate over multi-pair excitations for the kinematics of interest to neutrino scattering and absorption, it appears to be sufficient to determine the vector and axial-vector response, in a first step, within the Random Phase Approximation (RPA). Assuming that the interaction is short-ranged compared to the wavelength of the excitations, it is justified to retain only s-wave components in the interaction which in turn can be related to Fermi-liquid parameters. It is found that the repulsive nature of the parameter $G_0'$, which is related to the isovector spin-flip or giant Gamow-Teller resonances in nuclei, induces a collective state in the region $\omega/q \sim
v_F$ ($v_F$ is the Fermi velocity), while the cross section is reduced at smaller energies. However, these smaller energies are important for the neutrino mean free path at nuclear matter densities ($\rho_0$) or smaller densities. Assuming a typical neutrino energy $E_\nu
\approx 3 T$ (corresponding to a Fermi-Dirac distribution with temperature $T$ and zero chemical potential) RPA correlations increase the neutral-current neutrino mean-free path (see figure \[fig:nufree\]) at low temperatures and for $\rho = \rho_0$, compared with the mean-field result. An enhancement due to RPA correlations is also found for neutrino absorption mean-free paths for neutrino-trapped matter. Like in the case of neutrino-induced reactions on finite nuclei (see above), finite-temperature effects allow that nuclear excitation energy is transferred to the neutrino in inelastic scattering processes. This contributes to the cooling of the nuclear matter and increases the neutrino energy in the final state.
Neutrino heating is maximal in the layer just above the gain radius. The energy transport from this region to the shock, which is stalled further out, can be supported by convective overturn and might lead to successful explosions, as has been demonstrated in several simulations with two-dimensional hydrodynamics treatment of the region between the gain radius and the shock . The effect of convection is twofold [@Janka.Kifonidis.Rampp:2001]: At first, heated matter is transported outwards to cooler regions where the energy loss due to neutrino emission is reduced (the neutrino production rate for electron and positron captures on nucleons depends strongly on temperature). Second, cooler matter is brought down closer to the gain radius where the neutrino fluxes are larger and hence the heating is more effective. While this picture is certainly appealing, it is not yet clear whether multi-dimensional simulations will indeed lead to explosions as the two-dimensional studies did not include state-of-the-art Boltzmann neutrino transport, but treated neutrino transport in an approximate manner. In fact, in a simulation with an improved treatment of neutrino transport the convective overturn was found not to be strong enough to revive the stalled shock [@Mezzacappa.Calder.ea:1998].
The shock revival can also be supported by convection occuring inside the protoneutron star where it is mainly driven by the negative lepton gradient which is established by the rapid loss of leptons in the region around the neutrinosphere [@Burrows:1987; @Keil.Janka.Mueller:1996]. By this mode, lepton-rich matter will be transported from inside the protoneutron star to the neutrinosphere which increases the neutrino luminosity and thus is expected to help the explosion. The simulation of protoneutron star convection is complicated by the fact that neutrinos and matter are strongly coupled. In fact, two-dimensional simulations found that neutrino transport can equilibrate otherwise convective fluid elements [@Mezzacappa.Calder.ea:1998]. Such a damping is possible in regions where neutrinos are still strongly coupled to matter, but neutrino opacities are not too high to make neutrino transport insufficient. In the model of @Keil.Janka.Mueller:1996 the convective mixing occurs very deep inside the core where the neutrino opacities are high; no damping of the convection by neutrinos is then found.
Wilson and Mayle attempted to simulate convection in their spherical model by introducing neutron-fingers and found successful explosions [@Wilson.Mayle:1993]. This idea is based on the assumption that energy transport (by three neutrino flavors) is more efficient than lepton number transport (only by electron neutrinos). However, this assumption is under debate [@Bruenn.Dineva:1996].
Nucleosynthesis beyond iron
===========================
While the elements lighter than mass number $A\sim60$ are made by charged-particle fusion reactions, the heavier nuclei are made by neutron captures, which have to compete with $\beta$ decays. Already [@Burbidge.Burbidge.ea:1957; @Cameron:1957] realized that two distinct processes are required to make the heavier elements. This is the slow neutron-capture process (s-process), for which the $\beta$ lifetimes $\tau_\beta$ are shorter than the competing neutron capture times $\tau_n$. This requirement ensures that the s-process runs through nuclei in the valley of stability. The rapid neutron-capture process (r-process) requires $\tau_n \ll \tau_\beta$. This is achieved in extremely neutron-rich environment, as $\tau_n$ is inversely proportional to the neutron density of the environment. The r-process runs through very neutron-rich, unstable nuclei, which are far-off stability and whose physical properties are often experimentally unknown.
Weak-interaction processes play interesting, but different roles in these processes. The half-lives of the $\beta$-unstable nuclei along the s-process path are usually known with good precision. However, the nuclear half-life in the stellar environment can change due to the thermal population of excited states in the parent nucleus. This is particularly interesting if the effective lifetime is then comparable to $\tau_n$, leading to branchings in the s-process path, from which the temperature and neutron density of the environment can be determined. Several recent examples are discussed in the next subsection. For the r-process, $\beta$-decays are probably even more crucial. They regulate the flow to larger charge numbers and determine the resulting abundance pattern and duration of the process. Except for a few key nuclei, $\beta$-decays of r-process nuclei have to be modelled theoretically; we will briefly summarize the recent progress below. Although the r-process site is not yet fully determined, it is conceivable that it occurs in the presence of extreme neutrino fluxes. As we will discuss, neutrino-nucleus reactions can have interesting effects during and after the r-process, perhaps allowing for clues to ultimately identify the r-process site.
S-process
---------
The analysis of the solar abundances have indicated that two components of the s-process have contributed to the synthesis of elements heavier than $A\sim60$. The weak component produces the elements with $A\lesssim 90$. Its site is related with helium core burning of CNO material in more massive stars [@Couch.Schmiedekamp.Arnett:1974; @Kaeppeler.Wiescher.ea:1994]. The main component, which is responsible for the heavier s-process nuclides up to Pb and Bi, is associated with helium flashes occurring during shell burning in low-mass (asymptotic giant branch) stars [@Busso.Gallino.Wasserburg:1999]. The $^{22}$Ne($\alpha,n)^{25}$Mg and $^{13}$C($\alpha,n)^{16}$O reactions are believed to be the supplier of neutrons for the weak and main components, respectively. The s-process abundances $N_s$ are found to be inversely proportional to the respective (temperature averaged) neutron capture cross sections $\langle \sigma \rangle$, as expected for a steady-flow picture [@Burbidge.Burbidge.ea:1957], which, however, breaks down for the extemely small cross sections at the magic neutron numbers. As a consequence, the product $N_s \cdot \langle \sigma \rangle$ exhibits almost constant plateaus between the magic neutron numbers, separated by pronounced steps, [e.g. @Kaeppeler:1999].
![The s-process reaction path in the Nd-Pm-Sm region with the branchings at $A=147$, 148, and 149. Note that $^{148}$Sm and $^{150}$Sm are shielded against r-process $\beta$-decays [adapted from @Kaeppeler:1999].\[fig:ssm\]](sm150.eps){width="0.9\linewidth"}
The neutron density of the stellar environment during the main s-process component can be determined from branching points occuring, for example, in the $A=147$–149 mass region (see figure \[fig:ssm\]). Here the relative abundances of the two s-process-only isotopes $^{148}$Sm and $^{150}$Sm ($Z=62$), which are shielded against r-process contributions by the two stable Nd ($Z=60$) isotopes $^{148}$Nd and $^{150}$Nd, is strongly affected by branchings occuring at $^{147}$Nd and, more importantly, at $^{148}$Pm and at $^{147}$Pm. As the neutron captures on these branching nuclei will bypass $^{148}$Sm in the flow pattern, the $^{150}$Sm $N_s \langle
\sigma \rangle$ value will be larger for this nucleus than for $^{148}$Sm. Furthermore, the neutron capture rate $\lambda_n$ is proportional to the neutron density $N_n$. Thus, $N_n$ can be determined from the relative $^{150}$Sm/$^{148}$Sm abundances, resulting in $N_n = (4.1 \pm 0.6) \times 10^8$ cm$^{-3}$ [@Toukan.Debus.ea:1995]. A similar analysis for the weak s-process component yields neutron densities of order $(0.5\text{--}1.3) \times
10^8$ cm$^{-3}$ [@Walter.Beer.ea:1986a; @Walter.Beer.ea:1986b]. We stress that a 10% determination of the neutron density requires the knowledge of the involved neutron capture cross sections with about 1% accuracy [@Kaeppeler.Thielemann.Wiescher:1998], which has yet not been achieved for unstable nuclei. Improvements are expected from new time-of-flight facilities like LANSCE at Los Alamos or NTOF at CERN.
![The s-process neutron capture path in the Yb-Lu-Hf region (solid lines). For $^{176}$Lu the ground state and isomer are shown separately. Note that $^{176}$Lu and $^{176}$Hf are shielded against r-process $\beta$-decays [adapted from @Doll.Boerner.ea:1999].\[fig:slu176\]](lu176.eps){width="0.9\linewidth"}
![The figure shows an schematic energy-level diagram of $^{180}$Ta and its daughters illustrating the possibility for thermally enhanced decay of $^{180}$Ta$^m$ in the stellar environment of the s-process. The inset shows the photon density at s-process temperature [adapted from @Belic.Arlandini.ea:1999]. \[fig:mediate\]](ta180.eps){width="0.9\linewidth"}
The temperature of the s-process environment can be ‘measured’, if the $\beta$ half-life of a branching point nucleus is very sensitive to the thermal population of excited nuclear levels. A prominent example is $^{176}$Lu [@Beer.Kaeppeler.ea:1981]. For mass number $A=176$, the $\beta$-decays from the r-process terminate at $^{176}$Yb ($Z=70$), making $^{176}$Hf ($Z=72$) and $^{176}$Lu ($Z=71$) s-only nuclides (see figure \[fig:slu176\]). Besides the long-lived ground state ($t_{1/2}=4.00(22) \times 10^{10}$ y), $^{176}$Lu has an isomeric state at an excitation energy of 123 keV ($t_{1/2}=3.664(19)$ h). Both states can be populated by $^{175}$Lu$(n,\gamma)$ with known partial cross sections. At $^{176}$Lu, the s-process matter flow is determined by the competition of neutron capture on the ground state and $\beta$-decay of the isomer. But importantly, the ground and isomeric states couple in the stellar photon bath via the excitation of an intermediate state at 838 keV (figure \[fig:mediate\] shows a similar process taking place in $^{180}$Ta), leading to a matter flow from the isomer to the ground state, which is very temperature-dependent. A recent analysis of the $^{176}$Lu s-process branching yields an environment temperature of $T=(2.5\text{--}3.5)
\times 10^8$ K [@Doll.Boerner.ea:1999].
Similar finite-temperature effects play also an important role in the s-process production of $^{180}$Ta. This is the rarest isotope (0.012%) of nature’s rarest element. It only exists in a long-lived isomer ($J^\pi=9^-$) at an excitation energy of 75.3 keV and with a half-life of $t_{1/2} \ge 1.2 \times 10^{15}$ y. The $1^+$ ground state decays with a half-life of 8.152(6) h, mainly by electron capture to $^{180}$Hf. While potential s-process production paths of the $^{180}$Ta isomer have been pointed out, the survival of this state in a finite-temperature environment has long been questionable. While a direct electromagnetic decay to the ground state is strongly suppressed due to angular momentum mismatch, the isomer can decay via thermal population of intermediate states with branchings to the ground state (see figure \[fig:mediate\]). By measuring the relevant electromagnetic coupling strength, the temperature-dependent half-life, and thus the $^{180}$Ta survival rate, has been determined by @Belic.Arlandini.ea:1999 under s-process conditions as $t_{1/2} \lesssim 1$ y, i.e. more than 15 orders of magnitude smaller than the half-life of the isomer! Accompanied by progress in stellar modeling of the convective modes during the main s-process component (which brings freshly produced $^{180}$Ta to cooler zones, where it can survive more easily, on timescales of days) it appears now likely that $^{180}$Ta is partly made within s-process nucleosynthesis [@Wisshak.Voss.ea:2001]. The p-process [@Rayet.Arnould.ea:1995] and neutrino nucleosynthesis [@Woosley.Hartmann.ea:1990] have been proposed as alternative sites for the $^{180}$Ta production.
The two neighboring isotopes $^{186}$Os and $^{187}$Os are s-process only nuclides, shielded against the r-process by $^{186}$W and $^{187}$Rh. The nucleus $^{187}$Rh has a half-life of 42 Gy which is comparable with the age of the universe. As the $^{187}$Rh decay has contributed to the observed $^{187}$Os abundance, the Os/Rh abundance ratio can serve as a sensitive clock for the age of the universe, once the s-process component is subtracted from the $^{187}$Os abundance [@Clayton:1964]. To determine the latter precise measurements of the neutron capture cross sections on $^{186}$Os and $^{187}$Os are required, which are in progress at CERN’s NTOF facility. A potential complication arises from the fact that $^{187}$Os has a low-lying state at 9.8 keV which is populated equally with the ground state at s-process temperatures. The respective neutron capture cross sections on the excited state can be indirectly determined from (n,n’) measurements. Furthermore one has to consider that the half-life of $^{187}$Rh is strongly temperature-dependent: If stripped of its electrons, the half-life is reduced by 9 orders of magnitude to $t_{1/2}=32.9\pm2$ y as measured at the GSI storage ring [@Bosch.Faestermann.ea:1996]. However, the GSI data can be translated into a $\log ft$ value from which the $^{187}$Rh half-life can be deduced for every ionization state.
The decay of totally ionized $^{187}$Rh is an example for a bound state $\beta$-decay, i.e., the decay of bare $^{187}$Rh$^{75+}$ to continuum states of $^{187}$Os$^{76+}$ is energetically forbidden, but it is possible if the decay electron is captured in the K-shell ($Q_\beta$=73 keV) or in the L-shell ($Q_\beta$=9.1 keV) [@Johnson.Soff:1985]. We note that the decay of neutral $^{187}$Rh is energetically allowed for a first-forbidden transition to the $^{187}$Os ground state; the long half-life results from the small matrix element and $Q_\beta$ value related to this transition. Another example of a bound state $\beta$-decay with importance for the s-process occurs for $^{163}$Dy. This nucleus is stable as a neutral atom, but, if fully ionized, can decay to $^{163}$Ho with a half-life of $47^{+5}_{-4}$ d; the measurement of this half-life at the GSI storage ring was the first observation of bound state $\beta$-decay [@Jung.Bosch.ea:1992]. The consideration of the $^{163}$Dy decay at s-process conditions has been found to be essential to explain the abundance of the s-process only nuclide $^{164}$Er, which is produced by neutron capture on $^{163}$Ho, the daughter of the bound state $\beta$-decay of $^{163}$Dy, and the subsequent $\beta$-decay of $^{164}$Ho to $^{164}$Er.
R-process
---------
Phenomenological parameter studies indicate that the r-process occurs at temperatures around $T \sim 100$ keV and at extreme neutron fluxes (neutron number densities $n> 10^{20}$ cm$^{-3}$) [e.g. @Cowan.Thielemann.Truran:1991]. It has also been demonstrated that not all r-process nuclides can be made simultaneously at the same astrophysical conditions (constant temperature, neutron density), i.e. the r-process is a dynamical process with changing (different) conditions and paths [@Kratz.Bitouzet.ea:1993]. In a good approximation, the neutron captures proceed in $(n,\gamma) \rightleftarrows (\gamma,n)$ equilibrium, fixing the reaction paths at rather low neutron separation energies of $S_n \sim 2$–3 MeV [@Cowan.Thielemann.Truran:1991], implying paths through very neutron-rich nuclei in the nuclear chart as is shown in figure \[fig:pfad\]. While the general picture of the r-process appears to be well accepted, its astrophysical site is still open. The extreme neutron fluxes point to explosive scenarios and, in fact, the neutrino-driven wind above the nascent neutron star in a core-collapse supernova is the currently favored model . But also shock-processed helium shells in type II supernovae [@Truran.Cowan.Fields:2001] and neutron star mergers [@Freiburghaus.Rosswog.Thielemann:1999; @Rosswog.Davies.ea:2000] are investigated as possible r-process sites. Recent meteoritic clues [@Wasserburg.Busso.Gallino:1996; @Qian.Vogel.Wasserburg:1998a] as well as observations of r-process abundances in low-metallicity stars [@Sneden.Cowan.ea:2000] point to more than one distinct site for the solar r-process nuclides.
In an important astronomical observation @Cayrel.Hill.ea:2001 have recently detected the r-process nuclides thorium and uranium in an old galactical halo star. As these two nuclei have halflives, which are comparable to the expected age of the universe, their measured abundance ratio serves as a sensitive clock to determine a lower limit to this age, provided their initial r-process production abundance ratio can be calculated with sufficient accuracy. @Cayrel.Hill.ea:2001 have used their observed $^{238}$U/$^{232}$Th abundance ratio from the star CS31082-001 and the r-process model predictions from [@Cowan.Pfeiffer.ea:1999; @Goriely.Clerbaux:1999] to deduce a value of $12.5\pm3.3$ Gyr for the age of the star. Recently, a refined analysis of the CS31082-001 spectra has led to a significant improvement in the derived abundances which provides now an age estimate of $14.0\pm2.4$ Gyr [@Hill.Plez.ea:2002]. The effect of different nuclear physics input on the r-process production of U and Th have been studied by @Goriely.Arnould:2001 and by @Schatz.Toenjes.ea:2002
![The r-process occurs under dynamically changing astrophysically conditions which affect the reaction pathway. The figure shows the range of r-process paths, defined by their waiting point nuclei. After decay to stability the abundance of the r-process progenitors produce the observed solar r-process abundance distribution. The r-process paths run generally through neutronrich nuclei with experimentally unknown masses and halflives. In this calculation a mass formula based on the Extended Thomas Fermi model with Strutinski Integral (ETFSI) and special treatment of shell quenching (see text) has been adopted. (courtesy of Karl-Ludwig Kratz and Hendrik Schatz). \[fig:pfad\]](rprocess.eps){width="0.9\linewidth"}
As relevant nuclear input, r-process simulations require neutron separation energies (i.e. masses), half-lives and neutron capture cross sections of the very neutron-rich nuclei on the various dynamical r-process paths. As currently only few experimental data are known for r-process nuclei, these quantities have to be modelled. Traditionally this has been done by global models which, by fitting a certain set of parameters to known experimental data, are then being used to predict the properties of all nuclei in the nuclear landscape. Arguably the most important input to r-process simulations are neutron separation energies as they determine, for given temperature and neutron density of the astrophysical environment, the r-process paths. The most commonly used mass tabulations are based on the microscopic-macroscopic finite-range droplet model (FRDM) approach [@Moeller.Nix.Kratz:1997] or the Extended Thomas-Fermi model with Strutinski Integral (ETFSI) approach [@Aboussir.Pearson.ea:1995]. In more recent developments mass tabulations have been developed adopting parametrizations inspired by shell model results [@Duflo.Zuker:1995] or calculated on the basis of nuclear many-body theories like the Hartree-Fock model with a BCS treatment of pairing [@Goriely.Tondeur.Pearson:2001]. Special attention has been paid recently also to the ‘shell quenching’, i.e. the observations made in HFB calculations that the shell gap at magic neutron numbers is less pronounced in very neutron-rich nuclei than in nuclei close to stability [@Dobaczewski.Hamamoto.ea:1994]. Such a vanishing of the shell gap has been experimentally verified for the magic neutron number $N=20$ [@Guillemaud-Mueller.Detraz.ea:1984; @Motobayashi.Ikeda.ea:1995]. The confirmation of the predicted quenching at the $N=82$ shell closure is the aim of considerable current experimental activities [@Kratz.Moeller.ea:2000]. Recent theoretical studies on this topic are reported in [@Sharma.Farhan:2001; @Sharma.Farhan:2002]. The potential importance of shell quenching for the r-process rests on the observation [@Chen.Dobaczewski.ea:1995; @Pfeiffer.Kratz.Thielemann:1997] that it can correct the strong trough just below the r-process peaks in the calculated r-process abundances encountered with conventional mass models. Neutron capture cross sections become important, if the r-process flow drops out of $(n,\gamma) \rightleftarrows (\gamma,n)$ equilibrium, which happens close to freeze-out when the neutron source ceases. They can also be relevant for nuclides with small abundances for which no flow equilibrium is built up [@Surman.Engel.ea:1997; @Surman.Engel:2001]. In the following we will summarize recent progress in calculating half-lives for nuclei on the r-process paths.
### Half-Lives
The nuclear half-lives determine the relative r-process abundances. In a simple $\beta$-flow equilibrium picture the elemental abundance is proportional to the half-life, with some corrections for $\beta$-delayed neutron emission [@Kratz.Thielemann.ea:1988]. As r-process half-lives are longest for the magic nuclei, these waiting point nuclei determine the minimal r-process duration time; i.e. the time needed to build up the r-process peak around $A\sim 200$ via matter-flow from the seed nucleus. We note, however, that this time depends also crucially on the r-process path.
Pioneering experiments to measure half-lives of neutron-rich isotopes near the r-process path succeeded in determining the half-lives of two $N=50$ ($^{80}$Zn, $^{79}$Cu) and two $N=82$ ($^{129}$Ag, $^{130}$Cd) waiting point nuclei . @Pfeiffer.Kratz.ea:2001 reviewed the experimental information on r-process nuclei. These data play crucial roles in constraining and testing nuclear models, which are still necessary to predict the bulk of half-lives required in r-process simulations. It is generally assumed that the half-lives are determined by allowed Gamow-Teller (GT) transitions. The calculations of $\beta$-decays require usually two ingredients: the GT strength distribution in the daughter nucleus and the relative energy scale between parent and daughter, i.e. their mass difference. However, the $\beta$ decays only probe the weak low-energy tail of the GT distributions. Only a few percent of the $3(N-Z)$ Ikeda sum rule [@Ikeda.Fujii.Fujita:1963] lie within the $Q_\beta$ window (i.e. at energies accessible in $\beta$-decay), the rest being located in the region of the Gamow-Teller resonance at higher excitation energies. Due to the strong $E^5$ energy dependence of the phase space $\beta$-decay rates are very sensitive to the correct description of the detailed low-energy GT distribution and its relative energy scale to the parent nucleus. This explains why different calculations of the $\beta$-decay half-lives present large deviations among them.
{width="0.45\linewidth"} {width="0.45\linewidth"}\
{width="0.45\linewidth"} {width="0.45\linewidth"}
Because of the huge number of nuclei relevant for the r-process, the estimates of the half-lives are so far based on a combination of global mass models and the quasiparticle random phase approximation (QRPA), the latter to calculate the GT matrix elements. Examples of these models are the FRDM/QRPA [@Moeller.Nix.Kratz:1997] and the ETFSI/QRPA [@Borzov.Goriely:2000]. Recently, calculations based on the self-consistent Hartree-Fock-Bogoliubov plus QRPA model became available for r-process waiting point nuclei with magic neutron numbers $N=50$, 82 and 126 [@Engel.Bender.ea:1999]. The presence of a closed neutron shell has allowed also for the study of these nuclei by shell-model calculations [@Martinez-Pinedo.Langanke:1999; @Martinez-Pinedo:2001], which is the method of choice to determine the $\beta$-decay matrix elements. However an adequate calculation of the nuclear masses for heavy nuclei is yet prohibitive in the shell model. In @Martinez-Pinedo.Langanke:1999, and @Martinez-Pinedo:2001 the masses were adopted from the global model of @Duflo.Zuker:1995. Figure \[fig:n82n126\] compares the half-lives predicted by the different approaches. For $N=82$, the half-lives of $^{131}$In, $^{130}$Cd and $^{129}$Ag are known experimentally [@Kratz.Gabelmann.ea:1986; @Pfeiffer.Kratz.ea:2001]. The comparison of the predictions of the different models with the few experimental r-process benchmarks reveals some of their insufficiencies. For example, the FRDM/QRPA half-lives show a significant odd-even staggering which is not present in the data, while the ETFSI/QRPA half-lives appear globally too long. The HFB/QRPA and SM approaches obtain half-lives in reasonable agreement with the data and predict shorter half-lives for the unmeasured waiting point nuclei with $N=82$ than the global FRDM/QRPA and ETFSI/QRPA approaches. For $N=126$ there is no experimental information so the different models remain untested. While HFB calculations for the half-lives of all the r-process nuclei are conceivable, similar calculations within the shell-model approach are still not feasible due to computer memory limitations.
While the $Q_\beta$ value for the decay of neutron-rich r-process nuclei is large, the neutron separation energies are small. Hence $\beta$ decay can lead to final states above neutron threshold and is accompanied by neutron emission. If the r-process proceeds by an $(n,\gamma)\rightleftarrows(\gamma,n)$ equilibrium the $\beta$-delayed neutron emission probabilities, $P_{\beta,n}$, only play a role at the end of the r-process when the neutron source has ceased and the produced nuclei decay to stability. The calculated $P_{\beta,n}$ values are very sensitive to both the low-energy GT distribution and the neutron threshold energies. No model describes currently both quantities simultaneously with sufficient accuracy. Figure \[fig:n82n126\] compares the $P_{\beta,n}$ computed in the FRDM/QRPA and SM approaches for $N=82$ and 126. The predictions between different models can be quite different in experimentally non-determined mass regions. For the SM approach the error bars indicate the sensitivity of the computed $P_{\beta,n}$ values to a change of $\pm 0.5$ MeV in the neutron separation energies of the daughter nucleus; the effect can be large.
It has been pointed out that first-forbidden transitions might have important contributions in some nuclei close to magic numbers [@Blomqvist.Kerek.Fogelberg:1983; @Homma.Bender.ea:1996; @Korgul.Mach.ea:2001]. A systematic inclusion of first-forbidden transitions in the calculation of r-process beta-decay half-lives in any of the many-body methods used to describe the GT contributions has not been done. However, a first attempt towards this goal [@Moeller.Pfeiffer.Kratz:2002] has combined first-forbidden transitions estimated in the Gross Theory [@Takahashi.Yamada.Kondoh:1973] with GT results taken from QRPA calculations. No significant changes compared to r-process studies, which consider only the GT contributions to the halflives, have been observed [@Kratz.pvt].
The presence of low-lying isomeric states in r-process nuclei might change the effective half-lives in the stellar environment. Currently no estimates of these effect exists except for odd-A nuclei with $N=82$. Here shell-model calculations predict half-lives for the isomeric states very similar to the ground state half-lives [@Martinez-Pinedo.Langanke:1999]. The halflives of the isomeric state in $^{129}$Ag has also been estimated within the QRPA approach and by a second shell model calculation finding values about a factor of 2 larger than the $^{129}$Ag ground state half-life [@Kratz:2001]. This reference also reports about the first attempt to measure the half-life of the isomeric state.
For heavy nuclei ($Z \ge 84$) some final states populated by $\beta$-decay in the daughter nucleus can also decay by fission [@Cowan.Thielemann.Truran:1991]. The relevant beta-delayed fission probabilities depend sensitively on the modelling of the fission barriers [@Howard.Moeller:1980; @Mamdouh.Pearson.ea:2001].
![Abundances of r-process nuclides calculated in a dynamical r-process model and for different global sets of $\beta$-decay half-lives. In the dynamical r-process the matter flow timescale competes with the nuclear timescale, set by the $\beta$-decay half-lives. As a consequence the magic neutron numbers (here $N=82$) are reached at different astrophysical conditions and hence at different proton numbers, which is reflected in the shifts of the abundance peaks [from @Borzov.Goriely:2000]. \[fig:gor2\]](goriely2.eps){width="0.9\linewidth"}
Borzov and Goriely have studied the influence of the $\beta$ half-lives on the r-process abundances within two distinct scenarios: the canonical r-process picture with a exposure of the seed nucleus $^{56}$Fe by a constant neutron density and temperature for a fixed duration time (2.4 s) and the neutrino-driven wind model. In the canonical model the location of the r-process abundance peaks depends on the masses, but not on the $\beta$ half-lives, which act only as bottlenecks for the matter-flow to more massive nuclei. In the dynamical neutrino-driven wind model the half-lives affect the abundance distribution. This comes about as at later times in this model the environmental conditions shift the r-process to more neutron-rich nuclei. Long half-lives then imply that the matter-flow reaches the magic neutron numbers later, i.e. for more neutron-rich nuclei. Consequently the abundance peaks are shifted to smaller $A$-values [see figure \[fig:gor2\], @Borzov.Goriely:2000]. Similar studies have been presented by @Kratz.Pfeiffer.Thielemann:1998.
### The possible role of neutrinos in the r-process
Among the various possible astrophysical sites for the r-process, the neutrino-driven wind model has attracted most attention in recent years. Here it is assumed that the r-process occurs in the layers heated by neutrino emission and evaporating from the hot protoneutron star after core collapse in a type II supernova [@Thompson.Burrows.Meyer:2001]. Adopting the parameters of a supernova simulation by Wilson [@Wilson:1985] Woosley and collaborators obtained quite satisfying agreement between an r-process simulation and observation [@Woosley.Wilson.ea:1994]. In the classical picture the r-process nuclides are made by successive capture of neutrons, starting from a seed nucleus with mass number $A_{\text{seed}}$. Thus, to make the third r-process peak around $A \sim
200$ requires a large neutron-to-seed ratio of $n/s \sim 200 -
A_{\text{seed}}$. In the Wilson supernova models [@Wilson:1985; @Wilson:2001] this is achieved due to a high entropy found in the neutrino wind at late times (a few seconds after the bounce). However, other models with a different equation of state and treatment of diffusion do not obtain such high entropies; in these models the r-process fails to make the $A=200$ peak. To explain the strong sensitivity of the r-process nucleosynthesis on the entropy of the environment @Qian:1997 noted that the slowest reaction in the nuclear network, which transforms protons, neutrons and $\alpha$ particles into r-process seed nuclei, is the 3-body $\alpha+\alpha+n
\rightarrow {}^9$Be reaction. Due to its low binding energy ($E_b =
1.57$ MeV), $^9$Be can be easily destroyed in a hot thermal environment and thus the matter flow to nuclei heavier than $^9$Be depends strongly on the entropy of the surrounding. The larger the entropy, the smaller the abundance of surviving $^9$Be nuclei, which are then transformed into seed nuclei, and the larger the neutron-to-seed ratio. @Meyer:1995 pointed out that in a very strong neutrino flux the slow 3-body $\alpha+\alpha+n \rightarrow
{^9}$Be reaction can be potentially bypassed by a sequence of two-body reactions started by the neutrino-induced spallation of an $\alpha$ particle; e.g. $^4$He($\nu,\nu^\prime
p)^3$H$({^4}$He,$\gamma)^7$Li($^4$He,$\gamma$)$^{11}$B and $^4$He($\nu,\nu^\prime
n)^3$He$({^4}$He,$\gamma)^7$Be($^4$He,$\gamma$)$^{11}$C. This would speed up the mass flow to the seed nuclei and thus reduce the neutron-to-seed ratio.
Systematic studies by @Hoffman.Woosley.Qian:1997 and @Freiburghaus.Rembges.ea:1999 have shown that a successful r-process requires either large entropies at the $Y_e$ values currently obtained in supernova models, or smaller values for $Y_e$.
In the neutrino-driven wind model the extreme flux of $\nu_e$ and $\bar{\nu}_e$ neutrinos from the protoneutron star interacts with the free protons and neutrons in the shocked matter by charge-current reactions, setting the proton-to-neutron ratio $n/p$ or equivalently the $Y_e$ value of the r-process matter. As shown by Fuller and Qian one has the simple relation [@Qian.Fuller:1995; @Qian:1997]
$$\frac{n}{p} \approx \frac
{L_{\bar{\nu}_e} \langle E_{\bar{\nu}_e} \rangle}
{L_{\nu_e} \langle E_{\nu_e}
\rangle}$$
As the neutrino energy luminosities are about equal for all species ($L_\nu \sim 10^{52}$ erg s$^{-1}$) the n/p-ratio is set by the ratio of average energies for the antineutrino and neutrino. As discussed above, their different opacities in the protoneutron star ensure that $\langle E_{\bar{\nu}_e} \rangle > \langle E_{\nu_e} \rangle$ and the matter is neutron-rich, as is required for a successful r-process. When the matter reaches cooler temperatures, nucleosynthesis starts and the free protons are, in the first step, assemblied into $\alpha$-particles, with some extra neutrons remaining. If these neutrons are still exposed to a large neutrino flux, it will change some of the neutrons into protons, which will then, together with additional neutrons, also be bound very quickly into $\alpha$-particles. Thus, this so-called $\alpha$ effect [@Meyer.Mclaughlin.Fuller:1998] would severely reduce the final neutron-to-seed ratio and is therefore very counter-productive to a successful r-process. As mentioned above, the neutrino-nucleon cross sections are only considered to lowest order in supernova simulations. The correction, introduced by the weak magnetism, acts to reduce the neutron-to-proton ratio in the neutrino-driven wind [@Horowitz.Li:1999].
There are possible ways out of this dilemma: One solution is to remove the matter very quickly from the neutron star in order to reduce the neutrino fluxes for the $\alpha$ effect. Such a short dynamical timescale of the material in the wind is found in neutrino-driven wind models studied by Kajino and collaborators. These authors also observe that relativistic effects as well as nuclear reaction paths through neutron-rich light elements might be helpful for a successful r-process . Another intriguing cure is discussed by @Mclaughlin.Fetter.ea:1999 and @Caldwell.Fuller.Qian:2000 invoking matter-enhanced active-sterile neutrino oscillations to remove the $\nu_e$ from the r-process site.
A simple estimate for the duration of the r-process can be had by adding up the half-lives of the waiting point nuclei, which results in about 1–2 seconds. However, in the neutrino-driven wind model it appears that the ejected matter passes through the region with the conditions suited for an r-process in shorter times ($\sim 0.5$ s), implying that there might not be enough time for sufficient matter-flow from the seed to nuclides in the $A\sim200$ mass region. Such a ‘time problem’ is avoided if, as indicated above, the half-lives of the waiting point nuclei are shorter than conventionally assumed, or if, in a dynamically changing environment, the matter, which freezes out making the $A\sim200$ r-process peak, breaks through the $N=50$ and 82 waiting points closer to the neutron dripline, i.e. through nuclei with shorter half-lives, than the matter which freezes out at these lower magic neutron numbers.
![Half-Lives of r-process waiting point nuclei with $N=50$ (top panel), 82 (middle panel) and 126 (bottom panel) against charged-current $(\nu_e,e^-)$ reactions. For the neutrinos a Fermi-Dirac distribution with $T=4$ MeV and zero chemical potential (circles) and a luminosity of ($L_\nu \sim
10^{52}$ erg s$^{-1}$) has been adopted. It is assumed that the reactions occur at a radius of 100 km, measured from the center of the neutron star. The half-lives can be significantly shorter if $\nu_e \rightleftarrows \nu_{\mu,\tau}$ oscillations occur. The squares show the half-lives for neutrinos with a Fermi-Dirac distribution with $T=8$ MeV and zero chemical potential, which corresponds to complete $\nu_e \rightleftarrows \nu_{\mu,\tau}$ oscillations.\[fig:cctau\]](nu_tau.eps){width="0.9\linewidth"}
In an environment with large neutrino fluxes the matter-flow to heavier nuclei can also be sped up by charged-current $(\nu_e,e^-)$ reactions [@Nadyozhin.Panov:1993; @Qian.Haxton.ea:1997] which can compete with $\beta$-decays. This is particularly important at the waiting point nuclei associated with $N=50, 82$, and 126. Figure \[fig:cctau\] shows the $(\nu_e,e^-)$ half-lives [@Hektor.Kolbe.ea:2000; @Langanke.Kolbe:2001] for these waiting point nuclei, considering reasonable supernova neutrino parametrizations and assuming that the ejected matter has reached a radius of 100 km. Due to the dependence on $L_\nu$ the $(\nu_e,e^-)$ half-lives scale with $r^2$. Indeed, a comparison with the $\beta$ half-lives (figure \[fig:cctau\]) shows that $(\nu_e,e^-)$ reactions can be faster than the longest $\beta$-decays of the $N=50,82$, and 126 waiting point nuclei, thus speeding up the breakthrough of the matter at the waiting points, if the r-process occurs rather close to the surface of the neutron star. This is even further enforced if $\nu_e \rightleftarrows \nu_{\mu,\tau}$ oscillations occur due to the higher energy spectrum of the supernova $\nu_{\mu,\tau}$ neutrinos. However, the presence of charged-current reactions on nuclei in the neutrino-driven wind model implies also neutrino reactions on nucleons strengthening the $\alpha$ effect [@Meyer.Mclaughlin.Fuller:1998].
![Schematic view of the $(\nu_e,e^-)$ reaction on r-process nuclei. Due to the high neutrino energies the cross sections are dominated by transitions to the Fermi (IAS) and Gamow-Teller resonances. \[fig:nureso\]](nures.eps){width="0.9\linewidth"}
Under a strong neutrino flux the weak flow is determined by an effective weak rate given by the addition of the charged-current $(\nu_e,e^-)$ and the nuclear beta-decay rates [@Mclaughlin.Fuller:1997]. It has been argued that the solar system r-process abundances provide evidence for the weak steady-flow approximation, which implies that the observed abundances should be proportional to the half-lives of their progenitor nuclei on the r-process path [@Kratz.Thielemann.ea:1988; @Kratz.Bitouzet.ea:1993]. If this is so, the r-process freeze-out must occur at conditions (i.e. radii) at which $\beta$-decay dominates over $(\nu_e,e^-)$ reactions, at least for late times. The reason is that neutrino capture on magic nuclei with $N=50,82,126$ is not reduced if compared to the neighboring nuclides, as the capture occurs from a reservoir of neutrinos with sufficiently high energies to allow for transitions to the IAS and GT resonant states (see figure \[fig:nureso\]). Consequently $(\nu_e,e^-)$ cross sections scale approximately like the neutron excess $(N-Z)$, reflecting the Fermi and Ikeda sumrules (see figure \[fig:cctau\]). However, the observed solar abundances around the $A=130$ ($N=82$) and 195 ($N=126$) peaks do not show such a smooth dependence with $A$ as will be the case if the effective weak rate is dominated by neutrino reactions. If we require that the $\beta$ decay half-lives dominate over the $(\nu_e,e^-)$ reactions we can put constraints on the neutrino fluence in the neutrino-driven wind scenario, which is particularly strict if neutrino oscillations occur. If, as an illustrative example, we apply the constraint to the heavy waiting point nuclei with $N=126$ (e.g. the nuclei with $A \sim
199$) and adopt the neutrino and beta half-lives from figure \[fig:cctau\], $\beta$-decay is only faster if the neutrino-nucleus reactions occur at distances larger than $\sim 500$ km.
![Effect of post-processing by neutrino-induced reactions on the r-process abundance. The unprocessed distribution (solid line) is compared with the distribution after post-processing (dashed line). A constant fluence of ${\cal F}=0.015$ has been assumed, which provides a best fit to the observed abundances for $A=183-87$ (see inset). The observed abundances are plotted as filled circles with error bars [from @Qian.Haxton.ea:1997]). \[fig:postpr\]](post.ps){width="0.9\linewidth"}
As the $Q_\beta$ values in the very neutron-rich r-process nuclei are large, the IAS and GT resonant states reside at rather high excitation energies ($\sim 20$–30 MeV), in the daughter nuclei for $(\nu_e,e^-)$ reactions. This fact, combined with the small neutron separation energies in these nuclei, ensures that $(\nu_e,e^-$) reactions, and also neutral-current $(\nu,\nu^\prime$) reactions, spall neutrons out of the target nuclei [@Haxton.Langanke.ea:1997; @Qian.Haxton.ea:1997] ($\sim 5$–7 neutrons for nuclei in the $A=195$ mass region [@Haxton.Langanke.ea:1997; @Hektor.Kolbe.ea:2000]). During the r-process, i.e. as long as the neutron source is strong enough to establish $(n,\gamma) \rightleftarrows (\gamma,n)$ equilibrium, the neutrons will immediately be recaptured leading to no effect on the abundance distribution. However, once the neutron source has ceased, e.g. after freeze-out, and if the r-process matter in the neutrino-driven wind model is still subject to strong neutrino fluxes, neutrons liberated during this phase by neutrino-induced reactions, will not be recaptured and this post-processing [@Qian.Haxton.ea:1997; @Haxton.Langanke.ea:1997] leads to changes in the r-process abundance distribution. It is argued [@Haxton.Langanke.ea:1997] that due to the smooth dependence of the neutrino cross sections on the mass number, the post-processing, in general, shifts abundances from the peaks to the wings at lower $A$-values (figure \[fig:postpr\]). This shift depends on the neutrino exposures and allows constraints to be put on the total neutrino fluence in the neutrino-driven wind model [@Qian.Haxton.ea:1997; @Haxton.Langanke.ea:1997]. The limits obtained this way are compatible with the values predicted by supernova models. Whether $\beta$-delayed neutron emission, which has been neglected in [@Haxton.Langanke.ea:1997] might affect the post-processing is an open question [@Kratz:2001; @Kratz.pvt].
Attempts to include neutrino-induced reactions in the r-process network within the neutrino-driven wind model have been reported in . In particular @Meyer.Mclaughlin.Fuller:1998 studied the competition of the $\alpha$-effect with the possible speed-up of the matter-flow by charged-current reactions on nuclei. These authors estimated the respective charged-current cross sections for allowed transitions on the basis of the independent particle model. Improving on this treatment, Borzov and Goriely calculated $(\nu_e,e^-)$ cross sections for supernova neutrinos (Fermi-Dirac distribution with $T=4$ MeV) within the ETFSI method, consistently with their most recent estimates for the $\beta$ half-lives [@Borzov.Goriely:2000]. RPA-based neutrino-nucleus cross sections for selected nuclei have been reported in [@Surman.Engel:1998; @Hektor.Kolbe.ea:2000]. Very recently a tabulation with charged- and neutral-current total and partial neutron spallation cross sections have become available for the neutron-rich r-process nuclei [@Langanke.Kolbe:2001; @Langanke.Kolbe:2002]. This tabulation is based on the RPA and considers allowed and forbidden transitions. Furthermore, the cross sections are tabulated for various supernova neutrino distributions, thus also allowing study of the influence of complete neutrino oscillations on the r-process.
![The lower panel shows the average mass number of heavy seed nuclei $\langle A \rangle$ and the neutron-to-seed ratio (n/Seed) in a neutrino-driven wind simulation with an exponential time dependence of the matter flow, determined by the parameter $\tau_{\text{dyn}}$. The open squares show the results for a simulation including neutrino-nucleus reactions, while the full triangles refer to a study in which the neutrino-reactions on nuclei have been switched off. The upper panel shows the sum of the mass number of the seed nucleus plus the neutron-to-seed ratio. This quantity shows up to which mass number the r-process can produce nuclides. The horizontal lines indicate the second ($A\sim 130$) and third ($A\sim 200$) r-process peaks as well as the position of the smaller r-process peak related to the deformed nuclei in the rare-earth region (REE peak), (courtesy of M. Terasawa). \[fig:Terasawa\]](terasawa.eps){width="0.9\linewidth"}
@Terasawa.Sumiyoshi.ea:2001a [@Terasawa.Sumiyoshi.ea:2001b] have performed studies similar to the pioneering work of @Meyer.Mclaughlin.Fuller:1998, however, using their neutrino-driven wind model and the complete set of RPA neutrino-nucleus reaction rates [@Langanke.Kolbe:2001; @Langanke.Kolbe:2002]. A typical result is shown in figure \[fig:Terasawa\], where, however, the simplifying, but reasonable assumption of an exponential time-dependence of the matter-flow, governed by the parameter $\tau_{\text{dyn}}$, away from the neutron star has been assumed. The quantity $\langle A \rangle$ defines the average mass of the heavy seed nuclei present at the beginning of the r-process, defined at $T = 2.5 \times 10^9$ K. The neutron-to-seed ratio $n/s$ is very sensitive to the dynamical evolution time. This comes about as the shorter $\tau_{\text{dyn}}$, the less time is available to assemble the seed nuclei from $\alpha$-particles and neutrons. Consequently the abundance of seed nuclei decreases for shorter $\tau_{\text{dyn}}$, increasing the $n/s$ ratio. If the neutrino flux is artificially switched off, matter-flow to the 3rd r-process peak at $A \sim 200$ (2nd r-process peak at $A\sim
130$) is achieved if $\tau_{\text{dyn}} \le 0.01$ s ($\tau_{\text{dyn}} \le 1$ s). The consistent inclusion of neutrino reactions is counter-productive to a successful r-process. This effect becomes more dramatic if the matter-flow is slow as then the $\alpha$-effect strongly suppresses the availability of free neutrons at the beginning of the r-process [see also @Meyer.Mclaughlin.Fuller:1998]. No r-process, i.e. no production of nuclides in the second r-process peak at $A\sim130$, is observed if $\tau_{\text{dyn}} \gtrsim 0.05$ s.
![Total $(\nu_e,e^-)$ (circles) and partial $(\nu_e,e^- n)$ (triangles) and neutrino-induced fission cross sections (squares) for selected thorium and uranium isotopes. The calculations have been performed for Fermi-Dirac $\nu_e$ distributions with temperature $T=4$ MeV and 8 MeV. The first reflects a typical supernova $\nu_e$ spectrum, while the latter assumes complete $\nu_{\mu,\tau} \rightarrow \nu_e$ neutrino oscillations.[]{data-label="fig:nufission"}](nufission.eps){width="0.9\linewidth"}
Recently @Qian:2002 has suggested that, within the neutrino-driven wind r-process scenario, charged-current neutrino reactions can induce fission reactions on r-process progenitor nuclei heavier than lead and that the fission products account for the observed r-process abundance in metal-poor stars [@Sneden.Cowan.ea:2000]. These observed abundances show a peak around mass number $A \sim 195$, which follows the solar r-process distribution, and enhanced structures at around $A \sim 90$ and $\sim
132$ which, due to the suggestion of @Qian:2002, are fission products, which do not follow a solar r-process pattern. First calculations of neutrino-induced fission cross sections have been performed by [@Kolbe.Fuller.Langanke:2002] using a combination of RPA model, to calculate the $(\nu_e,e^-)$ excitation function, and statistical model to determine the final branching probabilities. The neutrino-induced fission cross sections (see figure \[fig:nufission\]) are quite large as the progenitor nuclei are neutronrich, which increases the Fermi and Gamow-Teller contributions to the total cross sections and places their strengths at energies above the fission barrier in the daughter nucleus. The calculations shown in figure \[fig:nufission\] use the fission barriers derived by @Howard.Moeller:1980. Modern evaluations predict larger fission barriers [e.g. @Mamdouh.Pearson.ea:2001] which would reduce the fission cross section. The cross sections can be significantly enlarged if $\nu_{\mu,\tau} \rightarrow \nu_e$ neutrino oscillations occur during the neutrino-driven wind r-process scenario.
The Neutrino Process
====================
When the flux of neutrinos generated by the cooling of the neutron star passes through the overlying shells of heavy elements interesting nuclear transmutations are induced, despite the small neutrino-nucleus cross sections. Of particular interest here are neutral-current reactions as they can be induced by $\nu_\mu,\nu_\tau$ neutrinos and their antiparticles with the higher energy spectra ($\langle E_\nu
\rangle \sim 25$ MeV). These neutrinos are energetic enough to excite the GT resonant state, and more importantly, also the giant dipole resonant states. These states usually reside above particle thresholds and hence decay mainly by proton or neutron emission, generating new nuclides. The neutrino reaction rates are too small to affect the abundances of the parent nuclei, but they can noticeably contribute to the production of the (sometimes much less abundant) daughter nuclei. As a rule-of-thumb, the neutrino process, i.e. the synthesis of nuclides by neutrino-induced spallation in a supernova, can become a significant production process for the daughter nuclide if one wants to explain abundance ratios of parent-to-daughter which exceed about $10^3$ [@Woosley.pvt].
The most interesting neutrino nucleosynthesis occurs in the outer burning shells of the massive star which have not been affected by the fatal core collapse in the center when the neutrinos pass through. However, a little later the shock reaches these shells and the matter will be subjected to rather high temperatures initiating fast nuclear reactions involving also the nuclides just produced by the neutrino-induced reactions. Hence studies of the neutrino process depend on various neutrino-nucleus reaction rates and on the neutrino spectra and fluxes, especially of the $\nu_\mu,\nu_\tau$ neutrinos, and they require a rather moderate nuclear network to simulate the effects of the re-processing by the shock. The first investigation of the neutrino process has been reported in [@Woosley.Hartmann.ea:1990]. A more recent study [@Woosley.Weaver:1995] confirmed the main result that a few specific nuclei ($^7$Li, $^{11}$B, $^{19}$F) are being made by the neutrino process in significant fractions. For example, $^{11}$B and $^{19}$F are being made by $(\nu,\nu'p)$ and $(\nu,\nu'n)$ (followed by a $\beta$-decay) reactions on the abundant $^{12}$C and $^{20}$Ne, respectively. As noted by @Woosley.Hartmann.ea:1990 neutrino nucleosynthesis can also contribute to the production of $^{180}$Ta (see above) and $^{138}$La. First calculations of the relevant total and partial neutrino-induced cross sections have been reported in [@Heger.Kolbe.ea:2002], [see also @Belic.Arlandini.ea:2002]. The nucleus $^{138}$La is special as it is mainly made by the charged-current reaction $^{138}$Ba$(\nu_e,e^-){}^{138}$La, while the lighter nuclei ($^7$Li, $^{11}$B, $^{19}$F) and dominantly also $^{180}$Ta, are being produced by neutral-current reactions induced by $\nu_\mu,\nu_\tau$ neutrinos. The neutrino process is therefore sensitive to $\nu_e$ and ($\nu_\mu,\nu_\tau$) neutrinos, which are expected to have different spectra in Type II supernovae (see section V.C), and hence it can test this prediction. Moreover, the $^{138}$La nucleosynthesis is sensitive to neutrino oscillations, as this nuclide would be significantly overproduced if supernova $\nu_\mu,\nu_\tau$ neutrinos have a noticeably larger average energy and if they oscillate into $\nu_e$ neutrinos before reaching the $^{138}$La production site (helium shell) in massive stars.
Neutrino-induced nucleon spallation on $^{12}$C can also knock-out a deuteron or a proton-neutron pair, in this way producing $^{10}$B. The expected $^{10}$B/$^{11}$B abundance ratio in neutrino-nucleosynthesis is $\sim 0.05$, which is significantly smaller than the observed abundance ratio, 0.25 [@Haxton:2000]. Thus there must be a second process which contributes to the production of $^{10,11}$B. These are reactions of energetic protons on $^{12}$C in cosmic rays, which yield a ratio of $^{10}$B/$^{11}$B of about 0.5, larger than the observed value. A solution might be that the two nuclides are being produced by both mechanisms, neutrino-nucleosynthesis and cosmic ray spallation. It is interesting to note that the first process, being associated with supernovae, is a primary process, while the latter is a secondary process, as it requires the existence of protons and $^{12}$C in the interstellar medium. As a consequence the $^{10}$B/$^{11}$B abundance ratio should have changed during the history of the galaxy. This can be tested once observers are able to distinguish between the abundances of the two different boron nuclides in stellar spectra [@Haxton:2000].
Binary Systems
==============
Weak processes can also play interesting roles in the evolution and nucleosynthesis processes in close binary systems where one component is a compact object (white dwarf or neutron star) and the other a massive star. If the latter expands during helium core burning, mass flow from the hydrogen envelopment of the star onto the surface of the compact object sets in. If the respective mass accretion rate is rather low ($10^{-8}$–$10^{-10}$ $M_\odot$ y$^{-1}$), the hydrogen, accreted on the surface of the compact object, burns explosively under degenerate conditions leading to a nova (if the compact object is a white dwarf) or an x-ray burst (neutron star). This means that the energy released by the nuclear reactions is used to heat the matter rather than for expansion. The rise in temperature increases the nuclear reaction rates, triggering a thermonuclear runaway until the degeneracy is finally lifted and an outer layer of matter is ejected. In a type Ia supernova the faster mass accretion rate on to the surface of a white dwarf (likely a carbon-oxygen white dwarf with sub-Chandrasekhar mass $\sim 0.7\ M_\odot$) leads to steady hydrogen and subsequently helium burning. If the growing mass of the white dwarf exceeds the Chandrasekhar mass, contraction sets in and the carbon in the center ignites by fusion reactions with screening enhancement. As the environment is highly degenerate, a thermonuclear runaway is triggered which eventually will explode the entire star.
Novae
-----
The main energy source of a nova is the CNO cycle, with additional burning from proton reactions on nuclei between neon and sulfur, if the white dwarf also contained some $^{20}$Ne [@Truran:1982]. A nova expels matter which is enriched in $\beta$-unstable $^{14,15}$O for carbon-oxygen white dwarfs (leading to the production of stable $^{14,15}$N nuclides) or can contain nuclides up to the sulfur mass region for neon-oxygen novae . The most important role of weak-interaction processes in novae is their limitation of the energy generation during the thermonuclear runaway. An interesting branching occurs at $T \sim 8 \times 10^7$ K. For lower temperatures the $\beta$-decay of $^{13}$N with a half-life of $\sim 10$ m dominates over the $^{13}$N$(p,\gamma)$ reaction and sets the timescale for the nuclear burning. As charged-particle fusion reactions are sensitively dependent on temperature, their reaction rates strongly increase with rising temperature and for $T \ge 8
\times 10^7$ K and densities of order $10^4$ g cm$^{-3}$ the proton capture on $^{13}$N is faster than the $\beta$-decay. The CNO cycle turns into the *hot* CNO cycle and now the positron decay of $^{15}$O with a half-life of 122 s is the slowest reaction occuring in the CNO nova network. It turns out that, once the degeneracy is lifted, the dynamical expansion timescale is faster than the one for nuclear energy generation, set by the $^{15}$O half-life. As a consequence the runaway is quenched [@Truran:1982]. We mention that the determination of the dominant resonant contribution to the $^{13}$N$(p,\gamma){}^{14}$O rate has been the first successful application of the Coulomb dissociation technique in nuclear astrophysics [@Motobayashi.Takei.ea:1991].
Weak-interaction rates relevant for nova studies can be derived from either experimental data or shell-model calculations.
X-ray Bursts
------------
An x-ray burst is explained as a thermonuclear runaway in the hydrogen-rich envelope of an accreting neutron star [@Lewin.Paradijs.Taam:1993; @Taam.Woosley.ea:1993]. Due to the higher gravitational potential of a neutron star, the accreted matter on the surface reaches larger densities than in a nova (up to a few $10^6$ g cm$^{-3}$) and the temperature during the runaway can rise up to $2 \times 10^9$ K [@Schatz.Aprahamian.ea:1998]. Under these conditions hydrogen burning is explosively fast. The trigger reactions of the runaway are the triple-alpha reactions and the break-out from the hot CNO cycle, mainly by $\alpha$-capture on $^{15}$O and $^{18}$Ne. We note that these two reaction rates are yet insufficiently known due to uncertain resonant contribution at low energies [@Mao.Fortune.Lacaze:1996; @Goerres.Wiescher.Thielemann:1995]. The thermonuclear runaway is driven first by the $\alpha p$ process, a sequence of $(\alpha,p)$ and $(p,\gamma)$ reactions which shifts the ashes of the hot CNO cycle to the Ar and Ca mass region, and then by the rp-process (short for rapid proton capture process). The rp-process represents proton-capture reactions along the proton dripline and subsequent $\beta$-decays of dripline nuclei processing the material from the Ar-Ca region up to $^{56}$Ni [@Schatz.Aprahamian.ea:1998]. The $\beta$ half-lives on the rp-process path up to $^{56}$Ni are known sufficiently well.
The runaway freezes out in thermal equilibrium at peak temperatures of $(2\text{--}3)\times 10^9$ K with an abundance distribution rich in $^{56}$Ni, forming Ni oceans on the surface of the neutron star. Further matter flow at these temperatures is suppressed due to the low proton separation energy of $^{57}$Cu and the long half-life of $^{56}$Ni. Re-ignition of the rp-process then takes place during the cooling phase, starting with proton capture on $^{56}$Ni and potentially shifting matter up to the $^{100}$Sn region where the rp-process ends in a cycle in the Sn-Te-I range [@Schatz.Aprahamian.ea:2001]. A matter flow to even heavier nuclei is possible, if the rp-process operates in a repetitive mannor; i.e. a new rp-process is ignited after the ashes of the previous process have decayed to stability and before these nuclei have sunk too deep into the crust of the neutron star (see below). Such repetitive rp-process models, shortly called $(\rm rp)^2$-process, have been studied by [@Boyd.Hencheck.Meyer:2002].
The reaction path beyond $^{56}$Ni runs through the even-even $N=Z$ nuclei which due to their known long half-lives ($^{64}$Ge has a half-life of 63.7(25) s) represent a strong impedance to the matter-flow. This can also not be overcome always by proton captures as for some of the $\alpha$-like nuclei the resulting odd-$A$ nucleus is proton-unbound and exists only as a resonance. Such situations occur, for example, for the $^{68}$Se$(p,\gamma){}^{69}$Br and $^{72}$Kr$(p,\gamma){}^{73}$Rb reactions. It has been suggested [@Goerres.Wiescher.Thielemann:1995] that the gap in the reaction path can be bridged by two-proton captures, with the resonance serving as intermediate state (like in the triple-$\alpha$ reaction). The reaction rate for such a two-step process depends crucially on the resonance energy, with some appropriate screening corrections.
Most half-lives along the rp-process path up to the $^{80}$Zr region are known experimentally. This region of the nuclear mass chart is known for strong ground state deformations, caused by coupling of the $pf$-shell orbitals to the $g_{9/2}$ and $d_{5/2}$ levels. The strong deformation makes theoretical half-life predictions quite inaccurate, mainly due to uncertainties in the $Q$-values stemming from insufficiently wellknown mass differences. We note that the effective half-life of a nucleus along the rp-process path could be affected by the feeding of isomeric states in the proton captures or by thermal population of excited states in general. Again, deformation plays a major role as then even in $\alpha$-like nuclei excited states are at rather low energies (e.g. the first $2^+$ state in $^{80}$Zr is at 290 keV). Apparently a measurement of the half-lives is indispensable for rp-process studies beyond $A=80$. An important step has recently been taken by measuring the half-life of $^{80}$Zr at the Holifield Facility in Oak Ridge [@Ressler.Piechaczek.ea:2000]. The experimental value of $4.1^{+0.8}_{-0.6}$ s reduces the previous (theoretical) uncertainty considerably and, in fact, it is shorter than the value adopted previously in x-ray bursts simulations. Fast proton captures on the daughter products $^{80}$Y and $^{80}$Sr allow matter-flow to heavier nuclei, with the $\alpha$-nucleus $^{84}$Mo ($N=Z=42$) being the next bottleneck. Experiments to measure this important half-life are in progress.
The nucleosynthesis during the cooling phase in an x-ray burst alters considerably the abundance distribution in the atmosphere, ocean and crust of the neutron star. For example, the rp-process may be a possible contributor to the presently unexplained relatively high observed abundance of light p-nuclei like $^{92}$Mo and $^{96}$Ru [@Schatz.Aprahamian.ea:1998]. This assumes that the matter produced in the x-ray burst gets expelled out of the large gravitational potential of the neutron star, which is still questionable. Due to continuing accretion the rp-process ashes are pressed into the ocean and crust of the neutron star, replacing there the neutron star’s original material. When the ashes sink into the crust, they reach regions of higher densities, and relatedly, larger electron chemical potentials. Thus, consecutive electron captures will become energetically favorable and make the ashes more neutron-rich. At densities beyond neutron drip ($\rho \sim (4\text{--}6) \times
10^{11}$ g cm$^{-3}$) neutron emissions become possible and at even higher densities pycnonuclear reactions can set in [@Bisnovatyi-Kogan.Chechetkin:1979; @Sato:1979; @Haensel.Zdunik:1990]. Importantly, these processes (electron capture, pycnonuclear reactions) generate energies which can be locally stored in the neutron star’s ocean and crust and will affect their thermal properties [@Brown.Bildsten:1998]. Previous studies of these processes in accreting neutron stars have assumed that iron is the endproduct of nuclear burning and the sole nucleus reaching the crust of the neutron star [e.g. @Haensel.Zdunik:1990]. But clearly the rp-process produces a wide mixture of heavy elements [@Schatz.Bildsten.ea:1999], where the abundance distribution depends on the accretion rate.
The ashes consist mainly of even-even $N=Z$ nuclei, for which electron capture at neutron star conditions occur always in steps of two. At first, the capture on the even-even nucleus sets in once sufficiently high-energy electrons are available to effectively overcome the $Q_{\text{EC}}$-value to the odd-odd daughter nucleus. Due to nuclear pairing, which favors even-even nuclei, the $Q_{\text{EC}}$-value for the produced daughter nucleus is noticeably lower so that electron capture on the daughter readily follows at the same conditions. The energy gain of the double-electron capture is of order the difference of the two $Q_{\text{EC}}$-values; this gain is split between the emitted neutrino and a local heating. Considering a blob of accreted matter initially consisting solely of $^{56}$Ni and assuming temperature $T=0$, the evolution of this blob was followed on the neutron star surface until neutron-drip densities and beyond [@Haensel.Zdunik:1990]. The $rp$-process simulations, however, indicate a finite temperature of the ashes of a few $10^8$ K, allowing electron capture already from the high-energy tail of the electron distribution and significantly reducing the required densities.
Type Ia Supernovae
------------------
Type Ia supernovae at high redshifts serve currently as the standard candles for the largest distances in the universe. Importantly recent surveys of such distant supernovae provide evidence for an accelerating expansion of the universe over the last several $10^9$ years [@Riess.Filippenko.ea:1998; @Perlmutter.Aldering.ea:1999].
Type Ia supernovae have been identified as thermonuclear explosions of accreting white dwarfs with high accretion rates in a close binary system. While the general explosion mechanism is probably understood, several issues are still open like the masses of the stars in the binary or the carbon/oxygen ratio and distribution in the white dwarf. The probably most important problem yet is the modelling of the matter transport during the explosion and the velocity of the burning front, both requiring multidimensional simulations .
![$Y_e$ profile as a function of radial mass for the standard type Ia supernova model WS15 [@Nomoto.Thielemann.Yokoi:1984] using the FFN and the shell-model weak-interaction rates (LMP), (courtesy of F. Brachwitz).\[fig:yeprof\]](yews15ffnlmp.eps){width="0.9\linewidth"}
{width="45.00000%"} {width="45.00000%"}
Type Ia supernovae contribute about half to the abundance of Fe-group nuclides in galactic evolution. Thus one can expect that type Ia supernovae should not overproduce abundances of nuclides in the iron group, such as $^{54}$Cr or $^{50}$Ti, relative to $^{56}$Fe by more than a factor of two compared with the relative solar abundances. This requirement puts stringent constraints on models, in particular on the central density of the progenitor white dwarf and the flame speed [@Iwamoto.Brachwitz.ea:1999]. When the flame travels outwards, it heats the matter to temperatures of a few $10^9$ K and brings its composition close to nuclear statistical equilibrium (NSE). As the original matter ($^{16}$O, $^{12}$C) had an electron-to-baryon ratio of $Y_e=0.5$, the NSE composition is dominated by $^{56}$Ni, which after being expelled decays to $^{56}$Fe. However, behind the flame front, which travels with a few percent of the local sound speed [@Niemeyer.Hillebrandt:1995], electron captures occur, which lower $Y_e$ and drive the matter composition more neutron-rich. This effect is larger the greater the central density of the white dwarf (which increases the electron capture rates) and the slower the flame speed (which allows more time for electron captures). Figure \[fig:yeprof\] shows the $Y_e$ profile obtained in a standard type Ia model WS15 [@Nomoto.Thielemann.Yokoi:1984], with slow deflagration flame speed (1.5% of sound velocity), central ignition density $\rho=2.1 \times 10^9$ g cm$^{-3}$ and a transition from deflagration to detonation at density $\rho = 2.1 \times
10^7$ g cm$^{-3}$. The calculations have been performed by @Brachwitz:2001 with the FFN [@Fuller.Fowler.Newman:1982a] and shell model [@Langanke.Martinez-Pinedo:2001] weak interaction rate sets. The differences are quite significant, even if one considers that about $60\%$ of the captures occur on free protons, which are unaffected by the differences in these rate sets. Under otherwise identical conditions the slower shell model rates yield a central $Y_e$ value of 0.45, which is about 0.01 larger than for the FFN rates. Consequently very neutron-rich nuclei with $Y_e \le 0.45$ are significantly suppressed (see figure \[fig:abundIa\]). In fact, no nuclide is significantly overproduced in this model compared to the solar abundance. The net effect of the new rates is that, for an otherwise unchanged model, it increases the central density by about a factor of 1.3 [@Brachwitz.Dean.ea:2000; @Woosley.pvt]. This can have quite interesting consequences if one wants to use the nucleosynthesis constraint to distinguish between two quite distinct type Ia models. On the basis of recent models it has been concluded that the majority of type Ia progenitors grow towards the Chandrasekhar mass through steady hydrogen and helium burning [@Hachisu.Kato.Nomoto:1999]. Such systems would lead to rather low central densities $\rho \le 2
\times 10^9$ g cm$^{-3}$. In these models, only a small fraction of progenitors would deviate from steady hydrogen burning at the end of the accretion history experiencing weak hydrogen flashes; such cases correspond to the W-model discussed above yielding higher central densities. We stress, however, that changes in the nucleosynthesis caused by differences in the central densities in the models can be counterbalanced by changes of the flame speed.
Conclusions and future perspectives {#sec:concl-future-persp}
===================================
It has long been recognized that nuclear weak-interaction processes play essential roles in many astrophysical scenarios. In a few cases these are specific reactions, which are particularly important, and these reactions have then been studied with increasingly refined models. Examples are the various weak-interaction processes in the solar hydrogen burning chains including the initial $p+p$ fusion reaction or the very challenging $^3\text{He}+p$ reaction , which generates the highest-energy neutrinos in the sun. Another typical example is the solar electron capture rate on $^7$Be, where the nuclear matrix element can be determined from the experimental lifetime of atomic $^7$Be, while the proper description of the solar plasma effects on the capture rate with the desired accuracy has been quite demanding.
However, most astrophysical applications require the knowledge of weak-interaction rates for a huge body of nuclei. If, like for s-process nucleosynthesis, the nuclei involved are close to the valley of stability and hence quite long-lived, the needed rates (usually half-lives) have been determined experimentally in decade-long efforts. For the s-process the challenge now focuses on the branching point nuclei where the reaction flow branches into two (or more) paths and where, in some cases, the observed relative abundances of nuclides along the different paths depend on the stellar conditions (temperature, neutron density) and hence allow determination of these quantities inside the star. Again, the approach is to measure the necessary data, e.g. half-lives of excited nuclear states.
Other astrophysical scenarios involve nuclei far-off stability, often under extreme conditions (high density, neutron flux, temperature). These astrophysical sites include core-collapse (type II) and thermonuclear (type Ia) supernovae, r-process nucleosynthesis and explosive hydrogen burning; and the interest in all of these has been boosted recently by novel observations and data (supernova 1987A, high-redshift supernova survey, Hubble Space Telescope, …). A direct experimental determination of the respective stellar weak-interaction rates has been possible in a few cases, like the half-lives of r-process and rp-process waiting point nuclei. However, in nearly all cases the weak-interaction processes had to be theoretically modeled so far – a very demanding job if one considers that often results for many hundreds of nuclei for a large range of stellar conditions are needed. These data were then derived globally based on parametrized nuclear structure arguments, as an appropriate treatment of the involved nuclear structure problem was prohibited by both the available computational capabilities and the lack of experimental guidance. Although evaluation of rate sets for astrophysical purposes often appears to be a theoretical problem, the second point – experimental guidance – is crucial and often overlooked. It is Willy Fowler’s strategy and legacy that nuclear models used to derive nuclear ingredients in astrophysical applications, should be consistent, but more importantly they should be accurate and, as a consequence, experimental data are to be used whenever available. Therefore the role of experimental data in nuclear astrophysics is twofold: If possible, they supply the needed information directly, or equally important, they constitute constraints and guidance for the nuclear models from which the needed information is obtained. Thus, the renaissance of nuclear structure, which we have witnessed in recent years, has two consequences in nuclear astrophysics. The recent development of new facilities, techniques and devices brought a large flood of new experimental information, in particular for the proton- and neutron-rich nuclei away from stability. These data indicate that the nuclear structure models adopted to derive the global data and rate sets were usually too simple and improvements were warranted. The experimental renaissance went hand-in-hand with decisive progress in nuclear structure theory, made possible by the development of new models and better computer hardware and software. Due to both experimental and theoretical advances, it becomes now possible to calculate nuclear data sets for astrophysical applications on the basis of realistic models rather than on crude and often oversimplified parametrizations. This review presents a summary of recent theoretical calculations.
The advances in nuclear structure modelling have also lead to progress in astrophysically important nuclear input other than weak-interaction processes. Typical examples are the equation of state, derived on the basis of the relativistic mean-field model guided by the relativistic Brückner-Hartree-Fock theory [@Shen.Toki.ea:1998a; @Shen.Toki.ea:1998b], which serves as an alternative to the standard Lattimer-Swesty EOS [@Lattimer.Swesty:1991], or the nuclear mass table and level density parametrizations determined within the framework of the Hartree-Fock model with BCS pairing [@Goriely.Tondeur.Pearson:2001; @Demetriou.Goriely:2001b]. Such mass tables, or equivalently neutron separation energies, play an essential role in r-process nucleosynthesis and are conventionally derived by parametrizations constrained to known masses.
Other astrophysical areas, loosely or indirectly related to the topic of this review, have also benefitted from the progress in modelling nuclear weak-interaction processes. A field with rapidly growing importance is Gamma-ray Astronomy with beta-unstable nuclei. Due to $\gamma$-ray observatories in space it has been possible in recent years to search the sky for sources of known $\gamma$-rays which can then be associated with recent nucleosynthesis activities. A highlight has been the observation of the 1.157 MeV $\gamma$-line, produced in the $\beta$-decay scheme of $^{44}$Ti, in the Cassiopeia supernova remnant [@Iyudin.Diehl.ea:1994]. Knowing the date of the supernova, the $^{44}$Ti half-life, the distance to the source and the measured intensity of the $\gamma$-line allows determination of how much $^{44}$Ti has been ejected into the interstellar medium by the supernova event. Furthermore, as $\gamma$-rays can, in contrast to optical wave lengths, escape from the galactic bulk, $\gamma$-ray observation allows to detect historical supernovae which have not been observed optically [@Iyudin.Schonfelder.ea:1998], [also observed in x-rays by @Aschenbach:1998]. Such searches for historical supernovae and an improved determination of the supernova frequency in our galaxy will be one of the main missions of future $\gamma$-ray observatories in space, like INTEGRAL[^3]. A longer-lived radioactive nuclide produced in supernovae is $^{60}$Fe. Investigations of rock samples taken from the ocean floor, by precision accelerator mass spectroscopy, found a significant increase of $^{60}$Fe abundance compared to other iron isotopes pointing to a close-by supernova about 5 million years ago [@Knie.Korschinek.ea:1999].
Weak-processes on nuclei and electrons are the means to observe solar and supernova neutrinos [@Balantekin.Haxton:1999]. Solar neutrinos have rather low energies ($E_\nu \le 14$ MeV) and hence induce specific low-lying transitions which are theoretically modelled best by shell-model calculations. Applications have been performed, for example, for $^{37}$Cl [@Aufderheide.Bloom.ea:1994], $^{40}$Ar [@Ormand.Pizzochero.ea:1995] and $^{71}$Ga [@Haxton:1998], the detector material in the Homestake [@Cleveland.Daily.ea:1998], ICARUS[^4], and GALLEX/SAGE/GNO detectors , respectively. Supernova neutrinos have higher energies; in particular, the energies of $\mu,\tau$ neutrinos and antineutrinos are expected to be high enough to excite the giant dipole resonances in nuclei. Studies for detector materials like Na, Fe, Pb have been performed in hybrid approaches combining shell model calculations for allowed transitions with the RPA model for forbidden transitions or treating the forbidden transitions on the basis of the Goldhaber-Teller model [@Fuller.Haxton.Mclaughlin:1999; @Kolbe.Langanke:2001].
Despite the experimental and theoretical progress, lack of knowledge of relevant or accurate weak-interaction data still constitutes a major obstacle in the simulation of some astrophysical scenarios today. This refers mainly to type II supernovae and r-process nucleosynthesis.
Core-collapse supernovae, as the breeding places of carbon and oxygen, and hence life, in the universe, attract currently significant attention and the quest to definitely identify the explosion mechanism is on the agenda of several international and interdisciplinary collaborations. While most of the efforts concentrate on computational developments towards a multidimensional treatment of the hydrodynamics and the neutrino transport, some relevant and potentially important nuclear problems remain. The improved description of weak-interaction rates in the iron mass range has lead to significant changes in the presupernova models. Evaluation of electron capture rates for heavier nuclei will be available in the near future. First results imply that capture on heavy nuclei, usually ignored in collapse simulations, can compete with the capture on free protons. Whether the inclusion of this process then leads to changes in the collapse trajectory has yet to be seen. Further, finite temperature neutrino-nucleus rates will also soon be available. This will then test whether inelastic neutrino scattering on nuclei, yet not modelled in the simulations, influences the collapse dynamics or supports the revival of the shock wave by preheating of the infalling matter after the bounce. The largest nuclear uncertainty in collapse simulations are likely associated with the description of nuclear matter at high density, extreme isospin and finite temperature. In particular a reliable description of the neutrino opacity in nuclear matter might very well be what is needed for successful explosion simulations. Nucleon-nucleon correlations have been identified to strongly influence the neutrino opacities and nuclear models like the RPA are quite useful to guide the way. Ultimately one would like to see the many-body Monte Carlo techniques, which have been so successfully applied to few-body systems or the nuclear shell model, extended to the nuclear matter problem. First steps along these lines have been reported by @Schmidt.Fantoni:1999 and @Fantoni.Sarsa.Schmidt:2001, who proposed a novel constrained-path Diffusion Monte Carlo model to study nuclear matter at temperature $T=0$. The Shell Model Monte Carlo model, formulated in momentum space and naturally at finite temperature, constitutes an alternative approach. First attempts in this direction have been taken by the Caltech group [@Zheng.pvt] and by @Rombouts.pvt. Whether the notorious sign-problem in the SMMC approach can also be circumvented for nuclear matter has still to be demonstrated. @Mueller.Koonin.ea:2000 have investigated whether nuclear matter can be formulated on a spatial lattice with nearest neighbor interactions, similar to the Hubbard model for high-$T_c$ superconductors. Besides these theoretical efforts it is equally important to improve our understanding of the nuclear interaction in extremely neutron-rich nuclei (matter).
Despite four decades of intense research, the astrophysical site of the r-process nucleosynthesis has yet not been identified. Recent astronomical and meteoric evidence points now to more than one source for the solar r-process nuclides, and it is clearly a major goal in the astrophysics community to solve this cosmic riddle. However, the puzzle will not be definitely solved if the nuclear uncertainties involved are not removed. This is even more necessary as recent research shows that the r-process is a dynamical process under changing astrophysical conditions. This implies dynamically changing r-process paths. To determine the paths one needs to know the masses of nuclei far-off stability accurately, besides the condition of the astrophysical environment. In a dynamical r-process, the astrophysical timescale will compete with the nuclear timescale, i.e. with the time needed for the matter flow from the seed nuclei to the heavier r-process nuclides. This nuclear timescale is set by the half-lives of the nuclei along the paths, in particular by those of the longer-lived waiting point nuclei associated with the magic neutron numbers. The half-lives of waiting point nuclei are a very illustrative example for the need of reliable experimental data: Modern global theoretical models predict half-lives at the waiting points with a spread of nearly one order of magnitude and only data can decide. For the $N=50$ and $N=82$ waiting points such data exist for a few key nuclides [e.g. @Pfeiffer.Kratz.ea:2001], but not for the $N=126$ nuclei.
R-process nucleosynthesis as well as other astrophysical processes will tremendously benefit from future experimental developments in nuclear physics. Worldwide radioactive ion-beam facilities, with important and dedicated programs in nuclear astrophysics, have just started operation or are under construction or in the proposal stage. These new facilities will boost our knowledge about nuclei far-off stability, they will determine astrophysically relevant nuclear input directly (e.g. masses and half-lives for the r-process, half-lives and cross sections for the rp-process, etc). But equally important, the radioactive ion-beam facilities will guide and constrain the nuclear models, in this way indirectly contributing to the reduction of the nuclear inaccuracies in astrophysical models. Ultimatively nuclear physics can and will then become a stringent test and guidance for astrophysical theories and ideas.
We like to thank many colleagues and friends for helpful collaborations and discussions: D. Arnett, S. M. Austin, J. N. Bahcall, R. N. Boyd, F. Brachwitz, R. Canal, E. Caurier, J. Christensen-Dalsgaard, J. J. Cowan, D. J. Dean, R. Diehl, J. Dobaczewski, J. L. Fisker, G. M. Fuller, E. García-Berro, S. Goriely, J. G[ö]{}rres, W. C. Haxton, A. Heger, M. Hernanz, W. Hillebrandt, R. Hix, J. Isern, H.-Th. Janka, J. José, T. Kajino, E. Kolbe, S. E. Koonin, K.-L. Kratz, F. K[ä]{}ppeler, M. Liebendörfer, A. Martínez-Andreu, G. J. Mathews, A. Mezzacappa, Y. Mochizuki, E. M[ü]{}ller, W. Nazarewicz, F. Nowacki, I. Panov, B. Pfeiffer, A. Poves, Y.-Z. Qian, G. Raffelt, R. Reiferth, A. Richter, C. E. Rolfs, J. M. Sampaio, H. Schatz, M. Terasawa, F.-K. Thielemann, J. W. Truran, P. Vogel, M. Wiescher, S. E. Woosley, and A. P. Zuker. Special thanks are due to our two referees, R.N. Boyd and anonymous, for very constructive and helpful comments on the manuscript. Our work has been supported by the Danish Research Council and the Schweizerische Nationalfonds. Computational resources for our work were provided by the Center for Advanced Computational Research at Caltech, by the Danish Center for Scientific Computing and by the Center for Computational Sciences at Oak Ridge National Laboratory.
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[^1]: The Gallex detector has been recently upgraded and has changed its name to GNO [@Altmann.Balata.ea:2000]
[^2]: http://www-boone.fnal.gov/
[^3]: http://astro.estec.esa.nl/SA-general/Projects/Integral/integral.html
[^4]: http://www.aquila.infn.it/icarus
|
---
abstract: 'By matching Wilson twisted mass lattice QCD determinations of pseudoscalar meson masses to Wilson Chiral Perturbation Theory we determine the low-energy constants ${W_6''}$, ${W_8''}$ and their linear combination $c_2$. We explore the dependence of these low-energy constants on the choice of the lattice action and on the number of dynamical flavours.'
address:
- |
PRISMA Cluster of Excellence, Institut f[ü]{}r Kernphysik\
Johannes Gutenberg-Universit[ä]{}t, D-55099 Mainz, Germany
- |
Departamento de Física Teórica and Instituto de Física Teórica UAM/CSIC,\
Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain
- 'NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany'
- |
Department of Physics, University of Cyprus\
P.O.Box 20537, 1678 Nicosia, Cyprus
- |
Theoretical Physics Division, Department of Mathematical Sciences\
The University of Liverpool, Liverpool L69 3BX, UK
- 'Helmholtz-Institut f[ü]{}r Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universit[ä]{}t Bonn, 53115 Bonn, Germany'
author:
- Gregorio Herdoíza
- Karl Jansen
- Chris Michael
- Konstantin Ottnad
- Carsten Urbach
bibliography:
- 'bibliography.bib'
title: 'Determination of Low-Energy Constants of Wilson Chiral Perturbation Theory'
---
{width="0.13\linewidth"}
,
,
,
,
Lattice QCD, Wilson Fermions, Chiral Perturbation Theory. 12.38.Gc ,12.39Fe\
Preprint-No: DESY 13-043, FTUAM-13-127, IFT-UAM/CSIC-13-015, MITP/13-015, SFB/CPP-13-18\
Introduction
============
Lattice QCD simulations employing a discretisation of the Dirac operator based on the original proposal by Wilson [@Wilson:1974sk] are currently being performed with light dynamical fermions [@DelDebbio:2006cn; @Boucaud:2007uk; @Lin:2008pr; @Durr:2008zz; @Aoki:2009ix; @Baron:2010bv; @Durr:2010aw; @Bietenholz:2011qq]. When decreasing the light quark mass at a fixed value of the lattice spacing, a subtle interplay between mass and discretisation effects can take place due to the explicit breaking of chiral symmetry by the Wilson term. In simulations with light values of the quark mass it is, therefore, vital to understand and monitor the discretisation effects and to obtain a quantitative measure of their size.
Close to the continuum limit, a useful way to determine the discretisation effects in the regime of light quark masses is provided by Wilson chiral perturbation theory (W$\chi$PT), an extension of the continuum chiral effective theory including additional terms proportional to powers of the lattice spacing [@Sharpe:1998xm]. Depending on the order of the expansion, additional low energy constants (LECs) appear, whose values are not known a priori: they depend on the lattice action and can only be computed from a simulation.
Knowing the values of the LECs of W$\chi$PT is of particular interest, because W$\chi$PT predicts a non-trivial phase structure for Wilson type fermions in the lattice spacing and quark mass plane [@Aoki:1984qi; @Creutz:1996bg; @Sharpe:1998xm]. Depending on the sign of a particular combination of Wilson LECs – commonly denoted as $c_2\propto -(2{W_6'}+{W_8'})$ – either the Aoki-scenario [@Aoki:1984qi] or a first order, so called Sharpe-Singleton [@Sharpe:1998xm] scenario is realised. Numerical evidence for both scenarios has been observed in lattice QCD simulations and dedicated studies of the associated phase diagrams have been performed by several groups [@Aoki:1992nb; @Blum:1994eh; @Aoki:1995ft; @Aoki:1996af; @Aoki:2001xq; @Ilgenfritz:2003gw; @Aoki:2004iq; @Farchioni:2004us; @Farchioni:2004fs; @Farchioni:2005tu; @Farchioni:2005bh; @Chiarappa:2006ae].
The Aoki scenario with a positive $c_2$ was found to be realised in quenched simulations. In dynamical simulations the Sharpe-Singleton scenario with negative $c_2$ was observed when using Wilson twisted mass fermions at maximal twist [@Farchioni:2004us; @Farchioni:2004fs; @Farchioni:2005tu; @Farchioni:2005bh]. This manifests itself in the fact that the neutral pion mass ${M_{\pi^0}}$ is lighter than the charged one, ${M_{\pi^\pm}}$, where the splitting of the squared masses is proportional to $c_2$. In turn, a measurement of the pion mass-splitting in Wilson twisted mass lattice QCD provides a way to measure $c_2$ and, hence, the LECs of the corresponding chiral effective theory. However, this way of computing $c_2$ is challenging since for the neutral pion mass disconnected contributions need to be evaluated.
There are also alternative ways to determine the LECs of W$\chi$PT. In Refs. [@Deuzeman:2011dh; @Damgaard:2011eg; @Damgaard:2012gy; @Damgaard:2013xi] they have been studied by matching the analytical predictions [@Damgaard:2010cz; @Akemann:2010em; @Splittorff:2011bj; @Splittorff:2011zk; @Splittorff:2012gp; @Kieburg:2012fw; @Damgaard:2012gy] for the spectrum of the Wilson Dirac operator – with fixed index in a finite volume – to lattice data.[^1] Determinations of the Wilson LECs have also been carried out via the spectral density of the Hermitian Wilson-Dirac operator [@Necco:2011vx; @Necco:2011jz; @Necco:2013sxa]. Lattice determinations of the pion scattering lengths have been used in Refs. [@Aoki:2008gy; @Bernardoni:2011fx], an approach that was extended to a partially quenched setup in Ref. [@Hansen:2011mc]. In a mixed action with Wilson-type sea fermions and chirally invariant valence quarks, a mixed action chiral Lagrangian [@Bar:2002nr; @Bar:2003mh] can be constructed. The corresponding LECs – in particular ${W_8'}$ – have been recently determined in the case of overlap valence quarks on a Wilson twisted mass sea [@Cichy:2012vg].
In this paper we are going to determine the LECs using a method recently proposed in Ref. [@Hansen:2011kk]. It relies on the measurement of pseudoscalar meson masses involving Wilson twisted mass fermions. In this approach, the LEC ${W_8'}$ is related to the mass-splitting between the charged pion mass ${M_{\pi^\pm}}$ and the “connected neutral pion” mass ${M_{\pi^{(0,{\rm c})}}}$. The latter is determined from the quark-connected correlation, which contributes to the neutral pion in twisted mass QCD. The “connected neutral pion” correlation function thus differs from the complete correlation function needed to determine the neutral pion by the absence of disconnected diagrams. This has the advantage that in numerical studies the “connected pion mass-splitting” should be accessible with good statistical precision. In addition, the mass-splitting, ${M_{\pi^{(0,{\rm c})}}}^2 - {M_{\pi^0}}^2$, between the connected and full neutral pion mass, provides an estimate of ${W_6'}$. We are also going to study the mass and lattice spacing dependence of these splittings arising at higher order in the W$\chi$PT expansion.
The paper is structured as follows: in the next section we collect the W$\chi$PT expressions relating the pion mass-splittings to the Wilson LECs. In Section \[sec:actions\] we present the lattice actions used in our study. The determination of the LECs from two of those lattice setups is described in Section \[sec:num\]. Finally, a qualitative comparison of the values of these LECs from different choices of the lattice action is reported in Section \[sec:discussion\].
Wilson Chiral Perturbation Theory (W$\chi$PT)
=============================================
In this section, we briefly discuss Wilson chiral perturbation theory and introduce the expressions used in our study. For a recent review on the applications of $\chi$PT to lattice QCD, we refer to Ref. [@Golterman:2009kw].
Our study will be based on the computation of pseudoscalar meson masses involving Wilson twisted mass fermions. We are therefore interested in a chiral Lagrangian involving a mass matrix $M =
m_0^{\rm R} + i\mu_\ell^{\rm R}\tau_3$, where $m_0^{\rm R}$ and $\mu_\ell^{\rm R}$ are the renormalised untwisted and twisted quark masses, respectively. The masses $m_0$ and $\mu_\ell$ appear in the Wilson twisted mass action of eq. (\[eq:sl\]).
At leading order (LO) in the power counting, $m_0\sim \mu_\ell\sim
a^2\Lambda_{\rm QCD}^3$, and after a shift of the quark mass to remove a term of ${\cal O}(a)$, the partially quenched chiral Lagrangian reads [@Sharpe:1998xm; @Bar:2003mh] $$\begin{aligned}
{\cal L}_\chi &=&
\frac{f^2}{8} \,{\textrm{Str}}\left(
\partial_\mu \Sigma \partial_\mu \Sigma^\dagger\right)
-
\frac{f^2\,B_0}{4} \,{\textrm{Str}}\left( M^\dagger \Sigma+\Sigma^\dagger M\right)
\nonumber\\
&& - \hat{a}^2 W_6' \,\left[{\textrm{Str}}\left(\Sigma+\Sigma^\dagger\right)\right]^2
- \hat{a}^2 W_7' \,\left[{\textrm{Str}}\left(\Sigma-\Sigma^\dagger\right)\right]^2
\nonumber\\
&&
- \hat{a}^2 W_8' \,{\textrm{Str}}\left(\Sigma^2+[\Sigma^\dagger]^2\right)
\,,
\label{eq:LPQWChPT}\end{aligned}$$ where $\Sigma$ parametrises the vacuum manifold and thus characterises the Nambu-Goldstone bosons arising from the spontaneous breaking of chiral symmetry. The traces over the flavour indices are denoted by Str and $\hat a = 2 W_0 a$. In addition to the continuum LECs $B_0$ and $f$ (defined with the convention giving $f_\pi\approx 130\;$MeV), the chiral Lagrangian also includes $W_0$, ${W_6'}$, $W'_7$ and ${W_8'}$, which are Wilson LECs describing discretisation effects.
In this work we are interested in the determination of the Wilson LECs by matching lattice QCD calculations of pseudoscalar meson masses to their PQW$\chi$PT expressions. As already mentioned, we consider Wilson twisted mass fermions at maximal twist. This is achieved in the chiral Lagrangian by setting $m_0=0$. At non vanishing values of the lattice spacing, the breaking of flavour symmetry by the twisted mass term in eq. (\[eq:sl\]) implies that the mass of the charged pion ${M_{\pi^\pm}}$ differs from that of the neutral pion ${M_{\pi^0}}$ by O($a^2$) effects. A similar pattern holds for the mass of the “connected neutral pion” mass, ${M_{\pi^{(0,{\rm c})}}}$. The PQW$\chi$PT expressions for these three meson masses at LO read [@Munster:2004am; @Scorzato:2004da; @Sharpe:2004ps; @Hansen:2011kk], $$\begin{aligned}
{M_{\pi^\pm}}^2 \ &=&\ 2 B_0 \mu_\ell \,,
\label{eq:mpc}
\\
{M_{\pi^0}}^2 \ &=&\ 2 B_0 \mu_\ell - 8a^2 \,(2 w'_6 + w'_8) \,,
\label{eq:mpn}
\\
{M_{\pi^{(0,{\rm c})}}}^2 \ &=&\ 2 B_0 \mu_\ell - 8a^2\,w'_8
\,,
\label{eq:mpinc}\end{aligned}$$ where $w'_{k}$ is related to the Wilson LEC $W'_{k}$ by $$\begin{aligned}
w'_k \ =\ \frac{16 W_0^2\,W'_k}{f^2} \qquad (k=6,8)
\,. \label{eq:wk}\end{aligned}$$ To determine the individual values of the LECs it is, therefore, possible to consider the following mass-splittings: $$\begin{aligned}
{M_{\pi^\pm}}^2 - {M_{\pi^{(0,{\rm c})}}}^2 \ &=& \ 8a^2\, w'_8\,, \label{eq:w8ms}\\
\frac{1}{2} \, \left( {M_{\pi^{(0,{\rm c})}}}^2 - {M_{\pi^0}}^2 \right) \ &=& \ 8a^2\, w'_6\,. \label{eq:w6ms}\end{aligned}$$ From eq. (\[eq:mpn\]), it appears that the linear combination of LECs which controls the mass-splitting between charged and neutral pions is given by $$c_2 \ = \ -\frac{32 W_0^2}{f^2}(2 {W_6'}+ {W_8'})\,.
\label{eq:c2}$$ This can also be re-expressed as $$\begin{aligned}
c_2 \ &=& \ \frac{1}{4a^2}\, \left( {M_{\pi^0}}^2 - {M_{\pi^\pm}}^2 \right) \,,
\label{eq:c2ms}\\ ~ \nonumber \\
&=& \ -2 \, ( 2 w'_6 + w'_8) \,.
\label{eq:c2w}\end{aligned}$$ In the W$\chi$PT expressions presented above, two light mass-degenerate flavours were assumed to be present in the sea sector. When considering also other dynamical flavours, such as the strange and the charm quarks, the same expressions hold when assuming that these heavier flavours sufficiently decouple from the light quark sector. In this case, the values of the Wilson LECs will have a further residual dependence on the heavier quark masses.
Before closing this section, we mention that W$\chi$PT calculations at NLO have been carried out for the pion mass and decay constant with ${N_\mathrm{f}=2}$ [@Colangelo:2010cu; @Bar:2010jk; @Ueda:2011ib] and ${N_\mathrm{f}=2+1+1}$ [@Munster:2011gh] flavours of twisted mass fermions.
Lattice actions {#sec:actions}
===============
The complete lattice action can be written as $$S= S_f + S_g\, ,$$ where $S_f$ is the fermionic action and $S_g$ is the pure gauge action. As we shall see, in this work we will consider a few alternatives for both the fermionic and the gauge actions in order to explore the dependence of the Wilson LECs on the details of the lattice action. As discussed below, we will use for $S_F$ a few variants of Wilson twisted mass fermions.
Wilson Twisted Mass Fermions
----------------------------
The Wilson twisted mass (Wtm) lattice action for the mass degenerate light doublet $(u,d)$ in the so-called twisted basis reads [@Frezzotti:2000nk; @Frezzotti:2003ni], $$\label{eq:sl}
S_l\ =\ a^4\sum_x\left\{ \bar\chi_l(x)\left[ D[U] + m_{0,l} +
i\mu_\ell\gamma_5\tau_3\right]\chi_l(x)\right\}\, ,$$ where $m_{0,l}$ is the untwisted bare quark mass, $\mu_\ell$ is the bare twisted light quark mass, $\tau_3$ is the third Pauli matrix acting in flavour space and $$\label{eq:DW}
D[U] = \frac{1}{2}\left[\gamma_\mu\left(\nabla_\mu +
\nabla^*_\mu\right) -a\nabla^*_\mu\nabla_\mu \right]\,,$$ is the massless Wilson-Dirac operator. $\nabla_\mu$ and $\nabla^*_\mu$ are the forward and backward gauge covariant difference operators, respectively. Twisted mass light fermions are said to be at maximal twist if the bare untwisted mass $m_{0,l}$ is tuned to its critical value, $m_{\rm crit}$. The quark doublet $\chi_l=(\chi_u, \chi_d)$ in the twisted basis is related by a chiral rotation to the quark doublet in the physical basis $$\psi_{l}^{phys} = e^{\frac{i}{2}\omega_{l}\gamma_{5}\tau_{3}} \, \chi_{l},
\qquad \bar{\psi}_{l}^{phys}=\bar{\chi}_{l} \, e^{\frac{i}{2}\omega_{l}\gamma_{5}\tau_{3}}\, ,$$ where $\omega_{l}$ is the twist angle.
The twisted mass parameter $\mu_\ell$ provides an infrared regulator avoiding the presence of accidental zero-modes in the Wilson-Dirac operator. An important property of Wtm fermions is that at maximal twist physical observables are O($a$) improved [@Frezzotti:2003ni]. In numerical simulations, maximal twist is achieved by tuning the value of the hopping parameter $\kappa=1/(2m_{0,l}+8)$ to its critical value $\kappa_{\rm crit}$ by tuning the PCAC quark mass ${m_\mathrm{PCAC}}$ to zero. The expected O($a^2$) scaling of physical observables when performing the continuum limit extrapolation has been confirmed in the quenched approximation [@Jansen:2003ir; @Jansen:2005gf; @Jansen:2005kk; @Abdel-Rehim:2005gz] and with ${N_\mathrm{f}=2}$ [@Urbach:2007rt; @Dimopoulos:2007qy; @Alexandrou:2008tn; @Baron:2009wt] and ${N_\mathrm{f}=2+1+1}$ [@Baron:2010bv; @Drach:2010hy; @Herdoiza:2011gp] dynamical quarks.
A peculiar lattice artifact can appear in observables made out of Wtm quarks due to the breaking of isospin and parity by the twisted mass term in eq. (\[eq:sl\]). This effect, which is expected to vanish in physical quantities at a rate of O($a^2$) when approaching the continuum limit, has been observed to be numerically small in most of the observables which have been analysed [@Urbach:2007rt; @Dimopoulos:2007qy; @Alexandrou:2008tn; @Baron:2009wt; @Baron:2010bv; @Drach:2010hy]
An exception to this observed small isospin breaking effects is found in the case of the neutral pseudoscalar mass. Indeed, isospin breaking induces a mass-splitting between charged and neutral pion masses. Dedicated numerical studies indicate that while in the charged pion mass only very mild cutoff effects are present, the neutral pion mass is instead affected by significant ${\mathcal{O}(a^2)}$ effects [@Jansen:2005cg; @Urbach:2007rt; @Baron:2009wt; @Baron:2010bv; @Herdoiza:2011gp]. An analysis based on the Symanzik expansion indicates that isospin breaking affects only a limited set of observables in a sizeable way, namely the neutral pion mass and kinematically related quantities [@Frezzotti:2007qv; @Dimopoulos:2009qv]. This analysis is complementary to that based on W$\chi$PT where, as previously mentioned, the mass-splitting between charged and neutral pions is parametrised by the combination of LECs appearing in $c_2$ defined in eq. (\[eq:c2\]).
The determination of the neutral pion mass ${M_{\pi^0}}$ involves both connected and disconnected contributions. The computation of quark-disconnected diagrams is challenging and requires the employment of specific techniques in order to achieve a statistically significant determination of ${M_{\pi^0}}$ [@Michael:2007vn; @Boucaud:2008xu].
### ${N_\mathrm{f}=2+1+1}$ Wtm fermions {#n_mathrmf211-wtm-fermions .unnumbered}
In addition to a doublet of mass-degenerate light quarks ($u$,$d$), a heavier doublet with strange and charm quarks – ($s$,$c$) – can be incorporated in lattice QCD studied with Wilson twisted mass fermions [@Frezzotti:2004wz; @Frezzotti:2003xj]. Also in this ${N_\mathrm{f}=2+1+1}$ setup, a tuning to maximal twist by imposing ${m_\mathrm{PCAC}}=0$, allows to achieve the automatic O($a$) improvement of physical observables. We refer to Ref. [@Baron:2010bv] for a complete description of the lattice action for the heavier quark doublet and for further details on the lattice setup.
### Mixed action with Osterwalder-Seiler valence quarks {#mixed-action-with-osterwalder-seiler-valence-quarks .unnumbered}
Osterwalder-Seiler (OS) valence quarks [@Osterwalder:1977pc] can be viewed as the building blocks of Wilson twisted mass fermions at maximal twist. The OS action for an [*individual*]{} quark flavour $\chi_f$ reads $$\label{eq:os}
S_f^\mathrm{OS}\ =\ a^4\sum_x\left\{ \bar\chi_f(x)\left[ D[U] + m_\mathrm{crit} +
i\mu_f\gamma_5 r_f\right]\chi_f(x)\right\}\, ,$$ where $r_f$ (here $|r_f|=1$) is the Wilson parameter and $m_{\rm
crit}$ the critical mass. By combining two flavours of OS quarks with opposite signs of $r_f$, e.g. $r_2 = -r_1$, the action of a doublet of maximally Wtm fermions of mass $\mu_f=\mu_1=\mu_2$ can be recovered. The benefits of the OS action are that ${\mathcal{O}(a)}$ improved physical observables [@Frezzotti:2004wz] can be obtained by using the same estimates of $m_{\rm crit}$ as in the Wtm case and, thus, avoiding further tuning effort. OS and Wtm fermions coincide with Wilson fermions in the massless limit and consequently share the same renormalisation factors. This simplifies the matching of sea and valence quark masses in the context of a mixed-action with Wtm sea and OS valence quarks.
The pseudoscalar correlation function obtained when considering only the connected contribution to the neutral pion correlation function ([*i.e.*]{} when ignoring disconnected diagrams), is precisely the pion correlator with OS fermions. The mass-splittings in eqs.(\[eq:w8ms\])-(\[eq:w6ms\]) can hence be interpreted as involving sea and valence quarks in a mixed action setup.
In this work, we aim at determining the Wilson LECs in a lattice theory with ${N_\mathrm{f}=2+1+1}$ Wtm fermions. A particular effort will be devoted to addressing the main systematic effects present in these determinations. We furthermore aim at exploring the qualitative change on the values of these LECs when varying the details of the lattice action. Below, we briefly summarise the alternative lattice setups used in this work. We will consider variants of the action differing by the presence of smearing of the gauge links in the covariant derivative, the inclusion of the Sheikholeslami-Wohlert term or by a change in the number of dynamical flavours.
### Stout Smearing. {#stout-smearing. .unnumbered}
A smearing procedure can be applied to the gauge links entering in the covariant derivatives in eq. (\[eq:DW\]). The stout smearing [@Morningstar:2003gk] procedure is analytic in the un-smeared link variables and hence well suited for simulations with the HMC algorithm. The smearing can be iterated several times, with the price of extending the coupling of fermions to the gauge links over a larger region.
### Sheikholeslami-Wohlert term. {#sheikholeslami-wohlert-term. .unnumbered}
In our comparison of the values of the mass-splittings in eqs. (\[eq:w8ms\])-(\[eq:w6ms\]) from different lattice setups, we will also consider results available in the literature from quenched lattice simulations with Wtm quarks including the Sheikholeslami-Wohlert term [@Sheikholeslami:1985ij].
Gauge action {#sec:gauge}
------------
The lattice gauge actions considered in this work have a generic form which includes a plaquette term $U^{1\times1}_{x,\mu,\nu}$ and rectangular $(1\times2)$ Wilson loops $U^{1\times2}_{x,\mu,\nu}$, $$\label{eq:Sg}
S_g = \frac{\beta}{3}\sum_x\Biggl( b_0\sum_{\substack{
\mu,\nu=1\\1\leq\mu<\nu}}^4\{1-{\operatorname{Re}}{\operatorname{Tr}}(U^{1\times1}_{x,\mu,\nu})\}\Bigr.
\Bigl.\,+\,
b_1\sum_{\substack{\mu,\nu=1\\\mu\neq\nu}}^4\{1
-{\operatorname{Re}}{\operatorname{Tr}}(U^{1\times2}_{x,\mu,\nu})\}\Biggr)\, ,$$ with $\beta=6/g_0^2$ the bare inverse coupling and the normalisation condition $b_0=1-8b_1$. We will consider the case of the Wilson plaquette [@Wilson:1974sk] action ($b_1=0$), the tree-level Symanzik improved [@Weisz:1982zw; @Weisz:1983bn] action ($b_1=-1/12$) and the Iwasaki [@Iwasaki:1985we; @Iwasaki:1996sn; @Iwasaki:2011jk] action ($b_1=-0.331$). In Wtm simulations, the strength of the phase transition has been found [@Farchioni:2004fs; @Farchioni:2005tu] to depend on the value of the parameter $b_1$ in eq. (\[eq:Sg\]).
Numerical Studies {#sec:num}
=================
${N_\mathrm{f}=2+1+1}$ Wtm fermions with Iwasaki gauge action
-------------------------------------------------------------
The purpose of this study is to determine the mass-splittings in eqs. (\[eq:w8ms\]) and (\[eq:w6ms\]), which are directly related to the Wilson LECs ${W_8'}$ and ${W_6'}$, respectively. The lattice action is composed of the Iwasaki gauge action and ${N_\mathrm{f}=2+1+1}$ flavours of Wilson twisted mass fermions.
The simulations [@Baron:2010bv] were performed at three values of the lattice gauge coupling, $\beta =1.90$, $1.95$ and $\beta=2.10$, corresponding to values of the lattice spacing $a\approx 0.09$fm, $0.08$fm and $0.06$fm, respectively. The charged pion mass ${M_{\pi^\pm}}$ approximately ranges from $230$MeV to $510$MeV. Simulated volumes correspond to values of ${M_{\pi^\pm}}L$ larger than $3.3$. Physical spatial volumes range from $(1.9\,\mathrm{fm})^3$ to $(2.8\,\mathrm{fm})^3$.
The values of the pseudoscalar meson masses for the ${N_\mathrm{f}=2+1+1}$ ensembles are collected in Table \[tab:nf211\].
[1.0]{}[@lccccccc]{} Ens. & $\beta$ & $L/a$ & $a\mu_\ell$ & $a{M_{\pi^\pm}}$ & $a{M_{\pi^{(0,{\rm c})}}}$ & $a{M_{\pi^0}}$ & $r_0/a$\
A30.32 & 1.90 & 32 & 0.0030 & 0.1234(03) & 0.2111(33) & 0.0611(036) & 5.23(4)\
A40.32 & & & 0.0040 & 0.1415(04) & 0.2274(31) & 0.0811(050) &\
A40.24 & & 24 & 0.0040 & 0.1445(06) & 0.2375(25) & 0.0694(065) &\
A60.24 & & & 0.0060 & 0.1727(06) & 0.2544(26) & 0.1009(113) &\
A80.24 & & & 0.0080 & 0.1987(06) & 0.2659(25) & 0.1222(157) &\
A100.24 & & & 0.0100 & 0.2215(04) & 0.2883(14) & 0.1570(178) &\
A80.24s & 1.90 & 24 & 0.0080 & 0.1982(04) & 0.2649(16) & 0.1512(115) &\
A100.24s & & & 0.0100 & 0.2215(04) & 0.2841(16) & 0.1863(141) &\
B25.32 & 1.95 & 32 & 0.0025 & 0.1064(07) & 0.1836(21) & 0.0605(036) & 5.71(4)\
B35.32 & & & 0.0035 & 0.1249(07) & 0.1919(17) & 0.0710(061) &\
B55.32 & & & 0.0055 & 0.1540(04) & 0.2177(19) & 0.1323(080) &\
B75.32 & & & 0.0075 & 0.1808(05) & 0.2360(12) & 0.1557(126) &\
B85.24 & & 24 & 0.0085 & 0.1931(08) & 0.2480(11) & 0.1879(180) &\
D15.48 & 2.10 & 48 & 0.0015 & 0.0695(03) & 0.1124(15) & 0.0561(031) & 7.46(6)\
D20.48 & & & 0.0020 & 0.0797(05) & 0.1170(16) & 0.0651(042) &\
D30.48 & & & 0.0030 & 0.0978(04) & 0.1296(15) & 0.0860(046) &\
D45.32sc & 2.10 & 32 & 0.0045 & 0.1198(05) & 0.1480(09) & 0.0886(095) &\
The mass-splittings in eqs. (\[eq:w8ms\])-(\[eq:w6ms\]), which are directly proportional to ${W_8'}$ and ${W_6'}$ respectively, are illustrated in Fig. \[fig:MS\_con\_211\]. We observe that the conditions ${W_8'}<0$ and ${W_6'}>0$ are fulfilled by the lattice data, in agreement with the bounds derived in Refs. [@Damgaard:2010cz; @Hansen:2011kk; @Kieburg:2012fw]. In order to quote the values of the Wilson LECs $W'_{6,8}$, the systematic effects from quark-mass dependence, residual lattice artifacts and finite volume effects have to be addressed.
In the “large cut-off effects” power counting and at LO in the W$\chi$PT chiral Lagrangian, the mass-splittings in eqs. (\[eq:w8ms\])-(\[eq:w6ms\]) are expected to be independent of the lattice spacing and the light-quark mass. The possible presence of such effects might thus signal effects entering at higher orders in the W$\chi$PT chiral expansion. Since the NLO expressions for these mass-splittings is currently not available in the literature, we rely in our systematic error analysis on a separate study of (i) the continuum-limit of the mass-splittings at a reference mass and (ii) the comparison of a constant and a linear chiral extrapolation in ${M_{\pi^\pm}}^2$.
Starting with point (i), we show in Fig. \[fig:MS\_con\_211\] the mass splittings eq. (\[eq:w8ms\]) and (\[eq:w6ms\]) relevant for ${W_8'}$ and ${W_6'}$, respectively, as a function of ${M_{\pi^\pm}}^2$. We first observe that data points with similar values of $({M_{\pi^\pm}}r_0)^2$, but coming from different lattice spacings, tend to be compatible with each other, in particular when considering the larger lattice sizes, represented by the filled symbols. The lattice spacing dependence of the two aforementioned mass splittings at a reference mass $({M_{\pi^\pm}}r_0)^2 \approx 0.55$ is illustrated in Fig.\[fig:MS\_a\_c2\_211a\]. Note that only the largest lattice sizes $L$ are considered in this figure. Although the lattice size slightly varies when changing $\beta$, the lattice data fulfils $L \gtrsim 2.5$fm and ${M_{\pi^\pm}}L \gtrsim 4$ and, therefore, we do not expect a large effect from a small mismatch in the physical volume. Fig.\[fig:MS\_a\_c2\_211a\] suggests that the residual lattice spacing effects are small. We remind that these lattice artifacts appear beyond the leading O($a^2$) effects. As an estimate of these effects, we include in our systematic error analysis the difference between the values of the mass-splittings from the two finer lattice spacings.
Concerning the light-quark mass dependence in point (ii), we can expect that the mass terms appearing at NLO can contain a linear term in ${M_{\pi^\pm}}^2$ but also a term of the form ${M_{\pi^\pm}}^2 \log({M_{\pi^\pm}}^2)$. Indeed, such terms are present in the W$\chi$PT expression of the mass splitting between the charged and neutral pion masses at NLO [@Bar:2010jk]. Since the precise form of the these logarithmic terms is yet unknown for the mass-splittings considered here, we limit ourselves to a linear chiral extrapolation in ${M_{\pi^\pm}}^2$. Note that our data is not precise enough to disentangle possible logarithmic contributions. We take as our central values the linearly extrapolated mass-splittings and use the difference with respect to the constant fit as an estimate of the systematic error from the light-quark mass dependence.
Finite volume effects are taken into account by adding to the systematic error the difference between the values of the mass splittings from two ensembles – A40.24 and A40.32 – differing only by a change of lattice size from $L \approx 2.1$fm to $2.8$fm. This is expected to be a conservative choice since these ensembles correspond to a rather small light-quark mass – and therefore finite size effects can be non-negligible. Also, these ensembles were obtained at the coarsest lattice spacing, where possible finite size effects (FSE) from the neutral pion mass should be larger.
As already mentioned, the determination of the Wilson LECs $W'_{6,8}$ from lattice data with ${N_\mathrm{f}=2+1+1}$ flavours assumes that the strange and the charm sea-quarks decouple sufficiently from the light-quark dynamics. The residual heavy quark mass dependence present in $W'_{6,8}$ can be studied by varying the strange and charm quark masses in the neighbourhood of their physical values. This effect is illustrated in Fig. \[fig:MS\_con\_211\] by the points labelled by “(s,c)” in the legend. We use the difference between the mass-splittings from ensembles A80.24 and A80.24s – which only differ by a change of the strange and charm quark masses – to estimate this systematic effect. We expect that this is a conservative choice because (a) the ensemble A80.24s has a strange quark mass which is very close to the physical point, (b) the change in the strange quark mass is largest for A80.24 and A80.24s and (c) the effect of strange sea-quarks should be larger than that of charm quarks.
After combining the previously discussed systematic uncertainties in quadrature, we obtain the following values for the mass-splittings for the case of a lattice setup with ${N_\mathrm{f}=2+1+1}$ Wtm fermions and the Iwasaki gauge action, $$\begin{aligned}
\left( \frac{ {M_{\pi^\pm}}^2 - {M_{\pi^{(0,{\rm c})}}}^2 }{ a^2 } \right) r_0^4\ &=&~-23.0
\pm 0.7 \pm 3.0\,, \label{eq:vw8msnf211}\\
\left( \frac{ {M_{\pi^{(0,{\rm c})}}}^2 - {M_{\pi^0}}^2 }{ 2a^2 } \right) r_0^4\ &=&~+13.8
\pm 0.6 \pm 5.6\,, \label{eq:vw6msnf211}\end{aligned}$$ where the first error is statistical and the second systematic. The corresponding values of the Wilson LECs are collected in Tab. \[tab:WLECnf211\]. As already anticipated, the results for $w'_{6,8}$ are precise enough to identify a definite sign for these LECs.
[0.8]{}[@cccc]{} & $w'_8\, r_0^4$ & $w'_8$ & ${W_8'}\, (r_0^6 W_0^2)$\
syst. & -2.9(4) & $-[571(32)\,{\rm MeV}]^4$ & -0.0138(22)\
& $w'_6\, r_0^4$ & $w'_6$ & ${W_6'}\, (r_0^6 W_0^2)$\
syst. & +1.7(7) & $+[502(58)\,{\rm MeV}]^4$ & +0.0082(34)\
& $c_2\,r_0^4$ & $c_2$ & $-2\, (2{W_6'}+{W_8'})\, (r_0^6 W_0^2)$\
lin. & -1.1(2) & $-[444(28)\,{\rm MeV}]^4$ & -0.0050(10)\
cst. & -2.3(1) & $-[541(24)\,{\rm MeV}]^4$ & -0.0111(10)\
The combination of LECs $c_2$, can be determined directly from the mass-splitting ${M_{\pi^0}}^2 - {M_{\pi^\pm}}^2$ as indicated in eq. (\[eq:c2ms\]). The measurements of this mass-splitting are illustrated in Fig. \[fig:MS\_a\_c2\_211b\] where the results of a chiral extrapolation by using a constant and a linear fit in ${M_{\pi^\pm}}^2$ are also shown. For the case of $c_2$, we provide in Tab. \[tab:WLECnf211\] the results from both these chiral extrapolations and quote in the individual numbers only the statistical error. The values arising from these extrapolations are both compatible with a negative sign of $c_2$.
The W$\chi$PT expressions at NLO relevant for $c_2$ have been derived in Ref [@Bar:2010jk]. In addition to the Wilson LECs appearing at LO and to the usual Gasser-Leutwyler LECs, other parameters also appear at NLO. A complete determination of these parameters is beyond the scope of this study. We postpone such an analysis to a future dedicated study of the W$\chi$PT description of both the pion mass and decay constant. First results for the Gasser-Leutwyler LECs, from fits based on continuum $\chi$PT, have been presented in Refs. [@Baron:2010bv; @Baron:2011sf].
The connected neutral pion mass, ${M_{\pi^{(0,{\rm c})}}}$, is an important ingredient in order to isolate the individual values of the LECs $W'_{6,8}$. As already pointed out, the connected neutral pion can be interpreted as the pion of a mixed action with OS fermions. Such a mixed action has been used to determine observables in the Kaon sector [@Constantinou:2010qv; @Farchioni:2010tb; @Herdoiza:2011gp]. We note that extensions of the analytical expressions to SU(3) W$\chi$PT is currently not available in the literature. Contrary to the pion case, the absence of disconnected diagrams in correlations functions in the Kaon sector could possibly allow to consider quantities from which the Wilson LECs can be determined with good accuracy.
${N_\mathrm{f}=2}$ Wtm fermions with tlSym gauge action
-------------------------------------------------------
In this section, we again determine the LECs $W'_{6,8}$, but this time using ${N_\mathrm{f}=2}$ flavours of Wilson twisted mass fermions and the tree-level Symanzik improved gauge action [@Boucaud:2007uk; @Boucaud:2008xu; @Baron:2009wt]. We already anticipate that a smaller set of ensembles and of measurements of the relevant pion masses are available in this case, in comparison to the ${N_\mathrm{f}=2+1+1}$ case discussed previously. Therefore, the resulting determinations and comparisons might suffer from insufficient control of systematic effects. However, we think that already a qualitative comparison can provide useful information to parametrise the size of cutoff effects from different lattice setups.
[1.0]{}[@ccccccc]{} $\beta$ & $L/a$ & $a\mu_\ell$ & $a{M_{\pi^\pm}}$ & $a{M_{\pi^{(0,{\rm c})}}}$ & $a{M_{\pi^0}}$ & $r_0/a$\
3.90 & 32 & 0.0040 & 0.1338(02) & 0.2080(30) & 0.1100(080) & 5.35(4)\
& 24 & 0.0040 & 0.1362(07) & 0.2120(30) & 0.1090(070) &\
& 16 & 0.0040 & 0.1596(30) & 0.2226(95) & - &\
& 24 & 0.0064 & 0.1694(04) & - & 0.1340(100) &\
& & 0.0085 & 0.1940(05) & - & 0.1690(110) &\
& 16 & 0.0074 & 0.1963(17) & 0.2541(55) & - &\
4.05 & 32 & 0.0030 & 0.1038(06) & 0.1500(30) & 0.0900(060) & 6.71(4)\
& 20 & 0.0030 & 0.1191(41) & 0.1571(62) & - &\
& 32 & 0.0060 & 0.1432(06) & 0.1800(20) & 0.1230(060) &\
4.20 & 24 & 0.0020 & 0.0941(31) & 0.1157(61) & - & 8.36(6)\
The simulations considered in this work [@Baron:2009wt; @Cichy:2010ta] were performed at three values of the lattice gauge coupling $\beta =3.90$, $4.05$ and $\beta=4.20$, corresponding to values of the lattice spacing $a\approx 0.08$fm, $0.07$fm and $0.05$fm, respectively. The charged pion mass ${M_{\pi^\pm}}$ approximately ranges from $310$MeV to $460$MeV. Physical spatial volumes range from $(1.3\,\mathrm{fm})^3$ to $(2.6\,\mathrm{fm})^3$ and ensembles which differ only by the lattice size have been considered in order to address the size of finite volume effects in the determination of the Wilson LECs.
The values of the pseudoscalar meson masses [@Baron:2009wt] for the ${N_\mathrm{f}=2}$ ensembles are collected in Table \[tab:nf2\]. The mass-splittings in eqs. (\[eq:w8ms\])-(\[eq:w6ms\]) are illustrated in Fig. \[fig:MS\_con\_2\].
In order to explore the systematic effects present in these determinations, we follow a similar path to that described for the case of ${N_\mathrm{f}=2+1+1}$ ensembles. The availability of ensembles differing only by the physical volume allows to address the size of FSE in the mass-splittings. This is illustrated in Fig. \[fig:MS\_L\_c2\_2a\]. At $\beta=3.90$, a set of three ensembles with $L/a=16,~24$ and $32$ could be used for the case of $({M_{\pi^\pm}}^2 - {M_{\pi^{(0,{\rm c})}}}^2)/a^2$. With the current statistical uncertainties, no clear signs of FSE can be observed in the data. Furthermore, data from two different lattice spacings – $\beta=3.90$ and $4.05$ – agree within errors, indicating the absence of large residual lattice artifacts in these mass-splittings.
However, the lack of sufficient data does not allow to address the mass dependence of the mass-splitting $({M_{\pi^{(0,{\rm c})}}}^2 - {M_{\pi^0}}^2)/a^2$ in a satisfactory way. In analogy to the ${N_\mathrm{f}=2+1+1}$ case, we include the deviation between a constant and a linear extrapolation in ${M_{\pi^\pm}}^2$ in the estimate of the systematic uncertainties. The central value is taken from the result of the linear fit.
For the case of a lattice setup with ${N_\mathrm{f}=2}$ Wtm fermions at maximal twist and the tlSym gauge action, we obtain the following values for the mass-splittings $$\begin{aligned}
\left( \frac{ {M_{\pi^\pm}}^2 - {M_{\pi^{(0,{\rm c})}}}^2 }{ a^2 } \right) r_0^4\ &=&~-20.1
\pm 2.3 \pm 1.7\,, \label{eq:vw8msnf2}\\
\left( \frac{ {M_{\pi^{(0,{\rm c})}}}^2 - {M_{\pi^0}}^2 }{ 2a^2 } \right) r_0^4\ &=&~~+8.4
\pm 3.3 \pm 5.5\,, \label{eq:vw6msnf2}\end{aligned}$$ where the first error is statistical and the second systematic. The corresponding values of the Wilson LECs are collected in Tab. \[tab:WLECnf2\].
[0.6]{}[@cccc]{} $w'_8\, r_0^4$ & $w'_8$ & ${W_8'}\, (r_0^6 W_0^2)$\
-2.5(4) & $-[552(025)\,{\rm MeV}]^4$ & -0.0119(17)\
$w'_6\, r_0^4$ & $w'_6$ & ${W_6'}\, (r_0^6 W_0^2)$\
+1.0(8) & $+[443(138)\,{\rm MeV}]^4$ & +0.0049(38)\
The LEC $W'_{8}$ has recently been determined from a mixed action involving the same ${N_\mathrm{f}=2}$ lattice action in the sea sector as that described here, but with Neuberger overlap valence quark [@Cichy:2012vg]. The value quoted in Ref. [@Cichy:2012vg], ${W_8'}\, (r_0^6 W_0^2)=-0.0064(24)$, differs from the estimate in Tab. \[tab:WLECnf2\] at the 2-sigma level. However, we stress once more that our present ${N_\mathrm{f}=2}$ estimate does not include a complete assessment of the systematic errors.
The determination of $c_2$ from ${N_\mathrm{f}=2}$ ensembles is illustrated in Fig.\[fig:MS\_L\_c2\_2b\]. More data would be needed to isolate the residual mass-dependence present in $c_2$. For this reason we opt for quoting separately the results of a constant and a linear chiral extrapolation in ${M_{\pi^\pm}}^2$, $$\begin{aligned}
c_2\,r_0^4\,[\,{\rm cst.}\,]\ &=&~-3.1
\pm 0.4\,, \label{eq:c2cstnf2}\\
c_2\,r_0^4\,[\,{\rm lin.}\,]\ &=&~-0.5
\pm 1.5\,, \label{eq:c2linnf2}\end{aligned}$$ where the errors are statistical only. These values are consistent with those arising from the measurements of pseudoscalar meson masses in Ref. [@Baron:2009wt] and are also very similar to the result obtained in Ref. [@Colangelo:2010cu] for $K=-4c_2$ from twisted mass finite volume effects. All the lattice measurements favour a negative sign of $c_2$. However, as already mentioned, more data would be needed to properly address the residual mass dependence.
We refer to Refs. [@Baron:2009wt; @Blossier:2010cr] for more details about the description of the pion mass and decay constant by means of $\chi$PT expressions including discretisation effects.
Discussion {#sec:discussion}
==========
In this section we collect a few comments concerning the extraction of the Wilson LECs. The LECs ${W_6'}$ and $c_2$ depend on the neutral pion mass ${M_{\pi^0}}$. The statistical error in ${M_{\pi^0}}$ is dominated by the contribution of disconnected diagrams.[^2] We observe that the precise form of the light-quark mass dependence of the mass-splittings related to ${W_6'}$ and $c_2$ – see e.g. Figs. \[fig:MS\_con\_211b\] and \[fig:MS\_a\_c2\_211b\] – cannot be addressed within the present uncertainties. As previously discussed this mass dependence can arise at NLO in the W$\chi$PT expansion. We can, however, not exclude at this stage that this higher-order effects are negligible. This issue is particularly relevant for the case of $c_2 \propto -(2 w'_6+ w'_8)$, where a partial cancellation of the effect of $w'_6$ and $w'_8$ is present. The presence of higher order effects in the determination of $c_2$ has also been discussed in Ref. [@Bernardoni:2011fx]. We recall that the sign of $c_2$ controls the appearance of an Aoki ($c_2
> 0$) or of a Sharpe-Singleton ($c_2 < 0$) scenario for the phase structure of Wilson fermions.
From the previously discussed determinations, a comparison of the values of the Wilson LECs from the lattice actions (i) ${N_\mathrm{f}=2+1+1}$ flavours of Wtm fermions at maximal twist and Iwasaki gauge action and (ii) ${N_\mathrm{f}=2}$ flavours of Wtm quarks and tlSym gauge action, suggests that $W'_{6,8}$ and $c_2$ do not vary significantly in between these two setups. Let us extend this observation by performing a comparison of the pion mass-splittings as determined from different lattice actions. We stress that a complete assessment of the overall uncertainty is not available for most of these measurements and therefore the comparison remains at the qualitative level.
Lattice simulations with ${N_\mathrm{f}=2+1+1}$ Wtm fermions and the Iwasaki gauge action, but including in addition one iteration of stout smearing – labelled 1-stout – have been reported in Ref. [@Baron:2010bv]. A qualitative comparison of the effect of the stout smearing on the size of the mass-splittings is shown in Fig. \[fig:comparison\]. Note that a single ensemble is used in the estimate of the mass-splittings for the case of stout smearing. The used smearing seems to help in reducing the magnitude of the splitting $({M_{\pi^\pm}}^2 - {M_{\pi^{(0,{\rm c})}}}^2)/a^2$ while – given the current uncertainties –it does not introduce a significant change in $({M_{\pi^{(0,{\rm c})}}}^2 - {M_{\pi^0}}^2)/a^2$.
Fig. \[fig:comparison\](a) also includes the estimates of the mass-splitting $({M_{\pi^\pm}}^2 - {M_{\pi^{(0,{\rm c})}}}^2)/a^2$ as determined from quenched ensembles either with or without the presence of the Sheikholeslami-Wohlert term. Maximally twisted-mass fermions and the plaquette gauge action are used in both cases. The values of the mass-splitting are derived from studies available in the literature. For the case without the Sheikholeslami-Wohlert term, results are taken from Ref. [@Jansen:2005cg]. We use the information from different lattice spacings and quark masses to estimate the size of the systematic effects in $({M_{\pi^\pm}}^2 -
{M_{\pi^{(0,{\rm c})}}}^2)/a^2$. For the case in which the Sheikholeslami-Wohlert term is included – labelled $c_{\rm SW}$ – we follow Ref. [@Dimopoulos:2009es], where the non-perturbative determination of $c_{\rm SW}$ was used.
For cases other than those involving the Sheikholeslami-Wohlert term or stout-smearing, Fig. \[fig:comparisona\] suggests that, with the current precision, the value of the mass-splitting $({M_{\pi^\pm}}^2 -
{M_{\pi^{(0,{\rm c})}}}^2)/a^2$ does not significantly depend on a simultaneous change of the number of flavours $N_{\rm f}$ and of the gauge action. Note however that we cannot exclude that a change only in $N_{\rm f}$ or only in the parameter $b_1$ of the gauge action, leads to a different conclusion. It would be very desirable, if further actions are investigated and the precision could be increased.
One important observation arising from the measurements in Ref. [@Dimopoulos:2009es] is that the introduction of the Sheikholeslami-Wohlert term in quenched studies significantly reduces the size of O($a^2$) effects by lowering the value of the mass-splitting $({M_{\pi^\pm}}^2 - {M_{\pi^{(0,{\rm c})}}}^2)/a^2$. It would thus be interesting to study whether this result still holds for simulations with dynamical fermions and whether the value of $({M_{\pi^{(0,{\rm c})}}}^2 - {M_{\pi^0}}^2)/2a^2$ is also reduced in that case.
Conclusions {#sec:concl .unnumbered}
===========
We have presented the determination of the Wilson LECs $W'_{6,8}$ and $c_2$, parametrising the size of O($a^2$) lattice artifacts in W$\chi$PT, from simulations with a lattice action composed out of the Iwasaki gauge action and ${N_\mathrm{f}=2+1+1}$ flavours of Wilson twisted mass fermions at maximal twist. The values of $W'_{6,8}$ include a rather complete account of the systematic uncertainties. Our measurements satisfy the recently derived bounds [@Damgaard:2010cz; @Hansen:2011kk; @Kieburg:2012fw], ${W_8'}<0$ and ${W_6'}> 0$.
We have also explored the dependence of the mass-splittings $({M_{\pi^\pm}}^2 -
{M_{\pi^{(0,{\rm c})}}}^2)/a^2$ and $({M_{\pi^{(0,{\rm c})}}}^2 - {M_{\pi^0}}^2)/2a^2$ on the choice of the lattice action. From this qualitative comparison, it is tempting to conjecture that a lattice action made of dynamical twisted mass fermions including the Sheikholeslami-Wohlert and smearing might lead to a reduction of the mass-splitting $({M_{\pi^\pm}}^2 -
{M_{\pi^{(0,{\rm c})}}}^2)/a^2$. Further studies are needed to clarify this point and to extend it to the case of $({M_{\pi^{(0,{\rm c})}}}^2 - {M_{\pi^0}}^2)/2a^2$.
A partial cancellation of the contributions from $W'_{6}$ and $W'_{8}$ implies that the residual mass-dependence of $c_2$ is more sensitive to higher order effects in the W$\chi$PT expansion. While this potential reduction of $c_2$ would certainly be beneficial, the precise determination of its value cannot be achieved with the currently available data.
The determination of the Wilson LECs can help to quantify the size of O($a^2$) terms in a given lattice action. Knowing these LECs, in particular with better precision, can significantly contribute to design a lattice fermion action with small lattice artifacts, thus allowing to reach the continuum limit in a better controlled way. In addition, an independent calculation of the Wilson LECs, as carried through here, can be used in chiral perturbation theory fits of light meson observables by constraining these fits to lattice data. Thus we think that the study of the (connected and full) neutral and charged pion masses of this work can be beneficial for many other groups working with Wilson-like lattice fermions.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Giancarlo Rossi for useful comments on the manuscript. The computer time for this project was made available to us by the John von Neumann-Institute for Computing (NIC) on the JUDGE and Jugene systems in J[ü]{}lich and the IDRIS (CNRS) computing center in Orsay. In particular we thank U.-G. Mei[ß]{}ner for granting us access on JUDGE. Falk Zimmermann cross-checked correlation functions for one of our ensembles, which we gratefully acknowledge. G. H. acknowledges the support by DFG (SFB 1044), the Spanish Ministry for Education and Science project FPA2009-09017, the Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), the Comunidad Autónoma de Madrid (HEPHACOS P-ESP-00346 and HEPHACOS S2009/ESP-1473) and the European project STRONGnet (PITN-GA-2009-238353). K. J. was supported in part by the Cyprus Research Promotion Foundation under contract $\Pi$PO$\Sigma$E$\Lambda$KY$\Sigma$H/EM$\Pi$EIPO$\Sigma$/0311/16. This work has been supported in part by the DFG Sonderforschungsbereich/ Transregio SFB/TR9. Two of the authors (K. O. and C. U.) were supported by the Bonn-Cologne Graduate School (BCGS) of Physics and Astronomie. This project was supported in parts by the DFG in SFB/TR16 and CRC 110.
[^1]: For a recent review, we refer to [@Splittorff:2012hz].
[^2]: It is interesting to note that Table \[tab:nf211\] indicates that the relative error on the neutral pion mass is roughly independent of the light-quark mass and that it decreases when increasing the volume. In practice, in the current simulation conditions, this implies that the measurements of ${M_{\pi^0}}$ are statistically more precise for the ensembles with lighter quark masses.
|
---
author:
- Fabrice Thalmann
date: 'Received: / Revised version: date'
title: A schematic model for molecular affinity and binding with Ising variables
---
Introduction {#sec:intro}
============
Molecular biology and biochemistry differ from chemistry and physics by the very high specificity of the interactions and the processes that they aim at describing. Words like *functions* or *shapes* are used instead of *molecules*, *atoms*, *forces* or *fields*. These biomolecules are perceived as capable of processing information (mutual recognition) and performing dedicated actions (switches or catalytic reactions). It takes only two orders of magnitude in size, from 0.1 nm to 10 nm, to abandon a world of thermal chaos and vibrations, and to enter a world of dedicated agents and reliable procedures [@Book_Alberts_Walter].
It was suggested by E. Fisher that selective interaction and binding between molecules were primarily a consequence of their complementary shapes, which has since been known as the lock and key paradigm. These specific interactions between highly complementary moieties accounts for the specialized and efficient action of enzymes and often explains how drugs work at the molecular level. The lock and key paradigm was then extended to include final conformational changes that may happen upon binding, a mechanism known as induced fit [@1958_Koshland]. Since a receptor and its ligand have two complementary shapes, any structural change results in decreasing the binding properties of the pair.
The tremendous increase of known 3d molecular structures (NMR and X-ray scattering) [@2000_Berman_Bourne; @2003_Berman_Nakamura] and the ever-growing computing capacities has made of the search for complementary ligand-receptor interaction a very intense and competitive research field. Numerical approaches based on the lock and key principle are commonly known as docking algorithms. The issue of these strategies depend on how successfully are Coulombic forces, hydrogen bonding, solvation properties, electronic densities…accounted for [@Book_Israelachvili; @2001_Leckband_Israelachvili].
As the size of the interacting bodies increases, it is natural to question whether statistical physics still plays a role in these highly optimized recognition processes. In cells, most molecular interactions are subtly balanced in order to achieve reversibility and to prevent the occurrence of irreversible aggregation. Entropic contributions may help to achieve this balance.
One of the current goal of computational biochemistry, still out of reach, is to predict realistic ligand binding stabilization energies with an accuracy of about 1 kcal.mol$^{-1}$ (1.6 $k_BT$), compared with experimental measurements [@2007_Gilson_Zhou]. The quenching of conformational degrees of freedom upon binding, and the subsequent entropic changes must be accounted for when computing thermodynamical association constants with such accuracy.
The immune system provides unrivaled cases of specific mutual recognition. For instance, antibodies are specialized proteins which recognize and bind in an extremely specific and accurate manner to foreign bodies invading a living organism. A recent bioinformatic study of interactions between T-cells and the major histocompatibility complex (MHC) supports the view that selective interactions between peptides may owe more to a delicate balance among many weak additive interactions, rather than to a strong complementary and exclusive mutual binding [@2008_Kosmrlj_Chakraborty].
Also striking is the phenomenon of allostery. Some proteins have their function subordinated to the presence of an effector. An historically famous example is the transcription regulation of *Lac*-operon, for which it was demonstrated by Monod and Jacob that the fabrication of the lactose digesting enzymes in *e.coli* was conditioned to the presence of a significant amount of lactose in the environment of the bacteria [@2005_Lewis]. Modern biology teaches us that in the absence of lactose, the protein *Lac-*repressor binds to a stretch of dna, and prevents the expression of the genes downstream. When lactose molecules bind to *Lac-*repressor, the protein shape changes, and it looses its ability of binding dna, enabling the expression of the genes under its control. The dna and lactose binding sites are located on distinct regions of the repressor protein, suggesting an *action at distance* caused by the presence of lactose molecules. Recently, the idea of a simple conformational change of allosteric molecules has been challenged. By studying a schematic mechanical model of *Lac*-repressor, R. Hawkins and T.C.B MacLeish estimated the contribution of internal, vibrational degrees of freedom, *i.e.* a change in protein stiffness induced by the ligand [@2004_Hawkins_McLeish]. They concluded that positive or negative binding entropy changes $\Delta\Delta S$ were associated to changes (hardening or softening) in the effective spring constants used in their mechanical model of repressor proteins. It is precisely this idea of stiffness and entropic modulation of the binding site efficiency that motivates the present work.
There is, as a matter of fact, a deep and formal connection between statistical physics, recognition, binding, and information theory. When a ligand binds selectively to a patterned substrate, it accomplishes some kind of *decoding* and reads a piece of information conveyed by its target, the more conspicuous case being the association of complementary base pairs in dna-dna or dna-rna duplexes which is the cornerstone of genetic information processing. In the following Sections, we intend to show that a simple Ising spin chain can be turned into an elementary model for the binding of a ligand molecule onto a patterned receptor, in the limit where both thermal fluctuations and internal entropic degrees of freedom are relevant. Within this framework, one can tune the interactions between ligand and binding site, as well as the internal stiffness of the ligand, and we also consider flexible binding sites. The binary nature of the receptors makes them natural information carriers. After defining the affinity and selectivity of a ligand for its receptor in this situation, we discuss how much dependent are the affinities and selectivities on the stiffness parameters. Then, we proceed by giving examples of non monotonic behaviors of the affinities with increasing stiffnesses. We exhibit a case of decreased affinity upon local stiffening of the ligand, reminiscent from Hawkins and McLeish results. We show that there are affinity biases between receptors with similar shapes but different local rigidities. We finally perform an exhaustive comparison of all pairs of patterns up to a length of 8 monomers, and discuss their intrinsic ability to reliably encode information, which is found to decrease with their length. In the following sections, the word receptor will be used with the same meaning as “binding site”, *i.e.* an object the size of the ligand that binds to it. This linguistic shortcut must not occult the fact that in many realistic cases, the receptor is a much bigger object than the ligand, and the binding site only constitutes a subpart of it. This schematic model does not pretend to accurately describe a realistic experimental situation. However, despite its simplicity, it already displays a fairly rich phenomenology which may find a counterpart in some real cases.
Using Ising-like models for modeling selective binding is not a new proposal. Indeed, Schmid, Behringer and coworkers introduced and performed intensive statistical studies of models with very similar Hamiltonians [@2006_Behringer_Schmid; @2007_Behringer_Schmid; @2008_Behringer_Schmid]. Their model describe the contact between hydrophobic and polar patches belonging to too opposite and complementary binding sites, and is presented as a coarse-grained approach to hydrophilic–hydrophobic interactions that are central to both protein folding and protein-protein interactions. Their spin variables describe the contact between the two rigid moieties and account for short range local rearrangements of the coarse-grained residues. A coupling $J$ between “spin variables” is interpreted as a cooperativity property, while we associate ours to a stiffness parameter. They model 2d small rectangular patches while we discuss more schematic 1d interacting chains; our attempt to encode information in the stiffness pattern in addition to the spatial shape is not found in their work. The model introduced is Section \[sec:affinity-selectivity\] is similar to their approach, but the general approach of Section \[sec:flexible\] is original.
Statistical physics of surface-bound receptors binding {#sec:association-constant}
======================================================
Let us consider a number of ligands ${ {\mathcal{L}} }$ in contact with a single binding site ${ {\mathcal{M}} }$ linked to a rigid surface. A balance is reached resulting in an equilibrium association constant ${ {\mathcal{K}} }^{a}_{{ {\mathcal{L}} },{ {\mathcal{M}} }}$ for the exchange: $${ {\mathcal{L}} }_{\mathrm{free}} + { {\mathcal{M}} }_{\mathrm{free}} \leftrightarrow
{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }_{\mathrm{bound}},$$ from which the probability $p_{{ {\mathcal{L}} },{ {\mathcal{M}} }}$ of observing a ligand bound to ${ {\mathcal{M}} }$ obeys: $${ {\mathcal{K}} }^{a}_{{ {\mathcal{L}} },{ {\mathcal{M}} }}\frac{[{ {\mathcal{L}} }_{\mathrm{free}}]}{C^0}(1- p_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }}) =
p_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }},
\label{eq:complexation-equilibrium}$$ with brackets $[.]$ denoting molar concentrations, and $C^0$ a molar standard reference concentration, making the equilibrium association constant dimensionless, for instance $C^0=1$mol.L$^{-3}$. This is a generalization of the law of mass action that one would write for the complexation equilibrium of two species ${ {\mathcal{L}} },{ {\mathcal{M}} }$ in solution. $${ {\mathcal{K}} }^{a}_{{ {\mathcal{L}} },{ {\mathcal{M}} }}[{ {\mathcal{L}} }_{\mathrm{free}}].[{ {\mathcal{M}} }_{\mathrm{free}}] =
[{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }_{\mathrm{bound}}].C^0.$$ The equilibrium constant depends a lot on the solvation (hydration) of ${ {\mathcal{L}} }$ and ${ {\mathcal{M}} }$, the ionic content of the solution and all the details of close range interactions between species.
As our current goal is to emphasize the role of the internal degrees of freedom of ${ {\mathcal{L}} }$, we disregard all solvent related interactions by making a kind of ideal solution assumption, and considering that all ligand-receptor interactions are short ranged. This assumption on the solvent is equivalent to saying that all bound and unbound conformations of the ligand receptor pair have the same solvation free-energy. The mutual affinity of such a pair comes entirely from shape and stiffness considerations.
It becomes possible to express the equilibrium constant ${ {\mathcal{K}} }^{a}_{{ {\mathcal{L}} },{ {\mathcal{M}} }}$ as a partition function ratio : $$\begin{aligned}
{ {\mathcal{K}} }^{a}_{{ {\mathcal{L}} },{ {\mathcal{M}} }}
&=&\frac{\mathcal{N}_Av}{8\pi^2 V^0}\frac{Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }\,\mathrm{bound}}}
{Z_{{ {\mathcal{L}} }\,\mathrm{free}}Z_{{ {\mathcal{M}} }\,\mathrm{free}}}.
\label{eq:equilibrium-constant-K}\\
&=& \frac{\mathcal{N}_A v}{8\pi^2 V^0} { {\mathcal{C}} }^{a}_{{ {\mathcal{L}} },{ {\mathcal{M}} }}
\label{eq:equilibrium-constant-C}
$$ Equation (\[eq:equilibrium-constant-K\]) is a particular instance of the equilibrium association constant derived and presented as eq. (13) in ref. [@1997_Gilson_McCammon]. $\mathcal{N}_A$ is the Avogadro number and $V^0$ is the volume occupied by one mole in the reference concentration state $C^0$. $Z_{{ {\mathcal{L}} }\,\mathrm{free}}$ stands for the Boltzmann-Gibbs sum over all the internal conformations of ${ {\mathcal{L}} }$, with fixed orientation and center of mass. $Z_{{ {\mathcal{M}} }\,\mathrm{free}}$ is a similar configuration integral over the internal conformations of ${ {\mathcal{M}} }$ when ${ {\mathcal{L}} }$ and ${ {\mathcal{M}} }$ are apart and not interacting. We assume that the configuration integral of the bound complex ${ {\mathcal{L}} }\cdot{ {\mathcal{M}} }$ reduces to a product $v Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }\,\mathrm{bound}}$, where $Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }\,\mathrm{bound}}$ represents the sum over all internal configurations of ${ {\mathcal{L}} }$ in contact with ${ {\mathcal{M}} }$ (with fixed centers of mass and orientations) and the volume $v$ corresponds to all the positions of the center of mass of the ligand ${ {\mathcal{L}} }$ relative to the receptor ${ {\mathcal{M}} }$ that are considered as forming a bound state. In the context of bimolecular associations, $v$ should range up to a few angstrom cube ($10^{-30}~\mathrm{m}^{3}$). Note that for rigid receptors $Z_{{ {\mathcal{M}} }\,\mathrm{free}}$ equals 1 by construction.
The association constant is obtained by setting $p_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }}=1/2$ in equation (\[eq:complexation-equilibrium\]). One can then argue that the partition sum per free ligand $$\frac{8\pi^2}{[{ {\mathcal{L}} }_{\mathrm{free}}]\mathcal{N}_A}
Z_{{ {\mathcal{L}} }\,\mathrm{free}}Z_{{ {\mathcal{M}} }\,\mathrm{free}}$$ equals the partition sum of the bound complex $$v Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} },\mathrm{bound}}.$$ In writing these expressions, we neglected the specific volume change occurring during the binding process. In other words, we assume that the Gibbs $\Delta G$ and Helmholtz $\Delta F$ thermodynamic quantities coincide.
Equation (\[eq:equilibrium-constant-K\]) links the thermodynamical affinity constant that can be experimentally determined and the configuration integrals $Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} },\mathrm{bound}}$, $Z_{{ {\mathcal{L}} }\mathrm{free}}$ and $Z_{{ {\mathcal{M}} }\mathrm{free}}$ that are the main focus of this work. Equation (\[eq:equilibrium-constant-K\]) can be also written as: $$\Delta G^{0}\simeq \Delta F^{0} = \langle\Delta U\rangle -T\Delta S_{\mathrm{c}},
\label{eq:connexion-GilsonZhou}$$ with $\Delta F^{(0)}=-k_BT\ln({ {\mathcal{K}} }^{a}_{{ {\mathcal{L}} },{ {\mathcal{M}} }})$ and where $\langle\Delta U\rangle$ designates the average enthalpic change of moieties ${ {\mathcal{L}} }$ and ${ {\mathcal{M}} }$ upon binding, caused by their change in conformation and mutual interaction, while $\Delta S_{\mathrm{c}}$ represents the corresponding change in configurational entropy. This expression stands a particular case of $$\Delta G^{0} = \langle\Delta U\rangle+\langle\Delta W\rangle -T\Delta S_{\mathrm{c}},
\label{eq:complete-GilsonZhou}$$ demonstrated in refs. [@2007_Gilson_Zhou; @1997_Gilson_McCammon; @2004_Mihailescu_Gilson], where $\langle\Delta
W\rangle$ accounts for the contribution of an explicit solvent to the formation of the ligand receptor pair. In aqueous solutions, both hydrogen bonding and hydration forces originate from specific interaction with the solvent. Strictly speaking, solvent mediated interactions are associated with $\langle \Delta W\rangle$ in eq. (\[eq:complete-GilsonZhou\]). However, it is to some extent possible to take them into account by means of an effective hamiltonian and treat these interactions as if they were part of the direct interaction term $\langle \Delta U\rangle$.
We define the **affinity** of ${ {\mathcal{L}} }$ for ${ {\mathcal{M}} }$ as ${ {\mathcal{C}} }^{(a)}_{\mathcal{L},\mathcal{M}}$ (eq. \[eq:equilibrium-constant-C\]). For a given number of ligands ${ {\mathcal{L}} }$ and binding sites ${ {\mathcal{M}} }$, the affinity controls the fraction of bound pairs of molecules (adsorption isotherms) and an increased affinity leads to an increase of bound pairs. However, when many patterns ${ {\mathcal{M}} }_1$, ${ {\mathcal{M}} }_2$…compete for the same ligand, the affinity is not a good assessment of how exclusive is the binding of ${ {\mathcal{L}} }$ for a given ${ {\mathcal{M}} }$. This is why one needs a **relative selectivity** parameter for comparing the respective behaviors of a ligand for a target (matching pattern ${ {\mathcal{M}} }$) and for a decoy (mismatching pattern ${ {\mathcal{W}} }$): $$\mathcal{S}_r({ {\mathcal{L}} },{ {\mathcal{W}} })= \frac{ Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }\,\mathrm{bound}} }
{Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{W}} }\,\mathrm{bound}}}\cdot
\frac{Z_{{ {\mathcal{W}} }\,\mathrm{free}}}{Z_{{ {\mathcal{M}} }\,\mathrm{free}}},
\label{eq:relative-selectivity}$$ or equivalently $\mathcal{S}_r = { {\mathcal{C}} }^{a}_{{ {\mathcal{L}} },{ {\mathcal{M}} }}/{ {\mathcal{C}} }^{a}_{{ {\mathcal{L}} },{ {\mathcal{W}} }}$. We will be also interested in the **absolute selectivity**, when comparing the affinity of the matching pattern with the affinity of a complete set of decoys: $$\mathcal{S}_a({ {\mathcal{L}} }) =
\frac{Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }\mathrm{bound}}}{Z_{{ {\mathcal{M}} }\mathrm{free}}}
\cdot\frac{1}
{\left[\displaystyle\sum_{{ {\mathcal{W}} }\neq{ {\mathcal{M}} }}
\frac{Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{W}} }\mathrm{bound}}}{Z_{{ {\mathcal{W}} }\mathrm{free}}}\right] },
\label{eq:absolute-selectivity}
$$ if the ligand is given a choice between all the possible patterns, or $$\mathcal{S}'_a({ {\mathcal{L}} }) = \inf_{{ {\mathcal{W}} }\neq { {\mathcal{M}} }} \left\lbrace
\frac{Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }\mathrm{bound}}}{Z_{{ {\mathcal{M}} }\mathrm{free}}}
\cdot\frac{Z_{{ {\mathcal{W}} }\mathrm{free}}}{Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{W}} }\mathrm{bound}}} \right\rbrace,
\label{eq:alternate-selectivity}
$$ if only one decoy ${ {\mathcal{W}} }$ is present at a time. The inf operator of eq. (\[eq:alternate-selectivity\]) runs over all the possible patterns ${ {\mathcal{W}} }$ that are different from the matching pattern ${ {\mathcal{M}} }$, and picks up the best competitor of ${ {\mathcal{M}} }$, *i.e.* the one for which the affinity difference is the lowest. Eq. (\[eq:absolute-selectivity\]) is better suited for assessing the affinity of a ligand for its matching pattern, while all the other possible competitors ${ {\mathcal{W}} }\neq{ {\mathcal{M}} }$ are simultaneously present. This definition of the selectivity depends crucially on the set ${ {\mathcal{P}} }$ of allowed patterns ${ {\mathcal{W}} }$. Changing this set ${ {\mathcal{P}} }$ results in changing the selectivity parameters $\mathcal{S}_{a,r}$. Using a subset ${ {\mathcal{P}} }'$ of ${ {\mathcal{P}} }$ naturally leads to higher selectivities. In practice, a large number of poor affinity decoys can eventually beat a good ligand-receptor pair [@1996_Janin].
A minimal model for flexible ligand and rigid receptor {#sec:affinity-selectivity}
======================================================
We now introduce a model containing the basic components of the above discussion: matching, internal degrees of freedom, stiffness, information, and we search for phenomenon such as stiffness dependent affinity and matching-decoding.
The ligand ${ {\mathcal{L}} }$ is modeled as an articulated chain of $n$ monomers. Each bead $i$ is allowed to occupy only two positions labelled by a binary variable $s_i = \pm 1$ and the ligand can adopt $2^n$ distinct internal conformations. Ligand stiffness is enforced by means of next nearest neighbors couplings $J_{i,i+1} = \pm J$, in the spirit of the original spin-glass model by Edwards and Anderson [@EdwAnd; @MezParVir].
The signs of the couplings $\lbrace J_{i,j} \rbrace$ define the ground state shape (native shape) of ${ {\mathcal{L}} }$, up to a trivial two-fold degeneracy, while the moduli $J=|J_{i,j}|$ describe the energetic cost associated to bending distortion of the ligand (strain). We assume that ligands have a well defined orientation, say from left to right, and we do not consider the possibility of left-right reversal. This can be justified considering that biopolymers (proteins, nucleic acids) always display such an orientation along the chain. Finally we also disregard the possibility of lateral shift between ligand and receptors, as occurring for instance in the hybridization of dna oligomers. This physically relevant situation increases significantly the combinatorics of the association and efficient algorithms have been designed to tackle these alignment problems [@Book_Durbin]. In this work we purposely focus more on the thermodynamical stability of two molecules that are prepositioned in the right conformation, or which possess a unique, non-degenerated optimal relative conformation. The outcome of this schematic model is expected to be relevant on length scales of about 1 nm between consecutive “monomers”, large enough to invoke some coarse-graining of the underlying molecules, but small enough to preserve the importance of entropic, conformational degrees of freedom.
We now introduce a symbolic notation to describe the ground state of these flexible molecules. For that purpose, one must distinguish between *open*, free end molecules and *cyclic*, closed end molecules. Open ligands with $n$ monomers have $2^n$ internal configurations. There are $2^{n-1}$ possible ground states, each one of them being doubly degenerated due to up-down symmetry. To fully describe the shape and ground state of a ligand, we denote by the letters `u` and `d` the position, respectively up or down, of the first monomer on the left. We then associate a `p` to each antiferromagnetic coupling constant $J>0$, and a `m` to each ferromagnetic one $J<0$. A symbol `(o)` is added at the end to signal an open chain. The ligand ${ {\mathcal{L}} }$ represented on the left of Figure \[fig:rigid-ligand-receptor\] is thus ascribed to the symbol ${ {\mathcal{L}} }=$`u/ppp(o)`. The corresponding Ising configuration reads $s_1 = +1$, $s_2 = -1$, $s_3 = +1$, $s_4 = -1$, and the coupling constants are $J_{12} = J_{23} = J_{34} = J$.
Cyclic ligands with $n$ monomers have spin $s_1$ and $s_n$ coupled with a term $J_{n1}s_1s_n$. Cyclization is not really justified by the initial molecular association problem, it is here just a convenient way to simplify calculations and get rid of boundary effects, giving to all monomers the same importance and simplifying the interpretation of results. Cyclic ligands of length $n$ have the same number of configurations as open ligands, but it requires $n$ coupling constants to fully determine their ground state. Cyclic ligands with an odd number of antiferromagnetic `p` couplings are “frustrated”, meaning that no ligand configuration can satisfy simultaneously all the constraints imposed by the $J_{i,j}$. The ground state conformation is then at least four-fold degenerated. Cyclic ligands with a even number of `p` have a well defined two-fold degenerated ground state. For instance, the unfrustrated cyclic ligand with the same shape as represented on the left of Figure \[fig:rigid-ligand-receptor\] is coded as ${ {\mathcal{L}} }'=$`u/pppp(c)`, where the suffix `(c)` reminds of the cyclic character of the molecule. Ligand ${ {\mathcal{L}} }''=$`u/pppm(c)` is frustrated with no well defined ground shape. In this schematic approach, the use of open or cyclic ligands is essentially a matter of convenience, as they generate qualitatively similar results.
In the same way, the patterned receptor (binding site) is represented by a sequence of binary values $b_i$, $1\leq i\leq n$, each one taking a value $\pm 1$. When the ligand comes in contact with the receptor, it gains a negative stabilizing energy $-A$ whenever the monomer position $s_i$ and the receptor value $b_i$ match. We do not give a penalty to a mismatch situation $b_i\neq s_i$, but this could just be done by shifting negatively of the total configurational energy. The coupling constant $A$ represents a short range interaction, possibly mimicking hydrogen bonding or hydrophobic patches. In both cases, the effective contact parameter $A$ may depend on temperature. The resulting total “Hamiltonian” describing the ligand and the receptor in close contact is: $${ {\mathcal{H}} }_r\lbrace s_i \rbrace = \sum_{i=1}^{n'} \Big( J_{i,i+1} s_i s_{i+1} \Big)
-A \sum_{i=1}^{n} \delta_{s_i b_i}.
$$ The sum runs until $n'=n-1$ for open chains, and $n'=n$, with coupling $J_{n,n+1}=J_{n,1}$ for cyclic chains. As $\delta_{sb}=(1+sb)/2$ for Ising variables, the Hamiltonian: $${ {\mathcal{H}} }_r\lbrace s_i \rbrace = \sum_{i=1}^{n} \Big( J_{i,i+1} s_i s_{i+1} \Big)
-\frac{A}{2} \sum_{i=1}^{n} \Big(s_i b_i\Big) -\frac{nA}{2}.
$$ assumes the form of a random field Ising spin glass with both quench bond disorder $J_{ij}$ and quenched random magnetic field $-A b_i/2$. However, contrary to usual disordered systems studies, we do not perform here any average over the quenched random bonds, as these couplings contain the relevant information. The related partition function, expressed with the inverse Boltzmann factor $\beta$, is $$\begin{gathered}
Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }\,\mathrm{bound}}=\\
\sum_{\{s_i=\pm 1\} }\exp\Bigg(\frac{n\beta A}{2}+
\sum_{i=1}^{n'}( -\beta J_{i,i+1} s_i
s_{i+1}) +\,\frac{\beta A}{2} \sum_{i=1}^{n} ( s_i b_i )\Bigg).
\label{eq:definition-ZLM-bound}\end{gathered}$$ Meanwhile, $Z_{{ {\mathcal{M}} }\,\mathrm{free}}=1$ and $Z_{{ {\mathcal{L}} },\mathrm{free}}$ is a special instance of (\[eq:definition-ZLM-bound\]) with $A=0$: $$\begin{aligned}
Z_{{ {\mathcal{L}} }\mathrm{free}} &=&
\sum_{\{s_i=\pm 1\} }\exp\Bigg(\sum_{i=1}^{n'}( -\beta J_{i,i+1} s_i s_{i+1}
)\Bigg), \label{eq:definition-ZL-free}\\
&=& 2^n \left\lbrack \cosh(\beta J)^n \pm \sinh(\beta J)^n \right
\rbrack,\nonumber
$$ result valid for cyclic chains, with a sign $+$ without bond frustration, and a sign $-$ otherwise.
Finally, from the binding free energy $\Delta F$, defined as: $$\Delta F = -{ \mathrm{k}_B }T\ln
\left[\frac{ Z_{{ {\mathcal{L}} }\cdot{ {\mathcal{M}} }\,\mathrm{bound}} }
{Z_{{ {\mathcal{L}} }\mathrm{free}} Z_{{ {\mathcal{M}} }\mathrm{free}}} \right],
\label{eq:binding-free-energy}$$ one deduces the partial enthalpic $\Delta U$ and entropic $-T\Delta
S_c$ contributions by numerically differentiating with respect to $\beta$. $$\begin{aligned}
\Delta U &=& \frac{\partial (\beta \Delta F)}{\partial \beta};
\label{eq:binding-energy}\\
-T\Delta S_c &=& \Delta F-\Delta U\label{eq:binding-entropy},\end{aligned}$$ where it is assumed that $A$ does not depend on temperature (enthalpic contribution).
To describe the shape of rigid receptors, it suffices, in principle, to enumerate the values $b_i$. However, anticipating the case of flexible receptors that will be considered in the coming section, we use for receptors the same convention as for ligands, *i.e.* a first letter `u` or `d`, followed by a list of couplings `p` and `m`. Receptors with matching ground state are called ${ {\mathcal{M}} }$, ${ {\mathcal{W}} }$ being associated with those with mismatching ground states.\
Let us illustrate the preceding section with ${ {\mathcal{L}} }$=`u/pppp(c)`, ${ {\mathcal{M}} }$=`u/pppp` and ${ {\mathcal{W}} }$=`u/pmpm`. The coupling constants of ${ {\mathcal{L}} }$ are $\lbrace
J_{1,2}=J_{2,3}=J_{3,4}=J_{4,1}=J\rbrace$, the matching motif corresponds to $b_1=b_3=1$, $b_2=b_4=-1$ and the mismatching motif to $b_1=b_4=1$, $b_2=b_3=-1$. The calculation of the partition functions by enumeration of the 16 configurations of ${ {\mathcal{L}} }$, or with a transfer matrix method gives: $$\begin{aligned}
Z_{{ {\mathcal{L}} }\,\mathrm{free}} & = & 12 + 2 e^{4\beta J} + 2 e^{-4\beta J};
\nonumber\\
Z_{{ {\mathcal{L}} }{ {\mathcal{M}} }\,\mathrm{bound}} & = & e^{4\beta J} ( 1 + e^{4\beta A}) +
4 (e^{\beta A} + e^{2\beta A}\\
& & + e^{3\beta A}) + 2e^{-4\beta J} e^{2\beta A}; \nonumber\\
Z_{{ {\mathcal{L}} }{ {\mathcal{W}} }\,\mathrm{bound}} & = & 2 e^{4\beta J} e^{2\beta A} + 4
e^{\beta A} + 2 e^{2\beta A}\nonumber\\
& & +4 e^{3\beta A} + 2 e^{2\beta A} e^{-4\beta J} + e^{4\beta A} + 1.\nonumber
$$ From now on, we assume that ${ \mathrm{k}_B }T$ sets the energy scale, and introduce the dimensionless coupling constants $a = \exp(\beta A/2)$, $j = \exp(\beta J)$. Affinity and selectivity are rational fractions of $a$ and $j$.
$$\mathcal{{ {\mathcal{C}} }}^{a}_{{ {\mathcal{L}} },{ {\mathcal{M}} }}(a,j) = \frac{(1+a^8)j^4 + 4(a^2 + a^4 + a^6) + 2 a^4
j^{-4}} {12 + 2j^4 + 2j^{-4}},
\label{eq:exact-quenched-match}$$
and $$\mathcal{S}_r(a,j) = \frac{(1+a^8)j^4 + 4(a^2 + a^4 + a^6) + 2
a^4 j^{-4}} {2a^4 j^4 + 4a^2 + 2a^4 + 4a^6 + 2a^4j^{-4} + 1 + a^8}.
\label{eq:exact-quenched-mismatch}
$$
We are interested in assessing the role of the stiffness parameter $j$. In Figure \[fig:affinity-rigid-match\], we observe that the affinity of the ligand ${ {\mathcal{L}} }$ for the matching pattern ${ {\mathcal{M}} }$ increases monotonically with $j$. Figure \[fig:thermo-rigid-match\] represents the variation with $j$ of the thermodynamic potentials $\Delta F$, $\Delta U$ and $-T\Delta S_{\mathrm{c}}$.
At the opposite, the stiffness $j$ reduces the affinity of ${ {\mathcal{L}} }$ for the mismatching pattern ${ {\mathcal{W}} }$, as represented in Figure \[fig:affinity-rigid-mismatch\], with the thermodynamic functions shown in Figure \[fig:thermo-rigid-mismatch\]. In addition, one notices that the special case $j=1$ represents a soft ligand which can adapt to any pattern. Quite naturally, the selectivity between ${ {\mathcal{W}} }$ and ${ {\mathcal{M}} }$ is 1 for $j=1$ and tend towards a finite value for $j\to \infty$ (Figure \[fig:selectivity-rigid-mismatch\]). The maximal selectivity depends on the short range contact parameter $a$ and is reached as soon as $j\geq a$.
We conclude that stiffness is always favorable when the shape of a ligand and a receptor agree, but becomes unfavorable is a mismatch is present. When ligand and receptor shapes almost agree but not perfectly, there must be an optimal compromise between a very soft ligand $j=1$, which precludes any selectivity at all, and a very hard ligand $j\gg 1$ which excessively penalizes the mismatches.
Flexible receptors {#sec:flexible}
==================
The next step is to consider flexible receptors ${ {\mathcal{M}} }$: we expect then soft ligands to beat stiff ligands, as they will better fit the various configurations of ${ {\mathcal{M}} }$. In our model, ${ {\mathcal{M}} }$ and ${ {\mathcal{L}} }$ play a dual role and it becomes possible to treat ligand and binding site (receptor) on the same footing, by inserting coupling constants $K$ between the “spins” $b_i$. This situation arises when two molecules with similar weight and structure bind together (Figure \[fig:flexible-ligand-receptor\]).
$$\begin{gathered}
{ {\mathcal{H}} }_r\lbrace s_i,b_i \rbrace=\\ \sum_{i=1}^{n'} \Big( J_{i,i+1} s_i
s_{i+1} \Big) -A \sum_{i=1}^{n} \delta_{s_i b_i} + \sum_{i=1}^{n'}
\Big( K_{i,i+1} b_i b_{i+1} \Big).
$$
The problem can be naturally solved for cyclic ligands and receptors with 4x4 transfer matrices. When all the coupling constants have same absolute magnitude, we define $K_{i,i+1} = \eta_i |K|$, $k=\exp(\beta
|K|)$, $J_{i,i+1} = \epsilon_i |J|$, $j=\exp(\beta |J|)$ where $\eta_i$ and $\epsilon_i$ are $\pm 1$, to find: $$\begin{aligned}
Z_{{ {\mathcal{L}} },{ {\mathcal{M}} }\,\mathrm{bound}} & = & a^{n}\mathrm{Tr} \left( \prod_{i=1}^{n}
T^{(\epsilon_i,\eta_i)}_{(a,j,k)} \right)\, ; \label{eq:numerical-trace}\\
Z_{{ {\mathcal{L}} }\,\mathrm{free}}Z_{{ {\mathcal{M}} }\,\mathrm{free}} & = & 4^n [\cosh(\beta J)^{n} \pm
\sinh(\beta J)^{n}]\\
& & \cdot [\cosh(\beta K)^{n} \pm \sinh(\beta K)^{n}]\, ;\nonumber
$$ (the sign $\pm$ depending on the bond frustration along the chains) with noncommuting matrices defined as: $$T^{(\epsilon,\eta)}_{(a,j,k)} = \left(
\begin{array}{cccc}
aj^{\epsilon}k^{\eta} & j^{-\epsilon}k^{\eta} & j^{\epsilon} k^{-\eta} &
a j^{-\epsilon}k^{-\eta} \\
j^{-\epsilon}k^{\eta} & a^{-1}j^{\epsilon}k^{\eta} &
a^{-1}j^{-\epsilon}k^{-\eta} & j^{\epsilon}k^{-\eta} \\
j^{\epsilon}k^{-\eta} & a^{-1}j^{-\epsilon}k^{-\eta} &
a^{-1}j^{\epsilon}k^{\eta} & j^{-\epsilon}k^{\eta} \\
aj^{-\epsilon}k^{-\eta} & j^{\epsilon}k^{-\eta} & j^{-\epsilon}k^{\eta}
& a j^{\epsilon}k^{\eta}
\end{array}
\right).
\label{eq:noncommuting-matrix}$$
In computing the trace, one actually performs a summation over the $4^n$ internal configurations. Thermodynamic quantities $\Delta F$, $\Delta S_c$ and $\Delta U$, given by equations (\[eq:binding-free-energy\]), (\[eq:binding-energy\]) and (\[eq:binding-entropy\]), are then obtained by numerically differentiating the transfer matrix results. One notices that the transfer matrix is entirely built from dimensionless parameters $a$, $j$ and $k$. The numerical differentiation of $\beta\Delta F$ with respect to $\beta$ assumes that $A$, $J$ and $K$ do not depend on temperature, leading to $\mathrm{d} a/\mathrm{d}\beta= \beta^{-1}
a\ln(a)$, $\mathrm{d} j/\mathrm{d}\beta= \beta^{-1} j\ln(j)$ and $\mathrm{d} k/\mathrm{d}\beta= \beta^{-1} k\ln(k)$. As a result, the thermodynamic quantities derived from equations (\[eq:binding-free-energy\]), (\[eq:binding-energy\]) and (\[eq:binding-entropy\]) are automatically expressed in units $k_BT$. Non purely enthalpic contributions to the contact energy parameter $A$ could also be included in this numerical scheme by using a different prescription for the derivative $\mathrm{d} a/\mathrm{d}
\beta$.
The external random field $b_i$ which was quenched for rigid receptors is now annealed (the random bonds still quenched), and we checked that for large values of $K$ (namely $k=\exp(\beta K)\geq 5$) the result for a flexible ligand receptor pair tends to the predictions for the rigid receptor. One can easily convince oneself that there is no difference between a short rigid receptor and a short flexible receptor with doubly degenerated ground state (*i.e.* no frustration) for which the condition $k\gg j\gg 1$ holds. In this limit, $Z_{{ {\mathcal{M}} },\mathrm{free}}\simeq 2$, a factor which also appears in $Z_{{ {\mathcal{L}} }.{ {\mathcal{M}} },\mathrm{bound}}$, leaving ${ {\mathcal{C}} }^a_{{ {\mathcal{L}} },{ {\mathcal{M}} }}$ unchanged. In practice, we regarded $k=10$ as sufficient to reach the rigid situation. This corresponds to an energy gap of $3~k_BT$ between the receptor ground state and its first distorted state. Indeed, all the results regarding rigid receptors presented in this study were actually obtained by setting $k$ to large values such as $k=10$ and applying eq. (\[eq:numerical-trace\]).
To calculate the partition function of a flexible, open, ligand receptor pair, one replaces the last matrix $T^{(\epsilon,\eta)}_{(a,j,k)}$ by a matrix $T_{(a,1,1)}$ representing the freely oscillating ends. Formula (\[eq:numerical-trace\]) becomes $$Z_{{ {\mathcal{L}} },{ {\mathcal{M}} }\,\mathrm{bound}} = a^{n}\mathrm{Tr} \left(T_{(a,1,1)} \prod_{i=1}^{n-1}
T^{(\epsilon_i,\eta_i)}_{(a,j,k)}\right)\,; \label{eq:numerical-trace-open}\\
$$
The transfer matrix formalism can be modified if one wishes to pick-up a particular bond coupling $J_{i,i+1}$ and assign to it a value different from the usual $J$. In our numerical implementation of the transfer matrix product, we use a 5-letters alphabet $\{p,m,P,M,.\}$ to describe a ligand pattern ${ {\mathcal{L}} }$.
- Characters $p$ and $m$ are respectively used for $\epsilon = 1$ and $\epsilon=-1$.
- Character $.$ denotes a vanishing coupling constant $J=0$ ($j=1$).
- Characters $P$ and $M$ represent $\epsilon = 1$ and $\epsilon=-1$, but with a different (usually larger) magnitude of $J$ or $K$.
Symbols $P$ and $M$ code for some localized hardening of ligand and receptors. Symbol $.$ loosely connects two adjacent and stiffer domains of ${ {\mathcal{L}} }$. More complex scenarios can be considered, but all are subject to the same limitation, which is that these transfer matrices can deal only with nearest neighbor couplings.
Ligand stiffness and affinity {#sec:stiffness-affinity}
=============================
We now provide a list of examples illustrating various behaviors. All curves represent the affinity variations when the ligand stiffness is increased from $j=1$ (soft ligand) till $j=8$, *i.e.* a 2 ${ \mathrm{k}_B }T$ activation barrier associated to local conformational change (spin reversal).
- **Uphill nonmonotonic affinity**:\
Ligand ${ {\mathcal{L}} }=$`u/ppp.ppp(o)` *vs* ${ {\mathcal{M}} }=$`u/ppmmppm` (Figure \[fig:M-ppp.ppp.\])\
The ligand ${ {\mathcal{L}} }$ is made of two loosely connected adjacent domains `ppp`. The resulting affinity is nonmonotonic, first increasing, then decreasing (Figure \[fig:p3.p3.-p2m2p2m2-k10\], a feature previously seen in Figure \[fig:affinity-rigid-mismatch\]). This behavior emerges from a competition between the matching subdomain, favored by large values of the stiffness parameter $j$ and the mismatching subdomain whose affinity decreases with $j$. In this particular case the mismatching domain forces the overall affinity to decrease below its starting point, a maximum being reached around $j\simeq 1.6$. Such a behavior illustrates the concept of optimal stiffness, where the association of a ligand with a (more) rigid receptor requires some tuning of its average rigidity.\
- **Downhill nonmonotonic affinity**:\
${ {\mathcal{L}} }=$`u/ppppppp(o)` *vs* ${ {\mathcal{W}} }=$`u/ppmmppm` (Figure \[fig:M-ppmmppmm\])\
For $a=2$ and $k=10$ (rigid pattern), the affinity curve is not monotonic with $j$, first decreasing, then increasing (Figure \[fig:allp-ppmmppmm-k10b\]). The affinity is minimal for a certain value $j\simeq 1.6$. This behavior in enhanced when $a$ is increased and reduced when the receptor stiffness $k$ is reduced, as seen in Figure \[fig:allp-ppmmppmm-a2b\]. The interest of this situation is to be the exact opposite of the preceding situation, with a local affinity minimum for $j\simeq 1.6$.\
- **Unbinding upon local hardening**:\
Ligand ${ {\mathcal{L}} }=$`u/ppPpp(o)` *vs* ${ {\mathcal{M}} }=$`u/ppmmp` (Figure \[fig:M-ppPppp\])\
The receptor shows a slight mismatch located under a region in ${ {\mathcal{L}} }$ which is locally more rigid (stiffness $\overline J > J$) than the average coupling. Increasing $\overline J$ while keeping $J$ constant should hamper the capacity of ${ {\mathcal{L}} }$ to accommodate the mismatch, and lead to lower affinity. This is precisely what is seen in Figure \[fig:ppPppp-ppmmpp-a3\]. The effect is seen for open chains ($j=1.5$, $j=2.$) and cyclic chains ($j=2$). It is reduced if the average chain stiffness is larger ($j=3$). If one admits that the stiff bond `P` is under control of an external agent or effector, the issue is that the affinity of the ligand for its target decreases even though their shape are not altered. Figures \[fig:DTS-uppPppx-ppmmpp\] and \[fig:DU-uppPppx-ppmmpp\] show respectively the energetic and entropic contributions to the changes in affinity associated with increasing the stiffness parameter $\overline{j}$. One can read from these data the associated relative entropy change $\Delta\Delta
S=\Delta S(\overline{j})-\Delta S(j)$. Ref [@2004_Hawkins_McLeish] suggests that a positive $\Delta\Delta S$ can be associated to the decreased affinity of *lac*-repressor for dna in the presence of lactose. This change in $\Delta\Delta S$ arises from increasing one spring constant and decreasing another one, in an harmonic model of *lac*-repressor. In our case, because our model is not harmonic, changes in $\Delta\Delta U$ and $\Delta\Delta S$ cannot be separated and both contribute to changing the affinity. Note that if the hard bond `P` was located on top of a matching subdomain, the opposite behavior of the affinity with $\overline J$ would occur. For instance the affinity of the pair `u/ppPpm(o)`-`u/ppppm` increases with large values of $\overline J$ (not shown).
To sum up, by combining matching and mismatching subdomains and bonds of adjustable stiffness, it is possible to obtain a variety of nonmonotonic behaviors of a ligand affinity for its target. It turns out to be possible to control the mutual affinity by acting on the rigidity of selected bonds, mimicking possibly the influence of a cofactor (or effector) involved in some allosteric mechanism.
Selectivity and information decoding {#sec:information}
====================================
Let us consider two persons $A$ and $B$ willing to communicate, and in possession of a number $N$ of specific ligand-receptors pairs. A sends a ligand $i$ to B, and B brings this (still unknown to him) ligand in contact with all the receptors available in his library. By using an analytical tool (fluorescence, quartz crystal microbalance, resonant surface plasmon absorption…) B determines the label $i$ of the ligand he has received. By referring to a preestablished codebook, shared with A, B determines the content of the message. Information is conveyed to B by means of selective binding [@Book_Lehn].
Information is coded in the shape of molecules, and especially biomolecules. When the biomolecule is in solution, this information about shape and chemical composition is made available to other molecules and a few of them will be able to selectively bind to it. Any selective adhesion process can be considered as information reading, or to be more precise, information decoding.
There are a number of issues raised by this medium of communication. One may be concerned by the reliability of the message transmission, directly connected to the absolute selectivity properties of the ligand-receptor pairs $\mathcal{S}_a$. One may also asks oneself whether there are physical bounds to the minimal binding energy or contact area necessary to ensure a transmission error rate lower than a predefined threshold value [@Book_MaxwellDemon2; @Book_Jones_Jones].
Coding and decoding information is every day’s concern for radio engineers, and many coding schemes have been developed in order to safely carry information around. The simplified model discussed in the previous section establishes a convenient connection with the realm of digital information, as ligand and receptors are already described in terms of binary chunks of information.\
Dependence in the number of mismatches
--------------------------------------
I now consider the selectivity of ${ {\mathcal{L}} }=$`u/pppp(c)`, in contact with altered configurations ${ {\mathcal{W}} }_1=$`u/pmmp` and\
${ {\mathcal{W}} }_2=$`u/pmpm`. Both altered patterns differ from the original one respectively by one and two values of the spin $s_i$ (Hamming distances $d_H=1$ and $d_H=2$, *cf* Figure \[fig:M-mismatch4letters\]). However the Hamming distance relative to the stiffness pattern is $d'_H=2$ for both ${ {\mathcal{W}} }_1$ and ${ {\mathcal{W}} }_2$ and is not directly related to the Hamming distance of configurations.
Results are shown in Figure \[fig:selectivity-mismatch4letters\]. It seems that the selectivity bias is indeed lower for $d_H=1$ than it is for $d_H=2$. One expects two qualitatively different limits for the selectivity dependence in the number of mismatches. If the stiffness $J$ is larger than the contact energy $A$, the selectivity is likely to correlate better with the Hamming distance $d'_H$ between stiffness patterns: one can flip an entire interval of spins by changing only two characters in the stiffness pattern. In the opposite limit, the distance between configurations $d_H$ should be dominant. Ref. [@2004_Bogner_Schmid] deals in details with related issues.\
Selectivity of finite length symbols
------------------------------------
We calculated the absolute selectivity (eq. \[eq:absolute-selectivity\]) among the subset of all $n$-long ligand and receptor symbols, to determine the true amount of information that such a family of ligands may carry.
To be more precise, each ground state conformation ${ {\mathcal{L}} }$ among the $2^{n-1}$ possible ones (due to up-down degeneracy) was compared with its matching counterpart ${ {\mathcal{M}} }$ along with all the $2^{n-1}-1$ other competitors ${ {\mathcal{W}} }$, to define a shape-dependent gap of selectivity $\mathcal{S}_a({ {\mathcal{L}} })$, eq. (\[eq:absolute-selectivity\]). In this procedure, all bonds have the same stiffness $j$ and the receptors are considered rigid, $k=10$. This shape-dependent gap of selectivity was subsequently minimized with respect to all different possible ${ {\mathcal{L}} }$, resulting in a quantity $\mathrm{Gap}(a,j,n)$ characteristic of this family of ligands. In particular, $\mathrm{Gap}(a,j,n)$ provides a lower bound for the occurrence of false positive if a ligand is put in contact with an equal number of all possible receptors. Using eq. (\[eq:alternate-selectivity\]) instead of eq. (\[eq:absolute-selectivity\]) leads to a different quantity $\mathrm{Gap}'(a,j,n)$ related to the noise to signal ratio when transmitting a message using a library of compounds as suggested earlier in this section.
Finally, the lower bound $\mathrm{Gap}(a,j,n)$ has to be optimized with respect to the internal parameters $a$ and $j$, in order to assess, in a context independent manner, the intrinsic coding capacity of a family of ligands with given length $n$, defining in this way $\mathrm{GAP}(n)$. The resulting formulas are: $$\begin{aligned}
\mathrm{Gap}(a,j,n)&=& \inf_{\mathrm{ligands\, { {\mathcal{L}} }\,of\,length}~n}
\Bigg(\mathcal{S}_a({ {\mathcal{L}} })\Bigg);\label{eq:Gap-definition}\\
\mathrm{GAP}(n) &=& \sup_{a,j}\mathrm{Gap}(a,j,n)
\label{eq:Select-definition}\\
&=&\sup_{a,j}\Bigg(\displaystyle\inf_{\mathrm{ligands\, { {\mathcal{L}} }\,of\,length}~n}
\Bigg(\mathcal{S}_a({ {\mathcal{L}} })\Bigg)\Bigg).\nonumber
$$ The inf operator defines the selectivity bias of a matching receptor competing against a league of all other receptors. The sup operator maximizes Gap with respect to $a$ and $j$.
In practice, cyclic chains were preferred to open chains, as they confer to all monomers the same importance. This optimization was performed by exhaustively scanning a rectangular grid of values $(a,j)\in [1,9]\times[1,9]$ with step 0.1, $k$ being kept equal to 10. Our findings are rendered as a 2d plot of Gap$(a,j,n=8)$ *vs* $(a,j)$ (Figure \[fig:Selectivity-a-j\]), and shows that the gap increases with both $a$ and $j$, but grows only marginally beyond $a,j\sim 2$. Thus, we took as representative the maximum obtained for ($a=8.8$, $j=8.8$) for estimating $\mathrm{GAP}(n)$. This arguably constitutes only a rough estimate of the $\sup_{a,j}$ operator of eq. (\[eq:Select-definition\]). Similar results were observed for all the values of $n=3,\ldots 8$ considered in this work. Values of $\ln[\mathrm{GAP}(n)]$ are reported in Table \[table:Selectivities\]. $\ln[\mathrm{GAP}]$ decreases almost linearly with $n$ (Table \[table:Selectivities\]) as the number of decoys grows exponentially. Extrapolating to large $n$, GAP$(n)$ vanishes for $n\ge n^*\simeq 20$, length for which, in the presence of an equal number of all receptors, a ligand has more chance to bind a mismatching receptor than its own complementary one.
If one uses eq. (\[eq:alternate-selectivity\]) as an alternative definition of $\mathrm{GAP}$, one observes a saturation of this quantity to a $n$ independent value, close to $e^{4.33}\simeq 76$ (for $a=8.8$, $j=8.8$, $k=10$, which are quite large values). This means that, irrespective of their length, any matching ligand receptor pair ${ {\mathcal{L}} }-{ {\mathcal{M}} }$ has an affinity larger by a factor 76 than any other mismatching pair ${ {\mathcal{L}} }-{ {\mathcal{W}} }$. Selectivity $\mathcal{S}'_a$ is not sensitive to the growing number of decoys (or “complexity”) with $n$.
One learns from these results that if one wishes to preserve a minimal selectivity while increasing the ligand length $n$, one must necessarily select a subset of shapes as valid ligands and disregard the other possibilities. This is tantamount to introducing “redundancy” in the “coding scheme”. Note that the numerical value of Table \[table:Selectivities\] depends on the details of the $a$,$j$ maximization and on the choice of $k$.\
Sensitivity to stiffness profiles
---------------------------------
The following step is to investigate the importance of stiffness patterns compared with shape patterns. Is it possible to encode information in the rigidity profile ? This situation is motivated, for instance, by the bioinformatic study of Sacquin-Mora *et al.* who investigated correlations between mechanical properties and binding location in proteins [@2007_SacquinMora_Lavery]. It would be indeed very interesting to determine whether self assembling pairs possess some kind of tactile sense and are sensitive to their respective local surface rigidity.
We consider four identically shaped ligands and receptors with patterns ${ {\mathcal{L}} }=$`u/PPppPP(c)` (ligand), `u/PPppPP`, `u/PPPPpp`,`u/PpPpPp` and `u/ppPPpp` (receptors), where `p` stands for a soft coupling and `P` for a stiff coupling (Figure \[fig:M-PPPp-pppP-ppPp\]).
The corresponding affinities are shown in Figure \[fig:affinity-stiffness-coding\]. The graph shows that the best affinity is observed when both patterns coincide (self) and the lowest when the stiffness pattern is opposite to the ligand (opposite). The mismatch `u/PPPPpp` is almost optimal and the intermediate case `u/PpPpPp` gives an intermediate value. The thermodynamic study shows that in both situations, the entropic contribution is dominated by the enthalpic contribution (Figures \[fig:DU-PPPp-pppP-ppPp\], \[fig:DF-PPPp-pppP-ppPp\] and \[fig:DTS-PPPp-pppP-ppPp\]). The stiffness patterns act on both enthalpic and entropic terms and the issue of the competition is not simple to predict. For instance, ligand ${ {\mathcal{L}} }=$`u/PpPpPp(c)` has a stronger affinity for ${ {\mathcal{W}} }=$`u/PPppPP` than for its self pattern ${ {\mathcal{M}} }$ (not shown).
If these results show that the affinity is sensitive to the stiffnesses, encoding information in stiffness profiles does not seem as straightforward as it is for shapes. Conveying information by means of selective binding, as sketched earlier in this section, would imply to design a quadruplet ${ {\mathcal{L}} }_1,{ {\mathcal{L}} }_2,{ {\mathcal{M}} }_3,{ {\mathcal{M}} }_4$ of ligands and receptors with identical shapes but distinct rigidity profiles, with affinities obeying: $$\begin{aligned}
{ {\mathcal{C}} }^a_{{ {\mathcal{L}} }_1,{ {\mathcal{M}} }_1}&\sim & { {\mathcal{C}} }^a_{{ {\mathcal{L}} }_2,{ {\mathcal{M}} }_2};\nonumber\\
{ {\mathcal{C}} }^a_{{ {\mathcal{L}} }_1,{ {\mathcal{M}} }_2}&\sim & { {\mathcal{C}} }^a_{{ {\mathcal{L}} }_2,{ {\mathcal{M}} }_1};\nonumber\\
{ {\mathcal{C}} }^a_{{ {\mathcal{L}} }_1,{ {\mathcal{M}} }_1},{ {\mathcal{C}} }^a_{{ {\mathcal{L}} }_2,{ {\mathcal{M}} }_2} & \gg &{ {\mathcal{C}} }^a_{{ {\mathcal{L}} }_1,{ {\mathcal{M}} }_2},
{ {\mathcal{C}} }^a_{{ {\mathcal{L}} }_2,{ {\mathcal{M}} }_1}. \end{aligned}$$ We could not come up with such a quadruplet. These quadruplets may not exist, or require longer lengths that the ones considered in this work (say $n\ge 8$). Stiffness profiles modulate the recognition process, but we were not able to prove that they could be substituted to shape profiles. We believe it is still an open issue to know for sure if stiffness profiles alone can bear some information.
Conclusion and Perspectives {#sec:conclusion-perspectives}
===========================
It is clear that a ligand with too few degrees of freedom cannot achieve good selectivity. In the top of Figure \[fig:lack-freedom\], a fine receptor is in contact with a coarse ligand, which averages out the details of the receptor. If one changes one bit of the receptor, the binding properties of the ligand are only marginally altered, and the selectivity ratio stays close to 1.
At the opposite, a ligand with many monomers in contact with each element of the receptor, will have a greater tolerance to the fluctuations of a particular monomer (Figure \[fig:lack-freedom\], bottom). Binding is enhanced by the addition of every monomer contribution. This case is a close analogous to the so-called “repetition code” in information theory. The repetition code consists in repeating many times every single bit of information to ensure the safe transmission of a code word (the message). It is a greedy procedure, as the length of the message is increased by the same factor. Here, a greater selectivity is achieved, at the expense of a greater complexity of the molecule ${ {\mathcal{L}} }$ which is thrice as long. As a rule, repetition is the easiest way one can come up with to amplify the trends observed in Figures \[fig:p3.p3.-p2m2p2m2-k10\] and \[fig:allp-ppmmppmm-a2b\]. By glueing together copies of the same patterns, one automatically enhances the characteristics of the ligand-receptor repeated unit.
To summarize a comparison with the results presented in [@2006_Behringer_Schmid; @2007_Behringer_Schmid; @2008_Behringer_Schmid], we can say that we focussed mostly on interesting characteristic trends exhibited by a few selected pairs of ligand and receptors, while the authors of [@2006_Behringer_Schmid; @2007_Behringer_Schmid; @2008_Behringer_Schmid] favor global and averaged trends running over the whole set of possible patterns. Our definition of the Gap and GAP indicators differ from the free energy difference $\Delta F$ which serve as a criterion in their work. Our transfer matrix approach treats exactly the binding statistics of a given pair of ligand and receptor, without need of Monte-Carlo sampling, nor mean-field or large $J$ approximations that are required in their 2d approach. Finally we believe that our model is the first one that makes it possible to investigate the role of local stiffness modulation, and the possibility of stiffness encoding of information.
The connection between error-correcting transmission codes and spin systems was recognized by N. Sourlas [@1989_Sourlas; @2001_Sourlas; @2001_Nishimori]. In particular, it was shown that the usual binary parity checks $b_1\oplus b_2 \oplus \ldots \oplus b_p$ which involves the sum of $p$ binary digits modulo 2 was in fact equivalent to coupling $p$ spins $s_1s_2\ldots s_p$, with $s_i=\pm 1$, leading to a formal connection between information theory and $p$-spin glasses. Our model is currently restricted to nearest neighbors coupling, and cannot account for long range couplings. Information redundancy is thus limited to simple repetition codes. It would be interesting to investigate how a second layer of spins and couplings could be added in order to better enforce robustness with respect to single bit mismatches. One also notices that a 2d generalization of the ligand shape would indeed be equivalent to a genuine 2d Edwards-Anderson spin glass. Spin glasses are well known for their long-lived or metastable states [@MezParVir; @1995_Monasson]. Each one of these states can be put in correspondence with a matching random-field representing a different receptor.
Another interesting connection between spin systems and pattern recognition, is the *Superparamagnetic clustering of data* [@1996_Blatt_Domany; @1998_Wiseman_Domany]. Wiseman, Blatt and Domanyi showed that it was possible to train a two dimensional array of Potts spins in order to recognize picture features and patterns (2d inhomogeneous distributions of points). This work could provide hints on how to train a 2d elastic network for shape and stiffness recognition.
The nonmonotonic behaviors of the affinity, or the decrease of affinity upon local stiffening of the chain are still modest, showing only a cut by half in the case illustrated in Figure (\[fig:ppPppp-ppmmpp-a3\]). One may be interesting in finding stronger effects by hardening more than a single bond.
The schematic model introduced and studied here is intended to guide us towards more realistic examples, such as simple molecules that could be designed and investigated with the help of *coarse grained* or *all atom* numerical models. This, we believe, should be the next step to endeavor.\
Acknowledgements {#sec:ackowledgements .unnumbered}
================
The author thanks Carlos Marques for discussions on this topic, and is indebted to the referees for many meaningful comments.
[30]{}
B. Alberts, A. Jonhson, J. Lewis, M. Raff, K. Roberts, P. Walter, *Molecular Biology of the Cell* (Garland Science, Taylor and Francis, 2002)
D. Koshland, Proc.Nat.Academy of Science USA **44**, 98 (1958)
H. Berman *et al.* Nucleic Acid Research **28**, 235 (2000), `http://www.rcsb.org/pdb/home`
H. Berman, K. Henrick, H. Nakamura, Nature Structural Biology **10**(12), 980 (2003), `http://www.wwpdb.org/`
J.N. Israelachvili, *Intermolecular and Surface Forces* (Elsevier, 1992)
D. Leckband, J. Israelachvili, Quarterly Review of Biophysics **34**(02), 105 (2001)
M.K. Gilson, H.X. Zhou, Annual Review of Biophysics and Biomolecular Structure **36**, 21 (2007)
A. Kosmrlj, A.K. Jha, E.S. Huseby, M. Kardar, A.K. Chakraborty, Proceding of the Natural Academy of Sciences USA **105**(43), 16671 (2008)
M. Lewis, Comptes Rendus Biologie **328**, 521 (2005)
R.J. Hawkins, T.C.B. McLeish, Phys. Rev. Lett. **93**(9), 098104 ( 4) (2004)
H. Behringer, A. Degenhard, F. Schmid, Phys. Rev. Lett. **97**(12), 128101 (2006)
H. Behringer, A. Degenhard, F. Schmid, Phys. Rev. E **76**(3), 031914 ( 12) (2007)
H. Behringer, F. Schmid, Phys. Rev. E **78**(3), 031903 (2008)
M.K. Gilson, J.A. Given, B.L. Bush, J. McCammon, Biophysical Journal **72**, 1047 (1997)
M. Mihailescu, M.K. Gilson, Biophysical Journal **87**, 23 (2004)
J. Janin, Proteins: Structure, Functions and Genetics **25**, 438 (1996)
S. Edwards, P. Anderson, J.of Physics F: Metal Physics **5**, 965 (1975)
M. Mézard, G. Parisi, M. Virasoro, *Spin glass theory and beyond.*, Vol. 9 (World Scientific, 1987)
R. Durbin, S. Eddy, A. Krogh and G. Mitchison, *Biological sequence analysis* (Cambridge University Press, 1998)
J.M. Lehn, *Supramolecular Chemistry: concepts and perspectives* (Wiley-VCH, 1995)
H. Leff, A.F. Rex, eds., *Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing* (Institute of Physics Publishing, 2003)
G.A. Jones, M.J. Jones, *Information and Coding Theory*, Springer Undergraduate Mathematics Series (Springer, 2000)
T. Bogner, A. Degenhard, F. Schmid, Phys. Rev. Lett. **93**, 268108 (2004)
S. Sacquin-Mora, E. Laforet, R. Lavery, Proteins: Structure, Functions and Bioinformatics **67**, 350 (2007)
N. Sourlas, Nature **339**, 693 (1989)
N. Sourlas, Physica A **302**, 14 (2001)
H. Nishimori, *Statistical Physics of Spin Glasses and Information Processing. An Introduction*, International Series of Monographs in Physics 111 (Clarendon Press Oxford, 2001)
R. Monasson, Phys. Rev. Lett. **75**(15), 2847 (1995)
M. Blatt, S. Wiseman, E. Domany, Phys. Rev. Lett. **76**, 3251 (1996)
S. Wiseman, M. Blatt, E. Domany, Phys. Rev. E **57**(4), 3767 (1998)
Length $n$ $\ln(\mathrm{GAP})$
------------ --------------------- --
2 3.65
3 3.22
4 2.91
5 2.67
6 2.47
7 2.30
8 2.15
: Table of selectivities for patterns with increasing sizes. See text for details.[]{data-label="table:Selectivities"}
|
---
abstract: 'Intergroup contact has long been considered as an effective strategy to reduce prejudice between groups. However, recent studies suggest that exposure to opposing groups in online platforms can exacerbate polarization. To further understand the behavior of individuals who actively engage in intergroup contact in practice, we provide a large-scale observational study of intragroup behavioral differences between members with and without intergroup contact. We leverage the existing structure of NBA-related discussion forums on Reddit to study the context of professional sports. We identify fans of each NBA team as members of a group and trace whether they have intergroup contact. Our results show that members with intergroup contact use more negative and abusive language in their affiliated group than those without such contact, after controlling for activity levels. We further quantify different levels of intergroup contact and show that there may exist nonlinear mechanisms regarding how intergroup contact relates to intragroup behavior. Our findings provide complementary evidence to experimental studies in a novel context, and also shed light on possible reasons for the different outcomes in prior studies.'
author:
- Jason Shuo Zhang
- Chenhao Tan
- Qin Lv
bibliography:
- 'references.bib'
title: 'Intergroup Contact in the Wild: Characterizing Language Differences between Intergroup and Single-group Members in NBA-related Discussion Forums'
---
[^1]
<ccs2012> <concept> <concept\_id>10010405.10010455</concept\_id> <concept\_desc>Applied computing Law, social and behavioral sciences</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003120.10003130</concept\_id> <concept\_desc>Human-centered computing Collaborative and social computing</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Introduction
============
Driven by the growing concerns of tribalism and polarization in world politics [@jamieson2008echo; @chua2018political; @sunstein2009going], it is increasingly important to understand intergroup contact as a straightforward yet potentially powerful strategy to reduce prejudice between groups. Intergroup contact refers to interactions between members of different groups, and groups can be defined using a variety of factors, including political ideology, place of origin, and race. A key hypothesis is that members with intergroup contact (henceforth “[intergroup members]{}”) behave differently, e.g., by showing sympathy towards other groups and voicing different opinions in their affiliated group [@pettigrew2006meta; @pettigrew2008does; @dovidio2017reducing; @dovidio2003intergroup; @pettigrew1997generalized; @pettigrew2011recent; @Bail201804840; @nyhan2010corrections; @bail2014terrified; @lee2018does; @gillani2018me]. However, prior studies have observed different effects of intergroup contact. For instance, self-reported surveys show that intergroup contact relates to reduced prejudice towards immigrants in European countries [@pettigrew1997generalized], while a recent experimental study finds that exposure to opposing groups on Twitter can exacerbate political polarization [@Bail201804840]. Although self-reported surveys and experimental studies have been the main methods for studying the effect of intergroup contact [@nyhan2010corrections; @bail2014terrified; @lee2018does; @camargo2010interracial; @boisjoly2006empathy; @green2009tolerance; @page2008little], we believe that observational study allows researchers to characterize intergroup contact in the wild and provide valuable complementary evidence . Indeed, with the emergence of online groups, it has become possible to observe intergroup contact and individual behavior at a massive scale for substantial periods. Our goal in this study is to investigate how individuals that choose to engage in intergroup contact behave differently from others without intergroup contact in their original affiliated group in online platforms.
We leverage the existing structure of NBA-related discussion forums on Reddit to identify the group affiliations of users and intergroup contact . We choose online fan groups of professional sports teams as a testbed for the following reasons. First, professional sports play a significant role in modern life [@wenner1989media; @cashmore2005making; @guttmann2004ritual]. People in the United States spent more than 31 billion hours watching sports games in 2015 [@nielsenreport], and the attendance of the 2017-2018 National Basketball Association (NBA) season reached 22 million [@nbaattend2018]. Second, professional sports teams are unambiguously competitive in nature. Similar to other common contexts for studies on intergroup contact (e.g., political ideology), fans of sports teams can treat fans of opposing teams as enemies and sometimes even engage in violence [@roadburg1980factors; @frosdick2013football]. Moreover, sports fans tend to think that the media and supporters from opposing teams are likely to have unfair opinions against their favored teams, just like people with different ideologies [@fansbiased].
![ Illustration of NBA-related discussion forums (also known as subreddits) on Reddit. We identify group affiliation (i.e., whether a person is a fan of an NBA team) and intergroup contact based on the existing structure of NBA-related subreddits on Reddit. Each team has its team subreddit. Here, we present 11 of the 30 NBA teams (with corresponding team logos) to cover subreddits of different sizes. The central [[/r/NBA]{}]{} logo represents [[/r/NBA]{}]{}, where intergroup contact happens. The radius of each logo is proportional to the number of subscribers in the corresponding subreddit. Users in each team logo represent fans of a team based on their indicated support in NBA-related subreddits. Red icons refer to [intergroup members]{}who have engaged in intergroup contact and are thus also in the [[/r/NBA]{}]{} logo, while blue icons refer to users without such behavior. We only put red and blue icons in the three largest team subreddits due to space limitations, but every team subreddit has these two categories of users (see [Figure \[fig:overallnumofusers\]]{} for the number of and [single-group members]{}of each team). Note that not all users who participated in these discussion forums qualify as a fan of an NBA team (grey icons). []{data-label="fig:illustration"}](illustration){width="75.00000%"}
[Figure \[fig:illustration\]]{} illustrates our framework. There are 30 teams in the NBA, and every team has its discussion forum (henceforth [*team subreddit*]{}) on Reddit, a place where fans of the corresponding team congregate and discuss news, games, and any other topics that are relevant to the team. The low-access barrier on the Internet also enables users to communicate easily with fans from opposing teams. In fact, [[/r/NBA]{}]{} is dedicated to interactions between fans of all NBA teams for any discussion related to the NBA. Contrary to each team’s “echo chamber,” which is dominated by fans that support the same team, [[/r/NBA]{}]{} represents an open and diverse environment where intergroup contact occurs. We can thus identify [intergroup members]{}and [single-group members]{}based on whether they have any activity in the intergroup setting ([[/r/NBA]{}]{}).
As posting comments is a major activity in online platforms such as Reddit, analyzing the language used in online platforms provides an opportunity for capturing individuals’ attitudes and emotions [@de2014characterizing; @park2015automatic; @schwartz2013personality; @quercia2012personality]. In particular, there has been growing concerns about hate speech and negative language in online communities [@hateoffensive; @cheng+dnm+leskovec:2015; @cheng2017anyone; @seering_shaping_2017; @chandrasekharan2017you]. We thus focus on characterizing the differences in language usage between intergroup and [single-group members]{}in [*their affiliated team subreddit*]{} (the intragroup setting), e.g., whether [intergroup members]{} swear more than [single-group members]{}in their affiliated team subreddit. As a result, we would be able to capture behavioral differences reflected in language use between intergroup and [single-group members]{}in the intragroup setting. Note that in this work, we do not claim that intergroup contact causes such differences due to endogenous factors that may lead to individuals choosing to engage in intergroup contact in the wild (i.e., individuals who choose to engage in intergroup contact in practice may be inherently different from those who do not). [[**Organization and highlights.**]{}]{} We start by summarizing related work to put our work in context ([Section \[sec:related\]]{}). We then introduce our dataset and provide an overview of the framework for identifying group affiliations and intergroup contact in [Section \[sec:setup\]]{}. With [intergroup members]{}and [single-group members]{}identified, We investigate two research questions in the rest of the paper (methods in [Section \[sec:methods\]]{} and results in [Section \[sec:results\]]{}):
[**RQ1:**]{} *How do members with intergroup contact differ from those without such contact in intragroup language usage in NBA fan groups?*
[**RQ2:**]{} *How do different levels of intergroup contact relate to intragroup language usage?*
For **RQ1**, we first apply matching techniques to make sure the and [single-group members]{}are comparable. We then analyze the behavioral differences between and [single-group members]{}by examining language usage of their comments in their affiliated team subreddit. We demonstrate intriguing contrasts between them: [intergroup members]{}tend to use more negative and swear words, and generate more hate speech comments compared to [single-group members]{}in their affiliated team subreddit.
For **RQ2**, we are able to quantify different levels of intergroup contact for each [intergroup member]{}based on the frequency of intergroup contact. Interestingly, we find varying mechanisms of how different levels of intergroup contact relate to intragroup behavioral differences. For instance, although intergroup contact mostly monotonically relates to differences in language usage, the trends are not necessarily linear. Such varying mechanisms provide complementary evidence to the seemingly conflicting results on intergroup contact in recent studies.
To explore the potential reasons behind the clear behavioral differences in language usage between and [single-group members]{}, we further compare the language usage of [intergroup members]{}between the intragroup setting (affiliated team subreddit) and the intergroup setting ([[/r/NBA]{}]{}) in [Section \[sec:twofaces\]]{}. This setup naturally controls for the subject because we compare the same person across two different environments. We find that [intergroup members]{}are even more negative and more likely to swear in the intergroup setting. Such negative intergroup contact may partly explain the observed differences in intragroup language usage.
Our work highlights the fact that individuals selectively choose to have intergroup contact in the wild, and in turn interact with people without intergroup contact in their original group. We further demonstrate a variety of ways in which intergroup contact levels can moderate intragroup behavior. These observations may reconcile recent conflicting results with respect to intergroup contact. Our findings indicate that observational studies can provide important complementary evidence to experimental studies on this topic because interventions can hardly result in deep and regular contacts. We offer discussions in [Section \[sec:discussion\]]{} and conclude our work in [Section \[sec:conclusion\]]{}.
Related Work {#sec:related}
============
In this section, we first discuss studies that use language as a lens to understand human behavior, especially recent studies on the use of negative language in the context of antisocial behavior. Next, we explain the growing concerns of tribalism, echo-chambers, and polarization, and highlight our specific context, sports, as a testbed for understanding these issues. We then discuss the role of intergroup contact in affecting individual opinions towards opposing groups, including recent work on its backfire effect in online platforms.
Language as a Lens of Human Behavior
------------------------------------
The proliferation of textual content online has inspired a vast body of literature to understand the language in online communication and its relationship with individual attributes. Prior research in CSCW and related communities has investigated how language can reflect properties of individuals [@toma2010reading; @naaman2010really; @de2014characterizing; @park2015automatic; @schwartz2013personality; @quercia2012personality; @tan2015all]. For instance, @toma2010reading show that linguistic emotions correlate with deception in online dating profiles; @de2014characterizing uses linguistics style features to show that mothers with post-partum depression are more likely to use first-person singular pronouns and swear words; @naaman2010really conduct a quantitative analysis of message content from over 350 Twitter users to characterize the type of messages posted on the platform and broadly classify users as self-broadcasters and informers. In general, users’ demographic information and personality can also be predicted based on linguistic features extracted from textual social media data [@minamikawa2011blog; @minamikawa2011personality; @markovikj2013mining; @golbeck2011predicting]. Recently, negative language use has been examined in the context of antisocial behavior in online communities [@cheng2017anyone; @blackburn2014stfu; @seering_shaping_2017; @chandrasekharan2017you; @cheng+dnm+leskovec:2015; @hateoffensive; @cheng2014community]. Conceptually related is a prior study on the effects of community feedback on user behavior, which reveals that negative feedback can lead to future antisocial behavior [@cheng2014community]. @cheng2017anyone further design an experiment that shows negative mood expressed from textual content increases the likelihood of trolling in online platforms. A supervised learning model proposed by @blackburn2014stfu indicates that negative sentiments are useful for predicting toxic players in online games. In this work, we compare the differences in language usage between intergroup and [single-group members]{}, with a focus on expressions of emotions.
Tribalism, Echo Chambers, and Polarization
------------------------------------------
A battery of studies in social sciences has shown that human behavior is shaped by our need to belong to a group and by our proclivity to hate rival groups [@pettigrew2006meta; @tajfel1982social; @cikara2011usversusthem]. Such behavior has been documented in a wide variety of contexts. In the political context, for instance, recent studies find that “liberal group” and “conservative group” on social media not only rarely talk to each other, but also use different hashtags and links to various websites within their tweets [@mason2018uncivil; @mapping2014smith; @gruzd2014investigating; @grevet2014managing; @pearce2014climate]. Another commonly studied context is brand communities in the marketing literature [@cova2006brand; @muniz2001brand; @hickman2007dark; @beal2001no; @guerra2013measure]. For example, @hickman2007dark show that in-groups are strongly motivated to develop negative views of out-groups and engage in “trash talking” about out-groups. In-group members will also gain pleasure at the misfortune of rival brands and their users.
In the sports context, the team sports literature focus on the negative consequences of rivalry, such as negative explicit and implicit attitudes towards the opposing team [@cikara2011usversusthem; @lehr2019outgroup], *schadenfreude* [@havard2014glory; @cikara2013their], and even riot [@guilianotti2013football]. These negative perceptions may even transfer to the sponsors of the rival team: @dalakas2005balance explore the negative sponsorship effects and find that sponsors of disliked NASCAR drivers are viewed less positively than sponsors of liked drivers. Similarly, @olson2018rival finds that brands faced a steep decline in sales among Manchester City fans when they announced the sponsorship of the soccer club Manchester United, a fierce rival of Manchester City. By examining the intergroup emotions of fans of the Boston Red Sox and New York Yankees, @lehr2019outgroup show that pleasure from a powerful rival’s losses can outstrip that from gains of the supported team. Given the competitive nature of professional sports and the importance of emotions in fan behavior, we believe that professional sports provide exciting opportunities for understanding polarization.
Intergroup Contact
------------------
Intergroup contact has long been considered as an effective strategy to reduce prejudice between groups [@dovidio2017reducing]. For instance, a seminal work by @pettigrew1997generalized shows that intergroup contact relates to reduced prejudice towards immigrants based on self-reported surveys in France, Great Britain, the Netherlands, and West Germany. @wright1997extended find correlational evidence that people who knew that an in-group member had an out-group friend had less negative intergroup attitudes. They also experimentally demonstrate that providing this information induces more positive attitudes. @abbott2014makes examine young people’s assertive bystander intentions in an intergroup (immigrant) name-calling situation and find that greater intergroup contact is related to higher levels of empathy, higher levels of cultural openness, and reduced intergroup bias. From the perspective of language usage online, a field experiment designed by @white2015emotion demonstrates that Muslim and Christian high-school students who have structured Internet intergroup interactions tend to use more affective and positive emotion words, and less anger and sadness words. @kim2018intergroup test online contact with two distinct out-groups, undocumented immigrants and gay people. They find that direct online contact improves attitudes towards both out-groups through positive and negative emotions, whereas extended online contact reduces negative emotions and improve attitudes towards undocumented immigrants.
However, recent studies on the “backfire” effect suggest that exposure to opposing groups in online platforms can exacerbate political polarization [@Bail201804840; @nyhan2010corrections; @bail2014terrified; @lee2018does]. [^2] For instance, @Bail201804840 introduce intergroup contact by following a Twitter bot that aggregates tweets of opinion leaders from the opposing political ideology and find that Republicans who follow a liberal Twitter bot become substantially more conservative. @lee2018does use panel data collected in South Korea to investigate the effects of social media usage on changing the political view. They highlight the role of social media in activating political participation and pushing users toward ideological poles.
A possible way to reconcile such differences in prior literature is to review the mechanisms that contribute to the positive effects of intergroup contact: (1) enhancing knowledge about other groups, (2) reducing anxiety when facing opposing groups, and (3) increasing empathy and perspective-taking [@pettigrew2006meta; @pettigrew2008does; @dovidio2017reducing; @dovidio2003intergroup; @pettigrew1997generalized; @pettigrew2011recent; @wright1997extended]. Depending on the motivations to engage in intergroup contact and the actual activities during the contact, intergroup contact in online platforms may not necessarily achieve these goals. We aim to conduct a large-scale observational study to understand the differences between intergroup and [single-group members]{}in their original affiliated group, and also provide some insights on the nature of intergroup contact in [[/r/NBA]{}]{}.
It is useful to point out that there is little work on intergroup contact in the CSCW community. In the meanwhile, several recent studies provide a characterization of intergroup conflict. @kumar2018community examine cases of intergroup conflict across 36,000 communities on Reddit where users of one community are mobilized by negative sentiment to comment in another community and show that less than 1% communities start 74% conflicts. At the community level, by constructing a conflict network between subreddits, @datta2019extracting find that larger subreddits are more likely to be involved in conflicts with a large number of subreddits, and the main “targets” change over time. However, intergroup contact is different from intergroup conflict as it may help improve intergroup attitudes and reduce intergroup tensions and conflicts. Approaching this topic from a CSCW lens raises additional questions about how socio-technical design decisions can influence the outcomes reported in traditional offline settings.
Despite the important role of sports in modern life, sports fan behavior in online communities for professional sports remains understudied. @yu2015world collect real-time tweets from US soccer fans during five 2014 FIFA World Cup games to examine soccer fans’ emotional responses in their tweets. The quantitative analyses show that fear and anger were the most common negative emotions and in general increased when the opponent team scored and decreased when the US team scored. @zhang+tan+lv:18 investigate the connection between online fan behavior and offline team performance, and @Leung2017Effect study the effect of NFL game outcomes on content contribution to Wikipedia. It is also worth noting that fan behavior can differ depending on the environment. @cottingham2012interaction demonstrates the difference in emotional energy between fans in sports bars and those attending the game in the stadium. We believe that there exist exciting opportunities in online sports discussion forums for understanding human behavior, including intergroup contact.
Professional sports as a testbed {#sec:setup}
================================
We focus on the professional sports context derived from NBA-related discussion forums ([[/r/NBA]{}]{} and 30 team subreddits) on Reddit, an active community-driven platform where users can submit posts and make comments. These user-created discussion forums are also called “subreddits”. Each subreddit has multiple moderators to make sure that posts are relevant to the subreddit’s theme. Over the years, basketball fans all over the world have flocked to [[/r/NBA]{}]{}, the site’s professional basketball subreddit, to discuss games in progress, seek meaning in the latest trade rumors, and debate the legality of calls by the referees. In fact, [[/r/NBA]{}]{} has become the largest single-sport subreddit with more than 1.9M subscribers [@redditlist] and one of the most active subreddits on Reddit [@topreddits]. NBA-related subreddits represent an ideal testbed for understanding how intergroup contact relates to intragroup behavior because their structure allows us to identify many users’ team affiliation. Moreover, [[/r/NBA]{}]{} is the place for all basketball fans to congregate, where intergroup contact between fans of different teams happens.
![The distribution of the number of activities made by users in NBA-related subreddits in the 2018, 2017, and 2016 seasons. []{data-label="fig:CDFNumofPosting"}](CDFNumofComments){width="70.00000%"}
Dataset and NBA Seasons
-----------------------
We obtain 2.1M posts and 61M comments in NBA-related subreddits from <https://pushshift.io> [@pushshift]. As pointed out in @zhang+tan+lv:18, offline NBA seasons are reflected in user behavior in these NBA-related subreddits. We organize our dataset according to the timeline of NBA seasons and focus on the most recent three seasons, i.e., from July 2015 to June 2018. For simplicity and clarity, we refer to a specific season by the calendar year when it ends. For instance, the official 2017-2018 NBA season is referred to as *the 2018 season* or *2018* in this paper.
Identifying Team Affiliation and Intergroup Contact
---------------------------------------------------
To identify the team affiliation of users in a season, we first define active users in NBA-related subreddits in a season as those who have at least five activities, where an activity refers to either submitting a post or making a comment. [Figure \[fig:CDFNumofPosting\]]{} shows the distribution of the number of activities by a user in NBA-related subreddits. These active users contribute over 95% of all the activities in NBA-related subreddits.
We identify the team affiliation of active users based on where their activities occur and by using a special mechanism on Reddit, known as flair. Flair appears as an icon next to the username in posts and comments. Every comment can have at most one flair. Before April 2018, flairs are represented by team logos. After that, Reddit adopted a new design to the entire platform, and the flairs are represented by team names in [[/r/NBA]{}]{}. An example is shown in [Figure \[fig:FlairChange\]]{}. In [[/r/NBA]{}]{}, fans can use flairs to indicate support of a team. $\sim$80% of the comments/posts in our [[/r/NBA]{}]{} dataset have been made with flairs even though flairs are optional. We use all the flairs that fans used in [[/r/NBA]{}]{} for the inference of their team affiliation [^3].
[0.48]{} ![An example of the flair usage in /r/NBA before and after the design change. Before the design change, flairs are represented by team logos while after the change, flairs are represented by the team name. []{data-label="fig:FlairChange"}](flairfig "fig:"){width="\textwidth"} \[fig:designbefore\]
[0.48]{} ![An example of the flair usage in /r/NBA before and after the design change. Before the design change, flairs are represented by team logos while after the change, flairs are represented by the team name. []{data-label="fig:FlairChange"}](flairtext "fig:"){width="\textwidth"} \[fig:designafter\]
We view posting/commenting in a team subreddit and using a team’s flair in [[/r/NBA]{}]{} as an indication of support towards that team. An active user is defined as a fan of a team if the user indicates support only for that team and such support sustains over all activities in an entire NBA season. In other words, all activities of a fan indicate support towards his/her affiliated team. It follows that not every active user in NBA-related subreddits is identified as a fan of some team.
We further determine whether a fan of a team is exposed to intergroup contact based on his/her (lack of) behavior in [[/r/NBA]{}]{}, which we refer to as intergroup status. To summarize, we categorize fans of a team into the following two categories:
- ****: Fans of a team who posted in both the affiliated team subreddit and [[/r/NBA]{}]{} in the season.
- ****: Fans of a team who had no activity in [[/r/NBA]{}]{} throughout the season.
2018 2017 2016
-- -------- -------- --------
6,023 5,941 4,843
28,296 24,528 20,467
: The number of and [single-group members]{}in the 2018, 2017, and 2016 seasons.
\[tab:stats\]
[Table \[tab:stats\]]{} shows the number of members in each category. Since our study is concerned with intragroup behavior, i.e., behavior in the affiliated team subreddit, we view these and fans as and members of the affiliated team and study their behavior in the affiliated team subreddit. [Figure \[fig:overallnumofusers\]]{} presents the number of and [single-group members]{}in all 30 team subreddits in the 2018 season (see [Figure \[fig:appoverallnumofusers\]]{} for the numbers in the 2017 and 2016 seasons). In every team subreddit, there are many more [intergroup members]{}than [single-group members]{}. Our definitions are based on user behavior in a single NBA season, and the label of a user can change across seasons. However, a single-group member rarely becomes an [intergroup member]{} in the next season in our dataset (6.0% of [single-group members]{}become [intergroup members]{}from 2016 to 2017, and 8.5% of [single-group members]{}become [intergroup members]{}from 2017 to 2018). This also confirms the tendency of [single-group members]{}to avoid intergroup contact.
![The number of and [single-group members]{} affiliated with each NBA team in the 2018 season. We rank 30 team subreddits by the number of subscribers each team has by the end of the 2018 season. (see [Figure \[fig:appoverallnumofusers\]]{} for the data statistics in the 2017 and 2016 seasons). []{data-label="fig:overallnumofusers"}](overall_num_of_users_by_team_subreddit_2018.pdf){width="70.00000%"}
We use both posting and commenting behavior to identify fans’ team affiliation, but we focus on analyzing comments in the rest of the paper since the posts are usually much longer and more formal, and are thus not comparable to comments.
Methods {#sec:methods}
=======
We study the behavioral differences between and [single-group members]{}in their affiliated team subreddit by examining the expression of emotions in their comments in team subreddits for two reasons: (1) emotion is a central theme in understanding sports fans and their opinions; (2) textual content constitutes the main observed behavior in NBA-related discussion forums [@zhang+tan+lv:18; @lehr2019outgroup; @sutton1997creating].
RQ1: Intragroup Behavior Differences {#sec:RQ1}
------------------------------------
[**Matched [intergroup members]{}.**]{} A naïve way to compare these two categories of users is to directly examine all users in each category. However, such an approach does not take into account other important confounding factors, such as how active a member is in the group. We thus seek to ensure that and [single-group members]{}are a priori balanced on any observable features in the affiliated team subreddit, which indicates similar loyalty to the team. To achieve this, we adopt matching techniques: for each [single-group member]{}, we match him/her with the most similar unmatched [intergroup member]{}from the same affiliated team, where similarity is based on all the observed features. Due to the observational nature, whether a member has intergroup contact or not is not randomly assigned. In other words, our study reflects the behavioral differences between those who engage in intergroup contact and those who do not.
Following prior studies on factors associated with fan behavior in online sports communities [@zhang+tan+lv:18; @mann1989sports; @psysportsfan; @Leung2017Effect], we consider the following observable feature set for matching: (1) the number of comments in the affiliated team subreddit, (2) the average time gap between comments, (3) the average length of comments, (4) the proportion of comments in the playoff season, and (5) the proportion of comments in the game threads (these game threads are created for discussions during a game). All the comments examined here are in the members’ affiliated team subreddit. We collect all of these feature values for each season. The similarities between fans are estimated using the nearest neighbor matching technique [@stuart2010matching]. Min-max normalization is applied to each feature before feeding it into the matching model so that no single feature dominates the matching. We do not include the feedback (upvotes/downvotes) that members received from the NBA subreddits for matching because it can be endogenous with the language used in the comments (e.g., comments with hate speeches may not get many upvotes).
To evaluate the outcome of our matching procedure, for each observable feature, we check distributional differences between the treatment group (intergroup members) and the control group (single-group members). We compare their empirical cumulative distributions before and after matching using the Mann-Whitney U test [@mann1947test]. The results of the 2018, 2017, and 2016 seasons are summarized in [Figure \[fig:matching\]]{}. A small p-value here indicates that there exists a significant difference between the treatment group and the control group. Prior to matching, the p-value for each feature is close to 0, implying that the distributions do differ between groups. After matching, we find no difference between the treatment group and the matched control group for any observable feature at the 5% significance level ($\alpha=0.05$) in all three seasons, indicating that the data is balanced across all the covariates after matching.
[**Language usage analysis.**]{} The proportion of emotional words (i.e., positive emotions, negative emotions, and swear words) in members’ comments are analyzed using the Linguistic Inquiry and Word Count software (LIWC [@pennebaker2007linguistic]), a word frequency-based text analysis tool (see [Table \[tab:negativeexamples\]]{} for examples of emotional words detected using this software). The hate speech comments are identified using an automated hate speech detection model [@hateoffensive]. It is a multi-class classifier that can reliably separate hate speech from other offensive language (see [Table \[tab:hatespeechexamples\]]{} for examples of hate speech comments detected using this detection model). According to @hateoffensive, the model achieved an overall precision of 0.91, recall of 0.90, and F1 score of 0.90 on detecting hate speech tweets.
[**[Fightin-Words]{}model.**]{} To identify a list of distinguishing keywords that are over-used by or [single-group members]{}, we apply the [Fightin-Words]{}algorithm [@monroe2008fightin] to compare the word frequencies to the background frequencies found in the other fan group’s corpora using the informative Dirichlet prior model. This method estimates the log-odds ratio of each word $w$ between two corpora $\alpha$ and $\beta$ given the frequencies obtained from the background corpus $\mathcal{D}$. Then the log-odds ratio $\delta_w^{(\alpha\text{-}\beta)}$ for word $w$ can be estimated as: $$\delta_w^{(\alpha\text{-}\beta)} = log\frac{c_w^{\alpha} + c_w^{\mathcal{D}}}
{c^{\alpha} + c^{\mathcal{D}} - c_w^{\alpha} + c_w^{\mathcal{D}}} \
- log\frac{c_w^{\beta} + c_w^{\mathcal{D}}}{c^{\beta} + c^{\mathcal{D}}
- c_w^{\beta} + c_w^{\mathcal{D}}},$$ where $c_w^{\alpha}$ and $c_w^{\beta}$ are the counts of word $w$ in corpora $\alpha$ and $\beta$, $c^{\alpha}$ and $c^{\beta}$ are the counts of all words in corpora $\alpha$ and $\beta$, $c_w^{\mathcal{D}}$ is the count of word $w$ in the background corpus $\mathcal{D}$, and $c^{\mathcal{D}}$ is the count of all words in corpus $\mathcal{D}$. The [Fightin-Words]{}algorithm also provides an estimation for the variance of the log-odds ratio, $$\sigma^2(\delta_w^{(\alpha\text{-}\beta)})
\backsim \frac{1}{c_w^{\alpha} + c_w^{\mathcal{D}}} \
+ \frac{1}{c_w^{\beta} + c_w^{\mathcal{D}}},$$ and the corresponding $z$-score can be calculated as follows: $$Z = \frac{\delta_w^{(\alpha\text{-}\beta)}}
{\sqrt{\sigma^2(\delta_w^{(\alpha\text{-}\beta)})}}.$$
The [Fightin-Words]{}model is known to outperform other traditional methods in detecting word usage differences between corpora, such as PMI (pointwise mutual information) [@manning1999foundations] and TF-IDF [@salton1986introduction], by not over-emphasizing fluctuations of rare words [@monroe2008fightin]. We use the comments made by [intergroup members]{}as the background corpus for [single-group members]{}and vice versa to identify differences in language usage by each fan group [^4]. We rank each word by averaging its z-scores calculated by the [Fightin-Words]{}model across all 30 teams. A higher positive z-score indicates this word is over-used by [single-group members]{}, and a higher negative z-score means this word is over-used by [intergroup members]{}.
RQ2: Different Levels of Intergroup Contact {#sec:level}
-------------------------------------------
[**Matched [intergroup members]{}with different levels of intergroup contact.**]{} We define a user’s level of intergroup contact based on the fraction of comments in the intergroup setting ([[/r/NBA]{}]{}). The fraction is calculated as the proportion of the number of comments the user made in the [[/r/NBA]{}]{} versus the total number of comments in NBA-related subreddits. Specifically, for each member, we again apply the nearest neighbor matching technique to find five closest members in the same affiliated team and assign a label of 1, 2, 3, 4, or 5 to them based on their fraction of comments in [[/r/NBA]{}]{} in a complete NBA season. Different from pairing members and members before, we do it with replacement because there are not enough members to conduct this matching uniquely. As such, an member can be matched to multiple members. We compare the empirical cumulative distributions before and after matching for each level using the Mann-Whitney U test \[35\]. The results of the 2018, 2017, and 2016 seasons are presented in Figure \[fig:matchinglevel2018\], \[fig:matchinglevel2017\], and \[fig:matchinglevel2016\], respectively. We also assign a label of 0 to members. A larger label indicates a higher level of intergroup contact that the member has in [[/r/NBA]{}]{}. We aggregate [intergroup members]{}at each level across all 30 team subreddits to compare their intragroup behavior. Note that the number of members at each level is the same, but some [intergroup members]{}may be counted more than once.
[**Regression analyses of the relationship between different levels of intergroup contact and language usage.**]{} To understand the relationship between members’ intergroup contact level and language usage, we also conduct OLS regression analyses after the above matching procedure. The independent variables considered in the regression model are the same set of features used in matching intergroup and [single-group members]{}([Section \[sec:RQ1\]]{}). We standardize all independent variables before feeding into the regression model. Our full linear regression model to test each language usage pattern is shown below: $$\begin{aligned}
{\text{\it\small Proportion of language usage}} \sim &
\beta_0 + \beta_1{\text{\it\small number of comments}} +
\beta_2{\text{\it\small average comment hours gap}} \\
&\quad + \beta_3{\text{\it\small average comment length}}
+ \beta_4{\text{\it\small proportion of playoff comments}} \\
&\quad + \beta_5{\text{\it\small proportion of game thread comments}}\\
&\quad + \beta_6{\text{\it\small fraction}} +
\beta_7{\text{\it\small fraction}}\times {\text{\it\small level1}}
+ \beta_8{\text{\it\small fraction}}\times {\text{\it\small level2}}\\
&\quad + \beta_9{\text{\it\small fraction}}\times {\text{\it\small level3}} +
\beta_{10}{\text{\it\small fraction}}\times {\text{\it\small level4}} +
\beta_{11}{\text{\it\small fraction}}\times {\text{\it\small level5}}\\\end{aligned}$$
The fraction in the linear regression model refers to the proportion of the number of comments the user made in the intergroup setting (/r/NBA) versus the total number of comments in NBA-related subreddits. All the control variables for matching [intergroup members]{}and [single-group members]{}are included. There are repeated measures in our regression model as an [intergroup member]{}can be matched to more than one [single-group member]{}. The average number of times for an [intergroup member]{}to be matched is 1.77 (excluding the [intergroup members]{}who never get matched). Among the [intergroup members]{}who are matched more than once, the average variance of their intergroup contact levels in different matches is 0.27. The small variance shows the consistency of our matching technique.
Results {#sec:results}
=======
In this section, we examine intragroup language differences between and [single-group members]{}(RQ1). We further discuss how different levels of intergroup contact relate to intragroup behavior (RQ2).
RQ1: Intragroup Language Differences {#sec:languagedifference}
------------------------------------
[0.32]{} ![The comparison of language usage between and [single-group members]{} in the 2018 season. [Intergroup members]{}use more negative words ([Figure \[fig:effectneg\]]{}; two-tailed t-test, $t=6.23$, $p<0.001$, 95% CI=0.08% to 0.16%; 26 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words ([Figure \[fig:effectswear\]]{}; two-tailed t-test, $t=3.51$, $p<0.001$, 95% CI=0.02% to 0.08%; 28 out of 30 teams, two-tailed binomial test $p<0.001$), and generate more hate speech comments ([Figure \[fig:effecthate\]]{}; two-tailed t-test, $t=10.44$, $p<0.001$, 95% CI=1.00% to 1.46%; 26 out of 30 teams, two-tailed binomial test $p<0.001$). “All” is based on concatenating the samples from all 30 NBA team subreddits, and we also show the top two and bottom two teams ranked by the number of subscribers that have at least 100 [single-group members]{}. We further show the scatter plot of all 30 teams in the top right to illustrate that the findings are robust across teams (the size of the dot is proportional to the number of subscribers). Error bars represent standard errors. The results are consistent in the 2017 and 2016 seasons (see [Figure \[fig:intrasentiment2017\]]{} and [Figure \[fig:intrasentiment2016\]]{}). []{data-label="fig:intrasentiment2018"}](UsersSubredditRNBACausal/users_subreddit_rnba_negative_2018.pdf "fig:"){width="\textwidth"}
[0.32]{} ![The comparison of language usage between and [single-group members]{} in the 2018 season. [Intergroup members]{}use more negative words ([Figure \[fig:effectneg\]]{}; two-tailed t-test, $t=6.23$, $p<0.001$, 95% CI=0.08% to 0.16%; 26 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words ([Figure \[fig:effectswear\]]{}; two-tailed t-test, $t=3.51$, $p<0.001$, 95% CI=0.02% to 0.08%; 28 out of 30 teams, two-tailed binomial test $p<0.001$), and generate more hate speech comments ([Figure \[fig:effecthate\]]{}; two-tailed t-test, $t=10.44$, $p<0.001$, 95% CI=1.00% to 1.46%; 26 out of 30 teams, two-tailed binomial test $p<0.001$). “All” is based on concatenating the samples from all 30 NBA team subreddits, and we also show the top two and bottom two teams ranked by the number of subscribers that have at least 100 [single-group members]{}. We further show the scatter plot of all 30 teams in the top right to illustrate that the findings are robust across teams (the size of the dot is proportional to the number of subscribers). Error bars represent standard errors. The results are consistent in the 2017 and 2016 seasons (see [Figure \[fig:intrasentiment2017\]]{} and [Figure \[fig:intrasentiment2016\]]{}). []{data-label="fig:intrasentiment2018"}](UsersSubredditRNBACausal/users_subreddit_rnba_swear_2018.pdf "fig:"){width="\textwidth"}
[0.32]{} ![The comparison of language usage between and [single-group members]{} in the 2018 season. [Intergroup members]{}use more negative words ([Figure \[fig:effectneg\]]{}; two-tailed t-test, $t=6.23$, $p<0.001$, 95% CI=0.08% to 0.16%; 26 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words ([Figure \[fig:effectswear\]]{}; two-tailed t-test, $t=3.51$, $p<0.001$, 95% CI=0.02% to 0.08%; 28 out of 30 teams, two-tailed binomial test $p<0.001$), and generate more hate speech comments ([Figure \[fig:effecthate\]]{}; two-tailed t-test, $t=10.44$, $p<0.001$, 95% CI=1.00% to 1.46%; 26 out of 30 teams, two-tailed binomial test $p<0.001$). “All” is based on concatenating the samples from all 30 NBA team subreddits, and we also show the top two and bottom two teams ranked by the number of subscribers that have at least 100 [single-group members]{}. We further show the scatter plot of all 30 teams in the top right to illustrate that the findings are robust across teams (the size of the dot is proportional to the number of subscribers). Error bars represent standard errors. The results are consistent in the 2017 and 2016 seasons (see [Figure \[fig:intrasentiment2017\]]{} and [Figure \[fig:intrasentiment2016\]]{}). []{data-label="fig:intrasentiment2018"}](UsersSubredditRNBACausal/users_subreddit_rnba_hatespeech_2018.pdf "fig:"){width="\textwidth"}
[Figure \[fig:intrasentiment2018\]]{} compares negative language usage between matched and [single-group members]{}. [Intergroup members]{}tend to use more negative language than [single-group members]{}, which is indicated by the use of more negative words (two-tailed t-test, $t=6.23$, $p<0.001$, 95% CI=0.08% to 0.16%; 26 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words (two-tailed t-test, $t=3.51$, $p<0.001$, 95% CI=0.02% to 0.08%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) based on lexicon analysis. We further compute the proportion of hate speech with an automated hate speech and offensive language detection model [@hateoffensive]. It is consistent that [intergroup members]{}also generate more hate speech (two-tailed t-test, $t=10.44$, $p<0.001$, 95% CI=1.00% to 1.46%; 26 out of 30 teams, two-tailed binomial test $p<0.001$). These results indicate that [intergroup members]{}are more emotionally charged in their intragroup behavior compared with the [single-group members]{}and are somewhat different from the hypothesis that intergroup contact enhances empathy and perspective thinking [@batson1997empathy; @pettigrew2008does; @stephan1999role]. Our results are consistent when excluding the NBA playoffs (see [Section \[sec:regular\]]{}). We also compare positive language usage between [intergroup members]{}and [single-group members]{}and do not find a consistent trend at the 5% significance level (see [Figure \[fig:positive\]]{}).
![ The top-10 over-represented words used by (red) and (blue) members in the 2018 season. For each word, we show the distribution of the z-scores for all 30 teams calculated by the [Fightin-Words]{} algorithm [@monroe2008fightin]. []{data-label="fig:fightinwords"}](FightinWords_2018.pdf){width="70.00000%"}
To further understand the difference between and [single-group members]{}in language usage, we identify a list of distinguishing words that are more likely to be used by or by [single-group members]{}, using the [Fightin-Words]{}algorithm with the informative Dirichlet prior model [@monroe2008fightin]. Figure \[fig:fightinwords\] lists the top-10 over-represented words used by and [single-group members]{}in the 2018 season. We rank each word by its average z-score calculated by the [Fightin-Words]{}algorithm across all 30 teams. A positive z-score indicates that this word is over-used by [single-group members]{}, while a negative z-score suggests that this word is over-used by [intergroup members]{}. Our results show that [single-group members]{}are more friendly and calm when commenting in the affiliated team subreddit and use more polite words, such as “agree”, “thanks”, and “help”. Also, “seats” suggest that some [single-group members]{}are local fans, as they frequently discuss information about attending live games. In comparison, [intergroup members]{}use more swear words and talk more about the referees (likely complaining).
RQ2: Different Levels of Intergroup Contact {#sec:differentlevel}
-------------------------------------------
In addition to identifying the intragroup behavioral differences, our observational study allows us to quantify different levels of intergroup contact, which can be difficult to operationalize in experimental studies. Here, we examine the mechanisms of how increased levels of intergroup contact relate to differences in intragroup behavior.
[0.32]{} ![Intragroup language usage differences of members with different intergroup contact levels in the 2018 season. x-axis represents intergroup levels determined by the fraction of comments in [[/r/NBA]{}]{}. We observe a consistent monotonic pattern in the proportion of negative words (mean = 1.69%, 1.69%, 1.74%, 1.81%, 1.85%, and 1.90%, respectively for labels from 0 to 6;), swear words (mean = 0.48%, 0.51%, 0.50%, 0.53%, 0.55%, and 0.60%, respectively for labels from 0 to 6), and hate speech comments (mean = 6.51%, 7.42%, 7.75%, 7.94%, 8.51%, and 9.08%, respectively for labels from 0 to 6). The monotonic trend is consistent in the 2017 and 2016 season (see [Figure \[fig:levellanguage2017\]]{} and [Figure \[fig:levellanguage2016\]]{}). Error bars represent standard errors. []{data-label="fig:levellanguage2018"}](IntergroupContactLevelCausalAnalysis/Intergroup_Contact_Level_Freq_negative_2018 "fig:"){width="\textwidth"}
[0.32]{} ![Intragroup language usage differences of members with different intergroup contact levels in the 2018 season. x-axis represents intergroup levels determined by the fraction of comments in [[/r/NBA]{}]{}. We observe a consistent monotonic pattern in the proportion of negative words (mean = 1.69%, 1.69%, 1.74%, 1.81%, 1.85%, and 1.90%, respectively for labels from 0 to 6;), swear words (mean = 0.48%, 0.51%, 0.50%, 0.53%, 0.55%, and 0.60%, respectively for labels from 0 to 6), and hate speech comments (mean = 6.51%, 7.42%, 7.75%, 7.94%, 8.51%, and 9.08%, respectively for labels from 0 to 6). The monotonic trend is consistent in the 2017 and 2016 season (see [Figure \[fig:levellanguage2017\]]{} and [Figure \[fig:levellanguage2016\]]{}). Error bars represent standard errors. []{data-label="fig:levellanguage2018"}](IntergroupContactLevelCausalAnalysis/Intergroup_Contact_Level_Freq_swear_2018 "fig:"){width="\textwidth"}
[0.32]{} ![Intragroup language usage differences of members with different intergroup contact levels in the 2018 season. x-axis represents intergroup levels determined by the fraction of comments in [[/r/NBA]{}]{}. We observe a consistent monotonic pattern in the proportion of negative words (mean = 1.69%, 1.69%, 1.74%, 1.81%, 1.85%, and 1.90%, respectively for labels from 0 to 6;), swear words (mean = 0.48%, 0.51%, 0.50%, 0.53%, 0.55%, and 0.60%, respectively for labels from 0 to 6), and hate speech comments (mean = 6.51%, 7.42%, 7.75%, 7.94%, 8.51%, and 9.08%, respectively for labels from 0 to 6). The monotonic trend is consistent in the 2017 and 2016 season (see [Figure \[fig:levellanguage2017\]]{} and [Figure \[fig:levellanguage2016\]]{}). Error bars represent standard errors. []{data-label="fig:levellanguage2018"}](IntergroupContactLevelCausalAnalysis/Intergroup_Contact_Level_Freq_hatespeech_2018 "fig:"){width="\textwidth"}
Figure \[fig:levellanguage2018\] shows language usage differences between members with different intergroup contact levels. Members of higher intergroup contact levels are generally more negative in language usage: They tend to use more negative words (mean = 1.69%, 1.69%, 1.76%, 1.80%, 1.85%, and 1.90%, respectively for labels from 0 to 6) and swear words (mean = 0.48%, 0.50%, 0.50%, 0.53%, 0.55%, and 0.59%, respectively for labels from 0 to 6 ), and generate more hate speech comments (mean = 6.51%, 7.61%, 7.72%, 7.77%, 8.43%, and 9.15%, respectively for labels from 0 to 6 ) in the affiliated team subreddit. However, the trends are not necessarily linear. For instance, [intergroup members]{}at level 1 do not show significant differences from [single-group members]{}in negative word usage, while [intergroup members]{}at level 5 present a significant jump from previous levels in negative words, swear words, and the use of hate speech. [Table \[tab:reg\]]{} shows the results of regression analyses. The fraction of intergroup contact has a statistically significant positive coefficient in regressions for the proportion of negative words, swear words, and hate speech comments. Moreover, the coefficients for some interaction terms with levels are also statistically significant (e.g., level 5, $\beta_{11}$, is statistically significant in regressions for the proportion of negative words, swear words, and hate speech comments), indicating that nonlinear corrections are required. Note that the BIC score is consistently better by incorporating the interaction terms, although adjusted $R^2$ remains the same due to the fact that this is a very challenging regression task.
[l|LL|LL|LL]{} & & &\
& & & & & &\
*Control* &&&&&&\
number of comments & 0.003\* & 0.003\* & 0.023\*\*\* & 0.022\*\*\* & 0.002\* & 0.002\*\
average comment hours gap &-0.004\*\*\*&0.004\*\*\*&-0.017\*\*\*& -0.017\*\*\*&-0.003\*\*\*&-0.003\*\*\*\
average comment length & -0.028\*\*\*&-0.028\*\*\*&-0.000&-0.000&-0.042\*\*\*&-0.042\*\*\*\
Prop. of playoff comments & 0.006\*\*\*&0.006\*\*\*&0.019\*\*\*& 0.018\*\*\*&0.004\*\*\*&0.004\*\*\*\
Prop. of game thread comments & 0.053\*\*\*&0.053\*\*\*&0.090\*\*\*&0.089\*\*\*&0.028\*\*\*&0.028\*\*\*\
*Fraction* &&&&&&\
fraction & 0.007\*\*\* & 0.005\*\*\*& 0.027\*\*\*&0.019\*\*\*& 0.004\*\*\*&0.003\*\*\*\
*Levels* &&&&&&\
fraction $\times$ level1 & & -0.002 & & -0.001 & & 0.001\
fraction $\times$ level2 & & 0.001 & & -0.001 & &-0.001\
fraction $\times$ level3 & & 0.002\*\* & & 0.003 & &0.000\
fraction $\times$ level4 & & 0.002\*\* & & 0.005\* & &0.001\
fraction $\times$ level5 & & 0.002\*\*\* & & 0.010\*\*\* & &0.002\*\*\*\
intercept & 0.052\*\*\* & 0.053\*\*\* & 0.040\*\*\* & 0.042\*\*\* & 0.028\*\*\*&0.028\*\*\*\
Adjusted $R^2$ & 0.172 & 0.172 & 0.033 & 0.033 & 0.138 & 0.138\
BIC & & & & & &\
\[tab:reg\]
Intragroup Behavior vs. Intergroup Behavior of the Same User {#sec:twofaces}
============================================================
[0.32]{} ![[Intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting in the 2018 season. Here we only consider the matched [intergroup members]{}in [Figure \[fig:intrasentiment2018\]]{} (i.e., the solid red bars in this figure are identical to the red bars in [Figure \[fig:intrasentiment2018\]]{} ). They use more negative words (two-tailed t-test, $t=9.39$, $p<0.001$, 95% CI=0.17% to 0.26%; 30 out of 30 teams, two-tailed binomial test $p=0.001$) and swear words (two-tailed t-test, $t=13.69$, $p<0.001$, 95% CI=0.22% to 0.29%; 29 out of 30 teams, two-tailed binomial test $p<0.001$), and generate more hate speech comments (two-tailed t-test, $t=2.97$, $p=0.003$, 95% CI=0.18% to 0.88%; 23 out of 30 teams, two-tailed binomial test $p=0.005$). “All” is based on concatenating the samples from all 30 NBA team subreddits, and we also show the top two and bottom two teams ranked by the number of subscribers that have at least 100 [single-group members]{}. We further show the scatter plot of all 30 teams in the top right to illustrate that the findings are robust across teams (the size of the dot is proportional to the number of subscribers). Error bars represent standard errors. The results are consistent in the 2017 and 2016 seasons (see [Figure \[fig:intersentiment2017\]]{} and [Figure \[fig:intersentiment2016\]]{}). []{data-label="fig:intersentiment2018"}](SameUsersSubredditRNBA/same_users_subreddit_rnba_negative_2018 "fig:"){width="\textwidth"}
[0.32]{} ![[Intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting in the 2018 season. Here we only consider the matched [intergroup members]{}in [Figure \[fig:intrasentiment2018\]]{} (i.e., the solid red bars in this figure are identical to the red bars in [Figure \[fig:intrasentiment2018\]]{} ). They use more negative words (two-tailed t-test, $t=9.39$, $p<0.001$, 95% CI=0.17% to 0.26%; 30 out of 30 teams, two-tailed binomial test $p=0.001$) and swear words (two-tailed t-test, $t=13.69$, $p<0.001$, 95% CI=0.22% to 0.29%; 29 out of 30 teams, two-tailed binomial test $p<0.001$), and generate more hate speech comments (two-tailed t-test, $t=2.97$, $p=0.003$, 95% CI=0.18% to 0.88%; 23 out of 30 teams, two-tailed binomial test $p=0.005$). “All” is based on concatenating the samples from all 30 NBA team subreddits, and we also show the top two and bottom two teams ranked by the number of subscribers that have at least 100 [single-group members]{}. We further show the scatter plot of all 30 teams in the top right to illustrate that the findings are robust across teams (the size of the dot is proportional to the number of subscribers). Error bars represent standard errors. The results are consistent in the 2017 and 2016 seasons (see [Figure \[fig:intersentiment2017\]]{} and [Figure \[fig:intersentiment2016\]]{}). []{data-label="fig:intersentiment2018"}](SameUsersSubredditRNBA/same_users_subreddit_rnba_swear_2018 "fig:"){width="\textwidth"}
[0.32]{} ![[Intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting in the 2018 season. Here we only consider the matched [intergroup members]{}in [Figure \[fig:intrasentiment2018\]]{} (i.e., the solid red bars in this figure are identical to the red bars in [Figure \[fig:intrasentiment2018\]]{} ). They use more negative words (two-tailed t-test, $t=9.39$, $p<0.001$, 95% CI=0.17% to 0.26%; 30 out of 30 teams, two-tailed binomial test $p=0.001$) and swear words (two-tailed t-test, $t=13.69$, $p<0.001$, 95% CI=0.22% to 0.29%; 29 out of 30 teams, two-tailed binomial test $p<0.001$), and generate more hate speech comments (two-tailed t-test, $t=2.97$, $p=0.003$, 95% CI=0.18% to 0.88%; 23 out of 30 teams, two-tailed binomial test $p=0.005$). “All” is based on concatenating the samples from all 30 NBA team subreddits, and we also show the top two and bottom two teams ranked by the number of subscribers that have at least 100 [single-group members]{}. We further show the scatter plot of all 30 teams in the top right to illustrate that the findings are robust across teams (the size of the dot is proportional to the number of subscribers). Error bars represent standard errors. The results are consistent in the 2017 and 2016 seasons (see [Figure \[fig:intersentiment2017\]]{} and [Figure \[fig:intersentiment2016\]]{}). []{data-label="fig:intersentiment2018"}](SameUsersSubredditRNBA/same_users_subreddit_rnba_hatespeech_2018 "fig:"){width="\textwidth"}
Given the clear intragroup behavioral differences in language usage between and [single-group members]{}, we end our study by exploring the potential reasons behind them. We study the differences in language usage of the same user in his/her affiliated team subreddit vs. in [[/r/NBA]{}]{}. We compare the same person in two different contexts and naturally control for most of the confounding factors, which is also connected with the personality vs. situation debate [@kenrick1988profiting].
[Figure \[fig:intersentiment2018\]]{} shows that [intergroup members]{} use even more negative language in the intergroup setting, as they use more negative words (two-tailed t-test, $t=9.39$, $p<0.001$, 95% CI=0.17% to 0.26%; 30 out of 30 teams, two-tailed binomial test $p=0.001$) and swear words (two-tailed t-test, $t=13.69$, $p<0.001$, 95% CI=0.22% to 0.29%; 29 out of 30 teams, two-tailed binomial test $p<0.001$), and generate more hate speech comments than in the intragroup setting (two-tailed t-test, $t=2.97$, $p=0.003$, 95% CI=0.18% to 0.88%; 23 out of 30 teams, two-tailed binomial test $p=0.005$). This indicates that fans are more hostile when facing fans from other teams than from the same team. This observation is robust after controlling for topics of discussion by only considering game threads (see [Section \[sec:appgamethreads\]]{} and [Figure \[fig:gamethread\]]{}). Our observation suggests that although [intergroup members]{}are more emotional than [single-group members]{}in the affiliated subreddit, they are not as “outrageous” as they are in the intergroup setting. In comparison, when going to the intergroup setting and confronting fans from other team groups, they tend to have more negative interactions and troll each other.
These observations may provide explanations for the characteristics of fans in intragroup behavior. Prior studies suggest that negative intergroup contact is more influential in shaping people’s attitudes and may curb the contact’s ability to reduce prejudice [@graf2014negative; @paolini2010negative; @stark2013generalization]. The emotionally charged intergroup contact from the intergroup setting may connect to fans’ more sentimental attitudes in their affiliated team subreddit. It requires further research to establish the causal link here, but the fact that we are able to observe these contrasts demonstrates the importance of such observational studies based on real interactions over substantial time periods.
Discussion {#sec:discussion}
==========
Although most previous studies have focused on the role of intergroup contact in changing attitudes of individual members, our study highlights the fact that users selectively become [intergroup members]{}, and and members in turn interact with each other in their affiliated group. Such interaction can potentially influence members’ language usage and shape the entire group. Moreover, we demonstrate a variety of ways in which intergroup contact levels can moderate intragroup behavior. This indicates that observational studies can provide important complementary evidence to experimental studies on this topic because interventions can hardly result in deep and regular contact. Novel methodologies are required to further bridge the gap between observational studies and experimental studies.
[[**Could social media be driving polarization?**]{}]{} Twitter, Reddit, and Facebook have become important platforms for political discussions as well as misinformation [@vosoughi2018spread; @grinberg2019fake]. Service providers are designing new features that would actively expose people to opposing views. For example, Twitter recently experimented with new algorithms that would promote alternative viewpoints in Twitter’s timeline to address misinformation and reduce the effect of echo chambers [@jackdorsey]. However, the proposed solution may increase polarization. Unlike decades of offline experiments which mostly indicate intimate contact between members of rival groups across an extended period can produce positive effects, the results in @Bail201804840 and our paper suggest that encountering views from opposing groups online may make them even more wedded to their own views. There are several possible explanations of this contrast by examining the possible mechanisms that intergroup contact affects individual attitudes. First, the comments created on social media are usually brief. These short messages without enough context may not enhance knowledge about opposing groups. Several studies suggest that people interpret short text-based messages inconsistently, which creates significant potential for miscommunication [@wu2014short; @miller2017understanding; @kelly2012s]. Second, the discussion structure may facilitate the spread of negative interaction. @cheng2017anyone examines the evolution of discussions on CNN.com and show that existing trolling comments in a discussion thread significantly increase the likelihood of future trolling comments. The spread of negativity will increase rather than reduce people’s anxiety levels when facing opposing groups. Third, the anonymous, spontaneous nature of communications on social media may not be conducive to cultivating empathy. In an experiment designed to examine the relationship between the presence of mobile devices and the quality of social interactions, results show that participants who have conversations in the absence of mobile devices report high levels of empathetic concern [@misra2016iphone]. In summary, intergroup contact may lead to diverging outcomes depending on the environment and the nature of the contact. Further research is required to examine these possibilities and understand how social and technical design decisions can influence the outcomes.
[[**Can we design better online discussion forums for different groups?**]{}]{} The findings in this work indicate that social platforms designers should consider strategies to shape intergroup contact online. As hinted above, it is insufficient to recommend users to follow members of opposing groups or opposing views. Better design strategies need to be experimented for encouraging civil and extended intergroup contact. It would also be useful to take into account how different levels of intergroup contact may moderate individual opinions differently. Content moderation can be a promising area for future studies in the context of intergroup contact [@kiesler+kraut+resnick+kittur:2011]. For instance, @matias2019preventing shows the displaying community rules can prevent harassment, but how to reduce negative intergroup contact remains an open question. Similarly, a powerful way of spreading online information is through social consensus cues and online endorsement (e.g., upvotes, likes). However, promoting content with the highest popularity can sometimes be problematic. Earlier research suggests that tweets with more sentiment-laden words are likely to be favorited or retweeted, and politicians may intentionally use this strategy to maximize impacts on Twitter [@brady2018ideological; @tan+lee+pang:14; @tan+etal:16b]. Our study also finds that [intergroup members]{}receive better feedback from their affiliated team subreddit even though they use more negative language ([Figure \[fig:feedback\]]{}). This type of behavior can generate negative reactions from opposing groups and push the whole discussion to cycle towards more emotionally-laden and potentially polarizing content. It is thus important to develop comment ranking systems that are cognizant of intergroup contact and prioritizes constructive interactions.
**Limitations.** Our findings are subject to the following limitations. First, the causal relationship between intergroup contact and negative language usage is not entirely clear. Due to the nature of our observational study, whether a member has intergroup contact is not randomly assigned. Though we match users based on a series of activity features, an important confounding factor could be that people who seek intergroup contact are inherently different from those who do not.
Second, our definition of intergroup contact entails that we focus on relatively active users. Thus, we cannot observe indirect intergroup contact, such as browsing [[/r/NBA]{}]{}. Prior studies have shown that indirect contact, such as imagining oneself interacting with an out-group member and observing an in-group member interacting with an out-group member [@turner2010imagining; @dovidio2011improving], may also shape human behavior. It also follows that [intergroup members]{}have more activities on NBA-related discussion forums as a whole than [single-group members]{}. We want to note that the nature of intergroup contact is that given the same amount of time in life, individuals with intergroup contact put more effort into intergroup contact than those without such contact.
Third, we use a coarse proxy to consider any users who have posted in our intergroup setting ([[/r/NBA]{}]{}) as [intergroup members]{}, and study the language differences in the intragroup setting at the user level instead of at the dyad level. However, some comments created by the fans in /r/NBA may be replies to the fans who are from the same team or do not have a team affiliation. More in-depth characterization of different types of discussions happened in the intergroup setting is required to further understand the differences observed in this study.
Fourth, the observations made in this study are limited to Reddit NBA fan groups. The sports context might be a strong case for understanding intergroup relations, as all the teams are created to compete with each other for the final championship. The expression of hostile attitudes towards opposing sides are culturally acceptable and even encouraged [@cikara2011usversusthem]. We should expect less negative intergroup contact between groups that do not contend for the same resources (e.g., music fans of different musicians may not have conflicts with each other at all). However, politics, especially in a polarized bipartisan situation, share common properties with the sports context. Examining the generalization of our results in other contexts is a promising avenue for future work.
Finally, the negative language observed in our study may not necessarily bring negative effects to the community. Prior studies suggest the main reason people use swearing words on the online platform is to express some strong emotions, such as anger and frustration [@wang2014cursing; @diakopoulos2011towards; @cheng2017anyone]. @heath2001emotional examine users’ emotional selection in memes when emotion is manipulated and observe that people prefer the version of the story that produced the highest levels of disgust and evoke strong sentiment. @jay2009offensive further argues that only when cursing occurs in the form of insults toward others, such as name-calling, harassment, and hate speech, it becomes harmful. In addition, earlier literature suggests that the reason people use swearing words on the online platform may relate to Internet humor, such as jokes and memes. Posting humorous content on the Internet has the potential to engage other users in art activities that are closely connected to their lives and receive online endorsements [@yoon2016not; @shifman2014memes]. Attempting to be funny could be another reason that [intergroup members]{}adopt a more negative language style than [single-group members]{}. However, as pointed out by @lockyer2005beyond, a significant proportion of Internet humor has offensive, sexism, and racism content, and its consequences are often overlooked.
Conclusion {#sec:conclusion}
==========
In this paper, by applying our computational framework to NBA-related discussion forums on Reddit, we identify clear language differences between intergroup and [single-group members]{}in their affiliated group (the intragroup setting). We find that in the affiliated team subreddit, [intergroup members]{}tend to use more negative and swear words, and generate more hate speech comments compared with single-group members. Moreover, we quantify different levels of intergroup contact for each intergroup member based on the fraction of their comments in the intergroup setting ([[/r/NBA]{}]{}). Interestingly, the level of intergroup contact can relate to differences in language usage in different ways, though the relationship is mostly monotonic. To further shed light on the behavior of intergroup members, we also compare the language usage of intergroup members between the intragroup setting and the intergroup setting. This setup naturally controls for the subject because we compare the same person across two different environments. We observe that intergroup members are even more negative and more likely to swear in the intergroup setting. As intergroup contact in online platforms becomes increasingly common and can play an important role in opinion formation, our work demonstrates how observational studies can provide complementary evidence to experimental studies on this topic.
Appendix
========
Positive credit\*, graced, attract\*, graceful\*, terrific\*, bonus\*, affection\*, humour\*, delicious\*, love, openness, sweetheart\*, bless\*, bold\*, madly, fine, friend\*, hurra\*, ready, trust\*, secur\*, won, improving, fiesta\*, dynam\*, toleran\*, sunniest, optimal\*, helpful\*, neat\*, enthus\*, joking, favour\*, giving, agreeab\*, easiness, supportive\*, frees\*, graces, gentler
---------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Negative ignor\*, aggravat\*, unattractive, scary, attack\*, offend\*, grief, fright\*, domina\*, unfriendly, violat\*, grave\*, nast\*, suck, shock\*, sucker\*, impatien\*, wept, heartless\*, shake\*, battl\*, moron\*, vanity, aggress\*, masochis\*, unsure\*, screw\*, lost, losing, mocker\*, envie\*, sadness, nag\*, timid\*, afraid, hateful\*, turmoil, agoniz\*, obnoxious\*, pain
Swear prick\*, dyke\*, tit, cock, dicks, butt, bloody, dick, sob, asshole\*, pussy\*, screw\*, suck, wanker\*, mofo, fucks, shit\*, bastard\*, arse, butts, darn, sucked, jeez, nigger\*, fucker\*, arses, ass, hell, crappy, dang, motherf\*, dumb\*, heck, crap, tits, queer\*, bitch\*, sonofa\*, titty, fuckin\*
: A sample of positive, negative and swear words in the Linguistic Inquiry and Word Count dictionary (LIWC [@pennebaker2007linguistic]). Words ending with “\*” match any string with the same prefix.
\[tab:negativeexamples\]
Kevin Sorbo’s Hercules was such a pussy magnet
-------------------------------------------------------------------------------------------------------------------------------------------------
Because I want losers like you to fuck off
Same reason KAT ass rapes our team everytime we play.
Holy shit Sabonis is stuntin’ like his daddy right now with these passes
That’s like me saying you’re a dumbass because your team is currently shit. Sorry pal, don’t talk about basketball until you make the playoffs.
Melo gets NO calls ever - I’ve always said he should bitch more
No defense, no rebounding. Same old shit. Embarrassing
Fuck the Celtics!
now since your dumb ass sees that it doesnt make difference like ive been saying what is your excuse?
Tyler Ennis you bum. Worst player in the league.
: Examples of hate speech comments detected with the automated detection model [@hateoffensive].
\[tab:hatespeechexamples\]
{#sec:regular}
[0.32]{} ![ []{data-label="fig:intrasentimentregular2018"}](UsersSubredditRNBACausal/users_subreddit_rnba_regular_negative_2018 "fig:"){width="\textwidth"}
[0.32]{} ![ []{data-label="fig:intrasentimentregular2018"}](UsersSubredditRNBACausal/users_subreddit_rnba_regular_swear_2018 "fig:"){width="\textwidth"}
[0.32]{} ![ []{data-label="fig:intrasentimentregular2018"}](UsersSubredditRNBACausal/users_subreddit_rnba_regular_hatespeech_2018 "fig:"){width="\textwidth"}
[0.32]{} ![[]{data-label="fig:intrasentimentregular2017"}](UsersSubredditRNBACausal/users_subreddit_rnba_regular_negative_2017 "fig:"){width="\textwidth"}
[0.32]{} ![[]{data-label="fig:intrasentimentregular2017"}](UsersSubredditRNBACausal/users_subreddit_rnba_regular_swear_2017 "fig:"){width="\textwidth"}
[0.32]{} ![[]{data-label="fig:intrasentimentregular2017"}](UsersSubredditRNBACausal/users_subreddit_rnba_regular_hatespeech_2017 "fig:"){width="\textwidth"}
[0.32]{} ![[]{data-label="fig:intrasentimentregular2016"}](UsersSubredditRNBACausal/users_subreddit_rnba_regular_negative_2016 "fig:"){width="\textwidth"}
[0.32]{} ![[]{data-label="fig:intrasentimentregular2016"}](UsersSubredditRNBACausal/users_subreddit_rnba_regular_swear_2016 "fig:"){width="\textwidth"}
[0.32]{} ![[]{data-label="fig:intrasentimentregular2016"}](UsersSubredditRNBACausal/users_subreddit_rnba_regular_hatespeech_2016 "fig:"){width="\textwidth"}
[Intergroup members]{}receive better feedback than [single-group members]{}
---------------------------------------------------------------------------
[Figure \[fig:feedback\]]{} shows that [intergroup members]{} receive better feedback in the intragroup setting than members in all three seasons (two-tailed t-test, $t=15.68$, $p<0.001$, 95% CI=0.048 to 0.062%, 29 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2018 season; two-tailed t-test, $t=12.56$, $p<0.001$, 95% CI=0.043 to 0.058%, 30 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2017 season; two-tailed t-test, $t=12.43$, $p<0.001$, 95% CI=0.046 to 0.064%, 30 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2016 season). Comment feedback is defined by whether the comment score (\#upvotes- \#downvotes) is above the median score of that team subreddit in that month, which accounts for the differences across subreddits. This observation suggests that using negative language is likely to draw attention in the corresponding team subreddits.
[0.32]{} ![The comparison of feedback received from affiliated team subreddit between and [single-group members]{}in the 2018, 2017, and 2016 seasons. [Intergroup members]{}receive better feedback than [single-group members]{}in all three seasons (two-tailed t-test, $t=15.68$, $p<0.001$, 95% CI=0.048 to 0.062%, 29 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2018 season; two-tailed t-test, $t=12.56$, $p<0.001$, 95% CI=0.043 to 0.058%, 30 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2017 season; two-tailed t-test, $t=12.43$, $p<0.001$, 95% CI=0.046 to 0.064%, 30 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2016 season). Error bars represent standard errors.[]{data-label="fig:feedback"}](UsersSubredditRNBACausal/users_subreddit_rnba_feedback_2018 "fig:"){width="\textwidth"}
[0.32]{} ![The comparison of feedback received from affiliated team subreddit between and [single-group members]{}in the 2018, 2017, and 2016 seasons. [Intergroup members]{}receive better feedback than [single-group members]{}in all three seasons (two-tailed t-test, $t=15.68$, $p<0.001$, 95% CI=0.048 to 0.062%, 29 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2018 season; two-tailed t-test, $t=12.56$, $p<0.001$, 95% CI=0.043 to 0.058%, 30 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2017 season; two-tailed t-test, $t=12.43$, $p<0.001$, 95% CI=0.046 to 0.064%, 30 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2016 season). Error bars represent standard errors.[]{data-label="fig:feedback"}](UsersSubredditRNBACausal/users_subreddit_rnba_feedback_2017 "fig:"){width="\textwidth"}
[0.32]{} ![The comparison of feedback received from affiliated team subreddit between and [single-group members]{}in the 2018, 2017, and 2016 seasons. [Intergroup members]{}receive better feedback than [single-group members]{}in all three seasons (two-tailed t-test, $t=15.68$, $p<0.001$, 95% CI=0.048 to 0.062%, 29 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2018 season; two-tailed t-test, $t=12.56$, $p<0.001$, 95% CI=0.043 to 0.058%, 30 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2017 season; two-tailed t-test, $t=12.43$, $p<0.001$, 95% CI=0.046 to 0.064%, 30 out of 30 teams, two-tailed binomial test $p<0.001$ for the 2016 season). Error bars represent standard errors.[]{data-label="fig:feedback"}](UsersSubredditRNBACausal/users_subreddit_rnba_feedback_2016 "fig:"){width="\textwidth"}
[Intergroup members]{}are more emotional in the intergroup setting than in the intragroup setting when controlling for the discussion topic {#sec:appgamethreads}
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In [Section \[sec:differentlevel\]]{}, we find that [intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting. This difference may occur due to the fact that more heated topics are discussed in [[/r/NBA]{}]{} than in team subreddits. To control for this factor, we further limit our comparison to the game threads in both settings. Game threads are important components of NBA-related team subreddits to facilitate game-related discussions during NBA games. In practice, each game has a game thread in the home-team subreddit, the away team subreddit, and the overall [[/r/NBA]{}]{}. [Figure \[fig:gamethread\]]{} shows the language usage difference of members in the game threads of the intergroup and intragroup setting. Only members who made comments in the game threads of both settings are included in this analysis (2118 members for the 2018 season, 1495 members for the 2017 season, and 1289 members for the 2016 season). In all three seasons, it is consistent that [intergroup members]{}use more negative words in the game threads of the intergroup setting than of the intragroup setting. We do not compare the language usage patterns per team in this analysis as there are teams with less than 20 members.
[0.32]{} ![The observation that [intergroup members]{}are more negative in the intergroup setting than in the intragroup setting is robust after controlling for topics of discussion by only considering game threads. [Intergroup members]{}use more negative words (two-tailed t-test, $t=2.43$, $p=0.015$, 95% CI=0.04% to 0.35% for the 2018 season; two-tailed t-test, $t=3.76$, $p<0.001$, 95% CI=0.16% to 0.52% for the 2017 season; two-tailed t-test, $t=3.04$, $p=0.002$, 95% CI=0.11% to 0.51% for the 2016 season) and swear words (two-tailed t-test, $t=3.99$, $p<0.001$, 95% CI=0.18% to 0.54% for the 2018 season; two-tailed t-test, $t=2.34$, $p=0.023$, 95% CI=0.04% to 0.32% for the 2017 season; two-tailed t-test, $t=1.96$, $p=0.048$, 95% CI=0.00% to 0.48% for the 2016 season) and generate more hate speech comments (two-tailed t-test, $t=3.10$, $p=0.002$, 95% CI=0.06% to 2.59% for the 2018 season; two-tailed t-test, $t=2.70$, $p=0.007$, 95% CI=0.05% to 3.05% for the 2017 season; two-tailed t-test, $t=1.42$, $p=0.155$, 95% CI=0.00% to 2.46% for the 2016 season) in the game threads of the intergroup setting than that of the intragroup setting. We do not compare the language usage patterns per team in this analysis, as there are teams with less than 20 members after this control. Error bars represent standard errors. []{data-label="fig:gamethread"}](SameUsersSubredditRNBA/same_users_game_thread_negative "fig:"){width="\textwidth"}
[0.32]{} ![The observation that [intergroup members]{}are more negative in the intergroup setting than in the intragroup setting is robust after controlling for topics of discussion by only considering game threads. [Intergroup members]{}use more negative words (two-tailed t-test, $t=2.43$, $p=0.015$, 95% CI=0.04% to 0.35% for the 2018 season; two-tailed t-test, $t=3.76$, $p<0.001$, 95% CI=0.16% to 0.52% for the 2017 season; two-tailed t-test, $t=3.04$, $p=0.002$, 95% CI=0.11% to 0.51% for the 2016 season) and swear words (two-tailed t-test, $t=3.99$, $p<0.001$, 95% CI=0.18% to 0.54% for the 2018 season; two-tailed t-test, $t=2.34$, $p=0.023$, 95% CI=0.04% to 0.32% for the 2017 season; two-tailed t-test, $t=1.96$, $p=0.048$, 95% CI=0.00% to 0.48% for the 2016 season) and generate more hate speech comments (two-tailed t-test, $t=3.10$, $p=0.002$, 95% CI=0.06% to 2.59% for the 2018 season; two-tailed t-test, $t=2.70$, $p=0.007$, 95% CI=0.05% to 3.05% for the 2017 season; two-tailed t-test, $t=1.42$, $p=0.155$, 95% CI=0.00% to 2.46% for the 2016 season) in the game threads of the intergroup setting than that of the intragroup setting. We do not compare the language usage patterns per team in this analysis, as there are teams with less than 20 members after this control. Error bars represent standard errors. []{data-label="fig:gamethread"}](SameUsersSubredditRNBA/same_users_game_thread_swear "fig:"){width="\textwidth"}
[0.32]{} ![The observation that [intergroup members]{}are more negative in the intergroup setting than in the intragroup setting is robust after controlling for topics of discussion by only considering game threads. [Intergroup members]{}use more negative words (two-tailed t-test, $t=2.43$, $p=0.015$, 95% CI=0.04% to 0.35% for the 2018 season; two-tailed t-test, $t=3.76$, $p<0.001$, 95% CI=0.16% to 0.52% for the 2017 season; two-tailed t-test, $t=3.04$, $p=0.002$, 95% CI=0.11% to 0.51% for the 2016 season) and swear words (two-tailed t-test, $t=3.99$, $p<0.001$, 95% CI=0.18% to 0.54% for the 2018 season; two-tailed t-test, $t=2.34$, $p=0.023$, 95% CI=0.04% to 0.32% for the 2017 season; two-tailed t-test, $t=1.96$, $p=0.048$, 95% CI=0.00% to 0.48% for the 2016 season) and generate more hate speech comments (two-tailed t-test, $t=3.10$, $p=0.002$, 95% CI=0.06% to 2.59% for the 2018 season; two-tailed t-test, $t=2.70$, $p=0.007$, 95% CI=0.05% to 3.05% for the 2017 season; two-tailed t-test, $t=1.42$, $p=0.155$, 95% CI=0.00% to 2.46% for the 2016 season) in the game threads of the intergroup setting than that of the intragroup setting. We do not compare the language usage patterns per team in this analysis, as there are teams with less than 20 members after this control. Error bars represent standard errors. []{data-label="fig:gamethread"}](SameUsersSubredditRNBA/same_users_game_thread_hatespeech "fig:"){width="\textwidth"}
![The number of and [single-group members]{} affiliated with each NBA team in the 2017 and 2016 seasons. We rank 30 team subreddits by the number of subscribers each team has by the end of the 2018 season. []{data-label="fig:appoverallnumofusers"}](overall_num_of_users_by_team_subreddit_2017 "fig:"){width="70.00000%"} ![The number of and [single-group members]{} affiliated with each NBA team in the 2017 and 2016 seasons. We rank 30 team subreddits by the number of subscribers each team has by the end of the 2018 season. []{data-label="fig:appoverallnumofusers"}](overall_num_of_users_by_team_subreddit_2016 "fig:"){width="70.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_num_of_comments_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_avg_comment_hours_gap_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_avg_comment_length_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_Nearest_2018 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_num_of_comments_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_avg_comment_hours_gap_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_avg_comment_length_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_Nearest_2017 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_num_of_comments_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_avg_comment_hours_gap_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_avg_comment_length_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique in the 2018, 2017, and 2016 seasons. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments. The corresponding p-values of the Mann-Whitney tests are also reported. Recall that a small p-value indicates that there is dependence between the treatment and control groups (relative frequencies are different). Prior to matching, each p-value is very close to 0.0. After the matching, at the 0.05 significance level ($\alpha=0.05$), we find no dependence on the group label for any activity feature observed in the matched dataset. These trends are consistent in all three seasons. []{data-label="fig:matching"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_Nearest_2016 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_num_of_comments_1_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_hours_gap_1_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_length_1_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_1_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_1_Nearest_2018 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_num_of_comments_2_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_hours_gap_2_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_length_2_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_2_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_2_Nearest_2018 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_num_of_comments_3_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_hours_gap_3_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_length_3_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_3_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_3_Nearest_2018 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_num_of_comments_4_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_hours_gap_4_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_length_4_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_4_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_4_Nearest_2018 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_num_of_comments_5_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_hours_gap_5_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_avg_comment_length_5_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_5_Nearest_2018 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2018 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2018"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_5_Nearest_2018 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_num_of_comments_1_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_hours_gap_1_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_length_1_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_1_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_1_Nearest_2017 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_num_of_comments_2_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_hours_gap_2_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_length_2_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_2_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_2_Nearest_2017 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_num_of_comments_3_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_hours_gap_3_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_length_3_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_3_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_3_Nearest_2017 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_num_of_comments_4_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_hours_gap_4_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_length_4_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_4_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_4_Nearest_2017 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_num_of_comments_5_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_hours_gap_5_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_avg_comment_length_5_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_5_Nearest_2017 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2017 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2017"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_5_Nearest_2017 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_num_of_comments_1_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_hours_gap_1_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_length_1_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_1_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_1_Nearest_2016 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_num_of_comments_2_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_hours_gap_2_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_length_2_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_2_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_2_Nearest_2016 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_num_of_comments_3_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_hours_gap_3_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_length_3_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_3_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_3_Nearest_2016 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_num_of_comments_4_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_hours_gap_4_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_length_4_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_4_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_4_Nearest_2016 "fig:"){width="19.00000%"}
![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_num_of_comments_5_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_hours_gap_5_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_avg_comment_length_5_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_playoff_comment_5_Nearest_2016 "fig:"){width="19.00000%"} ![Empirical cumulative distribution of each activity feature before and after the matching technique from level 1 to level 5 in the 2016 season. The activity features from left to right are the number of comments, the average hour gap between comments, the average comment length, the proportion of playoff comments, and the proportion of game thread comments.[]{data-label="fig:matchinglevel2016"}](MatchingResults/Matching_Results_proportion_of_game_thread_comment_5_Nearest_2016 "fig:"){width="19.00000%"}
[0.32]{} ![Comparison of language usage between and members in the 2017 season. [Intergroup members]{}use more negative words (two-tailed t-test, $t=4.04$, $p<0.001$, 95% CI=0.04% to 0.13%; 24 out of 30 teams, two-tailed binomial test $p=0.001$) and swear words (two-tailed t-test, $t=4.17$, $p<0.001$, 95% CI=0.03% to 0.10%; 29 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $t=11.01$, $p<0.001$, 95% CI=1.21% to 1.74%; 24 out of 30 teams, two-tailed binomial test $p=0.001$). Error bars represent standard errors.[]{data-label="fig:intrasentiment2017"}](UsersSubredditRNBACausal/users_subreddit_rnba_negative_2017 "fig:"){width="\textwidth"}
[0.32]{} ![Comparison of language usage between and members in the 2017 season. [Intergroup members]{}use more negative words (two-tailed t-test, $t=4.04$, $p<0.001$, 95% CI=0.04% to 0.13%; 24 out of 30 teams, two-tailed binomial test $p=0.001$) and swear words (two-tailed t-test, $t=4.17$, $p<0.001$, 95% CI=0.03% to 0.10%; 29 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $t=11.01$, $p<0.001$, 95% CI=1.21% to 1.74%; 24 out of 30 teams, two-tailed binomial test $p=0.001$). Error bars represent standard errors.[]{data-label="fig:intrasentiment2017"}](UsersSubredditRNBACausal/users_subreddit_rnba_swear_2017 "fig:"){width="\textwidth"}
[0.32]{} ![Comparison of language usage between and members in the 2017 season. [Intergroup members]{}use more negative words (two-tailed t-test, $t=4.04$, $p<0.001$, 95% CI=0.04% to 0.13%; 24 out of 30 teams, two-tailed binomial test $p=0.001$) and swear words (two-tailed t-test, $t=4.17$, $p<0.001$, 95% CI=0.03% to 0.10%; 29 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $t=11.01$, $p<0.001$, 95% CI=1.21% to 1.74%; 24 out of 30 teams, two-tailed binomial test $p=0.001$). Error bars represent standard errors.[]{data-label="fig:intrasentiment2017"}](UsersSubredditRNBACausal/users_subreddit_rnba_hatespeech_2017 "fig:"){width="\textwidth"}
[0.32]{} ![Comparison of language usage between and members in the 2016 season. [Intergroup members]{}use more negative words (two-tailed t-test, $t=3.93$, $p<0.001$, 95% CI=0.05% to 0.14%; 23 out of 30 teams, two-tailed binomial test $p=0.005$) and swear words (two-tailed t-test, $t=3.10$, $p=0.002$, 95% CI=0.02% to 0.09%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $t=7.95$, $p<0.001$, 95% CI=0.92% to 1.53%; 25 out of 30 teams, two-tailed binomial test $p<0.001$). Error bars represent standard errors.[]{data-label="fig:intrasentiment2016"}](UsersSubredditRNBACausal/users_subreddit_rnba_negative_2016 "fig:"){width="\textwidth"}
[0.32]{} ![Comparison of language usage between and members in the 2016 season. [Intergroup members]{}use more negative words (two-tailed t-test, $t=3.93$, $p<0.001$, 95% CI=0.05% to 0.14%; 23 out of 30 teams, two-tailed binomial test $p=0.005$) and swear words (two-tailed t-test, $t=3.10$, $p=0.002$, 95% CI=0.02% to 0.09%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $t=7.95$, $p<0.001$, 95% CI=0.92% to 1.53%; 25 out of 30 teams, two-tailed binomial test $p<0.001$). Error bars represent standard errors.[]{data-label="fig:intrasentiment2016"}](UsersSubredditRNBACausal/users_subreddit_rnba_swear_2016 "fig:"){width="\textwidth"}
[0.32]{} ![Comparison of language usage between and members in the 2016 season. [Intergroup members]{}use more negative words (two-tailed t-test, $t=3.93$, $p<0.001$, 95% CI=0.05% to 0.14%; 23 out of 30 teams, two-tailed binomial test $p=0.005$) and swear words (two-tailed t-test, $t=3.10$, $p=0.002$, 95% CI=0.02% to 0.09%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $t=7.95$, $p<0.001$, 95% CI=0.92% to 1.53%; 25 out of 30 teams, two-tailed binomial test $p<0.001$). Error bars represent standard errors.[]{data-label="fig:intrasentiment2016"}](UsersSubredditRNBACausal/users_subreddit_rnba_hatespeech_2016 "fig:"){width="\textwidth"}
[0.32]{} ![The comparison of positive language usage between and [single-group members]{}in the 2018, 2017, and 2016 seasons. We find no consistent trend at the 5% significance level ($\alpha$ = 0.05) (two-tailed t-test, $t=1.36$, $p=0.174$, 95% CI=0.019% to 0.107%, 16 out of 30 teams, two-tailed binomial test $p=0.856$ for the 2018 season; two-tailed t-test, $t=2.15$, $p=0.03$, 95% CI=0.000% to 0.144%, 16 out of 30 teams, two-tailed binomial test $p=0.856$ for the 2017 season; two-tailed t-test, $t=0.66$, $p=0.508$, 95% CI=-0.050% to 0.102%, 14 out of 30 teams, two-tailed binomial test $p=0.856$ for the 2016 season). Error bars represent standard errors.[]{data-label="fig:positive"}](UsersSubredditRNBACausal/users_subreddit_rnba_positive_2018 "fig:"){width="\textwidth"}
[0.32]{} ![The comparison of positive language usage between and [single-group members]{}in the 2018, 2017, and 2016 seasons. We find no consistent trend at the 5% significance level ($\alpha$ = 0.05) (two-tailed t-test, $t=1.36$, $p=0.174$, 95% CI=0.019% to 0.107%, 16 out of 30 teams, two-tailed binomial test $p=0.856$ for the 2018 season; two-tailed t-test, $t=2.15$, $p=0.03$, 95% CI=0.000% to 0.144%, 16 out of 30 teams, two-tailed binomial test $p=0.856$ for the 2017 season; two-tailed t-test, $t=0.66$, $p=0.508$, 95% CI=-0.050% to 0.102%, 14 out of 30 teams, two-tailed binomial test $p=0.856$ for the 2016 season). Error bars represent standard errors.[]{data-label="fig:positive"}](UsersSubredditRNBACausal/users_subreddit_rnba_positive_2017 "fig:"){width="\textwidth"}
[0.32]{} ![The comparison of positive language usage between and [single-group members]{}in the 2018, 2017, and 2016 seasons. We find no consistent trend at the 5% significance level ($\alpha$ = 0.05) (two-tailed t-test, $t=1.36$, $p=0.174$, 95% CI=0.019% to 0.107%, 16 out of 30 teams, two-tailed binomial test $p=0.856$ for the 2018 season; two-tailed t-test, $t=2.15$, $p=0.03$, 95% CI=0.000% to 0.144%, 16 out of 30 teams, two-tailed binomial test $p=0.856$ for the 2017 season; two-tailed t-test, $t=0.66$, $p=0.508$, 95% CI=-0.050% to 0.102%, 14 out of 30 teams, two-tailed binomial test $p=0.856$ for the 2016 season). Error bars represent standard errors.[]{data-label="fig:positive"}](UsersSubredditRNBACausal/users_subreddit_rnba_positive_2016 "fig:"){width="\textwidth"}
[0.32]{} ![Intragroup language usage differences of members with different intergroup contact levels in the 2017 season. x-axis represents intergroup levels determined by the fraction of comments in [[/r/NBA]{}]{}. We observe a consistent monotonic pattern in the proportion of negative words (mean = 1.92%, 1.91%, 1.96%, 1.98%, 2.05%, and 2.09%, respectively for labels from 0 to 6), swear words (mean = 0.45%, 0.48%, 0.48%, 0.50%, 0.55%, and 0.56%, respectively for labels from 0 to 6), and hate speech comments (mean = 6.27%, 7.05%, 7.81%, 8.97%, 8.67%, and 9.44%, respectively for labels from 0 to 6). []{data-label="fig:levellanguage2017"}](IntergroupContactLevelCausalAnalysis/Intergroup_Contact_Level_Freq_negative_2017 "fig:"){width="\textwidth"}
[0.32]{} ![Intragroup language usage differences of members with different intergroup contact levels in the 2017 season. x-axis represents intergroup levels determined by the fraction of comments in [[/r/NBA]{}]{}. We observe a consistent monotonic pattern in the proportion of negative words (mean = 1.92%, 1.91%, 1.96%, 1.98%, 2.05%, and 2.09%, respectively for labels from 0 to 6), swear words (mean = 0.45%, 0.48%, 0.48%, 0.50%, 0.55%, and 0.56%, respectively for labels from 0 to 6), and hate speech comments (mean = 6.27%, 7.05%, 7.81%, 8.97%, 8.67%, and 9.44%, respectively for labels from 0 to 6). []{data-label="fig:levellanguage2017"}](IntergroupContactLevelCausalAnalysis/Intergroup_Contact_Level_Freq_swear_2017 "fig:"){width="\textwidth"}
[0.32]{} ![Intragroup language usage differences of members with different intergroup contact levels in the 2017 season. x-axis represents intergroup levels determined by the fraction of comments in [[/r/NBA]{}]{}. We observe a consistent monotonic pattern in the proportion of negative words (mean = 1.92%, 1.91%, 1.96%, 1.98%, 2.05%, and 2.09%, respectively for labels from 0 to 6), swear words (mean = 0.45%, 0.48%, 0.48%, 0.50%, 0.55%, and 0.56%, respectively for labels from 0 to 6), and hate speech comments (mean = 6.27%, 7.05%, 7.81%, 8.97%, 8.67%, and 9.44%, respectively for labels from 0 to 6). []{data-label="fig:levellanguage2017"}](IntergroupContactLevelCausalAnalysis/Intergroup_Contact_Level_Freq_hatespeech_2017 "fig:"){width="\textwidth"}
[0.32]{} ![Intragroup language usage differences of members with different intergroup contact levels in the 2016 season. x-axis represents intergroup levels determined by the number of comments in [[/r/NBA]{}]{}. We observe a consistent monotonic pattern in the proportion of negative words (mean = 1.89%, 1.93%, 1.94%, 1.96%, 2.01%, and 2.04%, respectively for labels from 0 to 6), swear words (mean = 0.45%, 0.44%, 0.49%, 0.51%, 0.52%, and 0.58%, respectively for labels from 0 to 6), and hate speech comments (mean = 6.42%, 6.54%, 7.57%, 7.63%, 8.23%, and 8.53%, respectively for labels from 0 to 6). []{data-label="fig:levellanguage2016"}](IntergroupContactLevelCausalAnalysis/Intergroup_Contact_Level_Freq_negative_2016 "fig:"){width="\textwidth"}
[0.32]{} ![Intragroup language usage differences of members with different intergroup contact levels in the 2016 season. x-axis represents intergroup levels determined by the number of comments in [[/r/NBA]{}]{}. We observe a consistent monotonic pattern in the proportion of negative words (mean = 1.89%, 1.93%, 1.94%, 1.96%, 2.01%, and 2.04%, respectively for labels from 0 to 6), swear words (mean = 0.45%, 0.44%, 0.49%, 0.51%, 0.52%, and 0.58%, respectively for labels from 0 to 6), and hate speech comments (mean = 6.42%, 6.54%, 7.57%, 7.63%, 8.23%, and 8.53%, respectively for labels from 0 to 6). []{data-label="fig:levellanguage2016"}](IntergroupContactLevelCausalAnalysis/Intergroup_Contact_Level_Freq_swear_2016 "fig:"){width="\textwidth"}
[0.32]{} ![Intragroup language usage differences of members with different intergroup contact levels in the 2016 season. x-axis represents intergroup levels determined by the number of comments in [[/r/NBA]{}]{}. We observe a consistent monotonic pattern in the proportion of negative words (mean = 1.89%, 1.93%, 1.94%, 1.96%, 2.01%, and 2.04%, respectively for labels from 0 to 6), swear words (mean = 0.45%, 0.44%, 0.49%, 0.51%, 0.52%, and 0.58%, respectively for labels from 0 to 6), and hate speech comments (mean = 6.42%, 6.54%, 7.57%, 7.63%, 8.23%, and 8.53%, respectively for labels from 0 to 6). []{data-label="fig:levellanguage2016"}](IntergroupContactLevelCausalAnalysis/Intergroup_Contact_Level_Freq_hatespeech_2016 "fig:"){width="\textwidth"}
[0.32]{} ![[Intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting in the 2017 season. They use more negative words (two-tailed t-test, $t=9,36$, $p<0.001$, 95% CI=0.19% to 0.30%; 29 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words (two-tailed t-test, $t=13.80$, $p<0.001$, 95% CI=0.29% to 0.38%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $4.43$, $p<0.001$, 95% CI=0.48% to 1.25%; 26 out of 30 teams, two-tailed binomial test $p=0.005$). Error bars represent standard errors. []{data-label="fig:intersentiment2017"}](SameUsersSubredditRNBA/same_users_subreddit_rnba_negative_2017 "fig:"){width="\textwidth"}
[0.32]{} ![[Intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting in the 2017 season. They use more negative words (two-tailed t-test, $t=9,36$, $p<0.001$, 95% CI=0.19% to 0.30%; 29 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words (two-tailed t-test, $t=13.80$, $p<0.001$, 95% CI=0.29% to 0.38%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $4.43$, $p<0.001$, 95% CI=0.48% to 1.25%; 26 out of 30 teams, two-tailed binomial test $p=0.005$). Error bars represent standard errors. []{data-label="fig:intersentiment2017"}](SameUsersSubredditRNBA/same_users_subreddit_rnba_swear_2017 "fig:"){width="\textwidth"}
[0.32]{} ![[Intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting in the 2017 season. They use more negative words (two-tailed t-test, $t=9,36$, $p<0.001$, 95% CI=0.19% to 0.30%; 29 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words (two-tailed t-test, $t=13.80$, $p<0.001$, 95% CI=0.29% to 0.38%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $4.43$, $p<0.001$, 95% CI=0.48% to 1.25%; 26 out of 30 teams, two-tailed binomial test $p=0.005$). Error bars represent standard errors. []{data-label="fig:intersentiment2017"}](SameUsersSubredditRNBA/same_users_subreddit_rnba_hatespeech_2017 "fig:"){width="\textwidth"}
[0.32]{} ![[Intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting in the 2016 season. They use more negative words (two-tailed t-test, $t=7.21$, $p<0.001$, 95% CI=0.20% to 0.34%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words (two-tailed t-test, $t=12.22$, $p<0.001$, 95% CI=0.23% to 0.32%; 27 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $t=2.17$, $p=0.030$, 95% CI=0.04% to 0.89%; 22 out of 30 teams, two-tailed binomial test $p=0.016$). Error bars represent standard errors. []{data-label="fig:intersentiment2016"}](SameUsersSubredditRNBA/same_users_subreddit_rnba_negative_2016 "fig:"){width="\textwidth"}
[0.32]{} ![[Intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting in the 2016 season. They use more negative words (two-tailed t-test, $t=7.21$, $p<0.001$, 95% CI=0.20% to 0.34%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words (two-tailed t-test, $t=12.22$, $p<0.001$, 95% CI=0.23% to 0.32%; 27 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $t=2.17$, $p=0.030$, 95% CI=0.04% to 0.89%; 22 out of 30 teams, two-tailed binomial test $p=0.016$). Error bars represent standard errors. []{data-label="fig:intersentiment2016"}](SameUsersSubredditRNBA/same_users_subreddit_rnba_swear_2016 "fig:"){width="\textwidth"}
[0.32]{} ![[Intergroup members]{}use more negative language in the intergroup setting than in the intragroup setting in the 2016 season. They use more negative words (two-tailed t-test, $t=7.21$, $p<0.001$, 95% CI=0.20% to 0.34%; 28 out of 30 teams, two-tailed binomial test $p<0.001$) and swear words (two-tailed t-test, $t=12.22$, $p<0.001$, 95% CI=0.23% to 0.32%; 27 out of 30 teams, two-tailed binomial test $p<0.001$) and generate more hate speech comments (two-tailed t-test, $t=2.17$, $p=0.030$, 95% CI=0.04% to 0.89%; 22 out of 30 teams, two-tailed binomial test $p=0.016$). Error bars represent standard errors. []{data-label="fig:intersentiment2016"}](SameUsersSubredditRNBA/same_users_subreddit_rnba_hatespeech_2016 "fig:"){width="\textwidth"}
[^1]: We thank anonymous reviewers and members of the NLP+CSS research group at CU Boulder for their insightful comments and discussions; Maria Deslis for illustrating [Figure \[fig:illustration\]]{}; Scott Fredrick Holman from the CU Boulder Writing Center for his feedback and support during the writing process; Jason Baumgartner for sharing the dataset that enabled this research. This work is supported in part by the US National Science Foundation (NSF) through grant CNS 1528138.
[^2]: @wood2016elusive show that backfire in @nyhan2010corrections is stubbornly difficult to reproduce, which further demonstrates the varying results in recent studies.
[^3]: In our [[/r/NBA]{}]{} dataset, every comment’s JSON format has the “author\_flair\_css\_class” key, and the corresponding value represents a unique flair this comment uses. The value is a string with the team’s name and the flair id. For example, the flair values of the Log Angeles Lakers are phrased as “Lakers1”, “Lakers2”, “Lakers3”, and “Lakers4”.
[^4]: Part of our code is borrowed from Jack Hessel’s [Fightin-Words]{}model implementation [@fightinwords].
|
---
abstract: 'This paper deals with the analysis of sunspot number time series using the . We use the rescaled range ($R/S$) analysis to estimate the [Hurst exponent]{} for 259-year and $11\,360$-year sunspot data. The results show a varying degree of persistence over shorter and longer time scales corresponding to distinct values of the . We explain the presence of these multiple s by their resemblance to the deterministic chaotic attractors having multiple centers of rotation.'
author:
- 'Vinita $^{1}$, Awadhesh $^{1}$, Harinder P. $^{1}$'
title: Nonlinear Time Series Analysis of Sunspot Data
---
Introduction {#S-Introduction}
============
Strong magnetic field present in the sun’s outer regions is manifested by complex temporal dynamics, [*e.g.*]{}, sunspots, solar wind velocity and solar flares. Magnetic activity manifests itself most clearly in sunspots. It has been found that chromospheric flares show a very close statistical relationship with sunspots [@bray]. Long term variations of solar activity may cause climatic changes on earth, whereas short term variations may be accompanied by fluctuations of certain meteorological parameters [@wittmann]. Besides its 11-year fundamental periodicity, solar activity as measured by relative sunspot number shows quasiperiodic variations with period ranging from 2 to 1100 years [@michelson; @kimura; @turner; @kiral; @zhukov; @cole].
Alongside several periodicities, the solar activity also exhibits irregular fluctuations and these fluctuations were first assumed to be determined by the short term variation with a random distribution [@ruzmaikin]. The rediscovery of the grand minima of solar activity [@eddy] led to a re-examination of the nature of the non-periodic part of the variations of the sun’s activity. Solar activity in the frequency range from 100 to 3000 years includes an important continuum component in addition the well-known periodic variations [@ruzmaikin].
The [Hurst exponent]{} is a parameter that quantifies the persistent or anti-persistent (past trends tend to reverse in future) behavior of a time series. It determines whether the given time series is completely random or has some long-term memory. Ruzmaikin [[*et al.*]{} ]{}(1994) examined whether or not the nonperiodic variations in solar activity are caused by a white-noise, random process. They evaluated the [Hurst exponent]{} for a [time series]{} of $^{14}$C data from 6000 BC to 1950 AD. They find a [Hurst exponent]{} of $\approx 0.8$ indicating a high degree of persistence in the variations of solar activity. Kilcik [[*et al.*]{} ]{}(2009) used the monthly ISSN (international sunspot numbers) data for the last 3000 years to evaluate the [Hurst exponent]{} with a view to predict the sunspot activity for solar cycle 24. Xapsos [[*et al.*]{} ]{}(2009) used the reconstructed sunspot numbers for the past $11\,360$ years by Solanki [[*et al.*]{} ]{}(2004) to find the [Hurst exponent]{} of $\approx 0.8$ and also showed the evidence of 6000-year periodicity in the reconstructed sunspot numbers. In all the above studies involving the Hurst analysis to understand the persistent behavior of the sunspot data, a single [Hurst exponent]{} was estimated although there are many scaling regimes which give different s. In this paper, we use the Hurst analysis on 259-year and $11\,360$-year data sets and find multiple s in each [time series]{}. We explain the presence of multiple s for a single [time series]{} using systems from deterministic chaotic dynamics with multiple centers of rotation. Our results conclude that estimating a single [Hurst exponent]{} from the data where different linear scaling regimes exist may be improper.
In Section \[S-rs\] we review the $R/S$ method to calculate the Hurst exponent of a time series. The results for sunspot data are discussed in Section \[S-resultastro\]. Results for different chaotic models having one or more centers of rotation in phase space are given in Section \[S-resultmodels\]. The conclusions are given in Section \[S-conc\].
Hurst Analysis and $R/S$ Measure {#S-rs}
================================
The $R/S$ method to find the [Hurst exponent]{} was proposed by Mandelbrot and Wallis ([-@mandelbrot]b) which can be summarized as follows:
Let $X_{i},~ i=1,2,...,N$, be an observed time series whose [Hurst exponent]{} is to be computed. Let us now choose a parameter $w$ (temporal window such that $w_t \le {w} \le {N}$ where $w_t$ is the Theiler window [@thw]) and consider the subsets of the data $x_{i},~i=t_{0}, t_{0}+1, t_{0}+2,...,
t_{0}+w-1$, where $1 \le t_{0} < {N-w+1}.$
We then denote the average of these subsets as
$$\bar{x}(t_{0},w) = \frac{1}{w}\sum_{i=t_0}^{t_0+w-1}{x_i}.$$
Let $S(t_0,w)$ be the standard deviation of $x_i$, during the window $w$ [*[i.e., ]{}*]{}
$$S(t_0,w)=\left[ \frac{1}{w-1}\sum_{i=t_0}^{t_0+w-1}\left\lbrace
x_i -\bar{x}(t_0,w)\right\rbrace ^2\right] ^{\frac{1}{2}}.$$
Next, new variables $y_i,~ i=1,2,...,w, $ and *range* $R$ are defined as
$$y_i(t_0,w) = \sum_{k=t_0}^{t_0+i-1}[x_k-\bar{x}(t_0,w)],$$
$$R(t_0,w) = \max_{1\leqslant i \leqslant w}y_i(t_0,w) - \min_{1\leqslant i
\leqslant w}y_i(t_0,w),$$ which allows one to define *Rescaled range* measure $R/S$ as
$$(R/S) (t_0,w) = \frac{R(t_0,w)}{S(t_0,w)}.$$
Taking $ t_0 = 1, 2, ..., N-w+1$, and computing $(R/S)(t_0,w)$ for time lag $w$ the rescaled range for the time lag $w$ is finally written as the average of those values $$R/S = \frac{1}{N-w+1} \sum_{t_0} (R/S)(t_0,w).$$
It has been observed that the rescaled range $(R/S)$ over a time window of width $w$ varies as a power law: $$(R/S)_w = k\,w^H, \nonumber$$
where $k$ is a constant and $H$ is the Hurst exponent. To estimate the value of the Hurst exponent, $R/S$ is plotted against $w$ on log-log axes. The slope of the linear regression gives the value of the Hurst exponent. If the time series is purely random then the [Hurst exponent]{} $(H)$ comes out to be $0.5$. If $H > 0.5$, the [time series]{} covers more ‘distance’ than a random walk, and is a case of persistent motion, while if $H < 0.5$, the [time series]{} covers less ‘distance’ than a random walk it shows the anti-persistent behavior in [time series]{}. However, for periodic motions, the [Hurst exponent]{} is $1$.
Sunspot Data and $R/S$ Analysis {#S-resultastro}
===============================
Sunspot number (SSN) is continuously changing in time and constitutes a time series. Figure \[F-spots\](a) shows the monthly-averaged sunspot numbers from Sunspot Index Data Center SIDC (http://www.sidc.be/sunspot-data) from 1749 up to the present. The power spectrum of this data set is shown in Figure \[F-spower\](a). We further estimate the [Hurst exponent]{} for this [time series]{} using the $R/S$ method. Figure \[F-shurst\](a) shows the log-log plot for the 259-year data showing the presence of multiple Hurst exponents for this time series. The first bending is due to the prominent 11-year cycle and the estimated [Hurst exponent]{} 1.13 in the linear regime before this bending. The next prominent bending is at roughly 90 years showing the Wolf-Gleissberg cycle. The estimated [Hurst exponent]{} in the second linear regime between 11 years and 90 years is 0.83.
We further analyze the reconstructed sunspot [time series]{} of $11\,360$ years (Solanki [[*et al.*]{}, ]{}2004) shown in Figure \[F-spots\](b). Figure \[F-spower\](b) shows the power spectrum of this [time series]{} indicating the presence of two long-term periodicities at nearly 2300 and 6000 years. The $R/S$ analysis also shows two bendings around these periods (Figure \[F-shurst\](b)). Again, for this data set we get two linear regimes, the [Hurst exponent]{} of the first part being 0.77 and the second part being 1.26.
If we were to calculate a single [Hurst exponent]{} for these two [time series]{} we will get $H \approx 0.9$ for 259-year and $H \approx 0.8$ for $11\,360$-year data (Figure \[F-shurst\](a) and (b) respectively). These values of the [Hurst exponent]{} agree with the previously estimated values of [*H*]{} in the literature [@ruzmaikin; @alana; @xapsos]. However, as we have demonstrated, the log-log plots of sunspot data show two remarkably distinct scaling regimes and hence estimating only one [Hurst exponent]{} may be improper. In the next section we examine in detail the reasons for these different scaling regimes by using examples of a variety of [time series]{} of standard chaotic systems.
Chaotic Models and $R/S$ Analysis {#S-resultmodels}
=================================
In order to understand the behavior of different dynamical systems using the , we take two extreme examples of time series generated from random and periodic motion respectively. First we take a random time series with $100\,000$ data points distributed uniformly in the range $[0,1]$. Using the $R/S$ method we compute the Hurst exponent as discussed in Section 2 and find the [Hurst exponent]{} $H = 0.5$ as given by the slope of the line in Figure \[F-extreme\](a), which is expected for a purely random [time series]{} [@mandelbrot2].
We further consider the other extreme case of a purely periodic signal of period one generated from $\sin (x)$, with $x$ in the range $[0,1000]$ with step size $0.01$. Figure \[F-extreme\](b) shows the plot of [$\log (R/S)$]{} vs. §and the slope of the linear regime in this case is $1$. One prominent difference between these two extreme cases of a purely random signal and a periodic signal (Figure \[F-extreme\](a) and (b) respectively), is that for a purely random signal, the plot of [$\log (R/S)$]{} vs. §has a constant slope, while for a periodic signal it gets saturated and starts oscillating after a certain value of $w$ (marked by an arrow). The value where the bending begins corresponds to the period of oscillation $T=2\pi/0.01$, and has been verified by the power spectrum. The bending, therefore, gives us information about the frequency of the given signal.
Chaotic Time Series
-------------------
In the past few decades, chaotic motion, observed in deterministic systems, has received great attention due to its presence in systems from physical, chemical, biological, ecological, physiological to social sciences. These chaotic motions are temporally aperiodic and are strange because they have a fractal geometry. Since chaotic motion is neither periodic nor random we can expect its behavior to be in-between two extreme cases of randomness and periodicity. Therefore, we can also expect the [Hurst exponent]{} to be different from $0.5$ and $1$. An example of a system exhibiting chaotic motion is the celebrated [ Rössler]{} attractor [@rossler], $$\begin{aligned}
\dot x & =& -y-z ,\nonumber\\
\dot y &=& x+ay ,\nonumber\\
\dot z& =& b+z(x-c),\end{aligned}$$ where $a$, $b$ and $c$ are control parameters. A chaotic trajectory (with $a=0.1$, $b=0.1$ and $c=18.0$) in the $x-y$ plane is shown in Figure \[F-single\](a) which rotates around an unstable period-one fixed point $((c - d )/2,~ (-c + d)/{2a},~ (c - d)/{2a})$ where $d = \sqrt{c^{2}
-4ab}$ . The plot of [$\log (R/S)$]{} vs. §(Figure \[F-single\](c)) shows that there is linear regime (before marked arrow) after which it gets saturated. The slope of linear region gives $H \approx 1$. The first bending of the above-mentioned curve gives the frequency which matches with the frequency obtained from power spectrum or peak to peak analysis of the amplitude. This system is dissipative and has to be bounded in the sub-phase space [@rossler] and the bending here corresponds to the folding nature of the dynamics.
In order to see if the behavior is replicated in another similar system, we consider the Chua oscillator (Chua [*et al.*]{}, 1993), $$\begin{aligned}
\nonumber
\dot x &=& c_1(y-x-p(x)),\\
\nonumber
\dot y &=& c_2(x-y+z),\\
\dot z &=& -c_3y,
\label{eq:chua}\end{aligned}$$
where $p(x)$ is defined as, $$p(x)=m_1x+\left((m_0-m_1)(\mid x+1\mid - \mid x-1 \mid)\right)/2.$$ We take $c_1 = 15.6$, $c_2 = 1, m_0=-8/7,~m_1 = 5/7$ and first select the parameter $c_3=33$ such that the motion will be a single-scroll type as shown in Figure \[F-single\](b) (there is a symmetric attractor also, depending upon the initial conditions). The [$\log (R/S)$]{} vs. §plot for variable $x$ is shown in Figure \[F-single\](d) and the [Hurst exponent]{} is found to be $H \approx 1$. This also shows saturation at the average time period (shown by arrow). These two examples of chaotic dynamics, [ Rössler]{} and Chua systems, clearly demonstrate that whenever there is a single center of rotation the [$\log (R/S)$]{} vs. §plot shows a linear scale only up to the average period of the attractor after which it saturates.
In nature one may come across many dynamical systems which are chaotic and their trajectories rotate around more than one center of rotation. In order to see the behavior of [$\log (R/S)$]{} vs. §plot, we first consider the Lorenz system [@lorenz], $$\begin{aligned}
\dot x &=& \sigma(y-x),\nonumber\\
\dot y &=& x(\rho-z)-y,\nonumber\\
\dot z &=& xy-\beta z.
\label{eq:lor}\end{aligned}$$
Shown in Figure \[F-double\](a) is the chaotic trajectory in the $x-y$ plane at parameter values $\sigma=16$, $\rho=50$ and $\beta=4$. This clearly shows that the trajectory rotates for some time around the fixed points:\
$( \sqrt{\beta \left( \rho -1\right)} ,
\sqrt{\beta \left( \rho -1\right)},\left( \rho -1\right)) $ and $( -\sqrt{\beta \left( \rho -1\right)} ,-\sqrt{\beta \left
( \rho -1\right)},\left( \rho -1\right) )$,\
while some time it rotates around all of these fixed points (including $(0,0,0)$). The [$\log (R/S)$]{} vs. §plot in Figure \[F-double\](c) shows that there are two regimes of linear scaling. The slope of the first part of the linear region gives $H=0.93$ and the second part gives $H=0.64$. The first linear regime corresponds to the trajectory rotating about individual fixed points while the second is for all three fixed points. This indicates that the dynamics around individual fixed points is different from that of the combined one. Therefore taking the slope of individual regimes can give more details of the intrinsic dynamics. This method also allows us to estimate intrinsic frequencies of the system.
In order to see this behavior in another system where chaotic motion contains many unstable fixed points around which the trajectory revolves, we consider the Chua circuit represented by Equation (\[eq:chua\]). Figure \[F-double\](b) shows the trajectory having a double-scroll chaotic motion (Chua system at $c_3=28$). Similar to the [Lorenz]{} system, this system confirms the existence of two different regimes of linear scaling having the s, $H=1$ and $1.1$. These two examples confirm that for multiple centers of rotation, there are many regimes of linear scaling for which distinct s can be estimated.
Conclusions {#S-conc}
===========
In this paper, we use the Hurst analysis on 259-year and $11\,360$ -year data sets and find multiple s in each [time series]{}. We explain the presence of multiple s in a single [time series]{} using systems from deterministic chaotic dynamics with a single center of rotation ([ Rössler]{} and single-scroll Chua oscillators) as well as multiple centers of rotation ([Lorenz]{} and double-scroll Chua oscillators). We have shown that in the sunspot data, two distinct linear scaling regimes exist for which two distinct s could be estimated implying a variety of persistent behavior. The results show very high degree of persistent behavior till 11 years ($H \approx 1.1$), a slightly less persistent behavior till 100 years ($H \approx 0.8$), and comparatively less persistent behavior till 2300 years ($H \approx 0.77$).
VS thanks CSIR, India for a Junior Research Fellowship and AP acknowledges DST, India for financial support.
Bray, R.J., Loughhead, R.E.: 1979, [*Sunspots*]{}, New York, Dover, 256.
Chua, L.O, Itoh, M., Kocarev, L., Eckert, K.: 1993, [*J. Circ. Syst. Comput.*]{} [**3**]{}, 1.
Cohen, T.J, Sweetser, E.I.: 1975, [*Nature*]{} [**256**]{}, 295.
Cole, T.W.: 1973, [*Solar Phys.*]{} [**30**]{}, 103.
Eddy, J.: 1976, [*Science*]{} [**192**]{}, 1189.
Gil-Alana, L.A.: 2009, [*Solar Phys.*]{} [**257**]{}, 371.
Kilcik, A., Anderson, C.N.K., Rozelot, J.P., Ye, H., Sugihara, G., Ozguc, A.: 2009, [*Astrophys. J.*]{} [**693**]{}, 1173.
Kimura, H.: 1913, [*Mon. Not. Roy. Astron. Soc.*]{} [**73**]{}, 543.
Kiral, A.: 1961, [*Publ. Istanbul Univ. Observ.*]{} [**70**]{}.
Lorenz, E.N.: 1963, [*J. Atmos. Sci.*]{} [**20**]{}, 130.
Mandelbrot, B., Wallis, J.R.: 1969a, [*Water Resour. Res.*]{}. **5**, 228.
Mandelbrot, B., Wallis, J.R.: 1969b, [*Water Resour. Res.*]{}. **5**, 967.
Michelson, A.A.: 1913, [*Astrophys. J.*]{} **38**, 268.
[ Rössler]{}, O.E.: 1979, [ [*Phys. Rev. Lett.*]{}]{} [**71A**]{}, 155.
Ruzmaikin, A.A.: 1981, [*Comments Astrophys.*]{} **9**, 85.
Ruzmaikin, A., Feynman, J., Robinson, P.: 1994, [*Solar Phys.*]{} [**148**]{}, 395.
Schwabe, S.H.: 1843, [*Astron. Nachr.*]{} [**20**]{}, 234.
Solanki, S.K., Usoskin, I.G., Kromer, B., Schussler, M., Beer, J.: 2004, [*Nature*]{} [**431**]{},1084.
Theiler, J.: 1986, [[*Phys. Rev. A* ]{}]{}[**34**]{},2427.
Turner, H.H: 1913, [*Mon. Not. Roy. Astron. Soc.*]{} [**73**]{}, 549.
Wittmann, A.: 1978, [ [*Astron. Astrophys.*]{}]{}[**66**]{}, 93.
Xapsos, M.A, Burke, E.A.: 2009, [*Solar Phys.*]{} [**257**]{}, 363.
Zhukov, L.V., Muzalevskii, Y.S.: 1969, [*Astron. Zh.*]{} [**46**]{}, 600 ([*Soviet Astron.*]{} [**13**]{}, 473).
|
---
abstract: |
Fixed-parameter algorithms and kernelization are two powerful methods to solve $\mathsf{NP}$-hard problems. Yet, so far those algorithms have been largely restricted to *static* inputs.
In this paper we provide fixed-parameter algorithms and kernelizations for fundamental $\mathsf{NP}$-hard problems with *dynamic* inputs. We consider a variety of parameterized graph and hitting set problems which are known to have $f(k)n^{1+o(1)}$ time algorithms on inputs of size $n$, and we consider the question of whether there is a data structure that supports small updates (such as edge/vertex/set/element insertions and deletions) with an update time of $g(k)n^{o(1)}$; such an update time would be essentially optimal. Update and query times independent of $n$ are particularly desirable. Among many other results, we show that [Feedback Vertex Set]{} and admit dynamic algorithms with $f(k)\log^{O(1)}n$ update and query times for some function $f$ depending on the solution size $k$ only.
We complement our positive results by several conditional and unconditional lower bounds. For example, we show that unlike their undirected counterparts, [Directed Feedback Vertex Set]{} and [Directed $k$-Path]{} do not admit dynamic algorithms with $n^{o(1)}$ update and query times even for constant solution sizes $k\leq 3$, assuming popular hardness hypotheses. We also show that unconditionally, in the cell probe model, [Directed Feedback Vertex Set]{} cannot be solved with update time that is purely a function of $k$.
bibliography:
- 'dynamicfpt.bib'
title: Dynamic Parameterized Problems and Algorithms
---
Introduction {#sec:introduction}
============
Overview of the Algorithmic Techniques
======================================
Hitting Set-like problems
=========================
Vertex Cover
------------
Connected Vertex Cover
----------------------
Edge Dominating Set
-------------------
Point Line Cover {#sec:pointlinecover}
----------------
d-Hitting Set
-------------
Max Leaf Spanning Tree
======================
Undirected k-Path
=================
Dense Subgraph in Bounded-Degree Graphs
=======================================
Edge Clique Cover {#sec:edgecliquecover}
=================
Undirected Feedback Vertex Set
==============================
Conditional Lower Bounds
========================
Dynamic Algorithms from Prior Work {#sec:priorwork}
==================================
Conclusion
==========
We give dynamic fixed-parameter algorithms for a variety of classic fixed-parameter problems, and lower bounds suggesting that this is not possible for others. The next step, of course, is to expand such results to more problems. Two particular problems come to mind.
First, it would be exciting to extend Bodlaender’s algorithm for maintaining a tree decomposition [@Bodlaender1993] to work for any treewidth parameter $k$. This would open many problems parameterized by treewidth to our approach, as most fixed-parameter algorithms for such problems need to use the tree decomposition.
Second, we are only able to design algorithms in the promise model for some problems, but depending on the specific problem, the promise model can be substantially weaker than the full model. It would be interesting to explore which problems have dynamic fixed-parameter algorithms in the promise model but not the full model.
##### Acknowledgments. {#acknowledgments. .unnumbered}
The authors would like to thank Nicole Wein, Daniel Stubbs, Hubert Teo, and Ryan Williams for fruitful conversations.
|
---
abstract: 'The idea of a multiverse – an ensemble of universes or universe domains – has received increasing attention in cosmology, both as the outcome of the originating process that generated our own universe, and as an explanation for why our universe appears to be fine-tuned for life and consciousness. Here we review how multiverses should be defined, stressing the distinction between the collection of all possible universes and ensembles of really existing universes, which distinction is essential for anthropic arguments. We show that such realised multiverses are by no means unique, and in general require the existence of a well-defined and physically motivated distribution function on the space of all possible universes. Furthermore, a proper measure on these spaces is also needed, so that probabilities can be calculated. We then discuss several other physical and philosophical problems arising in the context of ensembles of universes, including realized infinities and the issue of fine-tuning – whether very special or generic primordial conditions are more fundamental in cosmology. Then we briefly summarise scenarios like chaotic inflation, which suggest how ensembles of universe domains may be generated, and point out that the regularities underlying any systematic description of truly disjoint multiverses must imply some kind of common generating mechanism, whose testability is problematic. Finally, we discuss the issue of testability, which underlies the question of whether multiverse proposals are really scientific propositions rather than metaphysical proposals.'
author:
- 'W. R. Stoeger$^{1,2}$, G. F. R. Ellis$^1$, and U. Kirchner$^1$.'
title: 'Multiverses and Cosmology: Philosophical Issues'
---
$^1$ Department of Mathematics and Applied Mathematics, University of Cape Town, 7700 Rondebosch, South Africa.\
$^2$ Permanent Address: Vatican Observatory Research Group, Steward Observatory, The University of Arizona, Tucson, Arizona 85721, USA.\
Key Words: Cosmology; inflation; multiverses; anthropic principle
Introduction
============
Over the past twenty years the proposal of a really existing ensemble of universes – a ‘multiverse’ – has gained prominence in cosmology, even though there is so far only inadequate theoretical and observational support for its existence. The popularity of this proposal can be traced to two factors. The first is that quite a few promising programs of research in quantum and very early universe cosmology suggest that the very processes which could have brought our universe or region of the universe into existence from a primordial quantum configuration, would have generated many other universes or universe regions as well. This was first modelled in a specific way by Vilenkin (1983) and was developed by Linde (Linde 1983, 1990) in his chaotic cosmology scenario. Since then many others, e. g. Leslie (1996), Weinberg (2000), Sciama (1993), Deutsch (1998), Tegmark (1998, 2003), Smolin (1999), Lewis (2000), Weinberg (2000), and Rees (2001) have discussed ways in which an ensemble of universes or universe domains might originate physically. More recently specific impetus has been given to this possibility by superstring theory. It is now claimed by some that versions of these theories provide “landscapes” populated by a large number of vacua, \* each of which could occur in or initiate a separate universe domain,\* with different values of the physical parameters, such as the cosmological constant, the masses of the elementary particles and the strengths of their interactions (Kachru, et al. 2003; Susskind 2003, 2005, and references therein).
So far, none of these proposals has been developed to the point of actually describing such ensembles of universes in detail, nor has it been demonstrated that a generic well-defined ensemble will admit life. Some writers tend to imply that there is only one possible multiverse, characterised by “all that can exist does exist” (Lewis 2000, see also Gardner 2003). This vague prescription actually allows a vast variety of different realisations with differing properties, leading to major problems in the definition of the ensembles and in averaging, due to the lack of a well-defined measure and the infinite character of the ensemble itself. Furthermore, it is not at all clear that we shall ever be able to accurately delineate the class of all possible universes.
The second factor stimulating the popularity of multiverses is that it is the only scientific way of avoiding the fine-tuning seemingly required for our universe. This applies firstly to the cosmological constant, which seems to be fine-tuned by 120 orders of magnitude relative to what is expected on the basis of quantum field theory (Weinberg 2000, Susskind 2005). If (almost) all values of the cosmological constant occur in a multiverse, then we can plausibly live in one with the very low observed non-zero value; indeed such a low value is required in order that galaxies, stars, and planets exist and provide us with a suitable habitat for life.
This is an example of the second motivation, namely the ‘anthropic principle’ connection: If any of a large number of parameters which characterize our universe – including fundamental constants, the cosmological constant, and and initial conditions – were slightly different, our universe would not be suitable for complexity or life. What explains the precise adjustment of these parameters so that microscopic and macroscopic complexity and life eventually emerged? One can introduce a “Creator" who intentionally sets their values to assure the eventual development of complexity. But this move takes us beyond science. The existence of a large collection of universes, which represents the full range of possible parameter values, though not providing an ultimate explanation, would provide a scientifically accessible way of avoiding the need for such fine-tuning. If physical cosmogonic processes naturally produced such a variety of universes, one of which was ours, then the puzzle of fine- tuning is solved. We simply find ourselves in one in which all the many conditions for life have been fulfilled.
Of course, through cosmology we must then discover and describe the process by which that collection of diverse universes, or universe domains, was generated, or at least could have been generated, with the full range of characteristics they possess. This may be possible. It is analogous to the way in which we look upon the special character of our Solar System. We do not agonize how initial conditions for the Earth and Sun were specially set so that life would eventually emerge – though at some level that is still a mystery. We simply realize that our Solar System in one of hundreds of billions of others in the Milky Way, and accept that, though the probability that any one of them is bio-friendly is very low, at least a few of them will naturally be so. We have emerged as observers in one of those. No direct fine-tuning is required, provided we take for granted both the nature of the laws of physics and the specific initial conditions in the universe. The physical processes of stellar formation throughout our galaxy naturally leads to the generation of the full range of possible stellar systems and planets.
Before going on, it is necessary to clarify our terminology. Some refer to the separate expanding universe regions in chaotic inflation as ‘universes’, even though they have a common causal origin and are all part of the same single spacetime. In our view (as ‘uni’ means ‘one’) *the Universe* is by definition the one unique connected[^1] existing spacetime of which our observed expanding cosmological domain is a part. We will refer to situations such as in chaotic inflation as a *Multi-Domain Universe*, as opposed to a completely causally disconnected *Multiverse*. Throughout this paper, when our discussion pertains equally well to disjoint collections of universes (multiverses in the strict sense) and to the different domains of a Multi-Domain Universe, we shall for simplicity simply use the word *ensemble*. When the universes of an ensemble are all sub-regions of a larger connected spacetime - the Universe as a whole- we have the multi-domain situation, which should be described as such. Then we can reserve multiverse for the collection of genuinely disconnected universes – those which are not locally causally related.
In this article, we shall critically examine the concept of an ensemble of universes or universe domains, from both physical and philosophical points of view, reviewing how they are to be defined physically and mathematically in cosmology (Ellis, Kirchner and Stoeger 2003, hereafter referred to as EKS), how their existence could conceivably be validated scientifically, and focusing upon some of the key philosophical problems associated with them. We have already addressed the physics and cosmology of such ensembles in a previous paper (EKS), along with some limited discussion of philosophical issues. Here we shall summarize the principal conclusions of that paper and then discuss in detail the more philosophical issues.
First of all, we review the description of the the set of possible universes and sets of realised (i. e., really existing) universes and the relationship between these two kinds of sets. It is fundamental to have a general provisionally adequate scheme to describe the set of all possible universes. Using this we can then move forward to describe potential sets of actually existing universes by defining distribution functions (discrete or continuous) on the space of possible universes. A given distribution function indicates which of the theoretically possible universes have been actualized to give us a really existing ensemble of universes or universe domains. It is obviously crucial to maintain the distinction between the set of all possible universes, and the set of all existing universes. For it is the set of all existing universes which needs to be explained by cosmology and physics – that is, by a primordial originating process or processes. Furthermore, it is only an [*actually existing*]{} ensemble of universes with the required range of properties which can provide an explanation for the existence of our bio-friendly universe without fine-tuning (see also McMullin 1993, p. 371). A conceptually possible ensemble is not sufficient for this purpose – one needs universes which actually exist, along with mechanisms which generate their existence. We consider in some depth how the existence of such an actually existing ensemble might be probed experimentally and observationally - this is the key issue determining whether the proposal is truly a scientific one or not.
Though the ensemble of all possible universes is undoubtedly infinite, having an infinite ensemble of actually existing universes is problematic – and furthermore blocks our ability to assign statistical measures to it, as we shall discuss in some detail later. For all these reasons, any adequate cosmological account of the origin of our universe as one of a collection of many universes – or even as a single realised universe – must include a process whereby the realised ensemble is selected from the space of all possible universes and physically generated. But it must also provide some metaphysical view on the origin of the set of possible universes as a subset of the set of conceivable universes - which is itself a very difficult set to define[^2].
Describing Ensembles: Possibility
=================================
To characterize an ensemble of existing universes, we first need to develop adequate methods for describing the class of all possible universes. This itself is philosophically controversial, as it depends very much on what we regard as “possible.” At the very least, describing the class of all possible universes requires us to specify, at least in principle, all the ways in which universes can be different from one another, in terms of their physics, chemistry, biology, etc. We have done this in EKS, which we shall review here.
The Set of Possible Universes
-----------------------------
Ensembles of universes, or multiverses, are most easily represented classically by the structure and the dynamics of a space $\mathcal{M}$ of all possible universes, each of which can be described in terms of a set of states $s$ in a state space $\mathcal{S}$ (EKS). Each universe $m$ in $\mathcal{M}$ will be characterised by a set $\mathcal{P}$ of distinguishing parameters $p$, which are coordinates on $\mathcal{S}$ (EKS). Each $m$ will evolve from its initial state to some final state according to the operative dynamics, with some or all of its parameters varying as it does so. The course of this evolution of states will be represented by a path in the state space $\mathcal{S}$. Thus, each such path (in degenerate cases, a point) is a representation of one of the universes $m$ in $\mathcal{M}$. The parameter space $\mathcal{P}$ has dimension $N$ which is the dimension of the space of models $\mathcal{M}$; the space of states $ \mathcal{S}$ has $N+1$ dimensions, the extra dimension indicating the change of each model’s states with time, characterised by an extra parameter, e.g., the Hubble parameter $H$ which does not distinguish between models but rather determines what is the state of dynamical evolution of each model. Note that $N$ may be infinite, and indeed will be so unless we consider only geometrically highly restricted sets of universes.
This classical, non-quantum-cosmological formulation of the set of all possible universes is obviously provisional and not fundamental. Much less should it provide the basis for adjudicating the ontology of these ensembles and their components.[^3] It provides us with a preliminary systematic framework, consistent with our present limited understanding of cosmology, within which to begin studying ensembles of universes and universe domains. It is becoming very clear that, from what we are beginning to learn from quantum cosmology, a more fundamental framework will have to be developed that takes seriously quantum issues such as entanglement. Additionally, there are serious unresolved problems concerning time in quantum cosmology. Already at the level of general relativity itself, as everyone recognizes, time loses its fundamental, distinct character. What is given is space-time, not space and time. Time is now intrinsic to a given universe domain and its dynamics and there is no preferred or unique way of defining it (Isham 1988, 1993; Smolin 1991; Barbour 1994; Rovelli 2004; and references therein). When we go to quantum gravity and quantum cosmology, time, while remaining intrinsic, recedes further in prominence and even seems to disappear. The Wheeler-de Witt equation for “wave function of the universe,” for example, does not explicitly involve time – it is a time-independent equation. However, our provisional classical formulation receives support from the fact that dynamics and an intrinsic time appear to emerge from it as the universe expands out of the Planck era (see, for instance, Isham 1988 and Rovelli 2004, especially pp. 296-301). Furthermore, as yet there is no adequate quantum gravity theory nor quantum cosmological resolution to this issue of the origin and the fundamental character of time – just tantalizing pieces of a much larger picture. The only viable approach at present is to proceed on the basis of the emergent classical description.
And then there are related issues connected with decoherence – how is the transition from “the wave function of the universe” to the classical universe, or an ensemble of universe domains, effected, and what emerges in this transition? What is crucial here is that as the wave function decoheres an entire ensemble of universes or universe domains may emerge. These would all be entangled with one another. This would provide the fundamental basis for the quantum ontology of the ensemble.[^4] Furthermore, it would provide a fundamental connection among a large number of the members of our classically defined $\mathcal{M}$ above. We have already stressed the difference between a multi-domain universe and a true multiverse. An entangled ensemble of universe domains decohering from a cosmological wave function would be an important example of that case. This process of cosmological decoherence, which we as yet do not understand and have not adequately modelled , may turn out to be a key generating mechanism for a really existing multiverse. In that case we would want to define a much more fundamental space of all possible cosmological wave functions. Each of these would generate an ensemble of classical universes or universe domains which we have represented individually in $\mathcal{M}$. We could then map the wavefunctions in that more fundamental space into the $m$ of $\mathcal{M}$. As yet, however, we do not have even a minimally reliable quantum cosmology that would enable us to implement that.
Despite our lack of understanding at the quantum cosmological level, and the less than fundamental character of our space $\mathcal{M}$, it enables us proceed with our discussion of cosmological ensembles at the non-quantum level - which is what cosmological observations relate to. While doing so, we must keep the quantum cosmological perspective in mind. Though we are without the resources to elaborate it more fully, it provides a valuable context within which to interpret, evaluate and critique our more modest classical discussion here.
Returning to our description of the space $\mathcal{M}$ of possible universes $m$, we must recognize that it is based on an assumed set of laws of behaviour, either laws of physics or meta-laws that determine the laws of physics, which all $m$ have in common. Without this, we have no basis for defining it. Its overall characterisation must therefore incorporate a description both of the geometry of the allowed universes and of the physics of matter. Thus the set of parameters $\mathcal{P}$ will include both geometric and physical parameters.
Among the important subsets of the space $\mathcal{M}$ are (EKS): $\mathcal{M}_{\mathrm{FLRW}}$, the subset of all possible Friedmann-Lemaître-Robertson-Walker (FLRW) universes, which are exactly isotropic and spatially homogeneous; $\mathcal{M}_{\mathrm{almost-FLRW}}$, the subset of all universes which deviate from exact FLRW models by only small, linearly growing anisotropies and inhomogeneities; $\mathcal{M}_{\mathrm{anthropic}}$, the subset of all possible universes in which life emerges at some stage in their evolution. This subset intersects $\mathcal{M}_{\mathrm{almost-FLRW}}$, and may even be a subset of $\mathcal{M}_{\mathrm{almost-FLRW}},$ but does not intersect $ \mathcal{M}_{\mathrm{FLRW}}$, since realistic models of a life-bearing universe like ours cannot be exactly FLRW, for then there is no structure.
If $\mathcal{M}$ truly represents all possibilities, as we have already emphasized, one must have a description that is wide enough to encompass *all* possibilities. It is here that major issues arise: how do we decide what all the possibilities are? What are the limits of possibility? What classifications of possibility are to be included? From these considerations we have the first key issue (EKS):
**Issue 1:** What determines $\mathcal{M}$? Where does this structure come from? What is the meta-cause, or ground, that delimits this set of possibilities? Why is there a uniform structure across all universes $m$ in $ \mathcal{M}$?
It should be obvious that these same questions would also have to be addressed with regard to the more fundamental space of all cosmological wave functions we briefly described earlier, which would probably underlie any ensembles of universes or universe domains drawn from $\mathcal{M}$. It is clear, as we have discussed in EKS, that these questions cannot be answered scientifically, though scientific input is necessary for doing so. How can we answer them philosophically?
Adequately Specifying Possible Anthropic Universes
--------------------------------------------------
When defining any ensemble of universes, possible or realised, we must specify all the parameters which differentiate members of the ensemble from one another at any time in their evolution. The values of these parameters may not be known or determinable initially in many cases – some of them may only be set by transitions that occur via processes like symmetry breaking within given members of the ensemble. In particular, some of the parameters whose values are important for the origination and support of life may only be fixed later in the evolution of universes in the ensemble.
We can separate our set of parameters $\mathcal{P}$ for the space of all possible universes $\mathcal{M}$ into different categories, beginning with the most basic or fundamental, and progressing to more contingent and more complex categories (see EKS). Ideally they should all be independent of one another, but we will not be able to establish that independence for each parameter, except for the most fundamental cosmological ones. In order to categorise our parameters, we can doubly index each parameter $p$ in $\mathcal{P}$ as $
p_j(i)$ such that those for $j=1-2$ describe basic physics, for $j=3-5$ describe the cosmology (given that basic physics), and $j=6-7$ pertain specifically to emergence of complexity and of life (see EKS for further details).
Though we did not do so in our first paper EKS, it may be helpful to add a separate category of parameters $p_8(i)$, which would relate directly to the emergence of consciousness and self-conscious life, as well as to the causal effectiveness of self-conscious (human) life – of ideas, intentions and goals. It may turn out that all such parameters may be able to be reduced to those of $p_7(i)$, just as those of $p_6(i)$ and $p_7(i)$ may be reducible to those of physics. But we also may discover, instead, that such reducibility is not possible.
All these parameters will describe the set of possibilities we are able to characterise on the basis of our accumulated scientific experience. This is by no means a statement that “all that can occur” is arbitrary. On the contrary, specifying the set of possible parameters determines a uniform high-level structure that is obeyed by all universes in $\mathcal{M}$.
In the companion cosmology/physics paper to this one (EKS), we develop in detail the geometry, parameters $p_5(i)$, and the physics, parameters $p_1(i)$ to $p_4(i)$, of possible universes. There we also examine in detail the FLRW sector $\mathcal{M}_{FLRW}$ of the ensemble of all possible universes $\mathcal{M}$ to illustrate the relevant mathematical and physical issues. We shall not repeat those discussions here, as they do not directly impact our treatment of the philosophical issues upon which we are focusing. However, since one of the primary motivations for developing the multiverse scenario is to provide a scientific solution to the anthropic fine-tuning problem, we need to discuss briefly the set of “anthropic” universes.
The Anthropic subset
--------------------
The subset of universes that allow intelligent life to emerge is of particular interest. That means we need a function on the set of possible universes that describes the probability that life may evolve. An adaptation of the Drake equation (Drake and Shostak 1998) gives for the expected number of planets with intelligent life in any particular universe $m$ in an ensemble (EKS), $$N_{\mathrm{life}}(m)=N_g*N_S*\Pi *F, \label{life1}$$ where $N_g$ is the number of galaxies in the model and $N_S$ the average number of stars per galaxy. The probability that a star provides a habitat for life is expressed by the product $$\Pi =f_S*f_p*n_e \label{life4}$$ and the probability of the emergence of intelligent life, given such a habitat, is expressed by the product $$F=f_l*f_i. \label{life5}$$ Here $f_S$ is the fraction of stars that can provide a suitable environment for life (they are ‘Sun-like’), $f_p$ is the fraction of such stars that are surrounded by planetary systems, $n_e$ is the mean number of planets in each such system that are suitable habitats for life (they are ‘Earth-like’), $
f_l $ is the fraction of such planets on which life actually originates, and $f_i $ represents the fraction of those planets on which there is life where intelligent beings develop. The anthropic subset of a possibility space is that set of universes for which $N_{\mathrm{life}}(m)>0.$
The quantities $\{N_g,N_S,f_S,f_p,n_e,f_l,f_i\}$ are functions of the physical and cosmological parameters characterised above. So there will be many different representations of this parameter set depending on the degree to which we try to represent such interrelations.
In EKS, following upon our detailed treatment of $\mathcal{M}_{FLRW}$ we identify those FLRW universes in which the emergence and sustenance of life is possible on a broad level[^5] – the necessary cosmological conditions have been fulfilled allowing existence of galaxies, stars, and planets if the universe is perturbed, so allowing a non-zero factor $ N_g*N_S*\Pi $ as discussed above. The fraction of these that will actually be life-bearing depends on the fulfilment of a large number of other conditions represented by the factor $F=f_l*f_i,\,$ which will also vary across a generic ensemble, and the above assumes this factor is non-zero.
The Set of Realised Universes
=============================
We have now characterised the set of possible universes. But in any given existing ensemble, many will not be realised, and some may be realised many times. The purpose of this section is to review our formalism (EKS) for specifying which of the *possible* universes (characterised above) occur in a particular *realised* ensemble.
A distribution function for realised universes
----------------------------------------------
In order to select from $\mathcal{M}$ a set of realised universes we need to define on $\mathcal{M}$ a distribution function $f(m)$ specifying how many times each type of possible universe $m$ in $\mathcal{M}$ is realised[^6]. The function $f(m)$ expresses the contingency in any actualisation – the fact that not every possible universe has to be realised. Things could have been different! Thus, $f(m)$ describes the *ensemble of universes* or * multiverse* envisaged as being realised out of the set of possibilities. In general, these realisations include only a subset of possible universes, and multiple realisation of some of them. Even at this early stage of our discussion we can see that the really existing ensemble of universes is by no means unique.
From a quantum cosmology perspective we can consider $f(m)$ as given by an underlying solution of the Wheeler-de Witt equation, by a given superstring model, or by some other generating mechanism, giving an entangled ensemble of universes or universe domains.
The class of models considered is determined by all the parameters which are held constant (‘class parameters’). Considering the varying parameters for a given class (‘member parameters’), some will take only discrete values, but for each one allowed to take continuous values we need a volume element of the possibility space $\mathcal{M}$ characterised by parameter increments $\mathrm{d} p_j(i) $ in all such varying parameters $p_j(i)$. The volume element will be given by a product $$\pi =\Pi _{i,j}\,m_{ij}(m)\,\mathrm{d}p_{j}(i) \label{measure}$$ where the product $\Pi _{i,j}$ runs over all continuously varying member parameters $i,j$ in the possibility space, and the $m_{ij}$ weight the contributions of the different parameter increments relative to each other. These weights depend on the parameters $p_{j}(i)$ characterising the universe $m$. The number of universes corresponding to the set of parameter increments $\mathrm{d}p_{j}(i)$ will be $\mathrm{d}N$ given by $$\mathrm{d}N=f(m)\pi \label{dist1}$$ for continuous parameters; for discrete parameters, we add in the contribution from all allowed parameter values. The total number of universes in the ensemble will be given by $$N=\int f(m)\pi \label{dist2}$$ (which will often diverge), where the integral ranges over all allowed values of the member parameters and we take it to include all relevant discrete summations. The probable value of any specific quantity $p(m)$ defined on the set of universes will be given by
$$P= \frac{\int p(m)f(m)\pi}{\int f(m)\pi} \label{prob}$$
Such integrals over the space of possibilities give numbers, averages, and probabilities.
Now it is conceivable that all possibilities are realised – that all universes in $\mathcal{M}$ exist at least once. This would mean that the distribution function $$f(m)\neq 0\mathrm{~for~all~}m\in \mathcal{M}.$$ But there are an infinite number of distribution functions which would fulfil this condition. So not even a really existing ‘ensemble of all possible universes’ is unique. In such ensembles, all possible values of each distinguishing parameter would be represented by its members in all possible combinations with all other parameters at least once. One of the problems is that this means that the integrals associated with such distribution functions would often diverge, preventing the calculation of probabilities.
From these considerations we have the second key issue:
**Issue 2:** What determines $f(m)$? What is the meta-cause that delimits the set of realisations out of the set of possibilities?
The answer to this question has to be different from the answer to * Issue 1*, precisely because here we are describing the contingency of selection of a subset of possibilities for realisation from the set of all possibilities – determination of the latter being what is considered in *Issue 1*. As we saw in EKS, and as we shall further discuss here (see Section 6), these questions can, in principle, be partially answered scientifically. A really existing ensemble of universes or universe domains demands the operation of a generating process, which adequately explains the origin of its members with their ranges of characteristics and their distribution over the parameters describing them, from a more fundamental potential, a specific primordial quantum configuration, or the decoherence of a specific cosmological wave function. That is, there must be a specific generating process, whatever it is, which determines $f(m)$. When it comes to the further question, what is responsible for the operation of this or that specific generating process rather than some other one which would generate a different ensemble, we see (EKS) that an adequate answer cannot be given scientifically. This is the question why the primordial dynamics leading to the given really existing ensemble of universes is of a certain type rather than of some other type. Even if we could establish $f(m)$ in detail, it is difficult to imagine how we would [*scientifically*]{} explain why one generating process was instantiated rather than some other one. The only possibility for an answer, if any, is via philosophical, or possibly theological, considerations.
Measures and Probabilities
--------------------------
From what we have seen above, it is clear that $f(m)$ will enable us to derive numbers and probabilities relative to the realisation it defines only if we also have determined a unique measure $\pi $ on the ensemble, characterised by a specific choice of the $\,$weights $m_{ij}(m)$ in (\[measure\]), where these weights will depend on the $p_{j}(i)$. There are a number of difficult challenges we face in doing this, including the lack of a “natural measure” on $\mathcal{M}$ in all its coordinates, the determination of $f(m)$, or its equivalent, from compelling physical considerations, and the possible divergence of the probability integrals (see Kirchner and Ellis, 2003). These issues have been discussed in EKS.
The Anthropic subset
--------------------
The expression (\[life1\]) can be used in conjunction with the distribution function $f(m)$ of universes to determine the expected number of planets bearing intelligent life arising in the whole ensemble: $$N_{\mathrm{life}}(E)=\int f(m)*N_g*N_S*f_S*f_p*n_e*f_l*f_i*\pi \label{life3}$$ (which is a particular case of (\[prob\]) based on (\[life1\])). An anthropic ensemble is one for which $N_{\mathrm{life}}(E)>0$. If the distribution function derives from a probability function, we may combine the probability functions to get an overall anthropic probability function - for an example see Weinberg (2000), where it is assumed that the probability for galaxy formation is the only relevant parameter for the existence of life. This is equivalent to assuming that $
N_S*f_S*f_p*n_e*f_l*f_i>0$.
This assumption might be acceptable in our physically realised Universe, but there is no reason to believe it would hold generally in an ensemble because these parameters will depend on other ensemble parameters, which will vary.
Anthropic Parameters, Complexity and Life
=========================================
The astrophysical issues expressed in the product $\Pi$ (the lower-$j$ parameters: $j \leq 6$) are the easier ones to investigate anthropically. We can in principle make a cut between those consistent with the eventual emergence of life and those incompatible with it by considering each of the factors in $N_g,$ $N_S,$ and $\Pi$ in turn, taking into account their dependence on the parameters $p_1(i)$ to $p_5(i),$ and only considering the next factor if all the previous ones are non-zero. In this way we can assign bio-friendly intervals to the possibility space $\mathcal{M}$. If $\ N_g*N_S*\Pi \,$ is non-zero we can move on to considering similarly whether $F$ is non-zero, based on the parameters $p_6(i)$ to $p_8(i)$, determining if true complexity is possible, which in turn depends on the physics parameters $p_1(i)$ in a crucial way that is not fully understood.
As we go to higher-level parameters we will narrow the number of the number of universes consistent with self-conscious life even more. Essentially, we shall have the sequence of inequalities: $$N_8 < N_7< N_6 < N_5 < N_4 < N_3 < N_2 < N_1,$$ where $N_j$ is the total number of universes specified by parameters of level $j$ which are compatible with the eventual emergence of self-conscious life.
This clearly fits very nicely with the Bayesian Inference approach to probability and provides the beginnings of an implementation of it for these multiverses. This approach also clearly keeps the distinction between necessary and sufficient conditions intact throughout. At each level we add to the necessary conditions for complexity or life, weeding out those universes which fail to meet any single necessary condition. Sufficiency is never really reached in our description – we really do not know the full set of conditions which achieve sufficiency. Life demands unique combinations of many different parameter values that must be realised simultaneously. Higher-order ($j \geq 6$) parameters $p_j(i)$ may not even be relevant for many universes or universe domains in a given ensemble, since the structures and processes to which they refer may only be able to emerge for certain very narrow ranges of the lower-$j$ parameters. It may also turn out, as we have already mentioned, that higher-level parameters may be reducible to the lower-level parameters.
It will be impossible at any stage to characterise that set of $\mathcal{M}$ in which *all* the conditions *necessary* for the emergence of self-conscious life and its maintenance have been met, for we do not know what those conditions are (for example, we do not know if there are forms of life possible that are not based on carbon and organic chemistry). Nevertheless it is clear that life demands unique combinations of many different parameter values that must be realised simultaneously, but do not necessarily involve all parameters (for example Hogan [@hogan] suggests that only 8 of the parameters of the standard particle physics model are involved in the emergence of complexity). When we look at these combinations, they will span a very small subset of the whole parameter space (Davies 2003, Tegmark 2003).
Problems With Infinity
======================
When speaking of multiverses or ensembles of universes – possible or realised – the issue of infinity inevitably crops up. Researchers often envision an *infinite* set of universes, in which all possibilities are realised. Can there be an infinite set of really existing universes? We suggest that the answer may very well be “No”. The common perception that this is possible arises from not appreciating the precisions in meaning and the restrictions in application associated with this profoundly difficult concept. Because we can assign a symbol to represent ‘infinity’ and can manipulate that symbol according to specified rules, we assume corresponding “infinite” entities can exist in practice. This is questionable[^7]. Furthermore, as we have already indicated, such infinities lead to severe calculational problems in the mathematical modelling of ensembles of universes or universe domains, blocking any meaningful application of probability calculus.
It is very helpful to recognize at the outset that there are two different concepts of “the infinite” which are often used: The [*metaphysical*]{} infinite, which designates wholeness, perfection, self-sufficiency; and the [*mathematical*]{} infinite, which represents that which is without limit (Moore 1990, pp.1-2, 34- 44; Bracken 1995, p.142, n.12). Here we are concerned with the mathematical infinite[^8]. But, now there are really two basic categories of the mathematical infinite: The potential or conceptual infinite and the actual, or realised, infinite. This distinction goes back to rather diffuse but very relevant discussions by Aristotle in his [*Physics*]{} and his [*Metaphysics*]{}. Basically, the potential or conceptual infinite refers to a process or set conceptually defined so that it has no limit to it – it goes on and on, e.g. the integers. The concept defining the set or process is without a bound or limit, and open, i. e. it does not repeat or retrace what is already produced or counted. The actual, or realized, infinite would be a concrete real object or entity, or set of objects, which is open and has no limit to its specifications (in space, time, number of components, etc.), no definite upper bound. Aristotle and many others since have argued that, though there are many examples of potential or conceptual infinities, actual realised infinities are not possible as applied to entities or groups of entities.[^9]
There is no conceptual problem with an infinite set – countable or uncountable – of *possible* or *conceivable* universes. However, as David Hilbert (1964) points out, the presumed existence of the actually infinite directly or indirectly leads to well-recognised unresolvable contradictions in set theory (e. g., the Russell paradox, involving the set of all sets which do not contain themselves, which by definition must both be a member of itself and not a member of itself!), and thus in the definitions and deductive foundations of mathematics itself (Hilbert, pp.141-142).
Hilbert’s basic position is that “Just as operations with the infinitely small were replaced by operations with the finite which yielded exactly the same results . . ., so in general must deductive methods based on the infinite be replaced by finite procedures which yield exactly the same results. (p.135) He strongly maintains that “the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. (p.142, see also pp.136-137) Further on he remarks, “Material logical deduction is indispensable. It deceives us only when we form arbitrary abstract definitions, especially those which involve infinitely many objects. In such cases we have illegitimately used material logical deduction; i.e., we have not paid sufficient attention to the preconditions necessary for its valid use.(p.142). Hilbert concludes, “Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the infinite to play is solely that of an idea . . . which transcends all experience and which completes the concrete as a totality . . .” (Hilbert, p.151).
Arguments against Actual Infinity
---------------------------------
What are we to make of these intuitions and arguments? There are many mathematicians and philosophers who espouse them. There are also a large number who maintain that they are flawed. From the point of view of cosmology itself, it would be very helpful if we could trust in the conclusion that an actualized mathematical infinity is physically impossible. For then this would provide a constraint on the scenarios we use in cosmology, and assure us that probability calculations using them could be successfully pursued. If, instead, there emerges a clear indication that actual infinite sets are possible, that would be mathematically disappointing. However, it still would be an important conclusion, providing guidance and reassurance in our quest to understand not only our observable universe, but the universe or multiverse as whole, even though we will never have *direct* access to all of it (see Section 7, below).
It is not possible to explore this issue conclusively here. Philosophers and philosophers of mathematics and science have proposed many arguments against the possibility of realized or actual mathematical infinities, and many others, arguments in their favor. A careful critical review is far beyond the scope of this paper. We need, however, to go beyond the general and somewhat unfocused reasonings of Aristotle, Hilbert and others we have summarized in the introduction to this section. Thus, we shall briefly but more carefully present several arguments against actual mathematical infinities which we consider the strongest. Then in the next section we shall explore in detail some of the mathematical and physical reasons for which we should avoid admitting actual infinities.
We begin by proposing several key definitions. An *actually existing set* is one which has concrete physical status in our extra-mental, intersubjective experience, and each of whose members has a determinate, phenomenally supported status in our experience, distinct from other members, with physically characterisable features and integrity (e. g., a certain mass or energy, etc.). If the members of the set are not distinct or determinate, then the set is either not well-defined or is not actually existing.[^10] Any such actually existing set, and the members constituting it, is contingent. They depend on something else for being the way they are. It came into existence at a certain moment, or within a certain series of moments, as the result of a certain process, and eventually dissipates or dissolves, gradually changing into something else over time. This is a pervasive feature of our experience and of our scientific investigation of physical reality.
An *infinite set* is, as we have already said, one the number of whose members is open, indeterminate and unbounded. By “indeterminate” we mean unspecified in terms of a definite number. Infinity, strictly speaking is not a number in any usual sense – it is beyond all specific numbers which might be assigned to a set or system – it is simply the code-word for it continues without end. This definition reflects how the term “infinity” is used in mathematical physics, and in most of mathematics. The key point is that there is no specific number which can be assigned to the number of elements in an infinite set. One could say that the number of its elements is one of Cantor’s transfinite numbers, but those are not numbers which specify a determinate bound.
An *infinite physically existing set* is an infinite set which is also an actually existing set – in other words it possesses an infinity of really existing objects. The number of its objects is therefore unbounded and indeterminate. That means in essence that, though it is unbounded and indeterminate in number, it is nevertheless physically realized as complete. One should note at this point that the definition is contradictory, not from a logical or mathematical point of view (see Stoeger 2004), but from the point of view of physics and metaphysics. Can what is essentially indeterminate and unbounded be physically or really complete, which seems to imply “bounded”? Can anything which physically exists be completely unbounded? It is clear from our definition of infinity that it is not a specific number we can determine. Not only is it beyond any number we can specify or conceive. It is unboundedly large and indeterminate. But an actual infinity is conceived as extra-mentally instantiated and therefore as completed. That means it must be determinate and in some definite sense bounded. But this contradicts the definition of what infinity is. Something cannot be bounded and determinate, and unbounded and indeterminately large, at the same time. Therefore, an actual physically realized infinity is not possible.
This appears, on the face of it, to be a compelling conceptual argument against an actual really existing infinite set of objects, whether they be universes, or something else. The only way to counter it would be to show that an indeterminate unboundedly large set is physically realisable. Conceptually this seems to be impossible, just from the point of view of what we mean by physically concrete or actual, which seems to demand specifications and some boundedness. Something which is unboundedly large, and therefore not specifiable or determinate in quantity or extent, is not materially or physically realisable. It is not just that we are incapable of knowing an actual infinity. It seems to involve a definite physical impossibility – the unbounded indeterminateness essential to infinity is inconsistent with what it means to be physically instantiated.
Another way of putting this is that the definition of infinity is an issue in mathematics, not in physics. The problem arises in linking the well-defined mathematical concept of infinity with attempts at its realisation in physics. Realisations in physics must have some determinateness – but infinity as such is not determinate!
This is recognized implicitly in scientific and applied mathematics practice. Whenever infinite values of physical parameters arise in physics – such as in the case of singularities – we can be reasonably sure, as is often indicated, that there has been a breakdown in our models. An achieved infinity in any physical parameter (temperature, density, spatial curvature) is almost certainly *not* a possible outcome of any physical process – simply because it is unboundly large, indefinite and indeterminate.
A second supporting argument against realized infinity can be constructed as follows. Since a realised infinite set of objects is actually existing as a physical set, it is contingent and therefore must have come into existence by some generating process.[^11] Then there are two possibilities: 1. it became an actual infinite set by some process of successive addition; or 2. it was produced as an infinite set all at once.
But 1. does not work, since one cannot achieve a physically infinite set by successive addition – we can never actually *arrive* at infinity that way (see Spitzer 2000 and Stoeger 2004, and references therein). There is no physical process or procedure we can in principle implement to complete such a set – they are simply incompletable. Some will concede that we can never physically arrive at infinity in a finite time (see Smith 1993), but maintain we can do so in an infinite time. But then we have the same problem again with time – that, for that to happen, we must *complete* an infinite number of events. But that seems to contradict the essentially unbounded and indeterminate character of “infinite time.”
So that leaves us with possibility 2., that the infinite set was physically produced all at once. This is the one possibility Bertrand Russell admits (Russell 1960). But, to produce an infinite realized set of physical objects all at once requires a process which makes an actually infinite amount of mass-energy available. Again this must be a real complete, specified, infinite amount of mass-energy. But this seems conceptually contradictory again, for similar reasons.
Furthermore, the specification “all at once” demands simultaneity, which is totally coordinate dependent. What is simultaneous with respect to one coordinate system is not simultaneous with respect to another. Thus, there is no assurance to begin with that one can avoid the temporal completion problem with 1. above; indeed one cannot do so with respect to all coordinate systems. Therefore, once again here on several counts it seems that a really infinite set of physical objects is not realisable or actualizable.
These arguments underscore the fact that the problem with a realised infinity is not primarily physical in the usual sense – it is primarily a conceptual or philosophical problem. “Infinity” as it is mathematically conceived and used, is not the sort of property that can be physically realised in an entity, an object, or a system, like a definite number can. It is indeterminately large, and really refers to a process rather than to an entity (Bracken 1995, pp. 11-24). And the process it refers to has no term or completion specified. No physically meaningful parameter really possesses an infinite value. It is true that cosmologists and physicists use infinities in ways which seem to border on realised infinities, such as an infinite number of points in a line segment, an infinite number of directions from any point in three-space, or an infinite dimensional Hilbert space.[^12] However, these are potential infinities, indicating possible directions, locations or states that could be taken or occupied. In no case are they all realised, occupied or taken by distinguishable, really existing entities.
Finally, it is worth emphasizing that actual physically realized infinities lead to a variety of apparently irresolvably paradoxical, if not contradictory results (see Craig 1993) in thought experiments, such as those involving adding to and borrowing books from a really infinite library, or putting up new guests in an already fully occupied hotel of an infinite number of rooms. In fact, just the notion of a completed infinite set seems to underlie some of the disturbing paradoxes of set theory (see Craig 1993 for a brief discussion and references).
This issue is distinct from the difference between an ontologically realised infinity or an epistemologically realised infinity. What we have presented above seems to undermine the possibility of the former, at least as a physical possibility. But it is a separate question whether or not, granted the existence of a physically realized infinity, it could ever be known or specified as such in a completed and determinate way.
Actual Infinities in Cosmology?
-------------------------------
Whether or not actual infinities are possible, they certainly need to be avoided on the physical level, in order to make progress in studying multiverses. As we have already discussed, actual infinities lead to irresolvable problems in making probability calculations; and their existence or non-existence is certainly not observationally provable. They are an untestable proposal.
In the physical universe spatial infinities can be avoided by compact spatial sections, either resultant from positive spatial curvature or from choice of compact topologies in universes that have zero or negative spatial curvature, (for example FLRW flat and open universes can have finite rather than infinite spatial sections). We argue that the theoretically possible infinite space sections of many cosmologies at a given time are simply unattainable in practice - they are a theoretical idea that cannot be realised. It is certainly unprovable that they exist, if they do. However one can potentially get evidence against such infinities - if either it is observationally proven that we live in a a ‘small universe’, where we have already seen round the universe because the spatial sections are compact on a scale smaller than the Hubble scale (Lachieze-Ray and Luminet 1995);[^13] or if we prove that the spatial curvature of the best-fit FLRW universe model is positive, which necessarily implies closed spatial sections (see Sec.\[small\] below).
Future infinite time also is never realised: rather the situation is that whatever time we reach, there is always more time available.[^14] Much the same applies to claims of a past infinity of time: there may be unbounded time available in the past in principle, but in what sense can it be attained in practice? The arguments against an infinite past time are strong - it is simply not constructible in terms of events or instants of time, besides being conceptually indefinite.[^15]
The same problem of a realised infinity appears in considering supposed ensembles of really existing universes. Aside from the strictly philosophical issues we have discussed above, conceiving of an ensemble of many ‘really existing’ universes that are totally causally disjoint from our own, and how that could come into being presents a severe challenge to cosmologists. There are two fundamental reasons for this. First, specifying the geometry of a generic universe requires an infinite amount of information because the quantities in $\mathcal{P}_{\mathrm{\
geometry}}$ are fields on spacetime, in general requiring specification at each point (or equivalently, an infinite number of Fourier coefficients) - they will almost always not be algorithmically compressible. This greatly aggravates all the problems regarding infinity and the ensemble itself. Only in highly symmetric cases, like the FLRW solutions, does this data reduce to a finite number of parameters. One can suggest that a statistical description would suffice, where a finite set of numbers describes the statistics of the solution, rather than giving a full description. Whether this suffices to describe adequately an ensemble where ‘all that can happen, happens’ is a moot point. We suggest not, for the simple reason that there is no guarantee that all possible models will be included in any known statistical description. That assumption is a major restriction on what is assumed to be possible.
Secondly, if many universes in the ensemble themselves really have infinite spatial extent and contain an infinite amount of matter, that entails certain deeply paradoxical conclusions (Ellis and Brundrit 1979). To conceive of the physical creation of an infinite set of universes (most requiring an infinite amount of information for their prescription, and many of which will themselves be spatially infinite) is at least an order of magnitude more difficult than specifying an existent infinitude of finitely specifiable objects.
The phrase ‘everything that can exist, exists’ implies such an infinitude, but glosses over all the profound difficulties implied. One should note here particularly that problems arise in this context in terms of the continuum assigned by classical theories to physical quantities and indeed to spacetime itself. Suppose for example that we identify corresponding times in the models in an ensemble and then assume that *all* values of the density parameter occur at each spatial point at that time. On the one hand, because of the real number continuum, this is an uncountably infinite set of models – and, as we have already seen, genuine existence of such an uncountable infinitude is highly problematic. But on the other hand, if the set of realised models is either finite or countably infinite, then almost all possible models are not realised – the ensemble represents a set of measure zero in the set of possible universes. Either way the situation is distinctly uncomfortable.
However, we might try to argue around this by a discretization argument: maybe differences in some parameter of less than say $10^{-10}$ are unobservable, so we can replace the continuum version by a discretised one, and perhaps some such discretisation is forced on us by quantum theory - indeed this is a conclusion that follows from loop quantum gravity, and is assumed by many to be the case whether loop quantum gravity is the best theory of quantum gravity or not. That solves the ‘ultraviolet divergence’ associated with the small-scale continuum, but not the ‘infrared divergence’ associated with supposed infinite distances, infinite times, and infinite values of parameters describing cosmologies.
Even within the restricted set of FLRW models, the problem of realised infinities is profoundly troubling: if all that is possible in this restricted subset happens, we have multiple infinities of realised universes in the ensemble. First, there are an infinite number of possible spatial topologies in the negative curvature case (see e.g. Lachieze-Ray and Luminet 1995), so an infinite number of ways that universes which are locally equivalent can differ globally. Second, even though the geometry is so simple, the uncountable continuum of numbers plays a devastating role locally: is it really conceivable that FLRW universes actually occur with *all* values independently of both the cosmological constant and the gravitational constant, and also all values of the Hubble constant, at the instant when the density parameter takes the value 0.97? This gives 3 separate uncountably infinite aspects of the ensemble of universes that are supposed to exist. Again, the problem would be allayed if spacetime is quantized at the Planck level, as suggested for example by loop quantum gravity. In that case one can argue that all physical quantities also are quantized, and the uncountable infinities of the real line get transmuted into finite numbers in any finite interval – a much better situation. We believe that this is a physically reasonable assumption to make, thus softening a major problem for many ensemble proposals. But the intervals are still infinite for many parameters in the possibility space. Reducing the uncountably infinite to countably infinite does not in the end resolve the problem of infinities in these ensembles. It is still an extraordinarily extravagant proposal, and, as we have just discussed, seems to founder in the face of careful conceptual analysis.
The argument given so far is based in the nature of the application of mathematics to the description of physical reality. We believe that it carries considerable weight, even though the ultimate nature of the mathematics-physics connection is one of the great philosophical puzzles. It is important to recognize, however, that arguments regarding problems with realised infinity arise from the physics side, independently of the mathematical and conceptual consideration we have so far emphasized.
On one hand, broad quantum theoretical considerations suggest that space-time may be discrete at the Planck scale, and some specific quantum gravity models indeed have been shown to incorporate this feature when examined in detail. If this is so, not only does it remove the real number line as a [*physics*]{} construct, but it [*inter alia*]{} has the potential to remove the ultraviolet divergences that otherwise plague field theory – a major bonus.
On the other hand, it has been known for a long time that there are significant problems with putting boundary conditions for physical theories actually at infinity. It was for this reason that Einstein preferred to consider universe models with compact spatial sections (thus removing the occurrence of spatial infinity in these models). This was a major motivation for his static universe model proposed in 1917, which necessarily has compact space sections. John Wheeler picked up this theme, and wrote about it extensively in his book [*Einstein’s Vision*]{} (1968). Subsequently, the book [*Gravitation*]{} by Misner, Thorne and Wheeler (1973) only considered spatially compact, positively curved universe models in the main text. Those with flat and negative spatial curvatures where relegated to a subsection on “Other Models.”
Thus, this concern regarding infinity has a substantial physics provenance, independent of Hilbert’s mathematical arguments and philosophical considerations. It recurs in present day speculations on higher dimensional theories, where the higher dimensions are in many cases assumed to be compact, as in the original Kaluza-Klein theories. Various researchers have then commented that “dimensional democracy” suggests all spatial sections should be compact, unless one has some good physical reason why those dimensions that remained small are compact while those that have expanded to large sizes are not. Hence we believe there is substantial support from physics itself for the idea that the universe may have compact spatial sections, thus also avoiding infra-red divergences – even though this may result in “non-standard” topologies for its spatial sections. Such topologies are commonplace in string theory and in M-theory – indeed they are essential to their nature.
On the origin of ensembles
==========================
Ensembles have been envisaged both as resulting from a single causal process, and as simply consisting of discrete entities. We discuss these two cases in turn, and then show that they are ultimately not distinguishable from each other.
Processes Naturally Producing Ensembles
---------------------------------------
Over the past 15 or 20 years, many researchers investigating the very early universe have proposed processes at or near the Planck era which would generate a really existing ensemble of expanding universe domains, one of which is our own observable universe. In fact, their work has provided both the context and stimulus for our discussions in this paper. Each of these processes essentially selects a really existing ensemble from a set of possible universes, often leading to a proposal for a natural definition of a probability distribution on the space of possible universes. Here we briefly describe some of these, and comment on how they fit within the framework we have been discussing.
The earliest explicit proposal for an ensemble of universes or universe domains was by Vilenkin (1983). Andrei Linde’s chaotic inflationary proposal (Linde 1983, 1990, 2003) is one of the best known scenarios of this type. The scalar field (inflaton) in these scenarios drives inflation and leads to the generation of a large number of causally disconnected regions of the Universe. This process is capable of generating a really existing ensemble of expanding FLRW-like regions, one of which may be our own observable universe region, situated in a much larger universe that is inhomogeneous on the largest scales. No FLRWapproximation is possible globally; rather there are many FLRW-like sub-domains of a single fractal universe. These domains can be very different from one another, and can be modelled locally by FLRW cosmologies with different parameters.
Vilenkin, Linde and others have applied a stochastic approach to inflation (Vilenkin 1983, Starobinsky 1986, Linde, *et al.* 1994, Vilenkin 1995, Garriga and Vilenkin 2001, Linde 2003), through which probability distributions can be derived from inflaton potentials along with the usual cosmological equations (the Friedmann equation and the Klein-Gordon equation for the inflaton) and the slow-roll approximation for the inflationary era. A detailed example of this approach, in which specific probability distributions are derived from a Langevin-type equation describing the stochastic behaviour of the inflaton over horizon-sized regions before inflation begins, is given in Linde and Mezhlumian (2003) and in Linde *et al.* (1994). The probability distributions determined in this way generally are functions of the inflaton potential.
As we mentioned in the introduction, over the past few years considerable progress has been achieved by theorists in developing flux stabilized, compactified, non-supersymmetric solutions to superstring/M theory which possess an enormous number of vacua (Susskind 2003, Kachru, et al. 2003, and references therein). Each of these vacua has the potential for becoming a separate universe or universe domain, with a non-zero cosmological constant. As such it is relatively easy to initiate inflation in many of them. Furthermore, the dynamics leading to these vacua also generate different values of the some of the other cosmological and physical parameters, and enable a statistical treatment of string vacua themselves.
These kinds of scenario suggests how overarching physics, or a law of laws(represented by the inflaton field and its potential), can lead to a really existing ensemble of many very different FLRW-like regions of a larger Universe. However these proposals rely on extrapolations of presently known physics to realms far beyond where its reliability is assured. They also employ inflaton potentials which as yet have no connection to the particle physics we know at lower energies. And these proposals are not directly observationally testable – we have no astronomical evidence that the supposed other FLRW-like regions exist, and indeed do not expect to ever attain such evidence. Thus they remain theoretically based proposals rather than provisionally acceptable models – much less established fact. There remains additionally the difficult problem of infinities, which we have just discussed: eternal inflation with its continual reproduction of different inflating domains of the Universe is claimed to lead to an infinite number of universes of each particular type (Linde, private communication). How can one deal with these infinities in terms of distribution functions and an adequate measure? As we have pointed out above, there is a philosophical problem surrounding a realised infinite set of any kind. In this case the infinities of really existent FLRW-like domains derive from the assumed initial infinite flat (or open) space sections - and we have already pointed out the problems in assuming such space sections are actually realised. If this is correct, then at the very least these proposals must be modified so that they generate a finite number of universes or universe domains.
Finally, from the point of view of the ensemble of all possible universes often invoked in discussions of multiverses, all possible inflaton potentials should be considered, as well as all solutions to all those potentials. They should all be represented in $\mathcal{M}$, which will include chaotic inflationary models which are stationary as well as those which are non-stationary. Many of these potentials may yield ensembles which are uninteresting as far as the emergence of life is concerned, but some will be bio-friendly.
In EKS we have briefly reviewed various proposals for probability distributions of the cosmological constant over ensembles of universe domains generated by the same inflaton potential, particularly those of Weinberg (2000) and Garriga and Vilenkin (2000, 2001). We shall not revisit this work here, except to mention the strong anthropic constraints on values of the cosmological constant, which is the primary reason for interest in this case. Galaxy formation is only possible for a narrow range of values of the cosmological constant, $\Lambda$, around $\Lambda =
0$ (one order of magnitude - hugely smaller than the 120 orders of magnitude predicted by quantum field theory as its natural value).
Testability of these proposals
------------------------------
In his popular book [*Our Cosmic Habitat*]{} Martin Rees (Rees 2001b, pp. 175ff) uses this narrow range of bio-friendly values of $\Lambda$ to propose a preliminary test which he claims could rule out the multiverse explanation of fine-tuning for certain parameters like $\Lambda$. This is what might be called a “speciality argument.” According to Rees, if “our universe turns out to be [*even more specially*]{} tuned than our presence requires,” the existence of a multiverse to explain such “over- tuning” would be refuted. The argument itself goes this way. Naive quantum physics expects $\Lambda$ to be very large. But our presence in the universe requires it to be very small, small enough so that galaxies and stars can form. Thus, in our universe $\Lambda$ must obviously be below that galaxy-forming threshold. This explains the observed very low value of $\Lambda$ as a selection effect in an existing ensemble of universes. Although the probability of selecting at random a universe with a small $\Lambda$ is very small, it becomes large when we add the prior that life exists. Now, in any universe in which life exists, we would not expect $\Lambda$ to be too far below this threshold. Otherwise it would be more fine-tuned than needed. In fact, data presently indicates that $\Lambda$ is not too far below the threshold, and thus our universe is not markedly more special than it needs to be, as far as $\Lambda$ is concerned. Consequently, explaining its fine-tuning by assuming a really existing multiverse is acceptable. Rees suggests that the same argument can be applied to other parameters.
Is this argument compelling? As Hartle (2004) has pointed out, for the first stage to be useful, we need an [*a priori*]{} distribution for values of $\Lambda$ that is very broad, combined with a very narrow set of values that allow for life. These values should be centred far from the most probable [*a priori*]{} values. This is indeed the case if we suppose a very broad Gaussian distribution for $\Lambda$ centred at a very large value, as suggested by quantum field theory. Then, regarding the second stage of the argument, the values allowing for life fall within a very narrow band centred at zero, as implied by astrophysics. Because the biophilic range is narrow, the [*a priori*]{} probability for $\Lambda$ will not vary significantly in this range. Thus, it is equally likely to take any value. Thus a [*uniform probability assumption*]{} will be reasonably well satisfied within the biophilic range of $\Lambda$.
As regards this second stage of the argument, because of the uniform probability assumption it is not clear why the expected values for the existence of galaxies should pile up near the biological limit. Indeed, one might expect the probability of the existence of galaxies to be maximal at the centre of the biophilic range rather than at the edges (this probability drops to zero at the edges, because it vanishes outside – hence the likelihood of existence of galaxies at the threshold itself should be very small). Thus there is no justification on this basis for ruling out a multiverse with any specific value for $\Lambda$ within that range. As long as the range of values of a parameter like $\Lambda$ is not a zero-measure set of the ensemble, there is a non-zero probability of choosing a universe within it. In that case, there is no solid justification for ruling out a multiverse and so no real testability of the multiverse proposal. All we can really say is that we would be less likely to find ourselves in a universe with a $\Lambda$ in that range in that particular ensemble. Indeed no probability argument can conclusively *disprove* any specific result - all it can state is that the result is improbable - but that statement only makes sense if the result is possible! What is actually meant by “more specially tuned than necessary for our existence”? In the end, any particular choice of a life-allowing universe will be more specially tuned for something. In our view “tuning” refers to parameters selected such that the model falls into a certain class, e.g., life-allowing. Any additional tuning would then just be the selection of sub-classes, and, after all, any particular model is “over-tuned” in such a way as to select uniquely the sub-class which contains only itself. Rees’s argument seems to imply that $\Lambda$ close to zero would be an over-tuned case, while $\Lambda$ close to the cut-off value would not be. However, would the reversed viewpoint be not just as legitimate?
Rees’s argument strongly builds on the predictions of quantum physics — a probability distribution peaked at very high values for $\Lambda$. Taking into account the unknown relation between general relativity and quantum physics we should treat the problem as a multiple hypothesis testing problem: The multiverse scenario can be true or false, and so can the quantum prediction for high values of $\Lambda$. An observed low value of $\Lambda$ would then strongly question the predictions for $\Lambda$, but say nothing about the multiverse scenario. We conclude that any observed value of $\Lambda$ does not rule out the multiverse scenario. It also seems questionable whether the life-allowing values for $\Lambda$ can be classified just by a simple cut-off value. It should be expected that there are more subtle and yet unknown constraints. Observing a cosmological constant far from the cut-off value might then just be the result of some unknown constraints.
Finally, probability arguments simply don’t apply if there is indeed only one universe - their very use assumes a multiverse exists. There might exist only one universe which just happens to have the observed value of $\Lambda$; then probabilistic arguments will simply not apply. Thus what we are being offered here is not in fact a proof a multiverse exists, but rather a consistency check as regards the nature of the proposed multiverse. It is a proposal for a necessary but not sufficient condition for its existence. As emphasized above, we do not even believe it is a necessary condition; rather it is a plausibility indicator.
The existence of regularities
-----------------------------
Consider now a genuine multiverse. Why should there be any regularity at all in the properties of universes in such an ensemble, where the universes are completely disconnected from each other? If there are such regularities and specific resulting properties, this suggests a mechanism creating that family of universes, and hence a causal link to a higher domain which is the seat of processes leading to these regularities. This in turn means that the individual universes making up the ensemble are not actually independent of each other. They are, instead, products of a single process, or meta-process, as in the case of chaotic inflation. A common generating mechanism is clearly a causal connection, even if not situated in a single connected spacetime – and some such mechanism is needed if all the universes in an ensemble have the same class of properties, for example being governed by the same physical laws or meta-laws.
The point then is that, as emphasized when we considered how one can describe ensembles, any multiverse with regular properties that we can characterise systematically is necessarily of this kind. If it did not have regularities of properties across the class of universes included in the ensemble, we could not even describe it, much less calculate any properties or even characterise a distribution function.
Thus in the end the idea of a completely disconnected multiverse with regular properties but without a common causal mechanism of some kind is not viable. There must necessarily be some pre-realisation causal mechanism at work determining the properties of the universes in the ensemble. What are claimed to be totally disjoint universes must in some sense indeed be causally connected together, albeit in some pre-physics or meta-physical domain that is causally effective in determining the common properties of the universes in the multiverse. This is directly related to the two key issues we highlighted above in Sections 2 and 3, respectively, namely how does the possibility space originate, and where does the distribution function that characterises realised models come from?
From these considerations, we see that we definitely need to explain (for Issue 2) what particular cosmogonic generating process or meta-law pre-exists, and how that process or meta-law was selected from those that are possible. Obviously an infinite regress lurks in the wings. Though intermediate scientific answers to these questions can in principle be given, it is clear that no ultimate scientific foundation can be provided.
Furthermore, we honestly have to admit that any proposal for a particular cosmic generating process or principle we establish as underlying our actually existing ensemble of universe domains or universes, after testing and validation (see Section 7 below), will always be at best provisional and imperfect: we will never be able to definitively determine its nature or properties. The actually existing cosmic ensemble may in fact be much, much larger – or much, much smaller – than the one our physics at any given time describes, and embody quite different generating processes and principles than the ones we provisionally settle upon. This is particularly true as we shall never have direct access to the ensemble we propose, or to the underlying process or potential upon which its existence relies (see Section 7 below), nor indeed to the full range of physics that may be involved.
The existence of possibilities
------------------------------
Turning to the prior question (Issue 1, see Section 2.1), what determines the space of all possible universes, from which a really existing universe or an ensemble of universes or universe domains is drawn, we find ourselves in even much more uncertain waters. This is particularly difficult when we demand some basic meta-principle which delimits the set of possibilities. Where would such a principle originate? The only two secure grounds for determining possibility are existence (“ab esse ad posse valet illatio”) and freedom from internal contradiction. The first really does not help us at all in exploring the boundaries of the possible. The second leaves enormous unexplored, and probably unexplorable, territory. There are almost certainly realms of the possible which we cannot even imagine. But at the same time, there may be, as we have already mentioned, universes we presently think are possible which are not. We really do not have secure grounds for determining the limits of possibility in this expanded cosmic context. We simply do not have enough theoretical knowledge to describe and delimit reliably the realm of the possible, and it is very doubtful we shall ever have.
Testability and Existence
=========================
The issue of evidence and testing has already been briefly mentioned. This is at the heart of whether an ensemble or multiverse proposal should be regarded as physics or as metaphysics.
Evidence and existence
----------------------
Given all the possibilities discussed here, which specific kind of ensemble is claimed to exist? Given a specific such claim, how can one show that this is the particular ensemble that exists rather than all the other possibilities?
There is no direct evidence of existence of the claimed other universe regions in an ensemble, nor can there be any, for they lie beyond the visual horizon; most will even be beyond the particle horizon, so there is no causal connection with them; and in the case of a true multiverse, there is not even the possibility of any indirect causal connection - the universes are then completely disjoint and nothing that happens in any one of them is causally linked to what happens in any other one (see Section 6.2). This lack of any causal connection in such multiverses really places them beyond any scientific support – there can be no direct or indirect evidence for the existence of such systems. We may, of course, postulate the existence of such a multiverse as a metaphysical assumption, but it would be a metaphysical assumption without any further justifiability – it would be untestable and unsupportable by any direct or indeed indirect evidence.
And so, we concentrate on possible really existing multiverses in which there is some common causal generating principle or process. What weight does a claim of such existence carry in this case, when no direct observational evidence can ever be available? The point is that there is not just an issue of showing a multiverse exists. If this is a scientific proposition one needs to be able to show which specific multiverse exists; but there is no observational way to do this. Indeed if you can’t show *which particular* one exists, it is doubtful you have shown *any* one exists. What does a claim for such existence mean in this context? Gardner puts it this way: “There is not the slightest shred of reliable evidence that there is any universe other than the one we are in. No multiverse theory has so far provided a prediction that can be tested. As far as we can tell, universes are not even as plentiful as even [*two*]{} blackberries” (Gardner 2003). This contrasts strongly, for example, with Deutsch’s and Lewis’s defence of the concept (Deutsch 1998, Lewis 2000).
Fruitful Hypotheses and evidence
--------------------------------
There are, however, ways of justifying the existence of an entity, or entities, like a multiverse, even though we have no direct observations of it. Arguably the most compelling framework within which to discuss testability is that of retroduction or abduction which was first described in detail by C.S.Peirce. Ernan McMullin (1992) has convincingly demonstrated that retroduction is the rational process by which scientific conclusions are most fruitfully reached. On the basis of what researchers know, they construct imaginative hypotheses, which are then used to probe and to describe the phenomena in deeper and more adequate ways than before. As they do so, they will modify or even replace the original hypotheses, in order to make them more fruitful and more precise in what they reveal and explain. The hypotheses themselves may often presume the existence of certain hidden properties or entities (like multiverses!) which are fundamental to the explanatory power they possess. As these hypotheses become more and more fruitful in revealing and explaining the natural phenomena they investigate, and their inter-relationships, and more central to scientific research in a given discipline, they become more and more reliable accounts of the reality they purport to model or describe. Even if some of the hidden properties or entities they postulate are never directly detected or observed, the success of the hypotheses indirectly leads us to affirm that something like them must exist.[^16] A cosmological example is the inflaton supposed to underlie inflation.
Thus, from this point of view, the existence of an ensemble of universes or universe domains would be a validly deduced – if still provisional – scientific conclusion if this becomes a key component of hypotheses which are successful and fruitful in the long term. By an hypothesis which manifests long-term success and fruitfulness we mean one that better enables us to make testable predictions which are fulfilled, and provides a more thorough and coherent explanation of phenomena we observe than competing theories.[^17] Ernan McMullin (1992; see also P. Allen 2001, p. 113) frames such fruitfulness and success as:
a\. accounting for all the relevant data (empirical adequacy);
b\. providing long-term explanatory success and stimulating fruitful lines of further inquiry (theory fertility);
c\. establishing the compatibility of previously disparate domains of phenomena (unifying power);
d\. manifesting consistency and correlation with other established theories (theoretical coherence).
The relevant example here would be a fruitful theory relying on a specific type of multiverse, all members of which would never be directly detectable except one. But, since its postulated existence renders the existence and the characteristic features of our own universe ever more intelligible and coherent over a period of time, this can be claimed to be evidence for the multiverse’s existence. If such indirect support for the existence of a given multiverse is inadequate in the light of other competing accounts, then from a scientific point of view all we can do is to treat it as a speculative scenario needing further development and requiring further fruitful application. Without that, espousing the existence of a given multiverse as the explanation for our life-bearing universe must surely be called metaphysics, because belief in its existence will forever be a matter of faith rather than proof or scientific support.
We do, of course, want to avoid sliding to the bottom of Rees’ (2001b, p. 169) slippery slope. In arriving at his conclusion that the existence of other universes is a scientific question, Rees (2001b, pp. 165-169) begins by considering first galaxies which are beyond the limits of present-day telescopes, and then galaxies which are beyond our visual horizon now, but will eventually come within it in the future. In both cases these galaxies are real and observable [*in principle*]{}. Therefore, they remain legitimate objects of scientific investigation. However, then he goes on to consider galaxies which are forever unobservable, but which emerged from the same Big Bang as ours did. And he concludes that, though unobservable, they are real, and by implication should be included as objects considered by science. Other universes, he argues, fall in the same category – they are real, and therefore they should fall within the boundaries of scientific competency. As articulated this is indeed “a slippery slope” argument – it can be used to place anything that we claim to be “real” within the natural sciences – unless we strengthen it at several points.
First, Rees shifts the criterion from “observable in principle” to being “real.” This is really an error. No matter how real an object, process, or relationship may be, if it is not observable in principle, or if there is not at least indirect support for its existence from the long-term success of the hypotheses in which it figures, then it simply falls outside serious scientific consideration. It may still temporarily play a role in scientific speculation, but, unless it receives some evidential support, that will not last. In mentioning that the forever unobservable galaxies he is considering are produced by the same Big Bang as ours, Rees may be intending to indicate that, though unobservable, they share a common causal origin and therefore [*figure*]{} in successful hypotheses, as would be required by McMullin’s retroductive inference discussed above. But Rees does not make that clear. Moving to other universes, the same requirement holds. Thus, the slippery slope is avoided precisely by implementing the “indirect evidence by fruitful hypotheses” approach that a careful application of retroduction requires.
Second, there is discontinuity in the argument as one moves from weaker and weaker causal relation to none at all. The slippery slope becomes a vertical precipice on one side of an unbridgeable gulf. An argument that relies on incremental continuity does not apply in this case.
Thus, if we are continually evaluating our theories and speculations with regard to their potential and actual fruitfulness in revealing and explaining the world around us, then we shall avoid the lower reaches of the slippery slope. The problem is that, in this case, the multiverse hypothesis is very preliminary and will probably always remain provisional. This should not prevent us from entertaining imaginative scenarios, but the retroductive process will subject these speculations to rigorous critique over time. The key issue then is to what degree will the multiverse hypothesis become fruitful. Unfortunately, as it stands now, it is not, because it can be used to explain anything at all – and hence does not explain anything in particular. You cannot predict something new from the hypothesis, but you can explain anything you already know. In order for it to achieve some measure of scientific fruitfulness, there must be an accumulation of at least indirect scientifically acceptable support for one particular well-defined multiverse. Indeed, from a purely evidential viewpoint, a multiverse with say $10^{120}$ identical copies of the one universe in which we actually live would be much preferred over one with a vast variety of different universes, – for then the probability of finding a universe like our own would be much higher. Such ensembles are usually excluded because of some hidden assumptions about the nature of the generating mechanism that creates the ensemble. But maybe that mechanism is of a different kind than usually assumed - perhaps once it has found a successful model universe, it then churns out innumerable identical copies of the same universe.
In the end belief in a multiverse may always be just that – a matter of faith, namely faith that the logical arguments discussed here give the correct answer in a situation where direct observational proof is unattainable and the supposed underlying physics is untestable, unless we are able to point to compelling reasons based on scientifically supportable evidence for a particular specifiable multiverse or one of a narrowly defined class of multiverses. One way in which this could be accomplished, as we have already indicated, would be to find accumulating direct or indirect evidence that a very definite inflaton potential capable of generating a certain type of ensemble of universe domains was operating in the very early universe, leading to the particular physics that we observe now. Otherwise, there will be no way of ever knowing which particular multiverse is realised, if any one is. We will always be able to claim whatever we wish about such an ensemble, provided it includes at least one universe that admits life.
Observations and Physics
------------------------
One way one might make a reasonable claim for existence of a multiverse would be if one could show its existence was a more or less inevitable consequence of well-established physical laws and processes. Indeed, this is essentially the claim that is made in the case of chaotic inflation. However the problem is that the proposed underlying physics has not been tested, and indeed may be untestable. There is no evidence that the postulated physics is true in this universe, much less in some pre-existing metaspace that might generate a multiverse.
Thus there are two further requirements which must still be met, once we have proposed a viable ensemble or multiverse theory. The first is to provide some credible link between these vast extrapolations from presently known physics to physics in which we have some confidence. The second is to provide some at least indirect evidence that the scalar potentials, or other overarching cosmic principles involved, really have been functioning in the very early universe, or before its emergence. We do not at present fulfil either requirement.
The issue is not just that the inflaton is not identified and its potential untested by any observational means - it is also that, for example, we are assuming quantum field theory remains valid far beyond the domain where it has been tested, and where we have faith in that extreme extrapolation despite all the unsolved problems at the foundation of quantum theory, the divergences of quantum field theory, and the failure of that theory to provide a satisfactory resolution of the cosmological constant problem.
Observations and disproof {#small}
-------------------------
Despite the gloomy prognosis given above, there are some specific cases where the existence of a chaotic inflation (multi-domain) type scenario can be disproved. These are when we either live in a universe with compact spatial sections because they have positive curvature, or in ‘small universe’ where we have already seen right round the universe (Ellis and Schreiber 1986, Lachieze-Ray and Luminet 1995), for then the universe closes up on itself in a single FLRW-like domain, and so no further such domains that are causally connected to us in a single connected spacetime can exist.
As regards the first case, the best combined astronomical data at present (from the WMAP satellite together with number counts and supernova observations) suggest that this is indeed the case: they indicate that $\Omega_0 = 1.02 \pm 0.02$ at a 2-$\sigma$ level, on the face of it favoring closed spatial sections and a spatially finite universe. This data does not definitively rule out open models, but it certainly should be taken seriously in an era of ‘precision cosmology.’
As regards the ‘small universe’ situation, this is in principle observationally testable, and indeed it has been suggested that the CBR power spectrum might already be giving us evidence that this is indeed so, because of its lack of power on the largest angular scales (Luminet et al, 2003). This proposal can be tested in the future by searching for identical circles in the CMB sky (Roukema, et al., 2004) and alignment of the CMB quadrupole and octopole planes (Katz and Weeks 2004). Success in this endeavour would disprove the usual chaotic inflationary scenario, but not a true multiverse proposal, for that cannot be shown to be false by any observation. Neither can it be shown to be true.
Special or Generic?
===================
When we reflect on the recent history of cosmology, we become aware that philosophical predilections have oscillated from assuming that the present state of our universe is very special (made cosmologically precise in contemporary cosmology as FLRW, or almost-FLRW, through the assumption of a Cosmological Principle – see Bondi 1960 and Weinberg 1972, for example), requiring very finely tuned initial conditions, to assuming it is generic, in the sense that it has attained its present apparently special qualities through the operation of standard physical processes on any of broad range of possible initial conditions (e. g., the “chaotic cosmology” approach of Misner (1968) and the now standard but incomplete inflationary scenario pioneered by Guth (1980)). This oscillation, or tension, has been described and discussed in detail, both in its historical and in its contemporary manifestations, by McMullin (1993) as a conflict or tension between two general types of principle – anthropic-like principles, which recognize the special character of the universe and tentatively presume that its origin must be in finely tuned or specially chosen initial conditions, and “cosmogonic indifference principles,” or just “indifference principles,” which concentrate their search upon very generic initial conditions upon which the laws of physics act to produce the special cosmic configuration we now enjoy. As McMullin portrays these two philosophical commitments, the anthropic-type preference inevitably attempts to involve mind and teleology as essential to the shaping of what emerges, whereas the indifference-type preference studiously seeks to avoid any direct appeal to such influences, relying instead completely upon the dynamisms (laws of nature) inherent in and emerging from mass-energy itself.
McMullin (1993), in a compelling historical sketch, traces the preference for the special and the teleologically suggestive from some of the earlier strong anthropic principle formulations back through early Big Bang cosmology to Clarke, Bentley, William Derham and John Ray in the 18th and 19th centuries and Robert Boyle in the 17th century and ultimately back to Plato and the Biblical stories of creation. The competing preference for indifferent initial conditions and the operation of purely physical or biological laws can be similarly followed back from the present appeal to multiverses to slightly earlier inflationary scenarios and Misner’s chaotic cosmology program to steady state cosmological models and then back through Darwin to Descartes and much earlier to the Greek atomists, such as Empedocles, Diogenes Laertius and Leucippus. Neither of these historical sequences involves clearly dependent philosophical influences, but the underlying basic assumptions and preferences of each of the two sets of thinkers and models are very similar, as are their controversies and interactions with the representatives of the competing approach.
Certainly it has become clear that the present preference among theoretical cosmologists for multiverse scenarios is the latest and most concerted attempt to implement the indifference principle in the face of the mounting evidence that, taken alone, our universe does require very finely tuned initial conditions. The introduction of inflation was similarly motivated, but has encountered some scepticism in this regard with the growing sense that initiating inflation itself probably requires special conditions (Penrose 1989; Ellis, et al. 2002). The appeal to multiverses, though first seriously suggested fairly early in this saga (by Dicke in 1961 and by Carter in 1968), has been reasserted as this failure of other indifference principle implementations seem more and more imminent. However, as we have just seen in our detailed discussion of realised ensembles of universes or universe domains, they are by no means unique, and accounting for their existence requires an adequate generating process or principle, which must explain the distribution function characterizing the ensemble.
Even though we are far from being able to connect specific types of ensembles with particular provisionally adequate cosmogonic generating processes in a compelling way, it is very possible that some fine-tuning of these processes may be required to mesh with the physical constraints we observe in our universe and at the same time to produce a realised ensemble which enbraces it. This would initiate another oscillation between the two types of principles. Whether or not that occurs, it is clear that the existence of a multiverse in itself does not support either the indifference principle nor the anthropic-type principle. What would do so would be the distribution function specifying the multiverse, and particularly the physical, pre-physical or metaphysical process which generates the multiverse with that distribution function, or range of distribution functions. Only an understanding of that process would ultimately determine which principle is really basic.
Whatever the eventual outcome of future investigations probing this problem, it is both curious and striking, as McMullin (1993, p. 385) comments, that “the same challenge arises over and over.” Fine- tuning at one level is tentatively explained by some process at a more fundamental level which seems at first sight indifferent to any initial conditions. But then further investigation reveals that that process really requires special conditions, which demands some fine-tuning. Meanwhile, “the universe” required for understanding and explanation “keeps getting larger and larger.”
It might seem that these competing philosophical or metaphysical preferences – for what is either basically special or basically generic – are choices without scientific or philosophical support. But that is an illusion. From what we have seen already, there is considerable physical and philosophical support for each preference – some of it observational and some of it theoretical – but there is no [*adequate*]{} or [*definitive*]{} support for one over against the other. Thus, either preference may be supported in various ways philosophically and scientifically, but neither the one nor the other is [*THE*]{} scientific approach. For example, the emergent universe model of Ellis, Maartens, et al. (2003) has fine-tuned initial conditions, but it still could be a good model – it may actually represent how the very early history of our universe unfolded, even though it does not explain how the special initial conditions were set. \[In fact it is not as fine-tuned as inflation with $k = 0$, which requires “infinite” fine-tuning, while being “asymptotic” to an Einstein static universe does not.\]
The issue is not so much which of the two principles or perspectives are correct – both seem to be important at different levels and in different heuristic and explanatory contexts. As far as we know, there has not been any resolution to the question of the epistemological or ontological status of either one. They function rather as contrary heuristic preferences which have both intuitive and experiential support. Perhaps the real question is: Which is more fundamental? It is possible that, from the point of view of physics, the indifference principle is more fundamental – relative to the explanations which are possible within the sciences – whereas from the point of view of metaphysics, an anthropic-type fine-tuning principle is more fundamental.
What does seem clear, in this regard, is that the effort to keep explanation and understanding completely within the realm of physics forces us to choose the indifference principle as more fundamental. This is simply because the need for fine-tuning threatens to take us outside of where physics or any of the other natural sciences can go. Furthermore, as we have also seen, physics and the other sciences cannot delve into the realm of ultimate explanation either.
Conclusion
==========
As we stressed in the conclusion of EKS, the introduction of the multiverse or ensemble idea is a fundamental change in the nature of cosmology, because it aims to challenge one of the most basic aspects of standard cosmology, namely the uniqueness of the universe (see Ellis 1991, 1999 and references therein). So far, research and discussion on such ensembles have not precisely specified what is required to define them, although some specific physical calculations have been given based on restricted low-dimensional multiverses.
Our fundamental starting point has been the recognition that there is an important distinction to be made between possible universes and realised universes, and a central conclusion is that a really existing ensemble or multiverse is not *a priori* unique, nor uniquely defined. It must somehow be selected for. We have defined both the ensemble of possible universes $\mathcal{M}$,and ensembles of really existing universes, which are envisioned as generated by a given primordial process or action of an overarching cosmic principle, physical or metaphysical. This effectively selects a really existing multiverse from $\mathcal{M}$, and, as such, effectively defines a distribution function over $\mathcal{M}$. Thus, there is a definite causal connection, or law of laws, relating all the universes in each of those multiverses. It is such a really existing ensemble of universes, one of which is our own universe, *not* the ensemble of all possible universes, which provides the basis for anthropic arguments. Anthropic universes lie in a small subset of $\mathcal{M}$, whose characteristics we understand to some extent. It is very likely that the simultaneous realisation of *all* the conditions for life will pick out only a very small sector of the parameter space of all possibilities: anthropic universes are fine-tuned in that sense. If cosmogonic processes or the operation of a certain primordial principle selected and generated an ensemble of really existing universes from $\mathcal{M}$, some of which are anthropic, then, though we would require some explanation for that process or principle, the fine-tuning of our universe would not require any other scientific explanation. It is, however, abundantly clear that really existing ensembles are *not* unique, and neither their properties nor their existence are directly testable. Arguments for their existence would be much stronger if the hypotheses employing them were fruitful in enabling new investigations leading to new predictions and understandings which are testable. However, so far this has not been the case. In our view these questions - Issues 1 and 2 in this paper – cannot be answered scientifically with any adequacy because of the lack of any possibility of verification of any proposed underlying theory. They will of necessity have to be argued with a mixture of careful philosophically informed science and scientifically informed philosophy. And, even with this, as we have just seen, we seem to fall short of providing satisfactory answers – so far!
Another philosophical issue we have emphasized which has a strong bearing on how we describe and delimit really existing multiverses is that of realised infinity. From our careful discussion of this concept, there is a compelling case for demanding that every really existing ensemble contain only a finite number of universes or universe domains.
There is strong support for both of two competing approaches – that which honors the special character of our universe by stressing the need for the fine-tuning of initial conditions and the laws of nature, and that which locates its emergence in the operation of primordial processes on a much more fundamental generic or indifferent configuration. Both are undoubtedly at work on different levels. The issues are: which is more fundamental, and whether the sciences themselves as they are presently conceived and practiced can deal with ultimate fundamentals. Must they yield that realm to metaphysics? Can metaphysics deal with them? The relative untestability or unprovability of the multiverse idea in the usual scientific sense is however problematic – the existence of the hypothesized ensemble remains a matter of faith rather than of proof, unless it comes to enjoy long-term fruitfulness and success. Furthermore in the end, the multiverse hypothesis simply represents a regress of causation. Ultimate questions remain: Why this multiverse with these properties rather than others? What endows these with existence and with this particular type of overall order? What are the ultimate boundaries of possibility – what makes something possible, even though it may never be realised? As we now see, the concept of a multiverse raises many fascinating issues that have not yet been adequately explored. The discussions here should point and guide research in directions which will yield further insight and understanding.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank A. Aguirre, C. Harper, A. Lewis, A. Linde, A. Malcolm, and J.P. Uzan for helpful comments and references related to this work, and two anonymous referees for comments and suggestions which have enabled us to correct errors and clarify our positions. GFRE and UK acknowledge financial support from the University of Cape Town and the NRF (South Africa).
[99]{}
Allen P 2001 *Philosophical Framework with the Science-Theology Dialogue: A Critical Reflection on the Work of Ernan McMullin*, unpublished Ph.D disseration, St. Paul University, Ottawa, Canada.
Barbour J 1994 The Timelessness of Quantum Gravity: I. Evidence from the Classical Theory; II. The Appearnace of Dynamics in Static Configurations, *Classical and Quantum Gravity* **11**, 2853.
Bracken J A 1995 *The Divine Matrix: Creativity as Link between East and West* (Maryknoll, N. Y.: Orbis Books), 179 pp.
Craig, W L 1993 Finitude of the Past and God’s Existence in *Theism, Atheism, and Big Bang Cosmology* (Oxford: Oxford University Press), pp. 3-76; Time and Infinity, ibid, pp. 93-107.
Davies P 2003 Multiverse or Design? Reflections on a ‘Third Way’, unpublished manuscript and talk given at the Centre for Theology and the Natural Sciences, Berkeley, California, March 22, 2003.
Drake F and Shostak S 1998 *Sharing the Universe: Perspectives on Extraterrestrial Life* (Berkeley Hills Books).
Deutsch D 1998 *The Fabric of reality: The science of parallel universes - and its implications* (Penguin).
Ellis G F R 1991 Major Themes in the relation between Philosophy and Cosmology *Mem. Ital. Ast. Soc.* **62** 553-605.
Ellis G F R 1999 Before the Beginning: Emerging Questions and Uncertainties in *Toward a New Millenium in Galaxy Morphology* ed D Block, I Puerari, A Stockton and D Ferreira (Kluwer, Dordrecht), *Astrophysics and Space Science* **269-279** 693-720.
Ellis G F R 2003 Emergent Reality, to appear in *The Re-emergence of Emergence*, edited by Philip Clayton, e al.
Ellis G F R and Brundrit G B 1979 Life in the infinite universe *Qu. Journ. Roy. Ast. Soc.* **20** 37- 41.
Ellis G F R, Kirchner U and Stoeger W R 2003 Multiverses and Physical Cosmology *Mon. Not. R. astr. Soc* **347**, 921; *Archive* astro-ph/0305292v3 (abbreviated in the paper as “EKS”).
Ellis G F R and Maartens Roy 2003, “The Emergent Universe: Inflationary Cosmology with No Singularity,” arXiv:gr-qc/0211082v3; Ellis, G F R, Murugan, J and Tsagas, C G The Emergent Universe: An Explicit Construction,” arXiv:gr- qc/0307112v1.
Ellis G F R, McEwan P, Stoeger W R, and Dunsby P 2002, “Causality in Inflationary Universes with Positive Spatial Curvature,” *Gen. Rel. & Grav.* **34** 1461.
Ellis G F R and Schrieber G 1986 *Phys. Lett.* A **115** 97. Gardner M 2003 *Are Universes Thicker than Blackberries?* (New York: Norton).
Garriga J Vilenkin A 2000 On likely values of the cosmological constant *Phys. Rev.* D **61** 083502 (*Archive* astro-ph/9908115).
Garriga J and Vilenkin A 2001 Many worlds in one *Phys. Rev.* D **64** 043511 (*Archive* gr-qc/0102010 and gr-qc/0102090).
Garriga J and Vilenkin A 2002 *Phys. Rev.* D **67** 043503 (*Archive* astro-ph/0210358).
Hartle, J (2004) “Anthropic Reasoning and Quantum Cosmlogy,” gr-qc/0406104.
Hilbert D 1964 On the Infinite in *Philosophy of Mathematics* ed P Benacerraf and H Putnam (Englewood Cliff, N. J.: Prentice Hall) pp 134-151.
Hogan C J 2000 Why the Universe is Just So *Rev. Mod. Phys*. **72** 1149-1161 (*Archive* astro- ph/9909295).
Isham C. J. 1988 Creation of the Universe as a Quantum Process, in *Physics, Philosophy and Theology: A Common Quest for Understanding*, ed. R J Russell, W R Stoeger, S.J., and G V Coyne, S.J. [Vatican City State: Vatican Observatory]{}, pp. 375-408.
Isham C. J. 1993 Quantum Theories of the Creation of the Universe in *Quantum Cosmology and the Laws of Nature: Scientific Perspectives on Divine Action*, ed. R J Russell, N Murphy and C J Isham [Vatican City State and Berkeley, California: Vatican Observatory Publications and the Center for Theology and the Natural Science]{}, pp. 49-89.
Kachru, S, Kallosh, R, Linde, A, Trivedi, S P 2003 *Phys. Rev.* **D68**, 046005, hep-th/0301240.
Katz G and Weeks J 2004 Polynomial Interpretation of Multipole Vectors *Archive* astro-ph/0405631.
Kirchner U and Ellis G F R 2003 A probability measure for FLRW models *Class. Quantum Grav.* **20** 1199-1213.
Lachieze-ray M, and Luminet J P 1995 *Phys. Rep.* **254** 135 (*Archive* gr-qc/9605010).
Leslie J 1996 *Universes* (Routledge).
Lewis D K (2000) *On the Plurality of Worlds* (Oxford: Blackwell).
Linde A D 1983 *Phys. Lett.* B **129** 177.
Linde A D 1990 *Particle Physics and Inflationary Cosmology* (Chur: Harwood Academic Publishers).
Linde A D 2003 Inflation, Quantum Cosmology and the Anthropic Principle, to appear in *Science and Ultimate Reality: >From Quantum to Cosmos*, ed J D Barrow, P C W Davies and C L Harper (Cambridge: Cambridge University Press) (*Archive* hep-th/0211048).
Linde A D, Linde D A, and Mezhlumian A 1994 *Phys. Rev*. D **49** 1783.
Linde A D, and Mezhlumian A 1993 Stationary Universe *Phys. Lett.* B **307** 25-33 (*Archive* gr-qc/9304015).
Luminet J-P, Weeks J R, Riazuelo A, Lehoucq R, and Uzan J-P 2003 Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background *Nature* (xxxx).
McMullin E 1992 *The Inference That Makes Science* (Milwaukee, WI: Marquette University Press), 112 pp.
McMullin E 1993 “Indifference Principle and Anthropic Principle in Cosmology,” *Stud. Hist. Phil. Sci.* **24** 359-389.
Misner, C W, Thorne, K S and Wheeler, J A 1973 *Gravitation* (New York: W. H. Freeman and Co.).
Moore, A W 1990 *The Infinite*(London: Routledge).
Penrose R 1989, “Difficulties with Inflationary Cosmology,” *Ann. N. Y. Acad. Sci.* **571** 249.
Rees M J 2001a *Just Six Numbers: The Deep Forces that Shape the Universe* (Basic Books). Rees M J 2001b *Our Cosmic Habitat* (Princeton: Princeton University Press). Rees M J 2001c ‘Concluding Perspective’ *Archive* astro-ph/0101268.
Roukema B F et al 2004 A Hint of Poincaré Dodecahedral Topology in the WMAP First Year Sky Map *Archive*astro-ph/0402608.
Rovelli C. 2004 *Quantum Gravity*(Cambridge: Cambridge University Press), 455 pp.
Russell, B 1960 *Our Knowledge of the External World* (New York: Mentor Books), as quoted in Quentin Smith, “Infinity and the Past.” in *Theism, Atheism, and Big Bang Cosmology*, edited by W. L. Craig and Q. Smith (Oxford: Oxford University Press), p. 88.
Sciama D 1993 *Is the universe unique?* in *Die Kosmologie der Gegenwart* ed G Borner and J Ehlers (Serie Piper).
Smith Q 1993 Infinity and the Past, in *Theism, Atheism, and Big Bang Cosmology*, edited by W. L. Craig and Q. Smith (Oxford: Oxford University Press), pp. 77-91.
Smolin L 1991 Space and Time in the Quantum Universe, in *Conceptual Problems in Quantum Gravity*, ed. A. Ashtekar and J. Stachel (Boston, Basil, Berklin: Birkhäuser), pp. 228-291.
Smolin L 1999 *The Life of the Universe* (Oxford: Oxford University Press).
Spitzer R J 2000 Definitions of Real Time and Ultimate Reality *Ultimate Reality and Meaning* **23** (No. 3) 260-276.
Stoeger W R 2003 What is ‘the Universe’ that Cosmology Studies, in *Fifty Years in Science and Religion: Ian Barbour and His Legacy*, ed. R J Russell (Burlington, VT: Ashgate Publishing Company), pp. 127-143.
Susskind, L. 2003 The Anthropic Landscape of String Theory, hep-th/0302219.
Susskind, L. 2005 *The Cosmic Landscape: String Theory and the Illusion of Intelligent Design*. (New York: Little, Brown and Company).
Tegmark M 1998 Is the Theory of Everything Merely the Ultimate Ensemble Theory? *Annal. Phys.* **270** 1-51 (*Archive* gr-qc/9704009).
Tegmark M 2003 Parallel Universes * Scientific American* (May 2003), 41-51 (*Archive* astro-ph/0302131).
Vilenkin A 1983 “The Birth of Inflationary Universes” *Phys Rev* **D27**, 2848.
Vilenkin A 1995 *Phys. Rev. Lett.* **74** 846.
Wallace D F 2003 *Everything and More: A Compact History of $\inf$*(New York: W. W. Norton and Company), 319 pp.
Weinberg S 2000 The Cosmological Constant Problems *Archive* astro-ph/0005265. Weinberg S 2000 A Priori Probability Distribution of the Cosmological Constant *Phys. Rev.* D **61** 103505 (*Archive* astro-ph/0002387).
Wheeler, J. A. 1968 *Einstein’s Vision* (New York: Springer-Verlag).
[^1]: Connected implies Locally causally connected, that is all universe domains are connected by $C^{0}$ timelike lines which allow any number of reversals in their direction of time, as in Feynman’s approach to electrodynamics. Thus, it is a union of regions that are causally connected to each other, and transcends particle and event horizons; for example all points in de Sitter space time are connected to each other by such lines.
[^2]: Science fiction and fantasy provide a rich treasury of conceivable universes, many of which will not be “possible universes” as outlined above.
[^3]: We thank and acknowledge the contribution of an anonymous referee who has pointed this out to us, and has stimulated this brief discussion of the important role of quantum cosmology in defining multiverses.
[^4]: Again, we thank the same referee for emphasizing the importance of the possibility.
[^5]: More accurately, perturbations of these models can allow life – the exact FLRW models themselves cannot do so.
[^6]: It has been suggested to us that in mathematical terms it does not make sense to distinguish identical copies of the same object: they should be identified with each other because they are essentially the same. But we are here dealing with physics rather than mathematics, and with real existence rather than possible existence, and then multiple copies must be allowed (for example all electrons are identical to each other; physics would be very different if there were only one electron in existence).
[^7]: Our discussion here follows EKS, with the addition of supporting philosophical material and references.
[^8]: For a fascinating and very readable, but somewhat eccentric, recent history of mathematical infinity and its connections with key mathematical developments, see David Foster Wallace (2003)
[^9]: Bracken (1995, pp.11-24) gives a recent critical summary of Aristotle’s treatment of these issues, and their later use by Thomas Aquinas, Schelling and Heidegger (Bracken, pp.25-51).
[^10]: In quantum theory, the members of a set (e. g. particles) may be virtual, going into and out of existence, but at any one time there are only “so many.” Furthermore, there is a number operator, even though the particles themselves cannot be physically distinguished.
[^11]: When contemplating mathematical concepts, it is debatable as to whether a procedure or process is needed. But we are talking physics, and the issue is precisely whether or not the concept is realisable in the physical sense.
[^12]: We thank an anonymous referee for pointing out these examples.
[^13]: There are some observational indications that this could be so (see Sec.(\[small\])), but they are far from definitive.
[^14]: Obviously this does not mean that we reject standard Big Bang cosmology – rejecting the really spatially infinite universes as unrealizable does not undermine the observational adequacy of these models, nor the essence of the Big Bang scenario, even in the cases of those which are flat or open. It just indicates that these models are incomplete, which we already recognized.
[^15]: One way out would be, as quite a bit of work in quantum cosmology seems to indicate, to have time originating or emerging from the quantum-gravity dominated primordial substrate only “later.” In other words, there would have been a “time” or an epoch before time as such emerged. Past time would then be finite, as seems to be demanded by philosophical arguments, and yet the timeless primordial state could have lasted “forever,” whatever that would mean. This possibility avoids the problem of constructibility.
[^16]: In light of discussions by McMullin elsewhere (McMullin 1993, pp. 381-382) more care and precision is needed here. He recommends separating explanation from proof of existence: “In science, the adequacy of a theoretical explanation is often regarded as an adequate testimony to the existence of entities postulated by the theory. But the debates that swirl around this issue among philosophers (the issue of scientific realism, as philosophers call it) ought to warn us of the risks of moving too easily from explanatory adequacy to truth-claims for the theory itself. This sort of inference depends sensitively on the quality of the explanation given, on the viability of alternatives, on our prior knowledge of beings in the postulated category, and on other more complex factors.”
[^17]: It is interesting to note that Rees (2001b, p. 172) hints at the use of a retroductive approach in cosmology, but does not develop the idea as an argument in any detail.
|
---
abstract: 'With the development of artificial intelligence and deep learning (DL) techniques, rotating machinery intelligent diagnosis has gone through tremendous progress with verified success and the classification accuracies of many DL-based intelligent diagnosis algorithms are tending to 100%. However, different datasets, configurations, and hyper-parameters are often recommended to be used in performance verification for different types of models, and few open source codes are made public for evaluation and comparisons. Therefore, unfair comparisons and ineffective improvement may exist in rotating machinery intelligent diagnosis, which limits the advancement of this field. To address these issues, we perform an extensive evaluation of four kinds of models, including multi-layer perception (MLP), auto-encoder (AE), convolutional neural network (CNN), and recurrent neural network (RNN), with various datasets to provide a benchmark study within the same framework. In this paper, we first gather most of the publicly available datasets and give the complete benchmark study of DL-based intelligent algorithms under two data split strategies, five input formats, three normalization methods, and four augmentation methods. Second, we integrate the whole evaluation codes into a code library and release this code library to the public for better development of this field. Third, we use the specific-designed cases to point out the existing issues, including class imbalance, generalization ability, interpretability, few-shot learning, and model selection. By these works, we release a unified code framework for comparing and testing models fairly and quickly, emphasize the importance of open source codes, provide the baseline accuracy (a lower bound) to avoid useless improvement, and discuss potential future directions in this field. The code library is available at <https://github.com/ZhaoZhibin/DL-based-Intelligent-Diagnosis-Benchmark>.'
address: 'School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China'
author:
- Zhibin Zhao
- Tianfu Li
- Jingyao Wu
- Chuang Sun
- Shibin Wang
- Ruqiang Yan
- Xuefeng Chen
bibliography:
- 'References/Reference.bib'
title: 'Deep Learning Algorithms for Rotating Machinery Intelligent Diagnosis: An Open Source Benchmark Study'
---
Deep learning ,machinery fault diagnosis ,open source codes ,benchmark study
Introduction {#S:1}
============
Prognostics health management (PHM) is one of the most essential systems in modern industrial equipment, such as helicopter, aero-engine, wind turbine, and high speed train. The main function of PHM systems used in rotating machinery is intelligent fault diagnosis for condition-based maintenance. Intelligent fault diagnosis is the key component of PHM systems and has been studied widely. Traditional intelligent diagnosis methods mainly consist of the feature extraction using various signal processing methods and the fault classification using various machine learning techniques. Although advanced signal processing methods (fast Fourier transform (FFT), spectrum kurtosis (SK), wavelet transform (WT), sparse representation, etc.) and machine learning algorithms (k-nearest neighbor (KNN), artificial neural network (ANN), support vector machine (SVM), etc.) have been successfully applied to intelligent diagnosis and have made considerable progress, it remains a challenging problem about how to perform diagnosis precisely and efficiently. With the development of online condition monitoring and data analysis systems, increasingly different kinds of real-time data are transferred from operating machines and the massive data are gained in the cloud. Facing with these heterogeneous massive data, feature extraction methods and mapping abilities from signals to conditions that are designed and chosen by experts, to a great extent depending on prior knowledge, are time-consuming and empirical.
Deep learning (DL) as a booming data mining technique has swept many fields including computer vision (CV) [@krizhevsky2012imagenet; @farabet2012learning], natural language processing (NLP) [@hirschberg2015advances; @sun2017review; @young2018recent], etc. In 2006, the concept of DL was first introduced through proposing the deep belief network (DBN)[@hinton2006reducing]. In 2013, MIT Technology Review ranked the DL technology as the top ten breakthrough technologies [@MIT2013]. In 2015, a review [@lecun2015deep] published in nature stated that DL allows computational models composed of multiple processing layers to learn data representations with multiple levels of the abstraction. Due to its strong representation learning ability, DL is well-suited to data analysis. Therefore, in the field of intelligent fault diagnosis, many researchers have applied DL-based techniques, such as multi-layer perception (MLP), auto-encoder (AE), convolutional neural network (CNN), deep belief network (DBN), and recurrent neural network (RNN) to various fields. A large number of DL-based intelligent diagnosis algorithms have been proposed in recent years and their classification accuracies have been tending to 100%. However, when different researchers design DL-based intelligent diagnosis algorithms, they often recommend to use different inputs (like time domain input, frequency domain input, time-frequency domain input, wavelet domain input, slicing image input, etc.) and set different hyper-parameters (like the dimension of the input, the learning rate, the batch size, the network architecture, etc.). In addition, few authors make their codes available for evaluation and comparison, and others are difficult to repeat the results completely and correctly. Therefore, unfair comparisons and ineffective improvement may exist in this field. Considering that this field lacks open source codes and benchmark study, it is crucial to evaluate and compare different DL-based intelligent diagnosis algorithms to provide the benchmark or the lower bound of their accuracies and performance, thereby helping further studies in this field for more persuasive and appropriate algorithms.
For comprehensive performance comparisons and evaluation, it is important to gather different kinds of datasets. Actually, there exist several datasets for intelligent fault diagnosis. However, not every dataset provides a detailed description and is suited for the fault classification. For some datasets, the category discrimination is relatively large, and even one simple classifier can achieve acceptable results. Therefore, to thoroughly perform data mining and assess the difficulty of datasets, it is necessary to collect different datasets in a library and evaluate the performance of algorithms for different datasets on a unified platform.
In addition, one common issue in intelligent fault diagnosis is that for splitting data, and researchers often use the random split strategy. This strategy is dangerous since if the preparation process exists any overlap for samples, the evaluation of classification algorithms will have test leakage [@riley2019three]. As for industrial data, they are rarely random and are always sequential (they might contain trends in time domain). Therefore, it is more appropriate to split data according to time sequences (we simply call it order split) [@riley2019three]. Actually, order split is closer to reality, because we always use historical data to predict the future condition in industry. Conversely, if we randomly split the data, it might be possible for the diagnosis algorithms to record the future patterns, and this might cause another pitfall with test leakage.
In this paper, we first collect most of the publicly available datasets and discuss whether it is suitable for intelligent fault diagnosis. Second, we release a code library of the data preparation for all datasets which are suitable for fault classification and the whole evaluation framework with different input formats, normalization methods, data split ways, augmentation methods, and DL-based models. Meanwhile, we also use some datasets to discuss the existing issues in intelligent fault diagnosis including class imbalance, generalization ability, interpretability, few-shot learning, and model selection. To the best of our knowledge, this is the first work to comprehensively perform the benchmark study and release the code library of DL-based intelligent algorithms. In summary, this work mainly focuses on evaluating various DL-based intelligent diagnosis algorithms for most of the publicly available datasets from several perspectives, providing the benchmark accuracy (it is worth mentioning that the results are just a lower bound of accuracy) to avoid useless improvement, and releasing the code library for complete evaluation procedures. Through these works, we hope to make comparing and testing models fairer and quicker, emphasize the importance of open source codes and the benchmark study in this field, and provide some suggestions and discussions of future studies.
The contributions of this paper are listed as follows:
1) *Various datasets and data preparing*. We gather most of the publicly available datasets and give the detailed discussion about its adaptability to DL-based intelligent diagnosis. For data preparing, we first discuss different kinds of input formats and different normalization methods for listed datasets. After that, we state that data augmentation which is a common step in CV and NLP might be important to make the training datasets more diverse, and we also try some kinds of data augmentation methods to clarify that they have not been fully investigated. Meanwhile, we also discuss the way of data split and state that it may be more appropriate to split data according to time sequences (also called order split).
2) *Benchmark accuracy and further studies*. We evaluate various DL-based intelligent diagnosis algorithms including MLP, AE, CNN, and RNN for different datasets and provide the benchmark accuracy to make the future studies in this field more comparable and meaningful. We also use the experimental examples to discuss the existing problems in intelligent fault diagnosis including class imbalance, generalization ability, interpretability, few-shot learning, and model selection problems.
3) *Open source codes*. For enhancing the importance and reproducibility of DL-based intelligent diagnosis algorithms, we release the whole evaluation codes in a code library for the better development of this field. At the same time, this is a unified intelligent fault diagnosis library, which retains an extended interface for everyone to load their own datasets and models by themselves to carry out new studies. The code library is available at <https://github.com/ZhaoZhibin/DL-based-Intelligent-Diagnosis-Benchmark>.
The outlines of the paper are listed as follows: In Section \[S:2\], we give a brief review of recent development of DL-based intelligent diagnosis algorithms. Then, Sections \[S:3\] to \[S:9\] discuss the evaluation algorithms, datasets, data preprocessing, data augmentation, data split, evaluation methodologies and evaluation results, respectively. After that, Section \[S:10\] makes some further discussions and the results, followed by conclusions in Section \[S:11\].
Brief Review {#S:2}
============
Recently, DL has become a promising method in a large scope of fields, and a huge amount of papers related to DL have been published since 2012. This paper mainly focuses on a benchmark study of intelligent fault diagnosis, rather than providing a comprehensive review on DL for other fields. Some famous DL researchers have published more professional references and interested readers can refer to [@lecun2015deep; @goodfellow2016deep].
In the field of intelligent fault diagnosis, due to the efforts of many researchers in recent years, DL has become one of the most popular data-driven methods to perform fault diagnosis and health monitoring. In general, DL-based methods can extract representative features adaptively without any manual intervention and can achieve higher accuracy than traditional machine learning algorithms in most of the tasks when the dataset is large enough. We conducted a literature search using Web of Science with a database called web of science core collection. As shown in [Fig. \[Fig1\]]{}, it can be observed that the number of published papers related to DL-based intelligent algorithms increases year by year.
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\[-5pt\]
Another interesting observation is that many review papers on this topic have been published in the recent four years. Therefore, in this paper, we only briefly review and introduce the main contents of different review papers to allow readers who just enter this field to find suitable review papers quickly.
In bearing fault diagnosis, Li et al. [@li2018systematic] provided a systematic review of fuzzy formalisms including combination with other machine learning algorithms. Hoang et al. [@hoang2019survey] provided a comprehensive review of three popular DL algorithms (AE, DBN, and CNN) for bearing fault diagnosis. Zhang et al. [@zhang2019machine] systematically reviewed the machine learning and DL-based algorithms for bearing fault diagnosis and also provided a comparison of the classification accuracy of CWRU with different DL-based methods. Hamadache et al. [@hamadache2019comprehensive] reviewed different fault modes of rolling element bearings and described various health indexes for PHM. Meanwhile, it also provided a survey of artificial intelligence methods for PHM including shallow learning and deep learning.
In rotating machinery intelligent diagnosis, Ali et al. [@ali2016acoustic] provided a review of AI-based methods using acoustic emission data for rotating machinery condition monitoring. Liu et al. [@liu2018artificial] reviewed Al-based approaches including KNN, SVM, ANN, Naive Bayes, and DL for fault diagnosis of rotating machinery. Wei et al. [@wei2019review] summarized early fault diagnosis of gears, bearings, and rotors through signal processing methods (adaptive decomposition methods, WT, and sparse decomposition) and AI-based methods (KNN, neural network, and SVM).
In machinery condition monitoring, Zhao et al. [@zhao2016research] and Duan et al. [@duan2018deep] reviewed diagnosis and prognosis of mechanical equipment based on DL algorithms such as DBN and CNN. Zhang et al. [@zhang2017comprehensive] reviewed computational intelligent approaches including ANN, evolutionary algorithms, fuzzy logic, and SVM for machinery fault diagnosis. Zhao et al. [@zhao2019deep] reviewed data-driven machine health monitoring through DL methods (AE, DBN, CNN, and RNN) and provided the data and codes (in Keras) about an experimental study.
In addition, Nasiri et al. [@nasiri2017fracture] surveyed the state-of-the-art AI-based approaches for fracture mechanics and provided the accuracy comparisons achieved by different machine learning algorithms for mechanical fault detection. Tian et al. [@tian2018review] surveyed different modes of traction induction motor fault and their diagnosis algorithms including model-based methods and AI-based methods. Khan et al. [@khan2018review] provided a comprehensive review of AI for system health management and emphasized the trend of DL-based methods with limitations and benefits. Stetco et al. [@stetco2018machine] reviewed machine learning approaches applied to wind turbine condition monitoring and made a discussion of the possibility for the future research. Ellefsen et al. [@ellefsen2019comprehensive] reviewed four well-established DL algorithms including AE, CNN, DBN, and LSTM for PHM applications and discussed the chances and challenges for the future studies, especially in the field of PHM in autonomous ships. AI-based algorithms (traditional machine learning algorithms and DL-based approaches) and applications (smart sensors, intelligent manufacturing, PHM, and cyber-physical systems) were reviewed in [@ademujimi2017review; @chang2018review; @wang2018deep; @sharp2018survey] for smart manufacturing and manufacturing diagnosis.
Although a large body of DL-based methods and many related reviews have been published in the field of intelligent fault diagnosis, few studies thoroughly evaluate various DL-based intelligent diagnosis algorithms for most of the publicly available datasets, provide the benchmark accuracy, and release the code library for complete evaluation procedures. For example, a simple code written in Keras was published in [@zhao2019deep], which is not comprehensive enough for different datasets and models. The accuracy comparisons were provided in [@zhang2019machine; @nasiri2017fracture] according to existing papers, but they were not comprehensive enough due to different configurations and test conditions. Therefore, this paper is intended to make up for this gap and emphasize the importance of open source codes and the benchmark study in this field.
Evaluation Algorithm {#S:3}
====================
A large amount of DL-based intelligent diagnosis methods have been published in the field of fault diagnosis and prognosis. It is impossible to cover all the published models since there is currently no open source community in this field. Therefore, we switch to test the performance of four categories of representative models (MLP, AE, CNN, and RNN) embedding some advanced techniques. It should be noted that DBN is also another commonly used DL methods for fault diagnosis, but we do not add it into this code library due to that the fact the training way of DBN is much different from those four categories.
MLP
---
Multilayer Perception (MLP) [@rumelhart1985learning], which was a fully connected network with one or more hidden layers, was proposed in 1987 as the prototype of an artificial neural network (ANN). With such a simple structure, MLP can complete some easy classification tasks such as MNIST. But as the task becomes more complex, it will be hard to train because of the huge amount of parameters. MLP with five fully connected layers and five batch normalization layers is used in this paper for the one dimension (1D) input data. The structure and parameters of the model are shown in [Fig. \[MLP\]]{}. Besides, in [Fig. \[MLP\]]{}, FC means the fully connected layer, BN means the Batch Normalization layer, and CE loss means the softmax cross-entropy loss.
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AE
--
Auto-encoder(AE) was first proposed in 2006 as a method for dimensionality reduction. It can reduce the dimensionality of the input data while retaining most of the information in the data. AE consists of an encoder and a decoder, which tries to reconstruct the input from the output of the encoder, and the reconstruction error is used as a loss function. The encoder and decoder are trained to generate the low-dimension representation of the input and reconstruct the input from low-dimension representation, respectively. Subsequently, various derivatives of AE were proposed by researchers, such as variational auto-encoder (VAE) [@kingma2013auto], denoising auto-encoder (DAE) [@vincent2008extracting], and sparse auto-encoder (SAE) [@ranzato2007efficient]. In this paper, we design the deep AE and its derivatives for 1D input data and two dimension (2D) input data, respectively. Considering different features of neural networks, the structures and hyper-parameters of them shown in [Fig. \[AE\]]{} change adaptively. Specifically, the network structures of DAE and SAE are the same with AE, and the differences are the loss function and inputs. During the training of AE and its derivatives, the encoder and decoder are trained jointly to get the low-dimensionality features of data. After that, the encoder and classifier are trained jointly for the classification task. Besides, in [Fig. \[AE\]]{}, the MSE loss means the mean square error loss, Conv means the convolutional layer, $ \text{Conv}^\text{T} $ means the transposed convolutional (e.g. inverse convolution) layer, and the KLP loss means the Kullback-Leibler divergence loss.
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\[-5pt\]
CNN
---
Convolutional neural network (CNN) [@lecun1995convolutional] was first proposed in 1997 and the proposed network was also called LeNet. CNN is a specialized kind of the neural network for processing data that have a known grid-like topology. Sparse interactions, parameter sharing, and equivalent representations are realized with convolution and pooling operations on CNN. In 2012, AlexNet [@krizhevsky2012imagenet] won the title in the ImageNet competition by far surpassing the second place, and deep CNN has attracted wide attention. Besides, in 2016, ResNet [@he2016deep] was proposed and its classification accuracy exceeded the human baseline. In this paper, we design 5 layers 1D CNN and 2D CNN for 1D input data and 2D input data, respectively, and also adapt three well known CNN models (LeNet, ResNet18, and AlexNet) for two types of input data. The details of them are shown in [Fig. \[CNN\]]{}. In [Fig. \[CNN\]]{}, MaxPool means the Max Pooling layer, AdaptiveMaxPool means the Adaptive Max Pooling layer, and Dropout means the Dropout layer.
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Recurrent Neural Network
------------------------
Recurrent neural network (RNN) can describe the temporal dynamic behavior and is very suitable to deal with the time series. However, RNN often exists the gradient vanishing and exploding problems during the training. To overcome these problems, Long Short-term Memory Network(LSTM) was proposed in 1997 [@hochreiter1997long] for processing continual input streams and has made great success in various fields such as NLP, etc. Bi-directional LSTM (BiLSTM) can capture bidirectional dependencies over long distances and learn to remember and forget information selectively. We utilize BiLSTM as the representation of RNN to deal with two types of input data (1D and 2D) for the classification task. The details of BiLSTM are shown in [Fig. \[LSTM\]]{}. Besides, in [Fig. \[LSTM\]]{}, Transpose means transposing the channel and feature dimensions of the input data, and BiLSTM Block means the BiLSTM layer.
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Datasets {#S:4}
========
In the field of intelligent fault diagnosis, publicly available datasets have not been investigated in depth. Actually, for comprehensive performance comparisons and evaluation, it is important to gather different kinds of representative datasets. We collected nine commonly used datasets which all have specific labels and explanations in addition to the PHM 2012 bearing dataset and IMS bearing dataset, so PHM 2012 and IMS are not suitable for fault classification that requires labels. To sum up, this paper uses seven datasets to verify the performance of models introduced in Section \[S:3\]. The description of all these datasets is listed as follows.
CWRU Bearing Dataset
--------------------
CWRU datasets were provided by the Case Western Reserve University Bearing Data Center [@CWRU]. Vibration signals were collected at 12 kHz or 48 kHz for normal bearings and damaged bearings with single-point defects under four different motor loads. Within each working condition, single-point faults were introduced with fault diameters of 0.007, 0.014, and 0.021 inches on the rolling element, the inner ring, and the outer ring, respectively. In this paper, we use the data collected from the drive end, and the sampling frequency is equivalent to 12 kHz. In Table \[Tab\_1\], one health state bearing and three fault locations, including the inner ring fault, the rolling element fault, and the outer ring fault, are classified into ten categories (one health state and 9 fault states) according to different fault sizes.
\[Tab\_1\]
Fault Mode Description:
------------------- -------------------------------------------------------
Health State the normal bearing at 1791 rpm and 0 HP
Inner ring 1 0.007 inch inner ring fault at 1797 rpm and 0 HP
Inner ring 2 0.014 inch inner ring fault at 1797 rpm and 0 HP
Inner ring 3 0.021 inch inner ring fault at 1797 rpm and 0 HP
Rolling Element 1 0.007 inch rolling element fault at 1797 rpm and 0 HP
Rolling Element 2 0.014 inch rolling element fault at 1797 rpm and 0 HP
Rolling Element 3 0.021 inch rolling element fault at 1797 rpm and 0 HP
Outer ring 1 0.007 inch outer ring fault at 1797rpm and 0 HP
Outer ring 2 0.014 inch outer ring fault at 1797rpm and 0 HP
Outer ring 3 0.021 inch outer ring fault at 1797rpm and 0 HP
: Detailed description of CWRU datasets
MFPT Bearing Dataset
--------------------
MFPT datasets were provided by Society for Machinery Failure Prevention Technology [@MFPT]. MFPT datasets consisted of three bearing datasets: 1) a baseline dataset sampled at 97656 Hz for six seconds in each file; 2) seven outer ring fault datasets sampled at 48828 Hz for three seconds in each file; 3) seven inner ring fault datasets sampled at 48828 Hz for three seconds in each file; 4) some other datasets which are not used in this paper (more detailed information can be referred to the website of MFPT datasets [@MFPT]). In Table \[Tab\_2\], one health state bearing and two fault bearings including the inner ring fault and the rolling element fault are classified into ten categories (one health state and nine fault states) according to different loads.
\[Tab\_2\]
Fault Mode Description:
-------------- ---------------------------------------------
Health State Fault-free bearing working at 270 lbs
Outer ring 1 Outer ring fault bearing working at 25 lbs
Outer ring 2 Outer ring fault bearing working at 50 lbs
Outer ring 3 Outer ring fault bearing working at 100 lbs
Outer ring 4 Outer ring fault bearing working at 150 lbs
Outer ring 5 Outer ring fault bearing working at 200 lbs
Outer ring 6 Outer ring fault bearing working at 250 lbs
Outer ring 7 Outer ring fault bearing working at 300 lbs
Outer ring 1 Inner ring fault bearing working at 0 lbs
Inner ring 2 Inner ring fault bearing working at 50 lbs
Inner ring 3 Inner ring fault bearing working at 100 lbs
Inner ring 4 Inner ring fault bearing working at 150 lbs
Inner ring 5 Inner ring fault bearing working at 200 lbs
Inner ring 6 Inner ring fault bearing working at 250 lbs
Inner ring 7 Inner ring fault bearing working at 300 lbs
: Detailed description of MFPT datasets
PU Bearing Dataset
------------------
PU datasets were provided by the Paderborn University Bearing Data Center [@PU; @lessmeier2016condition], and PU datasets consisted of 32 sets of bearing current signals and vibration signals. As shown in Table \[Tab\_3\], bearings are divided into: 1) six undamaged bearings; 2) twelve artificially damaged bearings; 3) fourteen bearings with real damages caused by accelerated lifetime tests. Each dataset was collected under four working conditions as shown in Table \[Tab\_PU\].
\[Tab\_3\]
[|m[0.07]{}<|m[0.18]{}<|m[0.14]{}<|m[0.07]{}<|m[0.18]{}<|m[0.14]{}<|]{} Bearing Code & Fault Mode & Description & Bearing Code & Fault Mode & Description\
K001 & Health state & Run-in 50 h before test &KI07 & Artificial inner ring fault (Level 2)&Made by electric engraver\
K002 & Health state &Run-in 19 h before test &KI08 & Artificial inner ring fault (Level 2) &Made by electric engraver\
K003 & Health state & Run-in 1 h before test &KA04 & Outer ring damage (single point + S + Level 1) &Caused by fatigue and pitting\
K004 & Health state & Run-in 5 h before test &KA15 & Outer ring damage (single point + S + Level 1) &Caused by plastic deform and indentation\
K005 & Health state & Run-in 10 h before test &KA16 & Outer ring damage (single point + R + Level 2) &Caused by fatigue and pitting\
K006 & Health state & Run-in 16 h before test &KA22 & Outer ring damage (single point + S + Level 1)&Caused by fatigue and pitting\
KA01 & Artificial outer ring fault (Level 1) & Made by EDM &KA30 & Outer ring damage (distributed + R + Level 1) &Caused by plastic deform and indentation\
KA03 & Artificial outer ring fault (Level 2) & Made by electric engraver&KB23 & Outer ring and inner ring damage (single point + M + Level 2) &Caused by fatigue and pitting\
KA05 & Artificial outer ring fault (Level 1) & Made by electric engraver&KB24 & Outer ring and inner ring damage (distributed + M + Level 3) &Caused by fatigue and pitting\
KA06 & Artificial outer ring fault (Level 2) &Made by electric engraver &KB27 & Outer ring and inner ring damage (distributed + M + Level 1) &Caused by plastic deform and indentation\
KA07 & Artificial outer ring fault (Level 1) & Made by drilling&KI04 & Inner ring damage (single point + M + Level 1) &Caused by fatigue and pitting\
KA08 & Artificial outer ring fault (Level 2) & Made by drilling&KI14 & Inner ring damage (single point + M + Level 1) &Caused by fatigue and pitting\
KA09 & Artificial outer ring fault (Level 2) & Made by drilling&KI16 & Inner ring damage (single point + S + Level 3) &Caused by fatigue and pitting\
KI01 & Artificial inner ring fault (Level 1) &Made by EDM &KI17 & Inner ring damage (single point + R + Level 1) &Caused by fatigue and pitting\
KI03 & Artificial inner ring fault (Level 1) & Made by electric engraver &KI18 & Inner ring damage (single point + S + Level 2) &Caused by fatigue and pitting\
KI05 & Artificial inner ring fault (Level 1) & Made by electric engraver &KI21 & Inner ring damage (single point + S + Level 1) &Caused by fatigue and pitting\
\[Tab\_PU\]
No. Rotating speed (rpm) Load torque (Nm) Radial force (N) Name of setting
----- ---------------------- ------------------ ------------------ -----------------
0 1500 0.7 1000 N15\_M07\_F10
1 900 0.7 1000 N09\_M07\_F10
2 1500 0.1 1000 N15\_M01\_F10
3 1500 0.7 400 N15\_M07\_F04
: Four working conditions of PU datasets
In this paper, since using all the data will cause huge computational time, we only use the data collected from real damaged bearings ( including KA04, KA15, KA16, KA22, KA30, KB23, KB24, KB27, KI14, KI16, KI17, KI18, and KI22) under the working condition N15\_M07\_F10 to carry out the performance verification. It is worth mentioning that since KI04 is the same as KI14 completely shown in Table \[Tab\_3\], we delete KI04 and the total number of classes is thirteen. Besides, only vibration signals are used for testing the models.
UoC Gear Fault Dataset
----------------------
UoC gear fault datasets were provided by the University of Connecticut [@UoC], and UoC datasets were collected at 20 kHz. In this dataset, nine different gear conditions were introduced to the pinions on the input shaft, including healthy condition, missing tooth, root crack, spalling, and chipping tip with 5 different levels of severity. All the collected datasets are used and classified into nine categories (one health state and eight fault states) to test the performance.
XJTU-SY Bearing Dataset
-----------------------
XJTU-SY bearing datasets were provided by the Institute of Design Science and Basic Component at Xi’an Jiaotong University and the Changxing Sumyoung Technology Co. [@XJTU; @wang2018hybrid]. XJTU-SY datasets consisted of fifteen bearings run-to-failure data under three different working conditions. Data were collected at 2.56 kHz. A total of 32768 data points were recorded for each sampling, and the sampling period is equal to one minute. The details of bearing lifetime and fault elements are shown in Table \[Tab\_4\]. In this paper, we use all the data described in Table \[Tab\_5\] and the total number of classes is fifteen. It should be noticed that we use collected data at the end of run-to-failure experiments.
\[Tab\_4\]
Condition File Lifetime Fault element
----------- -------------- ---------- ---------------------------------------------------
Bearing 1\_1 2h 3min Outer ring
Bearing 1\_2 2h 41min Outer ring
Bearing 1\_3 2h 38min Outer ring
Bearing 1\_4 2h 2min Cage
Bearing 1\_5 52 min Inner ring and Outer ring
Bearing 2\_1 8h 11min Inner ring
Bearing 2\_2 2h 41min Outer ring
Bearing 2\_3 8h 53min Cage
Bearing 2\_4 42min Outer ring
Bearing 2\_5 5h 39min Outer ring
Outer ring
Inner ring, Rolling element, Cage, and Outer ring
Inner ring
Inner ring
Outer ring
: Detailed description of XJTU-SY datasets
SEU Gearbox Dataset
-------------------
SEU gearbox datasets were provided by Southeast University [@SEU; @shao2018highly]. SEU datasets contained two sub-datasets, including a bearing dataset and a gear dataset, which are both acquired on Drivetrain Dynamics Simulator (DDS). There are two kinds of working conditions with rotating speed - load configuration (RS-LC) set to be 20 Hz - 0 V and 30 HZ - 2 V shown in Table \[Tab\_5\]. The total number of classes is equal to twenty according to Table \[Tab\_5\] under different working conditions. Within each file, there are eight rows of vibration signals, and we use the second row of vibration signals.
\[Tab\_5\]
Fault Mode RS-LC Fault Mode RS-LC
--------------- ------------- -------------------- -------------
Health Gear 20 Hz - 0 V Health Bearing 20 Hz - 0 V
Health Gear 30 Hz - 2 V Health Bearing 30 Hz - 2 V
Chipped Tooth 20 Hz - 0 V Inner ring 20 Hz - 0 V
Chipped Tooth 30 Hz - 2 V Inner ring 30 Hz - 2 V
Missing Tooth 20 Hz - 0 V Outer ring 20 Hz - 0 V
Missing Tooth 30 Hz - 2 V Outer ring 30 Hz - 2 V
Root Fault 20 Hz - 0 V Inner + Outer ring 20 Hz - 0 V
Root Fault 30 Hz - 2 V Inner + Outer ring 30 Hz - 2 V
Surface Fault 20 Hz - 0 V Rolling Element 20 Hz - 0 V
Surface Fault 30 Hz - 2 V Rolling Element 30 Hz - 2 V
: Detailed description of SEU datasets
JNU Bearing Dataset
-------------------
JNU bearing datasets were provided by Jiangnan University [@JNU; @li2013sequential]. JNU datasets consisted of three bearing vibration datasets with different rotating speeds, and the data were collected at 50 kHz. As shown in Table \[Tab\_6\], JNU datasets contained one health state and three fault modes which include inner ring fault, outer ring fault, and rolling element fault. Therefore, the total number of classes is equal to twelve according to different working conditions.
\[Tab\_6\]
Fault Mode Rotating Speed Fault Mode Rotating Speed Fault Mode Rotating Speed
----------------- ---------------- ----------------- ---------------- ----------------- ----------------
Health State 600 rpm Health State 800 rpm Health State 1000 rpm
Inner ring 600 rpm Inner ring 800 rpm Inner ring 1000 rpm
Outer ring 600 rpm Outer ring 800 rpm Outer ring 1000 rpm
Rolling Element 600 rpm Rolling Element 800 rpm Rolling Element 1000 rpm
: Detailed description of JNU datasets
PHM 2012 Bearing Dataset
------------------------
PHM 2012 bearing datasets were used for PHM IEEE 2012 Data Challenge [@PHM; @nectoux2012pronostia]. In PHM 2012 datasets, seventeen run-to-failure datasets were provided including six training sets and eleven testing sets. Three different loads were considered. Vibration and temperature signals were gathered during all those experiments. Since no label on the types of failures was given, it is not used in this paper.
IMS Bearing Dataset
-------------------
IMS bearing datasets were generated by the NSF I/UCR Center for Intelligent Maintenance Systems [@lee2007bearing]. IMS datasets were made up of three bearing datasets, and each of them contained vibration signals of four bearings installed on the different locations. At the end of the run-to-failure experiment, a defect occurred on one of the bearings. The failure occurred in the different locations of bearings. It is inappropriate to classify these failures simply using three classes, so IMS datasets are not evaluated in this paper.
Data Prepreocessing {#S:5}
===================
The reason why DL is superior in fault classification lies in its excellent feature extraction ability and feature space transformation ability. Although it is an end-to-end learning method, the type of input data and the way of normalization have a great impact on its performance. The type of input data determines the difficulty of feature extraction, and the normalization method determines the difficulty of calculation. So, in this paper, effects of five different input types and three different normalization methods on the performance of DL models are discussed.
Input Types
-----------
In the field of CV and NLP, commonly used input types consist of images and texts, while in intelligent fault diagnosis, what we collected directly is the time series. Therefore, many researchers use signal processing methods to map the time series to different domains to get a better input type. However, which input type is more suitable to the intelligent fault diagnosis is still an open question. In this paper, effects of different input types on model performance are discussed.
### Time Domain Input
For the time domain input, vibration signals are directly used as the input without data preprocessing. In this paper, the length of each sample is equivalent to 1024 and the total number of samples can be obtained from [Eq. \[eq1\]]{}. After generating samples, we take 80% of total samples as the training set and 20% of total samples as the testing set. $$\begin{aligned}
\label{eq1}
N = \text{floor}(\frac{L}{1024})
\end{aligned}$$ where $ L $ is the length of each signal, $ N $ is the total samples, and $ \text{floor} $ means rounding towards minus infinity.
### Frequency Domain Input
For the frequency domain input, FFT is used to transform each sample $ x_i $ from the time domain into the frequency domain shown in [Eq. \[eq2\]]{}. After this operation, the length of data will be halved and the new sample can be expressed as: $$\begin{aligned}
\label{eq2}
x_i^{\text{FFT}} = \text{FFT}(x_i)
\end{aligned}$$ where the operator $ \text{FFT}(\cdot) $ represents transforming $ x_i $ into the frequency domain and taking the first half of the result.
### Time-Frequency Domain Input
For the time-frequency domain input, Short-time Fourier Transform (STFT) is applied to each sample $ x_i $ to obtain the time-frequency representation shown in [Eq. \[eq3\]]{}. The Hanning window is used and the window length is set to 64. After this operation, the time-frequency representation (a 33x33 image) will be generated as: $$\begin{aligned}
\label{eq3}
x_i^{\text{STFT}} = \text{STFT}(x_i), \quad i=1,2,...,N
\end{aligned}$$ where the operator $ \text{SFFT}(\cdot) $ represents transforming $ x_i $ into the time-frequency domain.
### Wavelet Domain Input
For the wavelet domain input, continuous wavelet transform (CWT) is applied to each sample $ x_i $ to obtain the wavelet domain representation shown in [Eq. \[eq4\]]{}. Because CWT is time-consuming, the length of each sample $ x_i $ is set to 100. After this operation, the wavelet coefficients (an 100x100 image) will be obtained as: $$\begin{aligned}
\label{eq4}
x_i ^{\text{CWT}} = \text{CWT}(x_i), \quad i=1,2,...,N
\end{aligned}$$ where the operator $ \text{CWT}(\cdot) $ represents transforming $ x_i $ into the wavelet domain.
### Slicing Image Input
For slicing image input, each sample $ x_i $ is reshaped into a 32x32 image. After this operation, the new sample can be denoted as: $$\begin{aligned}
\label{eq5}
x_i ^{\text{Reshape}}= \text{Reshape}(x_i), \quad i=1,2,...,N
\end{aligned}$$ where the operator $ \text{Reshape}(\cdot) $ represents reshaping $ x_i $ into a 32x32 image.
However, the above data preprocessing method has some problems for training AE models and CNN models in the following two aspects: 1) if AE models input a large 2D signal, it will lead the decoder to have difficulty in the reconstruction procedure and the reconstruction error is very large; 2) if CNN models input a small 2D signal, it will make CNN unable to extract appropriate features.
Therefore, we have made a compromise on the data size obtained by the above data preprocessing methods. The size of the time domain and the frequency domain input are unchanged as shown in [Eq. \[eq1\]]{} and [Eq. \[eq2\]]{}. For the AE class, sizes of all 2D inputs are adjusted to 32x32, while for CNN models, sizes of signals after CWT, STFT, and slice image are adjusted to 300x300, 330x330, and 320x320, respectively. It should be noted that input sizes of CNN models can be different since we use the AdaptiveMaxPooling layer to adapt different input sizes.
Normalization
-------------
Input normalization can control values of data to a certain range. It is the basic step in data preparing, which can facilitate the subsequent data processing and accelerate the convergence of DL models. Therefore, we discuss effects of three normalization methods on the performance of DL models.
**Maximum-Minimum Normalization**: This normalization method can be implemented as
$$\begin{aligned}
\label{eq6}
x^{normalize-1}_i = \frac{x_i - x_i^{min}}{x_i^{max} - x_i^{min}}, \quad i=1,2,...,N
\end{aligned}$$
where $ x_i $ is the input sample, $ x_i^{min} $ is the minimum value in $ x_i $, and $ x_i^{max} $ is the maximum value in $ x_i $.
**\[-1-1\] Normalization**: This normalization method can be implemented as
$$\begin{aligned}
\label{eq7}
x^{normalize-2}_i = -1 + 2 * \frac{x_i - x_i^{min}}{x_i^{max} - x_i^{min}}, \quad i=1,2,...,N
\end{aligned}$$
**Z-score Normalization**: This normalization method can be implemented by as
$$\begin{aligned}
\label{eq8}
x^{normalize-3}_i = \frac{x_i - x_i^{mean}}{x_i^{std}}, \quad i=1,2,...,N
\end{aligned}$$
where $ x_i^{mean} $ is the mean value of $ x_i $, and $ x_i^{std} $ is the standard deviation of $ x_i $.
Data Augmentation {#S:6}
=================
Data augmentation, a common step in CV and NLP, might be important to make the training datasets more diverse and alleviate the learning difficulties caused by small sample problems. However, data augmentation for intelligent fault diagnosis has not been investigated in depth. It is also worth mentioning that the key challenge for data augmentation is to create the label-corrected samples from existing samples, and this procedure mainly depends on the domain knowledge. However, it is difficult to determine whether the generated samples are label-corrected. So, this paper provides some data augmentation techniques to reduce the concerns of other scholars. In addition, these data augmentation strategies are only a simple test and their applications still need to be studied in depth.
One Dimension Input Augmentation
--------------------------------
**RandomAddGaussian**: this strategy randomly adds Gaussian noise into the input signal formulated as follows: $$\begin{aligned}
\label{eq9}
x := x + n
\end{aligned}$$ where $ x $ is the 1D input signal, and $ n $ is generated by Gaussian distribution $\mathcal N (0, 0.01) $.
**RandomScale**: this strategy randomly multiplies the input signal with a random factor which is formulated as follows: $$\begin{aligned}
\label{eq10}
x := \sigma * x
\end{aligned}$$ where $ x $ is the 1D input signal, and $ \sigma $ is a scaler following the distribution $ \mathcal N (1, 0.01) $.
**RandomStretch**: this strategy resamples the signal into a random proportion and ensures the equal length by nulling and truncating.
**RandomCrop**: this strategy randomly covers partial signals which is formulated as follows: $$\begin{aligned}
\label{eq11}
x := mask * x
\end{aligned}$$ where $ x $ is the 1D input signal, and $ mask $ is the binary sequence whose subsequence of random position is zero. In this paper the length of subsequence is equal to 10.
Two Dimension Input Augmentation
--------------------------------
**RandomScale**: this strategy randomly multiplies the input signal with a random factor which is formulated as follows: $$\begin{aligned}
\label{eq12}
x := \sigma * x
\end{aligned}$$ where $ x $ is the 2D input signal, and $ \sigma $ is a scaler following the distribution $ \mathcal N (1, 0.01) $.
**RandomCrop**: this strategy randomly covers partial signals which is formulated as follows: $$\begin{aligned}
\label{eq13}
x := mask * x
\end{aligned}$$ where $ x $ is the 2D input signal, and $ mask $ is the binary sequence whose subsequence of random position is zero. In this paper the length of subsequence is equal to 20.
Due to the fact that 2D inputs in intelligent fault diagnosis often have clear physical meanings, data augmentation methods in the image processing are not suitable to directly transfer to intelligent fault diagnosis.
Data Split {#S:7}
==========
One common practice of data split in intelligent fault diagnosis is the random split strategy, and the diagram of this strategy is shown in [Fig. \[Fig-data-split0\]]{}. From this diagram, it can be observed that we stress the preprocessing step without overlap due to the fact that if the sample preparation process exists any overlap for samples, the evaluation of classification algorithms may have test leakage (it is also worth mentioning that if users split the training set and the testing set from the beginning of the preprocessing step, then they can use any processing to simultaneously deal with the training and testing sets, as shown in [Fig. \[Fig-data-split1\]]{}). In addition, many papers confuse the validation (val) set and the testing set. The formal way is that the training set is further splited into the training set and the validation set for the model selection. [Fig. \[Fig-data-split0\]]{} shows the condition of 4-fold cross validation, and we often use the average accuracy of 4-fold cross validation to represent the generalization accuracy, if there is no testing set. In this paper, for testing convenience and time saving, we only use 1-fold validation and use the last epoch accuracy to represent the testing accuracy (we also list the maximum accuracy in the whole epochs for comparison). It is worth noting that some papers use the maximum accuracy of the validation set, and this strategy is also dangerous because the validation set is used to select the parameters accidentally.
\
\[-5pt\]
\
\[-5pt\]
For industrial data from rotating machinery, they are rarely random and are always sequential (they might contain trends or other temporal correlation). Therefore, it is more appropriate to split data according to time sequences (order split). The diagram of data split strategy according to time sequences is shown in [Fig. \[Fig-data-split2\]]{}. From this diagram, it can be observed that we split the training and testing sets with the time phase instead of splitting the data randomly. In addition, [Fig. \[Fig-data-split2\]]{} also shows the condition of 4-fold cross validation with time. In the following study, we compare the results of this strategy with the random split strategy using the last epoch accuracy and the maximum accuracy in the whole epochs.
\
\[-5pt\]
Evaluation Methodology {#S:8}
======================
Evaluation Metrics
------------------
It is a rather challenging task to evaluate the performance of intelligent fault diagnosis algorithms with suitable evaluation metrics. In intelligent fault diagnosis, it has three standard evaluation metrics, which have been widely used, including the overall accuracy, the average accuracy, and the confusion matrix. In this paper, we only use the overall accuracy to evaluate the performance of algorithms. The overall accuracy is defined as the number of correctly classified samples divided by the total number of samples. The average accuracy is defined as the average classification accuracy of each category. It should be noted that each class in our datasets has the same number of samples, so the value of the overall accuracy is equivalent to that of the average accuracy.
Since the performance of DL-based intelligent diagnosis algorithms fluctuates during the training process, to obtain reliable results and show the best overall accuracy that the model can achieve, we repeated each experiment five times. Four indicators are used to assess the performance of models, including the mean and maximum values of the overall accuracy obtained by the last epoch (the accuracy in the last epoch can represent the real accuracy without any test leakage), and the mean and maximum values of the maximal overall accuracy (in fact, when we use the maximal accuracy, we also use the testing set to choose the best model). For simplicity, they can be denoted as Last-Mean, Last-Max, Best-Mean, and Best-Max.
Experimental Setting
--------------------
In preparation stage, we use two strategies, including random split and order split, to divide the dataset into training and testing sets. For random split, a sliding window is used to truncate the vibration signal without any overlap and each data sample contains 1024 points. After the preparation, we randomly take 80% of samples as the training set and 20% of samples as the testing set. For order split, the former 80% of time series is taken as the time series for dividing the training set, and then the last 20% is taken for dividing the testing set. Then, in two time series, a sliding window is used to truncate the vibration signal without any overlap, and each sample contains 1024 points.
In order to verify how input types, data normalization methods, and data split methods affect the performance of models, we set up three configurations of experiments (shown in Table \[Tab\_E1\], Table \[Tab\_E2\] and Table \[Tab\_E3\].) for each dataset. In model training, we use Adam as the optimizer and the softmax cross-entropy as the loss function. The learning rate and the batch size of each experiment are set to 0.001 and 64, respectively. Each model is trained for 100 epochs, and during the training procedure, model training and model testing are alternated. In addition, all the experiments are executed under Window 10 and Pytorch 1.1 through running on a computer with an Intel Core i7-9700K, GeForce RTX 2080Ti, and 16G RAM.
\[Tab\_E1\]
[c]{}
\[Tab\_E2\]
[c]{}
\[Tab\_E3\]
[c]{}
Evaluation Results {#S:9}
==================
In this section, we will discuss the experimental results in depth. Complete results are shown in **Appendix A.** (the accuracies which are larger than 95% are bold.)
Results of Datasets
-------------------
From the results, it can be observed that all datasets except the XJTU-SY dataset have some accuracies exceeding 95%. In addition, the accuracies of CWRU and SEU datasets can reach to 100%. The accuracy of XJTU-SY is much lower than others in all conditions, because XJTU-SY is a run-to-failure dataset and we only use the data at the end of the whole process (it may be hard to find the fail point easily and accurately). Besides, the diagnostic difficulty of seven datasets can be ranked according to the sum of the best accuracy and the worst accuracy in one certain condition. Results used for sorting come from samples with the randomly split strategy processed by FFT, the Z-score normalization, and data augmentation. As shown in [Fig. \[Datasets\]]{}, we can split the datasets into four levels of difficulty.
\
\[-5pt\]
Results of Input Types
----------------------
In all datasets, the frequency domain input always can achieve the highest accuracy followed by the time-frequency domain input since in the frequency domain, the noise is spread over the full frequency band and the fault information is much easier to be distinguished than that in the time domain. It is also worth mentioning that according to the computational load of CWT, we use the short length of samples to perform CWT and then upsample the wavelet coefficients. These steps may degrade the classification accuracies of CWT.
Results of Models
-----------------
From the results, it can be observed that models, especially ResNet18 belonging to CNN, can achieve the best accuracy in some datasets including CWRU, JNU, PU, and SEU. However, for MFPT, UoC, and XJTU-SY, models belonging to AE can perform better than other models. This phenomenon may be caused by the size of the datasets and the overfitting problem. Therefore, not every dataset can get better results using a more complex model.
Results of Data Normalization
-----------------------------
It is hard to conclude which data normalization method is the best one, and from the results, it can be observed that accuracies of different data normalization methods also depend on the used models and datasets. In general, Z-score normalization can make the models achieve the best accuracy.
Results of Data Augmentation
----------------------------
According to the results, we can conclude that when the accuracies of datasets are already high enough, data augmentation methods may slightly degrade the performance because models have already fitted original datasets well. More augmentation methods may change the distribution of original data and make the learning process harder. However, when the accuracies of datasets are not very high, data augmentation methods improve the performance of models, especially for the time domain input. It should be noted that data augmentation methods designed in this paper may be more suitable for the time domain input. Therefore, researchers can design other various data augmentation methods for their specific inputs.
Results of Splitting Data
-------------------------
When the datasets are easy to deal with (CWRU and SEU), the results between random split and order split are similar. However, the accuracies of some datasets (PU and UoC) decrease sharply under the order split. What we should pay more attention to is that whether randomly splitting these datasets has the risk of test leakage. Maybe it is more suitable for splitting the datasets according to time sequences to verify the performance of designed models.
Discussion {#S:10}
==========
Although intelligent diagnosis algorithms can achieve high classification accuracies in many datasets, there are still many issues that need to be discussed. In this paper, we further discuss the following five issues including class imbalance, generalization ability, interpretability, few-shot learning, and model selection.
Class Imbalance
---------------
During operation of the rotating machinery, most of measured signals are in the normal state, and only a few of them are in the fault state. Fault modes often have different probabilities of happening. Meanwhile, working conditions also have different probabilities of happening. For example, the samples generated by the helicopter hover, cruise, and other flight conditions are naturally unbalanced under the influence of the flight time, and thus the classification of helicopter flight conditions is a typical class imbalance issue. Therefore, the class imbalance issue will occur when using intelligent algorithms in real applications. Recently, although some researchers have published some related papers using traditional imbalanced learning methods [@zhang2018imbalanced] or generative adversarial networks [@mao2019imbalanced] to solve this problem, these studies are far from enough. In this paper, PU Bearing Datasets are used to simulate the class imbalance issue. In this experiment, we adopt ResNet18 as the experimental model and only use two kinds of input types (the time domain input and the frequency domain input). Besides, data augmentation methods are used and the normalization method is the Z-score normalization, while the dataset is randomly split. Three groups of datasets with different imbalance ratios are constructed, which are shown in Table \[Tab\_CI\].
\[Tab\_CI\]
------ -------- ----------------- -------- ------------
Testing samples
Group1 Group2 Group3 Group1/2/3
KA04 125 125 125 125
KA15 125 75 50 125
KA16 125 75 50 125
KA22 125 75 50 125
KA30 125 37 25 125
KB23 125 37 25 125
KB24 125 37 25 125
KB27 125 25 6 125
KI14 125 25 6 125
KI16 125 25 6 125
KI17 125 12 2 125
KI18 125 12 2 125
KI21 125 12 2 125
------ -------- ----------------- -------- ------------
: Number of samples in three groups of imbalanced datasets
As shown in Table \[Tab\_CI\], three datasets (Group1, Group2, and Group3) are constituted with different imbalanced ratios. Group1 is a balanced dataset, and there is no imbalance for each state. In real applications, it is almost impossible to let the number of data samples be the same. We reduce the training samples of some fault modes in Group1 to construct Group2, and then the imbalanced classification is simulated. In Group3, the imbalance ratio between fault modes increases further. Group2 can be considered as a moderately imbalanced dataset, while Group3 can be considered as a highly imbalanced dataset.
Experimental results are shown in [Fig. \[FIG1\]]{}, and it can be observed that the overall accuracy in Group3 is much lower than that of Group1, which indicates that the class imbalance will greatly degrade the performance of models. To address the problem of class imbalance, data-level methods and classifier-level methods can be used [@buda2018systematic]. Oversampling and undersampling methods are the most commonly used data-level methods in DL and some methods for generating samples based on generative adversarial networks (GAN) have also been studied recently. For the classifier-level methods, thresholding-based methods are applied in the test phase to adjust the decision threshold of tthe classifier. Besides, cost-sensitive learning methods assign different weights to different classes to avoid the suppression of categories with a small number of samples. In the field of fault diagnosis, other methods based on physical meanings and fault attention need to be explored.
\
\[-10pt\]
Generalization ability
----------------------
Many of the existing intelligent algorithms perform very well on one working condition, but the diagnostic performance tends to drop significantly on another working condition, and here, we call it the generalization problem. Recently, many researchers have used algorithms based on transfer learning strategies to solve this problem. To illustrate the weak generalization ability of the intelligent diagnosis algorithms, experiments are also carried out on the PU bearing dataset. Experiments use the data under three working conditions (N15\_M07\_F10, N09\_M07\_F10, N15\_M01\_F10). In these experiments, data under one working condition are used to train models, and data under another working condition are used to test the performance. A total of six groups of experiments are performed, and the detailed information is shown in Table \[Tab\_GP\].
\[Tab\_GP\]
Group Data for training Data for testing
-------- ------------------- ------------------
Group1 N15\_M07\_F10 N09\_M07\_F10
Group2 N15\_M07\_F10 N15\_M01\_F10
Group3 N09\_M07\_F10 N15\_M07\_F10
Group4 N09\_M07\_F10 N15\_M01\_F10
Group5 N15\_M01\_F10 N15\_M07\_F10
Group6 N15\_M01\_F10 N09\_M07\_F10
: Training data and testing data for each experiment
The experimental results are shown in [Fig. \[FIG2\]]{}. It can be concluded that in most cases, intelligent diagnosis algorithms trained on one working condition cannot perform well on another working condition, which means the generalization ability of algorithms is insufficient. In general, we expect our algorithms can adapt to the changes in working conditions or measurement situations since these changes occur frequently in real applications. Therefore, studies still need to be done on how to transfer the trained algorithms to different working conditions effectively.
Two excellent review papers [@zheng2019cross; @yan2019knowledge] and other applications [@han2019deep; @han2019learning] published recently pointed out several potential research directions which could be considered and studied further to improve the generalization ability.
\
\[-10pt\]
Interpretability
----------------
Although intelligent diagnosis algorithms can achieve high diagnostic accuracy in their tasks, the interpretability of these models is often insufficient and these black box models will generate high risk results [@rudin2019stop], which greatly reduces the reliability of results and limits their applications. Actually, some papers in intelligent fault diagnosis have noted this problem and attempted to propose some interpretable model [@li2019understanding; @li2019waveletkernelnet].
To point out that the intelligent diagnostic algorithm lacks interpretability, we perform three sets of experiments on the PU bearing dataset, and the datasets are shown in Table \[Tab\_IP\]. In each set of experiments, we use two different sets of data, which have the same fault mode and are acquired under the same condition.
\[Tab\_IP\]
Group Bearing code Training samples Testing samples
------- -------------- ------------------ -----------------
KA03 200 50
KA06 200 50
KA08 200 50
KA09 200 50
KI07 200 50
KI08 200 50
: The bearing code and the number of samples used in each experiment
The results, in which intelligent algorithms can get very high diagnosis accuracies in each set of experiments, are shown in [Fig. \[FIG3\]]{}. Nevertheless, for each binary classification task, since the fault mode and the working condition at the time of acquisition are same between two classes, theoretically, methods should not be able to achieve such high accuracy. These expected results are exactly contrary to those of the experiment, which shows that models only learn the discrimination of different collection points and do not learn how to extract the essential characteristics of fault signals. Therefore, it is very important to figure out whether models can learn essential fault characteristics or just classify the different conditions of collected signals.
\
\[-10pt\]
According to the development of interpretability in the computer science, we may be able to study the interpretability of DL-based models from the following aspects: (1) visualize the results of neurons to analyze the attention points of models [@zeiler2014visualizing]; (2) add physical constraints to the loss function [@tang2018regularized] to meet specific needs of fault feature extraction; (3) add prior knowledge to network structures and convolutions [@ravanelli2018interpretable] or unroll the existing optimization algorithms [@gregor2010learning] to extract corresponding fault features.
Few-Shot Learning
-----------------
The rapid development of deep learning is associated with the big data era. However, in intelligent diagnosis, the amount of data is far from big data because of preciousness of fault data and the high cost of fault simulation experiments, especially for the key components. To manifest the influence of the number of samples on the classification accuracy, we use the PU bearing dataset to design the few-shot training pattern with six groups of different sample numbers in each class for training.
Results of the time domain input and the frequency domain input are shown in [Fig. \[FIG4\]]{}. It is shown that with the decrease of the sample number, the accuracy decreases sharply. As shown in [Fig. \[FIG4\]]{}, for the time domain input, the Best-Max accuracy decreases from 91.46% to 20.39% as the sample number decreases from 100 to 1. Meanwhile, the Best-Max accuracy decreases from 97.73% to 29.67% as the sample number decreases from 100 to 1 with the frequency domain input.
Although the accuracy can be increased after using FFT, it is still too low to be accepted when the number of samples is extremely small. It is necessary to develop methods based on few-shot learning to copy with the application scenarios with limited samples.
\
\[-10pt\]
Many DL-based few-shot learning models have been proposed in recent years, most of these methods adopt a meta-learning paradigm by training networks with a large amount of tasks, which means that big data in other related fields are necessary for these methods. In the field of fault diagnosis, there is no relevant data with such a big size available, so methods embedding with physical mechanisms are required to address this problem effectively.
Model selection
---------------
For intelligent fault diagnosis, designing a neural network is not the final goal, and our task is applying the model to real industrial applications, while designing a neural network is only a small part of our task. However, to achieve a good effect, we have to spend considerable time and energy on designing the corresponding networks. Because building a neural network is an iterative process consisting of repeated trial and error, and the performance of models should be fed back to us to adjust models. The single trial and error cost multiplied by the number of trial and error can easily reach a huge cost. Besides, reducing this cost is also the partial purpose of this benchmark study which provides some guidelines to choose a baseline model.
Actually, there is another way called neural architecture search (NAS) [@elsken2019neural] to avoid the huge cost of trial and error. NAS can allow to design a neural network automatically through searching for a specific network based on a specific dataset. A limited search space of the network is first constructed according to the physical prior. After that, a neural network matching a specific dataset is sampled from the search space through reinforcement learning, the evolutionary algorithm or the gradient strategy. Besides, the whole construction process does not require manual participation, which greatly reduces the cost of building a neural network and allows us to focus on specific engineering applications.
Conclusion {#S:11}
==========
In this paper, we collect most of the publicly available datasets to evaluate the performance of MLP, AE, CNN, and RNN models from several perspectives. Based on the benchmark accuracies, we highlight some evaluation results which are very important for comparing or testing new models. First, not all datasets are suitable for comparing the classification effectiveness of the proposed methods since basic models can achieve very high accuracies on these datasets, like CWRU and SEU. Second, the frequency domain input can achieve the highest accuracy in all datasets, so researchers should first try to use the frequency domain as the input. Third, it is not necessary for CNN models to get the best results in all cases, and we should also consider the overfitting problem. Fourth, when the accuracies of datasets are not very high, data augmentation methods improve the performance of models, especially for the time domain input. Thus, more effective data augmentation methods need to be investigated. Finally, in some cases, maybe it is more suitable for splitting the datasets according to time sequences (order split) since random split may provide virtually high accuracies. It may be helpful to develop new models to take these evaluation results into consideration.
In addition, we release a code library for other researchers to test the performance of their own DL-based intelligent fault diagnosis models of these datasets. Through these works, we hope that the evaluation results and the code library can promote a better understanding of DL-based models, and provide a unified framework for generating more effective models. For further studies, we will focus on five listed issues (class imbalance, generalization ability, interpretability, few-shot learning, and model selection) to propose more customized works.
|
---
abstract: 'Some of the most interesting scenarios that can be studied in astrophysics, contain fluids and plasma moving under the influence of strong gravitational fields. To study these problems it is required to implement numerical algorithms robust enough to deal with the equations describing such scenarios, which usually involve hydrodynamical shocks. It is traditional that the first problem a student willing to develop research in this area is to numerically solve the one dimensional Riemann problem, both Newtonian and relativistic. Even a more basic requirement is the construction of the exact solution to this problem in order to verify that the numerical implementations are correct. We describe in this paper the construction of the exact solution and a detailed procedure of its implementation.'
author:
- 'F. D. Lora-Clavijo, J. P. Cruz-Pérez, F. S. Guzmán, J. A. González'
title: Exact solution of the 1D Riemann problem in Newtonian and relativistic hydrodynamics
---
Introduction {#sec:introduction}
============
High energy astrophysics has become one of the most important subjects in astrophysics because it involves phenomena associated to high energy radiation, modeled with sources traveling at high speeds or sources under the influence of strong gravitational fields like those due to black holes or compact stars. Current models involve a hydrodynamical description of the luminous source, and therefore hydrodynamical equations have to be solved.
In this scenario, due to the complexity of the system of equations it is required to apply numerical methods able to control the physical discontinuities arising during the evolution of initial configurations, for example the evolution of the front shock in a supernova explosion, the front shock of a jet propagating in space, the edges of an accretion disk, or any shock formed during a violent process. The study of these systems involve the implementation of advanced numerical methods, being two of the most efficient and robust ones the high resolution shock capturing methods and smooth particle hydrodynamics which are representative of Eulerian and Lagrangian descriptions of hydrodynamics, each one with pros and cons.
It is traditional that a first step to evaluate how appropriate the implementation of a numerical method is, requires the comparison of numerical results with an exact solution in a simple situation. The simplest problem in hydrodynamics is the 1D Riemann problem. This is an excellent test case because it has an exact solution in the Newtonian case (e.g. [@Toro]) and also in the relativistic regime [@MartiLR; @Marti], where codes dealing with high Lorentz factors are expected to work properly. From our experience we have found that the existent literature about the construction of the exact solution is not as explicit as it may be expected by students having their first contact with this subject. This is the reason why we present a paper that is very detailed in the construction and implementation of the solution. We focus on the solution of the problem and omit some of the mathematical background that is actually very well described in the literature.
The paper is organized as follows. In section \[sec:newtonian\] we present the Newtonian Riemann problem and how to implement it; in section \[sec:relativistic\] we present the exact solution to the relativistic case and how to implement it. Finally in section \[sec:final\] we present some final comments.
Riemann problem for the Newtonian Euler equations {#sec:newtonian}
=================================================
The Riemann problem is an initial value problem for a gas with discontinuous initial data, whose evolution is ruled by Euler’s equations. The set of Euler’s equations determine the evolution of the density of gas, its velocity field and either its pressure or total energy. A comfortable way of writing such equations involves a flux balance form as follows
$$\partial_t {\bf u} + \partial_x {\bf F}({\bf u}) = 0 \label{eq:Euler}$$
where ${\bf u}=(u_1,u_2,u_3)^T=(\rho,\rho v,E)^T$ is a set of conservative variables and ${\bf F}$ is a flux vector, where $\rho$ is the mass density of the gas, $v$ its velocity and $E=\rho(\frac{1}{2}v^2 + \varepsilon)$, with $\varepsilon$ the specific internal energy of the gas. The enthalpy of the system is given by the expression $H=\frac{1}{2}v^2 + h$, where $h$ is the specific internal enthalpy given by $h=\varepsilon + p/\rho$, where $p$ is the pressure of the gas. The fluxes are explicitly in terms of the primitive variables $\rho,v,p$ and the conservative variables [@Toro]
$${\bf F}({\bf u}) = \left(
\begin{array}{c}
\rho v \\
\rho v^2 + p \\
v(E+p)
\end{array}
\right) =
\left(
\begin{array}{c}
u_2 \\
\frac{1}{2}(3-\Gamma)\frac{u_2^2}{u_1}+(\Gamma-1)u_3 \\
\Gamma\frac{u_2}{u_1}u_3-\frac{1}{2}(\Gamma-1)\frac{u_2^3}{u_1^2}
\end{array}
\right).
\nonumber$$
The initial data of the Riemann problem is defined as follows
$${\bf u} = \left\{
\begin{array}{ll}
{\bf u}_L, & x<x_0 \\
{\bf u}_R, & x >x_0,
\end{array}
\right.\nonumber$$
where ${\bf u}_L$ and ${\bf u}_R$ represent the values of the gas properties on a chamber at the left and at the right from an interface between the two states at $x=x_0$ that exists only at initial time.
The evolution of the initial data is described by the characteristic information of the system of equations, and this is why the properties of the Jacobian matrix are important. The Jacobian matrix of the system of equations is $A({\bf u})=\frac{\partial{\bf F}}{\partial{\bf u}}$ and explicitly reads
$${\bf A} = \left(
\begin{array}{ccc}
0 &1 & 0\\
\frac{1}{2}(\Gamma-3)v^2 & (3-\Gamma)v & \Gamma -1\\
(\Gamma -1)v^3 - \frac{\Gamma v E}{\rho} & \frac{\Gamma E}{\rho} - \frac{3}{2}(\Gamma -1)v^2 & \Gamma v\\
\end{array}
\right).
\nonumber$$
Its eigenvalues satisfy the condition $\lambda_1({\bf u}) < \lambda_2 ({\bf u}) < \lambda_3 ({\bf u})$ and are given by
$$\begin{aligned}
\lambda_1 &=& v-a\label{eq:lambda_minus} \\
\lambda_2 &=& v\label{eq:lambda_0} \\
\lambda_3 &=& v+a\label{eq:lambda_plus}\end{aligned}$$
where $a=\sqrt{\frac{\partial p}{\partial \rho}}|_{s}$ is the speed of sound in the gas, which depends on the equation of state. For the ideal gas $p=\rho \varepsilon (\Gamma-1)$, where $\Gamma$ is the ratio between the specific heats at constant pressure and volume $\Gamma=c_p / c_v$, the speed of sound is $a=\sqrt{\frac{\Gamma p} {\rho}}$. On the other hand, the eigenvectors of the Jacobian matrix read
$${\bf r}_1 = \left(
\begin{array}{c}
1\\ v-a \\H-av
\end{array}
\right), ~
{\bf r}_2 = \left(
\begin{array}{c}
1\\ v \\\frac{1}{2}v^2
\end{array}
\right), ~
{\bf r}_3 = \left(
\begin{array}{c}
1\\v+a\\H+av
\end{array}
\right).
\nonumber$$
The eigenvectors ${\bf r}_1,~{\bf r}_2,~{\bf r}_3$ are classified in the following way:
- they are called genuinely non-linear when satisfy the condition $\nabla_u \lambda_i \cdot {\bf r}_i({\bf u}) \ne 0$.
- and linearly degenerate when $\nabla_u \lambda_i \cdot {\bf r}_{i}({\bf u}) = 0$.
It happens that ${\bf r}_2$ is linearly degenerate and represents a contact discontinuity, however the other two are genuinely non-linear.
Depending on the particular region of the solution we will use both the Riemann invariant conditions for rarefaction waves and the Rankine Hugoniot conditions for shocks and contact discontinuities. The Riemann invariants are based on the self-similarity property of the solution in some regions, in the sense that the solution depends on the spatial and time coordinates $(x,t)$ with the combination $(x-x_0)/t$; it can be seen that such behavior implies that the following conditions hold [@LeVeque]
$$\frac{d u_1}{{\bf r}^{i}_{1}} = \frac{d u_2}{{\bf r}^{i}_{2}} = \frac{d u_3}{{\bf r}^{i}_{3}}
\label{eq:riemann_invariants}$$
where $i$ indicates the component of a given eigenvector. On the other hand, the Rankine Hugoniot conditions relate states on both sides of a shock wave or a contact discontinuity
$$\Delta {\bf F} = V \Delta {\bf u},\label{eq:RHConditions}$$
which are simply jump conditions, where $\Delta {\bf u}$ is the size of the discontinuity in the variables, $V$ is the velocity of either the contact discontinuity or shock and $\Delta {\bf F}$ is the change of the flux across the discontinuity.
Contact discontinuity waves
---------------------------
The contact discontinuity is described by the second eigenvector and evolves with velocity $\lambda_2$. Let us then analyze the second eigenvector. In this case the Riemann invariant conditions read
$$\frac{d\rho}{1} = \frac{d(\rho v)}{v} = \frac{dE}{\frac{1}{2}v^2}.
\nonumber$$
These relations implies that $d(\rho \varepsilon)=dv=0$, further implying that $p=constant$ and $v=constant$ across the contact wave. In order to relate the two sides from the contact discontinuity we use the Rankine-Hugoniot conditions, which are given by
$$\begin{aligned}
\rho_L v_L- \rho_R v_R &=& V_c (\rho_L-\rho_R), \label{eq:1st_contact_newtonian}\\
\rho_L v^2_L + p^2_L - \rho_R v^2_R + p^2_R &=& V_c (\rho_L v_L - \rho_R v_R), \label{eq:2nd_contact_newtonian}\\
v_L (E_L +p_L ) -v_R(E_R+p_R) &=& V_c (v_L(E_L+p_L) \nonumber \\
&-&v_R(E_R+p_R)). \label{eq:3rd_contact_newtonian}\end{aligned}$$
Here $V_c$ is the velocity of propagation of the contact discontinuity.
The discontinuity travels at speed $\lambda^0=v$ therefore the $V_c=v$. For this reason from equation (\[eq:1st\_contact\_newtonian\]) follows that $v_L=v_R=V_c$. As a consequence of this, equation (\[eq:2nd\_contact\_newtonian\]) gives the condition $p_L=p_R$, which implies (\[eq:3rd\_contact\_newtonian\]) is satisfied. Notice that no condition on the density arises, which allows the density to be discontinuous.
Rarefaction waves
-----------------
At this point we do not know the nature of waves 1 and 3, and we can assume they may be rarefaction waves. Once again we use the Riemann invariant equalities, which for vectors 1 and 3 read
$$\begin{aligned}
\frac{d \rho}{1} &=& \frac{d(\rho v)}{v-a} = \frac{dE}{H-av}, \nonumber\\
\frac{d \rho}{1} &=& \frac{d(\rho v)}{v+a} = \frac{dE}{H+av}. \nonumber\end{aligned}$$
Manipulation of these equalities results in the following equations
$$\begin{aligned}
\frac{d\rho}{dv} &=& -\frac{\rho}{a}~~~ {\rm for} ~\lambda_1,\label{eq:rar_ri_1}\\
\frac{d\rho}{dv} &=& \frac{\rho}{a}~~~ {\rm for} ~\lambda_3,\label{eq:rar_ri_3}\\
\frac{d\varepsilon}{d\rho} &=& \frac{p}{\rho^2}~~~ {\rm for ~ both}~ \lambda_1 ~{\rm and}~
\lambda_3.\label{eq:rar_ri_eps}\end{aligned}$$
The next step is to integrate these equations assuming an equation of state, in our case the ideal gas. From (\[eq:rar\_ri\_eps\]) we obtain
$$p = K\rho^{\Gamma}\label{eq:isentropic}$$
where $K$ is a constant. A rarefaction process is isentropic (unlike a shock), and therefore the states at the left and at the right from the wave obey (\[eq:isentropic\]) with the same constant $K$.
Using this expression for $p$ in the speed of sound we have $a=\sqrt{K\Gamma \rho^{\Gamma-1}}=\sqrt{\Gamma p /\rho}$, which substituted into (\[eq:rar\_ri\_1\],\[eq:rar\_ri\_3\]) results in
$$v = \pm \int \sqrt{K\Gamma\rho^{\Gamma-3}} d\rho +k = \pm \frac{2a}{\Gamma-1}+k,\label{eq:vel_rarefaction}$$
where $+$ stands for the wave moving to the right (the case of $\lambda_3$ and ${\bf r}_3$ corresponding to a rarefaction wave) and $-$ when moving to the left (the case of $\lambda_1$ and ${\bf r}_1$ corresponding to a rarefaction wave), where $k$ is an integration constant and therefore the velocity is constant as well. This property allows us to set relations between the velocity of the gas on the state at the left and at the right from the rarefaction wave, explicitly there are two possible cases:
- When the wave is moving to the left, condition (\[eq:vel\_rarefaction\]) implies that
$$v_L + \frac{2a_L}{\Gamma-1} = v_R + \frac{2a_R}{\Gamma-1}.\label{eq:rar_vel_left}$$
- When the wave is moving to the right, condition (\[eq:vel\_rarefaction\]) implies
$$v_L - \frac{2a_L}{\Gamma-1} = v_R - \frac{2a_R}{\Gamma-1}\label{eq:rar_vel_right}$$
When the wave is moving to the left, we assume information from the left state is available and we look for expression of the variables on the state to the right from the wave. For the velocity of the fluid at the right state we then have from (\[eq:rar\_vel\_left\])
$$v_R = v_L - \frac{2}{\Gamma-1}[a_R-a_L],$$
now considering that the speed of sound on both sides obeys $a=\sqrt{K\Gamma \rho^{\Gamma-1}}=\sqrt{\Gamma p /\rho}$ (see (\[eq:isentropic\]))
$$a_R = a_L \left( \frac{p_R}{p_L} \right)^\frac{\Gamma-1}{2\Gamma},\label{eq:sound_rar_left}$$
a useful expression for $v_R$ arises
$$v_R = v_L - \frac{2a_L}{\Gamma-1} \left[ \left( \frac{p_R}{p_L}\right)^{\frac{\Gamma-1}{2\Gamma}}-1\right].
\label{eq:R_L}$$
The only unknown quantity is $p_R$.
On the other hand, when the wave is moving to the right we assume we know the information at the state at the right from the wave, then we search for expressions of the variables on the state at the left. For the velocity we find according to (\[eq:rar\_vel\_right\])
$$v_L = v_R - \frac{2}{\Gamma-1}[a_R-a_L],$$
and the speed of sound on both sides obeys
$$\label{eq:aaL}
a_L = a_R \left( \frac{p_L}{p_R} \right)^\frac{\Gamma-1}{2\Gamma},$$
which finally implies
$$v_L = v_R - \frac{2a_R}{\Gamma-1} \left[ 1-\left( \frac{p_L}{p_R}\right)^{\frac{\Gamma-1}{2\Gamma}}\right].
\label{eq:v_L_rarefactionR}$$
The only unknown quantity in this case is $p_L$.
The rarefaction zone has a finite size, bounded by two curves, the tail and the head. The head of the wave is the line of the front of the wave and the tail is the boundary left behind the wave. The region in the middle is called the fan of the rarefaction wave.
The velocity of all the particles between the head and the tail obeys the following expression
$$\label{eq:vfan}
\frac{x-x_0}{t} = v \pm a,$$
where + is used when the wave is propagating to the right and the $-$ when it is moving to the left. Then, when the wave is moving to the left, using this expression we have $a_R = v_R - (x-x_0)/t$, which substituted into (\[eq:R\_L\]) provides the following expression for the velocity of the gas on the state at the right from the wave is
$$v_R = \frac{2}{\Gamma+1}\left[ a_L + \frac{1}{2} (\Gamma-1) v_L + \frac{x-x_0}{t}\right].\label{eq:vR_rarL}$$
Then it is possible to calculate the pressure and density as well. Substituting (\[eq:vR\_rarL\]) into (\[eq:rar\_vel\_left\]) and (\[eq:sound\_rar\_left\]) we obtain an expression for the pressure also at the state to the right
$$p_R = p_L \left[ \frac{2}{\Gamma + 1} + \frac{\Gamma-1}{a_L (\Gamma+1)} \left( v_L - \frac{x-x_0}{t} \right)\right]^{\frac{2\Gamma}{\Gamma-1}}.\label{eq:pR_rarL}$$
Now, using this into (\[eq:isentropic\]) implies the expression for the density
$$\rho_R = \rho_L \left[ \frac{2}{\Gamma + 1} + \frac{\Gamma-1}{a_L (\Gamma+1)} \left( v_L - \frac{x-x_0}{t} \right)\right]^{\frac{2}{\Gamma-1}}.\label{eq:rhoR_rarL}$$
Then finally we have expressions for the velocity, pressure and density on the state at the right when the wave is moving to the left.
Similarly when the wave is moving to the right we have from (\[eq:vfan\]) that $a_L=v_L + (x-x_0)/t$, which substituted into (\[eq:v\_L\_rarefactionR\]) implies the following for the velocity on the state at the left from the wave
$$v_L = \frac{2}{\Gamma+1}\left[ -a_R + \frac{1}{2} (\Gamma-1) v_R + \frac{x-x_0}{t}\right].\label{eq:vL_rarR}$$
In order to obtain the expressions for the pressure and the density, we substitute this last expressions into (\[eq:rar\_vel\_right\]) in order to relate the speeds of sound, and then using (\[eq:aaL\]) we finally obtain the expression for the pressure at the left
$$p_L = p_R \left[ \frac{2}{\Gamma + 1} - \frac{\Gamma-1}{a_R (\Gamma+1)} \left( v_R - \frac{x-x_0}{t} \right)\right]^{\frac{2\Gamma}{\Gamma-1}}.\label{eq:pL_rarR}$$
Finally using the equation (\[eq:isentropic\]) we obtain the density
$$\rho_L = \rho_R \left[ \frac{2}{\Gamma + 1} - \frac{\Gamma-1}{a_R (\Gamma+1)} \left( v_R - \frac{x-x_0}{t} \right)\right]^{\frac{2}{\Gamma-1}}.\label{eq:rhoL_rarR}$$
In this way we have relations between the variables on to the state at the left and at the right from a rarefaction wave. These relations will be useful when solving the Riemann problem.
Shock waves
-----------
Similar to the previous case, the shock can move either to the right (if $\lambda_3$ and ${\bf r}_3$ correspond to a shock wave) or to the left (if $\lambda_1$ and ${\bf r}_1$ correspond to a shock wave), and for each of the two cases there is known and unknown information. When a shock is moving to the right one is expected to have information of the state at the right from the shock and conversely, when the shock is moving to the left one accounts with information of the state at the left.
Shocks require the use of Rankine Hugoniot conditions (\[eq:RHConditions\]). We express these conditions in terms of the primitive variables as follows
$$\begin{aligned}
\rho_L v_L - \rho_R v_R &=& S (\rho_L - \rho_R),\nonumber\\
\rho_L v_{L}^{2} + p_L - \rho_R v_{R}^{2} - p_R &=& S (\rho_L v_L - \rho_R v_R),\nonumber\\
v_L (E_L +p_L) - v_R(E_R +p_R) &=& S (E_L - E_R),\nonumber\end{aligned}$$
where $S$ is the speed of the wave, which may take the values $v-a$ or $v+a$ depending on whether the wave moves to the left or to the right respectively. Manipulating these equations one gets
$$\begin{aligned}
\rho_L \hat{v}_L &=& \rho_R \hat{v}_R,\label{eq:rh_1}\\
\rho_L \hat{v}_L^2 + p_L &=& \rho_R \hat{v}_R^2 + p_R,\label{eq:rh_2}\\
\hat{v}_L(\hat{E}_L + p_L) &=& \hat{v}_R(\hat{E}_R + p_R), \label{eq:rh_3}\end{aligned}$$
where $\hat{v}_L=v_L - S$, $\hat{v}_R=v_R - S$ are velocities in the rest frame of the shock and $\hat{E}_L=\rho_L\left(\frac{1}{2}\hat{v}_L^2 + \varepsilon_L \right)$ and $\hat{E}_R=\rho_R\left(\frac{1}{2}\hat{v}_R^2 + \varepsilon_R \right)$. These expressions correspond to the Rankine Hugoniot jump conditions measured by an observer located in the rest frame of the shock wave.
From equation (\[eq:rh\_1\]), we introduce the mass flux definition
$$j = \rho_L \hat{v}_L = \rho_R\hat{v}_R. \label{eq:MassFlux}$$
Then, from equation (\[eq:rh\_2\]) and the mass flux definition before mentioned , we can get an expression for $j$, which is given by
$$j=-\frac{p_R-p_L}{\hat{v}_R-\hat{v}_L}=-\frac{p_R-p_L}{v_R-v_L}, \label{eq:MassFlux2}$$
which is a consequence of $j$ being invariant under Galilean transformations. Considering the shock is moving to the left, we would be interested in constructing the variables on the state at the right from the shock and we can start with the velocity, which can be written as
$$\label{eq:thefantastic}
v_R = v_L-\frac{p_R-p_L}{j} .$$
Now, in order to express the velocity in terms of the pressure and the variables of the state at the left from the shock, we can rewrite (\[eq:MassFlux\]) as follows
$$\label{eq:pareja}
v_R-S=\frac{j}{\rho_R}, ~~~ v_L-S=\frac{j}{\rho_L}.$$
Thus, substituting this into (\[eq:MassFlux2\]) we obtain
$$\label{eq:j2}
j^2=-\frac{p_R-p_L}{\frac{1}{\rho_R}-\frac{1}{\rho_L}}.$$
On the other hand, using equation (\[eq:rh\_3\]) and the expression for the specific internal enthalpy $h$ we can easily get the following expression for the difference of internal specific enthalpies
$$\label{eq:hrmhl}
h_R-h_L=\frac{1}{2}\left[ \hat{v}_L^2-\hat{v}_R^2 \right],$$
where $h_L=\varepsilon_L+p_L/\rho_L$ and $h_R=\varepsilon_R+p_R/\rho_R$. Now, from equations (\[eq:rh\_1\]) and (\[eq:rh\_2\]) we give expressions for the velocitites measured by the observer located in the rest frame of the shock wave
$$\begin{aligned}
\hat{v}_R^2 &=& \frac{\rho_L}{\rho_R} \frac{p_L-p_R}{\rho_L-\rho_R}, \nonumber \\
\hat{v}_L^2 &=& \frac{\rho_R}{\rho_L} \frac{p_L-p_R}{\rho_L-\rho_R}. \nonumber\end{aligned}$$
With the substitution of these last equations into (\[eq:hrmhl\]) and considering the definitions for the specific internal enthalpy mentioned above, we obtain
$$\varepsilon_R-\varepsilon_L= \frac{1}{2} \frac{(p_L+p_R)(\rho_R-\rho_L)}{\rho_L \rho_R}.
\nonumber$$
Assuming the gas obeys an ideal equation of state we get an expression for the density as follows
$$\frac{\rho_R}{\rho_L}=\frac{p_L(\Gamma-1)+p_R(\Gamma+1)}{p_R(\Gamma-1)+p_L(\Gamma+1)}.
\label{eq:rhoR_shockL}$$
Notice that this expression relates the density among the two sides from the shock. Now, substituting this expression into (\[eq:j2\]) we obtain
$$\label{eq:pareja2}
j^2=\frac{p_R+B_L}{A_L}, ~~~A_L=\frac{2}{(\Gamma+1)\rho_L}, ~~~B_L=\frac{\Gamma-1}{\Gamma+1} p_L.$$
Thus, the expression for the velocity (\[eq:thefantastic\]) can be written as follows
$$v_R=v_L-(p_R-p_L) \sqrt{\frac{A_L}{p_R+B_L}}. \label{eq:vR_shockL}$$
From expression (\[eq:pareja\]) and using (\[eq:pareja2\]) we express the shock velocity as follows
$$S = v_L - \sqrt{\frac{p_R(\Gamma +1) + p_L (\Gamma -1)}{2\rho_L}}. \nonumber$$
Finally, using the sound speed expression $a_L=\sqrt{\frac{p_L \Gamma}{\rho_L}}$ we obtain the final expression for the shock velocity
$$S = v_L - a_L\sqrt{\frac{(\Gamma +1)p_R}{2p_L\Gamma}+\frac{\Gamma-1}{2\Gamma}} \label{eq:SL}.$$
Analogously, when the shock moves to the right, it is possible to construct the expressions for the variables for the state at the left from the shock
$$\begin{aligned}
v_L&=&v_R+(p_L-p_R) \sqrt{\frac{A_R}{p_L+B_R}}\label{eq:L_R},\\
\rho_L&=&\rho_R \frac{p_R(\Gamma-1)+p_L(\Gamma+1)}{p_L(\Gamma-1)+p_R(\Gamma+1)},\label{eq:rho_L_R}\\
S &=& v_R + a_R\sqrt{\frac{(\Gamma +1)p_L}{2p_R\Gamma}+\frac{\Gamma-1}{2\Gamma}}. \label{eq:SR}\end{aligned}$$
and we let this as an exercise to the reader.
Classical Riemann Problem
-------------------------
The Riemann problem is physically a tube filled with gas which is divided into two chambers separated by a removable membrane at $x=x_0$. At the initial time the membrane is removed and the gas begins to flow. Once the membrane is removed, the discontinuity decays into two elementary, non-linear waves that move in opposite directions.
Depending on the values of the thermodynamical variables in each chamber, four cases can occur. Considering the fluid is described on a one-dimensional spatial domain, rarefaction and shock waves can evolve toward the left or right from the location of the membrane.
In general the solution in all the cases can be studied in six following regions:
- Region 1: initial left state that has not been yet influenced by rarefaction or shock waves
- Region 2: wave traveling to the left (may be rarefaction or shock)
- Region 3: region between the wave moving to the left and the contact discontinuity, called region star-left
- Contact discontinuity
- Region 4: region between the contact discontinuity and the wave moving to the right, called region star-right
- Region 5: wave traveling to the right (may be rarefaction or shock)
- Region 6: initial right state that has not been yet influenced by rarefaction or shock waves
Regions 2 and 5 are special. If the wave propagating in such regions is a rarefaction wave the region involves a head-fan-tail structure, whereas if it is a shock the region becomes only a discontinuity. Counting from left to right on the spatial domain, the results can be reduced to the following four possible combinations of waves:
- rarefaction-shock
- shock-rarefaction
- rarefaction-rarefaction
- shock-shock
with a contact discontinuity between the two waves in all cases. It is worth noticing that these combinations can occur under a wide variety of possible combinations of the initial values of the thermodynamical variables. In this paper we illustrate each of these scenarios using particular sets of initial conditions.
### Case 1: Rarefaction-Shock
This case corresponds to the typical case used to test numerical codes, a test called the Sod’s shock tube problem [@SodShockTube]. A traditional set of initial values that produces this scenario corresponds to a gas with higher density and pressure in the left chamber than in the right chamber, and the velocity is set initially to zero in both.
A rarefaction wave travels into the high density region (moves to the left), whereas a shock moves into the low density region (moves to the right).
Summarizing, the problem then involves five regions only. Regions 1 correspond to the initial state to the left that has not been influenced by the evolution of the system. Region 2 corresponds to a rarefaction wave containing the head-fan-tail structure, region 3 and 4 are the left and right states separated by the contact discontinuity. Region 5 reduces to the shock. Finally region 6 is the initial state at the right chamber that has not been influenced by the evolution of the system.
The goal is to determine the state in all the regions using the relations between the thermodynamical quantities constructed before.
The starting point to construct the solution happens at the contact discontinuity, where the velocity and pressure obey the conditions $p_3=p_4=p^{*}$ and $v_3=v_4=v^{*}$.
Region 3 plays the role of the state at the right from the rarefaction wave and region 1 the state at the left. Then we can use (\[eq:R\_L\]) to obtain an expression for $v_3$
$$v_3 = v_1 - \frac{2a_1}{\Gamma-1}\left[ \left( \frac{p_3}{p_1} \right)^{\frac{\Gamma-1}{2\Gamma}} - 1\right].\label{eq:v_star_Raf_shock1}$$
On the other hand, region 4 plays the role of a state at the left from the shock wave and region 6 the role of the state at the right. Then we use (\[eq:L\_R\]) to calculate $v_4$:
$$v_4 = v_6 + (p_4 -p_6) \sqrt{\frac{A_6}{p_4+B_6}}.\label{eq:v_star_Raf_shock2}$$
where $A_6=2/\rho_6/(\Gamma+1)$ and $B_6=p_6(\Gamma-1)/(\Gamma+1)$. Given that $v_3=v_4=v^{*}$, equating both expressions one obtains a trascendental equation for $p^{*}$:
$$(p^{*} - p_6)\sqrt{\frac{A_6}{p^{*}+B_6}} + \frac{2a_1}{\Gamma-1}\left[ \left( \frac{p^{*}}{p_1} \right)^{\frac{\Gamma-1}{2\Gamma}} - 1\right]+ v_6-v_1 = 0. \label{eq:p_star_RarShock}$$
Unfortunately as far as we can tell, no exact solution is known for $p^{*}$, and then we proced to construct its solution numerically. Once this equation is solved, $p_3$ and $p_4$ are automatically known, and $v_3$ and $v_4$ can be calculated using (\[eq:v\_star\_Raf\_shock1\]) and (\[eq:v\_star\_Raf\_shock2\]) respectively.
Then, it is possible to calculate $\rho_3$ using (\[eq:isentropic\]) at both sides of the rarefaction zone, given $C$ is the same on both sides because it is an isentropic process:
$$\rho_3=\rho_1 \left( \frac{p_3}{p_1}\right)^{1/\Gamma}\label{eq:rho_3_Rar_Shock}$$
where now $p_1$, $\rho_1$ and $p_3$ are known. On the other hand one can also calculate $\rho_4$ using (\[eq:rho\_L\_R\])
$$\rho_4 = \rho_6 \left(\frac{p_6 (\Gamma-1)+p_4 (\Gamma+1)}{p_4(\Gamma-1) + p_6(\Gamma+1)}\right)\label{eq:rho_4_Rar_Shock}$$
also in terms of known information. With this information it is already possible to construct the solution in the whole domain. We explain how to do it region by region. A scheme of how the regions are distributed is shown in Fig. \[fig:regions\_RS\].
1. Region 1 is defined by the condition $x-x_0 < t V_{head}$, where $V_{head}$ is the velocity of the head of the rarefaction wave given by the characteristic value of the Jacobian matrix evaluated at the location next to the head from the left side, that is, considering (\[eq:lambda\_minus\]) $V_h = v_1-a_1$. The solution there is simply
$$\begin{aligned}
p_{exact} &=& p_1, \noindent \nonumber\\
v_{exact} &=& v_1, \noindent \nonumber\\
\rho_{exact} &=& \rho_1. \noindent \nonumber\end{aligned}$$
2. Region 2 is defined by the condition $t V_{head} < x-x_0 < t V_{tail}$, where $V_{tail}$ is the same characteristic value again, but this time evaluated at the tail curve, that is $V_{tail}=v_3-a_3$. This is the fan region for a rarefaction wave moving to the left, for which we simply use expressions (\[eq:vR\_rarL\],\[eq:pR\_rarL\],\[eq:rhoR\_rarL\]) that need only information from region 1 and obtain
$$\begin{aligned}
\rho_{exact} &=& \rho_1 \left[ \frac{2}{\Gamma + 1} + \frac{\Gamma-1}{a_1 (\Gamma+1)} \left( v_1 -\frac{x-x_0}{t} \right) \right]^{\frac{2}{\Gamma-1}},\nonumber\\
p_{exact} &=& p_1 \left[ \frac{2}{\Gamma + 1} + \frac{\Gamma-1}{a_1 (\Gamma+1)} \left( v_1 - \frac{x-x_0}{t} \right)\right]^{\frac{2\Gamma}{\Gamma-1}},\nonumber\\
v_{exact} &=& \frac{2}{\Gamma+1}\left[ a_1 + \frac{1}{2} (\Gamma-1) v_1 + \frac{x-x_0}{t}\right].\end{aligned}$$
3. Region 3 is defined by the condition $t V_{tail} < x-x_0 < t V_{contact}$, where $V_{contact}$ is the velocity of the contact discontinuity, which is the second eigenvalue (\[eq:lambda\_0\]) of the Jacobian matrix evaluated at this region, that is $V_{contact}= v_3=v_4$. The solution there finally reads
$$\begin{aligned}
p_{exact} &=& p_3, \noindent \nonumber\\
v_{exact} &=& v_3, \noindent \nonumber\\
\rho_{exact} &=& \rho_3. \noindent \nonumber\end{aligned}$$
4. Region 4 is defined by the condition $t V_{contact}< x-x_0 < t V_{shock}$, where according to (\[eq:SR\]), the velocity of a shock moving to the right separating regions 4 and 6 is $V_{shock}=v_6+a_6\sqrt{\frac{(\Gamma+1)p_4}{2\Gamma p_6} + \frac{\Gamma-1}{2\Gamma}}$, where $a_6=\sqrt{p_6 \Gamma / \rho_6}$. Then the solution in this region is
$$\begin{aligned}
p_{exact} &=& p_4, \noindent \nonumber\\
v_{exact} &=& v_4, \noindent \nonumber\\
\rho_{exact} &=& \rho_4. \noindent \nonumber\end{aligned}$$
as calculated
5. There is no region 5.
6. Region 6 is defined by $t V_{shock} < x-x_0$. In this region the solution is simply
$$\begin{aligned}
p_{exact} &=& p_6, \noindent \nonumber\\
v_{exact} &=& v_6, \noindent \nonumber\\
\rho_{exact} &=& \rho_6. \noindent \nonumber\end{aligned}$$
An example of how the solution looks like is shown in Fig. \[fig:Newtonian\_RS\] for initial data in Table \[tab:newtonian\].
Case $p_L$ $p_R$ $v_L$ $v_R$ $\rho_L$ $\rho_R$
------------------------- ------- ------- ------- ------- ---------- ----------
Rarefaction-Shock 1.0 0.1 0.0 0.0 1.0 0.125
Shock-Rarefaction 0.1 1.0 0.0 0.0 0.125 1.0
Rarefaction-Rarefaction 0.4 0.4 -1.0 1.0 1.0 1.0
Shock-Shock 0.4 0.4 1.0 -1.0 1.0 1.0
: Table with the initial data for the four different cases. We choose the spatial domain to be $x\in [0,1]$ and the location of the membrane at $x_0=0.5$. In all cases we use $\Gamma=1.4$.[]{data-label="tab:newtonian"}
![\[fig:Newtonian\_RS\] Exact solution for the Rarefaction-Shock case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/RS_rho.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_RS\] Exact solution for the Rarefaction-Shock case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/RS_p.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_RS\] Exact solution for the Rarefaction-Shock case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/RS_vel.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_RS\] Exact solution for the Rarefaction-Shock case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/RS_energy.eps "fig:"){width="4cm"}
### Case 2: Shock-Rarefaction
This case is identical to the previous one, except that we choose the initial pressure and density are higher on the right chamber. After initial time, the wave traveling to the left is a shock, while the one moving to the right is a rarefaction wave. This implies that region 2 plays the role of region 5 in the previous case and region 5 has the tail-fan-head structure of a rarefaction wave.
Starting from the contact discontinuity, the conditions $v_3=v_4=v^{*}$ and $p_3=p_4=p^{*}$ hold. The conditions on a shock wave moving to the left imply according to (\[eq:vR\_shockL\]) that the velocity of the state at the right is
$$v_3 = v_1 - (p_3-p_1)\sqrt{\frac{A_1}{p3+B_1}},\label{eq:v_3_SR}$$
and information from the rarefaction wave interface can be obtained from (\[eq:vL\_rarR\]) for $v_4$ as the velocity on the state at the left from a rarefaction wave moving to the right
$$v_4 = v_6 -\frac{2a_6}{\Gamma-1}\left[ 1 - \left( \frac{p_4}{p_6}\right)^{\frac{\Gamma-1}{2\Gamma}} \right]. \label{eq:v_4_SR}$$
Equating these two expression one obtains a trascendental equation for $p^{*}$:
$$-(p^{*} - p_1)\sqrt{\frac{A_1}{p^{*}+B_1}} + \frac{2a_6}{\Gamma-1}\left[ 1 - \left(\frac{p^{*}}{p_6} \right)^{\frac{\Gamma-1}{2\Gamma}} \right] + v_1 - v_6 =0 \label{eq:trascencental_SR}$$
that one solves numerically for $p^{*}$. This information provides the necessary information to construct the solution in the whole domain as described below. The different regions are illustrated in Fig. \[fig:regions\_SR\] and the exact solution region by region is as follows.
1. Region 1 is defined by $x-x_0 < t V_{shock}$, where the velocity of the shock is given by (\[eq:SL\]) because the shock is traveling to the left:
$$V_{s} = v_1 - a_1 \sqrt{\frac{(\Gamma+1)p_3}{2p_1 \Gamma} + \frac{\Gamma-1}{2\Gamma}}, \nonumber$$
and the exact solution here reads
$$\begin{aligned}
p_{exact} &=& p_1, \noindent \nonumber\\
v_{exact} &=& v_1, \noindent \nonumber\\
\rho_{exact} &=& \rho_1. \noindent \nonumber\end{aligned}$$
2. There is no region 2.
3. Region 3 is defined by the condition $t V_{s} < x - x_0 < t V_{contact}$. $V_{contact}$ is the characteristic value (\[eq:lambda\_0\]) evaluated at this region: $V_{contact} = v_3 = v_4 = v^{*}$. Using (\[eq:rhoR\_shockL\]) explicitly for the density and (\[eq:v\_3\_SR\]) for the velocity, the solution in this region reads
$$\begin{aligned}
p_{exact} &=& p_3,\nonumber\\
v_{exact} &=& v_3,\nonumber\\
\rho_{exact} &=& \rho_1 \frac{p_1 (\Gamma-1) + p_3 (\Gamma+1)}{p_3 (\Gamma-1) + p_1 (\Gamma+1)}.\nonumber\end{aligned}$$
4. Region 4 is defined by the condition $t V_{contact} < x - x_0 < t V_{t}$, where the velocity of the tail of the rarefaction wave $V_{t}$ is the third eigenvalue (\[eq:lambda\_plus\]) evaluated at the region behind the tail $V_{t} = v_4 + a_4$.
One uses (\[eq:v\_4\_SR\]) to calculate $v_4$ and (\[eq:isentropic\]) implies $p_4/p_6 = (\rho_4/\rho_6)^{\Gamma}$ for a constant value of $K$, which implies an expression for $\rho_4$. The resulting exact solution is
$$\begin{aligned}
p_{exact} &=& p_4,\nonumber\\
v_{exact} &=& v_4\nonumber\\
\rho_{exact} &=& \rho_6 \left( \frac{p_4}{p_6} \right)^{1/\Gamma}.\nonumber\end{aligned}$$
5. Region 5 is a fan region defined by the condition $t V_{t} < x - x_0 < t V_{h}$ where the velocity of the head of the wave is again the third eigenvalue, but this time evaluated at the head $V_{h}=v_6 +a_6$. One uses the expressions for a fan region of a rarefaction wave moving to the right (\[eq:vL\_rarR\],\[eq:pL\_rarR\],\[eq:rhoL\_rarR\]) to calculate the exact solution
$$\begin{aligned}
p_{exact} &=& p_6 \left[ \frac{2}{\Gamma+1} - \frac{\Gamma-1}{a_6(\Gamma+1)} \left(v_6 - \frac{x-x_0}{t}\right)\right]^{\frac{2\Gamma}{\Gamma-1}},\nonumber\\
v_{exact} &=& \frac{2}{\Gamma+1} \left[ -a_6 + \frac{1}{2}(\Gamma-1) v_6 + \frac{x-x_0}{t}\right],\nonumber\\
\rho_{exact} &=& \rho_6 \left[ \frac{2}{\Gamma+1}- \frac{\Gamma-1}{a_6(\Gamma+1)}\left( v_6 - \frac{x-x_0}{t}\right) \right]^{\frac{2}{\Gamma-1}}.\nonumber \end{aligned}$$
6. Region 6 is defined by the condition $t V_{h} < x-x_0$. The exact solution is given by the initial states at the right chamber.
$$\begin{aligned}
p_{exact} &=& p_6, \noindent \nonumber\\
v_{exact} &=& v_6, \noindent \nonumber\\
\rho_{exact} &=& \rho_6. \noindent \nonumber\end{aligned}$$
An example is shown in Fig. \[fig:Newtonian\_SR\] for initial data in Table \[tab:newtonian\].
![\[fig:Newtonian\_SR\] Exact solution for the Shock-Rarefaction case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/SR_rho.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_SR\] Exact solution for the Shock-Rarefaction case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/SR_p.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_SR\] Exact solution for the Shock-Rarefaction case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/SR_vel.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_SR\] Exact solution for the Shock-Rarefaction case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/SR_energy.eps "fig:"){width="4cm"}
### Case 3: Rarefaction-Rarefaction
A physical situation that provides this scenario is $p_L=p_R$, $\rho_L = \rho_R$ and $-v_L=+v_R >0$. In this case both, regions 2 and 5 correspond to rarefaction waves. In this particular case since one of the rarefaction waves moves to the left and the other one to the right, we distinguish them using the labels for each of their parts.
Again the contact discontinuity defines a relationship between velocity and pressure. In the present case, there is an expression for $v_3$ in terms of $v_1$ for a rarefaction wave moving to the left given by (\[eq:R\_L\]) and another one for $v_4$ in terms of $v_6$ for a rarefaction wave moving to the right (\[eq:v\_L\_rarefactionR\]):
$$\begin{aligned}
v_3 &=& v_1 -\frac{2a_1}{\Gamma-1}\left[ \left( \frac{p_3}{p_1}\right)^{\frac{\Gamma-1}{2\Gamma}} - 1\right],\label{eq:v_3_RR}\\
v_4 &=& v_6 -\frac{2a_6}{\Gamma-1}\left[ 1 - \left( \frac{p_4}{p_6}\right)^{\frac{\Gamma-1}{2\Gamma}} \right]. \label{eq:v_4_RR}\end{aligned}$$
The condition $v_3=v_4=v^{*}$ at the contact discontinuity implies a trascendental equation for $p^{*}=p_3=p_4$:
$$\frac{2a_6}{\Gamma-1}\left[ 1 - \left(\frac{p^{*}}{p_6}\right)^{\frac{\Gamma-1}{2\Gamma}} \right] -\frac{2a_1}{\Gamma-1} \left[ \left( \frac{p^{*}}{p_1}\right)^{\frac{\Gamma-1}{2\Gamma}} -1\right]+ v_1 - v_6 = 0
\label{eq:trascencental_RR}$$
Again, once $p^{*}$ is calculated numerically, the solution in all the regions of the domain can be calculated as follows. The first implication is that $p_3=p_4=p^{*}$, and thus $v_3$ and $v_4$ can be calculated using (\[eq:v\_3\_RR\]) and (\[eq:v\_4\_RR\]). The different regions are illustrated in Fig. \[fig:regions\_RR\].
1. Region 1 is defined by the condition $x-x_0 < t V_{h,2}$, where $V_{h,2}$ is the velocity of the head of the wave moving to the left, and is obtained from the characteristic value of such rarefaction wave evaluated at the left interface, that is $V_{h,2} = v_1 - a_1$. In this region the gas has not affected the initial state on the left, then the solution is
$$\begin{aligned}
p_{exact} &=& p_1, \nonumber\\
v_{exact} &=& v_1, \nonumber\\
\rho_{exact} &=& \rho_1. \nonumber\end{aligned}$$
2. Region 2 is a fan region defined by the condition $t V_{h,2} < x-x_0 < t V_{t,2}$, where the velocity of the tail $V_{t,2}$ is that of the state left behind by the wave, that is $V_{t,2} = v_3 - a_3$.
The exact solution is that of a fan region of a rarefaction wave moving to the left (\[eq:vR\_rarL\],\[eq:pR\_rarL\],\[eq:rhoR\_rarL\])
$$\begin{aligned}
p_{exact} &=& p_1 \left[ \frac{2}{\Gamma + 1} + \frac{\Gamma-1}{a_1 (\Gamma+1)} \left( v_1 - \frac{x-x_0}{t} \right)\right]^{\frac{2\Gamma}{\Gamma-1}},\nonumber\\
v_{exact} &=& \frac{2}{\Gamma+1}\left[ a_1 + \frac{1}{2} (\Gamma-1) v_1 + \frac{x-x_0}{t}\right],\nonumber\\
\rho_{exact} &=& \rho_1 \left[ \frac{2}{\Gamma + 1} + \frac{\Gamma-1}{a_1 (\Gamma+1)} \left( v_1 - \frac{x-x_0}{t} \right) \right]^{\frac{2}{\Gamma-1}}.\nonumber\end{aligned}$$
3. Region 3 is defined by the condition $tV_{t,2} < x-x_0 < t V_{contact}$. The velocity of the contact discontinuity is $V_{contact}=v_3=v_4=v^{*}$ according to the eigenvalue (\[eq:lambda\_0\]). In this region $p_3=p^{*}$ and $v_3=v^{*}$ are already known from $p^{*}$. Finally, the density is obtained from (\[eq:isentropic\]) for an isentropic process like the rarefaction wave for a constant $C$ on both sides of such wave as found in the previous two cases. Thus the solution is
$$\begin{aligned}
p_{exact} &=& p_3,\nonumber\\
v_{exact} &=& v_3.\nonumber\\
\rho_{exact} &=& \rho_1 \left( \frac{p_3}{p_1}\right)^{1/\Gamma},\nonumber\end{aligned}$$
4. Region 4 is defined by the condition $t V_{contact} < x-x_0 < t V_{t,5}$, where the velocity of the tail of the wave moving to the right $V_{t,5}$ is given by the eigenvalue (\[eq:lambda\_plus\]) evaluated at the state left behind the rarefaction wave moving to the right, that is $V_{t,5}=v_4+a_4$, where again we point out that $v_4=v^{*}$ and $p_4=p^{*}$ are known once $p^{*}$ is calculated. The solution is obtained in the same way as for the previous region, but now the wave relates states in regions 4 and 6:
$$\begin{aligned}
p_{exact} &=& p_4,\nonumber\\
v_{exact} &=& v_4.\nonumber\\
\rho_{exact} &=& \rho_6 \left( \frac{p_4}{p_6}\right)^{1/\Gamma},\nonumber\end{aligned}$$
5. Region 5 is defined by the condition $t V_{t,5} < x-x_0 < V_{h,5}$, where the velocity of the head of the wave moving to the right is $V_{h,5}=v_6+a_6$, and the solution is obtained using the values of the state variables for the fan of a rarefaction wave moving to the right (\[eq:vL\_rarR\],\[eq:pL\_rarR\],\[eq:rhoL\_rarR\]):
$$\begin{aligned}
p_{exact} &=& p_6 \left[ \frac{2}{\Gamma+1} - \frac{\Gamma-1}{a_6(\Gamma+1)} \left(v_6 - \frac{x-x_0}{t}\right)\right]^{\frac{2\Gamma}{\Gamma-1}},\nonumber\\
v_{exact} &=& \frac{2}{\Gamma+1} \left[ -a_6 + \frac{1}{2}(\Gamma-1) v_6 + \frac{x-x_0}{t}\right],\nonumber\\
\rho_{exact} &=& \rho_6 \left[ \frac{2}{\Gamma+1}- \frac{\Gamma-1}{a_6(\Gamma+1)}\left( v_6 - \frac{x-x_0}{t}\right) \right]^{\frac{2}{\Gamma-1}}.\nonumber \end{aligned}$$
6. Finally, region 6 is defined by the condition $V_{h,5} < x-x_0$. The exact solution is given by the initial values at the chamber on the right because in this region the gas has not been affected yet by the dynamics of the gas:
$$\begin{aligned}
p_{exact} &=& p_6, \nonumber\\
v_{exact} &=& v_6, \nonumber\\
\rho_{exact} &=& \rho_6. \nonumber\end{aligned}$$
An example is shown in Fig. \[fig:Newtonian\_RR\] for initial data in Table \[tab:newtonian\].
![\[fig:Newtonian\_RR\] Exact solution for the Rarefaction-Rarefaction case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/RR_rho.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_RR\] Exact solution for the Rarefaction-Rarefaction case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/RR_p.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_RR\] Exact solution for the Rarefaction-Rarefaction case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/RR_vel.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_RR\] Exact solution for the Rarefaction-Rarefaction case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/RR_energy.eps "fig:"){width="4cm"}
### Case 4: Shock-Shock
A physical situation that provides this scenario corresponds to two streams colliding with opposite directions. We choose in this case $p_L=p_R$, $\rho_L = \rho_R$ and $-v_L=+v_R <0$. In this case regions 2 and 5 are shock waves.
Again the contact discontinuity defines a relationship between velocity and pressure. In the present case there is an expression for $v_3$ in terms of $v_1$ for a shock-wave moving to the left given by (\[eq:vR\_shockL\]) and another one for $v_4$ in terms of $v_6$ for a shock-wave moving to the right (\[eq:L\_R\]):
$$\begin{aligned}
v_3 &=& v_1 - (p_3-p_1)\sqrt{\frac{A_1}{p_3+B_1}},\label{eq:v_3_SS}\\
v_4 &=& v_6 + (p_4-p_6)\sqrt{\frac{A_6}{p_4+B_6}}. \label{eq:v_4_SS}\end{aligned}$$
The condition $v_3=v_4=v^{*}$ at the contact discontinuity implies a trascendental equation for $p^{*}=p_3=p_4$:
$$-(p^{*} - p_1)\sqrt{\frac{A_1}{p^{*}+B_1}} - (p^{*}-p_6)\sqrt{\frac{A_6}{p^{*}+B_6}}+v_1-v_6=0.
\label{eq:trascencental_SS}$$
Again, once $p^{*}$ is calculated numerically, the solution in all the regions of the domain can be calculated as follows. Immediately one has that $p_3=p_4=p^{*}$ and $v_3$ and $v_4$ can be calculated using (\[eq:v\_3\_SS\]) and (\[eq:v\_4\_SS\]).
In this particular case regions 2 and 5 reduce to lines. The solution in each region reads as follows and the regions are illustrated in Fig. \[fig:regions\_SS\].
1. Region 1 is defined by the condition $x-x_0 < t V_{s,2}$, where the velocity of the shock moving to the left $V_{s,2}$ is given by (\[eq:SL\]) and reads $V_{s,2} = v_1 - a_1 \sqrt{\frac{(\Gamma+1)p_3}{2p_1 \Gamma} + \frac{\Gamma-1}{2\Gamma}}$. The solution there is that of the initial values of the variables on the left chamber:
$$\begin{aligned}
p_{exact} &=& p_1, \nonumber\\
v_{exact} &=& v_1, \nonumber\\
\rho_{exact} &=& \rho_1. \nonumber\end{aligned}$$
2. There is no region 2.
3. Region 3 is defined by the condition $t V_{s,2} < x - x_0 < t V_{contact}$, where $V_{contact} = v_3 = v_4 = v^{*}$. Once (\[eq:trascencental\_SR\]) is solved one can calculate all the required information. Using (\[eq:v\_3\_SS\]) for $v_3$ and (\[eq:rhoR\_shockL\]) for $\rho_3$ the solution in this region reads
$$\begin{aligned}
p_{exact} &=& p_3,\nonumber\\
v_{exact} &=& v_3\nonumber\\
\rho_{exact} &=& \rho_1 \frac{p_1 (\Gamma-1) + p_3 (\Gamma+1)}{p_3 (\Gamma-1) + p_1 (\Gamma+1)}.\nonumber\end{aligned}$$
4. Region 4 is defined by $t V_{contact} < x-x_0 < tV_{s,5}$, where the velocity of the shock moving to the right is given by (\[eq:SR\]) and reads $V_{s,5} =v_6 + a_6 \sqrt{\frac{(\Gamma+1)p_4}{2p_6 \Gamma} + \frac{\Gamma-1}{2\Gamma}}$. Finally, using (\[eq:v\_4\_SS\]) for $v_4$ and (\[eq:rho\_L\_R\]) for $\rho_4$ the solution in this region reads
$$\begin{aligned}
p_{exact} &=& p_4, \nonumber\\
v_{exact} &=& v_4, \nonumber\\
\rho_{exact} &=& \rho_6 \frac{p_6 (\Gamma-1) + p_4 (\Gamma+1)}{p_4 (\Gamma-1) + p_6 (\Gamma+1)}. \nonumber\end{aligned}$$
5. There is no region 5.
6. Finally region 6 is defined by the condition $V_{s,5} < x-x_0$. The exact solution is given by the initial values at the chamber on the right:
$$\begin{aligned}
p_{exact} &=& p_6, \nonumber\\
v_{exact} &=& v_6, \nonumber\\
\rho_{exact} &=& \rho_6. \nonumber\end{aligned}$$
An example is shown in Fig. \[fig:Newtonian\_SS\].
![\[fig:Newtonian\_SS\] Exact solution for the Shock-Shock case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/SS_rho.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_SS\] Exact solution for the Shock-Shock case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/SS_p.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_SS\] Exact solution for the Shock-Shock case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/SS_vel.eps "fig:"){width="4cm"} ![\[fig:Newtonian\_SS\] Exact solution for the Shock-Shock case at time $t=0.25$ for the parameters in Table \[tab:newtonian\].](Figs/SS_energy.eps "fig:"){width="4cm"}
FInal comments {#sec:final}
==============
In this academic article we have described in detail the implementation of the exact solution of the 1D Riemann in the newtonian and relativistic regimes, which according to our experience is not presented in a straightforward enough recipe in literature.
The contents in this article can be used in various manners, specially to: i) test numerical solutions of the Newtonian Riemann problem in basic courses of hydrodynamics, ii) test numerical implementations of codes solving hydrodynamical relativistic equations, iii) understand the different properties of the propagation of the different type of waves developing in a gas and the different conditions on the hydrodynamical variables in each case.
It is also helpful because with our approach it is possible to straightforwardly implement the exact solution, and this will save some time to a student starting a career in astrophysics involving hydrodynamical processes.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research is partly supported by grants: CIC-UMSNH-4.9,4.23 and CONACyT 106466. (J.P.C-P and F.D.L-C) acknowledge support from the CONACyT scholarship program.
[10]{}
E. F. Toro, Riemann solvers and numerical methods for fluid dynamics. Springer-Verlag Berlin-Heidelberg, 2009.
J. Ma. Martí, E. Müller, “Numerical Hydrodynamics in Special Relativity”, Living Rev. Relativity 6, (2003), 7. http://www.livingreviews.org/lrr-2003-7
J. Ma. Martí, E. Müller, The analytical solution of the Riemann problem in relativistic hydrodynamics, J. Fluid. Mech. (1994), vol. 258, pp. 317-333.
R. J. LeVeque, in [*Numerical methods for conservation laws*]{}. Birkhauser, Basel (1992).
G. A. Sod, J. Comp. Phys. [**27**]{} (1978) 1-31.
A. Taub, Relativistic Rankine-Hugoniot relations, Phys. Rev. (1948), vol. 74, pp. 328-334.
K. S. Thorne, Relativistic Shocks: the Taub adiabat, Astrophys. J. (1973), vol. 179, pp. 897-907.
|
---
abstract: 'We present a feasibility study of a simultaneous sub-percent extraction of the weak charge and the weak radius of the ${}^{12}$C nucleus using parity-violating electron scattering, based on a [largely]{} model-independent assessment of the uncertainties. The corresponding measurement is considered to be carried out at the future MESA facility in Mainz with $E_{\rm beam} = 155$ MeV. We find that a combination of a 0.3% precise measurement of the parity-violating asymmetry at forward angles with a 10% measurement at backward angles will allow to determine the weak charge and the weak radius of ${}^{12}$C with 0.4% and 0.5% precision, respectively. These values could be improved to 0.3% and 0.2% for a 3% backward measurement. This experimental program will have impact on precision low-energy tests in the electroweak sector and nuclear structure.'
author:
- 'Oleksandr Koshchii$^{a}$'
- 'Jens Erler$^{a,b,c}$'
- 'Mikhail Gorchtein$^{d}$'
- 'Charles J. Horowitz$^{e}$'
- 'Jorge Piekarewicz$^{f}$'
- 'Xavier Roca-Maza$^{g}$'
- 'Chien-Yeah Seng$^{h}$'
- 'Hubert Spiesberger$^{a}$'
title: ' Weak charge and weak radius of ${}^{12}$C at P2 '
---
Precise measurements of the parameters of the standard model (SM) are among the main tools to search for or constrain hypothetical contributions from physics beyond the SM. The central parameter of the electroweak sector of the SM is the weak mixing angle $\theta_W$ describing the mixing of the $SU(2)$ and $U(1)$ gauge boson fields, which results in the emergence of the physical fields, the massless photon and the massive $Z^0$. Its sine squared, $\sin^2\theta_W$, is related to the vector charge of SM fermions with respect to the weak neutral current and can be accessed in various processes and at different energy scales: from $Z$-pole measurements at colliders [@ALEPH:2005ab; @Aaltonen:2018dxj], including the LHCb [@Aaij:2015lka], ATLAS [@atlas] and CMS [@Sirunyan:2018swq] experiments, to deep inelastic scattering with electrons [@Prescott:1997; @Wang:2014] and neutrinos [@Zeller:2001hh], to parity violation in atoms [@Wood:1997; @Guena:2004sq] and to parity-violating electron scattering (PVES) off protons [@qweak] and electrons [@Anthony:2005pm]. To connect these measurements across the relevant energy scales, the SM running at the one-loop level needs to be taken into account [@Erler:2004in; @Erler:2017knj]. Currently, this running is theoretically known at the relative level of $\sim8\times10^{-5}$, which provides the basis for an ambitious experimental program at low energies: an ongoing effort in atomic parity violation [@Zhang:2016czr; @Antypas:2018mxf] has the goal to measure the weak charges of heavy nuclei and chains of nuclear isotopes at the per mille precision. The Qweak experiment [@qweak] has recently extracted $\sin^2\theta_W$ from low-energy PVES to 0.5% accuracy. P2@MESA [@Becker:2018ggl] and MOLLER [@Benesch:2014bas] aim at improving that result by a factor of 4 and 6, respectively. Further plans involve deep-inelastic electron scattering with SOLID [@solid].
Apart from tests of the SM, PVES has also been used to address aspects of nucleon and nuclear structure that are elusive to photons. PVES off heavy nuclei with a neutron excess is used to determine the neutron skin [@Thiel:2019tkm]—the difference in the radii of the neutron and proton distributions—with the goal of constraining the equation-of-state (EOS) of neutron rich matter[@Horowitz:2000xj]. The lead (Pb) Radius EXperiment (PREX) [@Abrahamyan:2012gp] has provided the first model-independent evidence in favor of a neutron-rich skin in ${}^{208}$Pb [@Horowitz:2012tj]. Further experiments with an improved precision are presently being analysed [@PREX-II], running [@CREX], or planned [@Becker:2018ggl]. PVES off the proton and light nuclei has been extensively used to determine the strange quark content of the nucleon [@Maas:2017snj].
In this letter we consider the parity-violating asymmetry which is defined as the difference between the cross sections for elastic scattering of [longitudinally]{} polarized electrons off an unpolarized target, A\^[PV]{}=, where $\sigma_{R} \, (\sigma_L)$ stands for the cross section with right-handed (left-handed) electron polarization. The asymmetry arises from the interference between the amplitudes due to the exchange of a virtual photon and [the corresponding one]{} for a virtual $Z^{0}$ boson, as shown in Fig. \[fig:tree\].
By conveniently factoring out the Fermi constant $G_F$, the fine structure constant $\alpha$, the 4-momentum transfer squared $Q^2$, and the ratio of the weak, $Q_W$, to the electric, $Z$, nuclear charge, the PV asymmetry [for a spinless]{} nucleus consisting of $Z$ protons and $N$ neutrons takes the following form: A\^[PV]{} = - (1+), \[eq:APV\] where a plane-wave Born (“tree-level") approximation was assumed. The weak nuclear charge is given by $Q_W(Z,N)\!=\!Z(1\!-\!4\sin^2\theta_W)\!-\!N$, so in the case of ${}^{12}$C it becomes proportional to the sine-squared of the weak mixing angle [@ftnt1]: $Q_W(6,6)\!=\!-24\sin^2\theta_W$.
Given that the interaction of the electron with the nucleus involves only the conserved hadronic vector current, the “correction" term $\Delta$ in Eq. (\[eq:APV\]) vanishes at $Q^2\!=\!0$. However, nuclear and hadronic structure contribute to $\Delta$ at non-zero $Q^2$. Indeed, to leading order in $\alpha$, F\_[wk]{}(Q\^2)/F\_[ch]{}(Q\^2) - 1 \[eq:DeltaDef\] is given by the ratio of the weak $F_{\rm wk}$ to the charge form factor $F_{\rm ch}$. Both form factors are normalized to unity at $Q^2\!=\!0$. Each of the form factors is related to the corresponding spatial distributions of charge by a three-dimensional Fourier transform, F(Q\^2) = (r) e\^[i[**q**]{}]{} d\^[3]{}r, |[**q**]{}|. Note that the normalization of the form factor at $Q^2\!=\!0$ implies that $\!\int\!\rho(r) d^{3}r\!=\!1$. At low $Q^2$, the form factors may be expanded in terms of various moments of their spatial distribution, F(Q\^2) = 1 - r\^[2]{}+ r\^[4]{}+ [O]{}(Q\^[6]{}), where the second term defines the root-mean-square radius of the spatial distribution, namely, R\^[2]{} r\^[2]{} = r\^[2]{} (r) d\^[3]{}r. \[eq:moments\] Thus, to lowest order in $Q^2$, $\Delta$ is proportional to the weak skin of the nucleus: = - R\_[wskin]{} R\_[ch]{} + [O]{}(Q\^2 R\_[wskin]{}\^2). \[eq:delta\_taylor\] The weak skin $R_{\rm wskin}\!\equiv\!R_{\rm wk}\!-\!R_{\rm ch}$, or, equivalently, = , \[eq:lambda\_def\] contains as much information as the neutron skin. However, unlike the neutron skin, the weak skin is a genuine physical observable.
Two terms in Eq. (\[eq:APV\]) are of great interest: the weak mixing angle $\theta_W$ encoded in the weak charge [@qweak; @Becker:2018ggl] and the ratio of nuclear form factors appearing in $\Delta$; to access the former one must constrain the latter. Conversely, to extract nucleon- or nuclear-structure information from PVES, such as the strange quark content of the nucleon [@Maas:2017snj] or the weak skin of heavy nuclei [@Horowitz:2012tj], one assumes that $Q_W$ is precisely known, so the measurement provides a constraint on $\Delta$. In this work we explore the possibility of a precise determination of both—the weak charge and the weak skin of ${}^{12}$C—within one single experiment.
The P2 experimental program at the MESA facility in Mainz [@Becker:2018ggl] includes a plan aiming for a $0.3\%$ determination of the weak charge of ${}^{12}$C. Given this ambitious goal, the tree-level formula of Eq. (\[eq:delta\_taylor\])—even when including higher-order terms in the $Q^2$ expansion—is not accurate enough. Order-$\alpha$ radiative corrections, particularly Coulomb distortions which scale as $Z\alpha$, should be included. To properly account for Coulomb distortions, we follow the formalism developed by one of us in Ref.[@Horowitz:1998vv]. The electron wave function $\Psi$ satisfies the Dirac equation ( + m + V(r) + \_[5]{}A(r))([**r**]{}) = E([**r**]{}), \[eq:dirac\] where $m$ is the electron mass, $\bm{\alpha}$, $\beta$, and $ \gamma_{5}$ are Dirac matrices, and $V(r)$ and $A(r)$ are the vector (Coulomb) and axial-vector components of the potential, respectively [@Horowitz:1998vv]. Here $E$ stands for the electron energy in the center of mass frame [@ftnt2] which is related (neglecting the electron mass) to the laboratory energy $E_{\rm beam}$ by $E_{\rm beam}/E\!=\!\sqrt{1\!+\!2E_{\rm beam}/M}$ with $M$ the nuclear mass.
The Coulomb potential is computed from the experimentally known nuclear charge distribution via V(r) = -Zd\^[3]{}r’. The axial-vector potential for a point-like weak charge is short range $A(r)\!\propto\!\delta^3({\bf r})$, but acquires a finite range due to the finite size of the nuclear weak charge distribution. That is, A(r) = \_[wk]{}(r). The Dirac equation displayed in Eq. (\[eq:dirac\]) is solved numerically using the ELSEPA package [@ELSEPA], properly modified to include the axial-vector potential[@RocaMaza:2011pm; @upcoming]. The intrinsic relative precision of the computation of $A^{\rm PV}$ is of the order $10^{-4}-10^{-5}$, and in a calculation at the per mille level there are only genuine uncertainties of $\rho_{\rm wk}$ itself.
For the nuclear charge distribution of ${}^{12}$C we use the parametrization of the world data on elastic electron-carbon scattering in the form of a sum of Gaussians [@DeJager:1987qc]. The fact that the charge density of ${}^{12}$C and its charge radius $R_{\rm ch}=2.4702(22)$fm [@Angeli:2013] are known with high precision serves as the basis for an accurate extraction of $\sin^2\theta_W$ and of $R_{\rm wk}$ from a measured $A^{\rm PV}$. A possible avenue is to rely on models to produce a range of predictions for $\rho_{\rm wk}$ which is then used to directly fit the PV asymmetry to determine the value of the weak radius, as was done in the case of the PREX. One choice for parametrizing the weak charge distribution is the two-parameter symmetrized Fermi distribution, $$\begin{aligned}
\rho_{\rm wk}(r)
= {{\rho}_{\raisebox{-2.50pt}{\!\tiny SF}}}(r,c,a)
& = \,{{\rho}_{\raisebox{-2.50pt}{\!\tiny 0}}}\,\frac{\sinh(c/a)}{\cosh{(r/a)}+\cosh(c/a)},
\nonumber \\
{{\rho}_{\raisebox{-2.50pt}{\!\tiny 0}}}
& = \frac{3}{4\pi c\left(c^{2}+\pi^{2}a^{2}\right)},
\label{eq:RhoSF}\end{aligned}$$ with $c$ and $a$ the half-density radius and surface diffuseness, respectively, and ${{\rho}_{\raisebox{-2.50pt}{\!\tiny SF}}}$ is normalized to unity. The advantage of the symmetrized Fermi parametrization, apart from its simplicity, is that its form factor and all of its moments are known analytically [@Piekarewicz:2016vbn]. In particular, the mean-square radius of the distribution is R\_[SF]{}\^2 = c\^[2]{} + \^[2]{}a\^[2]{}.
In Fig. \[fig:CDvsTL\] we show results for the PV asymmetry at a fixed electron beam energy of $E_{\rm beam}\!=\!155$MeV as a function of the laboratory scattering angle $\theta$ and momentum transfer $q$. Results are displayed in both a plane-wave (tree-level) approximation and with Coulomb distortions. The two distorted-wave calculations use $\rho_{\rm wk}\!=\!\rho_{\rm ch}$ (no skin) and $\rho_{\rm wk}\!=\!\rho_{\rm SF} (r, c\!=\!2.07\,\mathrm{fm},
a\!=\!0.494\,\mathrm{fm})$ with $R_{\rm wk}\!=\!2.44\,\mathrm{fm}$ which falls within the range of values of a representative set of nuclear-structure models [@Chen:2014mza; @Chen:2014sca; @RocaMaza:2012sj].We observe a strong dependence of the PV asymmetry on the value of the weak skin, especially at backward angles. We also find that it is important to include effects due to Coulomb distortions. Our results displayed in Fig. \[fig:CDvsTL\] are qualitatively similar to those obtained in Ref. [@Moreno:2013pna], but a quantitative comparison is difficult because of different perspectives adopted in the calculation of the weak charge density, and several kinematic approximations used in that work.
Unfortunately, the choice of a particular form for the weak charge distribution introduces model dependence that may be difficult to quantify when extracting weak charge and radius from a measurement of $A^{\rm PV}$: models that predict different values for the weak radius would generally differ in all the higher moments of the weak charge distribution, as well. To unambiguously disentangle the effect of the weak skin, we propose a different method. Given the $N\!=\!Z$ character of ${}^{12}$C, its weak charge distribution is expected to follow closely the electric charge distribution. We introduce the small difference between the two, the “weak-skin" distribution \_[wskin]{}(r) \_[wk]{}(r)-\_[ch]{}(r) . \[rhowk\] Note that $\rho_{\rm wskin}$ is the spatial analogue of the weak-skin form factor depicted in Figs. 3 and 6 of Ref. [@Thiel:2019tkm]. $\rho_{\rm wskin}(r)$ is normalized to zero and its second moment can be fixed to \_[wskin]{}(r) r\^[2]{} d\^[3]{}r = R\_[wk]{}\^[2]{} - R\_[ch]{}\^[2]{} = 2 R\_[ch]{}\^2 + [O]{}(\^2). \[eq:skinmom\] This allows us to write \_[wskin]{}(r) = |(r; ) , \[eq:rhobar\] where $\zeta$ is representative of the model dependence. This parametrization is advantageous because it allows to explicitly separate the dependence on $\lambda$ from the effects of the higher moments of the weak charge density encapsulated in a (set of) model parameter(s) $\zeta$. For example, assuming the symmetrized Fermi parametrization of $\rho_{\rm wk}$ as in Eq. (\[eq:RhoSF\]), one would find &&\_[wskin]{}(r)= (/[\_[SF]{}]{})([\_]{}(r,c,a) - \_[ch]{}(r) ). \[eq:rhoskin\]\
&&[with]{} \_[SF]{} =\_[SF]{}(c,a) =R\_[SF]{}(c,a)/R\_[ch]{}-1 . This parametrization corresponds to rewriting $\Delta$ in Eq. (\[eq:DeltaDef\]) as $$\begin{aligned}
\Delta
& = - \frac{\lambda}{3}Q^{2}R_{\rm ch}^2
+ \left(\frac{F_{\rm wk}}{F_{\rm ch}}-1
+ \frac{\lambda}{3}Q^{2}R_{\rm ch}^2\right), \nonumber \\
& = - \frac{\lambda}{3}Q^{2}R_{\rm ch}^2
+ \left[\frac{\lambda}{\lambda_{\rm SF}}\left(\frac{F_{{}_{\rm SF}}}{F_{\rm ch}}-1\right)
+ \frac{\lambda}{3}Q^{2}R_{\rm ch}^2\right],
\label{eq:delta_taylor2}\end{aligned}$$ and the low-$Q^2$ expansion of the term in the square brackets starts at the order $Q^4$ by construction. The nuclear models [@Chen:2014mza; @Chen:2014sca; @RocaMaza:2012sj] are used here—not to predict the distribution of weak charge in ${}^{12}$C, but rather—to determine the range of values that need to be explored to quantify the uncertainty in $\Delta$. These models, all informed by the charge radii and binding energies of a variety of nuclei including ${}^{12}$C, predict $|\lambda_{\rm SF}|\!\lesssim\!2\%$ with the central value $\lambda_{0}\!=\! - 0.90\%$.
To address the possibility to determine the weak charge of ${}^{12}$C with a precision of 0.3% in the P2 experiment, we study the sensitivity of the PV asymmetry to nuclear-structure uncertainties. In Fig. \[fig:Avslambda\] we display results for $A^{\rm PV}$ as a function of $\lambda$ for an incident electron energy of $E_{\rm beam}\!=\!155$MeV and two fixed scattering angles: $\theta\!=\!29^\circ$ (upper panel) and $\theta\!=\!145^\circ$ (lower panel). The central blue line corresponds with the second term in Eq.(\[eq:delta\_taylor2\]) fixed at a central value of $F_{\rm wk}$ consistent with the model predictions. The blue band around the central line indicates the spread of the model predictions for $F_{\rm wk}$. The pink-shaded band in the $\theta\!=\!29^\circ$ plot indicates the anticipated $0.3\%$ precision in $A^{\rm PV}$.
From the sensitivity of the forward angle measurement to $\lambda$ shown in Fig. \[fig:Avslambda\], one concludes that $\lambda$ should be known with a precision of 0.6% or better to constrain the weak charge of ${}^{12}$C to about 0.3%. Given that the nuclear models suggest a larger uncertainty in $\lambda$, we conclude that with a single measurement and theory input alone this task is not feasible.
Another option is to employ a second measurement of $A^{\rm PV}$ at $145^\circ$ (the lower panel of Fig. \[fig:Avslambda\]) to constrain the value of $\lambda$ to a narrower range. It is seen that varying $\lambda$ in the adopted range translates into a $\pm24\%$ variation in the asymmetry. Hence, a measurement at this backward kinematical setting with a higher precision will reduce the range of values of $\lambda$ and ultimately guarantee a precise extraction of the weak mixing angle from a combination of the two measurements. To a very good approximation the $\lambda$-dependence of $A^{\rm PV}$ seen in Fig.\[fig:Avslambda\] is linear and we can write A\^[PV]{} = - (1+ p\_0 + (p\_1 + p\_2 ) ) , \[eq:Deltapars\] so that the effect of varying $\zeta$ is depicted by the blue bands in Fig. \[fig:Avslambda\]. The parameter $\zeta$ can be chosen in such a way that $\zeta = \zeta_0 = 0$ corresponds to the central prediction and $\zeta = \pm 1$ to the upper and lower limits of the error band. Results of the distorted-wave calculation of the coefficients $p_0$, $p_1$, and $p_2$ are provided in Appendix. We perform a $\chi^2$-fit of the combined forward ($A^{\rm PV}_{f}$) and backward ($A^{\rm PV}_{b}$) measurements with respect to the three free parameters $\sin^2\!\theta_W$, $\lambda$ and $\zeta$. That is, $$\begin{aligned}
\label{eq:chi2}
& \chi^2(\sin^2\!\theta_W, \lambda, \zeta) =
\\ &
\sum_{i=f,b}
\left(
\frac{A_i^{\rm exp} - A_i^{\rm PV}(\sin^2\!\theta_W, \lambda, \zeta)}
{\delta A_i} \right)^2
+ \left(\frac{\zeta - \zeta_0}{\delta \zeta}\right)^2
\, .
\nonumber\end{aligned}$$ We assume that the experimental values $A_i^{\rm exp}$ agree with the SM prediction, for which we choose the central value $\lambda\!=\!\lambda_0$. The experimental uncertainties are given by $\delta A_i$. The last term in Eq. (\[eq:chi2\]) encodes our biases for the expected range of values of $\zeta$, and we have chosen $\delta \zeta\!=\!1$. The $1\sigma$-allowed range for $\sin^2\!\theta_W$ and $\lambda$ is obtained by solving $\chi^2(\sin^2\theta_W, \lambda, \zeta)\!=\!1$. In Fig. \[fig:chi2fit\] we show the projection of the $\chi^2\!=\!1$ solution onto the $\sin^2\!\theta_W$-$\lambda$-plane for three different choices of the precision of the backward-angle measurement. The accuracy of the forward measurement remains fixed at 0.3%.
The covariance ellipses in Fig. \[fig:chi2fit\] suggest that $\sin^2\!\theta_W$ and $\lambda$ are correlated and their correlation decreases with increasing accuracy of the backward measurement. Moreover, fractional uncertainties are given by: $\delta\sin^2\theta_W/\sin^2\!\theta_W\!=\!\pm 0.39\%$ for $\delta A_b/A^{\rm PV}_b\!=\!10\%$ and $\delta\sin^2\theta_W/\sin^2\theta_W\!=\!\pm 0.35\%$ for $\delta A_b/A^{\rm PV}_b\!=\!7\%$. An even higher precision of the backward measurement, $3\%$, results in a reduction in the uncertainty of the weak mixing angle to $\delta\sin^2\!\theta_W/\sin^2\theta_W\!=\!\pm 0.32\%$. At this point the uncertainty starts being dominated by the forward measurement, so further improvement to the backward measurement has no impact on the precision of $\sin^2\!\theta_W$.
In summary, we presented an ambitious proposal for a simultaneous sub-percent determination of the weak charge and weak radius of ${}^{12}$C using parity-violating electron scattering at the upcoming MESA facility in Mainz. We demonstrated that to take full advantage of an unprecedented 0.3% precision aimed for in the forward kinematical setting of P2 [@Becker:2018ggl], an additional $3\!-\!7\%$ measurement at a backward angle of 145$^\circ$ will ensure a largely model-independent extraction of $\sin^2\!\theta_W$ with a relative precision of $0.32\!-\!0.35\%$ and determination of $R_{\rm wk}$ within $0.19\!-\!0.35\%$ of $R_{\rm ch}$. Note that a similar combination of forward and backward measurements on the proton is planned as part of the P2 experiment [@Becker:2018ggl], which makes the proposal presented in this letter a viable and attractive possibility.
Whereas the weak skin of ${}^{12}$C and other symmetric nuclei does not constrain the nuclear EOS, its exact value will help quantifying generic isospin symmetry-breaking (ISB) effects. Coulomb repulsion among the protons inside a nucleus and other ISB mechanisms lead to a mismatch in the distribution of neutrons and protons therein. Along with generating the proton (and weak) skin of symmetric nuclei, ISB contributions play a major role in the analysis of superallowed nuclear $\beta$ decays and the extraction of $V_{ud}$ [@Hardy:2014qxa]. Importantly, in all pairs of nuclei involved in the known superallowed $\beta$ transitions, either the parent or the daughter nucleus is symmetric. Therefore, precise information on weak skins of the (nearly) symmetric parent and daughter nuclei will have an impact on the tests of unitarity of the Cabibbo-Kobayashi-Maskawa matrix and New Physics searches with superallowed nuclear $\beta$ decays.
[**Acknowledgments**]{} – The authors acknowledge useful discussions with K. Kumar and F. Maas. We thank T. W. Donnelly, R. Ferro-Hernández, and O. Moreno for helpful correspondence. The work of J. E., M. G., O. K., and H. S. is supported by the German-Mexican research collaboration grant No. 278017 (CONACyT) and No. SP 778/4-1 (DFG). M. G. is supported by the EU Horizon 2020 research and innovation programme, project STRONG-2020, grant agreement No. 824093. X. R-M. acknowledges funding from the EU Horizon 2020 research and innovation programme, grant agreement No. 654002. J. P acknowledges support from the U.S. Department of Energy Office of Nuclear Physics under Award Number DE-FG02-92ER40750. C. Y. S. is supported in part by the DFG (Grant No. TRR110) and the NSFC (Grant No. 11621131001) through the funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD”, and also by the Alexander von Humboldt Foundation through the Humboldt Research Fellowship. C. J. H. was supported by the U. S. Department of Energy grants No. DE-FG02-87ER40365 and DE-SC0018083.
Appendix: Supplemental Material {#appendix}
===============================
In Table \[tab:2\] we provide parameters of nuclear models used in our calculation of the parity-violating asymmetry and parameters $\zeta_f$ ($\zeta_b$) determined as a result of the calculation which accounts for Coulomb distortion effects at forward (backward) scattering angles.
[|l||\*[7]{}[l|]{}]{} & [$c, \mathrm{fm}$]{} & [$a, \mathrm{fm}$]{} & [$R_{\rm SF}, \mathrm{fm}$]{} & [$\lambda_{\rm SF}, \%$]{} & [$\zeta_f$]{} & [$\zeta_b$]{}\
RMF016 & $2.06065$ & $0.49389$ & $2.43274$ & $-1.52$ & $-1.00$ & $-1.00$\
RMF022 & $2.06849$ & $0.49445$ & $2.43830$ & $-1.29$ & $-0.80$ & $-0.30$\
RMF028 & $2.07585$ & $0.49544$ & $2.44482$ & $-1.03$ & $-0.44$ & $+1.00$\
RMF032 & $2.06421$ & $0.49433$ & $2.43578$ & $-1.39$ & $-0.89$ & $-0.62$\
SMC12 & $2.22693$ & $0.47318$ & $2.46358$ & $-0.27$ & $+1.00$ & $+0.59$\
The first four models listed in Table \[tab:2\] fall under the general rubric of covariant (or relativistic) energy density functionals. The models are based on an underlying Lagrangian density that includes nucleons interacting via the exchange of various mesons and the photon. In addition, nonlinear meson interactions are included to account for many-body forces. The calibration of the handful of model parameters is informed by ground-state properties of finite nuclei, their collective response, and constraints on the maximum neutron-star mass[@Chen:2014sca]. Incorporated in the ground state properties are charge radii of a variety of magic and semi-magic nuclei, including ${}^{12}$C. The outcome of the calibration procedure is an optimal set of parameters together with a covariance matrix that properly accounts for statistical uncertainties and correlations. The fitting protocol for all the models is identical save one important distinction: the assumed value for the yet to be determined neutron skin thickness of ${}^{208}$Pb ($R_{\rm skin}^{208}$). Indeed, the neutron skin thickness of ${}^{208}$Pb is allowed to vary over the range of $R_{\rm skin}^{208}\!=\!(0.16\!-\!0.32)\,{\rm fm}$[@Chen:2014mza].
The model named SMC12 is a non-relativistic energy density functional of the Skyrme type. SMC12 has been devised to reproduce the binding energy ($B$) and charge radii ($R_{\rm ch}$) of ${}^{12}$C without compromising the accuracy in the description of other observables along the nuclear chart. Specifically, the fitting protocol has been based on that of the SAMi interaction [@RocaMaza:2012sj] with the following modifications: i) inclusion of ${}^{12}$C data ($B$ and $R_{\rm ch}$); and ii) relaxation of the weight on the pure neutron matter equation of state. This allowed us to accommodate the new data within the presented model. As an example, the experimental charge radii and nuclear masses of ${}^{12}$C, ${}^{16}$O, ${}^{40}$Ca, ${}^{48}$Ca, ${}^{90}$Zr, and ${}^{208}$Pb are reproduced to better than 1% accuracy, except for the binding energy of the two Calcium isotopes which are accurate at the percent level. This example justifies the reliability of the model for the present study.
In Fig. \[fig:1\] we present an example of the calculation which accounts for Coulomb distortion effects at forward ($\theta\!=\!29^\circ$) and backward ($\theta\!=\!145^\circ$) scattering angles.
In Table \[tab:1\] we provide the coefficients $p_0$, $p_1$, and $p_2$ obtained as a result of the calculation which accounts for Coulomb distortion effects at forward and backward scattering angles. These coefficients are defined by Eq. (\[eq:Deltapars\]).
[|l||\*[3]{}[l|]{}]{} & [$\theta\!=\!29^\circ$]{} & [$\theta\!=\!145^\circ$]{}\
$p_0$ &$+0.04005$ &$+0.09586$\
$p_1$ &$-0.50612$ &$-29.5132$\
$p_2$ &$-0.06969$ &$-3.86420$\
[9]{}
S. Schael [*et al.*]{} \[ALEPH and DELPHI and L3 and OPAL and SLD Collaborations and LEP Electroweak Working Group and SLD Electroweak Group and SLD Heavy Flavour Group\], Precision electroweak measurements on the $Z$ resonance, [Phys. Rept. (2006) 257.](https://doi.org/10.1016/j.physrep.2005.12.006) T. A. Aaltonen [*et al.*]{} \[CDF and D0 Collaborations\], Tevatron Run II combination of the effective leptonic electroweak mixing angle, [Phys. Rev. D [**97**]{} (2018), 112007.](https://doi.org/10.1103/PhysRevD.97.112007) R. Aaij [*et al.*]{} \[LHCb Collaboration\], Measurement of the forward-backward asymmetry in $Z/\gamma^{\ast} \rightarrow \mu^{+}\mu^{-}$ decays and determination of the effective weak mixing angle, [JHEP [**1511**]{} (2015) 190.](https://doi.org/10.1007/JHEP11(2015)190) ATLAS Collaboration, Measurement of the effective leptonic weak mixing angle using electron and muon pairs from Z-boson decay in the ATLAS experiment at $\sqrt s = 8$ TeV, [Tech. Rep. ATLAS-CONF-2018-037, CERN (2018).](http://inspirehep.net/record/1681883/files/ATLAS-CONF-2018-037.pdf)
A. M. Sirunyan [*et al.*]{} \[CMS Collaboration\], Measurement of the weak mixing angle using the forward-backward asymmetry of Drell-Yan events in pp collisions at 8 TeV, [Eur. Phys. J. C [**78**]{} (2018) 701.](https://doi.org/10.1140/epjc/s10052-018-6148-7) C. Y. Prescott [*et al.*]{}, Further Measurements of Parity Nonconservation in Inelastic electron Scattering, [Phys. Lett., 524 (1979).](https://doi.org/10.1016/0370-2693(79)91253-X) D. Wang [*et al.*]{} \[PVDIS Collaboration\], Measurement of parity violation in electron–quark scattering, [Nature [**506**]{}, no. 7486, 67 (2014).](https://doi.org/10.1038/nature12964) G. P. Zeller [*et al.*]{} \[NuTeV Collaboration\], A Precise Determination of Electroweak Parameters in Neutrino Nucleon Scattering, [Phys. Rev. Lett. [**88**]{} (2002) 091802;](https://doi.org/10.1103/PhysRevLett.88.091802) Erratum: \[[Phys. Rev. Lett. [**90**]{} (2003) 239902.](https://doi.org/10.1103/PhysRevLett.90.239902)\] C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner and C. E. Wieman, Measurement of parity nonconservation and an anapole moment in cesium, [Science [**275**]{}, 1759 (1997).](https://doi.org/10.1126/science.275.5307.1759) J. Guena, M. Lintz and M. A. Bouchiat, Measurement of the parity violating $6S$-$7S$ transition amplitude in cesium achieved within $2 \times 10^{-13}$ atomic-unit accuracy by stimulated-emission detection, [Phys. Rev. A [**71**]{} (2005) 042108.](https://doi.org/10.1103/PhysRevA.71.042108) D. Androić [*et al.*]{} \[Qweak Collaboration\], Precision measurement of the weak charge of the proton, [Nature [**557**]{}, no. 7704, 207 (2018).](https://doi.org/10.1038/s41586-018-0096-0) P. L. Anthony [*et al.*]{} \[SLAC E158 Collaboration\], Precision measurement of the weak mixing angle in Moller scattering, [Phys. Rev. Lett. [**95**]{} (2005) 081601.](https://doi.org/10.1103/PhysRevLett.95.081601) J. Erler and M. J. Ramsey-Musolf, The Weak mixing angle at low energies, [Phys. Rev. D [**72**]{}, 073003 (2005).](https://doi.org/10.1103/PhysRevD.72.073003) J. Erler and R. Ferro-Hernández, Weak Mixing Angle in the Thomson Limit, [JHEP [**1803**]{}, 196 (2018).](https://doi.org/10.1007/JHEP03(2018)196) J. Zhang [*et al.*]{}, Efficient inter-trap transfer of cold francium atoms, [Hyperfine Interact. [**237**]{} (2016) 150.](https://doi.org/10.1007/s10751-016-1347-9) D. Antypas, A. Fabricant, J. E. Stalnaker, K. Tsigutkin, V. V. Flambaum and D. Budker, Isotopic variation of parity violation in atomic ytterbium, [Nature Phys. (2019) 120.](https://doi.org/10.1038/s41567-018-0312-8) D. Becker [*et al.*]{}, The P2 experiment, [Eur. Phys. J. A [**54**]{} (2018) 208.](https://doi.org/10.1140/epja/i2018-12611-6) J. Benesch [*et al.*]{} \[MOLLER Collaboration\], The MOLLER Experiment: An Ultra-Precise Measurement of the Weak Mixing Angle Using M[ø]{}ller Scattering, [arXiv:1411.4088 \[nucl-ex\].](https://arxiv.org/abs/1411.4088) P. A. Souder, Parity Violation in Deep Inelastic Scattering with the SoLID Spectrometer at JLab, [Int. J. Mod. Phys. Conf. Ser. [**40**]{}, 1660077 (2016).](https://doi.org/10.1142/S2010194516600776) M. Thiel, C. Sfienti, J. Piekarewicz, C. J. Horowitz and M. Vanderhaeghen, Neutron skins of atomic nuclei: per aspera ad astra, [J. Phys. G [**46**]{} (2019) 093003.](https://doi.org/10.1088/1361-6471/ab2c6d) C. J. Horowitz and J. Piekarewicz, Neutron Star Structure and the Neutron Radius of ${}^{208}{\rm Pb}$, [Phys. Rev. Lett. [**86**]{} (2001) 5647.](https://doi.org/10.1103/PhysRevLett.86.5647) S. Abrahamyan [*et al.*]{}, Measurement of the Neutron Radius of ${}^{208}$Pb through Parity Violation in Electron Scattering, [Phys. Rev. Lett. [**108**]{} (2012) 112502.](https://doi.org/10.1103/PhysRevLett.108.112502) C. J. Horowitz [*et al.*]{}, Weak charge form factor and radius of ${}^{208}$Pb through parity violation in electron scattering, [Phys. Rev. C [**85**]{} (2012) 032501.](https://doi.org/10.1103/PhysRevC.85.032501) K. Paschke et al., PREX-II: Precision parity-violating measurement of the neutron skin of lead. <http://hallaweb.jlab.org/parity/prex/prexII.pdf>, Proposal to Jefferson Lab PAC 38, 2011.
J. Mammei et al. \[CREX collaboration\], Parity-violating measurement of the weak charge distribution of Ca to 0.02 fm accuracy. <http://hallaweb.jlab.org/parity/prex/c-rex2013_v7.pdf>, Proposal to Jefferson Lab PAC 40, 2013.
F. E. Maas and K. D. Paschke, Strange nucleon form-factors, [Prog. Part. Nucl. Phys. [**95**]{} (2017) 209.](https://doi.org/10.1016/j.ppnp.2016.11.001) The discussion of electroweak radiative corrections is deferred to a future work.
C. J. Horowitz, Parity violating elastic electron scattering and Coulomb distortions, [Phys. Rev. C [**57**]{} (1998) 3430.](https://doi.org/10.1103/PhysRevC.57.3430) Other choices of the reference frame would introduce corrections depending on the nuclear recoil. Up to now, no fully consistent way of accounting for recoil corrections in the Dirac formalism is known.
F. Salvat, A. Jablonski, C. J. Powell, ELSEPA — Dirac partial-wave calculation of elastic scattering of electrons and positrons by atoms, positive ions and molecules, [Comp. Phys. Comm. [**165**]{} (2005) 157.](https://doi.org/10.1016/j.cpc.2004.09.006)
X. Roca-Maza, M. Centelles, X. Viñas, and M. Warda, Neutron Skin of $^{208}$Pb, Nuclear Symmetry Energy, and the Parity Radius Experiment, [Phys. Rev. Lett. 106, (2011) 252501.](https://doi.org/10.1103/PhysRevLett.106.252501) O. Koshchii et al., to be published.
H. De Vries, C. W. De Jager and C. De Vries, Nuclear charge and magnetization density distribution parameters from elastic electron scattering, [Atom. Data Nucl. Data Tabl. [**36**]{} (1987) 495.](https://doi.org/10.1016/0092-640X(87)90013-1) I. Angeli and K.P. Marinova, Table of experimental nuclear ground state charge radii: An update, [Atom. Data Nucl. Data Tabl. [**99**]{} (2013) 69.](https://doi.org/10.1016/j.adt.2011.12.006) J. Piekarewicz, A. R. Linero, P. Giuliani and E. Chicken, Power of two: Assessing the impact of a second measurement of the weak-charge form factor of $^{208}$Pb, [Phys. Rev. C [**94**]{}, 034316 (2016).](https://doi.org/10.1103/PhysRevC.94.034316)
W.-C. Chen and J. Piekarewicz, Building relativistic mean field models for finite nuclei and neutron stars, [Phys. Rev. C [**90**]{} (2014) 044305.](https://doi.org/10.1103/PhysRevC.90.044305) W.-C. Chen and J. Piekarewicz, Searching for isovector signatures in the neutron-rich oxygen and calcium isotopes, [Phys. Lett. B [**748**]{} (2015) 284.](https://doi.org/10.1016/j.physletb.2015.07.020
Get ) X. Roca-Maza, G Colò and H. Sagawa, A new Skyrme interaction with improved spin-isospin properties, [Phys. Rev. C [**86**]{} (2012) 031306.](https://doi.org/10.1103/PhysRevC.86.031306)
O. Moreno and T. W. Donnelly, Nuclear structure uncertainties in parity-violating electron scattering from ${}^{12}$C, [Phys. Rev. C [**89**]{}, 015501 (2014).](https://doi.org/10.1103/PhysRevC.89.015501) J. C. Hardy and I. S. Towner, Superallowed $0^+\!\to\!0^+$ nuclear $\beta$ decays: 2014 critical survey, with precise results for $V_{ud}$ and CKM unitarity, [Phys. Rev. C [**91**]{} (2015), 025501.](https://doi.org/10.1103/PhysRevC.91.025501)
|
---
abstract: 'Extreme environmental phenomena such as major precipitation events manifestly exhibit spatial dependence. Max-stable processes are a class of asymptotically-justified models that are capable of representing spatial dependence among extreme values. While these models satisfy modeling requirements, they are limited in their utility because their corresponding joint likelihoods are unknown for more than a trivial number of spatial locations, preventing, in particular, Bayesian analyses. In this paper, we propose a new random effects model to account for spatial dependence. We show that our specification of the random effect distribution leads to a max-stable process that has the popular Gaussian extreme value process (GEVP) as a limiting case. The proposed model is used to analyze the yearly maximum precipitation from a regional climate model.'
address:
- |
Department of Statistics\
North Carolina State University\
Raleigh, North Carolina 27560\
USA\
- |
Department of Statistics\
University of California—Berkeley\
Berkeley, Berkeley 94720\
USA\
author:
-
-
title: 'A hierarchical max-stable spatial model for extreme precipitation'
---
Introduction {#sintro}
============
Spatial statistical techniques are crucial for accurately quantifying the likelihood of extreme events and monitoring changes in their frequency and intensity. Extreme events are by definition rare, therefore, estimation of local climate characteristics can be improved by borrowing strength across nearby locations. While methods for univariate extreme data are well developed, modeling spatially-referenced extreme data is an active area of research. Max-stable processes \[@dehaan-2006a\] are the natural infinite-dimensional generalization of the univariate generalized extreme value (GEV) distribution. Just as the only limiting distribution of the scaled maximum of independent univariate random variables is the GEV, the scaled maximum of independent copies of any stochastic process can only converge to a max-stable process. Max-stable process models for spatial data may be constructed using the spectral representation of @dehaan-1984a. Max-stable processes built from this representation were first used for spatial analysis by @smith-1990a. Since then, a handful of subsequent spatial max-stable process models have been proposed, notably that of @schlather-2002a and @kabluchko-2009a, who proposed a more general construction that includes several other known models as special cases. Applications of spatial max-stable processes include @coles-1993a, @buishand-2008a, and @blanchet-2011.
Because closed-form expressions for the likelihoods associated with spatial max-stable processes are not available, parameter estimation and inference is problematic. Taking advantage of the availability of bivariate densities, @padoan-2010a suggest maximum pairwise likelihood estimation and asymptotic inference based on a sandwich matrix (composed of expected derivatives of the composite likelihood function) to properly account for using a pairwise likelihood when computing standard errors \[@godambe-1987a\]. Recently, @genton-2011 extend this approach using composite likelihood based on trivariate densities. The problem of spatial prediction, conditional on observations, for max-stable random fields (analogous to Kriging for Gaussian processes) has also proven difficult. The recent conditional sampling algorithm of @wang-2010a is capable of producing both predictions and prediction standard errors for most spatial max-stable models of practical interest, subject to discretization errors that can be made arbitrarily small.
Bayesian estimation and inference for max-stable process models for spatial data on a continuous domain has been elusive. Implementing these models in a fully-Bayesian framework has several advantages, including incorporation of prior information and natural uncertainty assessment for model parameters and predictions. Approximate Bayesian methods based on asymptotic properties of the pairwise likelihood function are possible. @ribatet-2010a use an estimated sandwich matrix to adjust the Metropolis ratio within an MCMC sampler, while @shaby-2010c rotates and scales the MCMC sample post-hoc and @smith-2009a use pairwise likelihoods without adjustment. Bayesian models that are not based on max stable processes have been used for analysis of extreme values with spatial structure. @cooley-2007a uses a hierarchical model with a conditionally-independent generalized Pareto likelihood, incorporating all spatial dependence through Gaussian process priors on the generalized Pareto likelihood parameters. Spatial dependence has also been achieved through Bayesian Gaussian copula models \[@sang-2010a\] and through a more flexible copula based on a Dirichlet process construction \[@fuentes-2010a\].
We develop a new hierarchical Bayesian model for analyzing max-stable processes. The responses are modeled as independent univariate GEV conditioned on spatial random effects with positive stable random effect distribution. Positive stable random effects have been used to model multivariate extremes with finite dimensions \[@fougeres-2009 [@stephenson-2009]\]. We extend this approach to accommodate data on a continuous spatial domain. We show that the resulting model is max-stable marginally over the random effects, and that a limiting case of this construction provides a finite-dimensional approximation to the well-known Gaussian extreme value process (GEVP) of @smith-1990a, often referred to as the “Smith process.” Lower-dimensional representations have previously been proposed for high-dimensional extremes in various settings \[@pickands-1981 [@Deheuvel-1983; @Schlather-2002; @Ehlert-2008; @wang-2010a; @wang-2010c; @oesting-2011a; @engelke-2011a]\]. Our construction permits analysis of the joint distribution of all observations, and thus can produce straightforward predictions at unobserved locations. Because we use a hierarchical model to represent the spatial max-stable process, a Bayesian implementation is a natural choice. This allows us to model underlying marginal structures as flexibly as we like, in addition to automatic pooling of information and uncertainty propagation. Also, the proposed framework permits representing the the spatial process using a lower-dimensional representation, which leads to efficient computing for large spatial data sets.
The remainder of the paper proceeds as follows. Section \[smodel\] describes the model, which is compared to the GEVP in Section \[ssmith\]. The method is evaluated using a simulation study in Section \[ssim\]. In Section \[stemp\] we use the proposed method to analyze yearly maximum precipitation using regional climate model output from the North American Regional Climate Change Assessment Program (NARCCAP) in the eastern US. Section \[sconc\] concludes.
The hierarchical max-stable process model {#smodel}
=========================================
Spatial random effects model
----------------------------
Let $Y({\mathbf{s}})$ be the extreme value at location ${\mathbf{s}}$, defined over the region ${\mathbf{s}}\in{\mathcal{D}}\subset{\mathcal{R}}^2$. Here we describe a max-stable model for $Y({\mathbf{s}})$ assuming that it is a block-maximum, that is, the maximum of many observations taken at location ${\mathbf{s}}$, such as the yearly maximum of daily precipitation levels. However, we note that max-stable models are increasingly being used to model extreme individual observations using a points above threshold approach \[@Huser-2012\], and the residual max-stable process model described here may be applicable to this type of analysis as well. We describe a model for a single realization of the process and extend to multiple independent realizations in Section \[sST\].
Assuming the process is max-stable, then the marginal distribution of $Y({\mathbf{s}})$ is $\operatorname{GEV}[\mu({\mathbf{s}}),\sigma({\mathbf{s}}),\xi({\mathbf{s}})]$, where $\mu
({\mathbf{s}})$ is the location, $\sigma({\mathbf{s}})>0$ is the scale, and $\xi({\mathbf{s}})$ is the shape (GEV distribution is described in Appendix \[appa1\]). Equivalently \[@resnick-1987\], we may express $Y({\mathbf{s}}) = \mu({\mathbf{s}}) + \frac{\sigma
({\mathbf{s}})}{\xi({\mathbf{s}})} [X({\mathbf{s}})^{\xi({\mathbf{s}})}-1 ]$, where $X({\mathbf{s}})$ is the residual max-stable process with unit ${\mbox{Fr\'{e}chet}}$ margins, that is, $X({\mathbf{s}})\sim \operatorname{GEV}(1,1,1)$. To allow for both nonspatial and spatial residual variability, we model $X({\mathbf{s}})$ as the product $X({\mathbf{s}})=U({\mathbf{s}})\theta({\mathbf{s}})$. Borrowing a term from geostatistics, we refer to $U({\mathbf{s}})$ as the nugget effect since it accounts for nonspatial variation due to measurement error or other small-scale features. The nugget is modeled as $U({\mathbf{s}}){\stackrel{\mathrm{i.i.d.}}{\sim}}\operatorname{GEV}(1,\alpha,\alpha)$, where, as described in detail below, the parameter $\alpha\in(0,1)$ controls the relative contribution to the nugget effect.
Residual spatial dependence is captured by $\theta({\mathbf{s}})$. We express the spatial process as a function of a linear combination of $L$ kernel basis functions $w_l({\mathbf{s}})\ge0$, scaled so that $\sum_{l=1}^Lw_l({\mathbf{s}})=1$ for all ${\mathbf{s}}$. The spatial process is $\theta({\mathbf{s}})= [\sum_{l=1}^LA_{l}w_{l}({\mathbf{s}})^{1/\alpha} ]^{\alpha}$, where $A_l$ are the basis function coefficients. To ensure max-stability and ${\mbox{Fr\'{e}chet}}$ marginal distributions, the random effects $A_l$ follow the positive stable distribution with density $p(A|\alpha)$ which has Laplace transformation $\int_0^\infty\exp(-At)p(A|\alpha)\,dA = \exp
(-t^\alpha
)$. We denote this as $A_l\sim\operatorname{PS}(\alpha)$. Although $p(A|\alpha)$ has no closed form, it possesses the essential feature that if $A_1,\ldots,A_T{\stackrel{\mathrm{i.i.d.}}{\sim}}\operatorname{PS}(\alpha)$, then $(A_1+\cdots+A_T)/T^{1/\alpha}\sim \operatorname{PS}(\alpha)$. Appendix \[appa2\] verifies that this model for $X({\mathbf{s}})$ is max-stable with unit ${\mbox{Fr\'{e}chet}}$ marginal distributions.
Marginalizing over the nugget terms $U({\mathbf{s}})$ gives the hierarchical model $$\begin{aligned}
\label{auxmodel} Y({\mathbf{s}}_i) | A_{1},\ldots,A_{L} &
{\stackrel{\mathrm{indep}}{\sim}}& \operatorname{GEV}\bigl[\mu^*({\mathbf{s}}_i),\sigma^*(
{\mathbf{s}}_i),\xi^*({\mathbf{s}}_i)\bigr],
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
A_{l} &{\stackrel{\mathrm{i.i.d.}}{\sim}}& \operatorname{PS}(\alpha),\end{aligned}$$ where $\mu^*({\mathbf{s}}) = \mu({\mathbf{s}}) + \frac{\sigma({\mathbf{s}})}{\xi({\mathbf{s}})} [\theta
({\mathbf{s}})^{\xi({\mathbf{s}})}-1 ]$, $\sigma^*({\mathbf{s}}) = \alpha\sigma({\mathbf{s}})\theta({\mathbf{s}})^{\xi({\mathbf{s}})}$, and $\xi^*({\mathbf{s}})=\alpha\xi({\mathbf{s}})$. The responses are conditionally independent given the random effects ${\mathbf{A}}$. The effect of conditioning on ${\mathbf{A}}= (A_{1},\ldots,A_{L})^T$, and thus the spatial process $\theta$, is to move spatial dependence from the residuals to a random effect in the GEV parameters. Marginalizing over the random effects induces spatial dependence. The joint distribution function of the residual process $X$ at $n$ locations ${\mathbf{s}}_1, \ldots, {\mathbf{s}}_n$ is $$\label{ASL} \mathrm{P}\bigl(X({\mathbf{s}}_i)<c_i, i=1,\ldots,n
\bigr) = \exp \Biggl\{-\sum_{l=1}^L \Biggl[
\sum_{i=1}^n \biggl(\frac{w_l({\mathbf{s}}_i)}{c_i}
\biggr)^{1/\alpha} \Biggr]^{\alpha
} \Biggr\}.$$ Therefore, although this is a process model defined on a continuous spatial domain, the finite-dimensional distributions are multivariate GEV (MGEV) with asymmetric logistic dependence function \[@Tawn-1990\].
Spatial dependence is often summarized by the extremal coefficient \[@smith-1990a\]. The pairwise extremal coefficient $\vartheta
({\mathbf{s}}_i,{\mathbf{s}}_j) \in[1,2]$ is defined by the relationship $$\label{ECdef} P\bigl(X({\mathbf{s}}_i)<c,X({\mathbf{s}}_j)<c\bigr) = P
\bigl(X({\mathbf{s}}_i)<c\bigr)^{\vartheta({\mathbf{s}}_i,{\mathbf{s}}_j)}.$$ If $X({\mathbf{s}}_i)$ and $X({\mathbf{s}}_j)$ are independent, then $\vartheta({\mathbf{s}}_i,{\mathbf{s}}_j) = 2$; in contrast, if $X({\mathbf{s}}_i)$ and $X({\mathbf{s}}_j)$ are completely dependent, then $\vartheta({\mathbf{s}}_i,{\mathbf{s}}_j) = 1$. The extremal coefficient introduced by (\[auxmodel\]) is $$\label{ECgeneral} \vartheta({\mathbf{s}}_i,{\mathbf{s}}_j) = \sum
_{l=1}^L \bigl(w_{l}(
{\mathbf{s}}_i)^{1/\alpha
}+w_{l}({\mathbf{s}}_j)^{1/\alpha}
\bigr)^{\alpha}.$$ Therefore, the extremal coefficient is the sum (over the $L$ kernels) of the $L^{1/\alpha}$ norms of the vectors $[w_l({\mathbf{s}}_i),w_l({\mathbf{s}}_j)]$.
To see how $\alpha$ controls the nugget effect, consider two observations at the same location, ${\mathbf{s}}_i={\mathbf{s}}_j$. The two observations share the same kernels, $w_l({\mathbf{s}}_i)=w_l({\mathbf{s}}_j)$ and thus $\theta({\mathbf{s}}_i)=\theta({\mathbf{s}}_j)$, but have different nugget terms $U({\mathbf{s}}_i)\ne
U({\mathbf{s}}_j)$. In this case, the extremal coefficient is $2^\alpha$. If $\alpha
=1$, then the nugget dominates and for all pairs of locations, regardless of their spatial locations \[since $\sum_{l=1}^Lw_l({\mathbf{s}})=1$ for all ${\mathbf{s}}$\]. If $\alpha=0$, then $\vartheta
({\mathbf{s}}_i,{\mathbf{s}}_j)=1$ when ${\mathbf{s}}_i={\mathbf{s}}_j$, and there is no nugget effect. The characteristics of the model are shown graphically in Figure \[fsamples\]. In these random draws from the model, we see the process is very smooth for $\alpha=0.1$ and has little discernable spatial pattern with $\alpha= 0.9$.
![Random draws on a $50\times50$ grid from the random effect model for various $\alpha$. Each sample has a knot at each data point, GEV parameters $\mu({\mathbf{s}})=\log
[\sigma({\mathbf{s}})]=0$ and $\xi({\mathbf{s}})=-0.1$, and bandwidth $\tau=2$.[]{data-label="fsamples"}](591f01.eps)
The parameter $\alpha$ clearly plays an important role in this model. It determines the magnitude of the nugget effect, the form of spatial dependence function in (\[ECgeneral\]), and the shape and scale of the conditional distributions in (\[auxmodel\]). To illustrate the links between the contribution of $\alpha$ to these aspects of the model, we consider the extreme cases with $\alpha=0$ and $\alpha=1$. With $\alpha= 1$, $p(A|\alpha)$ concentrates its mass on $A=1$, and thus $\theta({\mathbf{s}})= \sum_{l=1}^Lw_{l}({\mathbf{s}})=1$. In this case, the conditional and marginal GEV parameters are the same, for example, $\mu^*({\mathbf{s}})= \mu({\mathbf{s}})$, there is no residual dependence with $\theta
({\mathbf{s}}_i,{\mathbf{s}}_j)=2$, and thus $Y({\mathbf{s}}){\stackrel{\mathrm{indep}}{\sim}}\operatorname{GEV}[\mu({\mathbf{s}}),\sigma({\mathbf{s}}),\xi({\mathbf{s}})]$. On the other hand, if $\alpha\approx0$, then the conditional scale $\sigma^*({\mathbf{s}})\approx0$ and $Y({\mathbf{s}}) \approx\mu^*({\mathbf{s}})$, a continuous spatial process with strong small-scale spatial dependence $\theta({\mathbf{s}},{\mathbf{s}})\approx1$.
Spatial prediction (analogous to Kriging) at a new location ${\mathbf{s}}^*$ is straight-forward under this hierarchical model. Predictions are made by simply computing $\theta({\mathbf{s}}^*)=[\sum_{l=1}^LA_{l}w_{l}({\mathbf{s}}^*)^{1/\alpha
}]^{\alpha}$, and then sampling $Y({\mathbf{s}}^*)$ from the independent GEV in (\[auxmodel\]). Repeating this at every MCMC iteration gives samples from the posterior predictive distribution of $Y({\mathbf{s}}^*)$.
Kernel and knot selection {#knot}
-------------------------
Although other kernels are possible, we use a scaled version of the Gaussian kernel $$\label{K} K({\mathbf{s}}|{\mathbf{v}}_l,\tau) = \frac{1}{2\pi\tau^2}\exp \biggl[-
\frac
{1}{2\tau^2}({\mathbf{s}}-{\mathbf{v}}_l)^T({\mathbf{s}}-{\mathbf{v}}_l)
\biggr],$$ where ${\mathbf{v}}_1,\ldots,{\mathbf{v}}_L\in{\mathcal{R}}^2$ are spatial knots and $\tau>0$ is the kernel bandwidth. To ensure that the kernels sum to one at each location, the kernels are scaled as $$\label{w} w_l({\mathbf{s}}) = \frac{K({\mathbf{s}}|{\mathbf{v}}_{l},\tau)}{\sum_{j=1}^L K({\mathbf{s}}|{\mathbf{v}}_{j},\tau)}.$$ The knots are taken as a fixed and regularly-spaced grid of points covering the spatial domain. Section \[ssmith\] shows that this choice of kernel function and knot distribution gives the GEVP as a limiting case. Even with a regular grid of knots, the extremal coefficient is nonstationary, that is, $\vartheta({\mathbf{s}}_i,{\mathbf{s}}_j)$ is not simply a function of $\Vert {\mathbf{s}}_i-{\mathbf{s}}_j\Vert $. For example, $w({\mathbf{s}}_i)$ may not equal $w({\mathbf{s}}_j)$ if ${\mathbf{s}}_i$ is close to a knot and ${\mathbf{s}}_j$ is not. This discretization artifact dissipates for large $L$.
While the extremal coefficient does not fully characterize spatial dependence, it is useful for guiding knot selection. Knot selection poses a trade-off between computational burden with too many knots and poor fit with too few knots. Consider the case of a Gaussian kernel with bandwidth $\tau=1$ and knots on a large rectangular grid with grid spacing $d$. Figure \[fextremal\] plots the extremal coefficient for points $(0,0)$ and $(0,h)$ as a function of separation distance $h$. The extremal coefficient has nearly an identical shape for all $d$ less than or equal to $\tau$. For $d=1.25\tau$, the extremal coefficient differs slightly from the fine grids, and for $d>1.25\tau$ the extremal coefficient deviates considerably from the fine grids, especially for small $\alpha$. These results will scale for other $\tau$, therefore, a rule of thumb is to select the knots so that the grid spacing is approximately equal to the kernel bandwidth. Knot selection is discussed further in Section \[ssim\].
![Extremal coefficient $\vartheta({\mathbf{s}}_1,{\mathbf{s}}_2)$, where ${\mathbf{s}}_1 =
(0,0)$, ${\mathbf{s}}_2=(0,h)$, the kernel bandwidth is $\tau=1$, and the knots form a rectangular grid with grid spacing $d$.[]{data-label="fextremal"}](591f02.eps)
Adaption for the NARCCAP data {#sST}
-----------------------------
In Section \[stemp\] we analyze climate model output for $T>1$ years, which requires additional notation. Denote $Y_t({\mathbf{s}})$ as the response for year $t$ and site ${\mathbf{s}}$. Assuming the years are independent and identically distributed (over years, not space) gives $$\begin{aligned}
\label{auxmodel2} Y_t({\mathbf{s}}_i) | A_{1t},\ldots,A_{Lt}
&{\stackrel{\mathrm{indep}}{\sim}}& \operatorname{GEV} \bigl[\mu_t^*({\mathbf{s}}_i),
\sigma_t^*({\mathbf{s}}_i),\xi^*({\mathbf{s}}_i) \bigr],
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
A_{lt} &{\stackrel{\mathrm{i.i.d.}}{\sim}}& \operatorname{PS}(\alpha),\end{aligned}$$ where $\theta_t({\mathbf{s}})=[\sum_{l=1}^LA_{lt}w_{l}({\mathbf{s}})^{1/\alpha
}]^{\alpha
}$ is the spatial random effect for year $t$, $\mu_t^*({\mathbf{s}}) = \mu
({\mathbf{s}})
+ \frac{\sigma({\mathbf{s}})}{\xi({\mathbf{s}})} (\theta_t({\mathbf{s}})^{\xi({\mathbf{s}})}-1 )$, $\sigma_t^*({\mathbf{s}})=\alpha\sigma({\mathbf{s}})\theta_t({\mathbf{s}})^{\xi({\mathbf{s}})}$, and $\xi^*({\mathbf{s}})=\alpha\xi({\mathbf{s}})$. Note that while the GEV parameters conditioned on $\theta_t({\mathbf{s}})$ in (\[auxmodel2\]) vary by year, marginally, $Y_t({\mathbf{s}})\sim\operatorname{GEV}[\mu({\mathbf{s}}),\sigma({\mathbf{s}}),\xi({\mathbf{s}})]$ for all $t$.
Gaussian process priors are used for the GEV parameters $\mu({\mathbf{s}})$, $\gamma({\mathbf{s}}) = \log[\sigma({\mathbf{s}})]$, and $\xi({\mathbf{s}})$. The Gaussian process $\mu$ has mean $\mathbf{x}({\mathbf{s}})^T{\bolds{\beta}}_{\mu}$, where $\mathbf{x}({\mathbf{s}})$ includes the spatial covariates such as elevation. The spatial covariance of $\mu$ is ${ \mbox{Mat\'{e}rn}}$ \[@Banerjee03 [@Cressie93; @Handbook]\] with variance $\delta_{\mu}^2>0$, range $\rho_{\mu}>0$, and smoothness $\nu_{\mu}>0$. The other GEV parameters $\gamma({\mathbf{s}})$ and $\xi({\mathbf{s}})$ are modeled similarly. In some applications, it may also be desirable to allow for the GEV parameters to evolve over time, perhaps following a separate linear time trend at each location, which would be a straightforward modification of this model. The MCMC algorithm used to sample from this model is described in Appendix \[appa3\].
Connection with the Gaussian extreme value process {#ssmith}
==================================================
The GEVP of @smith-1990a is a well-known spatial max-stable process. In this section we show that the proposed positive stable random effects model in Section \[smodel\] contains this model as a limiting case. The GEVP construction for the residual process is $$\label{Smith} X({\mathbf{s}}) = \max\bigl\{h_{1}K({\mathbf{s}}|{\mathbf{u}}_{1},
\Sigma),h_{2}K({\mathbf{s}}|{\mathbf{u}}_{2},\Sigma ),\ldots\bigr\},$$ where $\{(h_1,{\mathbf{u}}_1), (h_2,{\mathbf{u}}_2),\ldots\}$ follows a Poisson process with intensity $\lambda(h,{\mathbf{u}}) = h^{-2}I(h>0)$, and $K$ is a kernel function standardized so that $\int K({\mathbf{s}}|{\mathbf{u}},\tau)\,d{\mathbf{u}}=1$ for all ${\mathbf{s}}$. The construction (\[Smith\]) is a special case of the de Haan \[@dehaan-1984a\] spectral representation. A useful analogy is to think of $X({\mathbf{s}})$ as the maximum rainfall at site ${\mathbf{s}}$, generated as the maximum over a countably-infinite number of storms. The $k$[th]{} storm has center ${\mathbf{u}}_{k}\in{\mathcal{R}}^2$, intensity $h_{k}>0$, and spatial range given by $K({\mathbf{s}}|{\mathbf{u}}_{k},\tau)$.
Under this model, the joint distribution at locations ${\mathbf{s}}_1,\ldots,{\mathbf{s}}_n$ is $$\label{joint1}
\quad\mathrm{P} \bigl[X({\mathbf{s}}_1)<c_1,\ldots,X(
{\mathbf{s}}_n)<c_n \bigr] = \exp \biggl[-\int\max_i
\biggl\{\frac{K({\mathbf{s}}_i|{\mathbf{u}},\Sigma
)}{c_i} \biggr\}\,d{\mathbf{u}}\biggr].$$ The GEVP has extremal coefficient $$\label{ECsmith} \vartheta({\mathbf{s}}_i,{\mathbf{s}}_j) = \int\max \bigl
\{K({\mathbf{s}}_i|{\mathbf{u}},\Sigma ),K({\mathbf{s}}_j|{\mathbf{u}},\Sigma) \bigr\}\,d{\mathbf{u}},$$ which simplifies to $\vartheta({\mathbf{s}}_i,{\mathbf{s}}_j)=2\Phi (\frac
{\Vert {\mathbf{s}}_i-{\mathbf{s}}_j\Vert }{2\tau} )$ for the Gaussian kernel (\[K\]). This does not include a nugget effect, since $\vartheta({\mathbf{s}}_i,{\mathbf{s}}_j)=1$ if $\Vert {\mathbf{s}}_i-{\mathbf{s}}_j\Vert =0$.
The connection to the model in Section \[smodel\] is made by restricting the storm locations to the set of $L$ knot locations $\{{\mathbf{v}}_1,\ldots,{\mathbf{v}}_L\}$ and rescaling the kernels to sum to one as in (\[w\]), giving $$\label{Smith3} X({\mathbf{s}}) = \max \bigl\{h_{1}w_1(
{\mathbf{s}}),\ldots,h_{L}w_L({\mathbf{s}}) \bigr\}.$$ This amounts to truncating the de Haan spectral representation. If $h_l\sim \operatorname{GEV}(1,1,1)$, then $X({\mathbf{s}})$ is max-stable with joint distribution $$\label{joint2} \mathrm{P} \bigl[X({\mathbf{s}}_1)<c_1,\ldots,X(
{\mathbf{s}}_n)<c_n \bigr] = \exp \Biggl[-\sum
_{l=1}^L\max_i \biggl\{
\frac{w_l({\mathbf{s}}_i)}{c_i} \biggr\} \Biggr],$$ which implies that the marginal distributions are unit ${\mbox{Fr\'{e}chet}}$. For equally-spaced knots, this distribution converges weakly to the full GEVP distribution function (\[joint1\]) as $L$ increases. We note that this finite approximation could be applied to other max-stable models such as those in @schlather-2002a and @kabluchko-2009a by allowing the functions $K$ to be suitably scaled Gaussian processes, unlike the current approach where $K$ is a kernel function.
Using the model described by (\[joint2\]) directly is problematic because it may not yield a proper likelihood. The process (\[Smith3\]) at $n$ locations $\{X({\mathbf{s}}_1),\ldots,\break X({\mathbf{s}}_n)\}$ is completely determined by the intensities $\{h_1,\ldots,h_L\}$. Therefore, the likelihood for $\{X({\mathbf{s}}_1),\ldots,X({\mathbf{s}}_n)\}$ requires a map from $\{
X({\mathbf{s}}_1),\ldots,X({\mathbf{s}}_n)\}$ to $\{h_1,\ldots,h_L\}$. This map may not exist, for example, if $L<n$, and generally does not have a closed form. This is common in dimension reduction methods for Gaussian process models \[e.g., @higdon-1998, @banerjee-2008a, and @cressie-2008a\].
As with the Gaussian process dimension reduction methods, the model in Section \[smodel\] includes both a spatial process ($\theta$) and a nonspatial nugget term ($U$). Comparing (\[ASL\]) and (\[joint2\]), we see a result of the nugget effect is that the $L^{\infty}$ norm (the maximum) in (\[joint2\]) is replaced with the $L^{1/\alpha}$ norm, and that (\[ASL\]) converges weakly to (\[joint2\]) as $\alpha$ goes to zero. Including a nugget aids in computation, as the likelihood becomes a simple product of univariate GEV densities. Including a nugget term also has advantages beyond computation. The GEVP has been criticized as unrealistically smooth \[@blanchet-2011\], and so a nugget may improve fit. Analogously, in the geostatistical literature for Gaussian data a nugget is not required, but is used routinely to account for small-scale phenomena that cannot be captured with a smooth spatial process \[@Cressie93 [@Banerjee03; @Handbook]\].
Simulation study {#ssim}
================
In this section we conduct a simulation study to verify that the MCMC algorithm produces reliable results, to investigate sensitivity to knot selection, and to determine which parameters are the most difficult to estimate. Data and knots are placed on $m\times m$ regular grids covering $[l,u]\times[l,u]$, denoted ${\mathcal{S}}(m,l,u)$. For each simulation design, we generate data from the model described in Section \[sST\] at the $n=49$ locations ${\mathcal{S}}(7,0,6)$ and $T=10$ independent years. The GEV location parameter varies by site following the Gaussian process with mean zero, variance one, and exponential spatial correlation $\exp(-\Vert {\mathbf{s}}_i-{\mathbf{s}}_j\Vert /2)$. Unlike the analysis of the NARCCAP data in Section \[stemp\], the GEV scale and shape parameters are assumed to be the same for all sites and fixed at $\sigma({\mathbf{s}})=1$ and $\xi({\mathbf{s}})=0.2$. We fix these parameters in the simulation study for computational purposes, and because these spatially-varying parameters will likely be hard to estimate for these moderately-sized simulated data sets. The simulations vary by the nugget effect ($\alpha$), the kernel bandwidth ($\tau$), and the number of knots used to generate the data ($L_0$). The simulation designs are numbered:
$L_0=49$ knots at ${\mathcal{S}}(7,0,6)$, $\alpha=0.3$, $\tau=3$,
$L_0=49$ knots at ${\mathcal{S}}(7,0,6)$, $\alpha=0.7$, $\tau=3$,
$L_0=25$ knots at ${\mathcal{S}}(5,0,6)$, $\alpha=0.3$, $\tau=3$,
$L_0=25$ knots at ${\mathcal{S}}(5,0,6)$, $\alpha=0.7$, $\tau=3$,
$L_0=10\mbox{,}000$ knots at ${\mathcal{S}}(100,-1,7)$, $\alpha=0.4$, $\tau=1$.
For the first four designs, the number of knots used to generate the data is small enough to permit fitting the model with the correct number of knots. We use these examples to explore sensitivity to knot selection. The final design with $L_0=10\mbox{,}000$ knots represents the limiting case with more knots than can be fit computationally. Here we fit several course grids of knots and compare performance as the number of knots increases to provide recommendations on the number of knots needed to provide a good approximation to the limiting process.
$M=50$ data sets are generated for each simulation design. For each simulated data set, we fit the model with a varying number of knots. For the first four designs we compare $L=25$ knots at ${\mathcal{S}}(5,0,6)$ and $L=49$ knots at ${\mathcal{S}}(7,0,6)$ to compare fits with the true knots and either too few ($L=25$ for designs 1 and 2) or too many ($L=49$ for designs 3 and 4) knots. For the final design we compare fits with 8 knot grids: $L=25$ knots at ${\mathcal{S}}(5,-1,7), \ldots, L=144$ knots at ${\mathcal{S}}(12,-1,7)$. The spatial covariance parameters for the GEV location have priors $\delta_{\mu}^2\sim \operatorname{InvGamma}(0.1,0.1)$ and range $\rho_{\mu}\sim \operatorname{InvGamma}(0.1,0.1)$; for this relatively small spatial domain we fix the smoothness parameter $\nu_{\mu}=0.5$, giving an exponential covariance. The design matrix ${\mathbf{X}}$ includes only the intercept with $\beta_{\mu}\sim\mathrm{N}(0,100^2)$. The GEV log scale and shape are constant across space and have $\mathrm{N}(0,1)$ and $\mathrm{N}(0,0.25^2)$ priors, respectively. The residual dependence parameters have priors $\tau\sim \operatorname{InvGamma}(0.1,0.1)$ and $\alpha\sim \operatorname{Unif}(0,1)$.
The results are presented in Figure \[fsim\]. For each data set, we compute the posterior mean of the GEV parameters at each location and the mean squared error (MSE) of the posterior means (averaged over the $n$ sites for the spatially-varying GEV location). Figure \[fsim\] plots the $M=50$ root MSEs and coverage probabilities (averaged over sites for the GEV location).
![Boxplots of root mean squared error (RMSE) for the GEV parameters in the simulation study. The horizontal lines in the boxplots are the 0.05, 0.25, 0.50, 0.75, and 0.95 quantiles of RMSE. Coverage percentages of posterior 95% intervals are given below the boxplots.[]{data-label="fsim"}](591f03.eps)
For the data generated with $L_0=25$ or $L_0=49$ knots in Figure \[fsim\](a), the coverage probabilities are generally near the nominal level. With $L=49$ knots, the coverage probabilities range from 0.90 to 0.94 for the GEV location. For the first two designs, the model with $L=25$ knots has fewer knots than were used to generate the data. This does not have a substantial impact on the estimation of the GEV location. However, using too few knots leads to increased RMSE and under-coverage for the GEV log scale, especially for design 1 with strong spatial dependence. For simulation designs 3 and 4, the model with $L=49$ knots has nearly twice as many knots than were used to generate the data. In these cases, the $L=49$ model performs nearly as well as the correct $L=25$ model. For these simulation settings, we conclude that using too few knots can lead to poor results, especially for the scale parameter, but that including too many knots does not degrade performance.
For the data generated with $L_0=10\mbox{,}000$ knots in Figure \[fsim\](b), we use knots grids with $L = 25,36,\ldots,144$ points. For comparison with the kernel bandwidth, rather than plotting the results by $L$, we plot results by the spacing between adjacent knots in the same column or row, which ranges from 0.70 for $L=144$ to 2.00 for $L=25$. The coverage probabilities are near or above the nominal level for all grid spacings at or below the bandwidth, $\tau=1.0$, and the RMSE appears to be fairly constant for all grid spacings at least as small as the bandwidth. Therefore, this appears to be a reasonable rule of thumb for selecting the number of knots.
We also computed RMSE for the spatial dependence parameters $\alpha$ and $\tau$ (not shown in Figure \[fsim\]) for this final case. For $\alpha$, the average (over data sets) RMSE was 0.049 (coverage percentage 96%), 0.060 (88%), and 0.101 (40%) for grid spacings 0.7, 1.0, and 2.0, respectively. For $\tau$, the average RMSE was 0.102 (88% ), 0.107 (90%), and 0.233 (38%) for grid spacings 0.7, 1.0, and 2.0, respectively. As with the GEV parameters, the approximation with the grid spacing at least as small as the bandwidth appears to provide reasonable estimation of the spatial dependence parameters. When too few knots are used, the bandwidth is often overestimated to compensate for the lack of knots and, thus, RMSE is high and coverage is far below the nominal level.=1
Analysis of regional climate model output {#stemp}
=========================================
To illustrate the proposed method, we analyze climate model output provided by the North American Regional Climate Change Assessment Program (NARCCAP). Our objective is to study changes in extreme precipitation under various climate scenarios in different spatial regions while accounting for residual spatial dependence remaining after allowing for spatially-varying GEV parameters. The data are downloaded from the website <http://www.narccap.ucar.edu/index.html>. We analyze output from two timeslice experiments. Both runs use the Geophysical Fluid Dynamics Laboratory’s AM2.1 climate model with 50 km resolution. The model is run separately under historical (1969–2000) and future conditions (2039–2070). Observational data is used for the sea-surface temperature and ice boundary conditions in the historical run. The boundary conditions for the future run are perturbations of the historical boundary conditions. The amount of perturbation is based on a lower resolution climate model. The perturbations assume the A2 emissions scenario \[Nakicenovic et al. ([-@Na00])\], which increases CO$_2$ concentration levels from the current values of about 380 ppm to about 870 ppm by the end of the 21st century.
![Grid cell centers for the NARCCAP output (left) and madogram extremal coefficient estimate (box width is proportional to the number of observations) for the historical run.[]{data-label="fdata"}](591f04.eps "fig:")\[t!\]
We analyze data for $n=697$ grid cells in eastern US as shown in Figure \[fdata\]. For grid cell $i$ with location ${\mathbf{s}}_i$ and year $t$, we take the annual maximum of the daily precipitation totals as the response, $Y_{t}({\mathbf{s}}_i)$. NARCCAP provides eight 3-hour precipitation rates each day, and we compute the daily total by summing these eight values and multiplying by three. To explore the form of residual spatial dependence, we use the madogram \[@cooley-2006a\] function in the `SpatialExtremes` package in $\mathtt{R}$ ([www.r-project.org](http://www.R-project.org)). The madogram converts the observations at each site to have unit ${\mbox{Fr\'{e}chet}}$ margins using a rank transformation, and then estimates the pairwise extremal coefficients. Figure \[fdata\] plots the estimated extremal coefficients against $\Vert {\mathbf{s}}_i-{\mathbf{s}}_j\Vert $. This plot clearly shows residual spatial dependence.
The data from the two runs are analyzed separately using the model described in Section \[smodel\]. We assume that the process is stationary in time during each period, that is, the GEV marginal density at each location is constant over time in each simulation period. We use $n=L$ terms with knots at the data points ${\mathbf{s}}_1,\ldots,{\mathbf{s}}_n$. The residual dependence parameters have priors $\tau\sim \operatorname{InvGamma}(0.1,0.1)$ and $\alpha\sim \operatorname{Unif}(0,1)$. For both scenarios, all three GEV parameters vary spatially following Gaussian process priors. The covariates for the mean of the GEV parameters, $\mathbf{x}({\mathbf{s}})$, include the intercept, grid cell latitude, longitude, elevation, and log elevation. The elements of ${\bolds{\beta}}_{\mu}$ have independent $\mathrm{N}(0,100^2)$ priors. The spatial covariance parameters have priors $\delta_{j}^2\sim
\operatorname{InvGamma}(0.1,0.1)$, range $\rho_{j}\sim \operatorname{InvGamma}(0.1,0.1$), and smoothness $\nu_{j}\sim \operatorname{InvGamma}(0.1,0.1)$ for $j\in\{\mu,\gamma
,\xi\}$.
![Posterior mean and standard deviation of the GEV location, log scale, and shape parameters for the historical simulation. All units are mm/h.[]{data-label="fgevhist"}](591f05.eps)
Figure \[fgevhist\] shows the estimated GEV parameters for the historical simulation. The estimated location and log scale parameters are highest in the southeast. The posterior mean of the GEV shape is generally positive, indicating a right-skewed distribution with no upper bound. The estimated shape is the largest in Florida. Comparing the posterior means and standard deviations, there is evidence that all three GEV parameters vary spatially. Figure \[fshapescale\] shows that there is strong positive dependence between the shape and scale as one might expect, since for shape in $(0,0.5)$ both the mean and variance of GEV includes the ratio of the scale and shape. For locations with large shapes, there is a negative dependence with the log scale.
![Plot of the posterior mean GEV scale versus posterior mean GEV shape at each site.[]{data-label="fshapescale"}](591f06.eps)
To formally assess the need for spatially-varying GEV parameters, we also refit the model for the historical simulation using the Bayesian variable selection prior of @reich-2010 to test whether the variance $\delta_{j}^2$ equals small constant $\Delta_j^2=0.01^2$. The test is carried out using the mixture prior $\delta_{j} = g_j\delta_j^*
+ (1-g_j)\Delta_0$, where $g_j\sim \operatorname{Bernoulli}(0.5)$ and $\delta_j^{*2}\sim \operatorname{InvGamma}(0.1,0.1$). The intuition behind this prior is that if $g_j=1$, then $\delta_{j}^2\sim \operatorname{InvGamma}(0.1,0.1)$ and the GEV parameter varies spatially; in contrast, if $g_0=0$, then $\delta_{j}^2=\Delta_j^2$, and spatial variation after accounting for spatial covariates $\mathbf{x}$ is negligible. Therefore, the posterior mean of $g_j$ can be interpreted as the posterior probability that the $j$[th]{} GEV parameter varies spatially, which can be used to approximate the Bayes factor comparing these models. In the separate mixture prior fit, the posterior probability that the GEV parameters vary spatially was at least 0.99 for all three parameters.
We also aim to quantify changes in extreme quantiles. The $q$[th]{} quantile at location ${\mathbf{s}}$ is $\mu({\mathbf{s}}) + \sigma({\mathbf{s}}) [1-\log
(1/q)^{-\xi({\mathbf{s}})} ]/\xi({\mathbf{s}})$, which is also called the $1/(1-q)$ year return level. Figure \[fquanthist\] plots the posterior of various pointwise quantile levels. The large location and scale parameters lead to large medians in the southeast, while the 0.95 quantile is the largest in Florida due to the large shape parameter.
![Posterior mean and standard deviation of the 0.10, 0.50, and 0.95 quantiles for the historical simulation. All units are mm/h.[]{data-label="fquanthist"}](591f07.eps)
The difference between the historical and future scenarios is summarized in Figures \[fgevchange\] and \[fquantchange\]. The estimated GEV location and log scale parameters are larger for the future scenario for the majority of the spatial domain. The increase is the largest in Alabama, Georgia, and New England. The shape parameter also shows an increase in Alabama, but statistically significant decrease in Florida. Figure \[fquantchange\](c) shows that these changes in GEV parameters lead to an increase in the 0.95 quantile for most of the spatial domain. With the exception of the midwest and southern Florida, the posterior probability of an increase in the 0.95 quantile is near one \[Figure \[fquantchange\](d)\], indicating that extremes have a different spatial pattern in the future scenario.
![Posterior mean change from historical to future time simulation and the posterior probability that this change is positive for the GEV location, log scale, and shape parameters. All units are mm/h.[]{data-label="fgevchange"}](591f08.eps)
![Posterior mean change from historical to future time simulations and the posterior probability that this change is positive for the 0.10, 0.50, and 0.95 quantiles. All units are mm/h.[]{data-label="fquantchange"}](591f09.eps)
![Ratio of the posterior variance of the GEV parameters for the models with and without residual spatial dependence.[]{data-label="fvarratio"}](591f10.eps)
Parameter estimates provide evidence of residual dependence: the posterior mean (standard deviation) of $\alpha$ is 0.483 (0.008) and the posterior mean of the spatial range $\tau$ is 41.6 (0.4) kilometers. To illustrate the effects of failing to account for residual spatial dependence, we compare these results with the model that ignores spatial dependence in the residuals, that is, sets $\alpha
=1$. One effect of accounting for residual dependence is an increase in posterior variance for the GEV parameters. Figure \[fvarratio\] shows that the posterior variance often doubles as a result of including residual dependence. Therefore, while spatial modeling of the GEV parameters reduces uncertainty by borrowing strength across space compared to analyzing all sites completely separately, it appears that spatial modeling of the GEV parameters without accounting for residual dependence underestimates uncertainty.=1
Discussion {#sconc}
==========
In this paper we propose a new modeling approach for spatial max-stable processes. The proposed model is closely related to the GEVP and permits a Bayesian analysis via MCMC methods. Applied to the climate data, we find statistically significant increases under the future climate scenario in the upper quantiles of precipitation for most of the spatial domain, with the largest increase in the southeast.
The proposed hierarchical model opens the door for several exciting research directions. The model could be made even more flexible by changing the form of the kernels. It should be possible to replace the Gaussian kernel with any other kernel that integrates to one, that is, any other two-dimensional density function. For large data sets, it may even be possible to estimate the kernel function nonparametrically from the data. @Zheng-2010 and @reich-2011 use Bayesian nonparametrics to estimate the spatial covariance function of a Gaussian process. This approach could be extended to the extreme data, using, say, a Dirichlet process mixture prior for the kernel function. The methods proposed in this paper could also be extended to more complicated dependency structures. For example, we have ignored the temporal dependence because the spatial association is far stronger than the temporal association for these data. However, using three-dimensional kernels (two for space, one for time) would give a feasible max-stable model for spatiotemporal data.
Appendix {#appendix .unnumbered}
========
Generalized extreme value (GEV) distribution {#appa1}
--------------------------------------------
The GEV distribution has three parameters: location $\mu$, scale $\sigma
>0$, and shape $\xi$. If $Y\sim\operatorname{GEV}(\mu,\sigma,\xi)$, then $Y$ has distribution function $P(Y<y) = \exp[-t(y)]$ and density $f(y) = \frac
{1}{\sigma}t(y)^{\xi+1}\exp[-t(y)]$, where $$t(y) = \cases{
\displaystyle\biggl[1+\frac{\xi}{\sigma}(y-
\mu) \biggr]^{-1/\xi}, &\quad $\xi\ne0,$
\vspace*{2pt}\cr
\displaystyle\exp\bigl[-(y-\mu)/\sigma\bigr], & \quad $\xi=0.$ }$$ The shape parameter determines the support, with $Y\in(-\infty,\mu
-\sigma/\xi]$ if $\xi<0$, $Y\in(-\infty,\infty)$ is $\xi=0$, and $Y\in
[\mu-\sigma/\xi,\infty)$ in $\xi>0$. The GEV has three well-known subfamilies defined by the shape: the Weibull ($\xi<0$), Gumbel ($\xi
=0$), and ${\mbox{Fr\'{e}chet}}$ ($\xi>0$) families.
Properties of the random effects model {#appa2}
--------------------------------------
Here we show that the hierarchical representation in (\[auxmodel\]) is max-stable and has GEV margins.
*GEV marginal distributions*: Since the margins are identical for all locations, we omit the notational dependence on ${\mathbf{s}}$. The marginal distribution function of $X$ is $$\begin{aligned}
P(X<c) &=& \int P(X|{\mathbf{A}})p({\mathbf{A}}|\alpha)\,d{\mathbf{A}}\nonumber\\
&=& \int\exp \biggl\{- \biggl[1+
\frac{\alpha}{\alpha\theta}(c-\theta) \biggr]^{-1/\alpha
} \biggr\}p({\mathbf{A}}|\alpha)\,d{\mathbf{A}}\nonumber\\
&=&\int\exp \Biggl\{-c^{-1/\alpha} \Biggl(\sum_{l=1}^LA_lw_l^{1/\alpha}
\Biggr) \Biggr\}p({\mathbf{A}}|\alpha)\,d{\mathbf{A}}\\
&=&\prod_{l=1}^L\int\exp \bigl
\{-c^{-1/\alpha}w_l^{1/\alpha
}A_l \bigr\}
p(A_l|\alpha)\,dA_l
\nonumber
\\
&=&\prod_{l=1}^L \exp \bigl\{-
\bigl(c^{-1/\alpha}w_l^{1/\alpha
} \bigr)^\alpha \bigr
\} =\exp \Biggl(-\frac{1}{c}\sum_{l=1}^Lw_l
\Biggr) =\exp (-1/c).
\nonumber\end{aligned}$$ This is the unit ${\mbox{Fr\'{e}chet}}$ distribution function.
*Max-stability*: The process is max-stable if for any set of locations $\{{\mathbf{s}}_1,\ldots,\break{\mathbf{s}}_n\}$ and any $t\,{>}\,0$, $\operatorname{Prob} [X({\mathbf{s}}_1)
\,{\leq}\,tc_1, \ldots, X({\mathbf{s}}_n)\,{\leq}\,tc_n ]^t\,{=}\,\operatorname{Prob} [X({\mathbf{s}}_1)
\,{\leq}\,c_1, \ldots,\break X({\mathbf{s}}_n) \leq c_n ]$ \[e.g., @zhang-2010\]. From (\[ASL\]), $$\begin{aligned}
&&\operatorname{Prob} \bigl[X({\mathbf{s}}_1) \leq tc_1, \ldots,
X({\mathbf{s}}_n) \leq tc_n \bigr]^t\\
&&\qquad= \exp \Biggl\{-
\sum_{l=1}^L \Biggl[\sum
_{i=1}^n \biggl(\frac{w_l({\mathbf{s}}_i)}{tc_i}
\biggr)^{1/\alpha} \Biggr]^{\alpha} \Biggr\}^t
\\
&&\qquad= \exp \Biggl\{-\frac{1}{t}\sum_{l=1}^L
\Biggl[\sum_{i=1}^n \biggl(
\frac
{w_l({\mathbf{s}}_i)}{c_i} \biggr)^{1/\alpha} \Biggr]^{\alpha} \Biggr
\}^t
\\
&&\qquad=\operatorname{Prob} \bigl[X({\mathbf{s}}_1) \leq c_1, \ldots,
X({\mathbf{s}}_n) \leq c_n \bigr].\end{aligned}$$
MCMC details {#appa3}
------------
A complication that arises when using positive stable random effects is that their density does not have a closed form. To overcome this problem, we use the auxiliary variable technique of @stephenson-2009 for the asymmetric logistic MGEV. @stephenson-2009 introduces auxiliary variables $B_{l}\in(0,1)$ so that $$\label{psu} p(A,B|\alpha) = \frac{\alpha A^{-1/(1-\alpha)}}{1-\alpha}c(B)\exp \bigl[-c(B)A^{-\alpha/(1-\alpha)}
\bigr],$$ where $c(B) = [\frac{\sin(\alpha\pi B)}{\sin(\pi B)}
]^{1/(1-\alpha)}\frac{\sin[(1-\alpha) \pi B]}{\sin(\alpha\pi B)}$. Then, marginally over $B_{l}$, $A_{l}\sim \operatorname{PS}(\alpha)$. This marginalization is handled naturally via MCMC. Incorporating the auxiliary variable gives $$\begin{aligned}
\label{auxmodel3} Y_t({\mathbf{s}}_i) | A_{1t},B_{1t}\ldots,A_{Lt},B_{Lt}
&{\stackrel{\mathrm{indep}}{\sim}}& \operatorname{GEV} \bigl[\mu_t^*({\mathbf{s}}_i),
\sigma_t^*({\mathbf{s}}_i),\xi^*({\mathbf{s}}_i) \bigr],
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
(A_{lt},B_{lt}) &{\stackrel{\mathrm{i.i.d.}}{\sim}}& p(A,B|\alpha),
\nonumber\end{aligned}$$ which is the model fit to the data.
We perform MCMC sampling for the model in (\[auxmodel3\]) using R (`http:// www.r-project.org/`). The Metropolis within Gibbs algorithm is used to draw posterior samples. This begins with an initial value for each model parameter, and then parameters are updated one-at-a-time, conditionally on all other parameters. The GEV parameters $\mu$, $\sigma=\exp(\gamma)$, and $\xi$, spatial dependence parameters $\tau$ and $\alpha$, and auxiliary variables $(A_l,B_l)$ are updated using Metropolis updates. To update the GEV location at site ${\mathbf{s}}_i$ for the $r$[th]{} MCMC iteration, we generate a candidate using a random walk Gaussian candidate distribution $\mu^{(c)}({\mathbf{s}}_i)\sim\mathrm{N}(\mu^{(r-1)}({\mathbf{s}}_i),s^2)$, where $\mu^{(r-1)}({\mathbf{s}}_i)$ is the value at MCMC iteration $r-1$ and $s$ is a tuning parameter. The acceptance ratio is $$\begin{aligned}
R&=& \biggl\{\frac{\prod_{t=1}^Tl[Y_t({\mathbf{s}}_i)|\mu^{(c)}({\mathbf{s}}_i), \exp
[\gamma
({\mathbf{s}}_i)], \xi({\mathbf{s}}_i), \theta_t({\mathbf{s}}_i)]} {
\prod_{t=1}^Tl[Y_t({\mathbf{s}}_i)|\mu^{(r-1)}({\mathbf{s}}_i), \exp[\gamma({\mathbf{s}}_i)], \xi
({\mathbf{s}}_i), \theta_t({\mathbf{s}}_i)]} \biggr\}\\
&&{}\times \biggl\{\frac{p[\mu^{(c)}({\mathbf{s}}_i)|\mu({\mathbf{s}}_j), j\ne i]} {
p[\mu^{(r-1)}({\mathbf{s}}_i)|\mu({\mathbf{s}}_j), j\ne i]} \biggr\},\end{aligned}$$ which is a function of the GEV likelihood of $Y_t({\mathbf{s}})$ in (\[auxmodel2\]), denoted as $l[Y_t({\mathbf{s}})|\mu({\mathbf{s}}), \exp[\gamma({\mathbf{s}})],
\xi
({\mathbf{s}}), \theta_t({\mathbf{s}})]$, as well as the full conditional prior of $\mu
({\mathbf{s}}_i)$ given $\mu({\mathbf{s}}_j)$ for all $j\ne i$, $p[\mu({\mathbf{s}}_i)|\mu
({\mathbf{s}}_j),
j\ne i]$, which is found using the usual formula for the conditional distribution of a multivariate normal. The candidate is accepted with probability $\min\{R,1\}$. If the candidate is accepted, then $\mu^{(r)}({\mathbf{s}}_i)= \mu^{(c)}({\mathbf{s}}_i)$, otherwise the previous value is retained, $\mu^{(r)}({\mathbf{s}}_i)= \mu^{(r-1)}({\mathbf{s}}_i)$. The other GEV parameters $\gamma({\mathbf{s}}_i)$ and $\xi({\mathbf{s}}_i)$ are updated similarly. GEV hyperparameters, such as ${\bolds{\beta}}_{\mu}$ and spatial covariance parameters, are updated conditionally on the GEV parameters and, thus, their updates are identical to the usual Bayesian geostatistical model.
The spatial dependence parameters $\tau$ and $\alpha$ and the auxiliary variables $A_{lt}$ and $B_{lt}$ are also updated using Metropolis sampling. These updates differ from $\mu({\mathbf{s}}_i)$ only in their acceptance ratios. For computing purposes, we transform to $\delta=
\log(\tau)$. The acceptance ratio for $\delta$ is $$\biggl\{\frac{\prod_{t=1}^T\prod_{i=1}^nl[Y_t({\mathbf{s}}_i)|\mu({\mathbf{s}}_i),
\exp
[\gamma({\mathbf{s}}_i)], \xi({\mathbf{s}}_i), \theta_t^{(c)}({\mathbf{s}}_i)]} {
\prod_{t=1}^T\prod_{i=1}^nl[Y_t({\mathbf{s}}_i)|\mu({\mathbf{s}}_i), \exp[\gamma
({\mathbf{s}}_i)], \xi({\mathbf{s}}_i), \theta_t^{(r-1)}({\mathbf{s}}_i)]} \biggr\} \biggl\{\frac{p[\delta^{(c)}]} {
p[\delta^{(r-1)}]} \biggr\},$$ where $\theta^{(c)}_{t}$ and $\theta^{(r-1)}_{t}$ are the values of $\theta_t$ evaluated with $\tau^{(c)}=\exp(\delta^{(c)})$ and $\tau^{(r-1)}=\exp(\delta^{(r-1)})$, respectively, and $p(\delta)$ is the log-gamma prior. The acceptance ratio for $\alpha$ is $$\begin{aligned}
&&\biggl\{\frac{\prod_{t=1}^T\prod_{i=1}^nl[Y_t({\mathbf{s}}_i)|\mu({\mathbf{s}}_i),
\exp
[\gamma({\mathbf{s}}_i)], \xi({\mathbf{s}}_i), \theta_t^{(c)}({\mathbf{s}}_i)]} {
\prod_{t=1}^T\prod_{i=1}^nl[Y_t({\mathbf{s}}_i)|\mu({\mathbf{s}}_i), \exp[\gamma
({\mathbf{s}}_i)], \xi({\mathbf{s}}_i), \theta_t^{(r-1)}({\mathbf{s}}_i)]} \biggr\} \\
&&\qquad{}\times \biggl\{\frac{\prod_{t=1}^T\prod_{l=1}^Lp(A_{lt},B_{lt}|\alpha^{(c)})} {
\prod_{t=1}^T\prod_{l=1}^Lp(A_{lt},B_{lt}|\alpha^{(r-1)})} \biggr\} I
\bigl(0<\alpha^{(c)}<1\bigr).\end{aligned}$$ We use a log-normal candidate distribution for $A_{lt}\sim$ LN$[\log
(A_{lt}^{(r-1)}),s_A^2]$, with density denoted $q(A^{(c)}_{lt}|A^{(r-1)}_{lt})$. The latent variables $A_{tl}$ and $B_{lt}$ have acceptance ratios $$\begin{aligned}
&&\biggl\{\frac{\prod_{i=1}^nl[Y_t({\mathbf{s}}_i)|\mu({\mathbf{s}}_i), \exp[\gamma
({\mathbf{s}}_i)],
\xi({\mathbf{s}}_i), \theta_t^{(c)}({\mathbf{s}}_i)]} {
\prod_{i=1}^nl[Y_t({\mathbf{s}}_i)|\mu({\mathbf{s}}_i), \exp[\gamma({\mathbf{s}}_i)], \xi
({\mathbf{s}}_i), \theta_t^{(r-1)}({\mathbf{s}}_i)]} \biggr\}\\
&&\qquad{}\times \biggl\{\frac{p(A^{(c)}_{lt},B_{lt}|\alpha)} {
p(A^{(r-1)}_{lt},B_{lt}|\alpha)} \biggr\} \biggl\{
\frac
{q(A^{(r-1)}_{lt}|A^{(c)}_{lt})}{q(A^{(c)}_{lt}|A^{(r-1)}_{lt})} \biggr\}\end{aligned}$$ for $A_{lt}$ and $$\frac{p(A_{lt},B^{(c)}_{lt}|\alpha)}{p(A_{lt},B^{(r-1)}_{lt}|\alpha
)}I\bigl(0<B_{lt}^{(c)}<1\bigr)$$ for $B_{lt}$.
The standard deviations of all candidate distributions are adaptively tuned during the burn-in period to give acceptance rates near 0.4. Note that after the burn-in, the candidate distribution is fixed and this defines a stationary Markov chain and satisfies the usual mixing conditions, generating samples from the true posterior distribution once convergence is reached. We generate two (one for the simulation study) chains of length 25,000 samples and discard the first 10,000 samples of each chain as burn-in. Convergence is monitored using trace plots and autocorrelation plots for several representative parameters.
Acknowledgments {#acknowledgments .unnumbered}
===============
We also wish to acknowledge several helpful discussions with Richard Smith of the University of North Carolina—Chapel Hill and Alan Gelfand of Duke University.
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---
abstract: 'The non-relativistic limit of the linear wave equation for zero and unity spin bosons of mass $m$ in the Duffin–Kemmer–Petiau representation is investigated by means of a unitary transformation, analogous to the Foldy–Wouthuysen canonical transformation for a relativistic electron. The interacting case is also analyzed, by considering a power series expansion of the transformed Hamiltonian, thus demonstrating that all features of particle dynamics can be recovered if corrections of order $1/m^{2}$ are taken into account through a recursive iteration procedure.'
author:
- |
P.Yu. Moshin${}^{a)}$[^1]and J.L. Tomazelli${}^{b)}$[^2]\
\
$^{a)}$Instituto de Física, Universidade de São Paulo,\
Caixa Postal 66318, CEP 05315-970, São Paulo, S.P., Brazil\
$^{b)}$Departamento de Física, Universidade Federal de Santa Catarina,\
Caixa Postal 476, CEP 88010-970, Florianópolis, S.C., Brazil
title: '[**On the Non-relativistic Limit of Linear Wave Equations for Zero and Unity Spin Particles**]{} '
---
Introduction
============
Recently, in view of the increasing technical complexity of string theories as the best candidates for the unification of the fundamental interactions, there is a renewed interest in the quantum field theory of higher spins as a natural covariant formalism for accommodating the particle spectra in the Standard Model and quantum gravity theories, as well as in their supersymmetric counterparts. Thus, from the phenomenological standpoint, it is mandatory to investigate such theories in the low-energy regime, by examining their non-relativistic formal properties and taking into account the interaction with external electromagnetic and/or metric fields as a starting point.
In relativistic quantum mechanics, one must seek for a relation between irreducible representations of the Poincaré group and wave equations. In Wigner’s standard form, non-trivial wave equations can only be presented for wave functions with a large number of components, simultaneously expressing constraints on redundant components and equations of motion for the physical ones. Considering general invariant equations, Gel’fand and Yaglom$^{{\scriptsize \cite{r1}}}$ expressed relativistic wave functions in terms of linear differential operators, simultaneously determining both these operators and the finite-dimensional representations of the homogeneous Lorentz group, according to which the components of the wave functions transform. However, such a procedure is not applicable to non-relativistic wave equations whose solutions transform according to the homogeneous Galilei group. Following another approach, relying upon the Bargmann–Wigner method, Lévi-Leblond$^{{\scriptsize \cite{r2}}}$ constructed a basis in a ten-dimensional representation space of the homogeneous Galilei group for free massive particles of spin $1$, by taking a complete set of independent linear combinations of symmetrical tensor products of two-component wave functions which describe non-relativistic particles of spin $1/2$, and arriving at a system of equations involving linear operators.
In order to investigate the physical properties of particles of zero and unity spin in the presence of electromagnetic external sources, instead of starting from Galilean-covariant wave equations, we start from a Lorentz-covariant linear wave equation in the Hamiltonian form and apply a canonical transformation, analogous to the Foldy–Wouthuysen (FW) transformation$^{{\scriptsize \cite{r3}}}$ for Dirac fermions, to a suitable reference frame in which one can recognize the different couplings of charged bosons with the electromagnetic field. In this sense, the covariant linear representation$^{{\scriptsize \cite{r4,r42,r43}}}$ of Duffin–Kemmer–Petiau (DKP) proves to be particularly useful, since all physical quantities are constructed from linear operators which obey convenient algebraic relations, in close similarity with the familiar Dirac operators. Notably, Darwin$^{{\scriptsize \cite{da}}}$ proposed a linear wave equation for the electromagnetic field some years before Petiau’s pioneering work, in close relation to the meson theory proposed by Kemmer, who referred to Dirac’s work$^{{\scriptsize \cite{di}}}$ on linear relativistic equations for particles with spins higher than one-half.
This work is organized as follows. In Section 2, we present the linear wave equation which describes bosons of spin zero and unity and the basic identities of the associated DKP algebra; we then rewrite this equation in the Hamiltonian form for non-interacting particles. In Section 3, we discuss the quantum canonical transformation for the free boson Hamiltonian, by analogy with the ordinary FW transformation. Next, in Section 4, we derive the non-relativistic limit of the Hamiltonian that describes charged bosons interacting with an external electromagnetic field. In Section 5, we make concluding remarks.
DKP Hamiltonian
===============
Let us briefly review the DKP formalism for non-interacting bosons of spin zero and one. The relativistic wave equation in such a representation reads $$\left( i{\partial\!\!\!\slash} -m\right) \psi=0\,,\label{1}$$ where ${\partial\!\!\!\slash} \equiv\beta_{\mu}\partial^{\mu}$ and $\psi$ is a five(ten)-row column associated with the zero (unity) spin field. The considerations of this work do not refer to any particular representation for $\psi$.
The $\beta$-matrices obey the algebra $$\beta_{\mu}\beta_{\nu}\beta_{\rho}+\beta_{\rho}\beta_{\nu}\beta_{\mu}=\beta_{\mu}g_{\nu\rho}+\beta_{\rho}g_{\nu\mu}\,,\label{2}$$ which implies the following consequences:$$\begin{aligned}
& \,\beta_{0}\beta_{k}\beta_{0}=0\,,\;k=1,2,3\,,\nonumber\\
& \,\beta_{0}^{3}=\beta_{0}\,,\label{4}\\
& \,{b\!\!\!\slash} \beta_{\nu}{b\!\!\!\slash} ={b\!\!\!\slash} b_{\nu}\,,\label{5}\\
& \,(\vec{\beta}\cdot\vec{b})\beta_{0}(\vec{\beta}\cdot\vec{b})=0\,,\label{6}$$ where $b_{\mu}=(b_{0},\vec{b})$ is a generic four-vector.
Multiplying (\[1\]) by ${\partial\!\!\!\slash} \beta_{\mu}$ and using (\[5\]), $$\left( i{\partial\!\!\!\slash} \beta_{\mu}{\partial\!\!\!\slash} -m{\partial\!\!\!\slash} \beta_{\mu
}\right) \psi=\left( i\partial_{\mu}{\partial\!\!\!\slash} -m{\partial\!\!\!\slash} \beta
_{\mu}\right) \psi=0\,,$$ and then (\[1\]), $$\left( m\partial_{\mu}-m{\partial\!\!\!\slash} \beta_{\mu}\right) \psi=0\,,$$ one obtains$$\partial_{\mu}\psi={\partial\!\!\!\slash} \beta_{\mu}\psi\,.\label{7}$$ Multiplying (\[1\]) by $\beta_{0}$ and taking the zero component of (\[7\]), times the imaginary unity, one obtains, upon adding the results, $$\left\{ i\left[ \partial_{0}+\partial^{k}\left( \beta_{0}\beta_{k}-\beta_{k}\beta_{0}\right) \right] -m\beta_{0}\right\} \psi=0\,,$$ or $$i\partial_{0}\psi=H\psi\,,\label{8}$$ where $$H=-i\vec{\alpha}\cdot\vec{\nabla}+\beta_{0}m=\vec{\alpha}\cdot\vec{p}+\beta_{0}m\label{9}$$ is the DKP Hamiltonian, and $\vec{\alpha}$ is defined by its spatial components:$$\alpha_{k}\equiv\beta_{0}\beta_{k}-\beta_{k}\beta_{0}\,,\;k=1,2,3\,.$$
FW Transformation
=================
As in the electron case, we now look for a unitary transformation,$$\begin{aligned}
\psi^{\prime} & =e^{iU}\psi\,,\\
H^{\prime} & =e^{iU}He^{-iU}\,,\end{aligned}$$ which should eliminate the term that involves the spatial components of the four-momentum. In case $H$ explicitly depends on time, equation (\[8\]) yields $$i\partial_{0}(e^{-iU}\psi^{\prime})=He^{-iU}\psi^{\prime}\,,$$ so that $$e^{-iU}\left( i\partial_{0}\psi^{\prime}\right) =\left( He^{-iU}-i\partial_{0}e^{-iU}\right) \psi^{\prime}\,,$$ or $$i\partial_{0}\psi^{\prime}=H^{\prime}\psi^{\prime}\,,$$ where $$H^{\prime}=e^{iU}\left( H-i\partial_{0}\right) e^{-iU}\,.\label{14}$$
Let us choose $$U=-i\frac{\vec{\beta}.\vec{p}}{|\vec{p}|}\theta\,.$$ The $\beta$-algebra (\[2\]) implies the identity$$\begin{aligned}
2(\vec{\beta}.\vec{p})^{3} & =\sum_{ijk}p_{i}p_{j}p_{k}\left( \beta
_{i}\beta_{j}\beta_{k}+\beta_{k}\beta_{j}\beta_{i}\right) \nonumber\\
& =-\sum_{ijk}p_{i}p_{j}p_{k}(\beta_{i}\delta_{jk}+\beta_{k}\delta
_{ji})\,,\nonumber\end{aligned}$$ so that$$(\vec{\beta}\cdot\vec{p})^{3}=-|\vec{p}|^{2}(\vec{\beta}\cdot\vec
{p})\,.\label{16}$$ Representing (\[16\]) in the form $$\lbrack(\vec{\beta}\cdot\vec{p})^{2}+\left| \vec{p}\right| ^{2}](\vec{\beta
}\cdot\vec{p})=0$$ and then, on the mass shell,$$(\vec{\beta}\cdot\vec{p})=\beta_{0}p_{0}-{p\!\!\!\slash} =\beta_{0}E+m\,,$$ we have $$\lbrack(\vec{\beta}\cdot\vec{p})^{2}+\left| \vec{p}\right| ^{2}]\left(
\beta_{0}E+m\right) \psi=0\,.\label{17}$$ On the other hand, due to the identity$$\begin{aligned}
(\vec{\beta}\cdot\vec{p})^{2}\beta_{0} & =\sum_{ij}p_{i}p_{j}\left(
\beta_{i}\beta_{j}\beta_{0}\right) \nonumber\\
& =-\sum_{ij}p_{i}p_{j}\left( \beta_{0}\beta_{j}\beta_{i}+\beta_{0}\delta_{ij}\right) =-\beta_{0}[(\vec{\beta}\cdot\vec{p})^{2}+|p|^{2}]\,,\nonumber\end{aligned}$$ equation (\[17\]) implies $$\left( m-\beta_{0}E\right) (\vec{\beta}\cdot\vec{p})^{2}\psi=-\left|
\vec{p}\right| ^{2}m\psi\,,$$ or $$(m^{2}-\beta_{0}^{2}E^{2})(\vec{\beta}\cdot\vec{p})^{2}\psi=-(m^{2}+\beta
_{0}Em)\left| \vec{p}\right| ^{2}\psi\,.$$ Then, due to (\[4\]), we obtain $$(\vec{\beta}\cdot\vec{p})^{2}=-\left| \vec{p}\right| ^{2}+\left( Em\right)
\beta_{0}+E^{2}\beta_{0}^{2}\,.\label{19}$$ Eq. (\[9\]) does not contain the complete information about the system because of multiplication by the singular matrix $\beta_{0}$.
Multiplying (\[1\]) by $(1-\beta_{0}^{2})$, one finds the additional constraint $$\left[ i\partial^{k}\beta_{k}\beta_{0}^{2}-(1-\beta_{0}^{2})m\right]
\psi=0\,,$$ or $$(\vec{\beta}\cdot\vec{p})\beta_{0}^{2}+(1-\beta_{0}^{2})m=0\,,\label{20}$$ on the mass shell. Also, left-multiplying (\[19\]) by $(\vec{\beta}\cdot
\vec{p})$, and using (\[16\]) and (\[20\]), one obtains$$(\vec{\beta}\cdot\vec{p})\beta_{0}=E(1-\beta_{0}^{2})\,.\label{21}$$ Now, multiplying (\[19\]) by $(\vec{\beta}\cdot\vec{p})^{2}$ and using (\[20\]), (\[21\]), one gets$$(\vec{\beta}\cdot\vec{p})^{4}=-(\vec{\beta}\cdot\vec{p})^{2}\left| \vec
{p}\right| ^{2}\,.\label{22}$$ Then $$e^{iU}=e^{(\vec{\beta}\cdot\vec{p}/|\vec{p}|)\theta}=1+\frac{(\vec{\beta}\cdot\vec{p})^{2}}{|\vec{p}|^{2}}\left( 1-\cos\theta\right) +\frac{(\vec
{\beta}\cdot\vec{p})}{|\vec{p}|}\sin\theta\,,$$ where (\[16\]) and (\[22\]) have been used in the series expansion. Hence, $$H^{\prime}=(\vec{\alpha}.\vec{p})\left( \cos\theta-\frac{m}{|\vec{p}|}\sin\theta\right) +\beta_{0}\left( |\vec{p}|\sin\theta+m\cos\theta\right)
\,.$$ Choosing $$\sin\theta=\frac{|\vec{p}|}{E}\,,\;\cos\theta=\frac{m}{E}\,,$$ one arrives at$$H^{\prime}=\frac{\beta_{0}}{E}\left( \vec{p}^{2}+m^{2}\right) =\beta_{0}E\,.$$
DKP Interaction Hamiltonian
===========================
In order to have a better understanding of the particle content of the theory, let us examine the behavior of charged bosons in the presence of an external electromagnetic field, transformed to a reference frame where particles carry low momenta. The electromagnetic interaction is introduced by means of the covariant derivative, so that$$(i{D\!\!\!\slash} -m)\psi=0\,,\label{27}$$ where the covariant derivative$$D_{\mu}=\partial_{\mu}+ieA_{\mu}$$ satisfies the commutation relation$$\left[ D_{\mu},D_{\nu}\right] =ieF_{\mu\nu}\,.$$
Multiplying (\[27\]) by ${D\!\!\!\slash}\beta_{\mu}$, one obtains,$^{{\scriptsize \cite{r43}}}$ by analogy with (\[7\]), $$D_{\mu}\psi={D\!\!\!\slash}\beta_{\mu}\psi+\frac{e}{2m}F^{\rho\sigma}(\beta_{\rho}\beta_{\mu}\beta_{\sigma}-\beta_{\rho}g_{\mu\sigma})\psi\,.\label{30}$$ Then, from equations (\[27\]) and (\[30\]),$$i\partial_{0}\psi=H\psi\,,$$ it follows that$$H=H^{(0)}+H^{(1)}\,,$$ where $$\begin{aligned}
& H^{(0)}=\vec{\alpha}\cdot\vec{\pi}+m\beta_{0}-eA_{0}\,,\\
& H^{(1)}=\frac{ie}{2m}F^{\rho\sigma}\left( \beta_{\rho}\beta_{0}\beta_{\sigma}-\beta_{\rho}g_{0\sigma}\right) \,,\end{aligned}$$ and $$\vec{\pi}=\vec{p}-e\vec{A}\,.$$
Using (\[14\]) and the Baker–Campbell–Hausdorf formula, one can write$^{{\scriptsize \cite{r3}}}$$$H{}^{\prime}=H+\frac{\partial U}{\partial t}+i\left[ U,H+\frac{1}{2}\frac{\partial U}{\partial t}\right] -\frac{1}{2!}\left[ U,\left[
U,H+\frac{1}{3}\frac{\partial U}{\partial t}\right] \right] +\dots\,.$$ By virtue of the nonrelativistic limit $\theta\sim\sin\theta\sim|\vec{p}|/m$, one can choose, in the first approximation, by analogy with the free case,$$U=-i\frac{\vec{\beta}{\cdot}\vec{\pi}}{m}\,.$$ From the commutation relations (\[A1\])–(\[A7\]) and the vector identities (\[B1\]), (\[B2\]), listed in the Appendix, one obtains $$\lbrack U,H^{(1)}]=-\frac{e}{m^{2}}[\vec{\beta}\cdot\vec{\pi},(\vec{\beta
}\cdot\vec{E})\beta_{0}^{2}]+\frac{e}{m^{2}}[\vec{\beta}\cdot\vec{\pi},\vec{\beta}\cdot\vec{E}]-\frac{e}{2m^{2}}[\vec{\beta}\cdot\vec{\pi},F^{ij}\beta_{i}\beta_{0}\beta_{j}]\,.$$ In addition,$$\lbrack\vec{\beta}\cdot\vec{\pi},(\vec{\beta}\cdot\vec{E})\beta_{0}^{2}]=i\vec{S}\cdot\lbrack\vec{\pi}\times\vec{E}]\beta_{0}^{2}+(\vec{\beta}\cdot\vec{E})[2(\vec{\beta}\cdot\vec{\pi})\beta_{0}^{2}-\vec{\beta}\cdot
\vec{\pi}]\,,$$ so that one arrives at$$\begin{aligned}
H{}^{\prime} & =m\beta_{0}-eA_{0}+\frac{{\vec{\pi}}^{2}}{2m}\left(
\vec{\alpha}\cdot\frac{\vec{\pi}}{m}-\beta_{0}\right) +\frac{e}{2m}(\vec
{S}\cdot\vec{H})\beta_{0}+\frac{e}{2m}(\vec{\beta}\times\vec{\alpha})\cdot
\vec{H}\nonumber\\
& +\frac{e}{2m^{2}}(\vec{S}\cdot(\vec{\pi}\times\vec{E}))\left( 1+2\beta
_{0}^{2}\right) +\frac{ie}{2m^{2}}[\vec{\beta}\cdot\vec{\pi},(\beta_{0}\vec{S}+\vec{\beta}\times\vec{\alpha})\cdot\vec{H}]\nonumber\\
& -\frac{ie}{m}(\vec{\beta}\cdot\vec{E})\beta_{0}^{2}-\frac{ie}{m^{2}}(\vec{\beta}\cdot\vec{E})[2(\vec{\beta}\cdot\vec{\pi})\beta_{0}^{2}-\vec
{\beta}\cdot\vec{\pi}]+\mathcal{O}\left( m^{-3}\right) ,\label{40}$$ where relation (\[B3\]) has been used. In the above expression, $\vec{S}$ corresponds to the spin operator of bosons,$$S_{ij}=i\left( \beta_{i}\beta_{0}\beta_{j}-\beta_{j}\beta_{0}\beta
_{i}\right) \,,\;i,j=1,2,3\,,$$ with eigenvalues $0$ or $1$, while $\vec{E}$ and $\vec{H}$ are the electric and magnetic fields, respectively.
Expression (\[40\]) is analogous to the Hamiltonian of the Pauli equation for spin-$1/2$ fermions in the case of charged bosons of spin $0$ and $1$ on the background of an external electromagnetic field. In (\[40\]), we can recognize each term individually. For example, the second term is related to the electrostatic potential, while the third one corresponds to the kinetic term of the non-relativistic interaction Hamiltonian. In fact, taking the same steps that led to equation (\[21\]) on the mass shell, and using the definition of the matrices $\alpha_{k}$, one can rewrite the kinetic term in the transformed Hamiltonian as the expression$$\frac{{\vec{\pi}}^{2}}{2m}\left[ \frac{\pi_{0}}{m}(2\beta_{0}^{2}-1)\right]
\,,$$ which is indeed diagonal and non-singular in the matrix realization of the DKP $\beta$-algebra, as one should expect by analogy with the disentangling property of the FW transformation.
In this approach, the most essential result is the appearance of the spin and orbital angular momentum couplings with the external magnetic field (the fourth and fifth terms, respectively), as well as the diagonal spin-orbital coupling (the sixth term) via the electric field; the last two terms may be interpreted as being similar to the Darwin term for spin-$1/2$ fermions in the presence of an electric field; the remaining terms represent higher-order corrections to such effects, as well as the (non-diagonal) corrections to the rest-energy (the first term).
Concluding Remarks
==================
In the preceding sections, we have investigated the non-relativistic limit of the Lorentz-invariant wave equation which describes scalar and vector mesons in the so-called Duffin–Kemmer–Petiau representation. By constructing unitary operators involving the spatial components of the relativistic $4$-momentum and those belonging to the associated DKP algebra, both for free particles and for charged bosons in an electromagnetic background, we performed a quantum canonical transformation to a reference frame where we succeeded in identifying the coupling terms with the electric and magnetic fields, in close similarity with the non-relativistic behaviour of interacting fermions described by the Pauli equation.
Our approach differs from that of Lévi-Leblond$^{{\scriptsize \cite{r2}}}
$ in the sense that he derived non-relativistic linear wave equations for particles of arbitrary spins which obey the Galilean invariance by construction, where the electromagnetic multipole moments are introduced on dimensional grounds. At the same time, in the case of massive particles of spin $1$ he settled the corresponding wave equations by employing the Bargmann–Wigner construction, not referring to the algebraic properties of the quantities involved, which we have done explicitly in our treatment.
In the context of the present work, it is relevant to mention the series of papers [@r51; @r52; @r53; @r54] by Fushchych, Nikitin et al., who first introduced non-relativistic Duffin–Kemmer–Petiau equations: in [@r51], the authors presented a discussion on the Galilean-invariant equations for free particles with arbitrary spins, with particular emphasis on spin $0$ and $1/2$; however, in the presence of external fields [@r52; @r53] their transformation operator, which partially diagonalizes the total Hamiltonian, differs from that of Foldy and Wouthuysen for spin $1/2$ particles, as they pointed out in [@r54]. Consequently, the transformation operator of [@r52] is not the same as ours, since we follow the steps of the Foldy–Wouthuysen original algebraic construction (see, e.g., eqs. (5.2), (5.12)–(5.14) of [@r53]), instead of appealing to formal group theoretical reasonings.
Yet in the framework of the DKP theory, other authors have recently investigated the non-relativistic wave equation for spinless bosons, also via the Galilean covariance, by introducing an extra degree of freedom into the free Lagrangian density$^{{\scriptsize \cite{r6}}}$, thus recovering the Schrödinger equation for a free particle. However, the introduction of electromagnetic potentials spoils the original structure of the associated Lie algebra on which the reasoning$^{{\scriptsize \cite{r6}}}$ is grounded.
An interesting issue related to the present work is a possible generalization of the above procedure to theories of higher spins, as well as to their non-Abelian counterparts$^{{\scriptsize \cite{r7}}}$.
**Acknowledgments** The authors are grateful to D.M. Gitman for useful discussions. The work was supported by CNPq.
Appendix
========
Below, we present some useful commutation and vector relations derived from the algebra (\[2\]) of the $\beta$-matrices:$$\begin{aligned}
& \left[ {U,\vec{\alpha}\cdot\vec{\pi}}\right] {=\frac{i}{m}\beta_{0}\vec{\pi}^{2}}\,\,,\label{A1}\\
& \left[ {U,\beta_{0}}\right] {=\frac{i}{m}\vec{\alpha}\cdot\vec{\pi}}\,\,,\label{A2}\\
& \left[ {U,A_{0}}\right] {=-\frac{i}{m}\vec{\beta}\cdot\vec{\nabla}A_{0}}\,,\label{A3}\\
& {\left[ U,\partial U/\partial t\right] =\frac{ie}{m^{2}}\vec{S}\cdot\left( \vec{\pi}\times\partial\vec{A}/\partial t\right) }\,,\label{A4}\\
& \left[ {U,}\left[ {U,\vec{\alpha}\cdot\vec{\pi}}\right] \right]
{=-\frac{\vec{\pi}^{2}}{m^{2}}}\left( {\vec{\alpha}\cdot\vec{\pi}}\right)
\,,\label{A5}\\
& \left[ {U,}\left[ {U,\beta_{0}}\right] \right] {=-\frac{1}{m^{2}}\beta_{0}\vec{\pi}^{2}}\,,\label{A6}\\
& \left[ {U,}\left[ {U,A_{0}}\right] \right] {=-\frac{1}{m^{2}}\vec
{S}\cdot}({\vec{\pi}\times\vec{\nabla}A_{0})}\,,\label{A7}\\
& {F^{\rho\sigma}\beta_{\rho}\beta_{0}\beta_{\sigma}=-2}({\vec{E}\cdot
\vec{\beta})\beta_{0}^{2}+\vec{E}\cdot\vec{\beta}+F^{ij}\beta_{i}\beta
_{0}\beta_{j}}\,,\label{B1}\\
& {F^{\rho\sigma}\beta_{\rho}g_{0\sigma}=-\vec{E}\cdot\vec{\beta}}\,,\label{B2}\\
& {F^{ij}\beta_{i}\beta_{0}\beta_{j}=-i\beta_{0}\vec{S}\cdot\vec{H}-i}({\vec{\beta}\times\vec{\alpha})\cdot\vec{H}}\,.\label{B3}$$
[9]{} I.M. Gel’fand, R.A. Minlos and Z.Y. Shapiro, *Representations of the Rotation and Lorentz Groups*, Pergamon Press, Oxford (1963).
J.M. Lévy-Leblond, Commun. Math. Phys. **6** (1967) 286.
L.L. Foldy and S.A. Wouthuysen, Phys. Rev. **78** (1950) 29.
G. Petiau, Acad. R. Belg. Cl. Sci. Mém. Collect. **8 16**, No. 2 (1936).
R.J. Duffin, Phys. Rev. **54** (1938) 1114.
N. Kemmer, Proc. Roy. Soc. **A 173** (1939) 91.
C.G. Darwin, Proc. Roy. Soc. **A 136** (1932) 36.
P.A.M. Dirac, Proc. Roy. Soc. **A 155** (1936) 447.
W.I. Fushchych and A.G. Nikitin, Lett. Nuovo Cimento **16**, No. 3 (1976) 81.
W.I. Fushchych and A.G. Nikitin, Lett. Nuovo Cimento **19**, No. 9 (1977) 347.
W.I. Fushchych, A.G. Nikitin and V.A. Salogub, Rept. Math. Phys. **13**, No. 2 (1978) 175.
A.G. Nikitin and W.I. Fushchych, Theor. Math. Phys. **34**, No. 3 (1978) 203.
M. de Montigny, F.C. Khanna, A.E. Santana, E.S. Santos and J.D.M. Vianna, J. Phys. **A 33** (2000) 273.
P.Yu. Moshin and J.L. Tomazelli, work in progress.
[^1]: Tomsk State Pedagogical University, 634041 Tomsk, Russia; e-mail: moshin@dfn.if.usp.br
[^2]: E-mail: tomazelli@fsc.ufsc.br
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---
abstract: 'We calculate the tunneling magnetoresistance (TMR) of Fe$\mid$ZnSe$\mid$Fe$\mid$ZnSe$\mid$Fe (001) double magnetic tunnel junctions as a function of the in-between Fe layer’s thickness, and compare these results with those of Fe$\mid$ZnSe$\mid$Fe simple junctions. The electronic band structures are modeled by a parametrized tight-binding Hamiltonian fitted to [*ab initio*]{} calculations, and the conductance is calculated within the Landauer formalism expressed in terms of Green’s functions. We find that the conductances for each spin channel and the TMR strongly depend on the in-between Fe layer’s thickness, and that in some cases they are enhanced with respect to simple junctions, in qualitative agreement with recent experimental studies performed on similar systems. By using a 2D double junction as a simplified system, we show that the conductance enhancement can be explained in terms of the junctions energy spectrum. These results are relevant for spintronics because they demonstrate that the TMR in double junctions can be tuned and enhanced by varying the in-between metallic layer’s thickness.'
author:
- 'J. Peralta-Ramos'
- 'A. M. Llois'
title: 'Enhanced tunneling magnetoresistance in Fe$\mid$ZnSe double junctions'
---
A magnetic tunnel junction (MTJ) consists of two ferromagnetic electrodes separated by a thin non-conducting barrier. It is experimentally observed that the conductance of a MTJ depends on the relative orientation of the electrodes’ magnetization, and because of this, during the last years a lot of attention has been paid to the investigation of MTJs as promising candidates for application in spintronic devices, such as read heads and magnetic random access memories (for reviews, see \[1\] and the references therein).
One of the challenges, that has to be overcome for practical applications, is to reach higher values of the tunneling magnetoresistance ratio (TMR), defined as TMR$=[(\Gamma_{P}-\Gamma_{AP})/\Gamma_P]\times 100 \%$, where $\Gamma_P$ and $\Gamma_{AP}$ are the conductances measured for the parallel (P) and antiparallel (AP) magnetization of the electrodes. Several possibilities are now being considered: to use highly polarized materials ([*half-metals*]{}) or diluted magnetic semiconductors as parts of MTJs, to produce junctions with almost perfect interfaces, and to use double magnetic tunnel junctions (DMTJs), in which metallic layers are inserted inside the semiconductor barrier of a MTJ. In this work we explore the latter alternative, and focus our attention on the dependence of the TMR on the in-between metallic layer’s thickness.
Since X. Zhang [*et al*]{} \[2\] suggested to use DMTJs, several groups \[3-6\] have theoretically shown that DMTJs exhibit richer spin-dependent transport properties than MTJs and that the TMR can be higher than that of MTJs, but only very recently could these DMTJs be fabricated \[7,8\]. T. Nozaki [*et al*]{} \[7\] have recently measured the tunnel magnetoresistance of epitaxial Fe$\mid$MgO$\mid$Fe$\mid$MgO$\mid$Fe (001) DMTJs at room temperature, and found an enhancement of the TMR with respect to MTJs (53 $\%$ for DMTJs versus 44 $\%$ for MTJs at low bias), indicating that DMTJs may present an advantage over simple junctions for their use in spintronics.
As far as we know, up to now the only theoretical studies of DMTJs [*with magnetic layers in between the semiconductor*]{} were made within the free electron model (that cannot reproduce the decay rates inside the semiconductor of evanescent states with different symmetry), and using rectangular potential profiles \[2,3,5,14\]. Moreover, these studies analyzed the dependence of TMR on the applied bias voltage and not on the in-between metallic layer’s thickness, as we do in this work. For this reason, in this paper transport through Fe($\infty$)$\mid$ZnSe($a$)$\mid$Fe($\infty$) (001) MTJs and through Fe($\infty$)$\mid$ZnSe($b$)$\mid$Fe($c$)$\mid$ZnSe($b$)$\mid$Fe($\infty$) (001) DMTJs is theoretically investigated using a realistic tight-binding (TB) Hamiltonian to obtain the electronic structure of the junctions. Fe($\infty$) are semi-infinite electrodes, and $a$, $b$ and $c$ denote thicknesses. The systems studied are epitaxial, and we restrict to zero temperature, infinitesimal bias voltage and elastic transport. We choose Fe$\mid$ZnSe because it can be grown epitaxially and there is very little interdiffusion at the interfaces, thus producing crystalline junctions in which there are no magnetically dead Fe layers \[9,10\]. Moreover, in contrast to what happens in Fe$\mid$MgO based junctions, there is no oxidation of the interfacial Fe layers, which is known to be detrimental to TMR \[11\]. To obtain a clearer insight into the physics involved in transport through double tunnel junctions, we also calculate the conductance through a simplified [*two-dimensional*]{} tunnel junction (2DDJ).
The conductances are calculated from the active region’s Green’s function $G_S^\sigma=[\hat{1}E-H_S^\sigma-\Sigma_L^\sigma-\Sigma_R^\sigma]^{-1}$, where $\hat{1}$ stands for the unit matrix, $H_S^\sigma$ is the Hamiltonian corresponding to the active region, $\Sigma_{L/R}^\sigma$ are the self-energies describing the interaction of the active region with the left (L) or right (R) electrodes ($\sigma$ corresponds to the majority or minority spin channels), and ’active region’ stands for whatever is sandwiched by the electrodes. For DMTJs, the active region consists of an ’in-between metal region’ (IBMR) sandwiched by [*two identical*]{} ’semiconductor regions’ (SCR), while for MTJs the active region is simply the SCR. The energy $E$ is actually $E_F+i\eta$, $E_F$ being the Fermi level of the system, and we take $\eta \rightarrow 0^+$. The self-energies are given by $\Sigma_L^\sigma=H_{LS}^{\dagger} g_L^\sigma H_{LS}$ and $\Sigma_R^\sigma=
H_{RS}^{\dagger} g_R^\sigma
H_{RS}$, where $H_{LS}$ and $H_{RS}$ describe the coupling of the active region with the electrodes, and $g_{L/R}^\sigma$ are the surface Green’s functions for each electrode. These surface Green’s functions are calculated using a semi-analytical method \[12\] and are exact within our TB approximation. The transmission probability $T^\sigma$ is given by \[13\] $T^\sigma(k_{//},E_F)=Tr~ [\Delta_L^\sigma G_S^\sigma \Delta_R^\sigma
G_S^{\sigma \dagger}]$ where $\Delta_{L/R}^\sigma=i (\Sigma_{L/R}^\sigma-\Sigma_{L/R}^{\sigma \dagger})$, while the conductance is given by $$\Gamma^\sigma(E_F)=\frac{e^2}{h}\frac{1}{N_{k_{//}}} \sum_{k_{//}} T^\sigma (k_{//},E_F)$$ where $N_{k_{//}}$ is the total number of wave vectors parallel to the interface that we consider (in our case 5000 is enough to achieve convergence in $\Gamma$).
We start our discussion with the 2DDJs case, which are of the type M($\infty$)$\mid$S$\mid$M$\mid$S$\mid$M($\infty$), where M($\infty$) are semi-infinite [*paramagnetic*]{} metallic electrodes, S is a semiconductor and M is a metal (the same as the electrodes). The metal and semiconductor have the same structure, a [*square*]{} Bravais lattice with two atoms per unit cell, and are periodic in the direction perpendicular to the transport direction. The 2DDJs electronic structure is modeled by a 2nd nearest neighbors TB Hamiltonian with one $s$ orbital per atom. The TB parameters are chosen to make $E_F$ fall in the middle of the semiconductor’s band gap (of 0.5 eV). The SC and IBM regions are varied between 3.2 Å and 32 Å .
It is found that for certain thicknesses of the IBMR the conductance presents peaks in which it is enhanced by 1 to 4 orders of magnitude, as can be seen in Fig. 1 for a 2DDJ with a SCR of 12.8 Å . This effect can be explained in terms of the active region’s density of states (DOS), obtained from its Green’s function $G_S$. When the conductance is enhanced, partial density of states (PDOS) calculations indicate that there exist states at $E_F$ extended throughout the whole junction, so in that situations transport occurs through resonant states. When this happens, the 2DDJs conductances are higher than the corresponding ones of simple 2D junctions. These results are consistent with those of Z. Zheng and coworkers for a DMTJ with a non-magnetic in-between metal \[14\].
Fig. 2 shows the maximum ratio between the conductance of 2DDJs and 2D simple junctions, as a function of the SCR thickness. The maximum attainable ratio is of 146 $\%$ and occurs for a SCR thickness of 9.6 Å and an IBMR thickness of 19.2 Å . For thinner SCRs the ratio is nearly constant and roughly 140 $\%$, but beyond 12.8 Å the enhancement effect is lost. Having mentioned the main results for the 2DDJs, we go on to discuss the details for the three-dimensional Fe$\mid$ZnSe DMTJs.
Fig. 3 shows schematically the structure of simple and double junctions, which are periodic in the [*x-y*]{} plane, and the different magnetic configurations considered, parallel (P) and antiparallel (AP). Since the coercive field of the electrode and the in-between Fe layers is different, the magnetic configurations shown are experimentally attainable \[7\].
Fig. 4 shows the structure of a simple Fe$\mid$ZnSe junction with a SCR of 5.67 Å , along the $z$ direction (which is the direction of transport). The BCC Fe lattice parameter is $2.87$ Å , and that of zincblende ZnSe is $5.67$ Å .
The electronic structure of the junctions is modeled by a parametrized 2nd nearest neighbors [*spd*]{} TB Hamiltonian fitted to [*ab initio*]{} calculations \[15,16\], in which the hoppings between the Fe atoms and the (Zn,Se) atoms are calculated using Shiba’s rules and Andersen’s scaling law \[17\]. The Fe $d$ bands are spin split by $\mu J_{dd}$, where $\mu=2.2 ~\mu_B$ is the experimental magnetic moment of Fe and $J_{dd}=1.16$ eV is the exchange integral between $d$ orbitals ($\mu_B$ is Bohr’s magneton). With these values for $\mu$ and $J_{dd}$, the Fe $d$ bands spin spitting is very well reproduced \[15\]. The ZnSe band structure is rigidly shifted to make the iron Fermi energy fall 1 eV above the ZnSe valence band and 1.1 eV below the conduction band, as indicated by photoemission experiments \[9\].
For simple junctions, we find that the conductances decay almost exponentially with semiconductor thickness, and that the TMR increases and is always positive (or direct), reaching a value of 90 $\%$ for a semiconductor thickness of 34 Å . Our results are in very good agreement with the [*ab initio*]{} results of MacLaren and coworkers \[18\].
For double junctions, we vary the SCR thickness between 5.67 Å and 28.35 Å , and the IBMR thickness between 2.87 Å and 22.96 Å . We find that the TMR and conductances strongly depend on the in-between metal thickness, and that for certain thickness combinations of the SC and IBM regions they can be higher than those corresponding to a MTJ, in agreement with the results of L. Sheng and coworkers \[3\]. The maximum ratio of DMTJs to simple MTJs conductances obtained is of 322 $\%$, and occurs for the P majority channel corresponding to SC regions of 22.7 Å and an IBM region of 21 Å . This large conductance ratio, which is pointing toward the existence of resonant states (confirmed by our DOS calculations), does not mean that the DMTJs TMR is going to be much larger than the MTJs one, although in general it is. In this particular case, the TMR value is of 97.9 $\%$ (the corresponding MTJ’s value is 63.8 $\%$), but in other cases the TMR values are greatly enhanced [*even in the absence of resonances*]{}. We find that the TMR enhancement can be a result of: (i) a drop in the conductance of some spin channels, while the conductances of other channels remain of the same order of magnitude as those in MTJs, or (ii) an increase in the conductance of one particular spin channel due to resonant tunneling. Both effects are produced by a change in the active region’s DOS near $E_F$, induced by the presence of the in-between Fe layers.
As an example of resonance conductance enhancement, we show in Fig. 5 the active region’s total DOS at $E_F$, as a function of the IBMR thickness and for a DMTJ with a SCR of 22.7 Å . An increase in one order of magnitude appears at an IBMR of 2.87 Å for the P majority channel and for the AP minority channel, and a smaller increase appears at an IBMR of 8.6 Å for the AP majority channel, while for the other cases the DOS is almost constant. These peaks coincide with a conductance enhancement in these three channels, as it can be seen in the lower panel of Fig. 5, indicating that the origin of the conductance enhancement is the same as in 2DDJs, namely resonant tunneling.
To visualize the interplay among the conductance values of the different channels and configurations, Fig. 6 shows the conductances and TMR values for a given DMTJ with a SCR of 17 Å and those of the corresponding simple MTJ, as a function of the IBMR thickness. It is seen that, already for 6 Å of Fe, the TMR is 1.5 times higher than that of a simple MTJ, although the conductances are, in general, a little bit smaller. This is similar to what happens in all the cases studied. It is noticeable that for very thin Fe layers the TMR obtained for this DMTJ is negative, and that for IBMR thicknesses in the range 12-23 Å the TMR is almost constant. This also happens for the other SCR thicknesses studied, and it is different to the damped oscillatory behavior that it is obtained using rectangular potential profiles and a non-magnetic in-between metal \[4\].
For a given SC region thickness, we look for the maximum attainable TMR value by sweeping over IBMR’s thicknesses. We find that for ZnSe regions with thicknesses below 20 Å , the DMTJs’ TMR values are 3 times higher than those of a simple junction, [*while the conductances of some spin channels remain of the same order of magnitude*]{}. Beyond this thickness, the DMTJs TMR can be 50 times higher but negative (inverse TMR), although in this case the conductances [*of all spin channels*]{} are 4 to 6 orders of magnitude smaller, and thus very hard to measure. There is one particular case in which this does not happen. For a DMTJ with a SCR of 22.7 Å and an IBMR of 8.6 Å , we obtain a drop in the conductances of the P and AP minority channels and the P majority channel, and an enhancement of 175 $\%$ in the AP majority channel with respect to the corresponding MTJ, which results in a negative TMR enhancement by a factor of -40.
In summary, we have investigated Fe$\mid$ZnSe double magnetic tunnel junctions within a realistic Hamiltonian model and found that the TMR values can be much higher than those of simple junctions. We should mention that temperature effects, interfacial roughness, and the presence of defects in the DMTJs active region may decrease the TMR values obtained in our calculations, but we believe that our results remain qualitatively valid. We conclude that the thickness of the in-between Fe layers in Fe$\mid$ZnSe DMTJs is an interesting degree of freedom, which may make it possible to tune and enhance the TMR of these systems, making them suitable for building future spintronic devices. To improve our understanding of these scarcely studied double junctions, it is highly desirable the experimental measurement of the TMR as a function of the in-between metallic layers thickness.
We are grateful to Julián Milano for useful discussions. This work was partially funded by UBACyT-X115, Fundación Antorchas and PICT 03-10698. Ana María Llois belongs to CONICET (Argentina).
$[1]$ X-G. Zhang and W. H. Butler, J. Phys.: Condens. Matter [**15**]{}, 1603 (2003); E. Y. Tsymbal, O. N. Mryasov, and P. R. LeClair [*ibid.*]{} [**15**]{}, 109 (2003)\
$[2]$ X. Zhang, B-Z Li, G. Sun, and F-C. Pu, Phys. Rev. B [**56**]{}, 5484 (1997)\
$[3]$ L. Sheng, Y. Chen, H. Y. Teng, and C. S. Ting, Phys. Rev. B [**59**]{}, 480 (1999)\
$[4]$ M. Chshiev, D. Stoeffler, A. Vedyayev, and K. Ounadjela, Europhys. Lett. [**58**]{}, 257 (2002)\
$[5]$ B. Wang, Y. Guo, and B-L. Gu, J. Appl. Phys. [**91**]{}, 1318 (2002)\
$[6]$ F. Giazotto, Fabio Taddei, Rosario Fazio, and Fabio Beltram, Appl. Phys. Lett. [**82**]{}, 2449 (2003)\
$[7]$ T. Nozaki [*et al*]{}, Appl. Phys. Lett. [**86**]{}, 082501 (2005)\
$[8]$ Z. Zeng [*et al*]{}, Phys. Rev. B [**72**]{}, 054419 (2005); J. H. Lee [*et al*]{}, J. Magn. Magn. Mater. [**286**]{}, 138 (2005)\
$[9]$ M. Eddrief [*et al*]{}, Appl. Phys. Lett. [**81**]{}, 4553 (2002)\
$[10]$ M. Marangolo [*et al*]{}, Phys. Rev. Lett. [**88**]{}, 217202-1 (2002)\
$[11]$ X-G. Zhang, W. H. Butler, and A. Bandyopadhyay, Phys. Rev. B [**68**]{}, 092402 (2003)\
$[12]$ S. Sanvito [*et al*]{}, Phys. Rev. B [**59**]{}, 11936 (1999)\
$[13]$ S. Datta, ’Electronic transport in mesoscopic systems’ (Cambridge University Press, United Kingdom, 1999)\
$[14]$ Z. Zheng, Y. Qi, D. Y. Xing, and J. Dong, Phys. Rev. B [**59**]{}, 14505 (1999)\
$[15]$ D. A. Papaconstantopoulos, ’Handbook of the band structure of elemental solids’ (Plenum Press, New York, 1986)\
$[16]$ R. Viswanatha, S. Sapra, T. Saha-Dasgupta and D. D. Sarma, cond-mat 0505451 v1, 18 May 2005\
$[17]$ O. K. Andersen, Physica B [**91**]{}, 317 (1977)\
$[18]$ J. M. MacLaren, X. G. Zhang, W. H. Butler and X. Wang, Phys. Rev. B [**59**]{}, 5470 (1999)\
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abstract: |
Freezing of the classical spin liquid of the Heisenberg kagomé antiferromagnet is studied. At low temperature, the coplanar spin configurations are known to dominate among all possible three-dimensional low-energy states because of the fluctuation interaction. It is shown that the statistical weight of ${\bf q} =0$ domains is negligible. This allows one to describe coplanar states in terms of $\sqrt{3}
\times \sqrt{3}$ domains with three possible spin orientations. The domain walls are mapped onto closed self-avoiding loops on the hexagonal lattice. The probability of domain formation is evaluated. It is shown that in FC (field cooled) regime a telescopic hierarchy of domains appears. The spatial and temporal behavior of the spin correlation function is estimated. The difference between FC and ZFC behavior is discussed.
address: |
School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel and\
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, 76100, Israel
author:
- 'V.B. Cherepanov'
date: 'July 13, 1994'
title: 'How do frustrations without disorder result in the spin-glass-like behavior of the kagomé antiferromagnet?'
---
It is usually assumed that the irreversibility of spin glasses results from interference of frustrations and randomness [@BY; @FH]. Nevertheless, the highly frustrated Heisenberg antiferromagnets, with the two-dimensional kagomé and the three-dimensional pyrochlore lattices, exhibit the macroscopic properties characteristic for spin glasses even when they have ordered structures and non-fluctuating exchange constants[@exp; @GRM]. The main purpose of this paper is to study the mechanism of the spin-glass-like behavior of the classical kagomé antiferromagnet with the nearest neighbor Heisenberg exchange interaction and to show that it is due to a domain hierarchy arising in this magnet at low temperature.
The nearest neighbor exchange interaction is known to dominate in the kagomé antiferromagnet [@exp; @CCR]. The Hamiltonian of the classical kagomé antiferromagnet can be represented as a sum of the total spins in the triangles of the nearest neighbors, ${\bf
S}_{\Delta}$: $$H = 1/2 \; J \sum_{\Delta} \left( {\bf S}_{\Delta}
\right)^{2},$$ where $\Delta$ numbers the triangles of the nearest neighbors. Each spin participates in two triangles and it contributes into two terms ${\bf S}_{\Delta}$. The ground state energy is equal to zero and there are infinitely many ground states with ${\bf S}_{\Delta} = 0 $.
In antiferromagnetic SCGO, which has been studied experimentally in some detail, the exchange interaction constant $J \approx 500$ K. The transition to the spin-glass-like state occurs at $T_{f}
\approx 4$ K, at the temperature two orders lower than the exchange interaction energy. One could relate this transition to some other relatively weak interaction which was not clearly indicated experimentally because of its weakness. However, such a “mechanism” for irreversibility unlikely takes place because the numerical simulations of the model (1) [@RB; @SH; @HR] with no additional interactions agree with the low temperature experimental results not only qualitatively but also quantitatively. In particular, the results of extensive MC simulations of the low temperature properties of the model (1) show that the classical kagomé antiferromagnet with the nearest neighbor interaction exhibits irreversible behavior when $T < T_{f}$, $T_{f}/J = 8 \cdot 10^{-3}$ [@RB]. The experimentally measured ratio of the transition temperature to the exchange interaction energy lies in the interval $(7.2 - 8.5)\cdot 10^{-3}$ [@SP]. This means that (i) the model (1) possesses the low temperature glass phase and (ii) the temperature of the experimentally observed transition is very close to that found in numerical simulations of the model. Therefore, relatively weak interactions neglected in Eq. (1) probably are not related to the spin-glass-like transition.
The classical kagomé antiferromagnet has a macroscopically large number of zero energy modes corresponding to continuous transitions between various ground states [@HKB; @CHS]. Since any ground state obeys the condition that the sum of the spins in each triad of the nearest neighbors is equal to zero, and it is the only condition for a ground state, one can consider all possible foldings of the triangular lattice in the three-dimensional spin space which leave the spin triangles rigid. Any folded configuration of the triangular lattice represents a ground state of the kagomé antiferromagnet [@we]. It can be shown that there exists one-to-one correspondence between the ground states and the foldings of the triangular lattice (“origami”) in the 3D spin space. The zero energy modes correspond to the foldings.
The statistical weight of thermally excited states near coplanar ground states dominates over the weight of all other low energy states [@CHS]. This means that the coplanar states and the near-coplanar states have the largest entropy because these states are most flexible states of the triangular tethered membrane. Now we introduce a geometrical classification of the coplanar ground states. Marking the axis of foldings on the triangular lattice yields a prescription how to get from the state ${\bf q} =0$ to any folded state. Each line depicts a boundary between different domains with ${\bf q} =0$ order. Thus, the map of folding lines on the triangular lattice provides a geometrical representation of the coplanar states in terms of boundaries of ${\bf q} =0$ domains on the plane. This map consists of a number of meetings and intersections of straight lines. There are only four possible configurations of the line meetings and intersections. They are shown in the fig. 1 as continuous lines: (a) meeting of four lines (“fork”), (b) straight line, (c) six radii star, and (d) no lines at all, it is the homogeneous state ${\bf q} =0$. The axis of folding depict the boundaries between domains with ${\bf q} =0$ order.
=
The state $ \sqrt{3} \times \sqrt{3}$ results from folding of the planar triangular lattice into one triangle, its folding map consists of the six radii stars located in each triangle junction, i. e. it is the whole triangular lattice of the folding axis. I will show further that the statistical weight of the state $ \sqrt{3} \times \sqrt{3}$ and the states close to it is much larger than that of the state ${\bf q} =0$. In order to describe the states close to $ \sqrt{3} \times \sqrt{3}$, another geometrical representation of the ground states, dual to the folding line map described above, is more convenient. Each coplanar state can be obtained by a sequence of $180^{\circ}$ unfoldings of the stack of the spin triangles which represents the state $ \sqrt{3} \times
\sqrt{3}$. If one depicts the axis of unfolding one obtains another geometrical representation of the coplanar states. There are four possible elements that form the dual map shown in fig. 1: (e) 120-degree angle, (f) cross constructed of two angles (fig. 1(e)), (g) the state $ \sqrt{3} \times \sqrt{3}$ with no unfolding lines, and (h) six radii star. These lines correspond to boundaries between domains of different states $ \sqrt{3} \times \sqrt{3}$. There are three kinds of domain orientations arising from unfolding along three axis on the triangular lattice. In the dual representation, a ${\bf q} =0$ domain is covered with the six radii stars, the smallest ${\bf q} =0$ domain is represented by one star (h).
The simplest spin configuration close to the homogeneous state $ \sqrt{3} \times \sqrt{3}$, the weather vane defect, is produced by rotation of a group of six neighboring spin triangles. There are three different ways to rotate the triangles along each of the triangular lattice axis. If $ \theta_{1},
\theta_{2}$, and $\theta_{3}$ are the the angles of those rotations the energy of a low-energy configuration for the small angles takes the form: $ V( \theta_{1}, \theta_{2}, \theta_{3}) =
\alpha_{0} J \left( \theta_{1}^{2} \theta_{2}^{2} +
\theta_{2}^{2} \theta_{3}^{2} + \theta_{3}^{2} \theta_{1}^{2} \right) $ where $\alpha_{0} $ is a dimensionless coefficient. At low temperature, only the states in the vicinity of the coplanar state with $ | \theta_{i} | \lesssim (T/\alpha_{0}J)^{1/4} $ contribute to the statistical sum [@CHS].
The energy of a state in the vicinity of an arbitrary coplanar ground state is $ V( \theta_{i} ) = J \sum_{i,j} \alpha_{i,j}
\theta_{i}^{2} \theta_{j}^{2}. $ For the state $ \sqrt{3} \times \sqrt{3}$ the coefficients $ \alpha_{i,j} $ are equal to zero for most pairs of the angles $ ( \theta_{i}, \; \theta_{j} ) $. They are nonzero for pairs of angles involved in the same weather-vane defects only. In the state ${\bf q} =0$, all the angles inside the domain, except foldings along parallel axis, are involved, i. e. $\alpha_{i,j} = \alpha_{1} $ for the majority of the pairs.
Comparing the statistical weights of the states near coplanar $ \sqrt{3} \times \sqrt{3}$ and ${\bf q} =0$ configurations one can find that the statistical sum $ Z_{{\bf q}=0} $ rapidly decreases with the number of spins $N$ and $$\frac{Z_{{\bf q}=0}}{ Z_{\sqrt{3} \times \sqrt{3}}} \approx
\left(\frac{\alpha_{0}}{\alpha_{1}} \right) ^{N/4} \frac{
\Gamma{(N/4)} /
\Gamma{(N/2)} }{\left[ \Gamma{(3/4)} / \Gamma{(3/2)}
\right] ^{N/3}}.$$ The smallest size of a ${\bf q} =0$ domain is $N = 6 $. In addition, such domains appear in pairs, therefore the statistical weight of ${\bf q} =0$ domains is strongly suppressed in comparison with that of $ \sqrt{3} \times \sqrt{3}$ domains. The weight a pair of domains ${\bf q} =0$ is $\epsilon \approx
2^{-12} (\alpha_{0}/\alpha_{1} )^{3}$ times smaller than the weight of the $ \sqrt{3} \times \sqrt{3}$ state. An analysis of results of MC simulations [@RB] shows that there was only one pair of smallest possible ${\bf q} =0$ domains, with $N=12$, on the lattice with 576 spins observed.
The small parameter $\epsilon $ allows one to neglect ${\bf q} =0$ domains. This provides a possibility for a simple geometrical classification of the domain boundaries: the domain boundaries form closed loops consisting of $120^{\circ}$ elements (fig. 1 (e)), and if we have two loops, only two alternatives exist: either one loop is entirely inside another loop or the loops are outside each other.
Now we evaluate the probability of a domain formation against the homogeneous $ \sqrt{3} \times \sqrt{3}$ background and the probability of dissipation of such a domain. This case corresponds to FC conditions because the thermal fluctuations select the state $ \sqrt{3} \times \sqrt{3}$ in a finite magnetic field [@SH].
Since no ground state dynamics may exist we employ the relaxation dynamics formalism[@FH; @ZJ] in order to describe the domain kinetics under influence of thermal fluctuations. We start with the Langevin equation for the angles $\theta_{i}$ and consider the case when a group of many spin triangles is rotated as a whole domain. In this case, only one $180^{\circ }$ rotation is necessary to overcome the free energy barrier between the initial and final coplanar configurations. (The case of a single weather vane defect rotation was considered in [@DH]). The potential of low energy states in the vicinity of such a trajectory is: $$V\{ \theta_{i} \} = J \sum_{j=1}^{L} f(\theta_{0})
\theta_{j}^{2},$$ where the angle $\theta_{0}$ goes from $0$ to $\pi $, it corresponds to the domain rotation, and the angles $\theta_{j}$ describe small fluctuations of the spin triangles in all other directions. The function $f(\theta_{0})$ is periodic, its expansion in series of $\theta_{0}$ starts from quadratic terms at both $\theta_{0} = 0$ and $\theta_{0} = \pi $.
The probability for the system to reach the configuration $\{ \theta_{i}^{(1)} \}$ at the moment $\tau_{1}$ starting from the configuration $\{ \theta_{i}^{(0)} \}$ at the moment $\tau_{0}$ is $$\begin{aligned}
P\{\theta_{i} \} = \int \exp{ \left( -
\int_{\tau_{0}}^{\tau_{1}} {\cal L}\{\theta_{i}(\tau ) \} d\tau \right) }
D\theta_{i}(\tau ), \nonumber \\
{\cal L} \{\theta_{i}(\tau ) \} = 1/2 \; (\partial \theta_{0} / \partial \tau
)^{2} + 1/2 \sum_{j=1}^{L} (\partial \theta_{j} / \partial \tau
)^{2} \nonumber \\
+ 1/2 \; (\beta J)^{2}\left( f^{2}(\theta_{0}) +
f'^{2}(\theta_{0}) \sum_{j} {\theta_{j}}^{2} \right)
\sum_{j} {\theta_{j}}^{2} \nonumber \\
+ 1/2 \; (\beta J) {f''(\theta_{0})} \sum_{j} {\theta_{j}}^{2}
+ 1/2 \; L (\beta J) {f(\theta_{0})}\end{aligned}$$ where $\beta = 1/T $ and $ \tau $ is the dimensionless time. The probability to get from a state in the vicinity of the coplanar ground state $
\theta_{0}^{(0)} =0 $ to the states near $ \theta_{0}^{(1)} = \pi $ is the product of the probability to overcome the free energy barrier between the initial and the final states and the statistical weight of the final state. At finite $\theta_{0}$, the $\theta_{j}$ degrees of freedom are stiff. After integrating over these fast degrees of freedom one obtains an effective potential for the slow rotation $\theta_{0}$. This effective potential is proportional to the domain perimeter $L$ in the large $L$ limit. This results in the effective action $\cal A$ along the trajectory proportional to $ \sqrt L$. The probability to overcome the barrier along the instanton path is proportional to $\exp{({\cal -A})}$. The ratio of the statistical weights of final and initial states $ W_{L} $ can be estimated as follows: $$W_{L} = \frac{\int \Pi d \theta_{i} \times \nonumber \\
\exp{\left[ - (\beta J )
\left( \alpha_{1}{ \sum^{L}_{j=1}} \theta_{0}^{2}
\theta_{j}^{2}
\right) \right]}}{
\int \Pi d \theta_{i} \times \nonumber \\
\exp{\left[ - (\beta J )
\left( \alpha_{0}{ \sum^{L}_{j=1}} \theta_{0}^{2}
\theta_{j}^{2}
\right) \right]}}
\label{W_L}$$ where summation runs over the rotating triangles. If all the angles $ \theta_{j} $ in the numerator (\[W\_L\]) were independent, the ratio $W_{L}$ (\[W\_L\]) would be equal to $ W_{L} = \exp{ \left[ - 1/2 \;
\ln{\left( \alpha_{0} /\alpha_{1} \right)} L \right]}.$ Actually, not all the angles $ \theta_{j} $ are independent. Nevertheless, it can be shown that $$W_{L} \propto \exp{ (- \rho L )}, \; \; \; \; \rho
\sim 1$$ in the $L \rightarrow \infty $ limit.
The probability of formation of any domain with the perimeter length $L$ is equal to the product of the probability for a single domain to appear, $W_{L}$, and the number of domains with the boundary length $L$, $N_{L}$. The latter coincides with the number of closed self-avoiding chains on the hexagonal lattice [@CJ] $$N_{L} \propto \exp {( \eta L )}, \; \; \; \eta \approx 0.609.$$
We arrive at an alternative: if $ \rho < \eta $ then a large domain arises against a homogeneous $ \sqrt{3} \times \sqrt{3}$ background most probably. In the opposite case, $ \rho > \eta $, smallest domains arise first, they cover all the lattice and prevent from growth of large domains.
A hierarchical structure of the phase space is a key component to the slow relaxation and glassy behavior[@PSAA]. In the case of the field cooled kagomé antiferromagnet, the hierarchy results from the instability of the homogeneous $ \sqrt{3} \times \sqrt{3}$ state against domain formation. If $ \rho < \eta $ then the largest domains are the most unstable ones. The instability leads to formation of the largest possible domain, of order of the sample size, with the perimeter length $ L_{1}$. All the area inside that domain is a homogeneous state $ \sqrt{3} \times \sqrt{3}$, the area outside the domain is also a homogeneous state $ \sqrt{3} \times \sqrt{3}$ of another orientation. The areas inside and outside the first domain are unstable with respect to domain formation. Let us concentrate on domain formation inside the first domain. If one waits for some time, a smaller domain inside the first one appears. The perimeter of the second domain is $ L_{2} \approx L_{1} - \Delta L $, where $\Delta L = \max \left[ (\eta-\rho)^{-1}, 12 \right] $. If one waits for a long time one observes formation of smaller and smaller domains inside larger ones. In the limit of large $L$, only one domain of the size $ \approx
L - \Delta L $ appears inside the domain of the previous generation, with the perimeter $L$. The hierarchy in such a domain structure is telescopic: a smaller domain is subordinated to a larger one whose boundary surrounds the former. In the opposite case, $ \rho > \eta $, no hierarchy arises and no irreversibility exists. The experiment [@exp] and especially MC simulations [@RB], where the telescopic domain hierarchy can be seen in a snap-shot picture, indicate that $\rho < \eta $ in the kagomé antiferromagnet.
Now we estimate the spatial and temporal dependencies of the spin correlation function. The spatial correlation function $C_{\sqrt{3} \times \sqrt{3}}({\bf r,r'})$ corresponding to the $\sqrt{3} \times \sqrt{3}$ order [@RB]. If there are domains in a sample the correlation function can be equal either to 1 if the spins belong to domains with the same orientation, or to $ - 1/2$ if the spins belong to domains with different orientations, and there are two possible choices of different domain orientations. In the sample filled with domains the correlation function $C_{\sqrt{3} \times \sqrt{3}}({\bf r,r'})$ is equal to 1 if there is a path connecting the spins that does not cross any domain boundary. If the spin $ {\bf S}({\bf r})$ belongs to a domain of the size $L$, the correlation function can be estimated as follows: it is $ \propto 1/|{\bf r - r'}|$ if $|{\bf r - r'}| < L $ and it is equal to zero if $|{\bf r - r'}| \gg L $. Averaging over the spin positions ${\bf r}$ yields the estimate $$C_{\sqrt{3} \times \sqrt{3}}({\bf r,r'}) \propto 1/|{\bf r - r'}|$$ This estimate is in a good agreement with the results of MC simulations [@RB] that $C_{\sqrt{3} \times
\sqrt{3}}({\bf r,r'}) \propto |{\bf r - r'}|^{-0.93}$ in the case of relaxation starting with the initial homogeneous state $\sqrt{3}
\times \sqrt{3}$.
Let us estimate the time dependence of the spin correlation function $ C(t) = \langle {\bf S}({\bf r},0) {\bf S}({\bf r},t)
\rangle $. If the spin $ {\bf S(r)}$ belongs to a domain of the size $L$, it rotates with a characteristic period $ t_{L}$. The product ${\bf S}({\bf r},t_{0}) {\bf S}({\bf r},t_{0} + \Delta t)$ changes its sign with the characteristic time $ \Delta t \sim t_{L}$. Therefore $$\frac{\Delta C(t)}{\Delta t} \propto
- \frac{C(t)}{N} \frac{A_{L + \Delta L} {\cal N}_{L + \Delta L}
- A_{L}{\cal N}_{L} }{\Delta L} \frac{ \Delta L}{\Delta t}.$$ Here $N$ is the number of the spins in the sample, the number of spins in a domain of the size $L$ is proportional to its area $A_{L}$, ${\cal N}_{L}$ is the number of domains of the size $L$. The size of the domains $L$ is connected with time $t$: $ t_{L} = w_{0}^{-1} \exp ({\alpha L})$. In general, one can expect that $$A_{L} \propto L^{2 \nu}, \; \;
{\cal N}_{L} \propto L^{\zeta}.$$ This results in in the following time decay of the spin correlation function: $$\ln \left(\frac{C({\bf r}, t)}{ C({\bf r}, 0)} \right)
\propto - \left[ \ln \left( w_{0} t \right)
\right]^{\zeta + 2 \nu}.$$ In the case of FC regime, we have the telescopic domain hierarchy with $\zeta = 0$. The exponent $2 \nu $ is the mean area of a random self-avoiding loop with the perimeter $L$ on the hexagonal lattice. The exponent $\nu $ is bounded by the exponent for the mean squared distance between elements of a random self-avoiding chain on the hexagonal lattice, $\nu_{low} = 3/4 $ [@CJ], and $\nu_{high}=1$. Thus the spin correlation exhibits a stretched exponential relaxation characteristic for glasses.
In the ZFC regime, the relaxation starts from a random domain distribution, with the exponent $\zeta \approx - 4/3 $ [@HR; @RCC]. The small domains do not allow new large domains to appear because of the no boundary crossing constraint. Qualitatively, this results in a faster decay of the spin correlation function $ C_{\sqrt{3} \times
\sqrt{3}}({\bf r}) $ with the distance and in a different susceptibility in comparison with the case of the FC regime.
In conclusion, it has been shown that: (i) The local $ \sqrt{3} \times
\sqrt{3}$ order dominates at low temperature, this explains results of experimental observations [@exp; @BAEC]. (ii) In the field cooled regime a telescopic hierarchy of domains arises. This hierarchy leads to slow spin relaxation, the estimates of the correlation function decay agree with results of MC simulations [@RB]. The reason for the difference between FC and ZFC spin correlations is justified. Thus, the origin of the spin-glass-like behavior of the pure kagomé antiferromagnet has been clarified.
I acknowledge numerous discussions with E. Shender on the initial stage of the research. I am grateful to A. Aharony, J. Berlinsky, B. Botvinnik, M. Chertkov, P. Coleman, V. Elser, C. Henley, D. Huber, D. Huse, A. Ruckenstein, M. Schwartz, and P. Young for useful discussions. I am grateful to J. Reimers and J. Berlinsky, D. Huse, R. Chandra and P. Coleman, and E. Shender and P. Holdsworth for sending me preprints of their papers. I also acknowledge financial support from the US — Israel Binational Science Foundation (GIF).
K. Binder and A.P. Young, Rev. Mod. Phys. [**58**]{}, 801 (1986).
K.H. Fisher and J.A. Hertz, Spin Glasses, Cambridge University Press, 1991.
For a brief review see A.P. Ramirez, J. Appl. Phys. [**70**]{}, 5952 (1991) and references cited therein.
B.D. Gaulin, J.N. Reimers and T.E. Mason, Phys. Rev. Lett., [**69**]{}, 3244 (1992).
P. Chandra, P. Coleman and I. Ritchey, J. Phys. I France, [**3**]{}, 591 (1993).
J.N. Reimers and A.J. Berlinsky Phys. Rev. [**B 48**]{}, 9539 (1993).
E.F. Shender and P.C.W. Holdsworth, preprint, to be published in Fluctuations and Order (ed. M.M. Milonas), MIT Press.
D.A. Huse and A.D. Rutenberg, Phys. Rev. [**B 45**]{}, 7536 (1992).
B. Martínez, F. Sandiumenge, A. Rouco, A. Labarta, J. Rodríguez-Carvajal, M. Tovar, M.T. Causa, S. Galí and X. Obradors, Phys. Rev. [**B 46**]{}, 10786 (1992).
A.B. Harris, C. Kallin and A.J. Berlinsky, Phys. Rev. [**B 45**]{}, 2899 (1992).
J. Chalker, P.C. Holdsworth and E.F. Shender, Phys. Rev. Lett., [**68**]{}, 855 (1992).
E.F. Shender, V.B. Cherepanov, P.C.W. Holdsworth, and A.J. Berlinsky, Phys. Rev. Lett., [**70**]{}, 3812 (1993).
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press, 1993.
J. von Delft and C.L. Henley, Phys. Rev. [**B 48**]{}, 965 (1993).
J. des Cloizeaux and G. Jannink, Polymers in Solution, Clarendon Press, Oxford, 1990.
R.G. Palmer, D.L. Stein, E. Abrahams and P.W. Anderson, Phys. Rev. Lett., [**53**]{}, 958 (1984).
I. Ritchey, P. Coleman and P. Chandra, Phys. Rev. [**B 47**]{}, 15342 (1993).
C. Broholm, G. Aeppli, G.P. Espinosa, and A.S. Cooper, Phys. Rev. Lett., [**65**]{}, 3173 (1990).
|
---
abstract: 'We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are $H(\zeta)=U+U^{-1}+V+\zeta V^{-1}$ and $H_{m,n}=U+V+q^{-mn}U^{-m}V^{-n}$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q={\mathrm{e}}^{{\mathrm{i}}\pi b^{2}}$, $b>0$ and $\zeta>0$, $m,n\in{\mathbb{N}}$. We prove that $H(\zeta)$ and $H_{m,n}$ are self-adjoint operators with purely discrete spectrum on $L^{2}({\mathbb{R}})$. Using the coherent state transform we find the asymptotical behaviour for the Riesz mean $\sum_{j\ge 1}(\lambda-\lambda_{j})_{+}$ as $\lambda\to\infty$ and prove the Weyl law for the eigenvalue counting function $N(\lambda)$ for these operators, which imply that their inverses are of trace class.'
address:
- |
Ari Laptev, Department of Mathematics\
Imperial College London, SW7 2AZ, London, UK\
Institut Mittag-Leffler, Djursholm, Sweden
- 'Lukas Schimmer, Department of Physics, Princeton University, Princeton, NJ 08544, USA'
- |
Leon A. Takhtajan, Department of Mathematics\
Stony Brook University\
Stony Brook, NY 11794-3651\
USA; Euler Mathematical Institute, Saint Petersburg, Russia
author:
- Ari Laptev
- Lukas Schimmer
- 'Leon A. Takhtajan'
bibliography:
- 'biblio\_weyl.bib'
title: 'Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves'
---
Introduction {#sec:intro}
============
Let $P$ and $Q$ be quantum-mechanical momentum and position operators on $L^{2}({\mathbb{R}})$, satisfying on their common domain the Heisenberg commutation relation $[P,Q]={\mathrm{i}}I$. Consider the corresponding Weyl operators $U={\mathrm{e}}^{-b P}$ and $V={\mathrm{e}}^{2\pi b Q}$, where $b>0$. The operators $U$ and $V$ are unbounded self-adjoint operators on $L^{2}({\mathbb{R}})$, satisfying on their common domain the Weyl relation $$UV=q^{2}VU,$$ where $q={\mathrm{e}}^{{\mathrm{i}}\pi b^{2}}$. In the coordinate representation $(P\psi)(x)={\mathrm{i}}\psi'(x)$ and $(Q\psi)(x)=x\psi(x)$, and the Weyl operators have the form $(U\psi)(x)=\psi(x+{\mathrm{i}}b)$ and $(V\psi)(x)={\mathrm{e}}^{2\pi b x}\psi(x)$. Their respective domains are $$\begin{aligned}
{D}(U)&={\left\{\psi\in L^2({\mathbb{R}}): \, {\mathrm{e}}^{-2\pi b k}\widehat{\psi}(k)\in L^2({\mathbb{R}})\right\}},\\
{D}(V)&={\left\{\psi\in L^2({\mathbb{R}}): \, {\mathrm{e}}^{2\pi b x}\psi(x)\in L^2({\mathbb{R}})\right\}},\end{aligned}$$ where $\mathcal{F}$ is the Fourier transform $$\begin{aligned}
\widehat{\psi}(k)=(\mathcal{F}\psi)(k)=\int_{\mathbb{R}}{\mathrm{e}}^{-2\pi{\mathrm{i}}kx}\psi(x){\, \mathrm{d}}x\end{aligned}$$ on $L^2({\mathbb{R}})$. Equivalently, ${D}(U)$ consists of those functions $\psi(x)$ which admit an analytic continuation to the strip ${\left\{z = x+{\mathrm{i}}y\in{\mathbb{C}}: \, 0<y <b\right\}}$ such that $\psi(x +{\mathrm{i}}y) \in L^2({\mathbb{R}})$ for all $0\leq y < b$ and there is a limit $\psi(x +{\mathrm{i}}b - {\mathrm{i}}0) = \lim_{{\varepsilon}\to 0^+}\psi (x + {\mathrm{i}}b - {\mathrm{i}}{\varepsilon})$ in the sense of convergence in $L^2({\mathbb{R}})$, which we will denote simply by $\psi(x+{\mathrm{i}}b)$. The domain of $U^{-1}$ can be characterized similarly.
Using the Weyl operators $U$ and $V$, one constructs the operator $$H=U+U^{-1}+V,$$ which in the coordinate representation becomes a functional-difference operator $$(H\psi)(x)=\psi(x+{\mathrm{i}}b) + \psi(x-{\mathrm{i}}b) + {\mathrm{e}}^{2\pi b x}\psi(x).$$ The operator $H$ first appeared in the study of the quantum Liouville model on the lattice [@Faddeev1986] and plays an important role in the representation theory of the non-compact quantum group $\mathrm{SL}_{q}(2,{\mathbb{R}})$. In the momentum representation it becomes the Dehn twist operator in quantum Teichmüller theory [@Kashaev2001]. In particular, in [@Kashaev2001] the eigenfunction expansion theorem for $H$ in the momentum representation was stated as formal completeness and orthogonality relations in the sense of distributions.
The spectral analysis of the functional-difference operator $H$ was done in [@Faddeev2014]. The operator $H$ was shown to be self-adjoint with a simple absolutely continuous spectrum $[2,\infty)$, and the eigenfunction expansion theorem for $H$, generalizing the classical Kontorovich-Lebedev transform, was proved.
It was discovered in [@ADKMV] that the functional-difference operators built from the Weyl operators $U$ and $V$, also appear in the study of local mirror symmetry as a quantization of an algebraic curve, the mirror to a toric Calabi-Yau threefold. The spectral properties of these operators were considered in [@GHM]. The typical example is a so-called local del Pezzo Calabi-Yau threefold, a total space of the anti-canonical bundle on a toric del Pezzo surface $S$. In the simplest case of the Hirzebruch surface $S=\mathbb{P}^{1}\times\mathbb{P}^{1}$ one gets the following operator $$\begin{aligned}
\label{eq:type1}
H(\zeta)={\mathrm{e}}^{-bP}+{\mathrm{e}}^{bP}+{\mathrm{e}}^{2\pi bQ}+\zeta{\mathrm{e}}^{-2\pi b Q}=U+U^{-1}+V+\zeta V^{-1}\,,\end{aligned}$$ where $\zeta>0$ is a “mass” parameter, so that $H=H(0)$. In case $S$ is a weighted projective space $\mathbb{P}(1,m,n)$, $m,n\in{\mathbb{N}}$, the corresponding operator is $$\begin{aligned}
\label{eq:type2}
H_{m,n}={\mathrm{e}}^{-bP}+{\mathrm{e}}^{2\pi b Q}+{\mathrm{e}}^{bmP-2\pi bnQ}=U+V+q^{-mn}U^{-m}V^{-n},\end{aligned}$$ and $H=H_{1,0}$ (see [@GHM] for details). It was conjectured in [@GHM] for the cases $\zeta>0$ and $m,n\in{\mathbb{N}}$ that these operators have a discrete spectrum, their inverses are of trace class and their Fredholm determinants can be explicitly evaluated in terms of enumerative invariants of the underlying Calabi-Yau threefolds. In a recent paper [@Kashaev2015] some of these conjectures were proved and the authors obtained a remarkable explicit formula for the operators $H(\zeta)^{-1}$ and $H^{-1}_{m,n}$ in terms of the modular quantum dilogarithm.
The present paper is devoted to the study of Weyl type asymptotics for the operators $H(\zeta)$ and $H_{m,n}$ as self-adjoint operators on $L^{2}({\mathbb{R}})$. Namely, we prove that they are operators with purely discrete spectrum and investigate the asymptotic behavior of their eigenvalues, from which it immediately follows that $H(\zeta)^{-1}$ and $H_{m,n}^{-1}$ are of trace class. Our main results are Theorems \[th:Hasymp\] and \[th:Gasymp\] on the asymptotic behaviour of the Riesz mean $\sum_{j\ge 1}(\lambda-\lambda_{j})_{+}$ and Corollaries \[cor:Hasymp\] and \[cor:Gasymp\] on the Weyl law for the eigenvalue counting function $N(\lambda)$ for these operators. Namely, $$\label{W-zeta}
\lim_{\lambda\rightarrow\infty}\frac{N(\lambda)}{\log^{2}\lambda}=\frac{1}{(\pi b)^{2}}$$ for the operator $H(\zeta)$ and $$\label{W-mn}
\lim_{\lambda\rightarrow\infty}\frac{N(\lambda)}{\log^{2}\lambda}=\frac{c_{m,n}}{(2\pi b)^{2}},\quad c_{m,n}=\frac{(m+n+1)^{2}}{2mn}$$ for the operator $H_{m,n}$. The proof follows ideas developed in [@Laptev1997], where the Fourier transform is replaced by the coherent state transform. The applied methods also mimic the derivation of the Berezin–Lieb inequality [@Berezin1972; @Berezin1972b; @Lieb1973].
Acknowledgements {#acknowledgements .unnumbered}
----------------
The work of L.S. was supported by the NSF grant PHY-1265118 at Princeton University. The work of L.T. was partially supported by the NSF grant DMS-1005769. A.L. is grateful to T. Weidl for useful discussions. Some results of the paper were obtained by using different methods by A.L.’s master student O. Mickelin [@Mickelin2015]. The research was supported by the Russian Foundation for Basic Research (grant no. 12-01-00203).
The Operator $H(\zeta)$ {#sec:H}
=======================
Let $H_0=U+U^{-1}$ and $W(\zeta)=V+\zeta V^{-1}$ so that $H(\zeta)=H_0+W(\zeta)$ and formally $$\begin{aligned}
\big(H(\zeta)\psi\big)(x)=\psi(x+{\mathrm{i}}b)+\psi(x-{\mathrm{i}}b)+({\mathrm{e}}^{2\pi b x}+\zeta{\mathrm{e}}^{-2\pi bx})\psi(x)\,.\end{aligned}$$ It is straightforward to show that $\mathcal{F}H_0\mathcal{F}^{-1}=W$, where we put $W=W(1)$, which yields $\sigma(H_0)=[2,\infty)$ and consequently $H\ge2I$. The operator $H(\zeta)$ is semi-bounded and symmetric on the common domain of $H_{0}$ and $W(\zeta)$, $$\langle H(\zeta)\psi,\psi\rangle=\langle\psi,H(\zeta)\psi\rangle\ge 2\Vert\psi\Vert^{2},$$ where $\langle~,~\rangle$ stands for the inner product in $L^{2}({\mathbb{R}})$. Thus we can define a self-adjoint Friedrichs extension of the operator $H(\zeta)$ (see e.g. [@Birman1987 Chapter 10.3]). It is this extension that we mean when we refer to the operator $H(\zeta)$. We first show that the spectrum of $H(\zeta)$ is purely discrete.
\[prop:Hspec\] Let $L(x)$ be a continuous, real-valued, bounded below function such that $L(x)$ tends to $+\infty$ as $|x|\rightarrow\infty$. Then the operator $T=H_{0}+L$ has purely discrete spectrum consisting of finite multiplicity eigenvalues tending to $+\infty$.
Indeed, by using the variational principle and the Birman–Schwinger principle we have $$\begin{aligned}
&\dim{\left\{\psi: \, {\langleT\psi,\psi\rangle} < \lambda {\langle\psi,\psi\rangle}\right\}}\\
&\le \dim{\left\{\psi: \, {\langleH_{0 }\psi, \psi\rangle} - \left(\lambda{\left\|\psi\right\|}^2 - {\langleL\psi,\psi\rangle}\right)_+ <0\right\}}\\
&=
\dim{\left\{\psi: \, {\langleW_\lambda H_{0}^{-1} W_\lambda \psi,\psi\rangle} > 1\right\}},\end{aligned}$$ where $W_\lambda= \sqrt{(\lambda-L)_+}$. The operator $K_\lambda =W_\lambda H_{0}^{-1} W_\lambda$ is an integral operator $$\begin{aligned}
(K_\lambda\psi)(x)= \int_{-\infty}^\infty\int_{-\infty}^\infty W_\lambda(x)\frac{{\mathrm{e}}^{2\pi {\mathrm{i}}(x-y)k}}{2\cosh( 2\pi b k)} W_\lambda(y) \psi(y) {\, \mathrm{d}}k {\, \mathrm{d}}y.\end{aligned}$$ Since $L(x)$ tends to $+\infty$ as $|x|\to \infty$, the support of $W_\lambda$ is compact. Therefore $K_\lambda$ is a compact operator and this proves that its spectrum above one is finite. This implies that the spectrum of $T$ below $\lambda$ is also finite for any fixed $\lambda>0$.
Clearly $T$ cannot have finite rank since it is the sum of two unbounded positive operators. Therefore the spectrum of the operator $T$ is discrete.
Let $\lambda_1\le\lambda_2\le\dots$ denote the eigenvalues of $H(\zeta)$ with the corresponding complete system of orthonormal eigenfunctions $\psi_j\in L^2({\mathbb{R}})$. We are interested in the asymptotic behaviour of the Riesz mean $\sum_{j\ge1}(\lambda-\lambda_j)_+$ as $\lambda\to\infty$. Here $x_+=(|x|+x)/2$ is defined as the positive part of a real number $x$. Our main result is the following.
\[th:Hasymp\] For any $\zeta>0$ the eigenvalues $\lambda_j$ of the operator $H(\zeta)$ have the following asymptotic behaviour $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+=\frac{\lambda\log^{2}\lambda}{(\pi b)^2}+O(\lambda\log\lambda)\quad\text{as}\quad\lambda\to\infty.
\label{eq:Hasymp} \end{aligned}$$
The following is an immediate consequence of Theorem \[th:Hasymp\].
\[cor:Hasymp\] For any $b>0$ the number $N(\lambda)=\#{\left\{j\in{\mathbb{N}}: \, \lambda_j<\lambda\right\}}$ of eigenvalues of $H(\zeta)$ less than $\lambda$ satisfies $$\begin{aligned}
\lim_{\lambda\to\infty}\frac{N(\lambda)}{\log^{2}\lambda}=\frac{1}{(\pi b)^2}\,.\end{aligned}$$
In particular, the operator $H(\zeta)^{-1}$ is of trace class since $$\sum_{j=1}^{\infty}\frac{1}{\lambda_{j}}=\int_{2}^{\infty}\frac{1}{\lambda}{\, \mathrm{d}}N(\lambda)=\left. \frac{N(\lambda)}{\lambda}\right|_{2}^{\infty}+\int_{2}^{\infty}\frac{N(\lambda)}{\lambda^{2}}{\, \mathrm{d}}\lambda<\infty.$$
\[Weyl law\] Theorem \[th:Hasymp\] and Corollary \[cor:Hasymp\] are Weyl type results that link the asymptotical behaviour of quantum mechanical expressions to classical phase space integrals. Namely, let $$\sigma(k,x)=2\cosh(2\pi bk)+{\mathrm{e}}^{2\pi b x}+\zeta{\mathrm{e}}^{-2\pi b x}$$ be the total symbol of the operator $H(\zeta)$. Then the term $\log^{2}\lambda/(\pi b)^2$ is precisely the leading term of the phase volume of the classical region $\{(k,x)\in{\mathbb{R}}^{2}: \sigma(k,x)\leq\lambda\}$ as $\lambda\to\infty$. Similarly, $\lambda\log^{2}\lambda/(\pi b)^2$ coincides with the leading term in the phase space integral $$\iint_{{\mathbb{R}}^{2}}(\lambda-\sigma(k,x))_{+}{\, \mathrm{d}}k{\, \mathrm{d}}x\quad\text{as}\quad\lambda\to\infty.$$
To prove Theorem \[th:Hasymp\], we establish lower and upper bounds on the Riesz mean $\sum_{j\ge1}(\lambda-\lambda_j)_+$ in Sect. \[subsec:upper\] and \[subsec:lower\] respectively. To this end we introduce the coherent state representation of $H(\zeta)$. To simplify notation and to keep focus on the arguments involved, we concentrate on the case $\zeta=1$, where $W=2\cosh(2\pi b x)$. Subsequently, we will be using notation $H=H_{0}+W$, not to be confused with the operator $H_{0}+V$. The general case $\zeta>0$ is a straightforward generalization, as is explained in Sect. \[subsec:Hgen\].
The Coherent State Representation {#subsec:coherent}
---------------------------------
Let $g$ be the Gaussian function $g(x)=(a/\pi)^{1/4}{\mathrm{e}}^{-\frac a2 x^2}$ with some $a>0$. Clearly $g$ satisfies ${\left\|g\right\|}=1$ in $L^2({\mathbb{R}})$. For $\psi\in L^2({\mathbb{R}})$ the classical coherent state transform (see e.g. [@Lieb2001 Chapter 12]) is given by $$\begin{aligned}
\widetilde{\psi}(k,y)=\int_{\mathbb{R}}{\mathrm{e}}^{-2\pi{\mathrm{i}}k x}g(x-y)\psi(x){\, \mathrm{d}}x\,.\end{aligned}$$ Denoting by $(f*g)(x)=\int_{\mathbb{R}}f(x-y)g(y){\, \mathrm{d}}y$ the convolution of $f$ and $g$, Plancherel’s theorem shows that $$\begin{aligned}
\int_{\mathbb{R}}|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k&=(|\psi|^2*|g|^2)(y)\,,\label{eq:Itrans1}\\
\int_{\mathbb{R}}|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}y&=(|\widehat{\psi}|^2*|\widehat{g}|^2)(k)\label{eq:Itrans2}\,.\end{aligned}$$ The proof of the second identity also uses the convolution theorem.
We aim to find representations of ${\langleH_0\psi,\psi\rangle}$ and ${\langleW\psi,\psi\rangle}$ in terms of coherent states. It follows from that $$\begin{aligned}
\iint_{{\mathbb{R}}^{2}} 2\cosh(2\pi b k)|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y
&=\iint_{{\mathbb{R}}^{2}} 2\cosh(2\pi b k)|\widehat{\psi}(k-q)|^2|\widehat{g}(q)|^2{\, \mathrm{d}}k{\, \mathrm{d}}q,\end{aligned}$$ and using $$\begin{aligned}
\cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y\end{aligned}$$ we obtain $$\begin{gathered}
\iint_{{\mathbb{R}}^{2}} 2\cosh(2\pi b k)|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y\\
=\iint_{{\mathbb{R}}^{2}} 2\cosh\big(2\pi b(k-q)\big)|\widehat{\psi}(k-q)|^2\cosh(2\pi b q)|\widehat{g}(q)|^2{\, \mathrm{d}}k{\, \mathrm{d}}q\\
+\iint_{{\mathbb{R}}^{2}} 2\sinh\big(2\pi b(k-q)\big)|\widehat{\psi}(k-q)|^2\sinh(2\pi bq)|\widehat{g}(q)|^2{\, \mathrm{d}}k{\, \mathrm{d}}q\,.\end{gathered}$$ Recalling that $\mathcal{F}H_0\mathcal{F}^{-1}=W$, the first integral on the right-hand side can be computed to be $\frac12{\langleH_0\psi,\psi\rangle}{\langleW\widehat{g},\widehat{g}\rangle}$. Since $g(x)=g(-x)$, it holds that $\widehat{g}(k)=\widehat{g}(-k)$ and consequently the second integral vanishes. Thus for $\psi\in{D}(H_0)$ we obtain the representation $$\begin{aligned}
{\langleH_0\psi,\psi\rangle}=d_1\iint_{{\mathbb{R}}^{2}} 2\cosh(2\pi b k)|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y
\label{eq:rep1} \end{aligned}$$ where $$d_1=\frac{2}{{\langleW\widehat{g},\widehat{g}\rangle}}={\mathrm{e}}^{-ab^2/4}<1.$$
Similarly, we can use to compute that $$\begin{aligned}
\iint_{{\mathbb{R}}^{2}} 2\cosh(2\pi by)|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y
&=\iint_{{\mathbb{R}}^{2}} 2\cosh(2\pi by)|{\psi}(y-q)|^2|{g}(q)|^2{\, \mathrm{d}}y{\, \mathrm{d}}q,\end{aligned}$$ which with the help of the same trigonometric identity as above can be simplified to $$\begin{aligned}
\iint_{{\mathbb{R}}^{2}} 2\cosh(2\pi by)|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y &=\frac12{\langleW\psi,\psi\rangle}{\langleWg,g\rangle}.\end{aligned}$$ Thus for $\psi\in{D}(W)$ we have the representation $$\begin{aligned}
{\langleW\psi,\psi\rangle}=d_2\iint_{{\mathbb{R}}^{2}} 2\cosh(2\pi b y)|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y,
\label{eq:rep2} \end{aligned}$$ where $$d_2=\frac{2}{{\langleWg,g\rangle}}={\mathrm{e}}^{-(\pi b)^2/a}<1.$$ Summarizing, we obtain $$\label{H-coherent}
{\langleH\psi,\psi\rangle}=\iint_{{\mathbb{R}}^{2}}2(d_1\cosh(2\pi b k)+d_2\cosh(2\pi by))|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y.$$
Deriving an Upper Bound {#subsec:upper}
-----------------------
We apply ideas that were used in [@Laptev1997] in investigation of the upper bounds on the eigenvalues of a general class of operators on sets of finite measure with Dirichlet boundary condition. While these results relied on the representation of the operators in Fourier space, we will use the representation in terms of the coherent states.
As a reminder, $\lambda_j$ denote the eigenvalues of $H$ and $\psi_j$ the corresponding orthonormal eigenfunctions which form a complete set. We first observe that representation yields $$\begin{gathered}
\sum_{j\ge1}(\lambda-\lambda_j)_+
=\sum_{j\ge1}(\lambda-{\langleH\psi_j,\psi_j\rangle})_+\\
=\sum_{j\ge1}\left(\lambda- \iint_{{\mathbb{R}}^{2}} 2\big(d_1 \cosh(2\pi bk)+ d_2 \cosh(2\pi by)\big)|\widetilde{\psi}_j(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y\right)_+.\end{gathered}$$ By Plancherel’s theorem it holds that $$\iint_{{\mathbb{R}}^{2}}|\widetilde{\psi}_j(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y={\left\|\psi_j\right\|}^2=1
\label{eq:coherentnorm}$$ and consequently we can apply Jensen’s inequality with the convex function $x\mapsto (\lambda-x)_+$ to obtain $$\begin{gathered}
\sum_{j\ge1}(\lambda-\lambda_j)_+\\
\le \iint_{{\mathbb{R}}^{2}} \big(\lambda-2d_1\cosh(2\pi bk)- 2d_2\cosh(2\pi by)\big)_+\sum_{j\ge1}|\widetilde{\psi}_j(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y\,.\end{gathered}$$ Put $e_{k,y}(x)={\mathrm{e}}^{2\pi{\mathrm{i}}kx}g(x-y)$. Since the eigenfunctions $\psi_j$ form an orthonormal basis in $L^{2}(\mathbb{R})$, $$\begin{aligned}
\sum_{j=1}^{\infty}|\widetilde{\psi}_j(k,y)|^2=\sum_{j=1}^{\infty}|(e_{k,y},\psi_{j})|^{2}=\Vert e_{k,y}\Vert^{2}=1\quad\text{for all}\quad k,y\in{\mathbb{R}},\end{aligned}$$ and we arrive at the upper bound $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+\le \iint_{{\mathbb{R}}^{2}} \big(\lambda-2d_1\cosh(2\pi bk)- 2d_2\cosh(2\pi by)\big)_+{\, \mathrm{d}}k{\, \mathrm{d}}y\,.\end{aligned}$$
To investigate the behaviour of the integral on the right-hand side as $\lambda\to\infty$, we first note that $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
&\le4\int_0^\infty\int_0^\infty \big(\lambda-2d_1\cosh(2\pi bk)- 2d_2\cosh(2\pi by)\big)_+{\, \mathrm{d}}k{\, \mathrm{d}}y\\
&\le4\int_0^\infty\int_0^\infty \big(\lambda-d_1 {\mathrm{e}}^{2\pi bk}- d_2 {\mathrm{e}}^{2\pi by}\big)_+{\, \mathrm{d}}k{\, \mathrm{d}}y\,,\end{aligned}$$ where we used that $2\cosh x>{\mathrm{e}}^{x}$ for $x>0$. Changing the variables $u_1=d_1{\mathrm{e}}^{2\pi bk}, u_2=d_2{\mathrm{e}}^{2\pi by}$ we arrive at $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
&\le \frac{1}{(\pi b)^2}\int_{d_1}^\infty\int_{d_2}^\infty\frac{(\lambda-u_1-u_2)_+}{u_1u_2}{\, \mathrm{d}}u_2{\, \mathrm{d}}u_1\\
&=\frac{1}{(\pi b)^2}\int_{d_1}^{\lambda-d_2}\int_{d_2}^{\lambda-u_1}\frac{\lambda-u_1-u_2}{u_1u_2}{\, \mathrm{d}}u_2{\, \mathrm{d}}u_1\,,\end{aligned}$$ where $\lambda\ge d_1+d_2$ since $\lambda\ge2$ and $d_1,d_2\le 1$. Now we immediately obtain $$\begin{aligned}
\int_{d_1}^{\lambda-d_2}\int_{d_2}^{\lambda-u_1}\frac{\lambda-u_1-u_2}{u_1u_2}{\, \mathrm{d}}u_2{\, \mathrm{d}}u_1&=\lambda\int_{d_1/\lambda}^{1-d_2/\lambda}\int_{d_2/\lambda}^{1-v_{1}}\frac{1-v_1-v_2}{v_1v_2}{\, \mathrm{d}}v_2{\, \mathrm{d}}v_1\\
&= \lambda\log^{2}\lambda+O(\lambda\log\lambda)\end{aligned}$$ as $\lambda\rightarrow\infty$, so that $$\begin{aligned}
\sum_{j\ge 1}(\lambda-\lambda_j)_+\le \frac{\lambda\log^{2}\lambda}{(\pi b)^2}+O(\lambda\log\lambda).\end{aligned}$$
Deriving a Lower Bound {#subsec:lower}
----------------------
To obtain a lower bound, we use a different argument. The ideas in this section are again taken from [@Laptev1997], where a lower bound on the eigenvalues of a general class of operators on sets of finite measure with Neumann boundary condition was obtained. Similarly to the previous subsection, the coherent state transform will replace the Fourier transform.
Recalling , we start from the identity $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
=\sum_{j\ge1}(\lambda-\lambda_j)_+\iint_{{\mathbb{R}}^{2}}|\widetilde{\psi}_j(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y,\end{aligned}$$ and observing that $$\widetilde{\psi}_j(k,y)=\int_{\mathbb{R}}\psi_j(x)\overline{e_{k,y}(x)}{\, \mathrm{d}}x={\langle\psi_j,e_{k,y}\rangle},$$ we obtain $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
&=\iint_{{\mathbb{R}}^{2}} \sum_{j\ge1}(\lambda-\lambda_j)_+{\langle\psi_j,e_{k,y}\rangle}\overline{{\langle\psi_j,e_{k,y}\rangle}}{\, \mathrm{d}}k{\, \mathrm{d}}y\\
&=\iint_{{\mathbb{R}}^{2}} \sum_{j\ge1}(\lambda-\lambda_j)_+\big\langle{\langlee_{k,y},\psi_j\rangle}\psi_j,e_{k,y}\big\rangle{\, \mathrm{d}}k{\, \mathrm{d}}y\,.\end{aligned}$$ Denoting by $\mathrm{d} E_{\mu}$ the projection-valued measure for $H$ on $[2,\infty)$, we conclude that $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
=\iint_{{\mathbb{R}}^{2}}\int_2^\infty(\lambda-\mu)_+{\langle\mathrm{d} E_{\mu} e_{k,y},e_{k,y}\rangle}{\, \mathrm{d}}k{\, \mathrm{d}}y\,.\end{aligned}$$ Since by the spectral theorem $$\begin{aligned}
\int_2^\infty {\langle\mathrm{d} E_{\mu} e_{k,y},e_{k,y}\rangle}={\langlee_{k,y},e_{k,y}\rangle}={\left\|g\right\|}^2=1,\end{aligned}$$ we can apply Jensen’s inequality with the convex function $x\mapsto(\lambda-x)_+$ and obtain the lower bound $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
\ge \iint_{{\mathbb{R}}^{2}}\left(\lambda-\int_2^\infty \mu{\langle\mathrm{d} E(\mu) e_{k,y},e_{k,y}\rangle}\right)_+{\, \mathrm{d}}k{\, \mathrm{d}}y\,.
\label{eq:lbound} \end{aligned}$$ Again it follows from the spectral theorem that $$\begin{aligned}
\int_2^\infty \mu{\langle\mathrm{d} E(\mu) e_{k,y},e_{k,y}\rangle}
={\langleHe_{k,y},e_{k,y}\rangle}={\langleH_0e_{k,y},e_{k,y}\rangle}+{\langleWe_{k,y},e_{k,y}\rangle}\,.\end{aligned}$$
The two terms on the right-hand side can be computed explicitly. We first consider ${\langleHe_{k,y},e_{k,y}\rangle}$ and note that $$\begin{aligned}
g(x-y\pm{\mathrm{i}}b)={\mathrm{e}}^{\frac{ab^2}{2}}g(x-y){\mathrm{e}}^{\mp a(x-y){\mathrm{i}}b},\end{aligned}$$ whence $$\begin{aligned}
{\langleH_0e_{k,y},e_{k,y}\rangle}
&=\int_{\mathbb{R}}\big({\mathrm{e}}^{-2\pi bk}g(x-y+{\mathrm{i}}b)+{\mathrm{e}}^{2\pi bk}g(x-y-{\mathrm{i}}b)\big)g(x-y){\, \mathrm{d}}x\\
&={\mathrm{e}}^{\frac{ab^2}{2}}\left({\mathrm{e}}^{-2\pi bk}\int_{\mathbb{R}}g(z)^2{\mathrm{e}}^{-{\mathrm{i}}abz}{\, \mathrm{d}}z+{\mathrm{e}}^{2\pi bk}\int_{\mathbb{R}}g(z)^2{\mathrm{e}}^{{\mathrm{i}}abz}{\, \mathrm{d}}z\right)\\
&=\frac{1}{d_1}2\cosh(2\pi bk)\,.\end{aligned}$$ For the second term, ${\langleWe_{k,y},e_{k,y}\rangle}$, we get $$\begin{aligned}
{\langleWe_{k,y},e_{k,y}\rangle}
&=\int_{\mathbb{R}}2\cosh(2\pi bx)g(x-y)^2{\, \mathrm{d}}x\\
&=\int_{\mathbb{R}}2\cosh\big(2\pi b(x-y)\big)\cosh(2\pi by) g(x-y)^2{\, \mathrm{d}}x
\\&\phantom{=}+\int_{\mathbb{R}}2\sinh\big(2\pi b(x-y)\big)\sinh(2\pi by)g(x-y)^2{\, \mathrm{d}}x\\
&=\frac{1}{d_2}2\cosh(2\pi by)\,.\end{aligned}$$
Combining these two results with we arrive at $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
&\ge\iint_{{\mathbb{R}}^{2}}\left(\lambda-\frac{2}{d_1}\cosh(2\pi bk)-\frac{2}{d_2}\cosh(2\pi by)\right)_+\!{\, \mathrm{d}}k{\, \mathrm{d}}y\\
&=4 \int_0^\infty\int_0^\infty\left(\lambda-\frac{2}{d_1}\cosh(2\pi bk)-\frac{2}{d_2}\cosh(2\pi by)\right)_+\!{\, \mathrm{d}}k{\, \mathrm{d}}y\,. \end{aligned}$$ Note that $2\cosh x\le 2{\mathrm{e}}^{x}$ for $x\ge0$ and thus $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
\ge 4 \int_0^\infty\int_0^\infty\left(\lambda-\frac{2}{d_1}{\mathrm{e}}^{2\pi bk}-\frac{2}{d_2}{\mathrm{e}}^{2\pi by}\right)_+{\, \mathrm{d}}k{\, \mathrm{d}}y\,.\end{aligned}$$ The integral on the right-hand side is computed in the same way as in the previous section. The only difference is that the numbers $d_1, d_2$ have been replaced by $2/d_1,2/d_2$. These coefficients have no influence on the leading term for large $\lambda$ as long as $\lambda\ge2/d_1+2/d_2$, and we conclude $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
\ge \frac{1}{(\pi b)^2}\lambda\log^{2}\lambda+O(\lambda\log\lambda)\quad\text{as}\quad\lambda\to\infty.\end{aligned}$$
The Number of Eigenvalues {#subsec:N}
-------------------------
We present two proofs of Corollary \[cor:Hasymp\]. One uses the Karamata–Tauberian theorem [@Karamata1931] to deduce it from Theorem \[th:Hasymp\], while the other consists in obtaining the optimal bounds for $N(\lambda)$ from the Riesz mean.
### Proof of Corollary \[cor:Hasymp\] with the Karamata–Tauberian Theorem
The Karamata–Tauberian theorem (for the proof see, e.g., [@Simon2005 Theorem 10.3]) connects the asymptotic behaviour of $N(\lambda)$ for large $\lambda$ to the divergence of ${\mathrm{tr}}\,{\mathrm{e}}^{-tH}$ for small $t$. In [@Laptev1999] a general method was discussed that allows to obtain asymptotics of the traces of convex functions of self-adjoint operators from the behaviour of their Riesz means. Namely, from the representation $$\begin{aligned}
{\mathrm{e}}^{-t\lambda}=t^2 \int_{\mathbb{R}}(s-\lambda)_+{\mathrm{e}}^{-ts}{\, \mathrm{d}}s\end{aligned}$$ for $\lambda\ge0$ and asymptotic behaviour we get the upper bound $$\begin{aligned}
{\mathrm{tr}}\,{\mathrm{e}}^{-tH}&=t^2 \int_0^\infty \sum_{j\ge1}(s-\lambda_j)_+{\mathrm{e}}^{-ts}{\, \mathrm{d}}s\\
&\le\frac{t^2}{(\pi b)^2}\int_0^\infty s(\log s)^2{\mathrm{e}}^{-ts}{\, \mathrm{d}}s +t^2C\int_0^\infty s(\log s){\mathrm{e}}^{-ts}{\, \mathrm{d}}s\end{aligned}$$ with some constant $C>0$, as well as a similar lower bound with a different constant. The two integrals on the right-hand side are computed explicitly and we obtain $$\begin{aligned}
\lim_{t\to0}\frac{{\mathrm{tr}}\,{\mathrm{e}}^{-tH}}{\log^{2} t}=\frac{1}{(\pi b)^2}\,.\end{aligned}$$ A slight modification of the Karamata–Tauberian theorem that allows for logarithmic terms [@Simon1983] implies that $$\begin{aligned}
\lim_{\lambda\to\infty}\frac{N(\lambda)}{\log^{2}\lambda}=\frac{1}{(\pi b)^2}\,.\end{aligned}$$
### Direct Proof of Corollary \[cor:Hasymp\]
To derive an upper bound on $N(\lambda)$, we let $\mu\ge\rho>0$ and note the that $$\sum_{j\ge1}(\mu-\lambda_j)_+=\sum_{\lambda_{j}<\mu}(\mu-\lambda_{j})\ge\sum_{\lambda_{j}<\mu-\rho}(\mu-\lambda_{j})>\rho N(\mu-\rho).$$ We can now use asymptotic behaviour of the Riesz mean to conclude that there exists a $C>0$ such that $$\begin{aligned}
N(\mu-\rho)\le\frac{\mu\log^{2}\mu}{\rho(\pi b)^2}+\frac{C}{\rho}\mu\log\mu\,.\end{aligned}$$ With $\tau>0$ we now choose $\mu=(1+\tau)\lambda$ and $\rho=\tau\lambda$ such that $\mu-\rho=\lambda$ and $$\begin{aligned}
N(\lambda)\le\frac{1}{(\pi b)^2}\left(1+\frac{1}{\tau}\right)\left(\log^{2}(\lambda+\lambda\tau)+C\log(\lambda+\lambda\tau)\right).\end{aligned}$$ It remains to optimize this upper bound with respect to $\tau>0$. The minimum is attained at $\tau_0$ defined by the equation $$\begin{aligned}
2\tau_0=\log(\lambda+\lambda\tau_0)\,.\end{aligned}$$ Since $2\tau-\log(1+\tau)$ is bijective as a function from $[0,\infty)$ to $[0,\infty)$, a unique solution $\tau_0$ exists for every $\lambda$. It clearly holds that $\tau_0\to\infty$ as $\lambda\to\infty$ and thus $\tau_0\le\log\lambda$ for sufficiently large $\lambda$. We can conclude that $$\begin{aligned}
\limsup_{\lambda\to\infty}\frac{N(\lambda)}{\log^{2}\lambda}\le\frac{1}{(\pi b)^2}\,.\end{aligned}$$ To find an analogous lower bound we note that again by for $\lambda\ge2$ $$\begin{aligned}
N(\lambda)\ge\sum_{j\ge1}\left(1-\frac{\lambda_j}{\lambda}\right)_+
=\frac{1}{\lambda}\sum_{j\ge1}(\lambda-\lambda_j)_+
\ge\frac{\log^{2}\lambda}{(\pi b)^2}+C\log\lambda\end{aligned}$$ with some constant $C>0$.
The General Case $\zeta>0$ {#subsec:Hgen}
--------------------------
It is straightforward to generalize the proof of Theorem \[th:Hasymp\] to any $\zeta>0$. The coherent state representation of $W(\zeta)=V+\zeta V^{-1}$ can be computed to be $$\begin{aligned}
{\langleW(\zeta)\psi,\psi\rangle}=d_2\iint_{{\mathbb{R}}^{2}}({\mathrm{e}}^{2\pi by}+\zeta{\mathrm{e}}^{-2\pi by})|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y\end{aligned}$$ for $\psi\in{D}(W(\zeta))$. Repeating calculations of Sect. \[subsec:upper\] leads to an upper bound of the Riesz $\sum_{j\ge1}(\lambda-\lambda_j)_+$, which can be written as a sum of four integrals of the form $\int_0^\infty\int_0^\infty (\lambda-c_1{\mathrm{e}}^{2\pi bk}-c_2{\mathrm{e}}^{2\pi by})_+{\, \mathrm{d}}k{\, \mathrm{d}}y$. The asymptotic behaviour of these integrals was discussed in Sect. \[subsec:upper\]. A lower bound of the Riesz mean can be established by repeating verbatim the computations in Sect. \[subsec:lower\], which proves Theorem \[th:Hasymp\] and Corollary \[cor:Hasymp\] for $\zeta>0$.
The Operator $H_{m,n}$
======================
The operator $H_{m,n}=U+V+q^{-mn}U^{-m}V^{-n}$ is given by the following formal functional-difference expression $$\begin{aligned}
(H_{m,n}\psi)(x)=\psi(x+{\mathrm{i}}b)+{\mathrm{e}}^{2\pi bx}\psi(x)+q^{-mn}{\mathrm{e}}^{-2\pi nbx}\psi(x-m{\mathrm{i}}b)\,.\end{aligned}$$ The operator $H_{m,n}$ is symmetric and non-negative on the domain $\psi\in\mathscr{D}$ consisting of linear combinations of the functions $p(x){\mathrm{e}}^{-x^{2}+cx}$, where $p(x)$ is a polynomial and $c\in{\mathbb{C}}$. Indeed, for $\psi\in\mathscr{D}$ it follows from the Weyl relation $$U^{-m}\widetilde{V}^{-n}=q^{mn}\widetilde{V}^{-n}U^{-m},\quad\text{where}\quad \widetilde{V}=V^{1/2}={\mathrm{e}}^{\pi bQ},$$ that $$\label{Weyl-m-n}
q^{-mn}\langle U^{-m}V^{-n}\psi,\psi\rangle=\langle U^{-m}\widetilde{V}^{-n}\psi,\widetilde{V}^{-n}\psi\rangle\ge 0.$$ Whence $H_{m,n}$ admits a Friedrichs extension and it what follows we will continue to denote it by $H_{m,n}$. The spectrum of this operator consists of positive eigenvalues $\lambda_j$ that converge to infinity, $\lim_{j\to\infty}\lambda_j=\infty$. The proof of this statement is deferred to the end of Sect. \[sec:coHmn\] since it makes use of the coherent state representation of $H_{m,n}$.
\[th:Gasymp\] For $m,n\in{\mathbb{N}}$ the eigenvalues $\lambda_j$ of the operator $H_{m,n}$ have the following asymptotic behaviour $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+=\frac{c_{m,m}}{(2\pi b)^2}\lambda\log^{2}\lambda+O(\lambda\log\lambda)\quad\text{as}\quad\lambda\to\infty,\end{aligned}$$ where $c_{m,n}=\dfrac{(m+n+1)^{2}}{2mn}$.
Having established Theorem \[th:Gasymp\], the exact same argument as in Sect. \[subsec:N\] proves the following corollary.
\[cor:Gasymp\] The number $N(\lambda)=\#{\left\{j\in{\mathbb{N}}: \, \lambda_j<\lambda\right\}}$ of eigenvalues of $H_{m,n}$ less than $\lambda$ satisfies $$\begin{aligned}
\lim_{\lambda\to\infty}\frac{N(\lambda)}{\log^{2}\lambda}=\frac{c_{m,n}}{(2\pi b)^2}\,.\end{aligned}$$
In particular, this implies that the operator $H_{m,n}^{-1}$ is of trace class since $$\sum_{j=1}^{\infty}\frac{1}{\lambda_{j}}=\int_{\lambda_{1}}^{\infty}\frac{1}{\lambda}{\, \mathrm{d}}N(\lambda)=\left. \frac{N(\lambda)}{\lambda}\right|_{\lambda_{1}}^{\infty}+\int_{\lambda_1}^{\infty}\frac{N(\lambda)}{\lambda^{2}}{\, \mathrm{d}}\lambda<\infty.$$
As in Remark \[Weyl law\], the term $c_{m,n}\log^{2}\lambda/(2\pi b)^{2}$ is precisely the leading term as $\lambda\to\infty$ of the phase volume of the classical region $\{(k,x)\in{\mathbb{R}}^{2} : {\mathrm{e}}^{-2\pi b k}+{\mathrm{e}}^{2\pi b x}+{\mathrm{e}}^{2\pi b(mk-nx)}\le \lambda\}$. Similarly, the term $c_{m,n}\lambda\log^{2}\lambda/(2\pi b)^{2}$ coincides with the leading term in the phase space integral $$\iint_{{\mathbb{R}}^{2}}(\lambda-{\mathrm{e}}^{-2\pi b k}-{\mathrm{e}}^{2\pi b x}-{\mathrm{e}}^{2\pi b(mk-nx)})_+{\, \mathrm{d}}k{\, \mathrm{d}}x\quad\text{as}\quad\lambda\to\infty.$$
As in Sect. \[sec:H\], we first obtain a representation of $H_{m,n}$ using the coherent state transform and then prove the upper and lower bounds. The computations will closely follow those in Sect. \[subsec:coherent\], \[subsec:upper\] and \[subsec:lower\], and we will just highlight the main points.
The Coherent State Representation {#sec:coHmn}
---------------------------------
Let $\widetilde{\psi}$ again denote the coherent state transform of a function $\psi\in L^2({\mathbb{R}})$ with respect to the Gaussian function $g$. In complete analogy with Sect. \[subsec:coherent\], identity , together with the facts that $U=\mathcal{F}^{-1}V^{-1}\mathcal{F}$ and ${\mathrm{e}}^{-2\pi bk}={\mathrm{e}}^{-2\pi b(k-q)}{\mathrm{e}}^{2\pi bq}$, leads to the representation $$\begin{aligned}
{\langleU\psi,\psi\rangle}=d_1\iint_{{\mathbb{R}}^{2}} {\mathrm{e}}^{-2\pi bk}|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y\,.\end{aligned}$$ Here, we have used the symmetries of the functions involved to conclude that $$\begin{aligned}
\frac{1}{{\langleV^{-1}\widehat{g},\widehat{g}\rangle}}=\frac{2}{{\langleW\widehat{g},\widehat{g}\rangle}}=d_1\,.\end{aligned}$$ Similarly, $$\begin{aligned}
{\langleU^{-m}\psi,\psi\rangle}=d^{m^{2}}_1\iint_{{\mathbb{R}}^{2}} {\mathrm{e}}^{2\pi bmk}|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y\,. \end{aligned}$$ In the same way identity yields the representation $$\begin{aligned}
{\langleV\psi,\psi\rangle}=d_2\iint_{{\mathbb{R}}^{2}} {\mathrm{e}}^{2\pi by}|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y\,,\end{aligned}$$ where we have used that $$\begin{aligned}
\frac{1}{{\langleVg,g\rangle}}=\frac{1}{{\langleV^{-1}g,g\rangle}}=\frac{2}{{\langleWg,g\rangle}}=d_2\,,\end{aligned}$$ since $g$ is even.
To derive of the representation of the mixed term $q^{-mn}U^{-m}V^{-n}$ we use to get $$q^{-mn}{\langleU^{-m}V^{-n}\psi,\psi\rangle}={\langleU^{-m}\psi_{1},\psi_{1}\rangle}=d^{m^{2}}_1\iint_{{\mathbb{R}}^{2}} {\mathrm{e}}^{2\pi bmk}|\widetilde{\psi}_{1}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y,$$ where $\widetilde{\psi}_{1}(k,y)$ is the coherent state transform of the function $\psi_{1}(x)=(\tilde{V}^{-n}\psi)(x)={\mathrm{e}}^{-\pi bnx}\psi(x)$. Completing the square, we obtain $$\begin{aligned}
\widetilde{\psi}_{1}(k,y)& =\int_{{\mathbb{R}}}{\mathrm{e}}^{-2\pi {\mathrm{i}}kx}g(x-y){\mathrm{e}}^{-\pi bnx}\psi(x){\, \mathrm{d}}x=
{\mathrm{e}}^{\frac{(\pi nb)^{2}}{2a}-\pi bny }\,\tilde{\psi}\left(k,y-\tfrac{\pi nb}{a}\right),\end{aligned}$$ so that $$\begin{aligned}
q^{-mn}{\langleU^{-m}V^{-n}\psi,\psi\rangle} &=d^{m^{2}}_1{\mathrm{e}}^{\frac{(\pi nb)^{2}}{a}}\iint_{{\mathbb{R}}^{2}} {\mathrm{e}}^{2\pi b(mk-ny)}|\widetilde{\psi}\left(k,y-\tfrac{\pi nb}{a}\right)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y \\
&=d^{m^{2}}_1d^{n^{2}}_{2}\iint_{{\mathbb{R}}^{2}} {\mathrm{e}}^{2\pi b(mk-ny)}|\widetilde{\psi}(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y. \end{aligned}$$ Summarizing, we obtain the coherent state representation of the operator $H_{m,n}$, $$\begin{gathered}
{\langleH_{m,n} \psi,\psi\rangle} =\nonumber \\
\iint_{{\mathbb{R}}^{2}} \left(d_1 {\mathrm{e}}^{-2\pi bk} + d_2 {\mathrm{e}}^{2\pi by} + d_3 {\mathrm{e}}^{2\pi b (m k-ny)}
\right) |\widetilde\psi (k,y)|^2 {\, \mathrm{d}}k {\, \mathrm{d}}y, \label{H-mn-coh}\end{gathered}$$ where we put $d_{3}=d^{m^{2}}_1d^{n^{2}}_{2}$.
Using representation , we can now prove that the spectrum of $H_{m,n}$ is discrete.
\[prop:Hmnspec\] The operator $H_{m,n}$ satisfies $H_{m,n}> cI$, where the constant $c>0$ depends on $m,n\in{\mathbb{N}}$, and has purely discrete spectrum consisting of finite multiplicity positive eigenvalues tending to infinity.
According to , the quadratic form of the operator $H_{m,n}$ is $$\begin{aligned}
{\langleH_{m,n} \psi,\psi\rangle} =\iint_{{\mathbb{R}}^{2}} \Psi(k,y) |\widetilde\psi (k,y)|^2 {\, \mathrm{d}}k {\, \mathrm{d}}y,\end{aligned}$$ where $$\begin{aligned}
\label{Psi}
\Psi(k,y) = d_1 {\mathrm{e}}^{-2\pi bk} + d_2 {\mathrm{e}}^{2\pi by} + d_3 {\mathrm{e}}^{2\pi b m k} {\mathrm{e}}^{- 2\pi bn y}.\end{aligned}$$ If $k\le 0$ and $y\ge0$, then omitting the last term in we obtain $$\begin{aligned}
\Psi(k,y) \ge \frac{d_1}{2} ({\mathrm{e}}^{-\pi bk} + {\mathrm{e}}^{\pi bk} ) + \frac{d_2}{2} ({\mathrm{e}}^{\pi by} + {\mathrm{e}}^{-\pi by}) . \end{aligned}$$ If $k\ge0$ and $y\le0$, then $$\begin{aligned}
{\mathrm{e}}^{2\pi b m k} {\mathrm{e}}^{- 2\pi bn y} \ge \frac12 \left( {\mathrm{e}}^{2\pi b m k} + {\mathrm{e}}^{- 2\pi bn y}\right)\end{aligned}$$ and therefore $$\begin{aligned}
\Psi(k,y) \ge d_1 {\mathrm{e}}^{-2\pi bk} + d_2 {\mathrm{e}}^{2\pi by} + \frac{d_3}{2} \left( {\mathrm{e}}^{2\pi b m k} + {\mathrm{e}}^{- 2\pi bn y}\right).\end{aligned}$$ Consider the case $k\ge 0$, $y\ge 0$. Assume that $\beta mk\ge ny$, where $\beta<1$. Then $$\begin{aligned}
{\mathrm{e}}^{2\pi b m k} {\mathrm{e}}^{- 2\pi bn y} \ge {\mathrm{e}}^{2\pi b m (1-\beta) k}.\end{aligned}$$ If now $k\ge 0$, $y\ge 0$ and $\beta mk \le ny$, then we omit the last term in and use $$\begin{aligned}
{\mathrm{e}}^{2\pi b y}\ge \frac12\left({\mathrm{e}}^{2\pi b y}+{\mathrm{e}}^{\beta m n^{-1} k}\right).\end{aligned}$$ Similarly we treat the case $k\le 0$, $y\le0$. Finally we conclude that there are positive constants $c_{1}$ and $c_{2}$ such that $$\begin{aligned}
\label{ineq}
\Psi(k,y) > \Phi(k,y) := c_{1}\left({\mathrm{e}}^{-c_{2}k} + {\mathrm{e}}^{c_{2}k} + {\mathrm{e}}^{-c_{2}y} + {\mathrm{e}}^{c_{2}y}\right).\end{aligned}$$ Denote by $A$ the operator defined by the quadratic form $$\begin{aligned}
{\langleA\psi, \psi\rangle} := \iint_{{\mathbb{R}}^2} \Phi(k,y) |\widetilde\psi(k,y)|^2{\, \mathrm{d}}k{\, \mathrm{d}}y.\end{aligned}$$ Then implies $H_{m,n} > A$ and it follows from the Plancherel theorem that $A\geq cI$, where $c=4c_{1}$. Obviously due to Proposition \[prop:Hspec\] the spectrum of $A$ is discrete. By the min-max principle we can conclude that the same holds for the spectrum of $H_{m,n}$ and the proof is complete.
Deriving an Upper Bound {#subsec:upperG}
-----------------------
Repeating the computation in Sect. \[subsec:upper\] and using Jensen’s inequality we obtain $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+\le\iint_{{\mathbb{R}}^{2}}\left(\lambda-d_1{\mathrm{e}}^{-2\pi bk}-d_2{\mathrm{e}}^{2\pi b y}-d_3{\mathrm{e}}^{2\pi b(mk-ny)}\right)_+{\, \mathrm{d}}k{\, \mathrm{d}}y\,.\end{aligned}$$ To find an upper bound on the right-hand side, we separately consider all four quadrants of ${\mathbb{R}}^{2}$.
If $k\le0, y\ge0$, an upper bound is obtained by omitting the mixed term $d_3{\mathrm{e}}^{2\pi b(mk-ny)}$. The double integral is then of the same form as the upper bound in Sect. \[subsec:upper\] and its leading term as $\lambda\to\infty$ is $\lambda\log^{2}\lambda/(2\pi b)^{2}$.
If $k\ge0, y\le0$, we omit two exponentially decaying terms $d_1{\mathrm{e}}^{-2\pi b k}$ and $d_2{\mathrm{e}}^{2\pi b y}$. Changing variables $u_1=d_3{\mathrm{e}}^{2\pi bmk}$ and $u_2={\mathrm{e}}^{-2\pi b ny}$, we obtain the double integral $$\begin{aligned}
\frac{1}{mn(2\pi b)^{2}}\int_{d_3}^{\lambda}\int_{1}^{\lambda/u_1}\frac{\lambda-u_1u_2}{u_1 u_2}{\, \mathrm{d}}u_2{\, \mathrm{d}}u_1=\frac{\lambda\log^{2}\lambda}{2mn(2\pi b)^2} +O(\lambda\log\lambda)
\label{eq:int2} \end{aligned}$$ as $\lambda\rightarrow\infty$, which can be easily verified by direct computation.
In case $k\ge0,y\ge0$ we omit the term $d_1{\mathrm{e}}^{-2\pi b k}$ and changing variables $u_1=d_3{\mathrm{e}}^{2\pi b(mk-ny)}$ and $u_2=d_2{\mathrm{e}}^{2\pi by}$ yields the integral $$\begin{aligned}
\frac{1}{m(2\pi b)^2}\int_{d_2}^{\tilde\lambda}\int_{d_4/u^{n}_2}^{\lambda-u_2}\frac{\lambda-u_1-u_2}{u_1u_2}{\, \mathrm{d}}u_1{\, \mathrm{d}}u_2\,,
\label{eq:int3} \end{aligned}$$ where $d_{4}=d_{3}d_{2}^{n}$ and $\tilde\lambda$ is the root of equation $\lambda=d_{4}u_{2}^{-n}+u_{2}$. It is easy to see that for $\lambda\rightarrow\infty$ one can replace $\tilde\lambda$ by $\lambda$ and obtain the leading term $(n+2)\lambda\log^{2}\lambda/2m(2\pi b)^{2}$. The case $k\le0,y\le0$ is treated similarly with the leading term $(m+2)\lambda\log^{2}\lambda/2n(2\pi b)^{2}$.
Summarizing, we arrive at the estimate $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+\le \frac{c_{m,n}}{(2\pi b)^2}\lambda\log^{2}\lambda+O(\lambda\log\lambda)\,.\end{aligned}$$
Deriving a Lower Bound {#subsec:lowerG}
----------------------
To derive a lower bound we repeat computations in Sect. \[subsec:lower\]. Denoting the projection-valued measure of $H_{m,n}$ on $[0,\infty)$ by $\mathrm{d}F_{\mu}$, we obtain upon an application of Jensen’s inequality that $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
\ge \iint_{{\mathbb{R}}^{2}}\left(\lambda-\int_0^\infty \mu{\langle\mathrm{d} F(\mu) e_{k,y},e_{k,y}\rangle}\right)_+{\, \mathrm{d}}k{\, \mathrm{d}}y\end{aligned}$$ for $\lambda\ge 0$. The spectral theorem implies that $$\begin{gathered}
\int_0^\infty \mu{\langle\mathrm{d} F(\mu) e_{k,y},e_{k,y}\rangle}
={\langleH_{m,n}e_{k,y},e_{k,y}\rangle}\\
={\langleUe_{k,y},e_{k,y}\rangle}+{\langleVe_{k,y},e_{k,y}\rangle}+q^{-mn}{\langleU^{-m}V^{-n}e_{k,y},e_{k,y}\rangle}\,.\end{gathered}$$ The three inner products on the right-hand side can be computed explicitly and we get the inequality $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+
\ge \iint_{{\mathbb{R}}^{2}}\left(\lambda-\frac{1}{d_1}{\mathrm{e}}^{-2\pi b k}-\frac{1}{d_2}{\mathrm{e}}^{2\pi b y}-\frac{1}{d_3}{\mathrm{e}}^{2\pi b(mk-ny)}\right)_+{\, \mathrm{d}}k{\, \mathrm{d}}y\,.\end{aligned}$$ To obtain a lower bound on the right-hand side, we again consider separately all four quadrants of ${\mathbb{R}}^{2}$.
If $k\le0, y\ge0$ we make the integrand smaller by replacing ${\mathrm{e}}^{2\pi b(mk-ny)}$ with ${\mathrm{e}}^{2\pi b y}$. The resulting double integral is of the form discussed in Sect. \[subsec:upper\] and its leading term as $\lambda\to\infty$ is $\lambda\log^{2}\lambda^2/(2 \pi b)^{2}$.
In case $k\ge0,y\le0$, we decrease the right-hand side by replacing both ${\mathrm{e}}^{-2\pi b k}$ and ${\mathrm{e}}^{2 \pi b y}$ with ${\mathrm{e}}^{2\pi b (mk-ny)}$. This yields a double integral of the same form as with the leading term $\lambda\log^{2}\lambda/2mn(2\pi b)^{2}$.
For $k\ge0, y\ge0$ we bound ${\mathrm{e}}^{-2\pi b k}$ from above by $1$. The integral takes the same form as with $\lambda$ replaced by $\lambda-1/d_1$. This does not affect the asymptotical behaviour $(n+2)\lambda\log^{2} \lambda/2m(2\pi b)^{2}$ as $\lambda\to\infty$.
The last case, $k\le 0,y\le0$, yields the leading term $(m+2)\lambda\log^{2} \lambda/2n(2\pi b)^{2}$ and we conclude that $$\begin{aligned}
\sum_{j\ge1}(\lambda-\lambda_j)_+\ge \frac{c_{m,n}}{(2\pi b)^2}\lambda\log^{2}\lambda+O(\lambda\log\lambda)\,.\end{aligned}$$ The proof of Theorem \[th:Gasymp\] is complete.
|
---
abstract: 'Let $N\ge 2$ and $\rho\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system $${\left\{f_i(x)=\rho x+\frac{i(1-\rho)}{N-1}: i=0,1,\ldots, N-1\right\}}.$$ Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $\mu=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, the pointwise lower $s$-density $\Theta_*^s(\mu,x)$ and upper $s$-density $\Theta^{*s}(\mu, x)$ of $\mu$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ for the sets $$E_*(a)={\left\{x\in E: \Theta_*^s(\mu, x)\ge a\right\}}\quad\textrm{and}\quad E^*(b)={\left\{x\in E: \Theta^{*s}(\mu, x)\le b\right\}}$$ respectively, such that $\dim_H E_*(a)>0$ if and only if $a<a_c$, and that $\dim_H E^*(b)>0$ if and only if $b>b_c$. We emphasize that both values $a_c$ and $b_c$ are related to the Thue-Morse type sequences, and our strategy to find them relies on ideas from open dynamics and techniques from combinatorics on words.'
address:
- 'College of Mathematics and Statistics, Chongqing University, 401331, Chongqing, P.R.China'
- 'Wenxia Li: Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200062, People’s Republic of China'
- 'Department of Mathematics, East China University of Science and Technology, Shanghai 200237, P.R. China'
author:
- Derong Kong
- Wenxia Li
- Yuanyuan Yao
title: Pointwise densities of homogeneous Cantor measure and critical values
---
Introduction {#s1}
============
Let $N\ge 2$ be an integer, and let $\rho\in(0, 1/N)$. The homogeneous Cantor set $E=E_{N,\rho}$ is the self-similar set generated by the *iterated function system* (IFS) $$f_i(x)=\rho x+i\; R:=\rho x+i\frac{1-\rho}{N-1},\quad i=0,1,\ldots, N-1.$$ When $N=2$ and $\rho=1/3$, $E_{2, 1/3}$ is the classical middle-third Cantor set. It is easy to see that the convex hull of $E$ is the unit interval $[0, 1]$, and the first level basic intervals $f_{0}([0,1]), f_1([0, 1]), \ldots$ and $f_{N-1}([0, 1])$ are located one by one from the left to the right (see Figure \[fig:1\]). These subintervals $f_i([0, 1]), i=0,1,\ldots, N-1$, are pairwise disjoint, and the gaps between any two neighboring subintervals are the same.
(0,0) – (1,0) node(xline)\[right\];
at (0, 0.05)[$0$]{};
at (1, 0.05)[$1$]{};
(0,-0.2) – (0.1,-0.2) node(xline)\[right\]; at (0.05, -0.15)[$f_0$]{}; at (0, -0.25)[$0$]{}; at (0.1, -0.25)[$\rho$]{};
(0.3,-0.21) – (0.4,-0.21) node(xline)\[right\]; at (0.35, -0.15)[$f_1$]{}; at (0.29, -0.25)[$R$]{}; at (0.41, -0.25)[$R+\rho$]{};
(0.6,-0.2) – (0.7,-0.2) node(xline)\[right\]; at (0.65, -0.15)[$f_{2}$]{}; at (0.58, -0.25)[$2R$]{}; at (0.72, -0.25)[$2R+\rho$]{};
(0.9,-0.2) – (1,-0.2) node(xline)\[right\]; at (0.95, -0.15)[$f_{3}$]{}; at (0.9, -0.25)[$3R$]{}; at (1, -0.25)[$1$]{};
Note that for each $x\in E$ there exists a (unique) sequence $(d_i)=d_1d_2\ldots\in{\left\{0, 1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ such that $$\label{eq:proj-map}
x=\lim_{n\to{\infty}}f_{d_1\ldots d_n}(0)=R\sum_{i=1}^{\infty}d_i\rho^{i-1}=:\pi((d_i)),$$ where $f_{d_1\ldots d_n}:=f_{d_1}\circ\cdots\circ f_{d_n}$ is the composition of maps. The infinite sequence $(d_i)$ is called a *coding* of $x$. Since $0<\rho< 1/N$, the self-similar IFS ${\left\{f_i\right\}}_{i=0}^{N-1}$ satisfies the strong separation condition (cf. [@Falconer_1990]). So the projection map $\pi: {\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}\to E$ defined in (\[eq:proj-map\]) is bijective. In other words, each $x\in E$ has a unique coding.
Accordingly, the self-similar set $E$ supports a unique measure $\mu=\mu_{N,\rho}$ satisfying $$\label{eq:mu}
\mu=\sum_{i=0}^{N-1}\frac{1}{N}\mu\circ f_i^{-1}.$$ The measure $\mu$ is called a *homogeneous Cantor measure*, which is a self-similar measure (cf. [@Hutchinson_1981]). In fact, the measure $\mu$ is the image measure of the uniform Bernoulli measure on the symbolic space ${\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ under the projection map $\pi$. It is also the $s$-dimensional Hausdorff measure $\mathcal H^s$ restricted to $E$, i.e., $\mu=\mathcal H^s|_E$, where $s=s_{N,\rho}:=-\log N/\log\rho$ is the Hausdorff dimension of $E$. For brevity, we always write $E, \mu$ and $s$ instead of $E_{N,\rho}, \mu_{N,\rho}$ and $s_{N, \rho}$ if no confusion arises.
Given $x\in E$, the *lower and upper $s$-densities* of $\mu$ at $x$ are defined by $$\Theta_*^s(\mu, x):=\liminf_{r\to 0}\frac{\mu(B(x, r))}{(2r)^s}=\liminf_{r\to 0}\frac{\mathcal H^s(B(x, r)\cap E)}{(2r)^s}$$ and $$\Theta^{*s}(\mu, x):=\limsup_{r\to 0}\frac{\mu(B(x,r))}{(2r)^s}=\limsup_{r\to 0}\frac{\mathcal H^s(B(x, r)\cap E)}{(2r)^s}$$ respectively, where $B(x, r)$ is the open interval $(x-r, x+r)$. The study of densities for a self-similar measure attracted a lot of attention in the literature (see [@Ayer-Strichartz-1999; @Bedford-Fisher-1992; @Falconer-1992; @Morters-Preiss-1998; @Olsen-2008; @Strichartz-Taylor-Zhang-1995] and the references therein). [When $N=2$ and $\rho\in(0, 1/3]$, Feng et al [@Feng-Hua-Wen-2000] explicitly calculated the pointwise densities $\Theta_*^s(\mu, x)$ and $\Theta^{*s}(\mu,x)$ for any $x\in E$. The upper bound $1/3$ for $\rho$ was later improved to $(\sqrt{3}-1)/2$ by Wang et al. [@Wang-Wu-Xiong-2011]. Motivated by their work, Li and Yao [@Li-Yao-2008] determined the pointwise densities of the self-similar measure for non-homogeneous self-similar IFSs. Dai and Tang [@Dai-Tang-2012] considered the same problem as in [@Feng-Hua-Wen-2000] but for $N=3$ and $\rho\in(0, 1/6]$.]{} In this paper we consider the pointwise lower and upper $s$-densities $\Theta_*^s(\mu, x)$ and $\Theta^{*s}(\mu, x)$ for $N\ge 2$ and $\rho\in(0, 1/N^2]$. To state our main results we first define a $N$-to-$1$ map $T: \bigcup_{i=0}^{N-1}f_i([0,1]) \to [0,1]$ such that $$T(x)=f_i^{-1}(x)=\frac{x-iR}{\rho}\quad\textrm{if}\quad x\in f_i([0,1])=[iR, iR+\rho].$$ Then $T(E)=E$ and $\mu\circ T^{-1}=\mu$. Furthermore, for $x=\pi(d_1d_2\ldots)\in E$ we have $T^n(x)=\pi(d_{n+1}d_{n+1}\ldots)$. So, $\pi$ is the isomorphic map from the symbolic dynamics $({\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}, {\sigma})$ to the expanding dynamics $(E, T)$, where ${\sigma}$ is the left shift map.
Our first result describes the pointwise densities of $\mu$.
\[main:densities\] Let $N\ge 2, 0<\rho\le 1/N^2$ and $s=-\log N/\log \rho$. Suppose $\mu$ is the self-similar measure supported on $E$ satisfying (\[eq:mu\]).
- For any $x=\pi(d_1d_2\ldots)\in E$ the pointwise lower $s$-density of $\mu$ at $x$ is given by $$\Theta_*^s(\mu, x)=\frac{1}{2^s}\frac{1}{(R/\rho-\liminf_{n\to{\infty}}{\gamma}_n(x))^s},$$ where $${\gamma}_n(x):=\left\{\begin{array}{lll}
T^n x&\textrm{if}& d_n=0,\\
\max{\left\{T^n x, 1-T^n x\right\}}&\textrm{if}& 1\le d_n\le N-2,\\
1-T^n x&\textrm{if}& d_n=N-1.
\end{array}\right.$$
- For any $x\in E$ the pointwise upper $s$-density of $\mu$ at $x$ is given by $$\Theta^{*s}(\mu, x)= \max{\left\{\frac{1}{2^s(\liminf_{n\to{\infty}}\eta_n(x))^s}, ~ \limsup_{n\to{\infty}}\frac{1+2\hat d_n}{2^s(\hat d_n R/\rho+\eta_n(x))^s}\right\}},$$ where $$\eta_n(x):=\max{\left\{T^n x, 1-T^n x\right\}}\quad\textrm{and}\quad \hat d_n:=\min{\left\{d_n, N-1-d_n\right\}}.$$
- For $\mu$-almost every $x\in E$ we have $$\Theta_*^s(\mu, x)=\left(2R/\rho\right)^{-s} \quad\textrm{and}\quad
\Theta^{*s}(\mu, x)=\left\{\begin{array}{lll}
1&\textrm{if}& N\textrm{ is odd},\\
(N R)^{-s}&\textrm{if}& N\textrm{ is even}.
\end{array}\right.$$
<!-- -->
- We point out that the upper bound $1/N^2$ for the contraction ratio $\rho$ in Theorem \[main:densities\] is not optimal. In fact, by a careful estimation one can improve the upper bound to $1/N^{\log3/\log2}$ for the lower density, and to $1/N^{\log 2/\log(3/2)}$ for the upper density.
- By Theorem \[main:densities\] (i) it follows that if $x$ is an endpoint of $E$, i.e., $x$ has a unique coding ending with $0^{\infty}$ or $(N-1)^{\infty}$, then ${\gamma}_n(x)=0$ for all sufficiently large $n$. So, $
\Theta_*^s(\mu, x)=(2R/\rho)^{-s},
$ which is equal to the typical value of the lower density by Theorem \[main:densities\] (iii). Furthermore, by Theorem \[main:densities\] (ii) it follows that if $x\in E$ is an endpoint, then $\hat d_n=0$ and $\eta_n(x)=1$ for all large integers $n$, and so $
\Theta^{*s}(\mu,x)={2^{-s}}.
$
Note by Theorem \[main:densities\] (iii) that for $\mu$-almost every $x\in E$ we have $\Theta_*^s(\mu, x)<\Theta^{*s}(\mu, x)$. This implies that $E$ is *irregular* (see [@Falconer_1990 Chapter 5] for its definition). In fact, the pointwise lower and upper densities are distinct at every point of $E$. Observe that ${\gamma}_n(x)\in[0, 1]$ and $\eta_n(x)\in[1/2, 1]$ for any $x\in E$. So, by Theorem \[main:densities\] it follows that $$\label{eq:bounds}
(2R/\rho)^{-s}\le \Theta_*^s(\mu,x)\le \left(2R/\rho-2\right)^{-s}\quad<\quad2^{-s}\le \Theta^{*s}(\mu, x)\le 1$$ for all $x\in E$, where the third inequality follows by Lemma \[lem:inequality\] (see below) and the last inequality is well-known (see [@Falconer_1990 Theroem 5.1] or [@Mattila_1995]). In fact, when $N$ is even, the upper bound for $\Theta^{*s}(\mu, x)$ can be refined to $(NR)^{-s}<1$ (see Lemma \[lem:upper-bound-density\] below). This motivates us to study the sets $$E_*(a):={\left\{x\in E: \Theta_*^s(\mu, x)\ge a\right\}}\quad\textrm{and}\quad E^*(b):={\left\{x\in E: \Theta^{*s}(\mu, x)\le b\right\}}$$ with $a, b\in{\ensuremath{\mathbb{R}}}$.
Clearly, the set-valued map $a\mapsto E_*(a)$ is non-increasing. By (\[eq:bounds\]) it follows that $E_*(a)=E$ if $a\le (2R/\rho)^{-s}$, and $E_*(a)=\emptyset$ if $a>(2R/\rho-2)^{-s}$. So, there must exist a critical value $a_c\in((2R/\rho)^{-s}, (2R/\rho-2)^{-s})$ such that $\dim_H E_*(a)>0$ if and only if $a<a_c$. Similarly, the set-valued map $b\mapsto E^*(b)$ is non-decreasing. Again by (\[eq:bounds\]) it gives that $E^*(b)=\emptyset$ if $b<2^{-s}$, and $E^*(b)=E$ if $b\ge 1$. So there exists a critical value $b_c\in(2^{-s}, 1)$ such that $\dim_H E^*(b)>0$ if and only if $b<b_c$.
Our next result is to describe the [critical values]{} $a_c$ and $b_c$ for the sets $E_*(a)$ and $E^*(b)$, respectively. Inspired by some works on the critical values in unique beta expansions (cf. [@Glendinning_Sidorov_2001; @Kong_Li_Dekking_2010]) and open dynamics (cf. [@Kalle-Kong-Langeveld-Li-18]), we introduce two Thue-Morse type sequences $(\theta_i)$ and $({\lambda}_i)$ in ${\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}$. For a word $\mathbf c=c_1\ldots c_n$ we define its *reflection* by $\overline{\mathbf c}:=(N-1-c_1)\ldots (N-1-c_n)$. Furthermore, if $c_n<N-1$, then we write $\mathbf c^+:=c_1\ldots c_{n-1}(c_n+1)$; and if $c_n>0$, then we set $\mathbf c^-:=c_1\ldots c_{n-1}(c_n-1)$.
\[def:sequences\] The sequences $({\lambda}_i)_{i=1}^{\infty}$ and $(\theta_i)_{i=1}^{\infty}$ are defined recursively as follows. Set ${\lambda}_1=\theta_1=N-1$, and if ${\lambda}_1\ldots {\lambda}_{2^n}$ and $\theta_1\ldots \theta_{2^n}$ are defined for some $n\ge 0$, then $${\lambda}_{2^n+1}\ldots{\lambda}_{2^{n+1}}=\overline{{\lambda}_1\ldots{\lambda}_{2^n}}^+\quad\textrm{and}\quad \theta_{2^n+1}\ldots \theta_{2^{n+1}}=\overline{\theta_1\ldots\theta_{2^n}}.$$
By Definition \[def:sequences\] it follows that the sequence $({\lambda}_i)\in{\left\{0,1, N-2, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ begins with $$({\lambda}_i)=(N-1)10(N-1)\,0(N-2)(N-1)1\; 0(N-2)(N-1)0(N-1)10(N-1)\ldots,$$ and the sequence $(\theta_i)\in{\left\{0, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ begins with $$(\theta_i)=(N-1)00(N-1)\, 0(N-1)(N-1)0\; 0(N-1)(N-1)0 (N-1)00(N-1)\ldots.$$ We emphasize that for $N=2$ the sequence $({\lambda}_i)$ is the shift of the classical Thue-Morse sequence, and the sequence $(1-\theta_i)$ is indeed the Thue-Morse sequence (cf. [@Alloche_Shallit_2003]).
Now we state our second result on the critical values of $E_*(a)$ and $E^*(b)$, respectively.
\[main:critical-values\] Let $N\ge 2$ and $0<\rho\le 1/N^2$.
1. The critical value for $E_*(a)$ is given by $$a_c=\left(2 \Big(\frac{R}{\rho}-1+R\sum_{i=1}^{\infty}{\lambda}_{i+1}\rho^{i-1}\Big)\right)^{-s}.$$ That is (i) if $a<a_c$, then $\dim_H E_*(a)>0$; (ii) if $a=a_c$, then $E_*(a)$ is uncountable; and (iii) if $a>a_c$, then $E_*(a)$ is at most countable.
2. The critical value for $E^*(b)$ is given by $$b_c=\left(2R\sum_{i=1}^{\infty}\theta_i\rho^{i-1}\right)^{-s}.$$ That is (i) if $b<b_c$, then $E^*(b)$ is at most countable; (ii) if $b=b_c$, then $E^*(b)$ is uncountable; and (iii) if $b>b_c$, then $\dim_H E^*(b)>0$.
Note by (\[eq:bounds\]) that $(2R/\rho)^{-s}\le a_c\le (2R/\rho-2)^{-s}<2^{-s}\le b_c\le 1$. By some numerical calculation we list the values of $(2R/\rho)^{-s}, a_c, (2R/\rho-2)^{-s}, 2^{-s}$ and $b_c$ for $2\le N\le 8$ and $\rho=1/N^2$ (see Table \[tab:1\]).
N 2 3 4 5 6 7 8
--------------------------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
$(2R/\rho)^{-s}\approx$ 0.408248 0.353553 0.316228 0.288675 0.267261 0.25 0.235702
$a_c \approx$ 0.422744 0.38039 0.340358 0.308856 0.284091 0.26419 0.247828
$(2R/\rho-2)^{-s}\approx$ 0.5 0.408248 0.353553 0.316228 0.288675 0.267261 0.25
$2^{-s}\approx$ 0.707107 0.707107 0.707107 0.707107 0.707107 0.707107 0.707107
$b_c\approx$ 0.809703 0.749479 0.730207 0.721665 0.717129 0.714431 0.712695
: The list of the values $(2R/\rho)^{-s}, a_c, (2R-2)^{-s}, 2^{-s}$ and $b_c$ with $N=2,3,\ldots, 8$ and $\rho=1/N^2$. In this case we have $s=-\log N/\log\rho=1/2$ and $R=(1-\rho)/(N-1)=N^{-1}+N^{-2}$. []{data-label="tab:1"}
The rest of the paper is arranged as follows. In Section \[s2\] we describe the pointwise lower and upper densities of $\mu$ at each $x\in E$ and prove Theorem \[main:densities\]. In Section \[sec:critical-values\] we determine the critical values of $E_*(a)$ and $E^*(b)$ respectively, and prove Theorem \[main:critical-values\].
Pointwise densities of $\mu$ {#s2}
============================
In this section we will describe the pointwise densities of $\mu$ at any point $x\in E$, and prove Theorem \[main:densities\]. Note that $E\subseteq \cup_{i=0}^{N-1}f_i([0, 1])$ and the union is pairwise disjoint. Then by (\[eq:mu\]) it follows that the measure $\mu$ has the same weight $1/N$ on each basic interval $f_i([0, 1])$. This is a special case of the following lemma.
\[lem:density-1\] Let $N\ge 2$ and $0<\rho\le 1/N^2$. Then for any measurable subset $A\subset (-1, 2)$ and any $d_1\ldots d_n\in{\left\{0,1,\ldots, N-1\right\}}^n$ we have $$\mu\big(f_{d_1\ldots d_n}(A)\big)=\frac{\mu(A)}{N^n}.$$
Let $A\subset(-1, 2)$ be a $\mu$-measurable set. Recall that $R=(1-\rho)/(N-1)$. Then for each $i\in{\left\{0,1,\ldots, N-1\right\}}$ we have $
f_i(A)\subset\left(-\rho+i R, 2\rho+i R\right).
$ Since $\rho\le 1/N^2$ and $f_j(E)\subset f_j([0, 1])$, one can easily verify that $$f_i(A)\cap f_j(E)=\emptyset\quad\forall~i\ne j.$$ Then by (\[eq:mu\]) this implies that $
\mu(f_i(A))={\mu(A)}/{N}.
$ So, by induction on $n$ it follows that $$\mu(f_{d_1\ldots d_n}(A))=\frac{\mu(A)}{N^n}\quad\textrm{for all }n\in{\ensuremath{\mathbb{N}}},$$ completing the proof.
In the following we give the bounds for $\mu([0, t])$ and $\mu([t, 1])$, which plays an important role in describing the densities of $\mu$.
\[lem:density-2\] Let $N\ge 2$ and $0<\rho\le 1/N^2$. Then for any $t\in [0, 1]$ we have $$\left(\frac{\rho}{R}\right)^s\cdot t^s\le \mu([0, t])\le t^s\quad\textrm{and}\quad \left(\frac{\rho}{R}\right)^s(1-t)^s\le \mu([t, 1])\le (1-t)^s.$$
Let $t\in [0, 1]$. We only prove the bounds for $\mu([0, t])$, since the bounds for $\mu([t,1])$ can be obtained by using the symmetry of $\mu$ that $\mu(B(x,r))=\mu(B(1-x, r))$ for any $x\in E$ and any $r>0$.
Observe that $\mu=\mathcal H^s|_E$. Then the upper bound for $\mu([0, t])$ follows directly from the work of Zhou [@Zhou-1998] that $\mathcal H^s(E\cap U)\le |U|^s$ for any $U\subset {\ensuremath{\mathbb{R}}}$. So it suffices to prove the lower bound. Clearly it holds for $t=0$. In the following we assume $t\in (0, 1]$. First we consider $t\in E$. Then by (\[eq:proj-map\]) there exists a sequence $(d_i)\in{\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ such that $
t=R\sum_{i=1}^{\infty}d_i\rho^{i-1}.
$ We claim that $$\mu([0,t])=\sum_{i=1}^{\infty}\frac{d_i}{N^i}.$$
To prove the claim it suffices to prove for any $k\in{\ensuremath{\mathbb{N}}}$ that $$\label{eq:kong-01}
\mu([0, t_k])=\sum_{i=1}^k\frac{d_i}{N^i}\quad\textrm{when}\quad t_k=R\sum_{i=1}^k d_i\rho^{i-1}.$$ We will prove (\[eq:kong-01\]) by induction on $k$. Clearly, for $k=1$ we have $t_1=d_1R$, and then $[0, t_1]$ contains exactly $d_1$ basic intervals of level $1$, i.e., $$\bigcup_{i=0}^{d_1-1}f_i(E)\subset [0, t_1].$$ So by Lemma \[lem:density-1\] it follows that $\mu([0, t_1])\ge d_1/N$. On the other hand, $[0, t_1]\cap f_{d_1}(E)$ is a singleton, and $[0, t_1]\cap f_j(E)=\emptyset$ for any $j>d_1$. Therefore, we conclude that $\mu([0, t_1])=d_1/N$. This proves (\[eq:kong-01\]) for $k=1$.
Now suppose (\[eq:kong-01\]) holds for $k=n$, and we consider $k=n+1$. Note that $$t_{n+1}=R\sum_{i=1}^{n+1}d_i\rho^{i-1}=t_n+R d_{n+1}\rho^n.$$ Since each basic interval of level $n$ has length $\rho^n$, this implies that $$[t_n, t_{n+1})\cap E=\bigcup_{j=0}^{d_{n+1}-1} f_{d_1\ldots d_n j}(E).$$ So, by the induction hypothesis and Lemma \[lem:density-1\] we obtain $$\begin{aligned}
\mu([0, t_{n+1}])&=\mu([0, t_n])+\mu([t_n, t_{n+1}])=\sum_{i=1}^{n}\frac{d_i}{N^i}+\frac{d_{n+1}}{N^{n+1}}=\sum_{i=1}^{n+1}\frac{d_i}{N^i}.\end{aligned}$$ This proves (\[eq:kong-01\]) for $k=n+1$, and then the claim follows by induction.
By the claim and using $N^{-1}=\rho^s$ it follows that $$\frac{\mu([0, t])}{t^s}=\frac{\sum_{i=1}^{\infty}d_i\rho^{i s}}{\left(\frac{R}{ \rho}\right)^s\left(\sum_{i=1}^{\infty}d_i\rho^i\right)^s}\ge \left(\frac{\rho}{R}\right)^s,$$ where the inequality follows by using $s\in(0, 1)$ and the basic inequality $$\left(\sum_{i=1}^{\infty}x_i\right)^s\le \sum_{i=1}^{\infty}x_i^s\quad\forall~ x_i\ge 0.$$
Now for $t\in(0,1]\setminus E$ let $t'$ be the smallest element in $E$ strictly larger than $t$. Then $\mu([0, t])=\mu([0, t'])$, and therefore, $$\mu([0, t])=\mu([0, t'])\ge \left(\frac{\rho}{R}\right)^s\cdot (t')^s>\left(\frac{\rho}{R}\right)^s\cdot t^s.$$ This completes the proof.
The following lemma is elementary but turns out to be useful in our proofs later.
\[lem:inequality\] Let $N\ge 2$ and $0<\rho\le 1/N^2$. Then $$0<\frac{\rho}{R}\le \frac{1}{N+1}.$$
1. Let $A, B\ge 0$. If $
\rho^{1-s}A<B,
$ then the function $$g(r)=\frac{A+(r-B)^s}{r^s}$$ is strictly increasing in $[B, B+\rho]$.
2. Let $m\ge 1$. Then for any $t\in[0, \rho]$ the sequences $$a_j:=\frac{j}{(j R-t)^s}\quad\textrm{and} \quad b_j:=\frac{1+m j}{(jR+t)^s}, \quad j\ge 1,$$ are both strictly increasing.
Note that $(N-1)R=1-\rho\ge N^2\rho-\rho>0$. This gives $$0<\frac{\rho}{R}\le \frac{1}{N+1}.$$ For (i) we observe that $$\label{eq:mar-28-1}
g'(r)>0\quad\Longleftrightarrow\quad (r-B)^{1-s}A<B.$$ Note that $r\in[B, B+\rho]$ and $s\in(0, 1)$. So, if $\rho^{1-s}A<B$, by (\[eq:mar-28-1\]) it follows that $g'(r)>0$ for all $r\in[B, B+\rho]$. This proves (i).
For (ii) we note for $j\ge 1$ that $a_{j+1}>a_j$ if and only if $$\label{eq:mar-28-2}
\frac{j+1}{j}>\left(\frac{(j+1)R-t}{jR-t}\right)^s.$$ Since $\rho\in(0, 1/N^2]$, we have $s=-\log N/\log \rho\in(0, 1/2]$. Thus, (\[eq:mar-28-2\]) holds if it holds for $s=1/2$, i.e., $$\frac{j+1}{j}>\left(\frac{jR-t+R}{jR-t}\right)^{1/2}.$$ By rearrangements and using $t\in[0, \rho]$ we only need to prove $$(2j+1)\left(j-\frac{\rho}{R}\right)>j^2.$$ Since $0<\rho/R\le 1/(N+1)$, it suffices to verify $(2j+1)(j-\frac{1}{N+1})>j^2$, which follows directly by using $j\ge 1$.
Similarly, one can verify that $b_{j+1}>b_j$ for all $j\ge 1$.
Lower density $\Theta_*(\mu, x)$
--------------------------------
In the following we give the lower bound for $\mu(B(x, r))/(2r)^s$, where $B(x, r)=(x-r, x+r)$ is the open ball of radius $r$ at center $x$. Based on Lemma \[lem:density-2\] we first consider $x\in f_i([0, 1])$ for $i=0$ and $i=N-1$.
\[lem:lower-bound\] Let $N\ge 2$ and $0<\rho\le 1/N^2$.
1. If $x\in[0, \rho]$, then for any $\max{\left\{x, \rho-x\right\}}\le r\le 1-x$ we have $$\frac{\mu(B(x, r))}{(2r)^s}\ge \frac{\rho^s}{2^s(R-x)^s},$$ where the equality holds for $r=R-x$.
2. If $x\in[1-\rho, 1]$, then for any $\max{\left\{1-x, \rho-(1-x)\right\}}\le r\le x$ we have $$\frac{\mu(B(x, r))}{(2r)^s}\ge \frac{\rho^s}{2^s(R-(1-x))^s},$$ where the equality holds for $r=R-(1-x)$.
Note that $\mu(B(x,r))=\mu(B(1-x, r))$ for any $x\in E$ and $r>0$. So, (ii) can be deduced from (i) by the symmetry of $\mu$. In the following we only prove (i).
Let $x\in[0, \rho]$, and take $\max{\left\{x, \rho-x\right\}}\le r\le 1-x$. Then $B(x, r)\subset(-1, 2)$, and $B(x, r)$ nearly contains the interval $[0, \rho]$ which means $B(x, r)$ contains $[0, \rho]$ up to a point. Recall that $R=(1-\rho)/(N-1)$. We split the proof into the following two cases.
Case I. $\max{\left\{x, (j-1)R+\rho-x\right\}}\le r\le j R-x$ for some $j\in{\left\{1, 2,\ldots, N-1\right\}}$. Then $(j-1)R+\rho\le x+r\le j R$. Note that $f_{i+1}(0)-f_i(0)=R$ for any $0\le i<N-1$. Then $B(x, r)$ nearly contains $f_i([0, 1])$ for $0\le i\le j-1$, and $B(x, r)$ has no intersect with any other basic interval of level $1$. So, by Lemma \[lem:density-1\] it follows that $$\mu(B(x, r))=j \rho^s.$$ This implies $$\begin{aligned}
\frac{\mu(B(x, r))}{(2r)^s}=\frac{j\rho^s}{(2r)^s}\ge \frac{j\rho^s}{2^s\left(j R-x\right)^s},
\end{aligned}$$ where the equality holds for $r=j R-x$.
Case II. $j R-x< r\le j R+\rho-x$ for some $j\in{\left\{1,2,\ldots, N-1\right\}}$. Then $B(x, r)$ contains the level-$1$ basic intervals $f_0([0, 1]),\ldots, f_{j-1}([0, 1])$ and intersect $f_j([0,1])$. But $B(x,r)$ has no intersect with any other level-$1$ basic interval. So, by Lemmas \[lem:density-1\], \[lem:density-2\] and using the uniformity of $\mu$ it follows that $$\begin{aligned}
\mu(B(x, r))=j \rho^s+\mu([j R, x+r])&=j \rho^s+\mu([0, x+r-j R])\ge j\rho^s+\left(\frac{\rho}{ R}\right)^s(x+r-j R)^s.
\end{aligned}$$ This implies $$\label{eq:feb-21-1}
\frac{\mu(B(x, r))}{(2r)^s}\ge \frac{\rho^s}{(2R)^s}g_1(r),\quad\textrm{where}\quad g_1(r):=\frac{j R^s+(x+r-j R)^s}{r^s}.$$ We claim that $g_1(r)$ is strictly increasing in $(jR-x, jR+\rho-x]$.
By Lemma \[lem:inequality\] (i) with $A=j R^s$ and $B=jR-x$ it suffices to check $\rho^{1-s}j R^s<jR-x$. Since $s\in(0, 1/2]$ and $x\in[0, \rho]$, it suffices to verify $$j\rho\left(\frac{R}{\rho}\right)^{1/2}<jR-\rho.$$ Dividing $R$ on both sides reduces to $$j\left(\frac{\rho}{R}\right)^{1/2}<j-\frac{\rho}{R}.$$ This can be easily verified by using $j\ge 1$ and $0<\rho/R\le 1/(N+1)$ (see Lemma \[lem:inequality\]). So, we establish the claim.
Therefore, by (\[eq:feb-21-1\]) and the claim it follows that $$\frac{\mu(B(x, r))}{(2r)^s}\ge \frac{\rho^s}{(2R)^s}g_1(r)> \frac{\rho^s}{(2R)^s}g_1(j R-x)= \frac{j\rho^s}{2^s\left(j R-x\right)^s}$$ for any $r\in(jR-x, jR+\rho-x]$. Finally, observe by Lemma \[lem:inequality\] (ii) that the sequence $$\hat a_j=\frac{j}{(j R-x)^s}, \quad j=1,2,\ldots, N-1,$$ is strictly increasing. This proves (i).
In the following we will determine the lower bound of $\mu(B(x, r))/(2r)^s$ for $x\in f_k([0, 1])$ with $k=1,2,\ldots, N-2$.
\[prop:lower-bound\] Let $N\ge 2$ and $0<\rho\le 1/N^{2}$.
1. If $x\in[k R, kR+\rho/2]$ for some $k\in{\left\{1,\ldots, N-2\right\}}$, then for any $k R+\rho-x \le r\le \max{\left\{x, 1-x\right\}}$ we have $$\frac{\mu(B(x, r))}{(2r)^s}\ge \frac{\rho^s}{2^s(R-(kR+\rho-x))^s},$$ where the equality holds when $r=R-(kR+\rho-x)$.
2. If $x\in[k R+\rho/2, k R+\rho]$ for some $k\in{\left\{1,\ldots, N-2\right\}}$, then for any $x-k R \le r\le \max{\left\{x, 1-x\right\}}$ we have $$\frac{\mu(B(x, r))}{(2r)^s}\ge \frac{\rho^s}{2^s(R-(x-k R))^s},$$ where the equality holds when $r=R-(x-k R)$.
Observe for $k\in{\left\{1,\ldots, N-2\right\}}$ that $x\in[k R+\rho/2, kR+\rho]$ if and only if $1-x\in[(N-1-k)R, (N-1-k)R+\rho/2]$. So, (ii) can be deduced from (i) by using the symmetry of $\mu$. In the following we only prove (i).
Take $x\in[k R, kR+\rho/2]$ with $k\in{\left\{1, 2,\ldots, N-2\right\}}$. Then for $kR+\rho-x \le r\le x$ the ball $B(x, r)$ nearly contains the basic interval $f_k([0, 1])$. Using the uniformity of $\mu$ it follows that $$\mu(B(x, r))\ge \mu(B((N-1-k)R+x, r)).$$ Observe that $(N-1-k)R+x\in[1-\rho, 1]$. By Lemma \[lem:lower-bound\] (ii) it follows that $$\label{eq:feb-1}
\begin{split}
\frac{\mu(B(x, r))}{(2r)^s}\ge \frac{\mu(B((N-1-k)R+x, r))}{(2r)^s}&\ge\frac{\rho^s}{2^s(R-(1-(N-1-k)R-x))^s}\\
&= \frac{\rho^s}{2^s(R-(kR+\rho-x))^s},
\end{split}$$ where the equalities hold for $r=R-(kR+\rho-x)$.
If $x\ge 1/2$, then $\max{\left\{x,1-x\right\}}=x$ and we are done. Now suppose $x<1/2$. Then for $x<r\le 1-x$ we have $\mu(B(x, r))=\mu(B(0, x+r))$. By Lemma \[lem:density-2\] it follows that $$\label{eq:mar-27-2}
\frac{\mu(B(x, r))}{(2r)^s}\ge\frac{\rho^s}{2^s}\cdot\frac{(x+r)^s}{(R r)^s}.$$ Using $x\ge kR, r\le 1-x$ and $0<\rho\le 1/N^2$, one can verify that $$\frac{x+r}{R r}>\frac{1}{R-(k R+\rho-x)}\quad \textrm{for all }1\le k\le N-2.$$ So, by (\[eq:feb-1\]) and (\[eq:mar-27-2\]) we establish (i).
By Lemma \[lem:lower-bound\] and Proposition \[prop:lower-bound\] it follows that for any $x\in E$, if $B(x, r)$ contains at least one level-$1$ basic interval but it does not contain the unit interval $[0, 1]$, then $$\label{eq:feb-19-1}
\frac{\mu(B(x, r))}{(2r)^s}\ge \frac{\rho^s}{2^s(R-S(x))^s},$$ where $$\label{eq:map-s}
S(x)=\left\{\begin{array}{lll}
x&\textrm{if}& 0\le x\le \rho,\\
\max{\left\{x-k R, kR+\rho-x\right\}}&\textrm{if}& k R\le x\le kR+\rho\textrm{ for }1\le k\le N-2,\\
1-x&\textrm{if}& 1-\rho\le x\le 1.
\end{array}\right.$$ Recall that the $N$-to-$1$ expanding map $T: \bigcup_{k=0}^{N-1}[kR, kR+\rho]\to [0, 1]$ satisfies $$T(x)= \frac{x-k R}{\rho}\quad\textrm{if}\quad k R\le x\le kR+\rho,$$ where $k=0,1,\ldots, N-1$. So for any $
x=\pi(d_1d_2\ldots)\in E
$ we have $
T^n(x)=\pi(d_{n+1}d_{n+2}\ldots).
$
Take $x=\pi(d_1d_2\ldots)\in E$ and $r\in(0, \rho)$. Then there exists $n\in{\ensuremath{\mathbb{N}}}$ such that $B(x, r)$ contains the level-$(n+1)$ basic interval $f_{d_1\ldots d_{n+1}}([0, 1])$, but it does not contain the level-$n$ basic interval $f_{d_1\ldots d_{n}}([0, 1])$. This implies that $$T^{n}(B(x, r) )=(f_{d_1\ldots d_{n}})^{-1}(B(x, r) )\supseteq f_{d_{n+1}}([0,1]),$$ but $T^{n}(B(x, r))$ does not contain $[0,1]$.
Let $y=T^{n}x=\pi(d_{n+1}d_{n+2}\ldots)$, and let $r'=\rho^{-n}r$. Then $y\in f_{d_{n+1}}([0,1])$, $\rho/2<r'<1$ and $T^{n}(B(x, r) )=B(y, r') $. By Lemma \[lem:density-1\] and (\[eq:feb-19-1\]) it follows that $$\label{eq:feb-19-2}
\begin{split}
\frac{\mu(B(x, r))}{(2r)^s}=\frac{(\rho^s)^{n}\mu(B(y, r'))}{(2r)^s}&=\frac{\mu(B(y, r'))}{(2r')^s}\\
&\ge \frac{\rho^s}{2^s(R-S(y))^s}\\
&=\frac{\rho^s}{2^s(R-S(T^{n}x))^s}=\frac{\rho^s}{2^s(R-\rho{\gamma}_{n+1}(x))},
\end{split}$$ where the last equality holds by the following observation: if $1\le d_{n+1}\le N-2$, then $$\begin{aligned}
S(T^n x)&=\max{\left\{T^n x-d_{n+1}R, d_{n+1}R+\rho-T^n x\right\}}\\
&=\rho\max{\left\{T^{n+1}x, 1-T^{n+1}x\right\}}=\rho{\gamma}_{n+1}(x);\end{aligned}$$ if $d_{n+1}=0$, then $S(T^n x)=T^n x=\rho T^{n+1}x=\rho{\gamma}_{n+1}(x)$; and if $d_{n+1}=N-1$, then $S(T^n x)=1-T^n x=\rho(1-T^{n+1}x)=\rho{\gamma}_{n+1}(x)$. Letting $r\to 0$ in (\[eq:feb-19-2\]), and then $n\to{\infty}$, we deduce that $$\label{eq:feb-19-3}
\Theta_*^s(\mu, x)=\liminf_{r\to 0}\frac{\mu(B(x, r))}{(2r)^s}\ge \frac{\rho^s}{2^s(R-\rho\liminf_{n\to{\infty}} {\gamma}_n(x))^s}.$$
On the other hand, observe that the equality holds in (\[eq:feb-19-2\]) when $r'=R-S(T^{n}x)$, or equivalently, when $r=\rho^{n}(R-S(T^{n}x)).$ So, let $r_n:=\rho^{n}(R-S(T^{n}x))$. Then $r_n\to 0$ as $n\to{\infty}$. Letting $r\to 0$ along the subsequence $(r_n)$ in (\[eq:feb-19-3\]) we conclude that $$\Theta_*^s(\mu, x)=\frac{\rho^s}{2^s(R-\rho\liminf_{n\to{\infty}}{\gamma}_n(x))^s}=\frac{1}{2^s(R/\rho-\liminf_{n\to{\infty}}{\gamma}_n(x))^s}.$$ This completes the proof.
Upper density $\Theta^*(\mu, x)$
--------------------------------
Using the same idea as in the previous subsection we are able to determine the pointwise upper density of $\mu$ at each $x\in E$. But the calculation is more involved. First we consider $x\in f_i([0,1])$ for $i=0$ and $i=N-1$.
\[lem:upper-density-1\] Let $N\ge 2$ and $0<\rho\le 1/N^2$.
1. Let $x\in[0, \rho]$. Then for any $\max{\left\{x, \rho-x\right\}}\le r\le 1-x$ we have
$$\frac{\mu(B(x,r))}{(2r)^s}\le\left\{ \begin{array}{lll}
\frac{\rho^s}{2^s(\rho-x)^s}&\textrm{if}&x\in[0, \rho/2],\\
\max{\left\{\frac{\rho^s}{2^s x^s}, \frac{1}{2^s(1-x)^s}\right\}}&\textrm{if}& x\in[\rho/2, \rho],
\end{array}\right.$$ where the equality is attainable.
2. Let $x\in[1-\rho, 1]$. Then for any $\max{\left\{1-x, \rho-(1-x)\right\}}\le r\le x$ we have $$\frac{\mu(B(x,r))}{(2r)^s}\le\left\{\begin{array}{lll}
\max{\left\{\frac{\rho^s}{2^s(1-x)^s}, \frac{1}{2^s x^s}\right\}}&\textrm{if}& x\in[1-\rho, 1-\rho/2],\\
\frac{\rho^s}{2^s (\rho-(1-x))^s}&\textrm{if}& x\in[1-\rho/2, 1],
\end{array}\right.$$ where the equality is attainable.
Since the measure $\mu$ is symmetric on $E$, (ii) can be deduced from (i). In the following we only prove (i). Let $x\in[0, \rho]$ and $\max{\left\{x,\rho-x\right\}}\le r\le 1-x$. Then $B(x, r)$ nearly contains $[0, \rho]$, and $B(x, r)\subset(-1, 2)$. Note that $R=(1-\rho)/(N-1)$. We consider the following two cases.
Case I. $\max{\left\{x, (j-1)R+\rho-x\right\}}\le r\le jR-x$ for some $j\in{\left\{1,2,\ldots, N-1\right\}}$. By the same argument as in the proof of Lemma \[lem:lower-bound\] we have $\mu(B(x,r))=j\rho^s$. So, if $x\le (j-1)R+\rho-x$, then $\max{\left\{x, (j-1)R+\rho-x\right\}}=(j-1)R+\rho-x$, and so $$\label{eq:feb-28-0}
\frac{\mu(B(x, r))}{(2r)^s}\le\frac{j\rho^s}{2^s((j-1)R+\rho-x)^s},$$ where the equality holds for $r=(j-1)R+\rho-x$. If $x>(j-1)R+\rho-x$, then $j=1$ and $x>\rho/2$. In this case we have $$\label{eq:feb-28-1}
\frac{\mu(B(x,r))}{(2r)^s}\le \frac{\rho^s}{2^s x^s}$$ where the equality holds for $r=x$.
Case II. $jR-x<r\le jR+\rho-x$ for some $j\in{\left\{1,2,\ldots, N-1\right\}}$. Then by Lemma \[lem:density-2\] one can verify that $$\mu(B(x, r))=j\rho^s+\mu([jR, x+r])\le j\rho^s+(x+r-jR)^s.$$ This implies [$$\frac{\mu(B(x,r))}{(2r)^s}\le\frac{1}{2^s}\cdot\frac{j\rho^s+(x+r-jR)^s}{r^s}=:\frac{1}{2^s} g_2(r).$$ By Lemma \[lem:inequality\] (i) with $A=j\rho^s$ and $B=jR-x$ one can easily verify that $\rho^{1-s}A<B$, and thus $g_2$ is strictly increasing in $(jR-x, jR+\rho-x]$.]{} So, for any $jR-x<r\le jR+\rho-x$, $$\label{eq:feb-28-2}
\frac{\mu(B(x,r))}{(2r)^s}\le\frac{1}{2^s}g_2(jR+\rho-x)=\frac{(j+1)\rho^s}{2^s(jR+\rho-x)^s},$$ where the equality holds for $r=jR+\rho-x$.
[Note by Lemma \[lem:inequality\] (ii) that the sequence $$\hat b_j:=\frac{j+1}{(jR+\rho-x)^s},\quad j=1,\ldots, N-1$$ is strictly increasing in $j$.]{} Therefore, by (\[eq:feb-28-0\]) and (\[eq:feb-28-2\]) it follows that for $x\in[0, \rho/2]$, $$\begin{aligned}
\frac{\mu(B(x,r))}{(2r)^s}&\le\max{\left\{\frac{\rho^s}{2^s(\rho-x)^s}, \frac{N\rho^s}{2^s((N-1)R+\rho-x)^s}\right\}}\\
&=\max{\left\{\frac{\rho^s}{2^s(\rho-x)^s}, \frac{1}{2^s(1-x)^s}\right\}} =\frac{\rho^s}{2^s(\rho-x)^s},\end{aligned}$$ where we have used $\rho^s=1/N$ and $R=(1-\rho)/(N-1)$. Moreover, by (\[eq:feb-28-1\]) and (\[eq:feb-28-2\]) it follows that for $x\in[\rho/2, \rho]$ we have $$\frac{\mu(B(x,r))}{(2r)^s}\le \max{\left\{\frac{\rho^s}{2^sx^s}, \frac{1}{2^s(1-x)^s}\right\}}.$$ This completes the proof.
In the next lemma we consider the upper bound of $\mu(B(x,r))/(2r)^s$ for $x\in f_k([0,1])$ with $k=1,\ldots, N-2$.
\[prop:upper-density\] Let $N\ge 2$ and $0<\rho\le 1/N^2$. For $k\in{\left\{0,\ldots, N-1\right\}}$ set $\hat k:=\min{\left\{k, N-1-k\right\}}$.
1. If $x\in[kR, kR+\rho/2]$ for some $k=1,\ldots, N-2$, then for any $kR+\rho-x\le r\le \max{\left\{x, 1-x\right\}}$ we have $$\frac{\mu(B(x, r))}{(2r)^s}\le \max{\left\{\frac{\rho^s}{2^s(kR+\rho-x)^s}, \frac{(1+2\hat k)\rho^s}{2^s(\hat k R+ (kR+\rho-x) )^s},\frac{1}{2^s(\max{\left\{x, 1-x\right\}})^s}\right\}},$$ where the equality is attainable.
2. If $x\in[kR+\rho/2, kR+\rho]$ for some $k=1,\ldots, N-2$, then for any $x-kR\le r\le \max{\left\{x, 1-x\right\}}$ we have $$\frac{\mu(B(x,r))}{(2r)^s}\le \max{\left\{\frac{\rho^s}{2^s(x-kR)^s}, \frac{(1+2\hat k)\rho^s}{2^s(\hat k R+ (x-kR) )^s},\frac{1}{2^s(\max{\left\{x, 1-x\right\}})^s}\right\}},$$ where the equality is attainable.
Note by the symmetry of $\mu$ that (ii) can be deduced from (i). In the following we only prove (i). Let $x\in[kR, kR+\rho/2]$ and $kR+\rho-x\le r\le \max{\left\{x, 1-x\right\}}$. Then the ball $B(x, r)$ contains the basic interval $f_k([0, 1])$ and is contained in $(-1,2)$. We first assume $x\le 1/2$. Then $1\le k\le (N-1)/2$, and thus $\hat k=k$. We will prove in the following two steps that $$\label{eq:mar-2-1}
\frac{\mu(B(x,r))}{(2r)^s}\le\max{\left\{\frac{\rho^s}{2^s(kR+\rho-x)^s}, \frac{(1+2k)\rho^s}{2^s(2kR+\rho-x)^s}, \frac{1}{2^s(1-x)^s}\right\}}$$ for any $x\in[0, 1/2]\cap[kR, kR+\rho/2]$ and $kR+\rho-x\le r\le 1-x$.
[**Step I**]{}. In this step we will prove that for each $0\le j\le k-1$ and for any $(k+j)R+\rho-x\le r\le (k+j+1)R+\rho-x$ we have $$\label{eq:step1}
\frac{\mu(B(x,r))}{(2r)^s}\le \max{\left\{\frac{(1+2j)\rho^s}{2^s(jR+(kR+\rho-x))^s}, \frac{(1+2(j+1))\rho^s}{2^s((j+1)R+(kR+\rho-x))^s}\right\}},$$ where the equality is attainable. Take $0\le j\le k-1$. Let $(k+j)R+\rho-x\le r\le (k+j+1)R+\rho-x$. Then $B(x, r)$ contains the basic intervals $f_i([0,1])$ with $k-j\le i\le k+j$ and it might have intersection with $f_{k-j-1}([0, 1])$ and $f_{k+j+1}([0,1])$ (see Figure \[fig:2\]). Since $x\in[kR, kR+\rho]$, we can partition the interval $[(k+j)R+\rho-x, (k+j+1)R+\rho-x]$ as follows: $$\label{eq:mar-29-0}
\begin{split}
(k+j)R+\rho-x&\le x-(k-j-1)R-\rho\\
&\le (k+j+1)R-x\\
&\le x-(k-j-1)R\le (k+j+1)R+\rho-x.
\end{split}$$ Now we prove (\[eq:step1\]) by considering the following four cases according to the partition.
(-0.5,-0.2) – (-0.3,-0.2) node(xline)\[right\]; at (-0.4, -0.15)[$f_{k-j-1}$]{}; at (-0.5, -0.25)[$(k-j-1)R$]{};
(-0.1,-0.2) – (0.1,-0.2) node(xline)\[right\]; at (0, -0.15)[$f_{k-j}$]{}; at (-0.12, -0.25)[$(k-j)R$]{}; at (0.25, -0.2)[$\cdots$]{};
(0.4,-0.21) – (0.6,-0.21) node(xline)\[right\]; at (0.5, -0.15)[$f_k$]{}; at (0.38, -0.25)[$kR$]{}; (0.45, -0.1)–(0.45,-0.21); at(0.45,-0.08)[$x$]{}; at (0.75, -0.2)[$\cdots$]{};
(0.9,-0.2) – (1.1,-0.2) node(xline)\[right\]; at (1, -0.15)[$f_{k+j}$]{}; at (0.88, -0.25)[$(k+j)R$]{}; (1.3,-0.2) – (1.5,-0.2) node(xline)\[right\]; at (1.4, -0.15)[$f_{k+j+1}$]{}; at (1.3, -0.25)[$(k+j+1)R$]{};
Case I. $(k+j)R+\rho-x\le r\le x-(k-j-1)R-\rho$. Then the ball $B(x, r)$ contains only the basic intervals $f_{i}([0,1])$ with $k-j\le i\le k+j$ and it has no intersect with any other basic interval of level-$1$. So, $$\frac{\mu(B(x,r))}{(2r)^s}=\frac{(1+2j)\rho^s}{(2r)^s}\le\frac{(1+2j)\rho^s}{2^s((k+j)R+\rho-x)^s},$$ where the equality holds for $r=(k+j)R+\rho-x$.
Case II. $x-(k-j-1)R-\rho\le r\le (k+j+1)R-x$. Then the ball $B(x,r)$ contains the basic interval $f_i([0,1])$ with $k-j\le i\le k+j$ and it intersects $f_{k-j-1}([0,1])$. But it has no intersect with any other basic interval of level-$1$. This gives $$\begin{aligned}
\mu(B(x,r))&=(1+2j)\rho^s+\mu([x-r, (k-j-1)R+\rho])\\
&\le (1+2j)\rho^s+((k-j-1)R+\rho-x+r)^s,\end{aligned}$$ where the inequality follows by Lemma \[lem:density-2\]. So, $$\label{eq:mar-29-1}
\frac{\mu(B(x,r))}{(2r)^s}\le\frac{(1+2j)\rho^s+((k-j-1)R+\rho-x+r)^s}{(2r)^s}=:\frac{1}{2^s}\hat g_1(r).$$ Let $A=(1+2j)\rho^s$ and $B=x-(k-j-1)R-\rho$. Note by Lemma \[lem:inequality\] that $0<\rho/R\le 1/(N+1)$. Then by using $x\ge kR$ one can easily verify that $\rho^{1-s}A<B$. So by (\[eq:mar-29-0\]) and Lemma \[lem:inequality\] (i) it follows that the function $\hat g_1$ is strictly increasing in $[x-(k-j-1)R-\rho, (k+j+1)R-x]$. Therefore, $$\frac{\mu(B(x,r))}{(2r)^s}\le\frac{1}{2^s}\hat g_1(r)\le \frac{1}{2^s}\hat g_1\big((k+j+1)R-x\big)=\frac{(1+2j)\rho^s+(2kR+\rho-2x)^s}{2^s((k+j+1)R-x)^s}.$$
Case III. $(k+j+1)R-x\le r\le x-(k-j-1)R$. Then the ball $B(x,r)$ not only contains the basic interval $f_{i}([0, 1])$ with $k-j\le i\le k+j$ but also intersect $f_{k-j-1}([0,1])$ and $f_{k+j+1}([0,1])$. However, it has no intersect with any other basic intervals of level-$1$. So, by Lemmas \[lem:density-2\] and \[lem:lower-bound\] it follows that $$\begin{aligned}
\frac{\mu(B(x, r))}{(2r)^s}&\le \frac{(1+2j)\rho^s+((k-j-1)R+\rho-x+r)^s+(x+r-(k+j+1)R)^s}{(2r)^s}\\
&=:\frac{1}{2^s}(\hat g_1(r)+\hat g_2(r)),\end{aligned}$$ where $\hat g_1$ is defined in (\[eq:mar-29-1\]) and $$\hat g_2(r):=\frac{(x+r-(k+j+1)R)^s}{r^s}.$$ Note by Case II that $\hat g_1(r)$ is increasing in $[x-(k-j-1)R-\rho, x-(k-j-1)R]$. Furthermore, by Lemma \[lem:inequality\] (i) one can easily show that $\hat g_2(r)$ is increasing in $[(k+j+1)R-x, (k+j+1)R-x+\rho]$. Therefore, by (\[eq:mar-29-0\]) it follows that $$\frac{\mu(B(x,r))}{(2r)^s}\le\frac{1}{2^s}\big(\hat g_1(x-(k-j-1)R)+\hat g_2(x-(k-j-1)R)\big)=\frac{(2+2j)\rho^s+(2x-2k R)^s}{2^s(x-(k-j-1)R)^s}$$ for any $(k+j+1)R-x\le r\le x-(k-j-1)R$.
Case IV. $x-(k-j-1)R\le r\le (k+j+1)R+\rho-x$. Then the ball contains the basic intervals $f_{i}([0, 1])$ with $k-j-1\le i\le k+j$, and it intersects the basic interval $f_{k+j+1}([0, 1])$. But it has no intersection with any other basic intervals of level-$1$. So, by Lemma \[lem:density-2\] it follows that $$\frac{\mu(B(x, r))}{(2r)^s}\le \frac{(2+2j)\rho^s+(x+r-(k+j+1)R)^s}{(2r)^s}=:\frac{1}{2^s}\hat g_3(r).$$ Let $A=(2+2j)\rho^s$ and $B=(k+j+1)R-x$. Note by Lemma \[lem:inequality\] that $\rho/R\in(0, 1/(N+1)]$. Then using $x\in[kR, kR+\rho/2]$ one can verify that $\rho^{1-s}A<B$. So, by (\[eq:mar-29-0\]) and Lemma \[lem:inequality\] (i) it follows that $\hat g_3(r)$ is strictly increasing in $[x-(k-j-1)R, (k+j+1)R-x+\rho]$. Therefore, $$\frac{\mu(B(x, r))}{(2r)^s}\le\frac{1}{2^s}\hat g_3(r)\le \frac{1}{2^s}\hat g_3((k+j+1)R+\rho-x)=\frac{(1+2(j+1))\rho^s}{2^s((k+j+1)R+\rho-x)^s},$$ where the equality holds for $r=(k+j+1)R+\rho-x$.
Observe that $\hat g_1$ and $\hat g_1+\hat g_2$ coincide at $r=(k+j+1)R-x$, and $\hat g_1+\hat g_2$ and $\hat g_3$ coincide at $r=x-(k-j-1)R$. Therefore, (\[eq:step1\]) follows by Cases I–IV.
[**Step II.**]{} Note that $kR+\rho-x\in[0, \rho]$. Then by Lemma \[lem:inequality\] (ii) it gives that the sequence $$\hat b_j:=\frac{1+2j}{(jR+(kR+\rho-x))^s},\quad j\ge 1$$ is strictly increasing. So, by (\[eq:step1\]) in Step I it follows that for any $r\in[kR+\rho-x, 2k R+\rho-x]$, $$\label{eq:mar-3-4}
\frac{\mu(B(x, r))}{(2r)^s}\le \max{\left\{\frac{\rho^s}{2^s(kR+\rho-x)^s}, \frac{(1+2k)\rho^s}{2^s(k R+(kR+\rho-x))^s}\right\}},$$ where the equality is attainable. If $k=(N-1)/2$, then $2k R+\rho-x=1-x$ and $(1+2k)\rho^s=N\rho^s=1$, and therefore (\[eq:mar-3-4\]) gives (\[eq:mar-2-1\]).
In the following it suffices to consider $k<(N-1)/2$. Note that for $2kR+\rho-x<r\le 1-x$ we have $x-r<0$. Then by the same argument as in the proof of Lemma \[lem:upper-density-1\] and using that the sequence $$\tilde b_j:=\frac{(1+2k+j)\rho^s}{2^s((k+j)R+(kR+\rho-x))^s},\quad j\ge 1$$ is strictly increasing, it follows that for any $2kR+\rho-x\le r\le 1-x$ we have $$\label{eq:mar-3-5}
\begin{split}
\frac{\mu(B(x,r))}{(2r)^s}&\le \max{\left\{\frac{(1+2k)\rho^s}{2^s(kR+(kR+\rho-x))^s}, \frac{N\rho^s}{2^s((N-1)R+\rho-x)^s}\right\}}\\
&=\max{\left\{\frac{(1+2k)\rho^s}{2^s(kR+(kR+\rho-x))^s}, \frac{1}{2^s(1-x)^s}\right\}},
\end{split}$$ where the equality is attainable. Therefore, (\[eq:mar-2-1\]) follows by (\[eq:mar-3-4\]) and (\[eq:mar-3-5\]).
If $x\in(1/2, 1]$, then $k> (N-1)/2$ and thus $\hat k=N-1-k$. By the same argument as above we can prove that for $kR+\rho-x\le r\le (k+\hat k)R+\rho-x$, $$\label{eq:mar-3-6}
\begin{split}
\frac{\mu(B(x,r))}{(2r)^s}&\le \max{\left\{\frac{\rho^s}{2^s(kR+\rho-x)^s}, \frac{(1+2\hat k)\rho^s}{2^s(\hat k R+(kR+\rho-x))^s} \right\}}.
\end{split}$$ Since $x>1/2$, for $1-x< r\le x$ we have $x+r>1$. By the same argument as in Lemma \[lem:upper-density-1\] it follows that $$\label{eq:may-6-1}
\frac{\mu(B(x,r))}{(2r)^s}\le \max{\left\{ \frac{(1+2\hat k)\rho^s}{2^s(\hat k R+(kR+\rho-x))^s}, \frac{1}{2^s x^s}\right\}}$$ for any $1-x\le r\le x$. Note that $(k+\hat k)R+\rho-x=(N-1)R+\rho-x=1-x$. Therefore, by (\[eq:mar-3-6\]) and (\[eq:may-6-1\]) it follows that for any $x\in[kR, kR+\rho/2]\cap(1/2, 1]$ and $kR+\rho-x\le r\le x$, $$\label{eq:mar-3-7}
\frac{\mu(B(x,r))}{(2r)^s}\le\max{\left\{\frac{\rho^s}{2^s(kR+\rho-x)^s}, \frac{(1+2\hat k)\rho^s}{2^s(\hat k R+(kR+\rho-x))^s}, \frac{1}{2^s x^s}\right\}},$$ where the equality is attainable.
Hence, by (\[eq:mar-2-1\]) and (\[eq:mar-3-7\]) we prove (i).
By Lemma \[lem:upper-density-1\] it follows that Proposition \[prop:upper-density\] also holds for $k=0$ and $N-1$. So, if $x\in[kR, kR+\rho]$ for some $k\in{\left\{0, 1,\ldots, N-1\right\}}$ and $\max{\left\{kR+\rho-x, x-kR\right\}}\le r\le \max{\left\{x, 1-x\right\}}$, then the ball $B(x, r)$ contains at least one basic interval $f_k([0, 1])$ and it does not contain $[0,1]$. In this case we conclude by Lemmas \[lem:upper-density-1\] and Proposition \[prop:upper-density\] that $$\label{eq:mar-4-1}
\frac{\mu(B(x, r))}{(2r)^s}\le \max{\left\{\frac{\rho^s}{2^s(M(k,x))^s}, \frac{(1+2\hat k)\rho^s}{2^s(\hat kR+M(k,x))^s}, \frac{1}{2^s(\max{\left\{x, 1-x\right\}})^s}\right\}},$$ where the equality is attainable, and $M(k,x):=\max{\left\{kR+\rho-x, x-kR\right\}}$.
The proof is similar to (i). Take $x=\pi(d_1d_2\ldots)\in E$ and $r\in(0, \rho)$. Then there exists $n\ge 0$ such that $B(x, r)$ contains the level-$(n+1)$ basic interval $f_{d_1\ldots d_{n+1}}([0, 1])$ but it does not contain the basic interval $f_{d_1\ldots d_{n}}([0,1])$. This implies that $$(-1, 2)\supseteq T^{n}(B(x,r) )=(f_{d_1\ldots d_{n}})^{-1}(B(x,r) )\supseteq f_{d_{n+1}}([0,1]),$$ but $T^{n}(B(x,r) )$ does not contain $[0,1]$.
Let $y=T^{n}x=\pi(d_{n+1}d_{n+2}\ldots)$ and let $r'=\rho^{-n}r$. Then $y\in f_{d_{n+1}}([0, 1])$, $\rho/2<r'<1$ and $T^{n}(B(x,r))=B(y,r')$. By (\[eq:mar-4-1\]) it follows that $$\label{eq:mar-4-2}
\begin{split}
&\frac{\mu(B(x,r))}{(2r)^s}=\frac{\mu(B(y, r'))}{(2r')^s}\\
&\le \max{\left\{\frac{\rho^s}{2^s(M(d_{n+1}, y))^s}, \frac{(1+2\hat d_{n+1})\rho^s}{2^s(\hat d_{n+1}R+M(d_{n+1}, y))^s}, \frac{1}{2^s(\max{\left\{y, 1-y\right\}})^s}\right\}}\\
&=\max\left\{\frac{\rho^s}{2^s(M(d_{n+1}, T^n x))^s}, \frac{(1+2\hat d_{n+1})\rho^s}{2^s(\hat d_{n+1}R+M(d_{n+1}, T^n x))^s},\right.\\
&\hspace{8cm}\left. \frac{1}{2^s(\max{\left\{T^n x, 1-T^n x\right\}})^s}\right\}.
\end{split}$$ Observe that $
T^{n}x-d_{n+1} R= \rho T^{n+1}(x).
$ Then $$\begin{aligned}
M(d_{n+1}, T^n x)&=\max{\left\{ T^nx-d_{n+1}R, d_{n+1}R+\rho-T^n x\right\}}\\
&=\rho\max{\left\{T^{n+1} x, 1-T^{n+1} x\right\}}=\rho\eta_{n+1}(x).\end{aligned}$$ Substituting this in (\[eq:mar-4-2\]) gives that $$\frac{\mu(B(x,r))}{(2r)^s}\le \max{\left\{\frac{1}{2^s(\eta_{n+1}(x))^s}, \frac{ 1+2\hat d_{n+1} }{2^s(\hat d_{n+1}R/\rho+\eta_{n+1}(x))^s}, \frac{1}{2^s(\eta_n(x))^s}\right\}}.$$ Letting $r\to 0$ in the above equation, and then $n\to{\infty}$, we obtain that $$\label{eq:mar-4-3}
\begin{split}
\Theta^{*s}(\mu, x)&=\limsup_{r\to 0}\frac{\mu(B(x,r))}{(2r)^s}\\
&\le \limsup_{n\to{\infty}}\max{\left\{\frac{1}{2^s(\eta_{n+1}(x))^s}, \frac{ 1+2\hat d_{n+1} }{2^s(\hat d_{n+1}R/\rho+\eta_{n+1}(x))^s}, \frac{1}{2^s(\eta_n(x))^s}\right\}}\\
&= \max{\left\{\frac{1}{2^s(\liminf_{n\to{\infty}}\eta_n(x))^s}, \limsup_{n\to{\infty}}\frac{1+2\hat d_n}{2^s(\hat d_n R/\rho+\eta_n(x))^s}\right\}}.
\end{split}$$
Observe that the equality in (\[eq:mar-4-2\]) is attainable. This implies that the equality holds in (\[eq:mar-4-3\]), completing the proof.
Typical values of the densities
-------------------------------
In the following we will prove Theorem \[main:densities\] (iii) for the typical pointwise densities of $\mu$. First we need the following upper bound.
\[lem:upper-bound-density\] If $N=2m$ for some $m\ge 1$, then $$\Theta^{*s}(\mu, x)\le \frac{1}{(NR)^s}$$ for any $x\in E$.
Observe that for any $x\in E$, $$\eta_n(x)=\max{\left\{T^n x, 1-T^n x\right\}}\ge \min_{y\in E}\max{\left\{y, 1-y\right\}}= mR=\frac{N}{2}R.$$ So, by Theorem \[main:densities\] (ii) it suffices to prove $$\label{eq:apr-25-2}
\frac{1+2 j}{(2 j R/\rho+N R)^s }\le \frac{1}{(NR)^s}\quad \textrm{for any }1\le j\le m-1.$$ By the same way as in the proof of Lemma \[lem:inequality\] (ii) one can verify that the sequence $$b_j':=\frac{1+2 j}{(2 j R/\rho+N R)^s }, \quad j\ge 1$$ is strictly increasing. So, to prove (\[eq:apr-25-2\]) we only need to prove it for $j=m-1$, i.e., $$\frac{N-1}{((N-2)R/\rho+NR)^s}\le \frac{1}{(NR)^s}.$$
Note that $s=-\log N/\log \rho$. Rearranging the above inequality gives $$\label{eq:apr-25-3}
N-1\le \left(1+\frac{N-2}{N\rho}\right)^{-\frac{\log N}{\log \rho}}=:\phi(\rho).$$ Clearly, (\[eq:apr-25-3\]) holds for $N=2$. When $N\ge 3$, note that $\phi(1/N^2)=N-1$. So it suffices to prove that the function $\rho\mapsto \phi(\rho)$ is strictly decreasing on $(0, 1/N^2]$. Write $t:=1/\rho$. Then $\phi(\rho)$ is strictly decreasing if and only if $$\label{eq:apr-29-1}
\phi_1(t)=\frac{1}{\ln t}\ln\left(1+\frac{N-2}{N}t\right)\quad\textrm{ is strictly increasing on }[N^2, +{\infty}),$$ where ‘$\ln$’ is the logarithm with the natural base $e$. Taking the derivative of $\phi_1$ gives that $\phi_1'(t)>0$ if and only if $$\phi_2(t)=\frac{N-2}{N}t\ln t-\left(1+\frac{N-2}{N}t\right)\ln\left(1+\frac{N-2}{N}t\right)>0.$$ Since $t\ge N^2$, one can easily verify that $\phi_2$ has positive derivative on $[N^2, +{\infty})$. So, $$\phi_2(t)\ge \phi_2(N^2)=2\left(N(N-2)\ln N-(N-1)^2\ln(N-1)\right).$$ Therefore, to prove (\[eq:apr-29-1\]) it suffices to prove $$\frac{N(N-2)}{(N-1)^2}\ge \frac{\ln(N-1)}{\ln N}.$$ Using one minus both sides of the above inequality it follows that $$\frac{1}{(N-1)^2}\le \frac{\ln(1+\frac{1}{N-1})}{\ln N}=\frac{\ln(1+\frac{1}{N-1})^N}{N\ln N}.$$ Since $(1+\frac{1}{N-1})^N$ decreases to $e$ as $N\to{\infty}$, we have $\ln(1+\frac{1}{N-1})^N>1$, and thus it suffices to prove $$N\ln N\le (N-1)^2\quad\textrm{for all }N\ge 3.$$ This can be easily verified by simple calculation.
Therefore, we establish (\[eq:apr-25-3\]), and thus (\[eq:apr-25-2\]). This completes the proof.
First we consider the typical value for $\Theta_*^s(\mu, x)$. By (i) it suffices to prove $$\label{eq:mar-5-1}
\liminf_{n\to{\infty}}{\gamma}_n(x)=0\quad\textrm{for }\mu-\textrm{almost every }x\in E.$$ For $\ell\ge 1$ let $$\begin{aligned}
A_\ell&:=\bigcap_{k=0}^{\infty}{\left\{\pi(d_1d_2\ldots)\in E: d_{k\ell+1}\ldots d_{(k+1)\ell}\ne 0^\ell\right\}},\\
\end{aligned}$$ Then for any $x\in E\setminus\bigcup_{\ell=1}^{\infty}A_\ell$ its unique coding must contain arbitrarily long length of consecutive zeros. This implies that $
\liminf_{n\to{\infty}}{\gamma}_n(x)=0
$ for any $x\in E\setminus\bigcup_{\ell=1}^{\infty}A_\ell$.
Therefore, to prove (\[eq:mar-5-1\]) it suffices to prove that $\mu(A_\ell)=0$ for all $\ell\ge 1$. Take $\ell\ge 1$. Observe that $A_\ell$ is the self-similar set generated by the IFS $${\left\{f_{i_1\ldots i_\ell}:~ i_1\ldots i_\ell\in{\left\{0,1,\ldots, N-1\right\}}^\ell\textrm{ but }i_1\ldots i_\ell\ne 0^\ell\right\}},$$ which satisfy the open set condition. So, $$\dim_H A_\ell=\frac{\log(N^\ell-1)}{-\ell\log\rho}<\frac{\log N}{-\log \rho}=s=\dim_H E.$$ This implies that $\mathcal H^s(A_\ell)=0$, and thus $\mu(A_\ell)=\mathcal H^s(E\cap A_\ell)=0$. Hence, we prove (\[eq:mar-5-1\]).
Next we turn to the typical value of the upper density. First we assume $N=2m+1$ for some $m\ge 1$. Note by [@Falconer_1990 Theroem 5.1] that $\Theta^{*s}(\mu, x)\le 1$ for all $x\in E$. Then by Theorem \[main:densities\] (ii) it suffices to prove $$\label{eq:apr-25-1}
\liminf_{n\to{\infty}}\eta_n(x)=\frac{1}{2}\quad\textrm{for }\mu-\textrm{almost every }x\in E.$$ For $\ell\ge 1$ let $$B_\ell:=\bigcap_{k=0}^{\infty}{\left\{\pi(d_1d_2\ldots)\in E: d_{k\ell+1}\ldots d_{(k+1)\ell}\ne m^\ell \right\}}.$$ Then for any $x\in E\setminus\bigcup_{\ell\ge 1}B_\ell$ its unique coding must contain arbitrarily long length of consecutive $m$’s. Note that $\eta_n(x)\ge 1/2$ for any $x\in E$. Therefore, $
\liminf_{n\to{\infty}}\eta_n(x) =1/2
$ for any $x\in E\setminus\bigcup_{\ell\ge 1} B_\ell$. Furthermore, by a similar argument as above one can verify that $\mu(B_\ell)=0$ for all $\ell\ge 1$. This proves (\[eq:apr-25-1\]).
Finally, we consider the typical value of the upper density with $N=2m$ for some $m\ge 1$. Define $$B_\ell':=\bigcap_{k=0}^{\infty}{\left\{\pi(d_1d_2\ldots)\in E: d_{k\ell+1}\ldots d_{(k+1)\ell}=m0^{\ell-1}\right\}}$$ for any $\ell\ge 2$. So, for any $x\in E\setminus\bigcup_{\ell=2}^{\infty}B_\ell'$ its unique coding contains the block $m 0^\ell$ with $\ell$ arbitrarily large. Note that $\eta_n(x)\ge mR=NR/2$ for any $x\in E$. This implies $
\liminf_{n\to{\infty}}\eta_n(x)=NR/2
$ for any $x\in E\setminus\bigcup_{\ell=2}^{\infty}B_\ell'$. Hence, by Lemma \[lem:upper-bound-density\] and Theorem \[main:densities\] (ii) it follows that $$\Theta^{*s}(\mu, x)=\frac{1}{(NR)^s}\quad\forall x\in E\setminus\bigcup_{\ell=2}^{\infty}B_\ell'.$$ By the same argument as above we can show that $\mu(B_\ell')=0$ for any $\ell\ge 2$. This completes the proof.
As a corollary of Theorem \[main:densities\] (iii) we have the precise packing measure of $E$ ( see [@Feng-Hua-Wen-2000 Theorem 1.2] for a proof).
\[cor:packing-measure\] Let $N\ge 2$ and $0<\rho\le 1/N^2$. Then the $s$-dimensional packing measure of $E$ is given by $$\mathcal P^s(E)=\left(\frac{2R}{\rho}\right)^s.$$
Critical values for the densities {#sec:critical-values}
=================================
In this section we will determine the critical values of the sets $$E_*(a)={\left\{x\in E: \Theta_*^s(\mu, x)\ge a\right\}}\quad\textrm{and}\quad E^*(b)={\left\{x\in E: \Theta^{*s}(\mu, x)\le b\right\}}$$ respectively, and prove Theorem \[main:critical-values\]. Recall that the critical values for $E_*(a)$ and $E^*(b)$ are defined by $$a_c:=\sup{\left\{a: \dim_H E_*(a)>0\right\}}\quad\textrm{and}\quad b_c:=\inf{\left\{b: \dim_H E^*(b)>0\right\}}.$$ Then by (\[eq:bounds\]) it follows that $a_c<b_c$. Motivated by some recent works on critical values of unique beta expansions (cf. [@Glendinning_Sidorov_2001; @Kong_Li_Dekking_2010]) and open dynamical systems [@Kalle-Kong-Langeveld-Li-18], we show that the critical values $a_c$ and $b_c$ are related to the Thue-Morse type sequences $({\lambda}_i)$ and $(\theta_i)$ defined in Definition \[def:sequences\].
Critical value of $E_*(a)$
--------------------------
For $t\in[0, 1]$ let $${\alpha}(t)={\alpha}_1(t){\alpha}_2(t)\ldots:=\pi^{-1}(t'),$$ where $t'$ is the smallest element of $E$ no less than $t$. So, if $t\in E$, then ${\alpha}(t)$ is indeed the unique coding of $t$. Since $E$ is a Cantor set, its complement $[0, 1]\setminus E$ is a countable union of open intervals. The definition of ${\alpha}(t)$ implies that in each connected component of $[0, 1]\setminus E$ the map $t\mapsto {\alpha}(t)$ is constant. Observe by Theorem \[main:densities\] (i) that $\Theta_*^s(\mu, x)$ is uniquely determined by ${\gamma}(x):=\liminf_{n\to{\infty}}{\gamma}_n(x)$. So it suffices to consider the critical value of the set $$E_{\gamma}(t):={\left\{x\in E: {\gamma}(x)\ge t\right\}}.$$ To describe the set $E_{\gamma}(t)$ it is convenient to study the corresponding set in the coding (sequence) space. For this reason we first recall some terminology from the symbolic dynamics (cf. [@Lind_Marcus_1995]).
For a sequence $(c_i)=c_1c_2\ldots\in{\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ we mean an infinite string of digits. Similarly, for a word $\mathbf c=c_1\ldots c_n$ with $n\in{\ensuremath{\mathbb{N}}}$ we mean a finite string of digits with each digit $c_i$ from ${\left\{0,1,\ldots, N-1\right\}}$. For two words $\mathbf c$ and $\mathbf d$ we denote by $\mathbf c\mathbf d$ the new word which is the concatenation of them. Also, for $n\in{\ensuremath{\mathbb{N}}}$ we write for $\mathbf c^n=\mathbf c\cdots\mathbf c$ the $n$ times concatenation of $\mathbf c$, and write for $\mathbf c^{\infty}$ the periodic sequence with period block $\mathbf c$. In this paper we use the lexicographical ordering ‘$\prec, {\preccurlyeq}, \succ$’ and ’${\succcurlyeq}$’ between sequences and words in the usual way. For example, for two sequences $(c_i), (d_i)\in{\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ we say $(c_i)\prec (d_i)$ if $c_1<d_1$, or there exists $n\in{\ensuremath{\mathbb{N}}}$ such that $c_1\ldots c_n=d_1\ldots d_n$ and $c_{n+1}<d_{n+1}$. Also, we write $(c_i){\preccurlyeq}(d_i)$ if $(c_i)=(d_i)$ or $(c_i)\prec (d_i)$. For two words $\mathbf c$ and $\mathbf d$ not necessarily of the same length, we say $\mathbf c\prec \mathbf d$ if $\mathbf c 0^{\infty}\prec \mathbf d 0^{\infty}$. Recall that for a word $\mathbf c=c_1\ldots c_n$ its reflection is defined by $\overline{\mathbf c}=(N-1-c_1)\ldots (N-1-c_n)$. If $c_n<N-1$, then we write $\mathbf c^+=c_1\ldots c_{n-1}(c_n+1)$; and if $c_n>0$ then we write $\mathbf c^-=c_1\ldots c_{n-1}(c_n-1)$. So, $\overline{\mathbf c}, \mathbf c^+$ and $\mathbf c^-$ are all words with each digit from ${\left\{0,1,\ldots, N-1\right\}}$. Analogously, for a sequence $(c_i)\in{\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ we denote its *reflection* by $\overline{(c_i)}:=(N-1-c_1)(N-1-c_2)\ldots$.
Now we define the symbolic analogue of $E_{\gamma}(t)$. For $t\in[0,1]$ let $E_{\gamma}'(t)$ be the set of sequences $(d_i)\in{\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ satisfying $$\left\{
\begin{array}{lll}
d_{n+1}d_{n+2}\ldots{\succcurlyeq}{\alpha}(t)&\textrm{if}& d_n=0,\\
d_{n+1}d_{n+2}\ldots {\succcurlyeq}{\alpha}(t)\textrm{ or }d_{n+1}d_{n+2}\ldots {\preccurlyeq}\overline{{\alpha}(t)}&\textrm{if}& 1\le d_n\le N-2,\\
d_{n+1}d_{n+2}\ldots {\preccurlyeq}\overline{{\alpha}(t)}&\textrm{if}& d_n=N-1.
\end{array}\right.$$
In the following proposition we show that $E_{\gamma}(t)$ and $\pi(E_{\gamma}'(t))$ have the same Hausdorff dimension for each $t\notin E$.
\[prop:dimension\] Let $N\ge 2$ and $0<\rho\le 1/N^2$. If $t\in[0, 1]\setminus E$, then $$\dim_H E_{\gamma}(t)=\dim_H\pi(E_{\gamma}'(t)).$$
Let $t\in[0, 1]\setminus E$. Since $E$ is compact and ${\left\{0, 1\right\}}\subset E$, there exists ${\varepsilon}>0$ such that $[t-{\varepsilon}, t+{\varepsilon}]\cap E=\emptyset$. Take $x\in E_{\gamma}(t)$. Then ${\gamma}(x)=\liminf_{n\to{\infty}}{\gamma}_n(x)\ge t$. So there exists a large integer $M$ such that $$\label{eq:apr-8-1}
{\gamma}_n(x)\ge t-{\varepsilon}\quad\forall ~n\ge M.$$ Write $x=\pi(d_1d_2\ldots)$. We claim that $
d_{M+1}d_{M+2}\ldots \in E_{\gamma}'(t).
$ Take $n\ge M+1$. We will prove the claim by considering the following three cases.
Case I. If $d_n=0$, then by (\[eq:apr-8-1\]) it follows that $$\label{eq:apr-8-2}
\pi(d_{n+1}d_{n+2}\ldots )=T^n x={\gamma}_n(x)\ge t-{\varepsilon}.$$ Since $[t-{\varepsilon}, t+{\varepsilon}]\cap E=\emptyset$, by (\[eq:apr-8-2\]) and the definition of ${\alpha}(t)$ it follows that $$\label{eq:apr-28-1}
\pi(d_{n+1}d_{n+2}\ldots )\ge \pi({\alpha}(t-{\varepsilon}))= \pi({\alpha}(t)).$$ Observe that the projection map $\pi: {\left\{0,1\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}\to E$ is bijective and strictly increasing with respect to the lexicographical ordering in ${\left\{0,1,\ldots, N-1\right\}}^{\ensuremath{\mathbb{N}}}$. We then conclude from (\[eq:apr-28-1\]) that $$d_{n+1}d_{n+2}\ldots {\succcurlyeq}{\alpha}(t)$$ as desired.
Case II. If $d_n\in{\left\{1,\ldots, N-2\right\}}$, then again by (\[eq:apr-8-1\]) we obtain that $$\max{\left\{\pi(d_{n+1}d_{n+2}\ldots ), \pi(\overline{d_{n+1}d_{n+2}\ldots})\right\}}=\max{\left\{T^n x, 1-T^n x\right\}}={\gamma}_n(x)\ge t-{\varepsilon}.$$ By the same argument as in Case I we deduce that $$d_{n+1}d_{n+2}\ldots {\succcurlyeq}{\alpha}(t)\quad\textrm{or}\quad \overline{d_{n+1}d_{n+2}\ldots }{\succcurlyeq}{\alpha}(t).$$
Case III. If $d_n=N-1$, then by (\[eq:apr-8-1\]) we also have $$\pi(\overline{d_{n+1}d_{n+2}\ldots})=1-T^n x={\gamma}_n(x)\ge t-{\varepsilon}.$$ Using the same argument as in Case I we obtain that $\overline{d_{n+1}d_{n+2}\ldots}{\succcurlyeq}{\alpha}(t)$. This establishes the claim.
Therefore, by the claim it follows that $$E_{\gamma}(t)\subset\bigcup_{k=0}^{\infty}\bigcup_{\mathbf i\in{\left\{0,1,\ldots, N-1\right\}}^k}f_{\mathbf i}(\pi(E_{\gamma}'(t))),$$ which gives $\dim_H E_{\gamma}(t)\le \dim_H\pi(E_{\gamma}'(t))$.
On the other hand, take $x=\pi(d_1d_2\ldots )\in\pi(E_{\gamma}'(t))$. Then by the same argument as above we can deduce that $${\gamma}_n(x)\ge \pi({\alpha}(t))\ge t\quad\forall n\ge 0.$$ This implies that ${\gamma}(x)=\liminf_{n\to{\infty}}{\gamma}_n(x)\ge t$. So, $\pi(E_{\gamma}'(t))\subset E_{\gamma}(t)$, and thus $\dim_H E_{\gamma}(t)=\dim_H\pi(E_{\gamma}'(t))$.
Recall from Definition \[def:sequences\] that $({\lambda}_i)_{i=1}^{\infty}$ is the Thue-Morse type sequence satisfying $${\lambda}_1=N-1,\quad\textrm{and}\quad {\lambda}_{2^n+1}\ldots {\lambda}_{2^{n+1}}=\overline{{\lambda}_1\ldots {\lambda}_{2^n}}^+\quad\forall n\ge 0.$$ Then $({\lambda}_i)$ begins with $
(N-1)10(N-1)0(N-2)(N-1)10\ldots.
$
\[lem:prop-lambda\]
1. For any $n\in{\ensuremath{\mathbb{N}}}$ we have $$\label{eq:inequality-lambda}
\overline{{\lambda}_1\ldots {\lambda}_{2^n-i}}\prec {\lambda}_{i+1}\ldots {\lambda}_{2^n}{\preccurlyeq}{\lambda}_1\ldots{\lambda}_{2^n-i}\quad \forall ~ 0\le i<2^n.$$
2. If ${\lambda}_i\in{\left\{1,\ldots, N-2\right\}}$ for some $i\ge 1$, then ${\lambda}_{i+1}\in{\left\{0, N-1\right\}}$.
First we prove (i). We will prove (\[eq:inequality-lambda\]) by induction on $n$. Note that ${\lambda}_1{\lambda}_2=(N-1)1$. Since $N\ge 2$, it is clear that (\[eq:inequality-lambda\]) holds for $n=1$. Now suppose (\[eq:inequality-lambda\]) holds for some $n\ge 1$, and we will prove (\[eq:inequality-lambda\]) with $n$ replaced by $n+1$, i.e., $$\label{eq:apr-28-2}
\overline{{\lambda}_1\ldots{\lambda}_{2^{n+1}-i}}\prec {\lambda}_{i+1}\ldots {\lambda}_{2^{n+1}}{\preccurlyeq}{\lambda}_1\ldots {\lambda}_{2^{n+1}-i}\quad\forall ~0\le i<2^{n+1}.$$ Clearly, (\[eq:apr-28-2\]) holds for $i=0$ since ${\lambda}_1>\overline{{\lambda}_1}$. So it suffices to prove (\[eq:apr-28-2\]) for $0<i<2^{n+1}$. We consider the following two cases.
Case I. $0< i<2^n$. Then by the induction hypothesis it follows that $$\label{eq:apr-7-11}
\overline{{\lambda}_1\ldots {\lambda}_{2^n-i}}\prec {\lambda}_{i+1}\ldots {\lambda}_{2^n}{\preccurlyeq}{\lambda}_1\ldots{\lambda}_{2^n-i}
\quad\textrm{and}\quad
\overline{{\lambda}_1\ldots {\lambda}_i}\prec {\lambda}_{2^n-i+1}\ldots {\lambda}_{2^n}.$$ Since ${\lambda}_{2^n+1}\ldots {\lambda}_{2^n+i}=\overline{{\lambda}_1\ldots {\lambda}_i}$, we then obtain by (\[eq:apr-7-11\]) that $$\overline{{\lambda}_1\ldots{\lambda}_{2^n}}\prec {\lambda}_{i+1}\ldots {\lambda}_{2^n+i}\prec {\lambda}_1\ldots {\lambda}_{2^n}\quad \textrm{for } 1\le i< 2^n.$$ This proves (\[eq:apr-28-2\]) for $0< i<2^n$.
Case II. $2^n\le i<2^{n+1}$. Write $i':=i-2^n$. Then by Definition \[def:sequences\] it follows that $$\label{eq:apr-7-13}
{\lambda}_{i+1}\ldots {\lambda}_{2^{n+1}}=\overline{{\lambda}_{i'+1}\ldots {\lambda}_{2^n}}^+=\overline{{\lambda}_{i'+1}\ldots {\lambda}_{2^n}^-}.$$ Note that $0\le i'<2^n$. By the induction hypothesis it follows that $$\overline{{\lambda}_1\ldots {\lambda}_{2^n-i'}}{\preccurlyeq}{\lambda}_{i'+1}\ldots {\lambda}_{2^n}^-\prec {\lambda}_1\ldots {\lambda}_{2^n-i'}.$$ Taking the reflection on both sides, and then by (\[eq:apr-7-13\]) it follows that $$\overline{{\lambda}_1\ldots {\lambda}_{2^{n+1}-i}}\prec {\lambda}_{i+1}\ldots {\lambda}_{2^{n+1}}{\preccurlyeq}{\lambda}_1\ldots {\lambda}_{2^{n+1}-i}.$$ This proves (\[eq:apr-28-2\]) for $2^n\le i<2^{n+1}$.
Therefore, by Cases I and II we establish (\[eq:apr-28-2\]). This completes the proof of (i) by induction.
Now we turn to prove (ii). We will prove by induction on $n$ that $$\label{eq:apr-7-21}
{\lambda}_i\in{\left\{1,\ldots, N-2\right\}}\quad\textrm{for some }1\le i<2^n\quad\Longrightarrow\quad {\lambda}_{i+1}\in{\left\{0, N-1\right\}}.$$ Note that (\[eq:apr-7-21\]) is trivial for $n=1$, since ${\lambda}_1{\lambda}_2=(N-1)1$. Now suppose (\[eq:apr-7-21\]) holds for some $n\ge 1$, and we consider it for $n+1$. Suppose ${\lambda}_{i}\in{\left\{1,\ldots, N-2\right\}}$ for some $1\le i<2^{n+1}$. We consider the following four cases.
Case 1. If $1\le i<2^n$, then by the induction hypothesis it follows that ${\lambda}_{i+1}\in{\left\{0, N-1\right\}}$.
Case 2. If $i=2^n$, then by Definition \[def:sequences\] we have ${\lambda}_{i+1}=\overline{{\lambda}_1}=0$.
Case 3. If $2^n< i<2^{n+1}-1$, then write $i':=i-2^n$, and by the definition of $({\lambda}_i)$ it follows that ${\lambda}_{i'}=\overline{{\lambda}_i}\in{\left\{1,\ldots, N-2\right\}}$. So by the induction hypothesis we have ${\lambda}_{i'+1}\in{\left\{0, N-1\right\}}$. This implies ${\lambda}_{i+1}=\overline{{\lambda}_{i'+1}}\in{\left\{0, N-1\right\}}$.
Case 4. If $i=2^{n+1}-1$, then ${\lambda}_{2^{n+1}-1}=\overline{{\lambda}_{2^n-1}}\in{\left\{1,\ldots, N-2\right\}}$. This gives ${\lambda}_{2^n-1}\in{\left\{1,\ldots, N-2\right\}}$. Note that ${\lambda}_{2^n-1}=\overline{{\lambda}_{2^{n-1}-1}}$. So we also have ${\lambda}_{2^{n-1}-1}\in{\left\{1,\ldots, N-2\right\}}$. Proceeding this argument we can deduce that ${\lambda}_1\in{\left\{1,\ldots, N-2\right\}}$, leading to a contradiction with ${\lambda}_1=N-1$. So, ${\lambda}_{2^{n+1}-1}\notin{\left\{1,\ldots, N-2\right\}}$.
By Cases 1–4 we prove (\[eq:apr-7-21\]) with $n$ replaced by $n+1$. By induction this proves (ii).
Now we are ready to determine the critical values of $E_{\gamma}(t)$, and show that it is equal to $$t_{\gamma}:=\pi(\overline{{\lambda}_2{\lambda}_3\ldots})=1-R\sum_{i=1}^{\infty}{\lambda}_{i+1}\rho^{i-1}.$$
\[lem:lower-bound-gamma-2\] If $t<t_{\gamma}$, then $\dim_H E_{\gamma}(t)>0$.
Let $s_n=\pi(\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}0^{\infty})$. Then $s_n\nearrow t_{\gamma}$ as $n\to{\infty}$. Note that the set-valued map $t\mapsto E_{\gamma}(t)$ is non-increasing and $E$ is a Cantor set. So by Proposition \[prop:dimension\] it suffices to prove $\dim_H \pi(E_{\gamma}'(s_n))>0$ for all $n\ge 1$. Fix $n\ge 1$ and write $$\xi:=\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}\quad \textrm{and}\quad \zeta:=\overline{{\lambda}_{2^n+2}\ldots {\lambda}_{2^{n+1}+1}}.$$ We claim that the words $
\xi\zeta, \xi\overline{\xi}$ and $\zeta\overline{\xi}
$ are all admissible in $E_{\gamma}'(s_n)$.
First we show that $\xi\zeta$ is admissible in $E_{\gamma}'(s_n)$. Note that $$\label{eq:apr-9-1}
\xi\zeta=\overline{{\lambda}_2\ldots {\lambda}_{2^{n+1}+1}}=\overline{{\lambda}_2\ldots {\lambda}_{2^n}}{\lambda}_1\ldots {\lambda}_{2^n}^-{\lambda}_1=:c_1\ldots c_{2^{n+1}}.$$ We will show for all $1\le i\le 2^n$ that $$\label{eq:apr-4-2}
\left\{
\begin{array}{lll}
c_{i+1}\ldots c_{i+2^{n}}{\succcurlyeq}\overline{{\lambda}_2\ldots {\lambda}_{2^{n}+1}}&\textrm{if}& c_i=0,\\
c_{i+1}\ldots c_{i+2^{n}}{\succcurlyeq}\overline{{\lambda}_2\ldots {\lambda}_{2^{n}+1}}\textrm{ or }c_{i+1}\ldots c_{i+2^{n}}{\preccurlyeq}{\lambda}_2\ldots {\lambda}_{2^n+1}&\textrm{if}&1\le c_i\le N-2,\\
c_{i+1}\ldots c_{i+2^{n}}{\preccurlyeq}{\lambda}_2\ldots {\lambda}_{2^n+1}&\textrm{if}& c_i=N-1.
\end{array}\right.$$ Observe that if $i=2^n$, then by (\[eq:apr-9-1\]) we have $c_{2^n}={\lambda}_1=N-1$, and thus $$c_{i+1}\ldots c_{i+2^n}={\lambda}_2\ldots {\lambda}_{2^n}^-{\lambda}_1\prec {\lambda}_2\ldots{\lambda}_{2^n+1}$$ as desired. In the following it suffices to prove (\[eq:apr-4-2\]) for $1\le i<2^n$. We consider the following three cases.
Case I. $c_i=0$ for some $1\le i< 2^n$. Then by (\[eq:inequality-lambda\]) and (\[eq:apr-9-1\]) it follows that $$c_{i}\ldots c_{i+2^n-1}=\overline{{\lambda}_{i+1}\ldots {\lambda}_{2^n}}{\lambda}_1\ldots {\lambda}_i\succ \overline{{\lambda}_1\ldots {\lambda}_{2^n}},$$ which together with $\overline{{\lambda}_1}=0=c_i$ implies $$c_{i+1}\ldots c_{i+2^n-1}\succ \overline{{\lambda}_2\ldots {\lambda}_{2^n}}$$ as required.
Case II. $c_i=N-1$ for some $1\le i<2^n$. Note by (\[eq:inequality-lambda\]) and (\[eq:apr-9-1\]) that $c_{2^{n}-1}=\overline{{\lambda}_{2^n}}<N-1$. So we have $1\le i<2^n-1$. By (\[eq:inequality-lambda\]) and (\[eq:apr-9-1\]) it follows that $$c_i\ldots c_{2^n-1}=\overline{{\lambda}_{i+1}\ldots {\lambda}_{2^n}}\prec {\lambda}_1\ldots {\lambda}_{2^n-i}.$$ This, together with ${\lambda}_1=N-1=c_i$, implies $c_{i+1}\ldots c_{2^n-1}\prec {\lambda}_2\ldots {\lambda}_{2^n-i}$.
Case III. $1\le c_i\le N-2$ for some $1\le i< 2^n$. Then $N\ge 3$, and by Lemma \[lem:prop-lambda\] (ii) we have $c_{i+1}\in{\left\{0, N-1\right\}}$. Since $\overline{{\lambda}_2}=N-2$, it gives that either $c_{i+1}>\overline{{\lambda}_2}$ or $c_{i+1}<{\lambda}_2$.
By Cases I–III we establish (\[eq:apr-4-2\]). Similarly, note that $$\xi\overline{\xi}=\overline{{\lambda}_2\ldots {\lambda}_{2^{n}+1}}{\lambda}_2\ldots {\lambda}_{2^n+1}=\overline{{\lambda}_2\ldots {\lambda}_{2^n}}{\lambda}_1\ldots {\lambda}_{2^n+1}=:c_1'\ldots c_{2^{n+1}}'$$ and $$\zeta\overline{\xi}=\overline{{\lambda}_{2^n+2}\ldots {\lambda}_{2^{n+1}+1}}{\lambda}_2\ldots {\lambda}_{2^n+1}={\lambda}_2\ldots {\lambda}_{2^n}^-{\lambda}_1\ldots {\lambda}_{2^n+1}=:c_1''\ldots c_{2^{n+1}}''.$$ Then by using Lemma \[lem:prop-lambda\] and the same argument as above we can prove the analogous inequalities in (\[eq:apr-4-2\]) with $c_1\ldots c_{2^{n+1}}$ replaced by $c_1'\ldots c_{2^{n+1}}'$ and $c_1''\ldots c_{2^{n+1}}''$, respectively.
Therefore, by using ${\alpha}(s_n)=\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}0^{\infty}$ it follows that the words $\xi\zeta, \xi\overline{\xi}$ and $\zeta\overline{\xi}$ are all admissible in $E'_{\gamma}(s_n)$, proving the claim. Observe that the set $E_{\gamma}'(t)$ is symmetric, i.e., $(d_i)\in E_{\gamma}'(t)$ if and only if $\overline{(d_i)}\in E_{\gamma}'(t)$. By the claim it follows that the words $$\xi\zeta,\quad \overline{\xi\zeta},\quad\xi\overline{\xi},\quad\overline{\xi}\xi,\quad \zeta\overline{\xi}\quad\textrm{and}\quad \overline{\zeta}\xi$$ are all admissible in $E_{\gamma}'(s_n)$, and hence $E_{\gamma}'(s_n)$ contains the subshift of finite type $X_A$ over the states ${\left\{\xi, \zeta, \overline{\xi}, \overline{\zeta}\right\}}$ with the adjacency matrix $$A=\left(
\begin{array}{llll}
0&1&1&0\\
0&0&1&0\\
1&0&0&1\\
1&0&0&0
\end{array}\right).$$ This implies that $$\dim_H \pi(E'_{\gamma}(s_n))\ge \dim_H\pi(X_A)=\frac{\log r_A}{-\log \rho^{2^n}}=\frac{\log \frac{1+\sqrt{5}}{2}}{-2^n\log \rho}>0,$$ where $r_A=(1+\sqrt{5})/2$ is the spectral radius of $A$. This completes the proof.
\[lem:upper-bound-gamma-2\] If $t>t_{\gamma}$, then the set $E_{\gamma}(t)$ is countable.
Let $t_n=\pi(\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}(N-1)^{\infty})$. Then $t_n\searrow t_{\gamma}$ as $n\to{\infty}$. Note that the set-valued map $t\mapsto E_{\gamma}(t)$ is non-increasing, and $E_{\gamma}(t)$ is a countable union of scaling copies of $\pi(E_{\gamma}'(t))$. So it suffices to prove $E_{\gamma}'(t_n)$ is countable for all $n\ge 1$. Observe that for $n=1$ we have ${\alpha}(t_1)=(N-2)(N-1)^{\infty}$. One can easily verify that any sequence in $E_{\gamma}'(t_1)$ must end with $(0(N-1))^{\infty}$, and thus $E_{\gamma}'(t_1)$ is countable. Since $E_{\gamma}'(t_{n+1})\supseteq E_{\gamma}'(t_n)$ for any $n\ge 1$, in the following we only need to prove that $E_{\gamma}'(t_{n+1})\setminus E_{\gamma}'(t_n)$ is countable for all $n\ge 1$.
Fix $n\ge 1$. Note that ${\alpha}(t_n)$ and ${\alpha}(t_{n+1})$ both begin with $\overline{{\lambda}_2{\lambda}_3}=(N-2)(N-1)$. Then $$\label{eq:apr-28-3}
\begin{array}{lll}
\overline{{\alpha}(t_n)}\prec \overline{{\alpha}(t_{n+1})}\prec {\alpha}(t_{n+1})\prec {\alpha}(t_n)&\quad\textrm{if}& N\ge 3,\\
{\alpha}(t_{n+1})\prec {\alpha}(t_n)\prec \overline{{\alpha}(t_n)}\prec \overline{{\alpha}(t_{n+1})}&\quad\textrm{if}& N=2.
\end{array}$$ Take $(d_i)\in E_{\gamma}'(t_{n+1})\setminus E_{\gamma}'(t_n)$. Then by (\[eq:apr-28-3\]) and the definition of $E_{\gamma}'(t)$ there must exist $k\in{\ensuremath{\mathbb{N}}}$ such that $$\label{eq:apr-4-4}
d_k\le N-2\quad\textrm{and}\quad {\alpha}(t_{n+1}){\preccurlyeq}d_{k+1}d_{k+2}\ldots \prec {\alpha}(t_n)$$ or $$\label{eq:apr-4-5}
d_k\ge 1\quad\textrm{and}\quad \overline{{\alpha}(t_{n})}\prec d_{k+1}d_{k+2}\ldots {\preccurlyeq}\overline{{\alpha}(t_{n+1})}.$$ Write $v_n:={\lambda}_2\ldots {\lambda}_{2^n+1}$. We will show in each case that the sequence $(d_i)$ must end with $(v_n\overline{v_n})^{\infty}$.
Suppose (\[eq:apr-4-4\]) holds. Then $d_k\le N-2$, and by using ${\alpha}(t_n)=\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}(N-1)^{\infty}$ it follows that $$\label{eq:apr-4-6}
\begin{split}
&d_{k+1}d_{k+2}\ldots\prec \overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}(N-1)^{\infty}, \\
&d_{k+1}d_{k+2}\ldots{\succcurlyeq}\overline{{\lambda}_2\ldots {\lambda}_2^{n+1}+1} (N-1)^{\infty}= \overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}{\lambda}_2\ldots {\lambda}_{2^n}^- (N-1)^{\infty}.
\end{split}$$ This implies $d_{k+1}\ldots d_{k+2^n}=\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}=\overline{v_n}$. In particular, $d_{k+2^n}=\overline{{\lambda}_{2^n+1}}={\lambda}_1=N-1$. So, by using $(d_i)\in E_{\gamma}'(t_{n+1})$ it follows that $$\label{eq:apr-4-7}
d_{k+2^n+1}\ldots d_{k+2^{n+1}}{\preccurlyeq}{\lambda}_2\ldots {\lambda}_{2^n+1}.$$ This together with (\[eq:apr-4-6\]) gives $d_{k+2^n+1}\ldots d_{k+2^{n+1}-2}={\lambda}_2\ldots {\lambda}_{2^n-1}$. If $d_{k+2^{n+1}-1}={\lambda}_{2^n}^-$, then (\[eq:apr-4-6\]) implies that $(d_i)$ must end with $(N-1)^{\infty}$, leading to a contradiction with $(d_i)\in E_{\gamma}'(t_{n+1})$. So, $d_{k+2^{n+1}-1}={\lambda}_{2^n}.$ By (\[eq:apr-4-7\]) and using ${\lambda}_{2^n+1}=\overline{{\lambda}_1}=0$ it follows that $d_{k+2^n+1}\ldots d_{k+2^{n+1}}={\lambda}_2\ldots {\lambda}_{2^n+1}$. Hence, $$\label{eq:apr-4-8}
d_{k+1}\ldots d_{k+2^{n+1}}=\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}{\lambda}_2\ldots {\lambda}_{2^n+1}=\overline{v_n}v_n.$$
Now, since $d_{k+2^n}=\overline{{\lambda}_{2^{n}+1}}=N-1$ and $(d_i)\in E_{\gamma}'(t_{n+1})$, by (\[eq:apr-4-8\]) it follows that $$\label{eq:apr-4-9}
d_{k+2^{n+1}+1}\ldots d_{k+2^{n+1}+2^n}{\preccurlyeq}{\lambda}_{2^n+2}\ldots {\lambda}_{2^{n+1}+1}=\overline{{\lambda}_2\ldots {\lambda}_{2^n}}^+ 0.$$ Also, by (\[eq:apr-4-8\]) that $d_{k+2^{n+1}}={\lambda}_{2^n+1}=0$ it follows that $$\label{eq:apr-4-10}
d_{k+2^{n+1}+1}\ldots d_{k+2^{n+1}+2^n}{\succcurlyeq}\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}=\overline{{\lambda}_1\ldots {\lambda}_{2^n}}(N-1).$$ By (\[eq:apr-4-9\]) and (\[eq:apr-4-10\]) it follows that $$\textrm{either }d_{k+2^{n+1}+1}\ldots d_{k+2^{n+1}+2^n}=\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}\quad\textrm{or}\quad d_{k+2^{n+1}+1}\ldots d_{k+2^{n+1}+2^n}=\overline{{\lambda}_2\ldots {\lambda}_{2^n}}^+ 0.$$ While in the second case we have by (\[eq:apr-4-8\]) and (\[eq:apr-4-9\]) that $$d_{k+2^n}=N-1\quad\textrm{and}\quad d_{k+2^n+1}\ldots d_{k+2^{n+1}+2^n}={\lambda}_2\ldots {\lambda}_{2^{n+1}+1}.$$ Using $(d_i)\in E_{\gamma}'(t_{n+1})$ this implies $(d_i)$ must end with $0^{\infty}$, leading to a contradiction. So, we have $d_{k+2^{n+1}+1}\ldots d_{k+2^{n+1}+2^n}=\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}=\overline{v_n}$. Proceeding this reasoning we conclude that $(d_i)$ must end with $(\overline{v_n}v_n)^{\infty}$.
Symmetrically, if (\[eq:apr-4-5\]) holds, then one can also show that $(d_i)$ ends with $(\overline{v_n}v_n)^{\infty}$. This completes the proof.
\[Proof of Theroem \[main:critical-values\] (A)\] Observe by Theorem \[main:densities\] that $$E_{\gamma}(t)={\left\{x\in E: {\gamma}(x)\ge t\right\}}={\left\{x\in E: \Theta_*^s(\mu, x)\ge \frac{1}{2^s(R/\rho-t)^{s}}\right\}}=E_*\left(\frac{1}{2^s(R/\rho-t)^{s}}\right).$$ So it suffices to prove that $t_{\gamma}$ is the critical values of $E_{\gamma}(t)$. By Lemmas \[lem:lower-bound-gamma-2\] and \[lem:upper-bound-gamma-2\] we only need to consider $E_{\gamma}(t)$ for $t=t_{\gamma}$. By the proof of Lemma \[lem:upper-bound-gamma-2\] it follows that the set $E_{\gamma}'(t_{\gamma})$ contains all sequences of the form $$((N-1)0)^{k_0}(v_1\overline{v_1})^{k_1}\cdots(v_n\overline{v_n})^{k_n}\cdots,\quad k_n\in{\left\{0,1,2,\ldots\right\}}\cup{\left\{{\infty}\right\}},$$and their reflections, where $v_n=\overline{{\lambda}_2\ldots {\lambda}_{2^n+1}}$. This implies that $\pi(E'_{\gamma}(t_{\gamma}))$ is uncountable. Since $\pi(E'_{\gamma}(t_{\gamma}))\subset E_{\gamma}(t_{\gamma})$, this proves the uncountability of $E_{\gamma}(t_{\gamma})$.
Critical value of $E^*(b)$
--------------------------
Now we turn to describe the critical value of $E^*(b)$. Let $\eta(x):=\liminf_{n\to{\infty}}\eta_n(x)=\liminf_{n\to{\infty}}\max{\left\{T^n x, 1-T^n x\right\}}$. Define $$E_\eta(t):={\left\{x\in E: \eta(x)\ge t\right\}},$$ and the corresponding symbolic set $$E_\eta'(t):={\left\{(d_i): d_{n+1}d_{n+2}\ldots{\succcurlyeq}{\alpha}(t)\quad\textrm{or}\quad d_{n+1}d_{n+2}\ldots{\preccurlyeq}\overline{{\alpha}(t)}~\forall n\ge 0\right\}}.$$ We first determine the critical value of $E_\eta(t)$, and then use it to determine the critical value of $E^*(b)$.
\[prop:dimension-1\] For any $t\in[0, 1]\setminus E$ we have $$\dim_H E_\eta(t)=\dim_H\pi(E_\eta'(t)).$$
The proof is similar to Proposition \[prop:dimension\]. Let $t\in[0, 1]\setminus E$. Choose ${\varepsilon}>0$ such that $[t-{\varepsilon}, t+{\varepsilon}]\cap E=\emptyset$. Take $x=\pi(d_1d_2\ldots)\in E_\eta(t)$. Then $\eta(x)=\liminf_{n\to{\infty}}\max{\left\{T^n x, 1-T^n x\right\}}\ge t$. So there exists a large integer $M$ such that $$\max{\left\{\pi(d_{n+1}d_{n+2}\ldots), \pi(\overline{d_{n+1}d_{n+2}\ldots})\right\}}=\max{\left\{T^n x, 1-T^n x\right\}}\ge t-{\varepsilon}\quad \forall ~n\ge M.$$ This implies that $$\max{\left\{\pi(d_{n+1}d_{n+2}\ldots), \pi(\overline{d_{n+1}d_{n+2}\ldots})\right\}}\ge \pi({\alpha}(t-{\varepsilon}))=\pi({\alpha}(t))\quad\forall n\ge M.$$ Thus, $$d_{n+1}d_{n+2}\ldots{\succcurlyeq}{\alpha}(t)\quad\textrm{or}\quad d_{n+1}d_{n+2}\ldots{\preccurlyeq}\overline{{\alpha}(t)}\quad\forall n\ge M,$$ implying $d_{M+1}d_{M+2}\ldots \in E_\eta'(t)$. So $\dim_H E_\eta(t)\le \dim_H \pi(E_\eta'(t))$.
On the other hand, for any $x=\pi(d_1d_2\ldots)\in \pi(E_\eta'(t))$ one can verify that $$\max{\left\{T^n x, 1-T^n x\right\}}\ge \pi({\alpha}(t))\ge t\quad\forall ~n\ge 0.$$ This implies $\eta(x)\ge t$, and thus $x\in E_\eta(t)$. Hence, $\dim_H E_\eta(t)= \dim_H\pi(E_\eta'(t))$.
Recall from Definition \[def:sequences\] that the sequence $(\theta_i)_{i=1}^{\infty}\in{\left\{0, N-1\right\}}^{\ensuremath{\mathbb{N}}}$ satisfies $$\theta_1=N-1,\quad\textrm{and}\quad \theta_{2^n+1}\ldots \theta_{2^{n+1}}=\overline{\theta_{1}\ldots \theta_{2^n}}\quad\forall n\ge 0.$$ Then the sequence $(\theta_i)$ begins with $(N-1)00(N-1)\, 0(N-1)(N-1)0\ldots.$ We will show that the critical value of $E_\eta$ is given by $$t_\eta:=\pi((\theta_i))=R\sum_{i=1}^{\infty}\theta_i\rho^{i-1}.$$ First we need the following property of the sequence $(\theta_i)$.
\[lem:property-th\] For any $n\in{\ensuremath{\mathbb{N}}}$ we have $$\theta_2\ldots\theta_{2^n-i+1}{\preccurlyeq}\theta_{i+2}\ldots \theta_{2^n+1}\prec\overline{\theta_2\ldots \theta_{2^n-i+1}}\quad\forall ~ 0\le i<2^n.$$
Recall from [@Allouche_Shallit_1999] that the classical Thue-Morse sequence $(\tau_i)_{i=0}^{\infty}$ is defined as follows. Set $\tau_0=0$, and if $\tau_0\ldots \tau_{2^n-1}$ is defined for some $n\ge 0$, then we set $\tau_{2^n}\ldots \tau_{2^{n+1}-1}=(1-\tau_0)(1-\tau_1)\ldots (1-\tau_{2^n-1})$. Comparing with Definition \[def:sequences\] it follows that the sequence $(\theta_i)_{i=1}^{\infty}$ is a variation of $(\tau_i)_{i=0}^{\infty}$, i.e., $$\theta_i=(N-1)(1-\tau_{i-1})\quad \textrm{for all }i\ge 1.$$ Therefore, the lemma follows from the property of $(\tau_i)$ (cf. [@Komornik-Loreti-1998]) that for each $n\in{\ensuremath{\mathbb{N}}}$, $$(1-\tau_1)\ldots(1-\tau_{2^{n}-i}){\preccurlyeq}(1-\tau_{i+1})\ldots (1-\tau_{2^n})\prec \tau_1\ldots \tau_{2^n-i}\quad \forall ~0\le i<2^n.$$
Now we show that $t_\eta$ is indeed the critical value of $E_\eta(t)$.
\[lem:eta-lower-bound\] If $t<t_\eta$, then $\dim_H E_\eta(t)>0$.
Let $s_n:=\pi(\theta_1\ldots \theta_{2^n} 0^{\infty})$ with $n\in{\ensuremath{\mathbb{N}}}$. Then $s_n\nearrow t_\eta$ as $n\to{\infty}$. Since the map $t\mapsto E_\eta(t)$ is non-increasing, by Proposition \[prop:dimension-1\] it suffices to prove $\dim_H \pi(E'_\eta(s_n))>0$ for any $n\in{\ensuremath{\mathbb{N}}}$. We do this now by showing that $$\label{eq:mar-31-1}
{\left\{\theta_1\ldots \theta_{2^n}, \overline{\theta_1\ldots\theta_{2^n}}\right\}}^{\ensuremath{\mathbb{N}}}\subseteq E_\eta'(s_n)\quad\forall~ n\in{\ensuremath{\mathbb{N}}}.$$
Fix $n\in{\ensuremath{\mathbb{N}}}$. Note that ${\alpha}(s_n)=\theta_1\ldots\theta_{2^n} 0^{\infty}$ begins with digit $N-1$. By the definition of $E_\eta'(s_n)$ it follows that $E_\eta'(s_n)\subseteq{\left\{0, N-1\right\}}^{\ensuremath{\mathbb{N}}}$. Since $E_\eta'(s_n)$ is symmetric, to prove (\[eq:mar-31-1\]) it suffices to prove that the words $\theta_1\ldots \theta_{2^n}\theta_1\ldots \theta_{2^n}$ and $\theta_1\ldots \theta_{2^n}\overline{\theta_1\ldots \theta_{2^n}}$ are both admissible in $E_\eta'(s_n)$. In other words, we only need to prove for any $0\le i<2^n$ that $$\label{eq:mar-31-2}
\begin{split}
\theta_{i+1}\ldots \theta_{2^n}\theta_1\ldots \theta_i{\preccurlyeq}\overline{\theta_1\ldots \theta_{2^n}}\quad\textrm{or}\quad \theta_{i+1}\ldots \theta_{2^n}\theta_1\ldots \theta_i{\succcurlyeq}\theta_1\ldots \theta_{2^n},
\end{split}$$ and $$\label{eq:mar-31-3}
\begin{split}
\theta_{i+1}\ldots \theta_{2^n}\overline{\theta_1\ldots \theta_i}{\preccurlyeq}\overline{\theta_1\ldots \theta_{2^n}}\quad\textrm{or}\quad \theta_{i+1}\ldots \theta_{2^n}\overline{\theta_1\ldots \theta_i}{\succcurlyeq}\theta_1\ldots \theta_{2^n}.
\end{split}$$ Note that $\theta_{i+1}\in{\left\{0, N-1\right\}}$. We will first prove (\[eq:mar-31-2\]) by considering the following two cases.
Case I. If $\theta_{i+1}=0$, then by Lemma \[lem:property-th\] it follows that $$\label{eq:apr-14-1}
\theta_{i+2}\ldots \theta_{2^n+1}\prec \overline{\theta_2\ldots \theta_{2^n-i+1}}.$$ Note that $\theta_{i+1}=\overline{\theta_1}$. If $\theta_{i+2}\ldots \theta_{2^n}\prec \overline{\theta_2\ldots \theta_{2^n-i}}$, then $\theta_{i+1}\ldots \theta_{2^n}\prec \overline{\theta_1\ldots \theta_{2^n-i}}$, and we are done. Otherwise, suppose $\theta_{i+2}\ldots \theta_{2^n}=\overline{\theta_2\ldots\theta_{2^n-i}}$. Then by (\[eq:apr-14-1\]) and $(\theta_i)\in{\left\{0,N-1\right\}}^{\ensuremath{\mathbb{N}}}$ it follows that $\overline{\theta_{2^n-i+1}}=N-1=\theta_1$. Again by Lemma \[lem:property-th\] it follows that $\theta_2\ldots \theta_i{\preccurlyeq}\overline{\theta_{2^n-i+2}\ldots \theta_{2^n}}$. So, $$\theta_{i+1}\ldots\theta_{2^n}\theta_1\ldots\theta_i{\preccurlyeq}\overline{\theta_1\ldots \theta_{2^n}},$$ proving (\[eq:mar-31-2\]).
Case II. If $\theta_{i+1}=N-1$, then by Lemma \[lem:property-th\] and using $\theta_1=N-1>0=\theta_{2^n+1}$ it follows that $$\theta_{i+2}\ldots \theta_{2^n}\theta_1\succ \theta_{i+2}\ldots \theta_{2^n+1}{\succcurlyeq}\theta_2\ldots \theta_{2^n-i+1}.$$ Since $\theta_{i+1}=\theta_1$, this again proves (\[eq:mar-31-2\]).
Therefore, by Cases I and II we establish (\[eq:mar-31-2\]). Similarly, we can prove (\[eq:mar-31-3\]). If $\theta_{i+1}=0$, then by Lemma \[lem:property-th\] it follows that $$\theta_{i+2}\ldots \theta_{2^n}\overline{\theta_1}=\theta_{i+2}\ldots \theta_{2^n+1}\prec \overline{\theta_2\ldots \theta_{2^n-i+1}},$$ proving (\[eq:mar-31-3\]). If $\theta_{i+1}=N-1$, then by Lemma \[lem:property-th\] we can deduce that $$\theta_{i+1}\ldots \theta_{2^n}\overline{\theta_1}=\theta_{i+1}\ldots\theta_{2^n+1}{\succcurlyeq}\theta_2\ldots \theta_{2^n-i+1}\quad\textrm{and}\quad
\overline{\theta_2\ldots \theta_i}{\succcurlyeq}\theta_{2^n-i+2}\ldots \theta_{2^n}.$$ This again proves (\[eq:mar-31-3\]).
\[lem:eta-upper-bound\] If $t>t_\eta$, then the set $E_\eta(t)$ is at most countable.
Let $t_n=\pi((\theta_1\ldots \theta_{2^n})^{\infty})$. Note by Definition \[def:sequences\] that $$\theta_1\ldots \theta_{2^{n+1}}=\theta_1\ldots \theta_{2^n}\overline{\theta_1\ldots \theta_{2^n}}^+\prec (\theta_1\ldots \theta_{2^n})^2$$ for any $n\ge 0$. This implies that $t_n\searrow t_\eta$ as $n\to{\infty}$. Observe that the set-valued map $t\mapsto E'_\eta(t)$ is non-increasing, and by the proof of Proposition \[prop:dimension-1\] that $E_\eta(t)$ is a countable union of scaling copies of $\pi(E_\eta'(t))$. It suffices to prove that $E'_\eta(t_n)$ is countable for any $n\ge 0$. Clearly, for $n=0$ we have $t_0=\pi((N-1)^{\infty})$. Then one can easily verify that any sequence in $E_\eta'(t_0)$ must end with $0^{\infty}$ or $(N-1)^{\infty}$, and thus $E_\eta'(t_0)$ is countable. Furthermore, note that $E_\eta'(t_{n+1})\supseteq E_\eta'(t_n)$ for all $n\ge 0$. So, it suffices to prove that $E_\eta'(t_{n+1})\setminus E_\eta'(t_n)$ is countable for all $n\ge 0$.
Fix $n\ge 0$. Then $$\overline{{\alpha}(t_n)}\prec \overline{{\alpha}(t_{n+1})}\prec {\alpha}(t_{n+1})\prec {\alpha}(t_n).$$Take $(d_i)\in E_\eta'(t_{n+1})\setminus E_\eta'(t_n)$. By the definition of $E_\eta'(t)$ there must exist $k\ge 0$ such that $$\label{eq:apr-1-1}
(\overline{\theta_1\ldots \theta_{2^n}})^{\infty}=\overline{{\alpha}(t_n)}\prec d_{k+1}d_{k+2}\ldots {\preccurlyeq}\overline{{\alpha}(t_{n+1})}=(\overline{\theta_1\ldots \theta_{2^n}}\theta_1\ldots\theta_{2^n})^{\infty},$$ or $$\label{eq:apr-1-2}
(\theta_1\ldots \theta_{2^n}\overline{\theta_1\ldots \theta_{2^n}})^{\infty}={\alpha}(t_{n+1}){\preccurlyeq}d_{k+1}d_{k+2}\ldots \prec{\alpha}(t_n)=(\theta_1\ldots\theta_{2^n})^{\infty}.$$ We consider the following two cases.
Case I. If (\[eq:apr-1-1\]) holds, then there exists $m\in{\ensuremath{\mathbb{N}}}$ such that $$\label{eq:apr-1-3}
d_{k+1}\ldots d_{k+m 2^n}=(\overline{\theta_1\ldots \theta_{2^n}})^m\quad\textrm{and}\quad d_{k+m 2^n+1}\ldots d_{k+(m+1)2^n}\succ \overline{\theta_1\ldots \theta_{2^n}}.$$ This, together with $(d_i)\in E_\eta'(t_{n+1})$, implies that $$\label{eq:apr-1-4}
d_{k+m 2^n+1}d_{k+m 2^n+2}\ldots {\succcurlyeq}{\alpha}(t_{n+1})=(\theta_1\ldots \theta_{2^n}\overline{\theta_1\ldots \theta_{2^n}})^{\infty}.$$ Note by (\[eq:apr-1-3\]) that $d_{k+(m-1)2^n+1}\ldots d_{k+m 2^n}=\overline{\theta_1\ldots \theta_{2^n}}$. Then by using $(d_i)\in E_\eta'(t_{n+1})$ it gives that $$\label{eq:may-7-1}
d_{k+(m-1)2^n+1}d_{k+(m-1)2^n+2}\ldots{\preccurlyeq}\overline{{\alpha}(t_{n+1})}=(\overline{\theta_1\ldots\theta_{2^n}}\theta_1\ldots \theta_{2^n})^{\infty}.$$ Hence, by (\[eq:apr-1-3\])–(\[eq:may-7-1\]) we conclude that $$d_{k+1}d_{k+2}\ldots =(\overline{\theta_1\ldots\theta_{2^n}})^m(\theta_1\ldots \theta_{2^n}\overline{\theta_1\ldots \theta_{2^n}})^{\infty}.$$
Case II. If (\[eq:apr-1-2\]) holds, then by the same argument as in Case I one can show that $(d_i)$ ends with $(\theta_1\ldots\theta_{2^n}\overline{\theta_1\ldots \theta_{2^n}})^{\infty}$.
Therefore, by Cases I and II it follows that $E_{\eta}'(t_{n+1})\setminus E_{\eta}'(t_n)$ is countable for any $n\ge 0$.
Observe by Theorem \[main:densities\] (ii) that $$\label{eq:apr-26-1}
E_\eta(t)={\left\{x\in E: \eta(x)\ge t\right\}}\supseteq {\left\{x\in E: \Theta^{*s}(\mu, x)\le \frac{1}{(2t)^s}\right\}}=E^*\left(\frac{1}{(2t)^s}\right).$$ Recall that $t_\eta=\pi((\theta_i))$. So it suffices to prove that the critical value $b_c$ of $E^*(b)$ is given by $(2 t_\eta)^{-s}$. First, by Lemma \[lem:eta-upper-bound\] and (\[eq:apr-26-1\]) it follows that $E^*(b)$ is at most countable for any $b<(2t_\eta)^{-s}$. This implies $b_c\ge (2t_\eta)^{-s}$.
Next, by Lemma \[lem:eta-lower-bound\] it follows that for $b>(2t_\eta)^{-s}$ we have $b^{-1/s}/2<t_\eta$, and thus $$\label{eq:apr-26-2}
\Lambda_n:=\pi\left({\left\{\theta_1\ldots \theta_{2^n}, \overline{\theta_1\ldots \theta_{2^n}}\right\}}^{\ensuremath{\mathbb{N}}}\right)\subseteq E_\eta(b^{-1/s}/2)$$ for sufficiently large integer $n$. Furthermore, $\dim_H \Lambda_n>0$. Since each $x\in \Lambda_n$ has a unique coding in ${\left\{0, N-1\right\}}^{\ensuremath{\mathbb{N}}}$, by Theorem \[main:densities\] (ii) and (\[eq:apr-26-2\]) it follows that $$\Theta^{*s}(\mu, x)=\frac{1}{2^s\eta(x)^s}\le b\quad\forall~ x\in \Lambda_n.$$ This implies that $$\Lambda_n \subseteq{\left\{x\in E: \Theta^{*s}(\mu, x)\ge b\right\}}=E^*(b),$$ and thus, $\dim_H E^*(b)>0$ for any $b>(2t_\eta)^{-s}$. This proves $b_c=(2t_\eta)^{-s}$.
Finally we consider $E^*(b)$ for $b=b_c=(2 t_\eta)^{-s}$. By the proof of Lemma \[lem:eta-upper-bound\] one can show that $E_\eta'(t_\eta)$ contains all sequences of the form $$(\theta_1\overline{\theta_1})^{k_0}\cdots(\theta_1\ldots\theta_{2^n}\overline{\theta_1\ldots\theta_{2^n}})^{k_n}\cdots\quad \textrm{with}\quad k_n\in {\left\{0,1,\ldots\right\}}\cup{\left\{{\infty}\right\}},$$ and their reflections. Since these sequences are all in ${\left\{0, N-1\right\}}^{\ensuremath{\mathbb{N}}}$, by Theorem \[main:densities\] (ii) it follows that these sequences also belong to $\pi^{-1}(E^*(b_c))$. So, $E^*(b_c)$ is uncountable, completing the proof.
Acknowlegements {#acknowlegements .unnumbered}
===============
The first author was supported by NSFC No. 11971079 and the Fundamental and Frontier Research Project of Chongqing No. cstc2019jcyj-msxmX0338. The second author was supported by NSFC No. 11671147, 11571144 and Science and Technology Commission of Shanghai Municipality (STCSM) No. 13dz2260400.
[10]{}
J.-P. Allouche and J. Shallit. The ubiquitous [P]{}rouhet-[T]{}hue-[M]{}orse sequence. In [*Sequences and their applications ([S]{}ingapore, 1998)*]{}, Springer Ser. Discrete Math. Theor. Comput. Sci., pages 1–16. Springer, London, 1999.
J.-P. Allouche and J. Shallit. . Cambridge University Press, Cambridge, 2003. Theory, applications, generalizations.
E. Ayer and R. S. Strichartz. Exact [H]{}ausdorff measure and intervals of maximum density for [C]{}antor sets. , 351(9):3725–3741, 1999.
T. Bedford and A. M. Fisher. Analogues of the [L]{}ebesgue density theorem for fractal sets of reals and integers. , 64(1):95–124, 1992.
M. Dai and L. Tang. The pointwise densities of 3-part [C]{}antor measure. , 35(17):2000–2016, 2012.
K. Falconer. . John Wiley & Sons Ltd., Chichester, 1990. Mathematical foundations and applications.
K. J. Falconer. Wavelet transforms and order-two densities of fractals. , 67(3-4):781–793, 1992.
D.-J. Feng, S. Hua, and Z.-Y. Wen. The pointwise densities of the [C]{}antor measure. , 250(2):692–705, 2000.
P. Glendinning and N. Sidorov. Unique representations of real numbers in non-integer bases. , 8:535–543, 2001.
J. E. Hutchinson. Fractals and self-similarity. , 30(5):713–747, 1981.
C. Kalle, D. Kong, N. Langeveld, and W. Li. The $\beta$-transformation with a hole at 0.
V. Komornik and P. Loreti. Unique developments in non-integer bases. , 105(7):636–639, 1998.
D. Kong, W. Li, and F. M. Dekking. Intersections of homogeneous [C]{}antor sets and beta-expansions. , 23(11):2815–2834, 2010.
W. Li and Y. Yao. The pointwise densities of non-symmetric [C]{}antor sets. , 19(9):1121–1135, 2008.
D. Lind and B. Marcus. . Cambridge University Press, Cambridge, 1995.
P. Mattila. , volume 44 of [*Cambridge Studies in Advanced Mathematics*]{}. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability.
P. Mörters and D. Preiss. Tangent measure distributions of fractal measures. , 312(1):53–93, 1998.
L. Olsen. Density theorems for [H]{}ausdorff and packing measures of self-similar sets. , 75(3):208–225, 2008.
R. S. Strichartz, A. Taylor, and T. Zhang. Densities of self-similar measures on the line. , 4(2):101–128, 1995.
J. Wang, M. Wu, and Y. Xiong. On the pointwise densities of the [C]{}antor measure. , 379(2):637–648, 2011.
Z. Zhou. The [H]{}ausdorff measure of the self-similar sets—the [K]{}och curve. , 41(7):723–728, 1998.
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abstract: 'Motivated by recent experiments on low-dimensional frustrated quantum magnets with competing nearest-neighbor exchange coupling $J_1$ and next nearest-neighbor exchange coupling $J_2$ we investigate the magnetic susceptibility of two-dimensional $J_1$-$J_2$ Heisenberg models with arbitrary spin quantum number $s$. We use exact diagonalization and high-temperature expansion up to order 10 to analyze the influence of the frustration strength $J_2$/$J_1$ and the spin quantum number $s$ on the position and the height of the maximum of the susceptibility. The derived theoretical data can be used to get information on the ratio $J_2$/$J_1$ by comparing with susceptibility measurements on corresponding magnetic compounds.'
address: |
$^1$Institut für Theoretische Physik, Otto-von-Guericke-Universität Magdeburg,\
PF 4120, D - 39016 Magdeburg, Germany\
$^2$Universität Osnabrück, Fachbereich Physik, Barbarastr. 7, D - 49069 Osnabrück, Germany\
$^3$ Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011
author:
- 'J. Richter$^1$, A. Lohmann$^1$, H.-J. Schmidt$^2$, D.C. Johnston$^3$'
title: 'Magnetic susceptibility of frustrated spin-$s$ $J_1$-$J_2$ quantum Heisenberg magnets: High-temperature expansion and exact diagonalization data '
---
Introduction
============
The investigation of frustrated magnetic systems is currently a field of active theoretical and experimental research [@1; @1a]. Systems with competing nearest-neighbor (NN) exchange coupling $J_1$ and next nearest-neighbor (NNN) exchange coupling $J_2$ can serve as model systems to study the interplay of quantum effects, thermal fluctuations and frustration. The quantum $J_1$-$J_2$ Heisenberg models on the square-lattice exhibit several ground-state phases including non-classical non-magnetic ground states, see, e.g., [@2]. The corresponding Hamiltonian reads $$\label{hamiltonian}
H=J_1\sum_{\langle i,j\rangle} {\bf S}_i \cdot {\bf S}_j+J_2\sum_{[i,j]} {\bf
S}_i\cdot {\bf S}_j ,$$ where $({\bf S}_i)^2=s(s+1)$, and $\langle i,j\rangle$ denotes NN and $[i,j]$ denotes NNN bonds. For antiferromagnetic NNN bonds, $J_2>0$, the spin system is frustrated irrespective of the sign of $J_1$. Due to frustration the theoretical treatment of this model is challenging.
The numerous theoretical studies of the ground state phase diagram so far did not lead to a consensus on the nature of the quantum ground state and on the nature of the quantum phase transitions present in the model, see, e.g., [@3] and references therein. Interestingly there are also various compounds well described by square-lattice $J_1$-$J_2$ Heisenberg models, such as oxovanadates [@4] and iron pnictides [@5]. In experiments, typically temperature-dependent quantities are reported. Hence reliable (and flexible) tools are desirable to calculate thermodynamic quantities such as the uniform magnetic susceptibility $\chi$. In this paper we present two methods, namely the full exact diagonalization, see, e.g., [@6], and the high-temperature expansion [@7; @8; @9] to calculate the temperature dependence of the magnetic susceptibility for the square-lattice $J_1$-$J_2$ spin-$s$ Heisenberg model with both ferromagnetic (FM) and antiferromagnetic (AFM) NN coupling $J_1$ and AFM NNN bonds $J_2$ for arbitrary spin quantum number $s$. In particular, we analyze the position and the height of the maximum in the susceptibility in dependence on $J_1$, $J_2$ and $s$.
![\[fig1\] Uniform susceptibility $\chi$ as a function of renormalized temperature $T/s(s+1)$ for NNN exchange $J_2=1$ and three values of the spin quantum number $s=1/2, 1$, and $7/2$. (a) Numerical exact data for a finite square lattice of $N=8$ sites and antiferromagnetic $J_1=1$. (b) Numerical exact data for a finite square lattice of $N=8$ sites and ferromagnetic $J_1=-1$. (c) \[6,4\] Padé approximant of the 10th order HTE series for an infinite square lattice and ferromagnetic $J_1=-1$. (d) \[6,4\] Padé approximant of the 10th order HTE series for an infinite square lattice and antiferromagnetic $J_1=1$. The order of labeling in each legend is the same as the order of the plots (top to bottom). ](chi_T.eps){width="37pc"}
Methods
=======
The full exact diagonaliazion (ED) yields numerical exact results at arbitrary temperature $T$, but it is typically limited to about $N=22$ sites for $s=1/2$ models. For larger spin quantum numbers $s$ the system size $N$ accessible for ED shrinks significantly. Hence, ED is used preferably for $s=1/2$ and $s=1$. In the present study we exploit the special symmetry properties of the finite square-lattice of $N=8$ sites and perform full ED for the $J_1$-$J_2$ model for $s=1/2,1,..,9/2$, thus allowing to study the role of the spin quantum number. Since the ED approach suffers from the finite-size effect, the ED calculations do not yield quantitatively correct results for the thermodynamic limit. Nevertheless, they will give insight into the qualitative behavior of the susceptibility. The high-temperature expansion (HTE) for the $J_1$-$J_2$ model up to 10th order was presented in [@8], however, restricted to $s=1/2$. This restriction can be overcome by using our general HTE scheme for Heisenberg models with arbitrary exchange patterns and arbitrary spin quantum number $s$ up to order 8 [@9]. The scheme is encoded in a simple C++-program and can be downloaded [@10] and freely used by interested researchers.
Very recently the present authors have extended this general HTE scheme up to 10th order [@11]. Here we use this 10th order HTE as an alternative method to the ED. We use here three different subsequent Padé approximants, namely Padé \[4,6\], \[5,5\], and \[6,4\], see e.g. [@7; @9]. Such a Padé approximant extends the region of validity of the HTE series down to lower temperatures. Since the HTE approach is designed for infinite systems the HTE data for the susceptibility maximum, in principle, can be quantitatively correct, if the maximum is not located at too low temperatures. Indeed, it was found [@9] that for the unfrustrated ($J_2=0$) square-lattice spin-$1/2$ Heisenberg antiferromagnet the Padé \[4,4\] approximant of the 8th order HTE series yields correct data for the susceptibility maximum located at $T \approx 0.94 J_1$. However, it may happen that a certain Padé approximant does not work for some particular values of $J_1$, $J_2$, and $s$, since Padé approximants may exhibit unphysical poles for temperatures in the region of interest. Hence we show in the next section only those Padé data not influenced by poles.
![\[fig2\]Position $T_{max}$ (a and c) and height $\chi_{max}$ (b and d) of $\chi(T)$ for the finite $N=8$ square-lattice $J_1$-$J_2$ model (left panels FM $J_1=-1$, right panels AFM $J_1=+1$).](FM_AFM_N8.eps){width="40pc"}
![\[fig3\] Position $T_{max}$ (a and c) and height $\chi_{max}$ (b and d) of $\chi(T)$ for an infinite square-lattice $J_1$-$J_2$ model obtained by 10th order HTE (left panels FM $J_1=-1$, right panels AFM $J_1=+1$). For comparison we show the ED data for $N=8$.](FM_AFM_HTE.eps){width="40pc"}
![\[fig4\]Height $\chi_{max}$ (a and c) and position $T_{max}$ (b and d) of $\chi(T)$ for the square-lattice $J_1$-$J_2$ model for various sets of parameters $J_1$ and $J_2$ as a function of the inverse spin quantum number $s$ obtained from Padé approximants of 10th order HTE series of an infinite system (a and b) and from ED for $N=8$ (c and d).](chimax_vs_s_HTE_and_ED.eps){width="38pc"}
Results
=======
First we present the temperature dependence of the susceptibility $\chi$ in Fig. \[fig1\] for a particular value of $J_2$ and both FM and AFM $J_1$. In this paper the symbol $\chi$ means $\chi|J_1|/Ng^2\mu_B^2$, where $N$ is the number of spins and $\mu_B$ is the Bohr magneton. The temperature is measured in terms of $|J_1|$, i.e. the symbol $T$ means $T/|J_1|$. The qualitative behavior of $\chi(T)$ shown in Figs. \[fig1\](a-d) is similar, there is the broad maximum in $\chi(T)$ that is typical for a two-dimensional antiferromagnet (note that for $J_2/|J_1|=1$ the system is in the AFM ground state irrespective of the sign of $J_1$). The various $\chi(T)$ curves give an impression on the finite-size effects, the effect of the sign of the NN exchange $J_1$, and the influence of spin quantum number $s$. The height, $\chi_{max}$, and the position, $T_{max}$, of the maximum in the $\chi(T)$ curve are interesting features for the comparison with experimental data, in particular to get information on the ratio $J_2/|J_1|$ from susceptibility measurements, see e.g. [@13]. Therefore we will discuss $\chi_{max}$ and the $T_{max}$ now in more detail.
We present our data for the susceptibility maximum for both FM and AFM NN exchange $J_1$ in Figs. \[fig2\] (ED data) and \[fig3\] (HTE and ED data). For FM $J_1$, $\chi_{max}$ ($T_{max}$) becomes larger (smaller) upon lowering $J_2$. Finally, when approaching the critical value $J_2^c$, where the transition to the ferromagnetic ground state takes place, $\chi_{max}$ diverges and $T_{max}$ goes to zero. The critical point for $s=1/2$ is $J^c_2=0.333\,|J_1|$ for $N=8$ (but it is $J^c_2 \approx 0.4|J_1|$ for $N \to \infty$ [@12]). It increases with growing $s$ and becomes $J^c_2=0.5
|J_1|$ for $s \to \infty$. The data for $N=8$ and $N \to \infty$ are in qualitative agreement. Although the finite-size effects are obviously large, the general features of $\chi_{max}$ and $T_{max}$ as functions of $J_2$ and $s$ are quite similar. Naturally the HTE fails when approaching $J^c_2$, since in this limit low temperatures become relevant. Note that the HTE data for FM $J_1$ and $s=1/2$ are also in qualitative agreement with recently reported data calculated by second-order Green’s function approach [@13]. We discuss now the case of AFM $J_1$ (right panels in Figs. \[fig2\] and \[fig3\]). For large $J_2$ the behavior of $\chi_{max}$ and $T_{max}$ is very similar to that for FM $J_1$, i. e. the sign of $J_1$ becomes irrelevant, cf. Ref. [@12]. On the other hand, for smaller values of $J_2$ naturally both cases behave completely different, since $J_1$ dominates the physics. We find a well pronounced minimum in $T_{max}$ in the region of strongest frustration around $J_2=0.5$. For the finite system $\chi_{max}$ exhibits a maximum in this region, whereas for the infinite system $\chi_{max}$ is almost constant in the region $0 \le J_2 \le 0.5$.
To take a closer look on the role of the spin quantum number $s$ we present in Fig. \[fig4\] the quantities $\chi_{max}$ and $T_{max}$ as a function of $1/s$ for particular values of $J_2$. Obviously, there is monotonous increase (decrease) of $\chi_{max}$ ($T_{max}/s(s+1)$) with growing $s$. For FM $J_1=-1$ the increase of $\chi_{max}$ is particular strong for $J_2=0.7$ (see the insets in panels a and c), since for large $s$ this value of $J_2$ becomes quite close to the transition point to the FM ground state. From Figs. \[fig4\](a-d) it is also seen that the position $T_{max}$ of the maximum for $J_2 \gtrsim 0.7|J_1|$ is almost independent of the sign of $J_1$, whereas the height $\chi_{max}$ strongly depends on the sign of the NN coupling. Let us finally mention the special $s$-dependence of the maximum in $\chi(T)$ for $J_1=1$ and $J_2=0.5$, where the classical ground state exhibits a large non-trivial degeneracy. The position $T_{max}/s(s+1)$ of the maximum shifts to zero in the limit $s
\to \infty$, whereas the height remains finite. This behavior is quite similar to that found for the pyrochlore AFM [@9; @moessner99; @huber01], where the classical ground state is also highly degenerate.
Summary {#secIV}
=======
Using high-temperature expansion and full exact diagonalization we have calculated the uniform susceptibility $\chi$ of the spin-$s$ $J_1$-$J_2$ square-lattice Heisenberg magnet in a wide parameter regime of FM and AFM $J_1$ and frustrating AFM $J_2$. Especially, we have studied the height and the position of the maximum in the $\chi(T)$ curve as functions of $J_2/J_1$ and the spin quantum number $s$. These data can be used to get information on the ratio $J_2/|J_1|$ from susceptibility measurements, e.g. on oxovanadates which are well described by the square-lattice $J_1$-$J_2$ model.
The work at Ames Laboratory was supported by the U.S. Department of Energy under Contract No. DE-AC02-07CH11358.
References {#references .unnumbered}
==========
[10]{}
2004 ed U. Schollwöck, J. Richter, D.J.J. Farnell, and R.F. Bishop, Lecture Notes in Physics [**645**]{} (Berlin: Springer) 2005 ed H. T. Diep (Singapore: World Scientific) D. Schmalfu[ß]{}, R. Darradi, J. Richter, J. Schulenburg, and D. Ihle 2006 Phys. Rev. Lett. [**97**]{} 157201; H.-C. Jiang, H. Yao, and L. Balents 2012 Phys. Rev. B [**86**]{} 024424 J. Sirker, Z. Weihong, O. P. Sushkov, and J. Oitmaa 2006 Phys. Rev. B [**73**]{} 184420; R. Darradi, O. Derzhko, R. Zinke, J. Schulenburg, S.E. Krüger, and J. Richter 2008 Phys. Rev. B [**78**]{} 214415; J. Richter and J. Schulenburg 2010 Eur. Phys. J. B [**73**]{} 117; L. Wang, D. Poilblanc, Z.-C. Gu, X.-G. Wen, and F. Verstraete 2013 Phys. Rev. Lett. [**111**]{} 037202; W.J. Hu, F. Becca, A. Parola, and S. Sorella 2013 Phys. Rev. B [**88**]{} 060402; Y.Z. Ren, N.H. Tong, X.C. Xie 2013 preprint arXiv:1308.2850
R. Nath, A.A. Tsirlin, H. Rosner, and C. Geibel 2008 Phys. Rev. B [**78**]{} 064422; L. Bossoni, P. Carretta, R. Nath, M. Moscardini, M. Baenitz, and C. Geibel 2011 Phys. Rev B [**83**]{} 014412; B. Roy, Y. Furukawa, R. Nath, and D. C Johnston 2011 J. Phys. Conf. Series [**320**]{} 012048
Q. Si and E. Abrahams 2008 Phys. Rev. Lett. [**101**]{} 076401; D. C. Johnston 2010 Adv. Phys. [**59**]{} 803 M. Härtel, J. Richter, D. Ihle, and S.-L. Drechsler 2008 Phys. Rev. B [**78**]{} 174412. J. Oitmaa, C.J. Hamer, and W.H. Zheng 2006 [*Series Expansion Methods*]{} (Cambridge: Cambridge University Press) H. Rosner, R.R.P. Singh, W.H. Zheng, J. Oitmaa, and W.E. Pickett 2003 Phys. Rev. B [**67**]{} 014416 H.-J. Schmidt, A. Lohmann, and J. Richter 2011 Phys. Rev. B [**84**]{} 104443 see http://www.uni-magdeburg.de/jschulen/HTE/ A. Lohmann, H.-J. Schmidt, and J. Richter 2014 Phys. Rev. B [**89**]{} 014415 N. Shannon, T. Momoi, and P. Sindzingre 2006 Phys. Rev. Lett. [**96**]{} 027213; J. Richter, R. Darradi, J. Schulenburg, D.J.J. Farnell, and H. Rosner 2010 Phys. Rev. B [**81**]{} 174429 M. Härtel, J. Richter, O. Götze, D. Ihle, and S.-L. Drechsler 2013 Phys. Rev. B [**87**]{} 054412 R. Moessner and A.J. Berlinsky 1999 Phys. Rev. Lett. [**83**]{} 3293 A.J. Garcia-Adeva and D.L. Huber 2001 Phys. Rev. B [**63**]{} 140404
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Introduction
============
In this paper we study the observational consequences of the presence of an approximate Galilean symmetry [@Nicolis:2008in] in the theory of inflation, or more precisely in the theory describing the perturbations around an inflating solution. While it is always important to study all the possible symmetries that can constrain the dynamics of a physical system, this is particularly true for single field inflation since, given its simplicity, there are not many symmetries one can impose. In a recent paper [@Burrage:2010cu] (see also [@Kobayashi:2010cm; @Mizuno:2010ag]) Burrage [*et al.*]{} studied inflaton models endowed with the Galilean symmetry + b\_x\^+ c , with $b_\mu$ and $c$ constant. They restricted their analysis to Galilean operators of the schematic form $(\partial\phi)^2(\partial^2\phi)^n$, $n \leq 3$, which are the ones that give second order equations of motion of the form $(\partial^2\phi)^{n+1}$ [@Nicolis:2008in] . In this way perturbations remain well behaved, even when these operators are important for the background solution, which is the most interesting regime, quite different from standard slow-roll. Interestingly this model can give rise to large non-Gaussianities and, even more interestingly, the cubic operators which generate them contain four derivatives, so that they are naively distinguishable from the ones generated in more standard models, where large non-Gaussianities are linked to a low speed of sound and the cubic operators have three derivatives [@Alishahiha:2004eh; @Chen:2006nt; @Cheung:2007st]. Unfortunately, all the operators with four derivatives can be rewritten, as we will explain below, in terms of the ones with three using the linear equations of motion; this implies that this class of models does not give rise to a distinctive shape of non-Gaussianity. This fact motivates us to look beyond the Galilean operators studied in [@Burrage:2010cu], in particular at the ones with two derivatives on each field. We will see that this is consistent – Galilean operators with less derivatives are not radiatively generated – and it gives new shapes for non-Gaussianity, with a maximum of six derivatives.
At first the idea of studying inflationary models where operators of the form $(\partial^2\phi)^n$ are important seems silly. First of all there is an infinite number of operators of this form, with all the possible contractions of indices. Moreover when these operators become important for the unperturbed solution – which is mandatory if we want them to be relevant for the dynamics of the perturbations – the equations of motion are of higher order and describe also ghost excitations. In other words the solution we are looking seems outside the regime of validity of the effective field theory (EFT). However this comes from a wrong approach to the problem. What matters is the existence of a good EFT describing fluctuations around a quasi de-Sitter background: this is what we call inflation and this is what is relevant for observations. This is the approach advocated in [@Cheung:2007st] (see also [@Creminelli:2006xe]) and we will review it in this context in Section \[sec:EFT\]. The construction of the theory and its radiative stability are discussed in Section \[sec:action\] and Appendix \[app:NR\]. The study of the three-point function is done in Sections \[sec:perturbations\] and \[sec:shapes\], while Section \[sec:NG4\] is devoted to the four-point function. Conclusions are drawn in Section \[sec:conclusions\].
\[sec:EFT\]The (Galilean) effective theory of inflation
=======================================================
Inflationary observables – the $n$-point functions of the conserved curvature perturbation on comoving slices, which we call $\zeta$ ([^1]) – are calculated within the theory describing small fluctuations around a quasi de Sitter background. As in any effective theory, we need the cut-off $\Lambda$ to be much larger than the typical energy scale of the problem, i.e. $H$, which is the scale of quantum fluctuations induced by the cosmological background. This is all we need and what we call effective approach to inflation [@Cheung:2007st; @Creminelli:2006xe].
However one may be more ambitious and require the validity of the theory to be much broader than what is needed to reproduce cosmological observations. Although we experimentally probe small perturbations (t+(t, x)),H = -\~10\^[-5]{}, one may wonder whether the regime of validity of the theory encompasses very different backgrounds: for example whether it can describe a solution that starts with zero velocity $\dot\phi =0 $ or that, after the end of inflation, oscillates around a local minimum. In other words whether the theory makes sense also for $\pi \sim t$. This is often implicitly assumed in the standard approach to inflation, where one starts from a Lagrangian, finds a classical inflating solution and then studies small perturbations around it. We stress that, although it is nice to have a single EFT describing quite different classical solutions, that can be used not only to derive the inflationary observables but also, for instance, to address the issue of initial conditions and reheating, this is not required and it may represent an unjustified restriction on the theory of inflation. For large values of $\pi$ the EFT may cease to make sense and the appropriate description may be in terms of completely different degrees of freedom.
Usually the issue is not appreciated as in the simplest model of inflation, obtained expanding around a classical slow-roll solution of a Lagrangian for a single scalar with minimal kinetic term and potential, one can indeed consistently describe very different backgrounds, assuming a sufficiently heavy UV completion. Things are a bit subtler for inflationary theories with a small speed of sound for perturbations. Although they can be described in the effective approach [@Cheung:2007st] in a simple way, these theories can be also explicitly derived from a Lagrangian of the form $P(X,\phi)$, $X \equiv - (\partial\phi)^2$. If $P$ is given, one can trust solutions quite different from the inflating one, at least as long as the kinetic term of perturbations remains healthy [@derivatively]. This of course is possible only at a price: while the theory of perturbations around the inflating solution just depends on few parameters, one must know the whole function $P(X,\phi)$ from some UV input if interested in very different backgrounds.
In the case under study, however, one is forced to give up the (unnecessary) ambition of describing solutions which are very far from the inflating one. We will be interested in a Lagrangian for perturbations of the schematic form [M\_]{}\^2 H()\^2 + M (\^2)\^3 + …and particularly in the regime of large non-Gaussianities, when the second term, evaluated at $\partial \sim H$ and $H \pi \sim 10^{-5}$, gives a non negligible correction to the kinetic term, say of order $10^{-3}$ to be around the present experimental limits. If we now try to describe solutions with $\partial \sim H$ and $\pi \sim t$ (which implies $H \pi \gg 1$ as $H \sim {\varepsilon}/t$ in terms of the slow-roll parameter ${\varepsilon}\equiv -\dot H/H^2$) the importance of the cubic term compared to the quadratic will increase by a factor $10^{5}/{\varepsilon}$, boosted by the large classical occupation number of $\pi$. In the EFT approach however, higher derivative terms [*must*]{} be small corrections and [*must*]{} be evaluated using the lowest order equations of motion (see for example [@Simon:1990ic; @Simon:1990jn]). If on the other hand higher derivative terms are not small, one should solve the complete equations of motion which would, in this case, contain higher time derivatives and need additional initial conditions. These new degrees of freedom are ghost-like, a clear sign that we are out of the regime of validity of the EFT we started with.
What are the rules to build the effective theory of inflation? In [@Cheung:2007st; @Creminelli:2006xe], the emphasis was put on a geometrical approach. Here this would not be very useful, because it is difficult to construct Galilean operators in the geometrical language[^2]. Moreover we will see that the mixing with gravity is negligible so that one is solely interested in the action for $\pi$. The straightforward approach is to write down all the possible $\pi$ operators which are Galilean invariant and realize nonlinearly the Lorentz symmetry. This last requirement is equivalent to build Lorentz invariant operators in terms of $\psi \equiv t + \pi(t,\vec x)$, as $\psi$ linearly realizes the Lorentz symmetry. Once the possible operators at each order are written in terms of $\pi$ the size of each term can be estimated by naturalness arguments.
\[sec:action\]Building up the action
====================================
The purpose of this paper is to study the theory of perturbations about an inflating background when they are endowed with the Galilean symmetry $\pi \to \pi + b_\mu x^\mu + c$; in particular we are interested in seeing whether we can generate new shapes for the three-point function. Galilean invariance implies that there are at least two derivatives for each $\pi$ in the equation of motion, which means that a cubic term in the action contains at least four derivatives (we are assuming that we can take the decoupling limit and just look at the action for $\pi$, neglecting the mixing with gravity; we will come back to this point later). One has cubic terms with only four derivatives if the action for $\pi$ is completed to Lorentz invariant operators for $\psi$ that have the minimum number of derivatives compatible with the Galilean symmetry, that is $(\partial\psi)^2(\partial^2\psi)^n$, because these terms give equations of motion with exactly two derivatives on each $\pi$ [@Nicolis:2008in].
It is easy to see, however, that cubic operators for $\pi$ with four derivatives do not generate new shapes of non-Gaussianity as they can be rewritten in terms of operators with three derivatives, which are the ones typical of models with a reduced speed of sound[^3] [@Alishahiha:2004eh; @Chen:2006nt; @Cheung:2007st]. Indeed the possible forms of the four-derivative operators are \[4deriv\] \^2 \^2\^2 ()\^2 \^2()\^2 , where with $\nabla$ we indicate spatial derivatives[^4]. To show that they can be rewritten as operators with fewer derivatives, we are going to impose the linear equation of motion (at leading order in slow-roll and with generic speed of sound $c_s$) $\ddot\pi + 3 H \dot\pi - c_s^2 \nabla^2\pi/a^2 = 0$ at the operator level; this is equivalent up to third order to perform a field redefinition. Since we want to compute equal-time correlation functions and not S-matrix elements, in principle one should take into account all the field redefinitions [@Maldacena:2002vr]. However it is easy to realize that the redefinition we need involves derivatives of the field and therefore, when the modes are much longer than the Hubble radius, it gives contributions of higher order in the slow-roll parameters.
The first operator in (\[4deriv\]) is obviously a total derivative, so that a time derivative can be moved to act on the $a^3$ of $\sqrt{-g}$ giving a term with three derivatives. The second is related to the first using the linear equation of motion, apart from terms with fewer derivatives. The third one can be written as $\frac12 \nabla\pi \nabla \dot\pi^2$ and it is thus related to the second integrating by parts. The fourth is the same as the third integrating by parts in time. And finally the fifth is related to the fourth using again the linear equation of motion. Therefore with four derivatives we cannot generate new forms of three-point function and we are hence motivated to consider terms with more derivatives on $\psi$.
Galilean action with more derivatives
-------------------------------------
Let us assume to start, besides the standard kinetic term, only with operators that have at least two derivatives on each $\pi$ (and therefore on $\psi$), i.e. setting to zero the lowest derivative Galilean terms $(\partial\psi)^2 (\partial^2\psi)^n$ ([^5]). In flat space this is consistent as loops will not generate them [@Luty:2003vm]. However, the Galilean symmetry cannot be defined in curved space, since there is no natural way of defining a constant vector $b_\mu$. Thus one may worry to generate terms containing the Riemann tensor, of the schematic form $(\partial\psi)^2 (\partial^2\psi)^n R$ and in general all terms in which a couple of derivatives is replaced by the curvature tensor. All these would give a contribution to the $\pi$ correlation functions which is of the same order as the one we started with. This is not the case as it is easy to prove.
Let us start from a 1PI graph for $\pi$ in flat space: this has (at least) two derivatives on each external leg. Now we add an external graviton line[^6]. If it couples to an internal line, it will not reduce the number of derivatives acting on each external $\pi$ line. If the graviton is attached to a vertex it can reduce to one the number of derivatives acting on an external $\pi$ line, when $g$ comes from the Christoffel symbol. But all the other external $\pi$ legs have still 2 derivatives. The loop can generate additional external derivatives, but it cannot reduce them, so we see that it is impossible to generate an operator of the form $(\partial\psi)^2 (\partial^2\psi)^n R$, which contains a term with a single external graviton and two $\pi$’s with a single derivative. One would have at least an additional suppression of the curvature: $(\partial\psi)^2 (\partial^2\psi)^n
R^2$, which makes these terms subleading for the $\pi$ correlation functions. Additional non-renormalization properties are discussed in Appendix \[app:NR\]. We conclude that, although the Galilean symmetry cannot be defined in curved space, we can consistently reduce to operators which contain two derivatives on each field.
Let us come to the explicit construction of the $\pi$ action. The $\pi$ operators at any order can be written starting from Lorentz invariant operators for $\psi$. These are traces of the double derivative matrix $\nabla_\mu\nabla_\nu \psi$. In the following we use $\Psi$ to denote this matrix and the square brackets to indicate the trace so that, for example, $[ \Psi \Psi]$ means $\nabla_\mu\nabla_\nu \psi \, \nabla^\mu\nabla^\nu \psi$. In order to parametrize the theory of perturbations around a given solution we would like to isolate operators that contain terms linear in $\pi$. Those operators change the background and therefore their coefficients are constrained once the background solution is chosen. On the other hand operators that start quadratic in $\pi$ do not affect the background solution for $\psi$ nor for the metric – as they have vanishing stress-energy tensor for $\pi = 0$ – so that their coefficients are not constrained [@Creminelli:2006xe; @Cheung:2007st].
If a term is composed by the product of several traces of products of $\Psi$, $[\Psi \Psi \ldots \Psi] \ldots [\Psi \Psi\ldots \Psi]$, then we can subtract to each trace its background value to make the term at least quadratic. If on the contrary an operator is composed by a single trace $[\Psi^n]$ this cannot be done and the operator will start with a linear term. We will show however that $[\Psi^n]$ can be written in term of operators, whose linear term is easy to isolate. To do this, consider the sum \[eq:total\] \_p (-1)\^p g\^[\_1 p(\_1)]{} …g\^[\_n p(\_n)]{} \_[\_1]{}\_[\_1]{} …\_[\_n]{}\_[\_n]{} , where $p$ is the parity of the permutation. This sum contains also the single trace of order $n$. For $n >4$ eq. trivially vanishes, as there are too many indices to antisymmetrize, so that the $[\Psi^n]$ can be rewritten in terms of products of two or more shorter traces. For $n \leq 4$ the sum does not vanish. In flat space it is a total derivative, as it easy to see calculating the equations of motion, which trivially vanish, but in curved space this does not occur since derivatives do not commute. Nevertheless the sum can be written—integrating by parts one of its derivatives, writing the commutator of derivatives in terms of the Riemann tensor and finally expressing the Riemann tensor of de Sitter in terms of the metric—as \[eq:totalpart\] H\^2 \_p (-1)\^p g\^[\_1 p(\_1)]{} …g\^[\_[n-1]{} p(\_[n-1]{})]{} \_[\_1]{}\_[\_1]{} …\_[\_n-1]{}\_[\_n-1]{} n4 , apart from overall combinatorial factors. Notice that these are the minimal Galilean terms, see Appendix A of [@Nicolis:2008in], up to the quartic. Therefore for $n \leq 4$ the single traces can be written in terms of shorter traces and minimal Galilean terms, cubic and quartic in $\psi$ ([^7]). We conclude that it is consistent to study the theory that contains all operators written in terms of $\nabla_\mu\nabla_\nu \psi$, except the single trace operators, plus the minimal Galilean operators of cubic and quartic order. These two will be suppressed by $H^2$ and therefore will give a contribution of the same order of magnitude as the other ones to the $\pi$ correlation functions.
It is straightforward to write the minimal Galilean operators of cubic and quartic order in such a way that they do not contain terms linear in $\pi$. The cubic (DGP-like) term $\Box\psi(\partial\psi)^2$ can be written as $(\Box\psi+3 H)[(\partial\psi)^2+1]$, just redefining the coefficients of the kinetic term and the cosmological constant term, which are anyway present in the Lagrangian. The procedure is slightly subtler for the quartic term and it is useful to rewrite it as [@Deffayet:2009wt]
& ()\^2 ()\^2 - 2 \_\_\_\_- (\_\_)\^2 ()\^2 + 2 \_\_\_\_\_\_\
= & ()\^2 - ()\^2 R\^ \_\_.
As we are in de Sitter the Ricci tensor is proportional to the metric, so that the second operator is just $(\partial\psi)^2(\partial\psi)^2$. In this form all terms can be treated similarly to what we did with the DGP one.
Now that we have the rules to build the action in terms of $\psi$, we move to study the third order action for $\pi$, which is relevant for the calculation of non-Gaussianity.
\[sec:perturbations\]Cubic action for perturbations
---------------------------------------------------
We are primarily interested in the 3-point function, i.e. in operators of the form $(\partial^2\pi)^3$. Some of these operators will be independent, while others will be related to quadratic operators by the non-linear realization of the Lorentz symmetry, i.e. once written in terms of $\psi$. In this second case the scale suppressing the cubic operators is related to the one suppressing the quadratic ones. Schematically we will have \[eq:ducks\] [M\_]{}\^2 H ()\^2 + M , where the factor of $H$ inside brackets comes from the fact that $\nabla_\mu\nabla_\nu\psi$ gives $H \delta_{ij}$ on the background. Going to canonical normalization we have \[eq:duckscan\] (\_c)\^2 + (\^2\_c)\^2 + (\^2\_c)\^3 + …where \^2 = and \^5 = \^5 \^5 , where the separation between the scales is stable only when $\tilde\Lambda > \Lambda$. Notice that $(\partial^2\pi)^2$ must be treated as a small perturbation to the standard kinetic term with $c_s=1$; therefore we have the usual normalization $H/(\sqrt{{\varepsilon}}{M_{\textrm{Pl}}}) \simeq 10^{-5}$. Cubic non-Gaussianities will be of order $(H/\Lambda)^5$, so that observable non-Gaussianity – i.e. larger than $10^{-5}$ corresponding, in the usual parametrization, to $f_{\rm NL} \gtrsim 1$ – implies $\tilde\Lambda \gg \Lambda$. We conclude that, assuming all independent operators are suppressed by a common scale $\Lambda$, cubic operators that are linked by symmetry to quadratic ones are suppressed by a much higher scale and can be neglected with respect to the independent ones. It is straightforward to see that this holds in general when an operator $(\partial^2\pi)^n$ is related by the Lorentz symmetry to a lower dimensional one: the one with the lowest dimension will be suppressed by the common scale $\Lambda$, while the others will be suppressed by powers of $\frac{H^3}{\Lambda^3} \cdot \sqrt{{\varepsilon}} \frac{{M_{\textrm{Pl}}}}{H}$. The same conclusion applies to the minimal Galilean operators of cubic and quartic order, where there is a relation between quadratic and cubic terms in $\pi$. Therefore these operators have a negligible contribution to the three-point functions in our setup.
Since we are interested in quadratic and cubic operators and we do not have linear terms, we need to consider only terms with two traces on $\psi$ and terms with three traces: (\[…\] - c\_1)(\[…\] - c\_2) (\[…\] - c\_3)(\[…\] - c\_4)(\[…\] - c\_5) where the constants $c_i$ are chosen to subtract from each trace its background value. Operators that contain a $\Box \pi$ are proportional to the linear equation of motion and can be neglected since they can be removed by a field redefinition, as we discussed above. Of course this is valid only at leading order in slow-roll, but still tells us that operators with a $\Box \pi$ are subdominant with respect to the others.
The only cubic operator without $\Box \pi$, coming from a term with three traces is of course (\^2)\^3 \^2 \^[ij]{} \_i\_j. Terms with two traces generate terms with 2 and 3 $\pi$’s and it is important to understand whether the latter are independent or not because this controls the scale that suppresses them, as we discussed in the previous paragraphs. If we restrict again to cubic operators without $\Box\pi$ the additional terms are \^2(\_i\_j)\^2 \^2(\_i\_)\^2 \^2(\_\_)\^2 . It is straightforward to see that there is enough freedom to make all of them independent from the 2 $\pi$’s operators[^8].
So far we have implicitly assumed that we can concentrate on the $\pi$ action, without taking into account its mixing with gravity. To check that this mixing is indeed negligible, one should follow [@Maldacena:2002vr]: in spatially flat gauge one solves the ADM constraints and plug the solution for the ADM variable $N$ and $N_i$ back into the action. To derive the cubic action, one needs to solve for $N$ and $N_i$ at first order only, as the second order result would multiply in the action the first order constraint equations, which vanish [@Maldacena:2002vr]. If the quadratic action is simply given by the standard kinetic term, then the mixing with gravity at energies of order $H$ is suppressed by the slow-roll parameters, so that the mixing with gravity just gives corrections to the 3-point function of the order of the slow-roll parameters. In our case we also have additional quadratic operators – see eq. – of the schematic form $(\partial^2\pi_c)^2/\Lambda^2$. However, neglecting the mixing with gravity, these terms are suppressed with respect to the canonical kinetic term by $H^2/\Lambda^2$, so that we expect they also give a suppressed contribution to the constraint equations. Indeed it is straigthforward to check that their effect is suppressed both by slow-roll and by $H^2/\Lambda^2$. Therefore we can safely concentrate on the $\pi$ action, without bothering about the mixing with gravity, to calculate the observable predictions.
At third order in perturbations, we have the following four independent operators: $$\begin{aligned}
S =& \int {\textrm{d}}^4 x a^3 \bigg[ {M_{\textrm{Pl}}}^2 \dot H {\partial}_\mu \pi {\partial}^\mu \pi
+ \tilde{M}_1 (g^{ij} {\nabla}_i {\nabla}_j \pi)^3
+ \tilde{M}_2 (g^{ij} {\nabla}_i {\nabla}_j \pi) ({\nabla}_i {\nabla}_j \pi)^2 \notag \\
&+ \tilde{M}_3 (g^{ij} {\nabla}_i {\nabla}_j \pi) ({\nabla}_i {\nabla}_\mu \pi)^2
+ \tilde{M}_4 (g^{ij} {\nabla}_i {\nabla}_j \pi) ({\nabla}_\mu {\nabla}_\nu \pi)^2 \bigg],
$$ in which we need to expand the covariant derivatives: $$\begin{gathered}
g^{ij} {\nabla}_i {\nabla}_j \pi = \frac{{\partial}_i^2 \pi}{a^2} - \frac{1}{a^2} {\delta}^{ij} {\Gamma}^0_{ij} \dot{\pi}
= \frac{{\partial}_i^2 \pi}{a^2} - 3 H \dot{\pi}\,, \\
({\nabla}_i {\nabla}_j \pi)^2 = \frac{1}{a^4} {\delta}^{ik} {\delta}^{jl} ({\partial}_i {\partial}_j \pi - H a^2 {\delta}_{ij} \dot{\pi}) ({\partial}_k {\partial}_l \pi - H a^2 {\delta}_{kl} \dot{\pi})
= \frac{({\partial}_i {\partial}_j \pi)^2}{a^4} - 2 H \dot{\pi} \frac{{\partial}_i^2 \pi}{a^2} + 3 H^2 \dot{\pi}^2 \,,\\
({\nabla}_i {\nabla}_0 \pi)^2 = - \frac{1}{a^2} ({\partial}_i \dot{\pi} - H {\partial}_i \pi)^2
$$ As discussed in the previous subsection, in order to calculate the non-Gaussian correlation functions at lowest order in slow-roll one can substitute the linear equation of motion $\ddot{\pi} + 3 H \dot{\pi} = \nabla^2 \pi / a^2$ in the action. Substituting the Laplacian in favour of the time derivatives (in order to recover the usual shapes involving the operators $\dot{\pi}^3$ and $\dot \pi ({\partial}_i \pi)^2/a^2$) yields $$\begin{gathered}
g^{ij} {\nabla}_i {\nabla}_j \pi = \ddot{\pi} \; , \qquad
({\nabla}_i {\nabla}_j \pi)^2 = \frac{({\partial}_i {\partial}_j \pi)^2}{a^4}
- 2 H \dot{\pi} \ddot{\pi} - 3 H^2 \dot{\pi}^2 \; ,\end{gathered}$$ which results in an action with only three, and not four, independent operators. Doing a further integration by parts, $a^3 \ddot{\pi} \dot\pi^2 = \frac{1}{3} a^3{\partial}_t \dot{\pi}^3 \rightarrow - a^3 H \dot{\pi}^3$, we finally get $$\begin{aligned}
S =& \int {\textrm{d}}^4 x \,a^3 \bigg[ -{M_{\textrm{Pl}}}^2 \dot H \left( \dot{
\pi}^2 - \frac{({\partial}_i \pi)^2}{a^2} \right)
+ M_1 \ddot{\pi}^3
+ M_2 \ddot\pi \frac{({\partial}_i \dot{\pi} - H {\partial}_i \pi)^2}{a^2}
\notag \\
& + M_3 \left(\ddot{\pi} \frac{ ({\partial}_i {\partial}_j \pi)^2}{a^4} - 2 H
\dot{\pi} \ddot{\pi}^2 + 3 H^3 \dot{\pi}^3 \right)
\bigg] \; .
\label{eq:Sint}\end{aligned}$$ This will be the starting point for the calculation of the three-point function.
\[sec:shapes\]Shapes of non-Gaussianity
=======================================
Using the cubic action we can calculate the tree-level contribution to the three-point correlation function for the $\pi$ field. With the prescriptions of the *in-in* formalism, this is given by the following formula [@Maldacena:2002vr; @Weinberg:2005vy]: [\^3(t\_0) ]{} = i \_[-]{}\^[t\_0]{} t a\^3 where $t_0$ is some late time when the physical modes of interest are well outside the Hubble radius, and $L_{\rm int}$ is the cubic Lagrangian to be read from Eq. .
Non-Gaussianity is usually computed in terms of the correlation functions of the Bardeen potential $\Phi$. It is customary to write the correlation functions for $\Phi$ isolating the Dirac’s delta that enforces the conservation of the three-momentum (implied by the traslational invariance). In the limit of exact scale invariance, the two- and three-point functions can be parametrized as $${\langle \Phi_{{\vec{k}}_1} \Phi_{{\vec{k}}_2} \rangle} = (2 \pi)^3 \delta^{(3)}({\vec{k}}_1+{\vec{k}}_2)
\frac{\Delta_\Phi}{k^3}$$ and $${\langle \Phi_{{\vec{k}}_1} \Phi_{{\vec{k}}_2} \Phi_{{\vec{k}}_3} \rangle} = (2 \pi)^3
\delta^{(3)}({\vec{k}}_1+{\vec{k}}_2+{\vec{k}}_3) F(k_1, k_2, k_3) \; ,$$ where $F$ (called the bispectrum of $\Phi$) is a homogeneous function of degree -6 which, because of rotational invariance, depends only on the length of the three-momenta. Therefore, in the same limit $k_1^6 F(k_1, k_2, k_3)$ becomes just a function of the ratios $r_2 = k_2/k_1$ and $r_3 = k_3/k_1$. As argued in [@Babich:2004gb], the most meaningful quantity to plot is r\_2\^2 r\_3\^2 F(1, r\_2, r\_3) , which is also commonly called the “shape” of the non-Gaussianity.
During matter dominance $\Phi$ is related to the curvature perturbation on comoving slices $\zeta$ by the simple relation $\Phi=\frac{3}{5}\zeta$. The curvature perturbation is constant outside the Hubble radius, regardless of the matter content of the Universe (and in particular it remains constant during reheating). At leading order in slow roll $\zeta$ is related to the $\pi$ field by $\zeta = - H \pi$. Shifting to conformal time ($\tau=-1/aH$), the bispectrum $F$ reads thus \[eq:bispectrum\] F(k\_1, k\_2, k\_3) = - ()\^3 \^\*(0,k\_1)\^\*(0,k\_2)\^\*(0,k\_3) \_[-]{}\^0 L\_(,k\_1,k\_2,k\_3) + + , where $\pi(\tau, k)$ is the classical Fourier mode in de Sitter space (see Appendix \[app:shapes\]) and the momenta are restricted to configurations forming a triangle in momentum space.
Doing this for each of the three independent cubic interactions in leads to three different bispectra, which are proportional to $M_1$, $M_2$ and $M_3$ respectively. As detailed in App. \[app:shapes\], their form is (setting $k_t\equiv k_1+k_2+k_3$) [ $$\begin{aligned}
F_{M_1}(k_1, k_2, k_3) =
&- \frac{20}{3} \, \frac{M_1H}{{\varepsilon}{M_{\textrm{Pl}}}^2} \, 6\Delta_\Phi^2
\frac{k_t^3 -6k_t(k_1k_2+k_2k_3+k_1k_3)
+15 k_1k_2k_3}{k_1 k_2 k_3 k_t^6}\;,
\label{eq:FM1} \\
F_{M_2}(k_1, k_2, k_3)
= &\frac{5}{6} \, \frac{M_2H}{{\varepsilon}{M_{\textrm{Pl}}}^2} \, \Delta_\Phi^2
\frac{k_1^2-k_2^2-k_3^2}{k_1 k_2^3 k_3^3 k_t}
\bigg[2 + \frac{2(k_2 + k_3) - k_1}{k_t}
\notag \\
&-2\frac{k_1(k_2 + k_3) + 2 k_2^2 - 2 k_2 k_3 + 2 k_3^2 }{k_t^2}
+ 6\frac{k_1(k_3^2 - k_2k_3 + k_2^2) - 2 k_2^2k_3 - 2 k_2k_3^2}{k_t^3}
\notag \\
&+ 24\frac{k_1 k_2 k_3(k_2 + k_3) +2 k_2^2k_3^2}{k_t^4}
- 120\frac{k_1 k_2^2 k_3^2}{k_t^5}
\bigg] + \mathrm{cyclic~permutations}\;,
\label{eq:FM2} \\
F_{M_3}(k_1, k_2, k_3)
= &\frac{5}{3} \,\frac{M_3H}{{\varepsilon}{M_{\textrm{Pl}}}^2} \,
\frac{\Delta_\Phi^2}{k_1 k_2 k_3 k_t^3}
\bigg[10-24\frac{k_2+k_3}{k_t}+48\frac{k_2k_3}{k_t^2} \notag \\
&+\bigg(\frac{k_1^2-k_2^2-k_3^2}{k_2k_3}\bigg)^{\!2}
\!\bigg(1 + 3\frac{k_2+k_3-k_1/2}{k_t} - 6\frac{k_1(k_2+k_3) - 2k_2k_3}{k_t^2}
-30\frac{k_1k_2k_3}{k_t^3}\bigg)
\bigg] \notag \\
&+ \mathrm{cyclic~permutations}\;.
\label{eq:FM3} \end{aligned}$$]{}
The simplest shape of non-Gaussianity is the so-called local shape, which peaks in the squeezed limit ($r_2\simeq1$ and $r_3\simeq 0$, *i. e. *when a mode is much larger than the other two). In single field inflation non-Gaussianity of this kind is proportional to the tilt of the power spectrum, and a measurable signal can be generated only within multi-field scenarios [@Creminelli:2004yq]. Single field inflation can generate large non-Gaussianity when derivative interactions are present, whose shape is enhanced in other regions of the $r_1-r_2$ plane than the local limit. The shape generated in scenarios like DBI inflations, for instance, peaks in the equilateral limit ($r_2\simeq r_3 \simeq 1$) [@Babich:2004gb]. Shapes peaking in the “flat” configuration ($r_2\simeq r_3 \simeq 0.5$, corresponding to flattened isosceles triangles) where obtained in Ref. [@Senatore:2009gt] as a difference of equilateral shapes, and directly from new higher derivative operators in Ref. [@Bartolo:2010bj]. Notice however that Ref. [@Bartolo:2010bj] did not identify any symmetry that allows to neglect lower dimensional operators and that operators containing higher time derivatives were not considered.
The shapes given by the three independent cubic operators in Eq. are shown in Fig. \[fig:shapes\].
![\[fig:shapes\]Bispectrum shapes generated by the cubic operators in Eq. . The ones proportional to $M_1$ and $M_2$ (upper panels) are very similar (with a cosine of nearly 1) to those generated by $\dot\pi^3$ and $\dot\pi({\partial}_i\pi)^2$ respectively, which where studied in [@Alishahiha:2004eh; @Chen:2006nt; @Cheung:2007st]. They have a cosine of 0.96 and 0.99 with the equilateral template. The third operator (lower left panel) is “surfing”, i.e. it is shaped like a wave with a maximum in the flat configuration, and has a large overlap with the enfolded template (lower right panel). The suppression in the equilateral regime is due to the presence of a higher number of scalar products of gradients.](M1.eps){width="\textwidth"}
![\[fig:shapes\]Bispectrum shapes generated by the cubic operators in Eq. . The ones proportional to $M_1$ and $M_2$ (upper panels) are very similar (with a cosine of nearly 1) to those generated by $\dot\pi^3$ and $\dot\pi({\partial}_i\pi)^2$ respectively, which where studied in [@Alishahiha:2004eh; @Chen:2006nt; @Cheung:2007st]. They have a cosine of 0.96 and 0.99 with the equilateral template. The third operator (lower left panel) is “surfing”, i.e. it is shaped like a wave with a maximum in the flat configuration, and has a large overlap with the enfolded template (lower right panel). The suppression in the equilateral regime is due to the presence of a higher number of scalar products of gradients.](M2.eps){width="\textwidth"}
![\[fig:shapes\]Bispectrum shapes generated by the cubic operators in Eq. . The ones proportional to $M_1$ and $M_2$ (upper panels) are very similar (with a cosine of nearly 1) to those generated by $\dot\pi^3$ and $\dot\pi({\partial}_i\pi)^2$ respectively, which where studied in [@Alishahiha:2004eh; @Chen:2006nt; @Cheung:2007st]. They have a cosine of 0.96 and 0.99 with the equilateral template. The third operator (lower left panel) is “surfing”, i.e. it is shaped like a wave with a maximum in the flat configuration, and has a large overlap with the enfolded template (lower right panel). The suppression in the equilateral regime is due to the presence of a higher number of scalar products of gradients.](M3.eps){width="\textwidth"}
![\[fig:shapes\]Bispectrum shapes generated by the cubic operators in Eq. . The ones proportional to $M_1$ and $M_2$ (upper panels) are very similar (with a cosine of nearly 1) to those generated by $\dot\pi^3$ and $\dot\pi({\partial}_i\pi)^2$ respectively, which where studied in [@Alishahiha:2004eh; @Chen:2006nt; @Cheung:2007st]. They have a cosine of 0.96 and 0.99 with the equilateral template. The third operator (lower left panel) is “surfing”, i.e. it is shaped like a wave with a maximum in the flat configuration, and has a large overlap with the enfolded template (lower right panel). The suppression in the equilateral regime is due to the presence of a higher number of scalar products of gradients.](enfolded.eps){width="\textwidth"}
The first operator $\ddot\pi^3$ has a clearly equilateral shape and is almost indistinguishable from the operator $\dot\pi^3$: indeed, in conformal time $\dot\pi_k\sim k^2\tau^2e^{i k \tau}$ is very similar to $\ddot\pi_k\sim k^2\tau^2(1+ik\tau/2) e^{i k \tau}$. For the same reason, $\ddot\pi({\partial}_i\dot\pi -H{\partial}_i\pi)^2$ is very close $\dot\pi({\partial}_i\pi)^2$. Both operators have therefore equilateral shapes of the kind already considered in Refs. [@Alishahiha:2004eh; @Chen:2006nt; @Cheung:2007st].
On the other hand, an asymmetric operator with more scalar product of space gradients like $\ddot\pi(\partial_i\partial_j\pi)^2/a^4$ will contain terms like $({\vec{k}}_2\cdot{\vec{k}}_3)^2=\frac{1}{4}(k_1^2-k_2^2-k_3^2)^2$. Moreover, each power of $1/a\sim\tau$ yields $1/(k_1+k_2+k_3)$ after the integral over conformal time; scalar products of gradients will therefore result in terms like $[(k_1^2-k_2^2-k_3^2)/2(k_1+k_2+k_3)^2]^n$, which tend to be suppressed in the equilateral limit. The precise scaling depends on the details of the symmetrization, but the general argument qualitatively holds. As a result, the shape generated by the operator proportional to $M_3$ looks like a wave reaching its maximum in the flat configuration.
In order to give a quantitative measure of the difference of the shapes generated by different interactions, it is very useful to define a scalar product between bispectra as follows [@Babich:2004gb]: F\_[(1)]{} F\_[(2)]{} = \_[triangles]{} \[eq:scalar\] where the sum is restricted to the vectors $k_i$ that form a triangle in momentum space[^9], from which it is natural to define the cosine (F\_[(1)]{}, F\_[(2)]{}) = . If the cosine between two shapes is large, they have a significant overlap and it is possible to use the same estimator to analyze CMB data and constrain the amplitude ${f_{\textrm{NL}}}$ of each shape. On the other hand, if the cosine is small this is no longer efficient and a different estimator is needed in order to set optimal constraints on ${f_{\textrm{NL}}}$.
Moreover, a crucial numerical boost in CMB data analysis is gained by using factorizable shapes, *i.e.* shapes which can be written as sums of monomials of $k_1$, $k_2$ and $k_3$ [@Creminelli:2005hu; @Wang:1999vf]. A good estimator to constrain the amplitude of a given model is then provided by a factorizable shape having a large cosine with the bispectrum generated by the model itself. Factorizable shapes which resemble model predictions are often called templates. Among the most popular templates are the local and equilateral ones [@Babich:2004gb], for which CMB constraints are usually cited (see for example [@Komatsu:2010fb]). Linear combinations of equilateral operators can however yield a significantly different shape for a quite wide range of coefficients, leading to the definition of an “orthogonal” template [@Senatore:2009gt]. Another template found in the literature is the one for the so-called “enfolded” shape, originally introduced in Ref. [@Meerburg:2009ys]. This template is a linear combination of the orthogonal and equilateral templates, as pointed out in Ref. [@Senatore:2009gt]: $$\begin{aligned}
F^\mathrm{enf}(k_1, k_2, k_3)
&= \frac{1}{2}\left[F^\mathrm{equil}(k_1, k_2, k_3)- F^\mathrm{orth}(k_1, k_2, k_3)\right] \notag \\
&= 6{f_{\textrm{NL}}}^\mathrm{enf}\Delta_\Phi^2\Bigg[\frac{1}{k_1^3 k_2^3} + \frac{1}{k_1^2 k_2^2 k_3^2} - \frac{1}{k_1 k_2^2 k_3^3} - \frac{1}{k_2 k_1^2 k_3^3} + \mathrm{cyclic\; permutations}\Bigg]\,.
\label{eq:enfoldedTemplate}\end{aligned}$$
We have computed the cosines of our shapes with the local, equilateral, orthogonal and enfolded templates. Our results are summarized in Table \[tab:cosines\].
Shape: local equilateral orthogonal enfolded
-------- ------- ------------- ------------ ----------
M1 0.49 0.96 -0.06 0.70
M2 -0.43 -0.99 -0.16 -0.53
M3 0.57 0.71 -0.52 0.94
: \[tab:cosines\] Cosines between different shapes of bispectra.
As seen in this table, a suitable template for the operators $\ddot\pi^3$ and $\ddot\pi({\partial}_i\dot\pi -H{\partial}_i\pi)^2$ is the equilateral one. The remaining operator has a large overlap with the enfolded template[^10]. Comparing with the templates allows us to compute the amplitude of non-Gaussianity of each type generated within our model, and hence constrain $M_1$, $M_2$ and $M_3$.
For nearly equilateral shapes, the amplitude of the non-Gaussianity is defined as $f_{\mathrm{NL}}^{\mathrm{eq}}\equiv k^6 F(k,k,k)/6\Delta_\Phi^2$ [@Creminelli:2005hu]. The amplitude induced by the two shapes $F_{M_1}$ and $F_{M_2}$ is thus $$f_{\mathrm{NL}}^{\mathrm{eq},1} = \frac{80}{729} \, \frac{M_1H}{{\varepsilon}{M_{\textrm{Pl}}}^2}
\,, \qquad
f_{\mathrm{NL}}^{\mathrm{eq},2} = - \frac{865}{2916} \, \frac{M_2H}{{\varepsilon}{M_{\textrm{Pl}}}^2}
\,;$$ using the most up-to date values $f_{\mathrm{NL}}^{\mathrm{eq}}=26\pm 140$ (68% CL) from Ref. [@Komatsu:2010fb] one gets $$\frac{M_1H}{{\varepsilon}{M_{\textrm{Pl}}}^2} = 240 \pm 1280\;, \qquad
\frac{M_2H}{{\varepsilon}{M_{\textrm{Pl}}}^2} = -80 \pm 470\;.$$
For the shape $F_{M_3}$ the amplitude must be defined starting from the enfolded template, $$f_{\mathrm{NL}}^{\mathrm{enf},3}
\equiv \frac{k^6 F_{M_3}(k,k/2,k/2)}{96\Delta_\Phi^2}
= \frac{35}{256} \, \frac{M_3H}{{\varepsilon}{M_{\textrm{Pl}}}^2} \, .$$ Since this template is a linear combination of the equilateral and orthogonal ones (which are slightly correlated), the errors on the value of ${f_{\textrm{NL}}}^{\mathrm{enf}}$ can be inferred from the errors on ${f_{\textrm{NL}}}^{\mathrm{eq}}$ and ${f_{\textrm{NL}}}^{\mathrm{orth}}$ given in Ref. [@Komatsu:2010fb], assuming that the correlation coefficient remained the same as in Ref. [@Senatore:2009gt]. The limit obtained in this way is $$f_\mathrm{NL}^\mathrm{enf} = 114 \pm 72 \quad (68\% \;\mathrm{CL}) \;,
\label{eq:limits}$$ leading to $$\frac{M_3H}{{\varepsilon}{M_{\textrm{Pl}}}^2} = 830 \pm 530\;.$$
A drawback of the enfolded template (as well as of the orthogonal one) is that it tends to a constant in the squeezed limit, while in the approximation of exact scale invariance any single-field three-point function vanishes in this regime [@Creminelli:2004yq]. This means that, while acceptable for CMB data analysis, such templates cannot give satisfactory results in contexts where all the signal is concentrated in the squeezed configurations as, for instance, with halo bias observations [@Catelan:1997qw; @Dalal:2007cu; @Verde:2009hy]. In order to improve the above constraint by combining it with LSS observations is therefore important to devise a different template, that peaks in the flattened limit but preserves the correct behavior when $k_1\ll k_2\sim k_3$. An example of such a template is given in App. \[app:template\].
In our theory there are 3 independent parameters which allow us to choose arbitrary linear combinations of our operators. Fixing the amplitude of non-Gaussianity leaves us with a 2-parameter family ($M_2/M_1$ and $M_3/M_1$) of different possible shapes. It is interesting to see whether one can generate a suitable linear combination that is nearly orthogonal to all of the existing templates, following the same reasoning that lead to the definition of the orthogonal template in Ref. [@Senatore:2009gt]. If this were the case, one should use a dedicated template for data analysis. If such a linear combination exists, the vector $(1,M_2/M_1,M_3/M_1)$ must be an approximate null eigenvector of the $3\times3$ matrix of the scalar products of the three shapes (with the $M_i$’s set to 1) with the local, equilateral and orthogonal templates (the enfolded is already a linear combination of two of them). Equivalently, the coefficients $(M_i/M_1)\|F_{M_i}\|$ must form an approximate null eigenvector of the $3\times3$ matrix of the cosines, which are just normalized scalar products. Since the determinant of the matrix is 0.0057 (much smaller than most of the entries), such approximate eigenvector can be found and this allows to solve for $M_2/M_1$ and $M_3/M_1$. For instance, a linear combination with $M_2=0.32 M_1$ and $M_3=-0.42 M_1$ is a good candidate, yielding cosines of -0.15, 0.03 and 0.06 with the local, equilateral and orthogonal template (and -0.03 with the enfolded template). The values of $M_2/M_1$ and $M_3/M_1$ yielding small (i.e. $<0.2$) cosines with each template are shown in Fig. \[fig:cosines\], where the shape of the combination nearly orthogonal to all templates is also given.
![\[fig:cosines\] *Left:* regions of the parameter space $M_2/M_1$ and $M_3/M_1$ for which the cosine of the resulting shape with the local, equilateral and orthogonal template is smaller than 0.2. *Right:* Non-Gaussian shape obtained with $M_2=0.32
M_1$ and $M_3=-0.42 M_1$. This shape is nearly orthogonal to all known templates (with cosines of -0.15, 0.03, 0.06 and -0.03 with the local, equilateral, orthogonal and enfolded template), and would require a dedicated template for data analysis.](cosines.eps){width=".9\textwidth"}
![\[fig:cosines\] *Left:* regions of the parameter space $M_2/M_1$ and $M_3/M_1$ for which the cosine of the resulting shape with the local, equilateral and orthogonal template is smaller than 0.2. *Right:* Non-Gaussian shape obtained with $M_2=0.32
M_1$ and $M_3=-0.42 M_1$. This shape is nearly orthogonal to all known templates (with cosines of -0.15, 0.03, 0.06 and -0.03 with the local, equilateral, orthogonal and enfolded template), and would require a dedicated template for data analysis.](smallCos.eps){width=".9\textwidth"}
\[sec:NG4\]Four-point function
==============================
Let us see what are the implications of our setup for the four-point function[^11]. The action, besides the quadratic and cubic operators we considered, will contain quartic operators (\_c)\^2 + (\^2\_c)\^2 + (\^2\_c)\^3 + (\^2\_c)\^4 + …The deviation from Gaussianity induced by the quartic terms can be quantified comparing these with the kinetic term at scales of order $H$ \[eq:NG4\] [NG]{}\_4 . |\_[E \~H]{} ()\^8 . The quartic non-Gaussianity comes from operators of dimension 12, and it is therefore suppressed by $(H/\Lambda)^8$. Notice that in terms of the parameter $\tau_{\rm NL}$ used in the literature, we have ${\rm NG}_4 \simeq \tau_{\rm NL} \Delta_{\zeta }$, where $\Delta_\zeta \simeq 2 \times 10^{-9}$ gives the normalization of the scalar power spectrum. As we saw, for the three-point function we have analogously \[eq:NG3\] [NG]{}\_3 . |\_[E \~H]{} ()\^5 ; indeed cubic operators are of dimension 9 and ${\rm NG}_3 \simeq f_{\rm NL} \Delta_{\zeta}^{1/2}$. Without additional ingredients, we see from eq.s and that the quartic non-Gaussianity is always smaller than the cubic one \_4 \~[NG]{}\_3\^[8/5]{} , so that the three-point function is much easier to detect. Notice, however, that the scaling of the quartic and cubic non-Gaussianities is different from the one one has in models with a low speed of sound, where the three-point function is given by operators of dimension 6 (schematically $(\partial\pi_c)^3/\Lambda^2$), while the four-point function is given by operators of dimension 8 (of the form $(\partial\pi_c)^4/\Lambda^4$), so that \_3 ()\^2 \_4 ()\^4 \_4 \~[NG]{}\_3\^[2]{} . This implies that, at the same level of cubic non-Gaussianity, our setup gives a larger four-point function signal, although still much smaller than the cubic one. This difference is somewhat important experimentally. If the three-point function is of the order of the present experimental limit, $f_{\rm NL} \sim 100$, equivalent to ${\rm NG}_3 \sim 5 \times 10^{-3}$, then in models with a small speed of sound one has $ {\rm NG}_4 \sim 2 \times 10^{-5}$, i.e. $\tau_{\rm NL} \sim 10^4$, so that the four-point function is not detectable by the Planck satellite, while it may be detectable in our setup, where $ {\rm NG}_4 \sim 2 \times 10^{-4}$, i.e. $\tau_{\rm NL} \sim 10^5$. Obviously these are just rough estimates as a proper forecast for the Planck capability on the 4-point function is still lacking.
The three-point function is usually considered the leading signal of non-Gaussianity as it typically dominates the higher order correlations as we saw above. However, it has been pointed out in [@Senatore:2010jy] that it is possible to have technically natural theories in which the leading source of non-Gaussianity is the four-point function, as a consequence of an approximate $\pi \to - \pi$ symmetry. We can straightforwardly apply the same arguments in our context. Let us consider the case in which, besides the standard kinetic term, we only have operators that start quartic in $\pi$. Of course both the kinetic term and these quartic interactions respect the symmetry $\pi \to - \pi$. However, as a consequence of the non-linear realization of the Lorentz symmetry, these operators will also contain terms of higher order in $\pi$, in particular quintic, that do not respect the symmetry[^12]. Obviously if these terms were unsuppressed the symmetry would be of no use. However, as we discussed above, operators with a given dimension that are related to lower dimensional ones by the Lorentz symmetry are not weighed by the cut-off scale $\Lambda$, but they are further suppressed. In particular the quintic operators will be of the form \^5 1 . The breaking of the $\pi \to -\pi$ symmetry will thus be weighted by the small parameter $\xi$. In particular loops will induce cubic operators of the form . These radiatively induced cubic interactions only give a completely negligible \_3 ()\^5 = ()\^2 ()\^2 \_\^[1/2]{} f\_[NL]{} ()\^2 . We conclude that it is possible in our setup to have a large four-point function, of order $(H/\Lambda)^8$, with a negligible three-point function signal. The number of independent operators – the quartic operators that are independent of the cubic ones – is quite large, even if we restrict to terms without $\Box\pi$. Their study is beyond the scope of this paper.
\[sec:conclusions\]Conclusions and future directions
====================================================
In this paper we explored a new class of single field inflationary models, endowed with an approximate Galilean symmetry. Their phenomenology is quite distinctive, with new shapes of the three-point function and potentially large four-point function.
The study of this class of theories can be extended in many directions. As we discussed, there are regions of parameter space in which the shape of the three-point function is quite different from all the ones for which a data analysis has been performed, so that the constraints on these regions are very loose. It would be important to extend the analysis of CMB maps to these shapes. Even more general shapes can be obtained considering, in addition to the $(\partial^2\pi)^3$ operators, operators with less derivatives (Galilean invariant or not). It is technically natural to keep these additional terms small, since they are not radiatively induced by the operators already included in the Lagrangian. Therefore they can give a contribution to the three-point function comparable to that of terms with more derivatives. In this paper we have just performed a quite preliminary and qualitative study of the four-point function signal. Since in these models this signal is larger than in models with a reduced speed of sound (for similar values of the three-point function) and it can become very large imposing an approximate $Z_2$ symmetry, we believe it is worthwhile to characterize the four-point signal and compare it with data.
From the point of view of model building, it is interesting to note that one may go a step further and study a theory in which interactions start with three or more derivatives per field. Although this cannot be dictated by any symmetry – the standard kinetic term is [*not*]{} invariant under a constant shift of the second derivative – terms with less derivatives in the interactions will not be generated. It may be interesting to study the phenomenology of these higher derivative models. Another possible direction of further study is the multi-field scenario: as in the case of models with reduced speed of sound [@Langlois:2008qf], multi-field models may have interesting signatures and it may be useful to explore the effects of the Galilean symmetry in the effective theory of multi-field inflation [@Senatore:2010wk].
Ackowledgements {#ackowledgements .unnumbered}
===============
It is a pleasure to thank Andrei Gruzinov, Lam Hui, Mehrdad Mirbabayi, Alberto Nicolis and Matias Zaldarriaga for useful discussions. P. C. thanks Leonardo Senatore and Matias Zaldarriaga for stressing the effective field theory approach to inflation. The work of G. D’A. is supported by a James Arthur fellowship.
\[app:NR\]Non-renormalization of Galilean operators
===================================================
In this Appendix we complete the analysis of the non-renormalization properties of Galilean interactions in the presence of gravity. In Section \[sec:action\] we discussed what happens when the leading Galilean terms of the form $(\partial \psi)^2 (\partial^2 \psi)^{n}$ are zero. In flat space these operators are not renormalized, but one may wonder whether in the presence of an external gravitational field generic terms of the form $(\partial \psi)^4 (\partial^2 \psi)^{n-2} R$ can be generated. This would be relevant in our construction, since Galilean terms with the minimum number of derivatives were introduced to get rid of terms linear in $\pi$. We show in the following that operators of the form $(\partial \psi)^4 (\partial^2 \psi)^{n-2} R$ cannot appear, because vertices generated by loop diagrams with $n$ fields $\psi$ must have at least $2n$ derivatives also on curved backgrounds.
To make clear the logic of our argument let us review first why this is the case in flat space. We consider 1PI graphs that involve the leading Galilean interactions. Each vertex has at least 2 internal lines, otherwise the graph is not 1PI. We want to show that we can always take the two lines with only one derivative to be the internal ones, so that the resulting graph has more derivatives than the minimal Galilean vertex. Following [@Endlich:2010zj], if we pick two internal lines and call them $\phi$ the vertices can be schematically of the form $$\begin{gathered}
(\partial^2 \psi)^{n-2} \partial \phi \, \partial \phi \label{didi} \\
\partial \psi (\partial^2 \psi)^{n-3} \, \partial^2 \phi \, \partial \phi \label{di2di} \\
\partial \psi \partial \psi (\partial^2 \psi)^{n-4} \, \partial^2 \phi \, \partial^2 \phi \, \label{di2di2}\end{gathered}$$ however it is easy to see that we can always rewrite the last two in the form of the first, namely with only one derivative acting on each $\phi$. For (\[di2di\]) it is immediate since \[intbyparts\] \_\_ \_ = \_[( ]{} (\_[)]{} \_ ) - \_ (\_ \_ ) and integrating by parts we can move one derivative to the other lines. The interaction (\[di2di2\]) can be also be put in the form (\[di2di\]). If we write it as $A^{\mu\nu\rho\sigma} \partial_\mu \partial_\nu \phi \, \partial_\rho \partial_\sigma \phi$, we see that it gives a contribution to the equation of motion of the form $A^{\mu\nu\rho\sigma} \partial_\mu \partial_\nu \partial_\rho \partial_\sigma \phi$ that must vanish because the e.o.m. obtained from minimal Galilean terms has 2 derivatives per field. This is possible only if the totally symmetric part of $A^{\mu\nu\rho\sigma}$ is zero. However, since the structure $ \partial_\mu \partial_\nu \phi \, \partial_\rho \partial_\sigma \phi$ has less symmetries under permutation of indices, it doesn’t imply that the vertex equally vanishes. In particular, $A$ can be antisymmetric in the exchange $[ \mu \rho ]$ and symmetric in all the others, thus giving a vanishing contribution to the e.o.m. but a non-zero vertex. In this case anyhow we can write \[trederiv\] \_\_[\[]{} \_[\]]{} \_ = \_[\[ ]{} (\_ \_[\]]{}\_ ) and integrating by parts we end up with a vertex of the form (\[di2di\]) and then use again (\[intbyparts\]) to complete the argument.
Now we want to see what changes if we include an external gravitational field. The starting point is the minimal covariantization of the leading Galilean interactions and we consider again 1PI graphs with $n$ external fields $\psi$ plus external graviton lines. There are two possibilities: when gravitons are attached to internal lines the vertices are the same as in flat space and we can proceed as before in order to finish with one derivative on each of the two $\phi$. This will simply add $g$ or $R$ or $\nabla R$ etc. to a $(\partial^2 \psi)^n$ graph. If instead the gravitons lines go in the vertices we can try to repeat the same steps we did before with the substitution of ordinary derivatives with covariant ones. The extra derivative in the structure $\nabla \nabla \phi \, \nabla \phi$ can be moved to the other lines using the identity (\[intbyparts\]) that is valid also with $\nabla$ instead of $\partial$ since it involves at most two derivatives on the same scalar field and so there is no problem in commuting covariant derivatives. The difference arises when we consider the vertex $A^{\mu\nu\rho\sigma} \nabla_\mu \nabla_\nu \phi \, \nabla_\rho \nabla_\sigma \phi$. Because $A$ contains at most two covariant derivatives on $\psi$ and no $R$, its symmetries are the same as in flat space: the only component different from zero can be taken to be antisymmetric in $[ \mu \rho ]$. Now though eq. (\[trederiv\]) gets an extra piece proportional to the Riemann tensor from the commutator of covariant derivatives: \_\_[\[]{} \_[\]]{} \_ = \_[\[ ]{} (\_ \_[\]]{}\_ ) + R\^\_ \_ \_ . The first term can be integrated by parts as before while from the second we see that it is possible to trade two external derivatives in the vertex for a factor of $R$. However, the minimal number of derivatives with $n$ fields $\psi$ is still $2n$ and because of that a term of the form $(\partial \psi)^4 (\partial^2 \psi)^{n-4} R$ cannot be generated.
\[app:shapes\]Calculation of the bispectra
==========================================
At leading order in slow roll the field $\pi_k(\tau)$ and its derivatives, which must be inserted in in order to compute the bispectrum, read in conformal time $$\pi_k(\tau) = i\frac{1-i k \tau}{2\sqrt{{\varepsilon}k^3}{M_{\textrm{Pl}}}}
e^{ik\tau} \;,\quad
\dot\pi_k(\tau) = -i \frac{H k^2\tau^2}{2\sqrt{{\varepsilon}k^3}{M_{\textrm{Pl}}}}
e^{ik\tau} \;, \quad
\ddot\pi_k(\tau) = i \frac{H^2 k^2\tau^2}{2\sqrt{{\varepsilon}k^3}{M_{\textrm{Pl}}}}
(2+ik\tau)e^{ik\tau} \;.$$ Hence, from all the modes in there will be a factor $$\frac{1}{2^6 {\varepsilon}^3 k_1^3k_2^3k_3^3 {M_{\textrm{Pl}}}^6}e^{i(k_1+k_2+k_3)\tau}$$ which is common to all interactions. Additional factors for the different fields in each interaction must be accounted for according to the following rules: $$\pi \longrightarrow 1 - ik_i\tau \;,\qquad
\dot\pi \longrightarrow -H k_i^2\tau^2 \;,\qquad
\ddot\pi \longrightarrow H^2 k_i^2\tau^2(2+ik_i\tau) \;, \qquad
\frac{\vec {\partial}}{a} \longrightarrow -iH\vec k_i\tau \;.$$ Since we are looking at interactions with six derivatives, there will also be an overall factor of $H^6$. Recalling the definition $\Delta_\Phi=\frac{9}{25}H^2/(4 {\varepsilon}{M_{\textrm{Pl}}}^2)$, the generic bispectrum will thus look like $$F_{M_i}(k_1, k_2, k_3) = -i
\frac{5 M_iH}{12 {\varepsilon}{M_{\textrm{Pl}}}^2}
\frac{\Delta_\Phi^2}{k_1^3k_2^3k_3^3} \int_{-\infty}^0 \mathrm{d}\tau \,
P_{M_i}(k_i,\tau)e^{ik_t\tau} + \mathrm{c.c.} + \mathrm{permutations} \,,$$ where the function $P(k_i,\tau)$ contains all the appropriate additional factors for the interaction as specified above, divided by $H^6\tau^4$ (because of the factor of $\tau^{-4}$ in and the factor of $H^6$ we extracted).
It is convenient to shift to the adimensional integration variable $y=k_t\tau$, and introduce a dummy parameter $\mu$ in the phase ($e^{iy}\rightarrow e^{i\mu y}$) to be set to one at the end of the computation. By doing this one can substitute $\tau\rightarrow -(i/k_t)\frac{\mathrm{d}}{\mathrm{d}\mu}$ in the function $P$, which can thus be pulled out from the integration: all is left to do is therefore the simple integral of a phase, $\int_{-\infty}^0e^{i\mu y} = -i/\mu$. After all these steps the final result is $$F_{M_i}(k_1, k_2, k_3) =
- \frac{5 M_iH}{6 {\varepsilon}{M_{\textrm{Pl}}}^2} \frac{\Delta_\Phi^2}{k_1^3k_2^3k_3^3 k_t}
\bigg[\hat P_{M_i} \bigg(k_i,-\frac{i}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\bigg) \frac{1}{\mu}\bigg]_{\mu=1}
+ \mathrm{permutations}\,,$$ where the factor of 2 comes from the complex conjugate (since the result is real). In other words, the specific momentum dependence of each interaction is contained in the operator $\hat P$ that must be applied to $1/\mu$ before setting $\mu=1$. The operator associated to a cubic interaction (but the argument can be generalized to any interaction) with $n$ space gradients is given by the composition of
i\) 3 differential operators according to the following substitutions: $$\pi \longrightarrow
1- \frac{k_i}{k_t} \frac{\mathrm{d}}{\mathrm{d}\mu}
\;,\qquad
\dot\pi \longrightarrow
\bigg(\frac{k_i}{k_t}\bigg)^{\!2} \frac{\mathrm{d^2}}{\mathrm{d}\mu^2}
\;,\qquad
\ddot\pi \longrightarrow
- \bigg(\frac{k_i}{k_t}\bigg)^{\!2} \bigg(2+\frac{k_i}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\bigg) \frac{\mathrm{d^2}}{\mathrm{d}\mu^2}\;;$$
ii\) $n$ operators $- (\vec k_i/k_t) \frac{\mathrm{d}}{\mathrm{d}\mu}$, one for each spatial derivative;
iii\) one operator $[(1/k_t)\frac{\mathrm{d}}{\mathrm{d}\mu}]^{-4}$ to account for $\tau^{-4}$.
The operators generated by the interactions studied in this paper are $$\begin{gathered}
-\frac{k_1^2k_2^2k_3^2}{k_t^2}
\left(2+\frac{k_1}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\right)\!
\left(2+\frac{k_2}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\right)\!
\left(2+\frac{k_3}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\right)
\frac{\mathrm{d}^2}{\mathrm{d}\mu^2}, \\
- k_1^2(\vec k_2\cdot\vec k_3)
\left(2+\frac{k_1}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\right)\!
\bigg[1-\frac{k_2}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}
-\bigg(\frac{k_2}{k_t}\bigg)^{\!2}\frac{\mathrm{d}^2}{\mathrm{d}\mu^2}\bigg]\!
\bigg[1-\frac{k_3}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}
-\bigg(\frac{k_3}{k_t}\bigg)^{\!2}\frac{\mathrm{d}^2}{\mathrm{d}\mu^2}\bigg]\end{gathered}$$ and $$-\frac{k_1^2k_2^2k_3^2}{k_t^2}
\bigg[\frac{(\vec k_2\cdot\vec k_3)^2}{k_2^2 k_3^2}\!
\left(2+\frac{k_1}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\right)\!
\left(1-\frac{k_2}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\right)\!
\left(1-\frac{k_3}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\right)\!
+ 2\left(2+\frac{k_2}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\right)\!
\left(2+\frac{k_3}{k_t}\frac{\mathrm{d}}{\mathrm{d}\mu}\right) - 3 \bigg]
\frac{\mathrm{d}^2}{\mathrm{d}\mu^2}$$ for $M_1$, $M_2$ and $M_3$ respectively. All of them must be applied to $1/\mu$ and evaluated at $\mu=1$.
The calculation of the different bispectra is now straightforward given the relation $$\bigg[\frac{\mathrm{d}^n}{\mathrm{d}\mu^n}\frac{1}{\mu}\bigg]_{\mu=1}
= (-1)^n n!$$ and gives the results quoted in Eqs. , and .
\[app:template\]A template that vanishes in the squeezed limit
==============================================================
For the analysis of CMB data the enfolded template, equation , is good enough for constraining the non-Gaussianity induced by the operator proportional to $M_3$ in . Since the enfolded template is a linear combination of the equilateral and orthogonal templates, its amplitude can be constrained using the recent limits on these shapes, as discussed in equation .
However, the enfolded template fails to capture some important qualitative behaviour of the physical shape as we will now explain. There are competitive constraints on the non-Gaussianity from Large Scale Structure observations [@Slosar:2008hx]. These arise from the fact that the non-Gaussian corrections to the halo bias $\Delta b$ have a characteristic scale dependence and one expects this scale dependence to be sensitive to the squeezed limit of the non-Gaussian shape. Both the orthogonal and enfolded templates go to a constant in the squeezed limit, $\lim_{r_2\rightarrow 0}r_2^2 r_3^2 F(1,r_2,r_3) = const.$, and one could thus expect some scale dependent effect in the halo bias induced by this non-Gaussianity. However, physical shapes generated by the single-field inflation models studied in the literature go to zero in the squeezed limit [@Creminelli:2004yq; @Cheung:2007st; @Senatore:2009gt]. Indeed this can be explicitly checked in [@Senatore:2009gt] for the model from which the orthogonal shape was inspired, and for the model under study in this paper. This means that one expects the scale dependence of $\Delta b$ to be negligible for these models, even if they have a large cosine with templates which do induce a scale dependence.
This motivates us to introduce a new factorizable template which apart from having a large cosine with the physical shape also behaves like the physical shape in the squeezed limit[^13]. Let us warn the reader though that it is not clear whether the templates introduced in this Appendix are useful for computing other quantities which are interesting for LSS studies, as they might be sensitive to specific limits where the physical shape and the template differ.
The new template will contain also monomials that go like $k^{-4}$: this will give us additional freedom to impose that it behaves as $k_l^{-1}$ in the squeezed limit, $k_l \to 0$. One way to build the template is to take a linear combination of the new monomials for which the leading divergencies $k_l^{-4}$ and $k_l^{-3}$ cancel. We do not have enough freedom to cancel also the $k_l^{-2}$ divergence, but we can at least require that the divergence is independent of the direction we approach the limit. This will allow a cancellation with the “standard” monomials. The only combination with such properties is given by $$F_1(k_1,k_2,k_3) = \frac{16}{9 k_1 k_2 k_3^4} + \frac{k_1^2}{9k_2^4 k_3^4} - \frac{1}{k_1^2 k_3^4} - \frac{1}{k_2^2 k_3^4} + \mathrm{cyclic\;perms.}\,.$$ Now we can combine this with the usual monomials diverging as $k^{-3}$ and $k^{-2}$ $$\begin{aligned}
F_2(k_1,k_2,k_3) &= \frac{1}{k_1^3 k_2^3} - \frac{1}{k_1 k_2^2 k_3^3} - \frac{1}{k_2 k_1^2 k_3^3} + \mathrm{cyclic\;perms.}\,,\\
F_3(k_1,k_2,k_3) &= \frac{1}{k_1^2 k_2^2 k_3^2}\,.\end{aligned}$$ The residual $k_l^{-2}$ divergence of $F_1$, which does not depend on the direction, can be cancelled by a proper addition of $F_3$. This gives a new template, besides the standard equilateral one, with the correct squeezed limit. We have thus a one-parameter family of templates going as $k_l^{-1}$ in the squeezed limit: $$F = A {f_{\textrm{NL}}}\Delta_\Phi^2\left[\alpha F_1 + F_2 + 2(1 + \alpha) F_3\right]\,,
\label{app:eq:template}$$ where $\alpha$ is a free coefficient which can be fixed by requiring the cosine with the physical shapes to be maximum, and $A$ is some normalization which can be fixed, for example, such that the template equals the local one in the equilateral limit[^14]. The equilateral template corresponds to $\alpha = 0$. In table \[tab:alpha\] we show the values of $\alpha$ that maximize the cosine with the shapes of different physical models. In figure \[app:fig:template\] we show the form of the template that approximates the shape of the non-Gaussianity generated by the operator proportional to $M_3$.
Model $\alpha$ $|\cos|$
-------------------------- ---------- ----------
$M_3$ 0.71 0.95
[@Senatore:2009gt], orth 0.55 0.98
[@Senatore:2009gt], flat 0.60 0.98
: \[tab:alpha\] Values of $\alpha$ maximizing the cosine of the template with different physical shapes, and corresponding value of the cosine. In the first line we give the values for $F_{M_3}$, in the second and third line we compare with the two shapes obtained in Ref. \[17\] as difference of equilateral shapes. Namely, the two shapes are obtained setting $\tilde{c}_3 = -5.4$ (orthogonal) and $\tilde{c}_3 = -6$ (flat) in Eq. (16) of that Reference.
![\[app:fig:template\]In the left panel this figure we show the template that goes to zero in the squeezed limit and has a large cosine with the physical bispectrum shape generated by the operator proportional to $M_3$, which is shown in the right panel for comparison.](plotImproved.eps){width="\textwidth"}
![\[app:fig:template\]In the left panel this figure we show the template that goes to zero in the squeezed limit and has a large cosine with the physical bispectrum shape generated by the operator proportional to $M_3$, which is shown in the right panel for comparison.](M3.eps){width="\textwidth"}
[99]{}
A. Nicolis, R. Rattazzi and E. Trincherini, “The galileon as a local modification of gravity,” Phys. Rev. D [**79**]{}, 064036 (2009) \[arXiv:0811.2197 \[hep-th\]\]. C. Burrage, C. de Rham, D. Seery and A. J. Tolley, “Galileon inflation,” arXiv:1009.2497 \[hep-th\]. T. Kobayashi, M. Yamaguchi and J. Yokoyama, “G-inflation: inflation driven by the Galileon field,” arXiv:1008.0603 \[hep-th\]. S. Mizuno and K. Koyama, “Primordial non-Gaussianity from the DBI Galileons,” arXiv:1009.0677 \[hep-th\]. M. Alishahiha, E. Silverstein and D. Tong, “DBI in the sky,” Phys. Rev. D [**70**]{}, 123505 (2004) \[arXiv:hep-th/0404084\]. X. Chen, M. x. Huang, S. Kachru and G. Shiu, “Observational signatures and non-Gaussianities of general single field inflation,” JCAP [**0701**]{}, 002 (2007) \[arXiv:hep-th/0605045\]. C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, “The Effective Field Theory of Inflation,” JHEP [**0803**]{}, 014 (2008) \[arXiv:0709.0293 \[hep-th\]\]. P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore, “Starting the universe: Stable violation of the null energy condition and non-standard cosmologies,” JHEP [**0612**]{}, 080 (2006) \[arXiv:hep-th/0606090\]. P. Creminelli and A. Nicolis, “On the validity of derivatively coupled theories", in preparation.
J. Z. Simon, “Higher derivative Lagrangians, nonlocality, problems and solutions,” Phys. Rev. D [**41**]{}, 3720 (1990). J. Z. Simon, “The Stability of flat space, semiclassical gravity, and higher derivatives,” Phys. Rev. D [**43**]{}, 3308 (1991). M. A. Luty, M. Porrati and R. Rattazzi, “Strong interactions and stability in the DGP model,” JHEP [**0309**]{}, 029 (2003) \[arXiv:hep-th/0303116\]. J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP [**0305**]{} (2003) 013 \[arXiv:astro-ph/0210603\]. C. Deffayet, G. Esposito-Farese and A. Vikman, “Covariant Galileon,” Phys. Rev. D [**79**]{}, 084003 (2009) \[arXiv:0901.1314 \[hep-th\]\]. S. Weinberg, “Quantum contributions to cosmological correlations,” Phys. Rev. D [**72**]{} (2005) 043514 \[arXiv:hep-th/0506236\]. D. Babich, P. Creminelli and M. Zaldarriaga, “The shape of non-Gaussianities,” JCAP [**0408**]{}, 009 (2004) \[arXiv:astro-ph/0405356\]. L. Senatore, K. M. Smith and M. Zaldarriaga, “Non-Gaussianities in Single Field Inflation and their Optimal Limits from the WMAP 5-year Data,” JCAP [**1001**]{}, 028 (2010) \[arXiv:0905.3746 \[astro-ph.CO\]\]. N. Bartolo, M. Fasiello, S. Matarrese and A. Riotto, “Large non-Gaussianities in the Effective Field Theory Approach to Single-Field Inflation: the Bispectrum,” JCAP [**1008**]{}, 008 (2010) \[arXiv:1004.0893 \[astro-ph.CO\]\]. P. Creminelli, A. Nicolis, L. Senatore, M. Tegmark and M. Zaldarriaga, “Limits on non-Gaussianities from WMAP data,” JCAP [**0605**]{} (2006) 004 \[arXiv:astro-ph/0509029\]. L. -M. Wang, M. Kamionkowski, “The Cosmic microwave background bispectrum and inflation,” Phys. Rev. [**D61** ]{} (2000) 063504. \[astro-ph/9907431\].
E. Komatsu, K. M. Smith, J. Dunkley [*et al.*]{}, “Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation,” \[arXiv:1001.4538 \[astro-ph.CO\]\].
P. D. Meerburg, J. P. van der Schaar, P. S. Corasaniti, “Signatures of Initial State Modifications on Bispectrum Statistics,” JCAP [**0905**]{}, 018 (2009). \[arXiv:0901.4044 \[hep-th\]\].
P. Creminelli and M. Zaldarriaga, “Single field consistency relation for the 3-point function,” JCAP [**0410**]{}, 006 (2004) \[arXiv:astro-ph/0407059\]. P. Catelan, F. Lucchin, S. Matarrese [*et al.*]{}, “The bias field of dark matter halos,” Mon. Not. Roy. Astron. Soc. [**297** ]{} (1998) 692-712. \[astro-ph/9708067\].
N. Dalal, O. Dore, D. Huterer [*et al.*]{}, “The imprints of primordial non-gaussianities on large-scale structure: scale dependent bias and abundance of virialized objects,” Phys. Rev. [**D77** ]{} (2008) 123514. \[arXiv:0710.4560 \[astro-ph\]\].
L. Verde and S. Matarrese, “Detectability of the effect of Inflationary non-Gaussianity on halo bias,” Astrophys. J. [**706**]{} (2009) L91 \[arXiv:0909.3224 \[astro-ph.CO\]\]. N. Bartolo, M. Fasiello, S. Matarrese and A. Riotto, “Large non-Gaussianities in the Effective Field Theory Approach to Single-Field Inflation: the Trispectrum,” JCAP [**1009**]{}, 035 (2010) \[arXiv:1006.5411 \[astro-ph.CO\]\]. L. Senatore and M. Zaldarriaga, “A Naturally Large Four-Point Function in Single Field Inflation,” arXiv:1004.1201 \[hep-th\]. D. Langlois, S. Renaux-Petel, D. A. Steer and T. Tanaka, “Primordial perturbations and non-Gaussianities in DBI and general multi-field inflation,” Phys. Rev. D [**78**]{}, 063523 (2008) \[arXiv:0806.0336 \[hep-th\]\]. L. Senatore and M. Zaldarriaga, “The Effective Field Theory of Multifield Inflation,” arXiv:1009.2093 \[hep-th\]. S. Endlich, K. Hinterbichler, L. Hui, A. Nicolis and J. Wang, “Derrick’s theorem beyond a potential,” arXiv:1002.4873 \[hep-th\].
A. Slosar, C. Hirata, U. Seljak [*et al.*]{}, “Constraints on local primordial non-Gaussianity from large scale structure,” JCAP [**0808** ]{} (2008) 031. \[arXiv:0805.3580 \[astro-ph\]\].
[^1]: This quantity is usually called ${\mathcal{R}}$. However, we follow the notation of [@Maldacena:2002vr], which has become common in the literature.
[^2]: For example one of the key point of the geometrical construction is that, given that there is a privileged slicing of the spacetime, one can project all tensors parallely or perpendicularly to it. However these projectors, once expressed in terms of the scalar degree of freedom, are not Galilean invariant as they contain a single derivative.
[^3]: We thank A. Gruzinov, M. Mirbabayi, L. Senatore and M. Zaldarriaga for pointing this out to us.
[^4]: Reference [@Mizuno:2010ag] claims that the shape induced by the fifth operator in is independent of the shapes given by operators with three derivatives. However it is straightforward to check that it can be written as a linear combination of the shapes induced by $\dot\pi^3$ and $\dot\pi (\nabla\pi)^2$.
[^5]: Notice that this implies that the kinetic term is standard, i.e. $c_s=1$.
[^6]: We do not consider internal graviton lines, as these would be suppressed by powers of ${M_{\textrm{Pl}}}$.
[^7]: This represents a nice consistency check of our statement before, that generic operators of the form $(\partial\psi)^2 (\partial^2\psi)^n R$ are not generated. As the operator is Galilean invariant, it will not generate non-Galilean terms, unless paying an additional power of $R$, i.e. $(\partial\psi)^2 (\partial^2\psi)^n R^2$, consistently with what we said above. See also Appendix \[app:NR\]. Moreover, as minimal Galilean terms are not renormalized, we are not going to generate the quintic one, which does not come in the process of rewriting the single trace operators.
[^8]: This independence is not general: not all the cubic operators $(\partial^2\pi)^3$ are independent from the quadratic ones. It is straightforward, however, to check that the ones which are dependent contain a $\Box\pi$ and are thus neglected here. Another example is given by the two minimal Galilean operators that we are considering: also these are thus neglected in the calculation of the three-point function.
[^9]: For CMB applications it is useful to introduce a 2d scalar product that quantifies the similarity of two shapes in a given CMB experiment [@Babich:2004gb]. Here we stick to the 3d definition.
[^10]: There exists a linear combination of the equilateral and orthogonal templates, $F \propto -0.5 F^{\mathrm{orth}} + 0.86 F^{\mathrm{equil}}$, which has an even larger cosine (0.98) with $F_{M_3}$. However, it is good enough for our purpose to stick to the enfolded template, which has a cosine of 0.94 and is already known in the literature.
[^11]: The contribution to the four-point function of operators with higher spatial derivatives was considered in [@Bartolo:2010di].
[^12]: Actually the operator $(\Box\psi+3H)^4$ does not contain quintic or higher order terms, so that the parity symmetry would be exact. However this operator gives a four-point function which is four times suppressed by slow-roll as all the legs are proportional to the linear equation of motion.
[^13]: A similar approach has been taken in Appendix B of [@Senatore:2009gt]. Notice that in the published version of the paper, the proposed template contains some typos, that have been corrected in the most recent version. We thank L. Senatore for correspondence about this point.
[^14]: The total number of monomials is $7$, so that we have 6 free parameters and an overall normalization. Requiring a $k_l^{-1}$ divergence implies 4 conditions, leaving a 2 parameter set of templates with the correct squeezed behaviour. The construction given in the text corresponds to a particular one-parameter subset of templates, which turns out not to contain the monomial $k_1/(k_2^4 k_3^3) +$perms. In the latest version of [@Senatore:2009gt] a different one-parameter set of templates is given.
|
---
author:
- |
E. van den Berg\
IBM Watson, Yorktown Heights, USA
bibliography:
- 'bibliography.bib'
title: Some Insights into the Geometry and Training of Neural Networks
---
Introduction {#Sec:Introduction}
============
Formation of decision regions {#Sec:DecisionRegions}
=============================
Region properties and approximation {#Sec:Regions}
===================================
Gradients {#Sec:Gradients}
=========
Optimization {#Sec:Training}
============
Conclusions {#Sec:Discussion}
===========
|
---
abstract: 'The ${\mathcal{L}}_2$ discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights were introduced by Sloan and Woźniakowski to take into account of the relative importance of the discrepancy of lower dimensional projections. As known under the name of quasi-Monte Carlo methods, point sets with small weighted ${\mathcal{L}}_2$ discrepancy are quite useful in numerical integration. In this study, we investigate the component-by-component construction of polynomial lattice rules over the finite field ${\mathbb{F}}_2$ whose scrambled point sets have small mean square weighted ${\mathcal{L}}_2$ discrepancy. We prove an upper bound on this discrepancy which converges at almost the best possible rate of $N^{-2+\delta}$ for all $\delta>0$, where $N$ denotes the number of points. Numerical experiments confirm that the performance of our constructed polynomial lattice point sets is comparable or even superior to that of Sobol’ sequences.'
author:
- |
Takashi Goda\
Graduate School of Information Science and Technology,\
The University of Tokyo,\
2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-0032\
[goda@iba.t.u-tokyo.ac.jp](goda@iba.t.u-tokyo.ac.jp)
title: 'Construction of polynomial lattice rules over ${\mathbb{F}}_2$ with small mean square weighted ${\mathcal{L}}_2$ discrepancy'
---
Key words: polynomial lattice rules, weighted ${\mathcal{L}}_2$ discrepancy, numerical integration, randomized quasi-Monte Carlo
Introduction {#intro}
============
In this paper, we study the approximation of an $s$-dimensional integral over the unite cube $[0,1)^s$ $$\begin{aligned}
I(f)=\int_{[0,1)^s}f({\boldsymbol{x}})d{\boldsymbol{x}},
\end{aligned}$$ by averaging function evaluations at $N$ points with equal weights $$\begin{aligned}
Q(f)=\frac{1}{N}\sum_{n=0}^{N-1}f({\boldsymbol{x}}_n) .
\end{aligned}$$ Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods choose the point set $P_{N,s}=\{{\boldsymbol{x}}_0,\ldots, {\boldsymbol{x}}_{N-1}\}$ randomly and deterministically, respectively. The aim of QMC methods is to distribute the quadrature points as uniformly as possible so as to yield a small integration error. This idea is supported by the general form of various integration error bounds $$\begin{aligned}
\label{eq:error_bound}
|I(f)-Q(f)|\le V(f)D(P_{N,s}) ,
\end{aligned}$$ where $V(f)$ is the variation of the integrand $f$ in a certain sense, which depends only on $f$, while $D(P_{N,s})$ is the corresponding discrepancy of the point set $P_{N,s}$, which measures the equidistribution properties of $P_{N,s}$ and depends only on $P_{N,s}$. Thus the smaller $D(P_{N,s})$ is, the smaller an integration error we can expect. The most well-known bound is the so-called Koksma-Hlawka inequality in which $V(f)$ is the variation of $f$ in the sense of Hardy and Krause and $D(P_{N,s})$ is the star discrepancy of $P_{N,s}$, see for example [@Lem09; @Nie92a].
Randomization of the QMC point set is helpful to obtain statistical information on the integration error and sometimes even enables us to improve the rate of convergence for numerical integration. There have been several methods introduced for randomization [@CP76; @Mat98; @Owe95; @TF03]. Using the linearity of expectation and (\[eq:error\_bound\]), the mean square integration error is upper-bounded by $$\begin{aligned}
{\mathbb{E}}\left[|I(f)-Q(f)|^2\right]\le V^2(f){\mathbb{E}}\left[D^2(\tilde{P}_{N,s})\right] ,
\end{aligned}$$ where the expectation is taken with respect to all the possible randomized point sets $\tilde{P}_{N,s}$ of $P_{N,s}$. Hence, the mean square discrepancy becomes a meaningful measure of the equidistribution properties of $P_{N,s}$ in this setting.
Among others for $D(P_{N,s})$, the ${\mathcal{L}}_2$ discrepancy is one of the popular measures of the equidistribution properties of point sets. The relationship between the ${\mathcal{L}}_2$ discrepancy and numerical integration has been often discussed in the literature, see for example [@Hic98; @NW10; @SW98; @Woz91; @Zar68]. Sloan and Woźniakowski [@SW98] introduced the concept of the weighted ${\mathcal{L}}_2$ discrepancy to take the relative importance of the discrepancy of lower dimensional projections into account. It provides a part of the reason why QMC methods are successful even for very large values of $s$, as often reported in the practical applications to financial problems [@CMO97; @NT96; @PT95]. This phenomenon is hard to explain by the classical integration error bounds. Hence, construction of point sets with small weighted ${\mathcal{L}}_2$ discrepancy is of particular interest to practitioners. Especially, in this paper, we focus on constructing randomized QMC point sets with small mean square weighted ${\mathcal{L}}_2$ discrepancy.
In order to give the definition of the weighted ${\mathcal{L}}_2$ discrepancy, we introduce some notations first. For a point set $P_{N,s}=\{{\boldsymbol{x}}_0,\ldots, {\boldsymbol{x}}_{N-1}\}$ in the unit cube $[0,1)^s$, the local discrepancy function is defined as $$\begin{aligned}
\Delta({\boldsymbol{t}}):=\frac{A_N([{\boldsymbol{0}},{\boldsymbol{t}}),P_{N,s})}{N}-t_1\cdots t_s ,
\end{aligned}$$ where ${\boldsymbol{t}}=(t_1,\ldots, t_s)$ is a vector from $[0,1)^s$, $[{\boldsymbol{0}},{\boldsymbol{t}})$ is the axis-parallel box of the form $[0,t_1)\times \cdots \times [0,t_s)$, and $A_N([{\boldsymbol{0}},{\boldsymbol{t}}),P_{N,s})$ denotes the number of indices $n$ with ${\boldsymbol{x}}_n\in [{\boldsymbol{0}},{\boldsymbol{t}})$. Let $I_s=\{1,\ldots,s\}$ and let $\gamma_u$ be a non-negative real number for $u\subseteq I_s$. We denote by $|u|$ the cardinality of $u$ and by ${\boldsymbol{t}}_u$ a vector from $[0,1)^{|u|}$ containing all the components of ${\boldsymbol{t}}\in [0,1)^s$ whose indices are in $u$. Further, let $d{\boldsymbol{t}}_u=\prod_{j\in u}dt_j$ and let $({\boldsymbol{t}}_u,{\boldsymbol{1}})$ denote a vector from $[0,1)^s$ with all the components whose indices are not in $u$ replaced by one. Then the weighted ${\mathcal{L}}_2$ discrepancy of the point set $P_{N,s}$ is given by $$\begin{aligned}
{\mathcal{L}}_{2,N,{\boldsymbol{\gamma}}}(P_{N,s})= \left(\sum_{\emptyset \ne u\subseteq I_s}\gamma_u \int_{[0,1]^{u}}|\Delta({\boldsymbol{t}}_u,{\boldsymbol{1}})|^2 d{\boldsymbol{t}}_u\right)^{1/2} .
\end{aligned}$$ We can recover the classical ${\mathcal{L}}_2$ discrepancy by choosing $\gamma_{I_s}=1$ and $\gamma_u=0$ for $u\subset I_s$. The most famous choice of $\gamma_u$ is the so-called product weights, that is, $\gamma_u=\prod_{j\in u}\gamma_j$ for all $u\subseteq I_s$. The following proposition generalizes the well-known formula for the classical ${\mathcal{L}}_2$ discrepancy introduced by Warnock, see for example [@DP10; @Mat99].
\[prop:L2\_disc\] For any point set $P_{N,s}=\{{\boldsymbol{x}}_0,\ldots, {\boldsymbol{x}}_{N-1}\}$ in $[0,1)^s$ and any sequence ${\boldsymbol{\gamma}}=(\gamma_u)_{u\subseteq I_s}$ of weights, we have $$\begin{aligned}
& {\mathcal{L}}_{2,N,{\boldsymbol{\gamma}}}^2(P_{N,s}) \\
= & \sum_{\emptyset \ne u\subseteq I_s}\gamma_u\left[ \frac{1}{3^{|u|}}-\frac{2}{N}\sum_{n=0}^{N-1}\prod_{j\in u}\frac{1-x^2_{n,j}}{2}+\frac{1}{N^2}\sum_{n,n'=0}^{N-1}\prod_{j\in u}(1-\max(x_{n,j},x_{n',j}))\right] ,
\end{aligned}$$ where $x_{n,j}$ is the $j$-th component of the point ${\boldsymbol{x}}_n$.
There are two prominent construction principles of QMC point sets: lattice rules [@Nie92a; @SJ94] and digital $(t,m,s)$-nets [@DP10; @Nie92a]. In this study, we are concerned with polynomial lattice rules which can be categorized into the latter, while its name comes from the analogy with lattice rules. Since first introduced by Niederreiter [@Nie92b], polynomial lattice rules have been extensively investigated, see for example [@DP10; @Lec04]. In the following, we give the definition of polynomial lattice rules for the case of base 2 because we will only deal with that case.
Let ${\mathbb{F}}_2:=\{0,1\}$ be the two element field and denote by ${\mathbb{F}}_2((x^{-1}))$ the field of formal Laurent series over ${\mathbb{F}}_2$. Every element of ${\mathbb{F}}_2((x^{-1}))$ has the form $$\begin{aligned}
L = \sum_{l=w}^{\infty}t_l x^{-l} ,
\end{aligned}$$ where $w$ is an arbitrary integer and all $t_l\in {\mathbb{F}}_2$. Further, we denote by ${\mathbb{F}}_2[x]$ the set of all polynomials over ${\mathbb{F}}_2$. For a given integer $m$, we define the map $v_m$ from ${\mathbb{F}}_2((x^{-1}))$ to the interval $[0,1)$ by $$\begin{aligned}
v_m\left( \sum_{l=w}^{\infty}t_l x^{-l}\right) =\sum_{l=\max(1,w)}^{m}t_l 2^{-l}.
\end{aligned}$$ We often identify a non-negative integer $k$ whose dyadic expansion is given by $k=\kappa_0+\kappa_1 2+\cdots +\kappa_a 2^a$ with the polynomial $k(x)=\kappa_0+\kappa_1 x\cdots +\kappa_a x^a \in {\mathbb{F}}_2[x]$. For ${\boldsymbol{k}}=(k_1\cdots, k_s)\in ({\mathbb{F}}_2[x])^s$ and ${\boldsymbol{q}}=(q_1\cdots, q_s)\in ({\mathbb{F}}_2[x])^s$, we define the inner product as $$\begin{aligned}
{\boldsymbol{k}}\cdot {\boldsymbol{q}}=\sum_{j=1}^{s}k_j q_j \in {\mathbb{F}}_2[x] ,
\end{aligned}$$ and we write $q\equiv 0 \pmod p$ if $p$ divides $q$ in ${\mathbb{F}}_2[x]$. Using these notations, the polynomial lattice point set is constructed as follows.
\[def:polynomial\_lattice\] Let $m, s \in {\mathbb{N}}$. Let $p \in {\mathbb{F}}_2[x]$ be an irreducible polynomial with $\deg(p)=m$ and let ${\boldsymbol{q}}=(q_1,\ldots,q_s) \in ({\mathbb{F}}_2[x])^s$. The polynomial lattice point set $P_{2^m,s}({\boldsymbol{q}},p)$ is the point set consisting of $2^m$ points given by $$\begin{aligned}
{\boldsymbol{x}}_n &:= \left( v_m\left( \frac{n(x)q_1(x)}{p(x)} \right) , \ldots , v_m\left( \frac{n(x)q_s(x)}{p(x)} \right) \right) \in [0,1)^s ,
\end{aligned}$$ for $0\le n<2^m$.
In the following, the notation $P_{2^m,s}({\boldsymbol{q}},p)$ implicitly means that $\deg(p)=m$ and the number of components for a vector ${\boldsymbol{q}}$ is $s$.
For randomization of the polynomial lattice point set, we consider to apply Owen’s scrambling [@Owe95; @Owe97a; @Owe97b]. It proceeds as follows. For ${\boldsymbol{x}}=(x_1,\cdots, x_s)\in [0,1)^s$, we denote the dyadic expansion by $x_j=x_{j,1}2^{-1}+x_{j,2}2^{-2}+\cdots$. Let ${\boldsymbol{y}}=(y_1,\cdots, y_s)\in [0,1)^s$ be the scrambled point of ${\boldsymbol{x}}$ whose dyadic expansion is represented by $ y_j=y_{j,1}2^{-1}+y_{j,2}2^{-2}+\cdots$. Each coordinate $y_j$ is obtained by applying permutations to each digit of $x_j$. Here the permutation applied to $x_{j,k}$ depends on $x_{j,l}$ for $1\le l\le k-1$. In particular, $y_{j,1}=\pi_j(x_{j,1}),\, y_{j,2}=\pi_{j,x_{j,1}}(x_{j,2}), y_{j,3}=\pi_{j,x_{j,1},x_{j,2}}(x_{j,3})$, and in general $$\begin{aligned}
y_{j,k}=\pi_{j,x_{j,1},\cdots, x_{j,k-1}}(x_{j,k}) ,
\end{aligned}$$ where $\pi_{j,x_{j,1},\cdots, x_{j,k-1}}$ is a random permutation of $\{0,1\}$. We choose permutations with different indices mutually independent from each other where each permutation is chosen with the same probability. Then, as shown in [@Owe95 Proposition 2], the scrambled point ${\boldsymbol{y}}$ is uniformly distributed in $[0,1)^s$.
Our aim here is to find a vector ${\boldsymbol{q}}$ with $p$ fixed, which yields a small mean square weighted ${\mathcal{L}}_2$ discrepancy. Restricting each $q_j\in {\mathbb{F}}_2[x]$ such that $q_j\ne 0$ and $\deg(q_j)<m$, the number of candidates for ${\boldsymbol{q}}$ is $(2^{m}-1)^s$, which is quite large. The component-by-component (CBC) construction can significantly reduce the computational burden by searching over all the candidates of $q_{j+1}$ while leaving the existing components ($q_1,\ldots, q_j$) unchanged. The CBC construction was first invented for lattice rules by Korobov [@Kor59] and re-discovered more recently by Sloan and Reztsov [@SR02]. It also has been applied to polynomial lattice rules. Without requiring exhaustive search, the CBC construction usually finds a good vector ${\boldsymbol{q}}$ as discussed in many previous studies, see for example [@BD11; @DKPS05; @DLP05; @KP07; @KP11]. Hence, we employ the CBC construction to find a vector ${\boldsymbol{q}}$ which gives a small mean square weighted ${\mathcal{L}}_2$ discrepancy.
We end this section with a brief outline of this paper. In the next section, we introduce Walsh functions and their useful properties. They play a central role in the analysis of the mean square weighted ${\mathcal{L}}_2$ discrepancy. In Section \[disc\], we study the mean square weighted ${\mathcal{L}}_2$ discrepancy of scrambled polynomial lattice rules. Next, in Section \[cbc\], we construct polynomial lattice rules which have small mean square weighted ${\mathcal{L}}_2$ discrepancy. We consider two cases for weights here: general weights and product weights. Our construction algorithm is extensible in $s$ for product weights, while it is not for general weights. Finding adequate construction algorithm extensible in $s$ for general weights is open for further research. We prove an upper bound on the root of this discrepancy which converges at a rate of $N^{-1+\delta}$ for all $\delta>0$, where $N=2^m$ denotes the number of points. As Roth [@Rot54] proved that the lower bound on the classical ${\mathcal{L}}_2$ discrepancy of $N$ points is given by $$\begin{aligned}
\label{eq:lower_bound}
{\mathcal{L}}_{2,N,{\boldsymbol{\gamma}}}(P_{N,s})\ge c_s\frac{(\log N)^{(s-1)/2}}{N} ,
\end{aligned}$$ where $c_s$ is a constant dependent only on $s$, our upper bound is almost best possible in the sense that a rate of $N^{-1}$ cannot be achieved. We further discuss strong tractability of our construction algorithm. Finally, in Section \[numer\], we show the performance of our constructed polynomial lattice rules and compare with that of the well-known Sobol’ sequences.
Walsh functions {#walsh}
===============
Walsh functions were first introduced by Walsh [@Wal23] and have been extensively studied for example in [@Chr55; @Fin49]. We refer to [@DP10 Appendix A] for more information on Walsh functions. In the following, ${\mathbb{N}}_0:={\mathbb{N}}\cup \{0\}$ denotes the set of non-negative integers. We first give the definition of dyadic Walsh functions for the one-dimensional case.
Let $k\in {\mathbb{N}}_0$ with dyadic expansion $k = \kappa_0+\kappa_1 2+\cdots +\kappa_{a}2^{a}$. Then, the $k$-th dyadic Walsh function ${\mathrm{wal}}_k: [0,1)\to \{-1,1\}$ is defined as $$\begin{aligned}
{\mathrm{wal}}_k(x) = (-1)^{x_1\kappa_0+\cdots+x_{a+1}\kappa_a} ,
\end{aligned}$$ for $x\in [0,1)$ with dyadic expansion $x=x_1 2^{-1}+x_2 2^{-2}+\cdots $ (unique in the sense that infinitely many of the $x_i$ are different from 1).
This definition can be generalized to the higher-dimensional case.
For $s\in {\mathbb{N}}$, let ${\boldsymbol{x}}=(x_1,\cdots, x_s)\in [0,1)^s$ and ${\boldsymbol{k}}=(k_1,\cdots, k_s)\in {\mathbb{N}}_0^s$. We define ${\mathrm{wal}}_{{\boldsymbol{k}}}: [0,1)^s \to \{-1,1\}$ by $$\begin{aligned}
{\mathrm{wal}}_{{\boldsymbol{k}}}({\boldsymbol{x}}) = \prod_{j=1}^s {\mathrm{wal}}_{k_j}(x_j) .
\end{aligned}$$
In the following, the operator $\oplus$ denotes the digitwise addition modulo $2$, that is, for $x, y\in [0,1)$ with dyadic representations $x=\sum_{i=1}^{\infty}x_i 2^{-i}$ and $y=\sum_{i=1}^{\infty}y_i 2^{-i}$, $\oplus$ is defined as $$\begin{aligned}
x\oplus y = \sum_{i=1}^{\infty}z_i 2^{-i} ,
\end{aligned}$$ where $z_i=x_i+y_i \pmod{2}$. We also define a digitwise addition for non-negative integers based on those dyadic representations. In case of vectors in $[0,1)^s$ or ${\mathbb{N}}_0^s$, the operator $\oplus$ is carried out componentwise. Further, we call $x\in [0,1)$ a dyadic rational if it can be represented by a finite dyadic expansion. The proposition below summarizes some basic properties of Walsh functions.
\[prop:walsh\] We have the following:
1. For all $k,l\in {\mathbb{N}}$ and all $x,y\in [0,1)$ with the restriction that if $x,y$ are not dyadic rationals, then $x\oplus y$ is not allowed to be a dyadic rational, we have $$\begin{aligned}
{\mathrm{wal}}_k(x){\mathrm{wal}}_l(x)={\mathrm{wal}}_{k\oplus l}(x) ,\ {\mathrm{wal}}_k(x){\mathrm{wal}}_k(y)={\mathrm{wal}}_k(x\oplus y) .
\end{aligned}$$
2. We have $$\begin{aligned}
\int_{0}^{1}{\mathrm{wal}}_0(x)dx=1 \quad \text{and} \quad \int_{0}^{1}{\mathrm{wal}}_k(x)dx=0 \quad \text{if} \ k\in {\mathbb{N}}.
\end{aligned}$$
3. For all ${\boldsymbol{k}}, {\boldsymbol{l}}\in {\mathbb{N}}_0^s$, we have $$\begin{aligned}
\int_{[0,1)^s}{\mathrm{wal}}_{{\boldsymbol{k}}}({\boldsymbol{x}}){\mathrm{wal}}_{{\boldsymbol{l}}}({\boldsymbol{x}})d{\boldsymbol{x}}= \left\{ \begin{array}{ll}
1 & \text{if} \ {\boldsymbol{k}}={\boldsymbol{l}}, \\
0 & \text{otherwise} . \\
\end{array} \right.
\end{aligned}$$
4. For $s\in {\mathbb{N}}$, the system $\{{\mathrm{wal}}_{{\boldsymbol{k}}}:\ {\boldsymbol{k}}=(k_1\ldots, k_s)\in {\mathbb{N}}_0^s\}$ is a complete orthonormal system in ${\mathcal{L}}_2([0,1]^s)$.
Furthermore, in order to introduce an important relation between Walsh functions and polynomial lattice rules as described below in Lemma \[lamma:dual\_walsh\], we add one more notation and introduce the concept of the so-called [*dual polynomial lattice*]{} of a polynomial lattice point set $P_{2^m,s}({\boldsymbol{q}},p)$. For $k\in {\mathbb{N}}_0$ with dyadic expansion $k=k_0+k_1 2+\cdots $, ${\,\mathrm{tr}}_m(k)$ gives a polynomial of degree at most $m$ by truncating the associated polynomial $k(x)\in {\mathbb{F}}_2[x]$ as $$\begin{aligned}
{\,\mathrm{tr}}_m(k)=k_0+k_1 x+\cdots +k_{m-1}x^{m-1}.
\end{aligned}$$ For a vector ${\boldsymbol{k}}=(k_1\cdots, k_s)\in {\mathbb{N}}_0^s$, we define ${\,\mathrm{tr}}_m({\boldsymbol{k}})=({\,\mathrm{tr}}_m(k_1),\ldots, {\,\mathrm{tr}}_m(k_s))$. With this notation, we introduce the following definition of the dual polynomial lattice $D^*_{{\boldsymbol{q}},p}$.
\[def:dual\_net\] The dual polynomial lattice for a polynomial lattice point set $P_{2^m,s}({\boldsymbol{q}},p)$ is given by $$\begin{aligned}
D^*_{{\boldsymbol{q}},p} = \{ {\boldsymbol{k}}\in {\mathbb{N}}_0^{s}:\ \mathrm{tr}_m({\boldsymbol{k}})\cdot {\boldsymbol{q}}\equiv 0 \pmod p \} .
\end{aligned}$$
Then, the following lemma relates the dual polynomial lattice of a polynomial lattice point set to the numerical integration of Walsh functions. It follows immediately from Definition \[def:dual\_net\], [@DP10 Lemma 10.6] and [@DP10 Lemma 4.75].
\[lamma:dual\_walsh\] Let $D^*_{{\boldsymbol{q}},p}$ be the dual polynomial lattice of a polynomial lattice point set $P_{2^m,s}({\boldsymbol{q}},p)$. Then we have $$\begin{aligned}
\frac{1}{2^m}\sum_{n=0}^{2^m-1}{\mathrm{wal}}_{{\boldsymbol{k}}}({\boldsymbol{x}}_n)=\left\{ \begin{array}{ll}
1 & \text{if} \ {\boldsymbol{k}}\in D^*_{{\boldsymbol{q}},p} , \\
0 & \text{otherwise} . \\
\end{array} \right.
\end{aligned}$$
Mean square weighted ${\mathcal{L}}_2$ discrepancy {#disc}
==================================================
In this section, we study the mean square weighted ${\mathcal{L}}_2$ discrepancy of scrambled polynomial lattice rules. In [@DPxx], Dick and Pillichshammer have derived the Walsh series expansion of the classical ${\mathcal{L}}_2$ discrepancy. By a slight modification, we can rewrite the expression of the square weighted ${\mathcal{L}}_2$ discrepancy given in Proposition \[prop:L2\_disc\] as follows.
\[prop:L2\_disc2\] For any point set $P_{N,s}=\{{\boldsymbol{x}}_0,\ldots, {\boldsymbol{x}}_{N-1}\}$ in $[0,1)^s$ and any sequence ${\boldsymbol{\gamma}}=(\gamma_u)_{u\subseteq I_s}$ of weights, we have $$\begin{aligned}
\label{eq:L2_disc}
{\mathcal{L}}_{2,N,{\boldsymbol{\gamma}}}^2(P_{N,s}) = \sum_{\emptyset \ne u\subseteq I_s}\gamma_u \sum_{{\boldsymbol{k}}_u,{\boldsymbol{l}}_u\in {\mathbb{N}}_0^{|u|}\setminus\{{\boldsymbol{0}}\}}r_u({\boldsymbol{k}}_u,{\boldsymbol{l}}_u)\frac{1}{N^2}\sum_{n,n'=0}^{N-1}{\mathrm{wal}}_{{\boldsymbol{k}}_u}({\boldsymbol{x}}_{n,u}){\mathrm{wal}}_{{\boldsymbol{l}}_u}({\boldsymbol{x}}_{n',u}) ,
\end{aligned}$$ where ${\boldsymbol{k}}_u=(k_j)_{j\in u}$, ${\boldsymbol{l}}_u=(l_j)_{j\in u}$, $r_u({\boldsymbol{k}}_u,{\boldsymbol{l}}_u)=\prod_{j\in u}r(k_j,l_j)$. Further, we have $r(k,l)=r(l,k)$, and for non-negative integers $0\le l\le k$ with dyadic expansions $k=2^{a_1-1}+\cdots+2^{a_v-1}$ with $a_1>\cdots > a_v>0$ and $l=2^{b_1-1}+\cdots+2^{b_w-1}$ with $b_1>\cdots > b_w>0$, we have $$\begin{aligned}
r(k,l) = \left\{ \begin{array}{ll}
\frac{1}{3} & \text{if}\ k=l=0 , \\
\frac{1}{2^{a_1+2}} & \text{if}\ v=1\ \text{and}\ l=0 , \\
-\frac{1}{2^{a_1+a_2+2}} & \text{if}\ v=2\ \text{and}\ l=0 , \\
-\frac{1}{2^{a_1+a_2+2}} & \text{if}\ v=w+2>2\ \text{and}\ a_3=b_1,\ldots, a_v=b_w , \\
\frac{1}{3\cdot 4^{a_1}} & \text{if}\ k=l>0 , \\
\frac{1}{2^{a_1+b_1+2}} & \text{if}\ v=w, a_1\ne b_1\ \text{and}\ a_2=b_2,\ldots, a_v=b_v ,\\
0 & \text{otherwise} .
\end{array} \right.
\end{aligned}$$
The next corollary provides an expression for the mean square weighted ${\mathcal{L}}_2$ discrepancy of scrambled polynomial lattice rules.
\[cor:L2\_disc\] For a polynomial lattice point set $P_{2^m,s}({\boldsymbol{q}},p)$, we have $$\begin{aligned}
{\mathbb{E}}[{\mathcal{L}}_{2,2^m,{\boldsymbol{\gamma}}}^2(\tilde{P}_{2^m,s}({\boldsymbol{q}},p))] = \sum_{\emptyset \ne v\subseteq I_s}\tilde{\gamma}_v\sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{{\boldsymbol{q}},p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) ,
\end{aligned}$$ where $$\begin{aligned}
\tilde{\gamma}_v:=\sum_{v \subseteq u\subseteq I_s}\frac{\gamma_u}{3^{|u|}} ,
\end{aligned}$$ and the expectation is taken with respect to all the possible scrambled point sets $\tilde{P}_{2^m,s}({\boldsymbol{q}},p)$ of $P_{2^m,s}({\boldsymbol{q}},p)$. Further, we denote by $({\boldsymbol{k}}_v,{\boldsymbol{0}})$ the vector from ${\mathbb{N}}_0^s$ with all the components whose indices are not in $v$ replaced by zero, and we have $\psi(k)=1/4^{a_1}$ for $k\in {\mathbb{N}}$ with dyadic expansion $k=2^{a_1-1}+\cdots+2^{a_v-1}$ with $a_1>\cdots > a_v>0$, $\psi(0)=1$ and $\psi({\boldsymbol{k}})=\prod_{j=1}^{s}\psi(k_j)$.
Let $y,y'\in [0,1)$ be two points obtained by applying Owen’s scrambling to the points $x,x'\in [0,1)$. From Owen’s lemma [@DP10 Lemma 13.3], we have $$\begin{aligned}
\label{eq:owen_lemma}
{\mathbb{E}}[{\mathrm{wal}}_k(y){\mathrm{wal}}_l(y')]=0 ,
\end{aligned}$$ whenever $k\ne l$. In the following, we denote by $y_{n,j}$ the point obtained by applying Owen’s scrambling to the point $x_{n,j}$. Using (\[eq:L2\_disc\]), (\[eq:owen\_lemma\]), Proposition \[prop:walsh\] and the linearity of expectation, we have $$\begin{aligned}
& {\mathbb{E}}[{\mathcal{L}}_{2,2^m,{\boldsymbol{\gamma}}}^2(\tilde{P}_{2^m,s}({\boldsymbol{q}},p))] \\
= & \sum_{\emptyset \ne u\subseteq I_s}\gamma_u \sum_{{\boldsymbol{k}}_u,{\boldsymbol{l}}_u\in {\mathbb{N}}_0^{|u|}\setminus\{{\boldsymbol{0}}\}}r_u({\boldsymbol{k}}_u,{\boldsymbol{l}}_u)\frac{1}{2^{2m}}\sum_{n,n'=0}^{2^m-1}\prod_{j\in u}{\mathbb{E}}[{\mathrm{wal}}_{k_j}(y_{n,j}){\mathrm{wal}}_{l_j}(y_{n',j})] \\
= & \sum_{\emptyset \ne u\subseteq I_s}\gamma_u \sum_{{\boldsymbol{k}}_u\in {\mathbb{N}}_0^{|u|}\setminus\{{\boldsymbol{0}}\}}r_u({\boldsymbol{k}}_u,{\boldsymbol{k}}_u)\frac{1}{2^{2m}}\sum_{n,n'=0}^{2^m-1}\prod_{j\in u}{\mathbb{E}}[{\mathrm{wal}}_{k_j}(y_{n,j}\oplus y_{n',j})] \\
= & \sum_{\emptyset \ne u\subseteq I_s}\gamma_u \sum_{\emptyset \ne v\subseteq u}\frac{1}{3^{|u\setminus v|}}\sum_{{\boldsymbol{k}}_v\in {\mathbb{N}}^{|v|}}r_v({\boldsymbol{k}}_v,{\boldsymbol{k}}_v)\frac{1}{2^{2m}}\sum_{n,n'=0}^{2^m-1}\prod_{j\in v}{\mathbb{E}}[{\mathrm{wal}}_{k_j}(y_{n,j}\oplus y_{n',j})] .
\end{aligned}$$
Now we need to introduce the following notations. For ${\boldsymbol{l}}_v=(l_j)_{j\in v} \in {\mathbb{N}}^{|v|}$, we define a set ${\mathcal{B}}_{{\boldsymbol{l}}_v}$ as $$\begin{aligned}
{\mathcal{B}}_{{\boldsymbol{l}}_v}=\{ (k_j)_{j\in v}\in {\mathbb{N}}^{|v|}: 2^{l_j-1} \le k_j< 2^{l_j}\ \text{for}\ j\in v\} .
\end{aligned}$$ We denote by $\sigma_{{\boldsymbol{l}}_v}$ the sum of $r_v({\boldsymbol{k}}_v,{\boldsymbol{k}}_v)$ over all ${\boldsymbol{k}}_v\in {\mathcal{B}}_{{\boldsymbol{l}}_v}$. We have $$\begin{aligned}
\sigma_{{\boldsymbol{l}}_v} & = \sum_{{\boldsymbol{k}}_v\in {\mathcal{B}}_{{\boldsymbol{l}}_v}}r_v({\boldsymbol{k}}_v,{\boldsymbol{k}}_v) \\
& = \sum_{{\boldsymbol{k}}_v\in {\mathcal{B}}_{{\boldsymbol{l}}_v}}\prod_{j\in v}r(k_j,k_j) \\
& = \prod_{j\in v}\sum_{k_j=2^{l_{j}-1}}^{2^{l_j}-1}r(k_j,k_j) \\
& = \prod_{j\in v}\frac{2^{l_j}-2^{l_{j}-1}}{3\cdot 4^{l_j}} \\
& = \frac{1}{3^{|v|}\cdot 2^{|v|+|{\boldsymbol{l}}_v|_1}} ,
\end{aligned}$$ where $|{\boldsymbol{l}}_v|_1:=\sum_{j\in v}l_j$. Further, we introduce a so-called gain coefficient, which is independent of the choice of ${\boldsymbol{k}}_v\in {\mathcal{B}}_{{\boldsymbol{l}}_v}$, $$\begin{aligned}
G_{{\boldsymbol{l}}_v} & = \frac{1}{2^{2m}}\sum_{n,n'=0}^{2^m-1}\prod_{j\in v}{\mathbb{E}}[{\mathrm{wal}}_{k_j}(y_{n,j}\oplus y_{n',j})] \\
& = 2^{|v|-|{\boldsymbol{l}}_v|_1}\sum_{\substack{{\boldsymbol{k}}_v\in {\mathcal{B}}_{{\boldsymbol{l}}_v}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{{\boldsymbol{q}},p}}}1 ,
\end{aligned}$$ where the last equality appeared in the proof of [@DP10 Corollary 13.7]. Using these notations and their results, we have $$\begin{aligned}
& {\mathbb{E}}[{\mathcal{L}}_{2,2^m,{\boldsymbol{\gamma}}}^2(\tilde{P}_{2^m,s}({\boldsymbol{q}},p))] \\
= & \sum_{\emptyset \ne u\subseteq I_s}\gamma_u \sum_{\emptyset \ne v\subseteq u}\frac{1}{3^{|u\setminus v|}}\sum_{{\boldsymbol{l}}_v\in {\mathbb{N}}^{|v|}}\sum_{{\boldsymbol{k}}_v\in {\mathcal{B}}_{{\boldsymbol{l}}_v}}r_v({\boldsymbol{k}}_v,{\boldsymbol{k}}_v)\frac{1}{2^{2m}}\sum_{n,n'=0}^{2^m-1}\prod_{j\in v}{\mathbb{E}}[{\mathrm{wal}}_{k_j}(y_{n,j}\oplus y_{n',j})] \\
= & \sum_{\emptyset \ne u\subseteq I_s}\gamma_u \sum_{\emptyset \ne v\subseteq u}\frac{1}{3^{|u\setminus v|}}\sum_{{\boldsymbol{l}}_v\in {\mathbb{N}}^{|v|}}G_{{\boldsymbol{l}}_v}\sigma_{{\boldsymbol{l}}_v} \\
= & \sum_{\emptyset \ne u\subseteq I_s}\frac{\gamma_u}{3^{|u|}} \sum_{\emptyset \ne v\subseteq u}\sum_{{\boldsymbol{l}}_v\in {\mathbb{N}}^{|v|}}\frac{1}{4^{|{\boldsymbol{l}}_v|_1}}\sum_{\substack{{\boldsymbol{k}}_v\in {\mathcal{B}}_{{\boldsymbol{l}}_v}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{{\boldsymbol{q}},p}}}1 \\
= & \sum_{\emptyset \ne u\subseteq I_s}\frac{\gamma_u}{3^{|u|}} \sum_{\emptyset \ne v\subseteq u}\sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{{\boldsymbol{q}},p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) .
\end{aligned}$$ The proof is complete by swapping the order of sums.
We denote the sum in Corollary \[cor:L2\_disc\] by $$\begin{aligned}
\label{eq:L2_disc2}
B({\boldsymbol{q}},{\boldsymbol{\gamma}}) = \sum_{\emptyset \ne v\subseteq I_s}\tilde{\gamma}_v \sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{{\boldsymbol{q}},p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) .
\end{aligned}$$ Using the property of dual polynomial lattice $D^*_{{\boldsymbol{q}},p}$ shown in Lemma \[lamma:dual\_walsh\], we can derive a more computable form of $B({\boldsymbol{q}},{\boldsymbol{\gamma}})$. In the following, we write $\log_2$ for the logarithm in base 2 and we set $2^{\lfloor \log_2 0\rfloor}=0$.
Let $B({\boldsymbol{q}},{\boldsymbol{\gamma}})$ be given by (\[eq:L2\_disc2\]). Then we have $$\begin{aligned}
\label{eq:criterion}
B({\boldsymbol{q}},{\boldsymbol{\gamma}}) = \frac{1}{2^m}\sum_{n=0}^{2^m-1}\sum_{\emptyset \ne v\subseteq I_s}\tilde{\gamma}_v\prod_{j\in v}\tilde{\phi}(x_{n,j}) ,
\end{aligned}$$ where for $x\in [0,1)$ we set $$\begin{aligned}
\tilde{\phi}(x) = \frac{1-3\cdot 2^{\lfloor \log_2 x\rfloor}}{2} ,
\end{aligned}$$ and $\tilde{\gamma}_v$ is defined as in Corollary \[cor:L2\_disc\]. In particular, in case of product weights, we have $$\begin{aligned}
\label{eq:criterion_product}
B({\boldsymbol{q}},{\boldsymbol{\gamma}}) = -\prod_{j=1}^{s}\left(1+\frac{\gamma_{j}}{3}\right)+\frac{1}{2^m}\sum_{n=0}^{2^m-1}\prod_{j=1}^{s}\left[1+\gamma_{j}\phi (x_{n,j})\right] ,
\end{aligned}$$ where for $x\in [0,1)$ we set $$\begin{aligned}
\phi(x) = \frac{1-2^{\lfloor \log_2 x\rfloor}}{2} .
\end{aligned}$$
Applying Lemma \[lamma:dual\_walsh\] to $B({\boldsymbol{q}},{\boldsymbol{\gamma}})$, we have $$\begin{aligned}
B({\boldsymbol{q}},{\boldsymbol{\gamma}}) & = \sum_{\emptyset \ne v\subseteq I_s}\tilde{\gamma}_v\sum_{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}})\frac{1}{2^m}\sum_{n=0}^{2^m-1}{\mathrm{wal}}_{({\boldsymbol{k}}_v,{\boldsymbol{0}})}({\boldsymbol{x}}_n) \\
& = \frac{1}{2^m}\sum_{n=0}^{2^m-1}\sum_{\emptyset \ne v\subseteq I_s}\tilde{\gamma}_v\prod_{j\in v}\left[ \sum_{k_j=1}^{\infty}\psi(k_j){\mathrm{wal}}_{k_j}(x_{n,j})\right] .
\end{aligned}$$ For the innermost sum, we have by following the similar line as the proof of [@Bal10 Theorem 7.3] $$\begin{aligned}
\sum_{k=1}^{\infty}\psi(k){\mathrm{wal}}_{k}(x) & = \sum_{l=1}^{\infty}\frac{1}{4^l}\sum_{k=2^{l-1}}^{2^l-1}{\mathrm{wal}}_{k}(x) \\
& = \frac{1-3\cdot 2^{\lfloor \log_2(x)\rfloor}}{2} \\
& = \tilde{\phi}(x) .
\end{aligned}$$ Thus the result for the first part of the lemma follows.
Next in case of $\gamma_v=\prod_{j\in v}\gamma_j$, by letting $\gamma_{\emptyset}=1$, we have $$\begin{aligned}
\tilde{\gamma}_v & = \prod_{j\in v}\frac{\gamma_j}{3}\left( \sum_{w\subseteq I_s\setminus v}\prod_{j'\in w}\frac{\gamma_{j'}}{3}\right) \\
& = \prod_{j\in v}\frac{\gamma_j}{3}\prod_{j'\in I_s\setminus v}\left(1+\frac{\gamma_{j'}}{3}\right) .
\end{aligned}$$ Inserting this result into (\[eq:criterion\]), we have $$\begin{aligned}
B({\boldsymbol{q}},{\boldsymbol{\gamma}}) & = \frac{1}{2^m}\sum_{n=0}^{2^m-1}\sum_{\emptyset \ne v\subseteq I_s}\prod_{j'\in I_s\setminus v}\left(1+\frac{\gamma_{j'}}{3}\right)\prod_{j\in v}\frac{\gamma_j}{3}\tilde{\phi}(x_{n,j}) \\
& = -\prod_{j=1}^{s}\left(1+\frac{\gamma_{j}}{3}\right)+\frac{1}{2^m}\sum_{n=0}^{2^m-1}\prod_{j=1}^{s}\left[ \left( 1+\frac{\gamma_j}{3}\right)+\frac{\gamma_j}{3}\tilde{\phi}(x_{n,j})\right] \\
& = -\prod_{j=1}^{s}\left(1+\frac{\gamma_{j}}{3}\right)+\frac{1}{2^m}\sum_{n=0}^{2^m-1}\prod_{j=1}^{s}\left[ 1+\gamma_{j}\phi(x_{n,j})\right].
\end{aligned}$$ Thus the proof for the second part of the lemma is complete.
Since we have the following recursion in the inner sum of (\[eq:criterion\]) $$\begin{aligned}
& \sum_{\emptyset \ne v\subseteq I_r}\tilde{\gamma}_v\prod_{j\in v}\tilde{\phi}(x_{n,j}) \\
= & \tilde{\gamma}_{\{r\}}\tilde{\phi}(x_{n,r})+\sum_{\emptyset \ne v\subseteq I_{r-1}}\left( 1+\frac{\tilde{\gamma}_{v\cup \{r\}}}{\tilde{\gamma}_v}\tilde{\phi}(x_{r,j})\right)\tilde{\gamma}_v\prod_{j\in v}\tilde{\phi}(x_{n,j}) ,
\end{aligned}$$ for $1\le r\le s$, the computational complexity of computing $B({\boldsymbol{q}},{\boldsymbol{\gamma}})$ with the general weights is $O(2^{ms})$. In case of the product weights, on the other hand, the computational complexity of computing $B({\boldsymbol{q}},{\boldsymbol{\gamma}})$ reduces to $O(s2^m)$.
Construction of polynomial lattice rules {#cbc}
========================================
In this section, we first show how to find a vector ${\boldsymbol{q}}$ by using the CBC construction algorithm for general weights and product weights respectively. We prove that an upper bound on $B({\boldsymbol{q}},{\boldsymbol{\gamma}})$ satisfied by our algorithm converges at almost the best possible rate of $N^{-2+\delta}$ for all $\delta >0$. Further in this section we discuss strong tractability of our algorithm.
We restrict each polynomial $q_j$ such that $q_j\ne 0$ and $\deg(q_j)<m$. In the following, we denote by $R_m$ the set of all the non-zero polynomials over ${\mathbb{F}}_2$ with degree less than $m$, i.e., $$\begin{aligned}
R_m = \{q\in {\mathbb{F}}_2[x]: \deg(q)<m \ \text{and}\ q\ne 0\} .
\end{aligned}$$ It is clear that $|R_m|=2^m-1$. Further, we write ${\boldsymbol{q}}_\tau=(q_1,\ldots, q_\tau)$ for $1\le \tau\le s$.
General weights
---------------
The CBC construction for general weights proceeds as follows.
\[cbc\_algor\](CBC construction for general weights) For $m,s\in {\mathbb{N}}$ and any sequence of weights ${\boldsymbol{\gamma}}=(\gamma_u)_{u\subseteq I_s}$, we proceed as follows.
1. Choose an irreducible polynomial $p\in {\mathbb{F}}_2[x]$ with $\deg(p)=m$.
2. Set $q_1^*=1$.
3. For $\tau=2,\cdots, s$, find $q^*_{\tau}$ by minimizing $B(({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),{\boldsymbol{\gamma}})$ as a function of $q_\tau\in R_{m}$ where $$\begin{aligned}
\label{eq:cbc_criterion_general}
B(({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),{\boldsymbol{\gamma}}) = \sum_{\emptyset \ne v\subseteq I_\tau}\tilde{\gamma}_v \sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) ,
\end{aligned}$$ in which $\tilde{\gamma}_v$ is defined as in Corollary \[cor:L2\_disc\].
Since computing $\tilde{\gamma}_v$ in Step 3. of Algorithm \[cbc\_algor\] requires $\gamma_v$ such that $v\nsubseteq I_\tau$, Algorithm \[cbc\_algor\] is not extensible in $s$. From the last line in the proof of Corollary \[cor:L2\_disc\], it is possible to replace $B(({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),{\boldsymbol{\gamma}})$ by $$\begin{aligned}
B(({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),{\boldsymbol{\gamma}}) & = \sum_{\emptyset \ne u\subseteq I_\tau}\frac{\gamma_u}{3^{|u|}} \sum_{\emptyset \ne v\subseteq u}\sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) .
\end{aligned}$$ Then we can make the construction algorithm extensible in $s$. Because of the technical difficulty in treating two outermost sums in the right-hand side, however, it is hard to prove that polynomial lattice rules constructed using this replaced criterion achieve almost the best possible rate of convergence.
The next theorem provides an upper bound on $B({\boldsymbol{q}}_\tau,{\boldsymbol{\gamma}})$ for the polynomials ${\boldsymbol{q}}^{*}_\tau$ for $1\le \tau\le s$ constructed according to Algorithm \[cbc\_algor\]. It converges at almost the best possible rate of $N^{-2+\delta}$ for all $\delta >0$. In the proof of the theorem, we use the following inequality, which states that for any sequence $(a_i)_{i\in {\mathbb{N}}}$ of non-negative real numbers we have $$\begin{aligned}
\label{eq:jensen}
\left( \sum a_i\right)^{\lambda} \le \sum a_i^{\lambda} ,
\end{aligned}$$ for any $0<\lambda \le 1$.
\[theorem1\] Let $p\in {\mathbb{F}}_2[x]$ be an irreducible polynomial with $\deg(p)=m$. Suppose that ${\boldsymbol{q}}^{*}_s\in R_m^s$ is constructed according to Algorithm \[cbc\_algor\]. Then for all $\tau=1,\ldots, s$ we have $$\begin{aligned}
\label{eq:theorem1}
B({\boldsymbol{q}}^{*}_\tau,{\boldsymbol{\gamma}}) \le \frac{1}{(2^m-1)^{1/\lambda}}\left[ \sum_{\emptyset \ne v\subseteq I_\tau}\tilde{\gamma}_v^{\lambda} \frac{1}{(2^{2\lambda}-2)^{|v|}} \right]^{1/\lambda} ,
\end{aligned}$$ for $1/2<\lambda \le 1$, where $\tilde{\gamma}_v$ is defined as in Corollary \[cor:L2\_disc\].
We prove the theorem by induction on $\tau$. For $\tau=1$, we have $$\begin{aligned}
B(q^{*}_1,{\boldsymbol{\gamma}}) & = \tilde{\gamma}_{\{1\}}\sum_{\substack{k=1\\ 2^m\mid k}}^{\infty}\psi(k) \\
& = \tilde{\gamma}_{\{1\}}\sum_{a=1}^{\infty}\sum_{\substack{k=2^{a-1}\\ 2^m\mid k}}^{2^a-1}\psi(k) \\
& = \tilde{\gamma}_{\{1\}}\sum_{a=m+1}^{\infty}2^{a-m-1}\cdot 2^{-2a} \\
& = \tilde{\gamma}_{\{1\}}\frac{1}{2^{2m+1}} \\
& \le \tilde{\gamma}_{\{1\}}\left[\frac{1}{(2^{m}-1)(2^{2\lambda}-2)}\right]^{1/\lambda} ,
\end{aligned}$$ for $1/2<\lambda \le 1$. Hence the result holds true for $\tau=1$.
Next, assume that the statement of the theorem is true for some $\tau\ge 1$. Then it is enough to show that the statement is also true for the ($\tau+1$)-th component. In the following, we classify each subset $u$ according to whether $u$ includes the component $\{\tau+1\}$ or not. Then we have $$\begin{aligned}
& B(({\boldsymbol{q}}^{*}_\tau,q_{\tau+1}),{\boldsymbol{\gamma}}) \\
= & \sum_{\emptyset \ne v\subseteq I_{\tau+1}}\tilde{\gamma}_v \sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_\tau,q_{\tau+1}),p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) \\
= & \sum_{\emptyset \ne v\subseteq I_{\tau}}\tilde{\gamma}_v \sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{{\boldsymbol{q}}^{*}_\tau,p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) + \sum_{v\subseteq I_{\tau}}\tilde{\gamma}_{v\cup \{\tau+1\}} \sum_{\substack{({\boldsymbol{k}}_v,k_{\tau+1}) \in {\mathbb{N}}^{|v|+1}\\ ({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_\tau,q_{\tau+1}),p}}}\psi({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}}) \\
= & B({\boldsymbol{q}}^{*}_\tau,{\boldsymbol{\gamma}})+\theta(q_{\tau+1}) ,
\end{aligned}$$ where we have defined $$\begin{aligned}
\theta(q_{\tau+1}) := \sum_{v\subseteq I_{\tau}}\tilde{\gamma}_{v\cup \{\tau+1\}} \sum_{\substack{({\boldsymbol{k}}_v,k_{\tau+1}) \in {\mathbb{N}}^{|v|+1}\\ ({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_\tau,q_{\tau+1}),p}}}\psi({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}}) .
\end{aligned}$$
In order to obtain an upper bound on $\theta(q_{\tau+1}^{*})$, we employ the averaging argument. Since we choose $q_{\tau+1}^*$ which minimizes $\theta_2(q_{\tau+1})$ in Algorithm \[cbc\_algor\], $\theta^\lambda(q_{\tau+1}^{*})$ has to be less than or equal to the average of $\theta^\lambda(q_{\tau+1})$ over $q_{\tau+1}\in R_{m}$ for any $1/2< \lambda \le 1$. We obtain $$\begin{aligned}
\theta^{\lambda}(q^{*}_{\tau+1}) & \le \frac{1}{2^m-1}\sum_{q_{\tau+1}\in R_{m}}\theta^{\lambda}(q_{\tau+1}) \\
& \le \frac{1}{2^m-1}\sum_{q_{\tau+1}\in R_{m}}\sum_{v\subseteq I_{\tau}}\tilde{\gamma}^\lambda_{v\cup \{\tau+1\}} \sum_{\substack{({\boldsymbol{k}}_v,k_{\tau+1}) \in {\mathbb{N}}^{|v|+1}\\ ({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_\tau,q_{\tau+1}),p}}}\psi^\lambda({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}}) \\
& = \sum_{v\subseteq I_\tau}\frac{\tilde{\gamma}^\lambda_{v\cup \{\tau+1\}}}{2^m-1}\sum_{q_{\tau+1}\in R_{m}} \sum_{\substack{({\boldsymbol{k}}_v,k_{\tau+1}) \in {\mathbb{N}}^{|v|+1}\\ ({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_\tau,q_{\tau+1}),p}}}\psi^\lambda({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}}) ,
\end{aligned}$$ where we have used (\[eq:jensen\]) in the second inequality. For a fixed $v\subseteq I_\tau$, we consider the condition $({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_\tau,q_{\tau+1}),p}$, that is, $$\begin{aligned}
{\,\mathrm{tr}}_m({\boldsymbol{k}}_v)\cdot {\boldsymbol{q}}_v + {\,\mathrm{tr}}_m(k_{\tau+1})\cdot q_{\tau+1} = 0 \pmod p .
\end{aligned}$$ If $k_{\tau+1}$ is a multiple of $2^m$, we always have ${\,\mathrm{tr}}_m(k_{\tau+1})=0$ and the above equation becomes independent of $q_{\tau+1}$. Otherwise if $k_{\tau+1}$ is not a multiple of $2^m$, we have ${\,\mathrm{tr}}_m(k_{\tau+1})\ne 0$ and the term ${\,\mathrm{tr}}_m(k_{\tau+1})\cdot q_{\tau+1}$ cannot be a multiple of $p$. Thus we have $$\begin{aligned}
& \frac{1}{2^m-1}\sum_{q_{\tau+1}\in R_{m}} \sum_{\substack{({\boldsymbol{k}}_v,k_{\tau+1}) \in {\mathbb{N}}^{|v|+1}\\ ({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_\tau,q_{\tau+1}),p}}}\psi^\lambda({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}}) \\
= & \sum_{\substack{k_{\tau+1}=1\\ 2^m\mid k_{\tau+1}}}^{\infty}\psi^{\lambda}(k_{\tau+1})\sum_{\substack{{\boldsymbol{k}}_v\in {\mathbb{N}}^{|v|}\\ {\,\mathrm{tr}}_m({\boldsymbol{k}}_v)\cdot {\boldsymbol{q}}_v=0 \pmod p}}\psi^{\lambda}({\boldsymbol{k}}_v) \\
& + \frac{1}{2^m-1}\sum_{\substack{k_{\tau+1}=1\\ 2^m\nmid k_{\tau+1}}}^{\infty}\psi^{\lambda}(k_{\tau+1})\sum_{\substack{{\boldsymbol{k}}_v\in {\mathbb{N}}^{|v|}\\ {\,\mathrm{tr}}_m({\boldsymbol{k}}_v)\cdot {\boldsymbol{q}}_v\ne 0 \pmod p}}\psi^{\lambda}({\boldsymbol{k}}_v) \\
\le & \frac{1}{2^{2\lambda m}}\sum_{k_{\tau+1}=1}^{\infty}\psi^{\lambda}(k_{\tau+1})\sum_{\substack{{\boldsymbol{k}}_v\in {\mathbb{N}}^{|v|}\\ {\,\mathrm{tr}}_m({\boldsymbol{k}}_v)\cdot {\boldsymbol{q}}_v=0 \pmod p}}\psi^{\lambda}({\boldsymbol{k}}_v) \\
& + \frac{1}{2^m-1}\sum_{k_{\tau+1}=1}^{\infty}\psi^{\lambda}(k_{\tau+1})\sum_{\substack{{\boldsymbol{k}}_v\in {\mathbb{N}}^{|v|}\\ {\,\mathrm{tr}}_m({\boldsymbol{k}}_v)\cdot {\boldsymbol{q}}_v\ne 0 \pmod p}}\psi^{\lambda}({\boldsymbol{k}}_v) \\
\le & \frac{1}{2^m-1}\sum_{k_{\tau+1}=1}^{\infty}\psi^{\lambda}(k_{\tau+1})\sum_{{\boldsymbol{k}}_v\in {\mathbb{N}}^{|v|}}\psi^{\lambda}({\boldsymbol{k}}_v) \\
= & \frac{1}{2^m-1}\left[ \sum_{k=1}^{\infty}\psi^{\lambda}(k)\right]^{|v|+1}\; = \; \frac{1}{(2^m-1)(2^{2\lambda}-2)^{|v|+1}} .
\end{aligned}$$ Using this result, we obtain an upper bound on $\theta(q_{\tau+1}^{*})$ as $$\begin{aligned}
\theta^{\lambda}(q^{*}_{\tau+1}) & \le \frac{1}{2^m-1}\sum_{v\subseteq I_\tau}\tilde{\gamma}^\lambda_{v\cup \{\tau+1\}}\frac{1}{(2^{2\lambda}-2)^{|v|+1}} .
\end{aligned}$$ Finally by applying (\[eq:jensen\]) we have $$\begin{aligned}
& B^{\lambda}(({\boldsymbol{q}}^{*}_{\tau},q_{\tau+1}^{*}),{\boldsymbol{\gamma}}) \\
= & \left( B({\boldsymbol{q}}^{*}_{\tau},{\boldsymbol{\gamma}})+\theta(q_{\tau+1}^{*}) \right)^{\lambda} \\
\le & B^{\lambda}({\boldsymbol{q}}^{*}_{\tau},{\boldsymbol{\gamma}})+\theta^{\lambda}(q_{\tau+1}^{*}) \\
\le & \frac{1}{2^m-1}\sum_{\emptyset \ne v\subseteq I_\tau}\tilde{\gamma}_v^{\lambda} \frac{1}{(2^{2\lambda}-2)^{|v|}} +\frac{1}{2^m-1}\sum_{v\subseteq I_\tau}\tilde{\gamma}^\lambda_{v\cup \{\tau+1\}}\frac{1}{(2^{2\lambda}-2)^{|v|+1}} \\
= & \frac{1}{2^m-1}\sum_{\emptyset \ne v\subseteq I_{\tau+1}}\tilde{\gamma}_v^{\lambda} \frac{1}{(2^{2\lambda}-2)^{|v|}} ,
\end{aligned}$$ for $1/2< \lambda \le 1$, from which (\[eq:theorem1\]) holds true for the $(\tau+1)$-th component. Hence the result follows.
\[remark:cbc\_one\_dimension\] For $\tau=1$, we have as in the proof of Theorem \[theorem1\] $$\begin{aligned}
B(q^{*}_1,{\boldsymbol{\gamma}}) = \tilde{\gamma}_{\{1\}}\frac{1}{2^{2m+1}}.
\end{aligned}$$ Since the lower bound on the ${\mathcal{L}}_2$ discrepancy is given as in (\[eq:lower\_bound\]), this achieves the best possible rate of convergence. As a one-dimensional polynomial lattice point set consists of the equidistributed points $x_n=n/b^m$, $n=0,\ldots, 2^m-1$, other QMC point sets such as Sobol’ and Niederreiter sequences constructed over ${\mathbb{F}}_2$ also give the same result.
\[remark:cbc\_bound\] For $\tau=s$, we further have $$\begin{aligned}
B(q^{*}_s,{\boldsymbol{\gamma}}) & \le \frac{1}{(2^m-1)^{1/\lambda}}\left[ \sum_{\emptyset \ne v\subseteq I_s}\tilde{\gamma}_v^{\lambda} \frac{1}{(2^{2\lambda}-2)^{|v|}} \right]^{1/\lambda} \\
& \le \frac{1}{(2^m-1)^{1/\lambda}}\left[ \sum_{\emptyset \ne v\subseteq I_s}\left(\sum_{v\subseteq u\subseteq I_s}\left( \frac{\gamma_u}{3^{|u|}}\right)^{\lambda} \right) \frac{1}{(2^{2\lambda}-2)^{|v|}} \right]^{1/\lambda} \\
& = \frac{1}{(2^m-1)^{1/\lambda}}\left[ \sum_{\emptyset \ne v\subseteq I_s}\left( \frac{\gamma_v}{3^{|v|}}\right)^{\lambda} \left( -1+\left(\frac{2^{2\lambda}-1}{2^{2\lambda}-2}\right)^{|v|}\right) \right]^{1/\lambda} ,
\end{aligned}$$ where we have used (\[eq:jensen\]) in the second inequality and swapped the order of sums in the last equality. Thus, we have obtained an upper bound on $B(q^{*}_s,{\boldsymbol{\gamma}})$ using not $\tilde{\gamma}_v$ but $\gamma_v$ for $v\subseteq I_s$.
In the following we discuss strong tractability of our construction algorithm. Let us consider the inverse of the mean square weighted ${\mathcal{L}}_2$ discrepancy which is defined as follows $$\begin{aligned}
N(s,\epsilon)=\min\{N\in {\mathbb{N}}: {\mathbb{E}}[{\mathcal{L}}_{2,N,{\boldsymbol{\gamma}}}^2(\tilde{P}_{N,s})]\le \epsilon {\mathbb{E}}[{\mathcal{L}}_{2,0,{\boldsymbol{\gamma}}}^2(\tilde{P}_{0,s})]\} .
\end{aligned}$$ We say that the mean square weighted ${\mathcal{L}}_2$ discrepancy is strongly tractable if there exist non-negative constants $C$ and $\beta$ such that $$\begin{aligned}
N(s,\epsilon)\le C\epsilon^{-\beta} ,
\end{aligned}$$ where $C$ depends neither on $\epsilon$ or $s$ and we call $\beta$ the exponent of tractability.
In the next corollary, we write $\gamma_{s,u}$ instead of $\gamma_u$ to emphasize the dependence on $s$ of our construction algorithm. We denote by ${\boldsymbol{\gamma}}$ a sequence of weights $(\gamma_{s,u})_{u\subseteq I_s}$ for $s\in {\mathbb{N}}$.
\[cor:tractability\] Assume that the weights ${\boldsymbol{\gamma}}$ satisfy the condition $$\begin{aligned}
B_{{\boldsymbol{\gamma}},\lambda} := \sup_{s\in {\mathbb{N}}}\frac{\left[\sum_{\emptyset \ne u\subseteq I_s}\left( \frac{\gamma_{s,u}}{3^{|u|}}\right)^{\lambda}\left(-1+\left( \frac{2^{2\lambda}-1}{2^{2\lambda}-2}\right)^{|u|} \right)\right]^{1/\lambda}}{\sum_{\emptyset \ne u\subseteq I_s}\frac{\gamma_{s,u}}{3^{|u|}}} < \infty ,
\end{aligned}$$ for some $\lambda$ such that $1/2<\lambda \le 1$. Then the mean square weighted ${\mathcal{L}}_2$ discrepancy is strongly tractable with the exponent of tractability at most $\lambda$.
For the empty point set $P_{0,s}$, we have $$\begin{aligned}
{\mathbb{E}}[{\mathcal{L}}_{2,0,{\boldsymbol{\gamma}}}^2(\tilde{P}_{0,s})] & = \sum_{\emptyset \ne u\subseteq I_s}\gamma_{s,u} \prod_{j\in u}\int_{0}^{1}t_j^2 dt_j \\
& = \sum_{\emptyset \ne u\subseteq I_s}\frac{\gamma_{s,u}}{3^{|u|}} .
\end{aligned}$$ For a polynomial lattice point set $P_{2^m,s}$ constructed by Algorithm \[cbc\_algor\], we have from Remark \[remark:cbc\_bound\] $$\begin{aligned}
{\mathbb{E}}[{\mathcal{L}}_{2,2^m,{\boldsymbol{\gamma}}}^2(P_{2^m,s})] & \le \frac{1}{(2^m-1)^{1/\lambda}}\left[ \sum_{\emptyset \ne u\subseteq I_s}\left( \frac{\gamma_{s,u}}{3^{|u|}}\right)^{\lambda}\left(-1+\left( \frac{2^{2\lambda}-1}{2^{2\lambda}-2}\right)^{|u|} \right)\right]^{1/\lambda} \\
& \le \frac{1}{(2^m-1)^{1/\lambda}}B_{{\boldsymbol{\gamma}},\lambda}\sum_{\emptyset \ne u\subseteq I_s}\frac{\gamma_{s,u}}{3^{|u|}} \\
& = \frac{1}{(2^m-1)^{1/\lambda}}B_{{\boldsymbol{\gamma}},\lambda}{\mathbb{E}}[{\mathcal{L}}_{2,0,{\boldsymbol{\gamma}}}^2(P_{0,s})] .
\end{aligned}$$ The last term is smaller than or equal to $\epsilon {\mathbb{E}}[{\mathcal{L}}_{2,0,{\boldsymbol{\gamma}}}^2(P_{0,s})]$ if $N=2^m\ge 1+B^{\lambda}_{{\boldsymbol{\gamma}},\lambda}\epsilon^{-\lambda}$. Thus the result follows.
Product weights
---------------
In case of product weights, we have for (\[eq:cbc\_criterion\_general\]) $$\begin{aligned}
B(({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),{\boldsymbol{\gamma}}) & = \sum_{\emptyset \ne v\subseteq I_\tau}\prod_{j\in v}\frac{\gamma_j}{3}\prod_{j'\in I_s\setminus v}\left(1+\frac{\gamma_{j'}}{3}\right) \sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) \\
& = \prod_{j''=\tau+1}^{s}\left(1+\frac{\gamma_{j''}}{3}\right)\sum_{\emptyset \ne v\subseteq I_\tau}\prod_{j\in v}\frac{\gamma_j}{3}\prod_{j'\in I_\tau\setminus v}\left(1+\frac{\gamma_{j'}}{3}\right) \sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) .
\end{aligned}$$ Omitting the term $\prod_{j''=\tau+1}^{s}\left(1+\frac{\gamma_{j''}}{3}\right)$ from criterion, computing $B(({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),{\boldsymbol{\gamma}})$ does not require $\gamma_{\tau+1},\ldots,\gamma_s$. Therefore we can make the construction algorithm extensible in $s$, while it is still possible to prove that constructed polynomial lattice rules achieve almost the best possible rate of convergence as shown below in Theorem \[theorem2\]. Thus the CBC construction for product weights proceeds as follows.
\[cbc\_algor\_product\](CBC construction for product weights) For $m,s\in {\mathbb{N}}$ and product weights $\gamma_u=\prod_{j\in u}\gamma_j$ with any non-negative numbers $\gamma_1,\ldots,\gamma_s$, we proceed as follows.
1. Choose an irreducible polynomial $p\in {\mathbb{F}}_2[x]$ with $\deg(p)=m$.
2. Set $q_1^*=1$.
3. For $\tau=2,\cdots, s$, find $q^*_{\tau}$ by minimizing $B(({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),{\boldsymbol{\gamma}})$ as a function of $q_\tau\in R_{m}$ where $$\begin{aligned}
B(({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),{\boldsymbol{\gamma}}) = \sum_{\emptyset \ne v\subseteq I_\tau}\tilde{\gamma}_{\tau,v} \sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) ,
\end{aligned}$$ where we define $$\begin{aligned}
\tilde{\gamma}_{\tau,v}:=\prod_{j\in v}\frac{\gamma_j}{3}\prod_{j'\in I_\tau\setminus v}\left(1+\frac{\gamma_{j'}}{3}\right).
\end{aligned}$$
\[theorem2\] Let $p\in {\mathbb{F}}_2[x]$ be irreducible polynomial with $\deg(p)=m$. Suppose that ${\boldsymbol{q}}^{*}_s\in R_m^s$ is constructed according to Algorithm \[cbc\_algor\_product\]. Then for any $\tau=1,\ldots, s$ we have $$\begin{aligned}
B({\boldsymbol{q}}^{*}_\tau,{\boldsymbol{\gamma}}) \le \frac{1}{(2^m-1)^{1/\lambda}}\left[ \sum_{\emptyset \ne v\subseteq I_\tau}\tilde{\gamma}_{\tau,v}^{\lambda} \frac{1}{(2^{2\lambda}-2)^{|v|}} \right]^{1/\lambda} ,
\end{aligned}$$ for $1/2<\lambda \le 1$, where $\tilde{\gamma}_{\tau,v}$ is defined as in Algorithm \[cbc\_algor\_product\].
We prove the theorem by induction on $\tau$ in a quite similar way as the proof of Theorem \[theorem1\]. For $\tau=1$, we have the result by replacing $\tilde{\gamma}_{\{1\}}$ with $\gamma_{\{1\}}/3$. For $\tau\ge 1$, assume that the statement of the theorem is true. Then we have $$\begin{aligned}
& B(({\boldsymbol{q}}^{*}_\tau,q_{\tau+1}),{\boldsymbol{\gamma}}) \\
= & \sum_{\emptyset \ne v\subseteq I_{\tau+1}}\tilde{\gamma}_{\tau+1,v} \sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_{\tau},q_{\tau+1}),p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) \\
= & \left(1+\frac{\gamma_{\tau+1}}{3}\right)\sum_{\emptyset \ne v\subseteq I_\tau}\tilde{\gamma}_{\tau,v} \sum_{\substack{{\boldsymbol{k}}_v \in {\mathbb{N}}^{|v|}\\ ({\boldsymbol{k}}_v,{\boldsymbol{0}})\in D^*_{{\boldsymbol{q}}^{*}_\tau,p}}}\psi({\boldsymbol{k}}_v,{\boldsymbol{0}}) \\
& + \sum_{v\subseteq I_\tau}\tilde{\gamma}_{\tau+1,v\cup \{\tau+1\}} \sum_{\substack{({\boldsymbol{k}}_v,k_{\tau+1}) \in {\mathbb{N}}^{|v|+1}\\ ({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_{\tau},q_{\tau+1}),p}}}\psi({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}}) \\
= & \left(1+\frac{\gamma_{\tau+1}}{3}\right)B({\boldsymbol{q}}^{*}_\tau,{\boldsymbol{\gamma}})+\theta(q_{\tau+1}) ,
\end{aligned}$$ where we have defined $$\begin{aligned}
\theta(q_{\tau+1}):=\sum_{v\subseteq I_\tau}\tilde{\gamma}_{\tau+1,v\cup \{\tau+1\}} \sum_{\substack{({\boldsymbol{k}}_v,k_{\tau+1}) \in {\mathbb{N}}^{|v|+1}\\ ({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}})\in D^*_{({\boldsymbol{q}}^{*}_{\tau},q_{\tau+1}),p}}}\psi({\boldsymbol{k}}_v,k_{\tau+1},{\boldsymbol{0}}) .
\end{aligned}$$ Following the same argument with the proof of Theorem \[theorem1\], we can obtain $$\begin{aligned}
\theta^\lambda(q^*_{\tau+1}) & \le \frac{1}{2^m-1}\sum_{v\subseteq I_\tau}\tilde{\gamma}_{\tau+1,v\cup \{\tau+1\}}^\lambda \frac{1}{(2^{2\lambda}-2)^{|v|+1}} .
\end{aligned}$$ Then using (\[eq:jensen\]), we have $$\begin{aligned}
& B^\lambda({\boldsymbol{q}}^{*}_{\tau+1},{\boldsymbol{\gamma}}) \\
= & \left(\left(1+\frac{\gamma_{\tau+1}}{3}\right)B({\boldsymbol{q}}^{*}_\tau,{\boldsymbol{\gamma}})+\theta(q^*_{\tau+1})\right)^\lambda \\
\le & \left(1+\frac{\gamma_{\tau+1}}{3}\right)^{\lambda}B^\lambda({\boldsymbol{q}}^{*}_\tau,{\boldsymbol{\gamma}})+\theta^\lambda(q^*_{\tau+1}) \\
= & \left(1+\frac{\gamma_{\tau+1}}{3}\right)^{\lambda}\frac{1}{2^m-1}\sum_{\emptyset \ne v\subseteq I_\tau}\tilde{\gamma}_{\tau,v}^\lambda \frac{1}{(2^{2\lambda}-2)^{|v|}}+\frac{1}{2^m-1}\sum_{v\subseteq I_\tau}\tilde{\gamma}_{\tau+1,v\cup \{\tau+1\}}^\lambda \frac{1}{(2^{2\lambda}-2)^{|v|+1}} \\
= & \frac{1}{2^m-1}\sum_{\emptyset \ne v\subseteq I_\tau}\tilde{\gamma}_{\tau+1,v}^\lambda \frac{1}{(2^{2\lambda}-2)^{|v|}}+\frac{1}{2^m-1}\sum_{v\subseteq I_\tau}\tilde{\gamma}_{\tau+1,v\cup \{\tau+1\}}^\lambda \frac{1}{(2^{2\lambda}-2)^{|v|+1}} \\
= & \frac{1}{2^m-1}\sum_{\emptyset \ne v\subseteq I_{\tau+1}}\tilde{\gamma}_{\tau+1,v}^\lambda \frac{1}{(2^{2\lambda}-2)^{|v|}} ,
\end{aligned}$$ for $1/2<\lambda \le 1$. Hence the result follows.
For any $\tau$ such that $1\le \tau\le s$, we have $$\begin{aligned}
B({\boldsymbol{q}}^{*}_\tau,{\boldsymbol{\gamma}}) & \le \frac{1}{(2^m-1)^{1/\lambda}}\left[ \sum_{\emptyset \ne v\subseteq I_\tau}\prod_{j\in v}\left(\frac{\gamma_j}{3}\right)^\lambda \prod_{j'\in I_\tau\setminus v}\left(1+\frac{\gamma_{j'}}{3}\right)^\lambda \frac{1}{(2^{2\lambda}-2)^{|v|}} \right]^{1/\lambda} \\
& = \frac{1}{(2^m-1)^{1/\lambda}}\left[ \prod_{j=1}^{\tau}\left(\left( \frac{1}{2^{2\lambda}-2}\cdot \frac{\gamma_j}{3}\right)^\lambda+\left(1+\frac{\gamma_j}{3}\right)^\lambda\right)-\prod_{j=1}^{\tau}\left(1+\frac{\gamma_j}{3}\right)^\lambda \right]^{1/\lambda} \\
& \le \frac{1}{(2^m-1)^{1/\lambda}}\left[ \prod_{j=1}^{\tau}\left(1+\frac{2^{2\lambda}-1}{2^{2\lambda}-2}\left( \frac{\gamma_j}{3}\right)^\lambda\right)-\prod_{j=1}^{\tau}\left(1+\frac{\gamma_j}{3}\right)^\lambda \right]^{1/\lambda},
\end{aligned}$$ where we have used (\[eq:jensen\]) in the last inequality. This expression gives an upper bound on $B({\boldsymbol{q}}^{*}_\tau,{\boldsymbol{\gamma}})$ using not $\tilde{\gamma}_{\tau,v}$ but $\gamma_j$ for $1\le j\le s$. Then as in Corollary \[cor:tractability\], assume that the sequence of weights $\gamma_1,\gamma_2,\ldots,$ satisfy the condition $$\begin{aligned}
B_{{\boldsymbol{\gamma}},\lambda} := \sup_{s\in {\mathbb{N}}}\frac{\left[ \prod_{j=1}^{s}\left(1+\frac{2^{2\lambda}-1}{2^{2\lambda}-2}\left( \frac{\gamma_j}{3}\right)^\lambda\right)-\prod_{j=1}^{s}\left(1+\frac{\gamma_j}{3}\right)^\lambda \right]^{1/\lambda}}{\prod_{j=1}^{s}\left(1+\frac{\gamma_j}{3}\right)-1},
\end{aligned}$$ for some $\lambda$ such that $1/2<\lambda \le 1$. Then the mean square weighted ${\mathcal{L}}_2$ discrepancy is strongly tractable with the exponent of tractability at most $\lambda$.
\[remark:fast\_cbc\] The criterion used in Algorithm \[cbc\_algor\_product\] can be simplified into $$\begin{aligned}
B(({\boldsymbol{q}}^{*}_{\tau-1},q_\tau),{\boldsymbol{\gamma}}) = -\prod_{j=1}^{\tau}\left(1+\frac{\gamma_{j}}{3}\right)+\frac{1}{2^m}\sum_{n=0}^{2^m-1}\prod_{j=1}^{\tau}\left[1+\gamma_{j}\phi (x_{n,j})\right] .
\end{aligned}$$ For this form, it is possible to reduce the computational cost of the CBC construction by using the fast Fourier transform as shown in [@NC06a; @NC06b].
Numerical experiments {#numer}
=====================
Finally, we demonstrate the performance of our constructed polynomial lattice rules. We focus on the case of product weights, that is, $\gamma_u=\prod_{j\in u}\gamma_j$, because of their importance in practice and the availability of the fast CBC construction algorithm using fast Fourier transform as mentioned in Remark \[remark:fast\_cbc\]. Three choices for $\gamma_j$ are considered here: $\gamma_j=1$ (unweighted), $\gamma_j=0.9^j$ and $\gamma_j=1/j^2$ for $j=1,\ldots,s$.
We compare the performance of our constructed polynomial lattice point sets with that of Sobol’ sequences, which is one of the most well-known digital sequences over ${\mathbb{F}}_2$ [@Sob67]. Since the weights emphasize the relative importance of the discrepancy of lower dimensional projections, we use Sobol’ sequences as constructed in [@JK08], which should work as a good competitor.
In Table \[tb:1\]-\[tb:3\], we show the values of the mean square weighted ${\mathcal{L}}_2$ discrepancy for Sobol’ sequences and our constructed polynomial lattice point sets, denoted by Sobol’ and PLR respectively, with $m=4,\ldots, 15$ and $s=1,5,50,100$ and different choices for the weights.
As expected from Remark \[remark:cbc\_one\_dimension\], we obtain exactly the same values for both the rules for $s=1$ and achieve the optimal rate of convergence, $2^{-2m}$, independent of the choice of the weights. In case of $s=5$, although Sobol’ sequence provide the slightly better results for large $m$, the values are comparable. For $s=50$ and $s=100$, we obtain almost the same values for both the rules in the unweighted case, while our constructed polynomial lattice point sets outperform Sobol’ sequences in other cases.
Acknowledgement {#acknowledgement .unnumbered}
===============
This research is supported by Grant-in-Aid for JSPS Fellows No.24-4020.
[99]{}
J. Baldeaux, Higher order nets and sequences, PhD thesis, The University of New South Wales, 2010.
J. Baldeaux and J. Dick, A construction of polynomial lattice rules with small gain coefficients, Numer. Math., [**119**]{}, 271–297, 2011.
R.E. Caflisch, W. Morokoff and A.B. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance, [**1**]{}, 27–46, 1997.
H.E. Chrestenson, A class of generalized Walsh functions, Pacific J. Math., [**5**]{}, 17–31, 1955.
R. Cranley and T.N.L. Patterson, Randomization of number theoretic methods for multiple integration, SIAM. J. Numer. Anal., [**13**]{}, 904–914, 1976.
J. Dick, F.Y. Kuo, F. Pillichshammer and I.H. Sloan, Construction algorithms for polynomial lattice rules for multivariate integration, Math. Comp., [**74**]{}, 1895–1921, 2005.
J. Dick, G. Leobacher, and F. Pillichshammer, Construction algorithms for digital nets with low weighted star discrepancy, SIAM. J. Numer. Anal., [**43**]{}, 76–95, 2005.
J. Dick and F. Pillichshammer, [*Digital nets and sequences. Discrepancy theory and quasi-Monte Carlo integration*]{}, Cambridge University Press, Cambridge, 2010.
J. Dick and F. Pillichshammer, Optimal ${\mathcal{L}}_2$ discrepancy bounds for higher order digital sequences over finite field ${\mathbb{F}}_2$, preprint. Available at [http://arxiv.org/abs/1207.5189]{}.
N.J. Fine, On the Walsh functions, Trans. Amer. Math. Soc., [**65**]{}, 372–414, 1949.
F.J. Hickernell, A generalized discrepancy and quadrature error bound, Math. Comp., [**67**]{}, 299–322, 1998.
S. Joe, and F.Y. Kuo, Constructing Sobol’sequences with better two-dimensional projections, SIAM J. Sci. Comput., [**30**]{}, 2635–2654, 2008.
N.M. Korobov, The approximate computation of multiple integrals/ approximate evaluation of repeated integrals, Dokl. Akad. Nauk SSSR, [**124**]{}, 1207–1210, 1959.
P. Kritzer, and F. Pillichshammer, Constructions of general polynomial lattices for multivariate integration, Bull. Anstral. Math. Soc., [**76**]{}, 93–110, 2007.
P. Kritzer, and F. Pillichshammer, On the component by component construction of polynomial lattice point sets for numerical integration in weighted Sobolev spaces, Unif. Distrib. Theory, [**6**]{}, 79–100, 2011.
P. L’Ecuyer, Polynomial integration lattices, In: Monte Carlo and Quasi-Monte Carlo Methods 2002, H. Niederreiter (Ed.), Springer, Berlin, 73–98, 2004.
C. Lemieux, [*Monte Carlo and quasi-Monte Carlo sampling*]{}, Springer Series in Statistics, Springer, New York, 2009.
J. Matoušek, On the ${\mathcal{L}}_2$-discrepancy for anchored boxes, J. Complexity, [**14**]{}, 527–556, 1998.
J. Matoušek, [*Geometric discrepancy. An illustrated guide*]{}, Algorithms and Combinatorics, Springer, Berlin, 1999.
H. Niederreiter, [*Random number generation and quasi-Monte Carlo methods*]{}, CBMS-NSF Series in Applied Mathematics, vol. 63, SIAM, Philadelphia, 1992.
H. Niederreiter, Low-discrepancy point sets obtained by digital constructions over finite fields, Czechoslovak Math. J., [**42**]{}, 143–166, 1992.
S. Ninomiya and S. Tezuka, Toward real-time pricing of complex financial derivatives, Appl. Math. Finance, [**3**]{}, 1–20, 1996.
E. Novak and H. Woźniakowski, [*Tractability of multivariate problems. Volume II: Standard informations for functionals*]{}, EMS, Zurich, 2010.
D. Nuyens and R. Cools, Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces, Math. Comp., [**75**]{}, 903–920, 2006.
D. Nuyens and R. Cools, Fast component-by-component construction, a reprise for different kernels, In: Monte Carlo and Quasi-Monte Carlo Methods 2004, H. Niederreiter and D. Talay (Eds.), Springer, Berlin, pp. 373–387, 2006.
A.B. Owen, Randomly permuted $(t,m,s)$-nets and $(t,s)$-sequences, In: Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. H. Niederreiter and J.-S. Shiue (Eds.), Springer, New York, pp. 299–317, 1995.
A.B. Owen, Monte Carlo variance of scrambled net quadrature, SIAM. J. Numer. Anal., [**34**]{}, 1884-1910, 1997.
A.B. Owen, Scrambled net variance for integrals of smooth functions, Ann. Statist., [**25**]{}, 1541-1562, 1997.
S.H. Paskov and J.F. Traub, Faster valuation of financial derivatives, J. Portfolio Manage., [**22**]{}, 113–120, 1995.
K.F. Roth, On irregularities of distribution, Mathematika, [**1**]{}, 73–79, 1954.
I.H. Sloan and S. Joe, [*Lattice Methods for Multiple Integration*]{}, Oxford University Press, Oxford, 1994.
I.H. Sloan and A.V. Reztsov, Component-by-component construction of good lattice rules, Math. Comp., [**71**]{}, 263–273, 2002.
I.H. Sloan and H. Woźniakowski, When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?, J. Complexity, [**14**]{}, 1–33, 1998.
I.M. Sobol’, Distribution of points in a cube and approximate evaluation of integrals (in Russian), Zh. Vycisl. Mat. i Mat. Fiz., [**7**]{}, 784–802, 1967.
S. Tezuka and H. Faure, $I$-binomial scrambling of digital nets and sequences, J. Complexity, [**19**]{}, 744–757, 2003.
J.L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math., [**45**]{}, 5–24, 1923.
H. Woźniakowski, Average case complexity of multivariate integration, Bull. Amer. Math. Soc. New Series, [**24**]{}, 185–194, 1991.
S.K. Zaremba, Some applications of multidimensional integration by parts, Ann. Polon. Math., [**21**]{}, 85–96, 1968.
$m$
----- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
Sobol’ PLR Sobol’ PLR Sobol’ PLR Sobol’ PLR
4 6.51E-04 6.51E-04 4.83E-02 3.79E-02 3.93E+07 3.91E+07 2.54E+16 2.54E+16
5 1.63E-04 1.63E-04 1.45E-02 1.37E-02 1.96E+07 1.94E+07 1.27E+16 1.27E+16
6 4.07E-05 4.07E-05 5.04E-03 4.29E-03 9.70E+06 9.64E+06 6.35E+15 6.35E+15
7 1.02E-05 1.02E-05 1.27E-03 1.32E-03 4.78E+06 4.77E+06 3.18E+15 3.18E+15
8 2.54E-06 2.54E-06 4.11E-04 4.69E-04 2.36E+06 2.35E+06 1.59E+15 1.59E+15
9 6.36E-07 6.36E-07 1.21E-04 1.38E-04 1.17E+06 1.16E+06 7.94E+14 7.94E+14
10 1.59E-07 1.59E-07 4.01E-05 4.47E-05 5.80E+05 5.70E+05 3.97E+14 3.97E+14
11 3.97E-08 3.97E-08 1.15E-05 1.28E-05 2.89E+05 2.80E+05 1.98E+14 1.98E+14
12 9.93E-09 9.93E-09 3.45E-06 4.41E-06 1.44E+05 1.37E+05 9.92E+13 9.91E+13
13 2.48E-09 2.48E-09 1.17E-06 1.39E-06 7.17E+04 6.69E+04 4.96E+13 4.95E+13
14 6.21E-10 6.21E-10 2.78E-07 4.05E-07 3.56E+04 3.27E+04 2.48E+13 2.48E+13
15 1.55E-10 1.55E-10 7.98E-08 1.31E-07 1.76E+04 1.59E+04 1.24E+13 1.24E+13
: The mean square weighted ${\mathcal{L}}_2$ discrepancy for $\gamma_j=1$.
\[tb:1\]
$m$
----- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
Sobol’ PLR Sobol’ PLR Sobol’ PLR Sobol’ PLR
4 5.86E-04 5.86E-04 2.13E-02 1.72E-02 1.43E+00 1.22E+00 1.48E+00 1.26E+00
5 1.46E-04 1.46E-04 6.25E-03 5.93E-03 6.27E-01 5.16E-01 6.47E-01 5.34E-01
6 3.66E-05 3.66E-05 2.07E-03 1.80E-03 2.47E-01 2.17E-01 2.56E-01 2.25E-01
7 9.16E-06 9.16E-06 5.25E-04 5.41E-04 9.81E-02 8.85E-02 1.02E-01 9.19E-02
8 2.29E-06 2.29E-06 1.64E-04 1.84E-04 3.94E-02 3.52E-02 4.11E-02 3.67E-02
9 5.72E-07 5.72E-07 4.73E-05 5.23E-05 1.60E-02 1.41E-02 1.66E-02 1.47E-02
10 1.43E-07 1.43E-07 1.52E-05 1.70E-05 6.73E-03 5.62E-03 7.02E-03 5.87E-03
11 3.58E-08 3.58E-08 4.29E-06 5.19E-06 2.97E-03 2.26E-03 3.10E-03 2.36E-03
12 8.94E-09 8.94E-09 1.25E-06 1.58E-06 1.25E-03 8.90E-04 1.31E-03 9.33E-04
13 2.24E-09 2.24E-09 4.01E-07 4.85E-07 5.61E-04 3.57E-04 5.86E-04 3.75E-04
14 5.59E-10 5.59E-10 9.89E-08 1.43E-07 2.13E-04 1.41E-04 2.24E-04 1.49E-04
15 1.40E-10 1.40E-10 2.79E-08 4.38E-08 7.84E-05 5.61E-05 8.30E-05 5.91E-05
: The mean square weighted ${\mathcal{L}}_2$ discrepancy for $\gamma_j=0.9^j$.
\[tb:2\]
$m$
----- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
Sobol’ PLR Sobol’ PLR Sobol’ PLR Sobol’ PLR
4 6.51E-04 6.51E-04 1.84E-03 1.73E-03 2.99E-03 2.47E-03 3.07E-03 2.53E-03
5 1.63E-04 1.63E-04 4.81E-04 4.76E-04 8.63E-04 7.31E-04 8.95E-04 7.50E-04
6 4.07E-05 4.07E-05 1.35E-04 1.28E-04 2.64E-04 2.10E-04 2.78E-04 2.17E-04
7 1.02E-05 1.02E-05 3.53E-05 3.43E-05 7.42E-05 5.98E-05 8.09E-05 6.24E-05
8 2.54E-06 2.54E-06 9.21E-06 9.43E-06 2.23E-05 1.75E-05 2.48E-05 1.84E-05
9 6.36E-07 6.36E-07 2.53E-06 2.51E-06 6.56E-06 4.94E-06 7.37E-06 5.24E-06
10 1.59E-07 1.59E-07 6.94E-07 6.86E-07 1.75E-06 1.41E-06 2.02E-06 1.51E-06
11 3.97E-08 3.97E-08 1.82E-07 1.90E-07 4.87E-07 4.12E-07 5.53E-07 4.43E-07
12 9.93E-09 9.93E-09 4.76E-08 5.00E-08 1.39E-07 1.16E-07 1.62E-07 1.26E-07
13 2.48E-09 2.48E-09 1.29E-08 1.35E-08 4.06E-08 3.40E-08 4.89E-08 3.70E-08
14 6.21E-10 6.21E-10 3.35E-09 3.80E-09 1.29E-08 1.01E-08 1.53E-08 1.10E-08
15 1.55E-10 1.55E-10 8.87E-10 1.01E-09 3.61E-09 2.97E-09 4.37E-09 3.27E-09
: The mean square weighted ${\mathcal{L}}_2$ discrepancy for $\gamma_j=1/j^2$.
\[tb:3\]
|
---
abstract: 'Starting from kinetic theory, we obtain a nonlinear dissipative formalism describing the nonequilibrium evolution of scalar colored particles coupled selfconsistently to nonabelian classical gauge fields. The link between the one-particle distribution function of the kinetic description and the variables of the effective theory is determined by extremizing the entropy production. This method does not rely on the usual gradient expansion in fluid dynamic variables, and therefore the resulting effective theory can handle situations where these gradients (and hence the momentum-space anisotropies) are expected to be large. The formalism presented here, being computationally less demanding than kinetic theory, may be useful as a simplified model of the dynamics of color fields during the early stages of heavy ion collisions and in phenomena related to parton energy loss.'
author:
- 'J. Peralta-Ramos'
- 'E. Calzetta'
title: Effective dynamics of a nonabelian plasma out of equilibrium
---
Introduction
============
The results of numerous experiments on ultrarelativistic heavy ion collisions carried out at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) clearly point to the conclusion that a hot, dense and opaque nuclear medium is created in such events [@RHIC-disc1; @RHIC-disc2; @RHIC-disc3; @RHIC-disc4; @RHIC-disc5; @RHIC-disc6]. By now, the standard picture for the evolution of such matter consists in a pre-equilibrium stage dominated by strong chromoelectromagnetic fields, followed by the formation of the Quark Gluon Plasma (QGP) in local thermal equilibrium. The QGP thus formed expands and cools under its own pressure, going through the deconfinement transition in which hadrons are formed and later on detected (for reviews see [@revhydro1; @revhydro2; @reviewIANCU] and references therein).
Before the collision, the occupation number of gluons at high energy is so large that an approximation in terms of classical gauge fields obeying Yang-Mills equations becomes more suitable than a description in terms of on-shell particles [@reviewIANCU; @GLASMA]. The time scale for this highly nonlinear regime occurring at the earliest stage of a heavy ion collision is $Q_s^{-1}\sim 0.2$ fm/c at RHIC, where $Q_s$ is the saturation scale.
The dense system appearing between first impact and the formation of the equilibrated QGP develops Chromo-Weibel instabilities (see e.g. [@strick-Weibel; @arn-Weibel]): the non-equilibrium anisotropic distribution of the partons is responsible for the fast growth of the chromomagnetic plasma modes, which in turn isotropize the system and speed up the thermalization process to yield a thermalization time $\sim 1$ fm/c. This is roughly the value of the thermalization time that can be inferred from a comparison of state-of-the-art hydrodynamic simulations to data. Moreover, a fast parton transversing the matter formed in heavy ion collisions excites color fields and loses energy, a phenomenon known as jet quenching [@loss1; @loss2]. For these reasons, the dynamics of classical gauge fields (both coupled or not to hard partons) has received a lot of attention in the context of heavy ion phenomenology [@reviewIANCU; @GLASMA; @romstrick; @mro93; @mro94; @mro97; @Selikhov; @nayak; @GLASMA; @muller96therm; @muller96lattice; @bergesgauge; @DuslingPhi; @florMHD; @Gale; @fujii; @strick-Weibel; @arn-Weibel; @mannakin; @fluct; @strick; @flor; @dum; @kurk; @mromull; @mullerloss; @losscomp; @losscomp2; @rom1; @rom2; @carrdiel; @mro89; @sch; @sch-WYM; @muller-pos; @Tani; @ipp; @stein; @ManMro; @manna; @jiang; @holm; @wakejiang; @white; @localeq].
The color fields “see" the evolving [*matter*]{} (which in our case is composed of colored partons) through the conserved color current, as dictated by Yang-Mills equations. The dynamics of the conserved currents, i.e. the color current and the total energy-momentum tensor of the combined system of matter plus gauge fields, is in general very complex, since in principle it must be computed from the dynamics of the matter fields (or particles in the kinetic limit).
In the semiclassical kinetic approach, the coupling of hard and soft modes is implemented through a nonabelian transport equation [@Heinz; @HeinzPRL] which determines the evolution of a one-particle distribution function $\mathbf{f}$, which is a matrix in color space (see Section \[kin\]), or equivalently through Wong’s equations [@wong]; see also Refs. [@libro; @elze; @Litim; @winter; @revBlaizot; @BlaizotIancu; @BraPis90a; @FreTay90; @Bod98; @Bod99; @ArSoYa99a; @ArSoYa99b]. Once $\mathbf{f}$ is known, the color current acting as the source in Yang-Mills equation is completely determined, so the dynamics of the gauge fields can be found.
Here, we take the view that it is a reasonable hypothesis to assume that the dynamics of the conserved currents is [*largely*]{} determined by the conservation laws themselves. This opens up the possibility of constructing an effective theory incorporating the conservation laws, that allows one to investigate relevant aspects of the dynamics of gauge fields (for example, the border between stability and instability, or the back reaction of hard particles on the evolution of color fields) in a simpler context as compared to the microscopic theory.
The degrees of freedom of the effective theory include hydrodynamic variables such as flow velocity. However, since we are working in a strongly out of equilibrium regime, it is not possible to obtain a closed dynamics based on hydrodynamic variables alone.
We obtain a closed theory by identifying a tensor $\lambda^{\mu\nu}$ (which is introduced in Section \[eff\]) through which the system couples to the hydrodynamic variables. One can think of the introduction of the nonhydrodynamic variable $\lambda^{\mu\nu}$ as a simple way of modeling the back reaction of the distribution function (which could correspond to a highly nonequilibrium situation) on the hydrodynamic modes (i.e. those modes associated to conservation laws and hence relaxing much more slowly). A somewhat similar situation is discussed in [@tanos; @tanos2; @tanos3; @anile; @muscato] in the context of the Entropy Maximum Principle (EMP), in [@GKpaper] (see also [@GKbook]) where the moments of the collision term of the classical Boltzmann equation are interpreted as independent variables rather than as infinite moment series, in [@mauricio] and [@aniso] in the context of “anisotropic hydrodynamics”, in [@nagy] in the framework of divergence-type theories [@geroch] (see also [@dev; @app; @linking]), in [@denicol] where the closure is obtained from an expansion of the distribution function in moments [*at all orders*]{}, and in [@christen; @christen2] in relation to the Entropy Production Principle (EPP).
Here, the tensor $\lambda^{\mu\nu}$ is identified with a Lagrange multiplier in a well-defined variational problem whose solution yields the distribution function that extremizes the entropy production (for a review of this method see [@epvm]). This procedure results in a closure for the distribution function of colored particles which has two satisfying properties: (i) it is nonlinear in the variables of the effective theory, thus generalizing Grad’s quadratic ansatz [@deGroot; @ferz; @is1; @liboff; @kremer] in a nontrivial way [@linking; @progress], and (ii) it does not rely in any way on the gradient expansion in fluid variables. The equation of motion for $\lambda^{\mu\nu}$ is then obtained from this closure by the method of moments.
The result is an effective theory for the dynamics of color fields coupled to colored particles that can handle highly nonequilibrium situations, for example the large momentum anisotropy present at early times in heavy ion collisions. Our formalism is a simplified model that shares with the true dynamics the conservation laws, Lorentz and gauge invariance, being causal, and satisfying the Second Law. In the spirit of finding the simplest possible theory incorporating the conservation laws, we postulate a simple BGK form for the collision term [@liboff; @kremer] in the Boltzmann equation, and obtain solutions at second order in the relaxation time. By going to second order in the relaxation time (instead of the usual first order treatment) we expect to broaden even more the range of applicability of the effective theory, to be able to describe better those situations with large momentum-space anisotropies.
The limitations of the model put forward in this work are discussed in the last section, but it is worth mentioning them here. The model is phenomenological, it is valid only when the gauge fields can be treated classically, and only when the relaxation time approximation is valid. Moreover, the collision term is linear. We would like to stress that our approach is not new. The idea of obtaining a closure from a variational principle whose Lagrange multipliers are identified with macroscopic variables is at the heart of the so-called Extended Thermodynamic Theories (see e.g. [@extended]), which use as a variational principle the EMP. However, in most applications of this theory the EMP is only used to express $f$ in terms of the Lagrange multipliers [*but not*]{} to obtain their dynamics. In Refs. [@tanos; @anile; @tanos2; @tanos3; @muscato] the EMP was used to accomplish both tasks in the context of electron transport through mesoscopic semiconductors in the nonlinear and nonequilibrium regime (i.e. under conditions of very strong electric fields and large gradients). The result was a closed effective theory which could reproduce well the results of kinetic Monte Carlo simulations.
In this paper, instead of using the EMP we rely on the EPP, which allows us to find the distribution function that extremizes the entropy production subject to constraints on the conserved currents [@epvm]. We note that the EPP was used in [@christen] to construct a model of nonequilibrium electron transport and in [@christen2] to describe radiative heat transport in a photon gas, obtaining results that agree well with more sophisticated approaches.
The motivation for the choice of the EPP is threefold. This principle is deeply connected to nonequilibrium statistical physics, in particular to the evolution of fluctuations around a stationary state (we shall not discuss this connection here; for details see [@epvm; @min1; @ferz; @prigo]). Moreover, as it is rigorously shown in Ref. [@epvm] for nonrelativistic systems (see also [@ferz]), the distribution function obtained from the EPP actually solves the [*linearized*]{} Boltzmann equation, which means that the closure provided by this variational principle is able, at least in principle, to capture some features of the microscopic dynamics. Finally, the transport coefficients obtained from the EMP can differ substantially from those computed from the EPP, the latter providing better agreement with the results obtained from kinetic theory (see for instance [@epvm; @tanos; @anile; @tanos2; @tanos3; @muscato; @christen2; @app]).
In relation to our developments, we note in particular Ref. [@florMHD] in which the so-called “anisotropic hydrodynamics” (that was developed in [@aniso; @mauricio]) is coupled to color fields in a way reminiscent of magnetohydrodynamics. In [@flor], an effective model for the dynamics of color fields coupled to particles is obtained from the Boltzmann-Vlasov equations by the method of moments and then applied to study chromoelectric oscillations in a dynamically evolving anisotropic background; see also [@strick]. As it will become clear in what follows, our approach resembles those adopted in these studies in that they are effective theories capable of dealing with large deviations from equilibrium.
The colorless version of the effective theory developed in this paper was investigated in [@dev] (although there it was obtained in a different way as the one followed here). It was then applied to study the evolution of matter created in heavy ion collisions at RHIC and to the calculation of hadronic observables in [@app], and compared to second order fluid dynamics [@hyd1; @hyd2], to which the colorless effective theory reduces when deviations from equilibrium are small. The connection between the (colorless) effective formalism and kinetic theory was established in [@linking].
Similarly to what happens in the colorless version of the effective theory [@dev; @app], the formalism presented here reduces to the so-called “chromohydrodynamics” [@Heinz; @ManMro; @jiang; @holm] when deviations from equilibrium are small (see Section \[matrix\]). Previous studies based on chromohydrodynamics include the calculation of the wake potential induced by a fast parton [@wakejiang] as well as collective excitations and instabilities in the QGP [@ManMro; @manna; @jiang; @fraga; @cse].
This paper is organized as follows. In Section \[setup\], we describe the basic theoretical setup for our developments, give a very brief overview of the kinetic theory of a nonabelian plasma, and introduce the conserved and entropy currents. In Section \[eff\] we obtain a closure for the distribution function from the entropy production principle and derive the evolution equations of the effective theory by the method of moments applied to the transport equation; this Section contains our main results. In Section \[matrix\] we compare our developments to the matrix approach to chromohydrodynamics, and as a simple illustrative example we compute the polarization tensor of the colored plasma including a finite relaxation time for the fluctuations. We conclude in Section \[summ\] with some comments on the possible application of the developed formalism to the dynamics of color fields and instabilities in heavy ion collisions.
Theoretical setup {#setup}
=================
The system
----------
We are interested in obtaining an effective theory for the dynamics of a system of colored particles interacting with nonabelian classical gauge fields. In this work we will deal with scalar particles coming in three colors, which in our simple model would represent massless and spinless quarks. We shall therefore consider a classical Yang-Mills field coupled to conformal scalar matter in the fundamental representation of $SU(3)$.
In what follows, we will use $(\mu,\nu,\ldots)$ to denote world indices and $(a,b,\ldots)$ to denote internal (color) indices. We shall denote with $N=3$ the dimension of the fundamental representation, and use $n$ to indicate a generic dimension ($n=3$ or $n=8$ for the fundamental or adjoint representations, respectively). The generators $\mathbf{T}_a$ are traceless hermitian $n\times n$ matrices with commutation relations
=iC\^c\_[ab]{}\_c \[0\] and trace
\_a\_b=12\_[ab]{} \[01\] The Yang-Mills field is $\mathbf{A}_{\mu}=A^a_{\mu}\mathbf{T}_a$. The field tensor
\_=\_\_-\_\_-ig\[1\] belongs to the adjoint representation of the gauge group. The equations of motion for the Yang-Mills field are
\_\^=-\[3\] where the covariant derivative is
\_=\_-ig\[4\] $\mathbf{Q}$ is a projection operator
=2\_a\_a\_a=-1n \[4.01\] Eq. (\[3\]) implies the Bianchi identity
=0 \[5\]
We also have the energy-momentum tensor
T\^\_[YM]{}=\^\_[YM]{} \[6\] where
\^\_[YM]{}=\^\_\^-14g\^\^\_ \[7\] is traceless in world indices. Using the identity $\mathbf{D}_{\left(\mu\right.}\mathbf{F}_{\left.\nu\lambda\right)}=0$ (where brackets mean symmetrization) we get
{\_\^\_[YM]{}+\_\^}=0 \[8\]
Kinetic theory {#kin}
--------------
In principle the scalar matter should be described by quantum field theory. The reduction of the nonequilibrium quantum field description to kinetic theory is fairly established by now, see e.g. [@libro] and references therein.
The kinetic equation which governs the evolution of the one-particle distribution matrix $\mathbf{f}$ reads [@HeinzPRL; @Heinz] (see also [@libro; @elze; @BlaizotIancu; @winter; @Litim]) p\^=(p\^0)\_[col]{} \[9\] where \_ = \_ - ig \[\_,\] with $\mathbf{A}_\mu$ expressed in the fundamental representation. $\mathbf{f}\left(X,p\right)$ is an $N\times N$ matrix ($N=3$ for quarks) and obeys $\mathbf{f}^{\dagger}\left(X,p\right)=\mathbf{f}\left(X,p\right)$.
The collision kernel on the right hand side encodes the interaction among the hard excitations of the matter field, and is, in general, a complicated functional of the self-energy [@libro]. However, to carry out our developments we really do not need to consider it in much detail, and we anticipate that later on we will use a phenomenological linear collision operator that will suffice for our present purposes.
Conserved currents {#currents}
------------------
We now introduce the matter entropy and the conserved currents of the microscopic theory. The nonabelian current reads \_=gDpp\_ \[10\] where $Dp=d^4p\delta\left(p^2\right)/\left(2\pi\right)^3$.
We have not written down an explicit equation for the matter stress-energy tensor $T^{\mu\nu}_{m}$, but we know that since the total stress-energy must be conserved, we must have T\^\_[m;]{}=-T\^\_[YM;]{}= \_\^ \[ecT\] We get this by writing T\^\_[m]{}=\^\_[m]{} \[11\]
\^\_[m]{}=Dpp\^p\^ \[12\] In Eq. (\[ecT\]), the semicolon stands for an ordinary derivative. We shall drop the subindex $m$ for $\mathbf{T}^{\mu\nu}_{m}$ and write $\mathbf{T}^{\mu\nu}$ in what follows.
Eqs. (\[5\]) and (\[8\]) are identically satisfied provided
&Dp(p\^0)(\_a\_[col]{})=\
&Dp(p\^0)p\^\_[col]{}=0 \[13\]
The entropy current is
S\^=Dpp\^(p\^0){(+)(+)-} \[14\] leading to the entropy production
S\^\_[;]{}=Dp{\_[col]{}\^[-1]{}(+)} \[15b\]
We note that to go from Eq. (\[14\]) to Eq. (\[15b\]) one must assume that $[\mathbf{f},\mathbf{D}_\mu \mathbf{f}]=0$ (see [@localeq]). If this condition is not imposed on the distribution function, the entropy production contains terms of the form $\rm{tr} \{\mathbf{F}_{\mu\nu},\mathbf{f}\}$ which contribute to entropy production even in mean field. Assuming $[\mathbf{f},\mathbf{D}_\mu \mathbf{f}]=0$ then corresponds to assuming that entropy is produced solely by collisions among the particles. We shall stick to this approximation in what follows.
Effective theory {#eff}
================
Obtaining a closure for $\mathbf{f}$ {#closure}
------------------------------------
We will now obtain an expression for the one-particle distribution matrix in terms of variables of the effective theory. To do so, we will rely on the EPP [@epvm] discussed in the Introduction.
This method allows one to find the distribution function which extremizes the entropy production $S^\mu_{;\mu}$, subject to the constraints that the conserved currents take on known values. It provides a prescription to associate a distribution function to given macroscopic currents, yielding a nonlinear closure that generalizes the well-known Grad’s quadratic ansatz [@deGroot; @is1; @ferz; @liboff; @kremer] in a nontrivial way [@linking; @progress].
The EPP does not rely on a gradient expansion, which is usually invoked when deriving hydrodynamics from kinetic theory. This results in effective theories capable of describing highly nonequilibrium and nonlinear situations quite reliably as compared to microscopic approaches [@epvm; @tanos; @anile; @tanos2; @tanos3; @muscato; @christen; @christen2].
### Deviations from the unperturbed state
For simplicity, in what follows we will neglect quantum statistics. Given $\rm{tr}(\mathbf{T}^{\mu\nu})=T^{\mu\nu}$, we can define a flow velocity $u^\mu$ and a temperature $T$ by using the Landau-Lifshitz prescription. We have u\_T\^ = (T) u\^ where $\rho(T)$ is the energy density as obtained from the equation of state. The prescription amounts to matching the local nonequilibrium state of the flowing real matter to a fiducial perfect fluid. We will show later that the stress tensor $\Pi^{\mu\nu}$ is transverse, i.e. $u_\mu\Pi^{\mu\nu}=0$, so the prescription is consistent. With $u^\mu$ and $T$, we can define $\beta^\mu=u^\mu/T$ and then construct f\_0(x\^,p\^) = e\^[-\_p\^]{} We write the distribution function as = f\_0 \[+(1+f\_0)\] f\_0 \[+\] \[fandchi\] The entropy production S\^\_[;]{}=-Dp{\_[col]{}} then reads S\^\_[;]{}=-Dp{\_[col]{}(+)} \[smumubis\] where we have used that Dp{\_[col]{}(f\_0)} = 0
Given the expression for the entropy production given by Eq. (\[smumubis\]) and taking into account that we will consider a linear collision operator, in order to satisfy the H-theorem we introduce a new variable $\mathbf{Z}$ such that e\^ = + The solution to the variational problem entails finding $\mathbf{Z}$ as a function of the Lagrange multipliers to be introduced shortly. The outcome is a closure for the distribution function $\mathbf{f}$, i.e., an expression for $\mathbf{f}$ in terms of the variables of the effective theory, which are the usual hydrodynamic variables and the Lagrange multipliers. The latter encode the back reaction of the distribution function on the hydrodynamic modes.
The basic plan we will follow is to divide relevant variables into colorless and colored pieces. Therefore, we parametrize $\mathbf{Z}$ as follows = + \^a \^a \[parz\] The quantities $\zeta^a$ can be identified with color fugacities $\zeta^a \equiv \mu^a/T$, where $\mu^a$ are the color chemical potentials needed to conserve color (see Eq. (\[13\])). We shall show later that the color chemical potentials must adjust to the flow of matter as well as to the evolving gauge fields in order to make the system globally colorless, resulting in a highly nontrivial dynamics for the system of colored particles and gauge fields. We will work to quadratic order in $\mathbf{Z}$, so we get
= + \^2 Using that \^a\^b = \^[ab]{} + K\^[ab]{}\_c \^c with K\^[ab]{}\_c (i C\_[abc]{}+ d\_[abc]{}) where $d_{abc}$ are the symmetric structure constants, we get \^2 = \^2 + \^a \^a + \^a \^a + \^a \^b d\^[ab]{}\_c \^c
Having $\mathbf{Z}$, we obtain
&= + \^a \^a +
### Currents, entropy production and collision term
To solve the variational problem we must express the currents and the entropy production in terms of $\mathbf{Z}$.
The shear tensor \^ = p\^p\^ reads \^ = p\^p\^\[shearfull\] where we have introduced the notation Dp f\_0 (…) (…)
From Eq. (\[10\]) we get the expression for the color currents \^= |[n]{}u\^+ p\^\^a \^a \[jdfull\] where we have defined $\bar{n}=g\left\langle \omega \right\rangle$.
For the entropy production, we have
S\^\_[;]{}= -Dp (\_[col]{} )
The collision operator contains color independent and dependent parts, so, similarly to the decomposition used for $\mathbf{Z}$, we put \_[col]{} = I\^[(0)]{}\_[col]{} + I\^[a]{}\_[col]{} \^[a]{} We then have (\_[col]{} ) = I\^[(0)]{}\_[col]{}+ I\^[a]{}\_[col]{}\^a where we have used Eq. (\[01\]), so S\^\_[;]{}= - Dp (I\^[(0)]{}\_[col]{}+ I\^[a]{}\_[col]{}\^a ) \[entaux\]
We shall write the collision operator as \_ = - R\[FR\[\]\] \[linea\] where we have put \_[col]{} f\_0 \_
In Eq. (\[linea\]), $\tau$ is the relaxation time, $F=F(\omega)$ is an arbitrary function of energy $\omega=-u_\mu p^\mu$, and $R$ is an operator enforcing the integrability conditions given in Eqs. (\[13\]). The projector $R$ explicitly reads R\[\] = - R\^[ab]{}(\_a ) \_b - R\_p\^() \[Rexplicit\] with R\_= u\_+ \^\_p\_and $R_{ab}=2\delta_{ab}$. Here, \^ = g\^+u\^u\^ is the spatial projector. Note that this form for $\mathbf{I}_{\beta}$ guarantees that the Second Law holds exactly. Moreover, it is flexible enough to include the important cases of Marle’s relativistic generalization of the BGK model [@liboff; @kremer], corresponding to $F(\omega)=T$, as well as the Anderson-Witting model corresponding to $F(\omega)=\omega$ [@kremer] (for a discussion of these models in connection to the freeze out stage in heavy ion collisions see [@linking; @progress]).
From Eq. (\[linea\]) we then have I\^[(0)]{}\_ = - R\[FR\[\]\] \[i0aux\] and I\^[a]{}\_ = - R\[FR\[\^[a]{}\]\] \[iaaux\] with R\[\] = - R\_p\^ and \_a R\[\_a\] = \_a \_a - R\^[bc]{} \_c \_b \[ra\]
The integrability conditions then read p\^I\^[(0)]{}\_ = 0 \[int0\] and I\^[a]{}\_ = 0 \[inta\] Note that there are no restrictions on $\left\langle p^\rho I^{a}_{\beta} \right\rangle$ or on $\left\langle I^{(0)}_{\beta} \right\rangle$.
### Variational equations
In our case the conserved currents are $J^{a\mu}$ and $T^{\mu\nu}$, so that the variational problem becomes \[S\^\_[;]{} - \_T\^-\_[a]{}J\^[a]{}\] = 0 where $\lambda_{\mu\nu}$ and $\lambda_{a\mu}$ are Lagrange multipliers forcing the energy momentum tensor and the color currents to take on their known values.
We then have
&= \_ + \_[a]{}\
&= \_ + \_[a]{}
\[varia\]
Using Eqs. (\[shearfull\]), (\[jdfull\]) and (\[entaux\]), together with Eqs. (\[i0aux\]) and (\[iaaux\]), the variational equations (\[varia\]) become R\[FR\[\]\] = \_ p\^p\^+ g\_\^a p\^\[variat1\] and R\[FR\[\^[b]{}\]\] = \_ p\^p\^\^b + g\_\^a p\^\[variat2\]
We will solve the equations (\[variat1\]) and (\[variat2\]) to second order in the relaxation time $\tau$. To this end, we will follow [@linking] and expand the nonequilibrium correction $\mathbf{Z}$ and the Lagrange multipliers as follows
&= \^[(1)]{} + \^[(2)]{}\
\^[a]{} &= \^[a(1)]{} + \^[a(2)]{}\
\_ &= \_\^[(1)]{} + \_\^[(2)]{}\
\^a\_ &= \_\^[a(1)]{} + \_\^[a(2)]{}
\[zetas2nd\]
We now go over to solve the equations at first and second order in $\tau$. We note that the solution given here closely follows the one given in [@linking]. For the reader’s convenience, a brief summary of the logical steps carried out to obtain the closure for the distribution function can be found in Section (\[sumclos\]).
### First order solution
At first order, the variational equations are R\[FR\[\^[(1)]{}\]\] = \^[(1)]{}\_ p\^p\^ \[var1\] and R\[FR\[\^[b(1)]{}\]\] = 0 \[var2\] Eqs. (\[ra\]) and (\[var2\]) imply that $\zeta^{b(1)}$ must be independent of $p^\nu$, as it must be given that $\zeta^{b(1)}$ are color fugacities.
From Eq. (\[var1\]) we get \^[(1)]{}\_ p\^p\^p\^= 0 \[l0var\] Setting $\rho=k$ we obtain $\lambda^{(1)}_{0k}=0$. Without loss of generality we can take $\lambda^{(1)}_{00}=0$. Since we are dealing with a conformal theory $\left\langle p^i p^j \right\rangle = \delta^{ij} \left\langle \omega^2 \right\rangle/3$, and we get $\lambda^{(1)i}_i = 0$. We thus obtain \^[(1)]{}= \^[(1)]{}\_[ij]{} p\^i p\^j \[zeta10\]
The first order shear tensor is then \_1\^ = p\^p\^\^[(1)]{} We find that $\Pi^{00}=\Pi^{k0}=0$. Using that G() p\^i p\^j p\^k p\^l = (\^[ij]{}\^[kl]{}+ \^[ik]{}\^[jl]{}+\^[il]{}\^[jk]{}) \[idenG\] for any function of energy $G(\omega)$, we get \_1\^[ij]{} = \^[(1)ij]{} \[pi1aux\] which is traceless and transverse. The first order color current is (recall that $\zeta^{a(1)}$ must be independent of $p^\nu$) J\_1\^[a]{} = |[n]{} \^[a(1)]{} u\^\[j1a\]
The lowest order nontrivial contributions to the entropy current and the entropy production are S\^0\_1 = - \^[(1)ij]{}\^[(1)]{}\_[ij]{} [and]{} S\^i\_1 = 0 \[ecur\] in the rest frame, and S\^\_[;]{} = \^[(1)ij]{}\^[(1)]{}\_[ij]{} \[eprod\] respectively. The entropy flux (\[ecur\]) and the entropy production (\[eprod\]) computed from the first order solutions $(\zeta^{(1)},\zeta^{a(1)})$ to the variational equations are already quadratic in deviations from equilibrium, so we do not need to consider higher order contributions. Note that, because of the first order equation (\[var2\]), $\zeta^{a(1)}$ does not contribute to the entropy production at quadratic order.
### Second order solution
The variational equations at second order read
R\[FR\[\^[(2)]{}\]\] &= \^[(2)]{}\_ p\^p\^+ \^[(1)]{}\_ p\^p\^\^[(1)]{}\
& + \_\^[a(1)]{}p\^\^[a(1)]{}
and R\[FR\[\^[b(2)]{}\]\] = g\_\^[b(1)]{} p\^\^[(1)]{} + \_\^[(1)]{} p\^p\^\^[b(1)]{}
The integrability conditions then read N \_\^[(2)]{}p\^p\^+ (\^[(1)]{})\^2 F + g\_0\^[a(1)]{}\^[a(1)]{} \^2 = 0 \[s1\] N \_[0j]{}\^[(2)]{}p\^i p\^j + g\_j\^[a(1)]{}\^[a(1)]{} p\^i p\^j = 0 \[s2\] corresponding to Eq. (\[int0\]) and \_[0]{}\^[b(1)]{}\^[(1)]{} = 0 \[s3\] corresponding to Eq. (\[inta\]). The latter equation shows that $\lambda_{0}^{b(1)}=0$.
Any term in $\lambda_{\mu\nu}^{(2)}$ which is not strictly required by the integrability conditions can be absorbed into $\lambda_{\mu\nu}^{(1)}$, so there is no loss of generality if we take $\lambda_{00}^{(2)}= 0$. Moreover, from Eq. (\[s2\]) we see that $\lambda_{0j}^{(2)}=0$ and $\lambda_{j}^{a(1)}=0$ is a solution to the integrability condition. Therefore, we can write \_[ij]{}\^[(2)]{}=-\_[ij]{} where we have put =
The quadratic equations then read R\[FR\[\^[(2)]{}\]\] = -N \^2 + (\^[(1)]{})\^2 F \[sec1\] and R\[FR\[\^[b(2)]{}\]\] = -\^2 \^[b(1)]{} \[secb\]
Note that the left hand side of Eq. (\[sec1\]) vanishes when integrated against $\omega$ (which means that the equation has a solution) but does not vanish when integrated agaist $\omega/F$. Therefore, the solution to Eq. (\[sec1\]) is \^[(2)]{} = (\^[(1)]{})\^2 - N - A \[zeta20\] where A= \^[-1]{}
Similarly, we obtain from Eq. (\[secb\]) \^[b(2)]{} = - \^[b(1)]{} + \[zeta2b\] where B\^b = \^[b(1)]{}\^2 Note that $\Lambda \propto (\zeta^{(1)})^2$ is already quadratic in $\tau$, which means that $\zeta^{b(2)}$ is actually third order and therefore can be neglected.
We are ready to compute $\Pi_2^{\mu\nu}$ and $J_2^{a\mu}$; see Eqs. (\[shearfull\]) and (\[jdfull\]). We have \_2\^ = p\^p\^Note that $\Pi_2^{0i}=0$, but $\Pi_2^{00}\neq 0$, so $\Pi_2^{\mu\nu}$ is not the true correction to the energy-momentum tensor, whereby the parameter $T$ in our equations is not the true temperature (that would be measured by an observed moving with the local rest frame). To obtain the physical correction to $T^{\mu\nu}$, which we call $\Pi_{2,phys}^{\mu\nu}$, we must perform a temperature shift. Putting $T=T_{phys}-\delta T$ with $\delta T d\rho/dT = \Pi_2^{00}$, the true correction reads \_[2,phys]{}\^[ij]{}= \_2\^[ij]{}-\^[ij]{}\_2\^[00]{} which is traceless. Taking into account the above we find
\_[2,phys]{}\^[ij]{} &= L \[\^[(1)i]{}\_k \^[(1)kj]{}-\^[ij]{}\^[(1)lm]{}\^[(1)]{}\_[lm]{}\]\
& +T\^[ij]{}\^[a(1)]{} \^[a(1)]{} \[piphys\]
with L = To obtain the last term in Eq. (\[piphys\]) we have used the identity $\left\langle \omega^2 \right\rangle = 2T\left\langle \omega \right\rangle $. Note also that $\Pi_{2,phys}^{ij}$ is transverse. For simplicity, in what follows we will drop the subindex “phys” in $\Pi_{2,phys}^{ij}$.
The quadratic contribution to the color current can be computed directly from Eqs. (\[jdfull\]), (\[zeta20\]) and (\[zeta2b\]). We get J\_[2c]{}\^ = u\^d\^[ab]{}\_c \^[a(1)]{} \^[b(1)]{} \[j2htot\] where we have used that $\lambda^{(1)j}_j = 0$ so that the last term in Eq. (\[jdfull\]) drops out.
Using the closure picked out by the EPP, we have completed the task of expressing the currents $\rm{tr}(\mathbf{T}^{\mu\nu})$ and $\mathbf{J}^\mu$ in terms of the variables of the effective theory.
### Summary of the EPP method {#sumclos}
For clarity, we now briefly summarize the main logical steps followed to obtain the closure for $\mathbf{f}$.
From a linear transport equation, we set up the variational problem given in Eqs. (\[varia\]). The solution to these equations gives the distribution function that extremizes the production of entropy subject to the constraints that the conserved currents take on known values. Using the expressions for the stress tensor, the color currents and the entropy production given in Eqs. (\[shearfull\]), (\[jdfull\]) and (\[entaux\]), respectively, together with the integrability conditions for the collision term, Eqs. (\[i0aux\]) and (\[iaaux\]), the variational equations become Eqs. (\[variat1\]) and (\[variat2\]).
We then expand the Lagrange multipliers and the nonequilibrium correction $\mathbf{Z}$ in powers of the relaxation time $\tau$ (Eq. (\[zetas2nd\])), and solve the variational equations to second order in $\tau$. The result is an expression for the correction $\mathbf{Z}$ in terms of the Lagrange multipliers, which is given by Eqs. (\[zeta10\]) and (\[zeta20\]). From these equations, we can express the shear tensor and the color currents in terms of the Lagrange multipliers, obtaining Eqs. (\[pi1aux\]), (\[j1a\]), (\[piphys\]) and (\[j2htot\]).
The distribution function then reads
&=f\_0\
&+ f\_0
\[closfin\] For convenience, we denote by $f$ and $f^a$ the colorless and colored parts of $\mathbf{f}$:
f&= f\_0 \[fexp\]
f\^a = f\_0 (\^[a(1)]{}+ \_[ij]{}\^[(1)]{}p\^i p\^j \^[a(1)]{} + d\_[ef]{}\^a \^[e(1)]{} \^[f(1)]{}) \[fa\]
As discussed in the Introduction, this method, or very similar ones based on maximizing the entropy, have been used in different contexts to obtain closures for the distribution function, which resulted in effective models whose dynamics compared well with kinetic theory (see e.g. [@epvm; @prigo] for a broad perspective and [@tanos; @anile; @tanos2; @muscato; @tanos3; @linking; @christen; @christen2] for specific applications).
To obtain a dynamical model, we must now find the equation of motion of the Lagrange multipliers.
Equations of motion {#motion}
-------------------
The EPP described above has provided us with an expression for the distribution function $\mathbf{f}$ in terms of the reduced set of variables of the effective theory: the usual hydrodynamic variables $u^\mu$, $\rho$, the nonhydrodynamic tensor $\lambda_{\mu\nu}^{(1)}$ encoding the back reaction of $\mathbf{f}$ on the hydrodynamic modes, and the color fugacities $\zeta^{a(1)}$. This expression is given in Eq. (\[closfin\]).
The equations of motion for $u^\mu$, $\rho$ and $\zeta^{a(1)}$ are the conservation equations for $T^{\mu\nu}$ and $J^{a\mu}$, respectively. However, the conservation laws are not enough to fully determine the dynamics of the system, and an equation governing the evolution of $\Pi^{\mu\nu}$ must be given. Usually, this is done within the gradient expansion for hydrodynamic variables [@deGroot; @ferz; @kremer; @revhydro1; @revhydro2; @hyd1; @hyd2]. Here, instead, $\Pi^{\mu\nu}$ is an algebraic function of $\lambda^{(1)ij}$ and $\zeta^{a(1)}$, so we must obtain the evolution equation for $\lambda^{(1)ij}$. We will get the latter from the kinetic equation.
It will prove convenient to express our results in terms of a new relaxation time $\tau_\pi$ (related by a constant to the previously introduced $\tau$) = \^[-1]{} and the shear viscosity = \^[-1]{} \^2 which naturally arise in the context of second order fluid dynamics [@revhydro1; @revhydro2; @hyd1; @hyd2]. For simplicity, we also introduce a new variable \^[ij]{}T \^[(1)ij]{}
We shall deal with the conservation equations first, and then go over to discuss the evolution equation for $\gamma^{ij}$.
### Conservation equations
Inserting the expression for $\mathbf{f}$ given by Eq. (\[closfin\]) into Eqs. (\[5\]) and (\[8\]), and using that $T^{\mu\nu}_{m,\nu}=-T^{\mu\nu}_{YM,\nu}=
\mathrm{tr}\mathbf{J}_{\lambda}\mathbf{F}^{\mu\lambda}$ in the latter, we get equations of motion for the velocity $u^\mu$, the energy density $\rho$ and $\zeta^{a(1)}$.
For the matter energy-momentum tensor $T^{\mu\nu} = T^{\mu\nu}_0 + \Pi^{\mu\nu}_1+\Pi^{\mu\nu}_2$, where T\^\_0 = (u\^u\^+\^) \[T0\] is the perfect-fluid energy-momentum tensor, \^\_1 = \^ \[pi1\] and
\^\_2 &= (\^\_\^ + \^\^\_ )\
& + T\^\^[a(1)]{} \^[a(1)]{} \[pi2\]
we get T\^\_[0;]{} + \^\_[1;]{} +\^\_[2;]{} = |[n]{}(\^[e(1)]{}+d\^[e]{}\_[cd]{}\^[c(1)]{} \^[d(1)]{}) u\_F\^[e]{} \[eomtfinal\] To avoid being cumbersome, we shall not write down the explicit expression for $T^{\mu\nu}_{;\nu}$, but it follows immediately from Eqs. (\[T0\])-(\[pi2\]).
The conservation equations for $u^\mu$ and $\rho$ as given in Eq. (\[eomtfinal\]) involve the time derivative of $\zeta^{a(1)}$ and $\gamma_{\mu\nu}$ (these quantities appear in the LHS of Eq. (\[eomtfinal\])). The evolution equation for $\zeta^{a(1)}$ is obtained from the (covariant) conservation of $\mathbf{J}^\mu$. We get
&\_[;]{}\
& + g|[n]{} u\^C\^f\_[ab]{} A\^a\_ ( \^[b(1)]{} + d\^b\_[cd]{}\^[c(1)]{} \^[d(1)]{} ) = 0 \[eomzeta1\]
### Evolution equation
We shall now deal with the evolution equation for the nonequilibrium tensor $\gamma_{\mu\nu}$. We will obtain this equation from the second moment of the singlet sector of the transport equation Eq. (\[9\]), i.e. from the equation that results from taking the trace of (\[9\]). The reason for considering the colorless part of the transport equation will be discussed after presenting the evolution equation.
The second moment of the colorless part of the kinetic equation reads
&Dp p\^p\^p\^\
&= Dp(p\^0) p\^p\^(\_[[col]{}]{})
\[3moment\]
We have that $p^\mu \rm{tr}(\mathbf{D}_\mu \mathbf{f}) = p^\mu\partial_\mu \rm{tr}(\mathbf{f})=p^\mu\partial_\mu f$, where $f$ is the colorless part of $\mathbf{f}$ given in Eq. (\[fexp\]) at quadratic order.
The second term in the left-hand side of Eq. (\[3moment\]) becomes -Dp p\^p\^p\^F\_\^a where $f^a$ is the colored part of $\mathbf{f}$ given in Eq. (\[fa\]) at quadratic order.
The right-hand side of Eq. (\[3moment\]) is N Dp p\^p\^[sign]{}(p\^0) I\_[[col]{}]{}\^[(0)]{}
Using the explicit expressions for $f$ and $f^a$ given in Eqs. (\[fexp\]) and (\[fa\]) in the moment equation (\[3moment\]) we obtain (recall that $\gamma^{ij}\equiv \eta T \lambda^{(1)ij}$ and that, for simplicity, we assume that $\eta$ is a constant)
& -M\^u\_[;]{} + ( N\_1\^[ij]{} + N\_2\^[ijlm]{} \_[lm]{} ) \_[ij;]{}\
&+ N\_0\^ \^[a(1)]{} \^[a(1)]{}\_[;]{} + \^F\^a\_ ( (\^[a(1)]{}+d\^a\_[bc]{}\^[b(1)]{} \^[c(1)]{})N\_0\^\
&+ \^[a(1)]{} \_[ij]{} N\_1\^[i j]{}) - F\^a\_[j]{}\^[a(1)]{} \_i\^[j]{}N\_1\^[i]{} =\
& -\_[ij]{}( N\_1\^[ij]{} + \_[lm]{} N\_2\^[ijlm ]{} ) \[eomtemp\]
For brevity, we have defined the following quantities
M\^ &= (1+\^[a(1)]{} \^[a(1)]{})N\_0\^ - \_[ij]{} N\_1\^[i j]{}\
&+ \_[ij]{} \_[lm]{} N\_2\^[i j l m]{}
with N\_\^ = p\^p\^p\^p\^\[Ns\]
In order to find an explicit evolution equation for $\gamma_{ij;\mu}$, we must be able to invert the tensor with which it is contracted, namely H\^[ij]{} N\_1\^[ij]{} + N\_2\^[ijlm]{} \_[lm]{} For our present purposes it is enough to display the equation of motion for the nonequilibrium tensor $\gamma^{ij}$ to linear order. Using suitable generalizations of Eq. (\[idenG\]) to compute the $N's$ explicitly we get (in the local rest frame)
\^[ij]{} &= -a\_1 \^[a(1)]{} \^[ij]{}\_1\^a -(1+\^[a(1)]{} \^[a(1)]{})\^[ij]{}\
&+ a\_2 \^[a(1)]{} F\^[a(i]{}\_\^[j)]{} -\^[ij]{} -\^[ij]{}\
&+ \^[i]{}\_k \^[kj]{}+\^[j]{}\_k \^[ki]{} -\^[ij]{}\_[kl]{}\^[kl]{}
\[finaleom\]
In Eq. (\[finaleom\]), $\sigma^{\rho\sigma}$ is the first order shear tensor \^ = \^[<]{}u\^[>]{} where $C^{<\mu\nu>}$ denotes taking the traceless and transverse part of a tensor $C$ and $\nabla^{\mu}=\Delta^{\mu\nu}\partial_\nu$ is the spatial gradient. We have denoted the convective derivative by an overdot, i.e. $\dot{C}=u^\alpha \partial_\alpha C$. The parenthesis around indices denote symmetrization.
The transport coefficients $a_k$ that appear in the evolution equation are
a\_[1]{}&= \^[-1]{} \^[-1]{}\^3\
a\_[2]{}&= \^[-1]{}
The transport coefficients $a_1$ and $a_2$ are novel coefficients that couple the nonequilibrium tensor $\gamma^{\rho\sigma}$ to color degrees of freedom. A nonvanishing $a_1$ implies that the varying color chemical potentials affect the evolution of $\gamma^{\rho\sigma}$. The term containing $a_2$ represents the coupling of $\gamma^{\rho\sigma}$ to the gauge fields.
If $\tau_\pi \rightarrow 0$ and $\zeta^{a(1)}=0$ in Eq. (\[finaleom\]), we recover the (colorless) Navier-Stokes limit with $\gamma^{\mu\nu} \rightarrow -\eta \sigma^{\mu\nu}$. On the other hand, we have already shown in [@dev] that if we expand $\gamma^{\mu\nu}$ to second order in velocity gradients, the above formalism (with $\zeta^{a(1)}=0$) goes over to the second order conformal hydrodynamics that was derived in Refs. [@hyd1; @hyd2].
The equations of motion presented in this section are the main result of this work. The essential features of the effective formalism developed here are that it is nonlinear and that it is not tied up in any way to a gradient expansion. Note that the nonequilibrium tensor $\gamma^{\mu\nu}$, from which the shear tensor $\Pi^{\mu\nu}$ is obtained [*a posteriori*]{} as a quadratic function, satisfies a differential equation (\[finaleom\]) instead of being an algebraic function of velocity gradients (as it is $\Pi^{\mu\nu}$ in fluid dynamics). In the colorless case, it was shown in [@dev; @app] that this feature results in a faster isotropization of the pressure as compared to second order hydrodynamics. Moreover, as opposed to the case of hydrodynamics, in the (colorless) effective theory the longitudinal pressure is positive throughout the entire evolution. We note that similar results were obtained in [@mauricio] within the so-called anisotropic hydrodynamics approach. We expect that these two results hold also in the present case including color degrees of freedom, although numerical simulations are needed to verify this.
The color fugacities enter nontrivially in both sides of the total stress-energy tensor conservation equation, Eq. (\[eomtfinal\]). In the left-hand side they enter through the expression for the matter stress-energy tensor, Eq. (\[pi2\]), while in the right-hand side they enter through the coupling to nonabelian fields. Moreover, both the hydrodynamic variables $(u^\mu, \rho)$ and the nonhydrodynamic variable $\gamma^{\mu\nu}$ couple to the color fields and to $\zeta^{a(1)}$ in the evolution equation (\[finaleom\]).
From Eq. (\[finaleom\]), it is seen that the dynamics of the system is highly nontrivial. In part, this is because the color chemical potentials must adjust to the flow as well as to the evolving gauge fields in order to make the system globally colorless (recall that we obtained the evolution equation from the singlet sector of the kinetic equation). We emphasize that a nonvanishing color current does not imply that the system as a whole carries a finite color charge, because the space-time dependence of the color chemical potentials can be such that the total color charge vanishes [@ManMro].
The information about the constituents of the microscopic theory is encoded in the transport coefficients of the effective theory, and should in principle be computed from the former. However, the transport coefficients can also be treated as adjustable parameters. A well-known example is the case of fluid dynamics, which is usually derived from kinetic theory in the weakly coupled limit and under the relaxation time approximation (to first order in $\tau$), but then used to describe strongly coupled matter [@revhydro1; @revhydro2]. This is done by replacing the transport coefficients of the kinetic theory by those corresponding to strong coupling, which must be computed by different means, for example, by using the AdS/CFT correspondence [@hyd1; @hyd2]. A similar route can be taken with the various transport coefficients that arise in our formalism (at linear order in $\gamma^{ij}$ these are $(\eta,\tau_\pi,a_1,a_2)$). In this regard, it is important to emphasize that the effective theory presented here is consistent [*by itself*]{}, independently of its derivation from kinetic theory (this is also true in the case of second order hydrodynamics [@hyd1; @hyd2]), because it satisfies the Second Law and it is expected to be causal (the colorless version was shown to be causal in [@dev]).
The reason for considering the colorless part of the transport equation is that, as discussed in [@Bod98; @white], the color currents can persist in the plasma significantly longer than the color charge density, which is neutralized rather fast. Therefore, a reasonable hypothesis is to assume that the dynamics of the system is determined by the singlet part of the distribution function. What this means is that, even though the particles carry color (and thus interact with nonabelian fields), what we are actually describing with the effective theory is the collective (or macroscopic) behavior of these particles, and it is this collective flow that is colorless. This hypothesis is physically well-motivated because one does not expect the plasma to be globally colorful, and due to this fact it has been adopted in previous studies dealing with chromohydrodynamics [@ManMro; @manna; @jiang; @wakejiang] and kinetic theory [@mannakin; @sch-WYM; @sch] (see also [@Heinz] for a related discussion of this issue in the context of the so-called “color hierarchy” transport equations).
### Yang-Mills equation
In order to obtain a selfconsistent system of equations for the variables $(\rho,u^\mu,\zeta^{a(1)},\gamma^{\mu\nu}, A^{a\mu})$, the conservation equations (\[eomtfinal\]) and (\[eomzeta1\]) and the evolution equation (\[finaleom\]) must be supplemented with Yang-Mills equation for the gauge fields. For completeness, we write it down
&(A\^[b]{})\^[;]{}\_[;]{} - (A\^[b]{})\^[;]{}\_[;]{} + gC\_[cd]{}\^b (A\^[c]{}A\^[d]{})\_[;]{} + g A\_\^e C\_[ef]{}\^b F\^[f]{}\
& = |[n]{} u\^( \^[b(1)]{} + d\^b\_[ad]{}\^[a(1)]{} \^[d(1)]{} ) \[ymsplit\]
Relation to matrix chromohydrodynamics {#matrix}
======================================
In this section we will compare the linearized versions of our approach to linearized matrix chromohydrodynamics of Ref. [@ManMro]. We remark that beyond the linear order it is not possible to establish a simple mapping between the approach of Ref. [@ManMro], which relies upon a gradient expansion, and the one presented here, which does not.
Although our approach involving nonhydrodynamic variables is not tied up to a gradient expansion, for a local equilibrium state $\lambda^{(1)}_{ij}$ vanishes and so the effective theory reduces to ideal fluid dynamics. The ideal fluid chromohydrodynamic approach has been discussed in [@Heinz; @ManMro; @holm], and later on applied to diverse studies mostly related to plasma instabilities and medium-jet interaction in the context of heavy ion collisions [@localeq; @manna; @jiang; @wakejiang].
In the matrix chromohydrodynamic approach [@ManMro], the standard procedure is to linearize the matrix equations \_\^=\_\^ \_\^=0 together with Yang-Mills equation in matrix fluctuations $\delta \mathbf{u}^\mu,\delta \mathbf{\rho}, \delta J^a_{\nu},\delta F^{a\nu\mu}$ with respect to a given background. Aditionally, a relation between fluctuations of the matrix energy density $\mathbf{\rho}$ and the matrix pressure $\mathbf{p}$ is used. To the best of our knowledge, in previous studies based on the matrix approach the relation used in all cases is $(\delta p)_a=c_s^2(\delta \rho)_a$. It is worth noting that the use of $\delta p = c_s^2\delta \rho$ in our approach is completely equivalent to the one adopted in the matrix formalism, as will be shown shortly. In studies dealing with chromohydrodynamics, usually the ideal fluid case is considered; for a recent extension to the Navier-Stokes case see [@jiang; @wakejiang].
Generalities
------------
To understand the connection between ideal or Navier-Stokes chromohydrodynamics and our effective theory, we first note that our development of the effective theory was ultimately based on Eq. (\[8\]), which determines the coupling of matter to fields. In Eq. (\[8\]), the gauge fields are coupled to the [*trace*]{} of the matter energy-momentum tensor. Therefore, we end up with evolution equations for $\rm{tr}(\mathbf{T}^{\mu\nu})$, and not for $\mathbf{T}^{\mu\nu}$ itself, coupled to the Yang-Mills equation. Instead, the chromohydrodynamics of [@ManMro] is based on $\mathbf{T}^{\mu\nu}$, which requires the introduction of color [*matrices*]{} $\mathbf{u}^\mu,\mathbf{\rho}$.
The color current, defined in Eq. (\[10\]), is then written as $\mathbf{J}^\mu=\mathbf{n}\mathbf{u}^\mu$. In our formalism, the quantity $\bar{n}\zeta^a$ can be interpreted (at least at linear order) as the [*average*]{} color charge of a stream of colored classical particles. This interpretation for $\zeta^a$ can be seen from its [*linearized*]{} equation of motion given in Eq. (\[eomzeta1\]) |[n]{}\^[f(1)]{} + g|[n]{} u\^C\^f\_[ab]{} A\^a\_ \^[b(1)]{} = 0 We see that it is identical to Wong’s equation [@wong] (see also [@libro; @winter; @Selikhov; @Litim]) for an [*average*]{} classical color charge $Q^a \equiv \bar{n}\zeta^{a(1)}$, with the time derivative of the particle’s trajectory replaced by $u^\mu$, i.e., the flow velocity.
Having the interpretation for $\zeta_a$ described above in mind, the linearized colored fluctuations of the velocity and the four-flow $\mathbf{n}$ of the matrix approach can be written in terms of the scalar fluctuations used in our approach as $\delta u^{a\mu} = \bar{\zeta}^{a(1)}\delta u^\mu$ and $\delta n^a = \bar{n}\zeta^{a(1)}$, where $\bar{\zeta}^{a(1)}$ stands for the background value, which must be nonzero to avoid ending up in the case of a truly colorless system, as opposed to one which [*has*]{} color degrees of freedom but is in a colorless equilibrium state. This mapping will be used in the next section to obtain the polarization tensor of the colored plasma.
A simple example: The polarization tensor {#polar}
-----------------------------------------
The polarization tensor characterizes the linear response of the system to external perturbations [@libro; @revBlaizot; @Litim; @ichi; @krall; @fluct; @Heinz; @romstrick; @mro89], and it is therefore an interesting quantity to compute in the formalism presented here. With the mapping between the matrix and our approach described in the previous section, it is straightforward to show that indeed the linearized equations of motion have the same structure, and thus a very similar polarization tensor is obtained. However, the linearized equations are the same only if we set $\gamma^{\mu\nu}=0$ or $\gamma^{\mu\nu}=-\eta \sigma^{\mu\nu}$, which were the only cases studied so far [@ManMro; @manna; @jiang].
Our formalism naturally incorporates higher order velocity gradients, since the dissipative tensor $\gamma^{\mu\nu}$ evolves according to a differential equation rather than being expressed as an algebraic function of velocity gradients, e.g. at first order as $\gamma^{\mu\nu}=-\eta \sigma^{\mu\nu}$. It is therefore interesting to investigate the role of higher order terms on the polarization tensor. Such terms are important, for instance, in the earliest stage of evolution of the QGP created in heavy ion collisions or in situations where a fast parton goes through the QGP [@neuf; @neufmach].
In order to quantify the impact of higher order viscous terms on the linear response of the system to an external gauge field, we compute the polarization tensor explicitly in our setting. The polarization tensor is defined through J\_a\^=-\^\_[ab]{}A\_[b]{} where $A_{b\nu}$ is a small external perturbation. For simplicity, we shall consider an homogeneous, stationary and colorless background described by $\bar{n}$, $\bar{u}^\mu$ and $\bar{\rho}$. Using the mapping described above, namely $\delta u^{a\mu} = \bar{\zeta}^{a(1)}\delta u^\mu$ and $\delta n^a = \bar{n}\zeta^{a(1)}$, together with $(\delta \rho)^a \equiv \bar{\zeta}^{a(1)}\delta \rho$ and $\gamma^{a\mu\nu} \equiv \bar{\zeta}^{a(1)} \gamma^{\mu\nu}$, our linearized equations become (covariant derivatives become ordinary derivatives at this order)
&|[n]{}\_u\^\_a + |[u]{}\^\_n\_a = 0\
&|[u]{}\^\_\_a + (1+c\_s\^2)|\_u\^\_a = 0\
&c\_s\^2 |\^\_\_a + (1+c\_s\^2)||[u]{}\^\_u\^\_a\
& - |[n]{}|[u]{}\_F\^\_a + \_\^\_a = 0\
&|[u]{}\^\_\^\_a = -(\^\_a + \^\_a)\
&\_F\^\_a = |[u]{}\^n\_a + |[n]{}u\^\_a
\[lineari\] where we have put $\bar{\Delta}^{\mu\nu}=g^{\mu\nu}+\bar{u}^\mu \bar{u}^\nu$. For reasons that will become clear soon, in the above equations we use a generic squared speed of sound $c_s^2$ instead of the conformal value $c_s^2=1/3$, i.e. the relation between pressure and energy density perturbations reads $\delta p = c_s^2 \delta \rho$.
Performing a Fourier transformation we can express $\delta J_a^\mu$ in terms of the gauge field perturbation and thus find the polarization tensor (the calculation is very similar to that presented in Ref. [@jiang]). We get
\_[ab]{}\^ &= -\_[ab]{} (\
& )
where
W\_1 &= -(k\^2+(c\_s\^[-2]{}-1)(k|[u]{}) )\^[-1]{}\
W\_2 &=\
W\_3 &= -\
W\_4 &= k\^2 - (k|[u]{})\^2\
W\_5\^ &= (k|[u]{})(|[u]{}\^k\^+ k\^|[u]{}\^)
and \_[[pl]{}]{}\^2 = is the plasma frequency. Note that $\Gamma_{ab}^{\mu\nu}$ is diagonal in color space, as expected since as shown by Eqs. (\[lineari\]) there is no mixing of colors at linear order, and transverse with respect to $k^\mu$.
The result for $\Gamma_{ab}^{\mu\nu}$ that we obtain is the same as that obtained in Ref. [@jiang] but with the shear viscosity $\eta$ replaced by an effective one \_[[eff]{}]{}= The appearance of $\eta_{\rm{eff}}$ in place of $\eta$ is quite natural since $\tau_\pi$ is precisely the relaxation time of the shear tensor $\Pi^{\mu\nu}$ towards its Navier-Stokes value. We emphasize that this similarity with the chromohydrodynamic result holds only when linearizing the equations. The fully nonlinear evolution equations of the effective theory developed here, which do not involve hydrodynamic gradients, are different from those of Navier-Stokes chromohydrodynamics.
To quantify the effect of higher order viscous terms on the linear response of the plasma to an external gauge field, we will work with the longitudinal part $\epsilon_L$ of the dielectric tensor $\epsilon^{ij}$. Similar analysis can be performed for the transverse part of the dielectric tensor, $\epsilon_T$, but for brevity we shall only consider $\epsilon_L$. We have (we suppress color indices and put $k^\mu=(\omega,\mathbf{k})$ in what follows) \^[ij]{}= \^[ij]{} + \^[ij]{} and \_L = In the rest frame $\bar{u}^\mu=(1,0,0,0)$ we get \_L(,k) = 1-()(1-(W\_1+W\_3)) where we can use that $(1+c_s^2)\bar{\rho}=sT$ ($\bar{s}$ is the entropy density) to rewrite $W_2$ as W\_2 =
For illustrative purposes, and following [@jiang], we focus on the soft modes $\omega,\sqrt{\mathbf{k}^2} \ll T$ and set $\sqrt{\mathbf{k}^2}=2 \omega_{\rm{pl}}$ and $T=10\omega_{\rm{pl}}$. The relaxation time is set to its value computed from the kinetic theory of a Boltzmann gas (without color), which is given by $\tau_\pi = 6 \eta/(sT)$. As a typical value for the temperature we shall use $T=200$ MeV.
The comparison made between ideal chromohydrodynamics and kinetic theory carried out in Ref. [@mannakin] in the context of jet-induced instabilities shows that in order to achieve reasonable agreement between both descriptions an effective speed of sound must be used in the former. To be consistent with previous studies we will show numerical results obtained with this effective speed of sound (instead of $c_s^2=1/3$), which is given by c\_s\^2 = \^[-1]{} + \[ceff\] where $y=\sqrt{\mathbf{k}^2}/\omega$. We note that this expression for $c_s^2$ is chosen to make the longitudinal dielectric function obtained from the ideal chromohydrodynamic approach of Ref. [@mannakin] identical to that obtained from kinetic theory in the leading-order HTL approximation [@libro; @reviewIANCU; @revBlaizot; @BraPis90a; @FreTay90; @Bod98; @Bod99; @ArSoYa99a; @ArSoYa99b; @Litim]. Specifically, in the soft limit $\omega,\sqrt{\mathbf{k}^2} \ll T$ and putting $\mathbf{k}=(k,0,0)$ for simplicity we have [@jiang]
\_L &= 1+( 1- )\
&- ( 1- | | + \^2\
&-\^2(k\^2-\^2)\
&+ i (k\^2-\^2) )
\[elfull\] where $\delta \rightarrow 0^+$ and we used that = | | - i(k\^2-\^2) with $\theta(x)$ the Heavyside step function. This shows that, in the soft limit, the modes with $\omega>k$ are undamped if $\tau_\pi = 0$. If the relaxation time does not vanish, the imaginary part of $\epsilon_L$ is nonzero even for $\omega>k$. We will now show some illustrative examples of this feature.
Figure \[p03\] shows the real and imaginary parts of $\epsilon_L$ as a function of $\omega/k$ for $\eta/s = 0.3$. This value for $\eta/s$ is on the high side in terms of fitting viscous fluid dynamics results to RHIC and LHC data [@revhydro1; @revhydro2; @reviewIANCU]. It is seen that there are significant differences between the longitudinal dielectric function computed with different values of the relaxation time. The most noticeable effects are seen on the imaginary part of $\epsilon_L$, which, for $\omega < k$, is smaller in the case with nonvanishing $\tau_\pi$. This indicates that, in this range of frequencies, the induced color excitations decay more slowly as compared to the case with $\tau_\pi=0$. This behavior can be understood by recalling the physical meaning of $\tau_\pi$ as the relaxation time of the shear tensor $\Pi^{\mu\nu}$ towards its Navier-Stokes value $-\eta \sigma^{\mu\nu}$. If the value of $\tau_\pi$ is increased, then hydrodynamic fluctuations will decay more slowly. Since hydrodynamic fluctuations are coupled to color fluctuations, the latter will decay more slowly as well. This result is in agreement with those of Ref. [@sch] obtained from kinetic theory with a BGK collision kernel, showing that the addition of hard-particle collisions slows the rate of growth of QCD plasma unstable modes.
As shown in Figure \[p03\], the results that we obtain for $\omega > k$ show that the damping of color excitations in this frequency range is completely different according to whether $\tau_\pi$ is zero or not. As expected, if $\tau_\pi=0$ there is no damping in this frequency range. This feature stems from the analytic structure of the longitudinal dielectric function in the regime where $\omega, k \ll T$, as given by Eq. (\[elfull\]). In this limit, the imaginary part of $\epsilon_L$ is proportional to the step function $\theta(k^2-\omega^2)$ (see e.g. [@Litim]). On the contrary, if $\tau_\pi \neq 0$ then those color excitations with $\omega \gtrsim k$ become considerably damped, with a damping rate which falls off steeply with increasing frequency. Similar results were obtained in Ref. [@carrdiel] for QED dispersion relations obtained from kinetic theory with a BGK collision term. There it was found that when collisions are included, the longitudinal dispersion intersects the light cone $\omega=k$, in contrast to the case of collisionless dispersion where $\omega > k$ for all $k$. In a collisionless plasma, Landau damping, which is possible only for $\omega < k$, is the only damping mechanism, and thus plasma waves are undamped. In contrast, collisions introduce an additional damping mechanism for plasma waves (see e.g. [@sch]). We emphasize that a first order hydrodynamic formalism, in which the shear tensor relaxes instantaneously to its Navier-Stokes value, can not completely account for such damping of plasma waves.
We now briefly discuss the influence of the value of $\eta/s$ on the longitudinal dielectric function. Figure \[p015\] shows the real and imaginary parts of $\epsilon_L$ for a smaller value of the viscosity-to-entropy ratio than before, namely $\eta/s = 0.15$. It is seen that the impact of a nonvanishing relaxation time on $\epsilon_L$ decreases with decreasing values of $\eta/s$. For the value $\eta/s = 0.15$, the effect of $\tau_\pi$ on $\epsilon_L$ is still significant, and we still see that excitations with $\omega \gtrsim k$ are damped due to collisions. Although not shown, we find that for $\eta/s \lesssim 0.08$ the difference between the longitudinal dielectric function obtained with a vanishing or a nonvanishing value of $\tau_\pi$ is hardly appreciable.
The effective formalism discussed in this work is phenomenological and involves several approximations. In spite of this, it is interesting to qualitatively discuss possible implications of our results for the phenomenon of jet quenching in heavy ion collisions. For this, and considering the already discussed limitations, we take the view that our formalism constitutes an appropriate model to understand some features of the response of the QGP to a fast moving parton that crosses it.
There are two main energy loss mechanisms which contribute to energy loss: radiation of soft gluons and collisions involving the exchange of [*hard*]{} ($\sim T$ or larger) and [*soft*]{} ($\sim 2\omega_{\rm{pl}}$) momenta [@loss1; @loss2]. The dominant source of energy loss is gluon bremsstrahlung, although the contribution of collisions to the total energy loss is significant, particularly when attempting to fit the results of theoretical models of jet quenching to data [@loss1; @loss2; @losscomp; @losscomp2]. In connection to our results, we note that the contribution of soft collisions to the energy loss can be directly calculated from $\epsilon_L$ and $\epsilon_T$ (see for example [@ichi; @bluhm; @mullerloss]). We shall not discuss in detail this issue here, but just mention that a smaller imaginary part of $\epsilon_L$ will result in a decrease in the energy loss. Our results then show that a nonvanishing value of the relaxation time $\tau_\pi$ is expected to lead to a sizeable reduction in the energy loss.
As a final remark, we note that the radiation spectrum and hence the energy loss of a hard parton crossing the QGP is also modified by the dielectric polarization of the medium (this is known as the Ter-Mikaelian effect [@TMorig] - see [@TM] for an extension to QCD). As noted recently [@bluhm], the effect of radiation damping occurring in an absorptive medium on the spectrum of radiated gluons is particularly interesting and might lead to sizeable effects on energy loss related phenomena. The polarization tensor derived from the formalism presented here naturally incorporates damping. A detailed study of the influence of $\tau_\pi$ on the energy loss of fast partons for the conditions prevailing in heavy ion collisions at RHIC and LHC is left for future work. We emphasize, however, that the richness of the effective formalism presented here lies in its nonlinear character, which is not reflected in the polarization tensor but may become relevant when dealing with parton energy loss phenomena .
Summary and outlook {#summ}
===================
In this work we have obtained from the kinetic theory of nonabelian plasmas an effective model describing the evolution of a system composed of colored particles interacting with nonabelian classical gauge fields. The link between the one-particle distribution function of colored particles in the kinetic description and the variables of the effective theory is determined by the entropy production variational method. The closure provided by this method does not rely in any way on a gradient expansion in macroscopic variables and can therefore be applied even when these gradients are large.
In order to compare the developed effective theory with chromohydrodynamic formalisms based on the usual gradient expansion, we have calculated the longitudinal dielectric function $\epsilon_L$ of the plasma. Using typical values of the plasma parameters appropriate for the QGP, together with an effective speed of sound chosen to reproduce the longitudinal dielectric function of hard-thermal loop kinetic theory, we have found that the relaxation time for the shear tensor has a strong influence on the dynamics of color fluctuations, in agreement with the results of kinetic theory including collisions among the hard partons. The implications of such changes on the evolution of color excitations on phenomena relevant to heavy ion collisions, particularly on jet quenching, deserve further investigation.
The formalism presented here is a simplified model of the dynamics of color fields during the early and intermediate stages of heavy ion collisions. It may be useful to shed light on issues which would require intensive simulations in a microscopic approach, for example the magnitude of the back reaction of the particles’ flow on the gauge fields. If the color fields eventually die out, the effective theory goes over to second order fluid dynamics if the velocity gradients are small, so that the effective theory could be used to describe (starting from suitable initial conditions) the evolution of the fireball created in a heavy ion collision from very early times ($\gtrsim 0.2$ fm/c) till freeze-out in a unified, albeit simplified, way.
Concerning the dynamics of color fields at early and intermediate times, our formalism could be used to study plasma instabilities and its effect on the evolution of matter created in heavy ion collisions. It would be particularly interesting to solve numerically the full [*nonlinear*]{} equations of the effective theory presented here for the conditions prevailing in heavy ion collisions, and to compare the results to those obtained by a microscopic approach. To carry out this program, the inclusion of hard gluons into the model is certainly required for a realistic description of the physical processes involved at those stages. Work is in progress along these lines.
At this point we would like to comment on the limitations of our approach in connection to possible applications to describe the early-time dynamics of color fields in heavy ion collisions (some of the issues discussed here are also relevant for parton energy loss). We think that it is clearer to distinguish between limitations inherent to our approach (that are either truly unsurmountable or else very difficult to address) and simplifying hypothesis that could be relaxed in the future.
We start by discussing those limitations that are inherent to our approach.
The effective theory presented here constitutes a simplified model of the true dynamics given by kinetic theory with a linear collision term, and therefore the equations derived in this paper should be applicable for similar time scales. The kinetic theory description of the early stage of heavy ion collisions is valid for times $\gtrsim Q_s^{-1}$ ($\sim 0.2$ fm/c at RHIC) when particles having transverse momenta greater than $Q_s$ are formed out of the color fields [@reviewIANCU; @GLASMA]. However, there are some limiting factors that should be considered.
The first one was already hinted to and refers to the use of a linear collision term. Although one could, in principle, write down the variational equations of the EPP for a transport equation with nonlinear collision kernels, there is not much prospect of being able to solve them. However, this may not be a serious issue, because several studies have shown that the Boltzmann-Vlasov equation coupled to Yang-Mills equation provides a fairly reliable description of the early stages of a heavy ion collision [@mro93; @mro89; @mro94; @mro97; @strick-Weibel; @arn-Weibel; @mannakin; @fluct; @strick; @rom1; @rom2; @muller-pos; @flor; @ipp]. The short-range interaction between hard partons has been incorporated in the kinetic models only recently [@sch; @sch-WYM; @dum; @nayak; @Gale].
The second point is that the range of applicability of the effective theory developed here is not determined by the magnitude of velocity gradients, but rather by whether the dynamics of the system as given by (linearized) kinetic theory can or can not be described by few variables (including nonhydrodynamic ones) coupled to classical gauge fields. As it happens with other approaches to the closure problem [@deGroot; @denicol; @ichi; @krall; @GKbook; @GKpaper; @anile; @tanos; @tanos2; @tanos3; @muscato; @christen; @christen2; @ferz; @nagy; @geroch; @liboff], it is difficult to precisely establish [*a priori*]{} the range of validity of the resulting effective theory. We believe that, as it happens with fluid dynamics, our formalism may prove useful to describe some stages of a heavy ion collision provided the transport coefficients are suitably chosen. In any case, this point should be settled by comparing the results obtained from the effective theory to those obtained from kinetic theory.
As an issue that can be improved in the future, we mention first the fact that here we deal with excitations of scalar fields (and not spinors), and second that we do not take into account the hard gluons. In the early stage of heavy ion collisions, the hard gluons are, as the hard quarks, coupled to the soft gluons and therefore should also be described by a kinetic equation with its corresponding collision term. This term would involve not only interactions among hard gluons themselves but also between hard gluons and the excitations of the scalar fields. The latter coupling should also be reflected in $\mathbf{I}_{col}$ of Eq. (\[9\]). The inclusion of hard gluons interacting with the classical gauge fields is mandatory for the model to be applicable to the early stage of a heavy ion collision, in which strong fields decay into gluons to eventually form the QGP.
Another point that can be addressed in the future is the possibility of performing a quadratic expansion around an anisotropic (in momentum space) distribution function instead of the isotropic distribution function we use here. This idea was exploited in [@mauricio; @aniso] to obtain an effective theory that can handle the very large anisotropies in momentum space that are present at early times in heavy ion collisions, and moreover reproduces both the ideal fluid and the free-streaming limits. The correct description of both regimes (which are ultimately determined by the value of the Knundsen number) may be a relevant issue when dealing with plasma instabilities. Although our formalism can handle very large anisotropies as well, at present it is not clear how well can it describe plasma instabilities, so further studies are certainly needed to settle this point.
Finally, we plan to include stochastic terms into the evolution equations of the effective theory [@cfluc; @nosjhep]. This would allow us to address in a simple model setup the important question on the fate of fluctuations and their impact on observables at RHIC and LHC.
We thank Gastão Krein and Mauricio Martinez for useful comments and discussions. This work has been supported in part by ANPCyT, CONICET and UBA under project UBACYT X032 (Argentina).
[99]{}
S. S. Adler, et al. (PHENIX Collaboration), Phys. Rev. Lett. [**91**]{}, 182301 (2003). J. Adams, et al. (STAR Collaboration), Phys. Rev. Lett. [**92**]{}, 112301 (2004). K. Adcox, et al. (PHENIX Collaboration), Phys. Rev. C [**69**]{}, 024904 (2004). J. Adams, et al. (STAR Collaboration), Phys. Rev. C [**72**]{}, 014904 (2005). I. Arsene, et al. (BRAHMS Collaboration), Phys. Rev. C [**72**]{}, 014908 (2005). B. B. Back, et al. (PHOBOS Collaboration), Phys. Rev. C [**72**]{}, 051901(R) (2005). U. W. Heinz, arXiv:0901.4355 \[nucl-th\]. P. Romatschke, Int. J. Mod. Phys. E [**19**]{}, 1 (2010). E. Iancu, arXiv:1205.0579 \[hep-ph\]. K. Fukushima and F. Gelis, Nucl. Phys. A [874]{}, 108 (2012). M. Strickland, J. Phys. G [**34**]{}, S429 (2007). P. Arnold, J. Lenaghan, and G. D. Moore, Journ. High Energy Phys. [**308**]{}, (2003) 002. A. Majumder and M. Van Leeuwen, Prog. Part. Nucl. Phys. A [**66**]{}, 41 (2011). U. A. Wiedemann, arXiv:0908.2306v1 \[hep-ph\].
R. S. Bhalerao and G. C. Nayak, Phys. Rev. C [**61**]{}, 054907 (2000). U. Heinz, C. R. Hu, S. Leupold, S. G. Matinyan, and B. Müller, Phys. Rev. D [**55**]{}, 2464 (1997). C. R. Hu and B. Müller, Phys. Lett. B [**409**]{}, 377 (1997). J. Berges, S. Scheffler, S. Schlichting, and D. Sexty, Phys. Rev. D [**85**]{}, 034507 (2012). K. Dusling, T. Epelbaum, F. Gelis, R. Venugopalan, arXiv:1206.3336 \[hep-ph\]. W. Florkowski and R. Ryblewski, Acta Phys. Polon. B [**40**]{}, 2843 (2009). C. Gale, S. Jeon, B. Schenke, P. Tribedy, and R. Venugopalan, arXiv:1209.6330 \[nucl-th\]. H. Fujii and K. Itakura, Nucl. Phys. A [**809**]{}, 88 (2008). W. Pöschl and B. Müller, Nucl. Phys. A [**661**]{}, 641 (1999). Y. Mehtar-Tani, Phys. Rev. C [**75**]{}, 034908 (2007). A. Selikhov and M. Gyulassy, Phys. Lett. B [**316**]{}, 373 (1993). B.-F. Jiang and J.-R. Li, Nucl. Phys. A [**856**]{}, 121 (2011). C. Manuel and S. Mrówczyński, Phys. Rev. D [**74**]{}, 105003 (2006). M. Mannarelli and C. Manuel, Phys. Rev. D [**76**]{}, 094007 (2007). B.-F. Jiang and J.-R. Li, Nucl. Phys. A [**847**]{}, 268 (2010). D. D. Holm and B. A. Kupershmidt, Phys. Lett. A [**105**]{}, 225 (1984). C. Manuel and S. Mrówczyński, Phys. Rev. D [**68**]{}, 094010 (2003). C. Manuel and S. Mrówczyński, Phys. Rev. D [**70**]{}, 094019 (2004). M. Mannarelli and C. Manuel, Phys. Rev. D [**77**]{}, 054018 (2008). S. Mrówczyński, Phys. Rev. D [**77**]{}, 105022 (2008). M. Attems, A. Rebhan, and M. Strickland, arXiv:1207.5795v1 \[hep-ph\]. W. Florkowski, R. Ryblewski, and M. Strickland, arXiv:1207.0344v1 \[hep-ph\]. A. Dumitru, Y. Nara, B. Schenke, and M. Strickland, Phys. Rev. C [**78**]{}, 024909 (2008). A. Kurkela and G. D. Moore, Journ. High Energy Phys. [**11**]{} (2011) 120. S. Mrówczyński and Berndt Müller , Phys. Rev. D [**81**]{}, 065021 (2010). A. Majumder, B. Müller, and S. Mrówczyński, Phys. Rev. D [**80**]{}, 125020 (2009). S. A. Bass, C. Gale, A. Majumder, C. Nonaka, G.-Y. Qin, T. Renk, and J. Ruppert, Phys. Rev. C [**79**]{}, 024901 (2009). G.-Y. Qin, J. Ruppert, C. Gale, S. Jeon, G. D. Moore, and M. G. Mustafa, Phys. Rev. Lett. [**100**]{}, 072301 (2008). P. Romatschke and A. Rebhan, Phys. Rev. Lett. [**97**]{}, 252301 (2006). P. Romatschke and R. Venugopalan, Phys. Rev. D [**74**]{}, 045011 (2006). P. Romatschke and M. Strickland, Phys. Rev. D [**68**]{}, 036004 (2003). S. Mrówczyński, Phys. Lett. B [**314**]{}, 118 (1993). S. Mrówczyński, Phys. Rev. C [**49**]{}, 2191 (1994). S. Mrówczyński, Phys. Lett. B [**393**]{}, 26 (1997). M. E. Carrington, T. Fugleberg, D. Pickering, and M. H. Thoma, Can. J. Phys. [**82**]{}, 671 (2004). S. Mrówczyński, Phys. Rev. D [**39**]{}, 1940 (1989). B. Schenke, M. Strickland, C. Greiner, and M. H. Thoma, Phys. Rev. D [**73**]{}, 125004 (2006). B. Schenke, Nucl. Phys. A [**830**]{}, 689 (2009). A. Ipp, A. Rebhan, and M. Strickland, Phys. Rev. D [**84**]{}, 056003 (2011). A. Rebhan and D. Steineder, Phys. Rev. D [**81**]{}, 085044 (2010).
U. Heinz, Ann. Phys. [**161**]{}, 48 (1985); ibid. [**168**]{}, 148 (1986). U. Heinz, Phys. Rev. Lett. [**51**]{}, 351 (1983). S. K. Wong, Nuovo Cim. A [**65**]{}, 689 (1970). E. Calzetta and B.-L. Hu, [*Nonequilibrium Quantum Field Theory*]{} (University Press, Cambridge, 2008). H. Elze and U. Heinz, Phys. Rep. [**183**]{}, 81 (1989). J.-P. Blaizot and E. Iancu, Nucl. Phys. B [**557**]{}, 183 (1999). J. Winter, Journal de Physique [**45**]{}, 53 (1984). D. F. Litim and C. Manuel, Phys. Rep. [**364**]{}, 451 (2002). J.-P. Blaizot and E. Iancu, Phys. Rep. [**359**]{}, 355 (2002). E. Braaten and R. Pisarski, Nucl. Phys. B [**337**]{}, 569 (1990). J. Frenkel and J. Taylor, Nucl. Phys. B [**334**]{}, 199 (1990). D. Bödeker, Phys. Lett. B [**426**]{}, 351 (1998). D. Bödeker, Nucl. Phys. B [**559**]{}, 502 (1999). P. Arnold, D. T. Son, and L. G. Yaffe, Phys. Rev. D [**59**]{}, 105020 (1999). P. Arnold, D. T. Son, and L. G. Yaffe, Phys. Rev. D [**60**]{}, 025007 (1999).
M. Trovato and P. Falsaperla, Phys. Rev. B [**57**]{}, 4456 (1998). A. M. Anile and M. Trovato, Phys. Lett. A [**230**]{}, 387 (1997). A. M. Anile and O. Muscato, Phys. Rev. B [**51**]{}, 16728 (1995). M. Trovato and L. Reggiani, J. Appl. Phys. [**85**]{}, 4050 (1999). O. Muscato, J. Appl. Phys. [**96**]{}, 1219 (2004). A. N. Gorban and I. V. Karlin, Physica A [**360**]{}, 325 (2006). A. N. Gorban and I. V. Karlin, [*Invariant Manifolds for Physical and Chemical Kinetics*]{}, Lecture Notes in Physics (Springer, Berlin, 2005). M. Martinez and M. Strickland, Nucl. Phys. A [**848**]{}, 183 (2010). W. Florkowski, Phys. Lett. B [**668**]{}, 32 (2008). G. Nagy and O. Reula, J. Phys. A [**30**]{}, 1695 (1997). R. Geroch and L. Lindblom, Phys. Rev. D [**41**]{}, 1855 (1990). J. Peralta-Ramos and E. Calzetta, Phys. Rev. D [**80**]{}, 126002 (2009). J. Peralta-Ramos and E. Calzetta, Phys. Rev. C [**82**]{}, 054905 (2010). E. Calzetta and J. Peralta-Ramos, Phys. Rev. D [**82**]{}, 106003 (2010). G. S. Denicol, J. Noronha, H. Niemi, D. H. Rischke, Phys. Rev. D [**83**]{}, 074019 (2011). T. Christen, Eur. Phys. Lett. [**89**]{}, 57007 (2010). T. Christen and F. Kassubek, J. Quant. Spectrosc. Radiat. Transf. [**110**]{}, 452, (2009).
L. M. Martyushev, and V. D. Seleznev, Phys. Rep. [**426**]{}, 1 (2006). S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, [*Relativistic Kinetic Theory*]{} (North-Holland, Amsterdam, 1980). J. H. Ferziger and H. G. Kaper, [*Mathematical Theory of Transport Processes in Gases*]{} (North-Holland, London, 1972). R. Liboff, [*Kinetic Theory: Classical, Quantum and Relativistic Descriptions*]{} (Springer, 2003, New York). C. Cergignani and G. Kremer, [*The Relativistic Boltzmann Equation: Theory and Applications*]{} (Birkhäuser, 2002, Basel). W. Israel, Ann. Phys. (NY) [**100**]{}, 310 (1976); W. Israel and J. M. Stewart, Phys. Lett. A [**58**]{}, 213 (1976); W. Israel and J. M. Stewart, Ann. Phys. (NY) [**118**]{}, 341 (1979).
J. Peralta-Ramos and E. Calzetta, [*A new closure from relativistic kinetic theory*]{}, in progress. D. Jou, J. Casas-Vazquez and G. Lebon, [*Extended Irreversible Thermodynamics*]{} (Springer, Berlin, 2001). D. Kondepudi and I. Prigogine, [*Modern Thermodynamics: From Heat Engines to Dissipative Structures*]{} (Wiley, Sussex, 1998). R. Kikuchi, Phys. Rev. [**124**]{}, 1682 (1961); E. I. Blount, Phys. Rev. [**131**]{}, 2354 (1963).
R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov, J. High Energy Phys. [**04**]{} (2008) 100. S. Bhattacharyya, V. E. Hubeny, S. Minwalla, and M. Rangamani, J. High Energy Phys. [**02**]{} (2008) 045.
C. E. Aguiar, E. S. Fraga, and T. Kodama, J. Phys. G [**32**]{}, 179 (2006). L. P. Csernai, D. D. Strottman, and Cs. Anderlik, Phys. Rev. C [**85**]{}, 054901 (2012).
N. A. Krall and A. W. Trivelpiece, [*Principles of Plasma Physics*]{} (McGraw-Hill, New York, 1973). S. Ichimaru, [*Statistical Plasma Physics, Volume I: Basic Principles*]{} (Addison-Wesley, New York, 1992).
R. B. Neufeld and T. Renk, Phys. Rev. C [**82**]{}, 044903 (2010). R. B. Neufeld and I. Vitev, arXiv:1105.2067v1 \[hep-ph\].
M. Bluhm, P. B. Gossiaux, and J. Aichelin, Phys. Rev. Lett. [**107**]{}, 265004 (2011).
M. L. Ter-Mikaelian, [*High-Energy Electromagnetic Processes in Condensed Media*]{} (Wiley, New York, 1972). M. Djordjevic and M. Gyulassy, Phys. Rev. C [**68**]{}, 034914 (2003).
E. Calzetta, Class. Quant. Grav. [**15**]{}, 653 (1998). J. Peralta-Ramos and E. Calzetta, J. High Energy Phys. [**02**]{} (2012) 085.
|
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bibliography:
- 'refs.bib'
title: ICDAR 2019 Historical Document Reading Challenge on Large Structured Chinese Family Records
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abstract: 'Traffic forecasting has emerged as a core component of intelligent transportation systems. However, timely accurate traffic forecasting, especially long-term forecasting, still remains an open challenge due to the highly nonlinear and dynamic spatial-temporal dependencies of traffic flows. In this paper, we propose a novel paradigm of Spatial-Temporal Transformer Networks (STTNs) that leverages dynamical directed spatial dependencies and long-range temporal dependencies to improve the accuracy of long-term traffic forecasting. Specifically, we present a new variant of graph neural networks, named spatial transformer, by dynamically modeling directed spatial dependencies with self-attention mechanism to capture real-time traffic conditions as well as the directionality of traffic flows. Furthermore, different spatial dependency patterns can be jointly modeled with multi-heads attention mechanism to consider diverse relationships related to different factors (e.g. similarity, connectivity and covariance). On the other hand, the temporal transformer is utilized to model long-range bidirectional temporal dependencies across multiple time steps. Finally, they are composed as a block to jointly model the spatial-temporal dependencies for accurate traffic prediction. Compared to existing works, the proposed model enables fast and scalable training over a long range spatial-temporal dependencies. Experiment results demonstrate that the proposed model achieves competitive results compared with the state-of-the-arts, especially forecasting long-term traffic flows on real-world PeMS-Bay and PeMSD7(M) datasets.'
author:
- 'Mingxing Xu, Wenrui Dai, Chunmiao Liu, Xing Gao, Weiyao Lin, Guo-Jun Qi, Hongkai Xiong'
bibliography:
- 'STTN.bib'
title: 'Spatial-Temporal Transformer Networks for Traffic Flow Forecasting'
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traffic flow predictions, spatial-temporal dependencies, dynamcial graph neural networks, transformer.
Introduction
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![(a) Traffic forecasting models with joint spatial temporal dependencies. (b) Evolution of the traffic conditions spatial distribution. []{data-label="fig0"}](model.pdf "fig:"){width="1\columnwidth" height="0.25\columnwidth"}
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![(a) Traffic forecasting models with joint spatial temporal dependencies. (b) Evolution of the traffic conditions spatial distribution. []{data-label="fig0"}](Dynamical_spatial.pdf "fig:"){width="1\columnwidth" height="0.6\columnwidth"}
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the deployment of affordable traffic sensor technologies over the last few years, the exploding traffic data have been bringing us to the era of big data of transportation. Intelligent Transportation System (ITS) [@mori2015review] is thus developed to leverage transportation big data for efficient urban traffic controlling and planning. As a core component of ITS, accurate traffic forecasting in a timely fashion has attracted increasing attentions.
In traffic forecasting, the future traffic conditions (e.g. speeds, volumes and density) of a node are predicted from its historical traffic data as well as its neighbors. Thus, it is important for a forecasting model to effectively and efficiently capture the spatial and temporal dependencies. Generally, traffic forecasting can be classified into two scales: short-term ($\leq 30$ min) and long-term ($\geq 30$ min). Previous approaches such as time-series models [@liu2011discovering] and Kilman filtering models [@lippi2013short] are mostly focus on short-term forecasting and perform quiet well. However, these models are typically based on the stationary assumption which is commonly impractical in long-term forecasting as the traffic flows are naturally highly dynamical. Furthermore, they fail to jointly capture the spatio-temporal correlations in the traffic flows. So that these model fail to effectively forecast long-term traffic flows.
Naturally, traffic networks can be represented as graphs in which the nodes represent traffic sensors and the edges together with its weights are determined by the connectivity as well as Euclidean distances among sensors. Thus traffic flows can be viewed as graph signals evolving with time. As graph neural networks [@atwood2016diffusion] [@defferrard2016convolutional] [@kipf2017semi] have show its power in processing graph-represented data, recent works tend to combine graph neural networks with sequence-learning model to jointly capture spatio-temporal correlations and improve the performance in both short-term and long-term forecasting. [@yu2018spatio] and [@li2018dcrnn_traffic] are the first two graph-based traffic forecasting models. They largely improve the model performance by introducing the inherent graph topology of a traffic network into sequence-learning models. They integrate spatial-based [@atwood2016diffusion] or spectral-based [@defferrard2016convolutional] Graph Convolutional Networks (GCNs) with convolution-based [@gehring2017convolutional] or Recurrent-neural-networks (RNNs)-based sequence learning models to jointly capture the spatial and temporal dependencies.
However, they have some significant limitations that could be further improved especially for long-term forecasting.
From the perspective of spatial dependencies, the spatial dependencies are dynamical and the dynamics result from two aspects. On one hand, the spatial dependencies between two sensors are influenced by their connectivity and distance. The former one indicates that whether there may be dependencies between them, while the latter one together with the real-time speed determines when the dependencies will be evoked, thus the spatial dependencies should be evolving with time. In fact, this kind of dynamic is a characteristic of traffic forecasting task. As shown in Fig.1 (a), for simplicity, we assume the traffic speeds keep the same across different time steps and all connected sensors are interacted with each other. Taking the central purple node for example, the spatial dependencies between the central node with its neighbor nodes are evolving with time, and the influence of remote nodes will provoked after several time steps while it is short for the nearest nodes to influence it. On the other hand, in real-world traffic networks, the spatial dependencies are inherently dynamical which are influenced by many factors such as traffic accidents, weather conditions as well as rush hours. As shown in Fig.1(b), the spatial patterns of traffic flows change significantly as time goes by. Thus we argue that an effective traffic forecasting model should be able to model dynamical spatial dependencies. Furthermore, the influence of upstream and downstream traffic flows is also quite different, thus the spatial dependencies should be modeled as directed.
To address this limitation, we propose a novel paradigm of self-attention based graph neural network, named [*spatial transformer*]{} to dynamically learn directed spatial dependencies by considering the real-time traffic conditions, connectivity and distance among sensors as well as traffic flow directions. We first enhance the node features with spatial and temporal positional embedding to incorporate the connectivity, distance as well as time information into each node, then several latent high-dimensional subspace are learned, in which spatial dependencies are dynamically computed to reflect the time-varying directed dependencies among nodes. Furthermore, through the self-attention mechanism, we can capture both local and global dependencies beyond adjacent nodes, thereby reflecting the hidden long-range patterns evolving over time, which enable our model to better capture sharp traffic flow changes. To further incorporate fixed spatial dependencies among sensors into our model, we employ fixed graph convolutions in each spatial transformer and balance it with the learned dynamical spatial dependencies through a gate mechanism.
Long-range temporal dependencies also play an indispensable role traffic forecasting which are usually ignored by previous models. As illustrated in Fig.1(a), in different time steps, there are different spatial dependencies that will be evoked, thus limited range of temporal dependencies will result in significant information loss, leading to poor performance. Furthermore, most existing methods forecast traffic flows in an auto-regressive fashion. Thus, the error-prone predictions are incorporated into the inputs along with previous observations to make further predictions, resulting in an quick error accumulation especially for long-term forecasting. As demonstrated in Fig.2 (a), short-range dependencies may not severely influence the performance of short-term forecasting, as all the observations used for predictions are error-free. However, the long-term forecasting will be significantly degraded, as predictions can only made based on error-prone previous predictions through short-range dependencies, thus resulting in the quick propagation of prediction errors. In this paper, we proposed to address this problem with an effective modeling of long-range temporal dependencies as well as predicting multi-step results at the same time, as shown in Fig.2(b). One one hand, more information is utilized to make predictions. On the other hand, past temporal context instead of error-prone predictions can be directly leveraged to make multi-step predictions, bypassing the auto-regressive manner, the error of earlier predictions will not be propagated to further predictions.
Inspired by the newly proposed transformer [@NIPS2017_7181] for efficiently and efficiently modeling long-range dependencies. We develop a temporal transformer to capture dynamical long-range temporal dependencies in traffic flows to directly forecast multi-steps traffic conditions, bypassing the auto-regressive process. Specifically, on one hand, in each temporal transformer layer, each step in a sequence can attend to the context of all other steps to customize its long-range dependencies in a time-varying fashion. Thus, we can model long-range temporal dependencies in each layer. In contrast to the RNNs-based or convolution-based models, it also enables more parallelization of long-range dependencies, facilitating a more efficient model training. One the other hand, we can directly scaled to long-sequence without increasing model depth and complexity.
In this paper, we seek to address several challenges facing the traffic forecasting problem, and propose a novel paradigm of Spatial Temporal Transformer Networks (STTNs). Our contributions are summarized below.
- We propose a novel paradigm of Spatial Temporal Transformer Networks (STTNs) that can dynamically model long-range spatial-temporal dependencies.
- A new variant of graph neural network named [*spatial transformer*]{} is developed to model the time-varying directed spatial dependencies by dynamically attending to hidden spatial patterns of traffic flows.
- A temporal transformer that enables an efficient parallelization of long-range temporal dependencies is also developed.
This paper is structured as follows. In section II, we briefly review the existing spatial and temporal dependencies modeling approaches. In section III, we formally formulate the traffic forecasting problem as a spatial-temporal graph prediction problem. Then, in section IV, we present the proposed Spatial temporal networks for traffic forecasting, and elaborate the components. In Section V, we conduct extensive experiments on real-world traffic datasets and make comparison with state-of-the-arts. Finally, we conclude our paper and present our further work in section VI.
Related Work
============
In traffic flows forecasting, how to effectively and efficiently model spatial and temporal dependencies is the core problem. In this section, we briefly review the existing approaches for spatial and temporal dependencies modeling in traffic flow forecasting.
Spatial dependencies modeling
-----------------------------
The earliest traffic flows forecasting models are statistic-based or neural-network-based models. For statistic-based model, such as autoregressive integrated moving average (ARIMA)[@min2011real] and Bayesian networks [@wang2014new], spatial dependencies are modeled from a probabilistic view.Although they help to analyze the uncertainty within traffic flows, their linear natures impedes them to effectively model the highly-nonlinearity within traffic flows. Neural networks based models are more capable to capture the nonlinearity of traffic flows, however, their fully connected structures are expensive in both computational and memory. Furthermore, the lack of assumptions make it impossible to capture the complicated spatial patterns in traffic flows.
With the development of convolutional neural networks (CNNs), they have shown powerful feature extraction abilities in many applications[@gehring2017convolutional] [@Krizhevsky:2012:ICD:2999134.2999257] [@long2015fully], thus attracting much interests to apply them into traffic forecasting area. [@ma2017learning] [@zhang2017deep] adopt CNNs to extract the spatial features in which traffic networks are converted to regular grids. However, traffic networks are inherently irregular, so the conversion will loss the inherent topology information within traffic networks. Graph neural networks (GNNs) [@scarselli2008graph] [@gilmer2017neural] are latter proposed to generalize the deep learning to non-Euclidean domain. As a variant of GNNs, Graph convolution networks (GCNs) [@atwood2016diffusion] [@defferrard2016convolutional] [@kipf2017semi] generalize classical convolutions to the graph domain, attracting increasing interests from both researchers and practitioners. Recently, GCNs are applied to model the spatial dependencies of traffic flows to explore the inherent traffic topology. For example, STGCN [@yu2018spatio] models spatial dependencies with spectral graph convolutions defined on an undirected graph, while [@li2018dcrnn_traffic] employs diffusion graph convolutions on a directed graph to incorporate the influence of traffic flow directions. However, they both have drawback that their spatial dependencies are fixed once trained, ignoring the dynamic changes of traffic conditions (e.g. rush hours and traffic accidents). Most recently, [@guo2019attention] also proposed to generate dynamical spatial dependencies but their spatial dependencies are not evolving time but the depth of spatial and temporal blocks. [@pan2019urban] models dynamical spatial dependencies by adopting a extra meta learner to summarize the geo-graph features and then the embedded geo-graph features together with GATs [@velivckovic2017graph] are used to generate dynamical spatial dependencies, however, their spatial dependencies are still limited to $k$ nearest neighbors of predefined graph topology that cannot discover hidden patterns of dependencies at various scales beyond local nodes. [@wu2019graph] capture hidden spatial patterns beyond predefined graph topology through a learnable embedding for each nodes in graph, which largely improve the forecasting accuracy but their spatial dependencies are still fixed once trained. Unlike these approaches, the proposed model provide a effective and efficient mechanism to model dynamical directed spatial dependencies. We could discover richer hidden dependencies beyond predefined graph structures and local nodes by dynamically computing spatial dependencies in different latent high-dimensional subspace.
Temporal dependencies modeling
------------------------------
As stated in [@ma2015long] [@wu2016short], previous models for temporal dependencies modeling especially RNNs suffer from two main limitations. On one hand, RNNs are limited in capture the long-range dependencies due to gradients exploding or vanishing in model training, resulting in their poor performance when the input sequences are long. On the other hand, the temporal dependencies are also highly dynamical so that it is difficult to determine the optimal sequence length for accurate traffic forecasting. To alleviate these drawbacks, Gated Recurrent Units (GRUs) [@chung2014empirical] [@pan2019urban] or Long-Short Term Memory (LSTM) [@Hochreiter:1997:LSM:1246443.1246450] are thus developed to model long-range dependencies and applied to traffic forecasting [@guo2019attention] [@ma2015long] [@wu2016short]. Unfortunately, they still suffer from their inherent sequential nature, making the training process time-consuming and limiting their scalability to model long sequences. Alternatively, [@yu2018spatio] [@guo2019attention] adopted Convolution-based sequence learning models [@gehring2017convolutional], however, the limited size of receptive fields requires multiple hidden layers to cover a sufficiently large context. [@wu2019graph] adopted WaveNet with dilation convolution to increase the receptive field to cover the whole sequence with smaller layers. However, the number of layers still increase linearly with input sequence length, thus limiting the model scalability to explore long input sequence. Furthermore, deep layers increases the length of the path between two components, and degrades its efficiency in capturing long-range dependencies [@hochreiter2001gradient] [@NIPS2017_7181]. The model needs to be redesigned if the input sequence length are changed, thus making it expensive to search for optimal input sequence length. In contrast, transformers [@NIPS2017_7181] are a newly proposed efficient sequence learning model that relies on a self-attention mechanism that is highly parallelizable. It enables to effectively and efficiently capture long-range dependencies over time with single layers and can be easily adapted for different input sequence length.
The Formulation
===============
A traffic network can naturally be represented as a graph ${\mathcal G}=({\mathcal V, \mathcal E }, A)$, where ${\mathcal V}$ is the set of nodes representing the sensors with $|{\mathcal V}|=N$, ${\mathcal E}$ is the set of edges reflecting the physical connectivity between sensors, and $A\in {\mathbb R^{N\times N}}$ is the adjacent matrix that is constructed based on Euclidean distances between sensors. Traffic forecasting is a classic spatial-temporal prediction problem. Formally, given the past $T'$ observed traffic conditions $[v^{\tau-T'+1},\cdots, v^\tau]$ from $N$ sensors and a traffic network $\mathcal G$, traffic forecasting seeks to predict $T$ future traffic conditions $[{\hat v}^{\tau+1},\cdots,{\hat v}^{\tau+T}]$ as shown in Fig. 2. In this paper, we focus on forecasting traffic speeds where $v^\tau\in {\mathbb R^N}$, and it can be easily adapted to forecast volume and density. The traffic forecasting problem can generally be formulated as $${\hat v}^{\tau+1},\cdots,{\hat v}^{\tau+T}={\mathcal F}(v^{\tau-T'+1},\cdots, v^\tau;{\mathcal G})$$ where ${\mathcal F}$ is the model we wish to learn. In this paper, we learn dynamical spatial dependencies for forecasting traffics as shown in Fig.1b(ii), where the spatial dependencies $S^\tau\in {\mathbb R^{N\times N}}$ change over time and are learned at each time $\tau$.
The Proposed Model
==================
In this section, we introduce the proposed spatial-temporal transformer network. Specifically, we first describe the overall architectures of the proposed model, and then elaborate its main components: [*spatial transformer*]{}, [*temporal transformer*]{} and prediction layer, respectively.
The Overall Architectures
-------------------------
The overall structures of the proposed spatial-temporal transformer network are demonstrated in Fig.3, which consists of stacked spatial-temporal blocks and a prediction layer which consists of two $1\times 1$ convolution layers. More specifically, a spatial-temporal block contains a spatial transformer and a temporal transformer to jointly learn spatial-temporal features in the context of dynamically changing dependencies. Several blocks can be stacked to form deep models for more complicated spatial-temporal features. Then, the prediction layer aggregates the learned deep spatial-temporal features for final predictions. Previous models usually make predictions in an auto-regressive fashion and output single-step prediction each time. There are usually two quite different schemes to train the model. In STGCN, only the single-step prediction error is adopted to train the model, while multi-step predictions are made in test. It ignores the dynamic of traffic flows, thus its performance is relatively limited especially in long-term predictions. In contrast, DCRNN adopted encoder-decoder scheme to incorporate multi-step prediction errors into the final loss and random sampling scheme is further adopted to alleviate the error accumulation problem in long-term predictions. As GraphWaveNet. we argue that multi-step results can be directly predicted with the powerful deep models without predicting in an auto-regressive manner. By make multi-step predictions directly, both short-term and long-term predictions can be predicted with the true observations without using the error-prone predictions. Thus error accumulation problem can be well addressed. Here, we output multi-step predictions with the the learned deep spatial-temporal features, and use the multi-step errors to train the model.
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Spatial Transformer
-------------------
In this subsection, we first elaborate the four components of the proposed spatial transformer: spatial position-temporal embedding layer, graph convolution layer, dynamical graph convolution layer as gate mechanism for information fusion. More specifically, the spatial-temporal embedding layer is learned to incorporate spatial-temporal position information (e.g. topology, connectivity, timesteps.) into each node. The graph convolution layer is adopted to explore the road topology information for some fixed spatial dependency patterns, and the dynamical graph convolution layer is used to capture dynamical directed spatial dependencies evolving with time. In final, the learned fixed spatial and dynamical features are fused with gate mechanism. We further present that the proposed spatial transformer can be viewed as a general message passing graph neural network for dynamical graph construction and feature learning.
### Spatial-temporal Position Embedding
As the transformer contains only fully feed-forward structures, and there is no convolutional or recurrent operation, thus the spatial position information of nodes as well as the temporal information of each observations are inherently lost. As a matter of fact, they both play an important role in modeling spatial dependencies, as illustrated in Fig 1(a), the distance and time determine whether a spatial dependency between two node should be utilized for future predictions. In transformer [@NIPS2017_7181], prior positional embedding is adopted to inject ’position’ information into the input sequences. Here, we adopt learnable spatial and temporal positional embedding layer to learn spatial temporal embedding into each node feature. Specifically, a dictionary ${\mathcal D}^{\mathcal S}\in {\mathcal R^{N\times N}}$ and ${\mathcal D}^{\mathcal T}\in {\mathcal R^{N\times T}}$ is learned as spatial position embedding matrix and temporal position embedding matrix respectively. We initialize the dictionaries with the graph adjacent matrix and one-hot time encode,respectively, as graph adjacent matrix contains the connectivity as well as distance information which is important for spatial dependencies modeling while one hot time encode can well inject the time step information into each node. The dictionaries are then updated during training. The input features are concatenated with the dictionaries as $X^{i}=[X^{\mathcal S}, {\mathcal D}^{\mathcal S}, {\mathcal D}^{\mathcal T}]$ before feeding into following graph convolution layers, where $X^{\mathcal S}$ is the input features to spatial-temporal blocks.
### Fixed Graph Convolution Layer
Graph convolution is a generalization of classical convolution to graph domain. It learns node features by aggregating its neighbors information according to the learned weights and predefined graph, thus it is effective to learn the structure-aware node features. In this paper, we adopt the chebyshev-based graph convolution to capture the fixed spatial dependencies from the prior road topology. Formally, we denote $A \in {\mathbb R^{N\times N}}$ as the adjacent matrix that are calculated via gaussian kernel according to the distances among sensors. Let $ X\in {\mathbb R}^{N\times d}$ denote the input node features that contain real-time traffic conditions of $N$ sensors, and $T_k$ is the order-$k$ Chebyshev polynomials. Also, $D$ is the degree matrix with $D_{ii}=\sum_i{A_{ij}}$, $L=I_n-D^{-1/2}AD^{-1/2}$ is the normalized Laplacian matrix, and ${\mathcal {\widetilde L}}=2L/\lambda_{max} -I_n$ is the scaled Laplacian matrix for chebyshev polynomials, where $\lambda_{max}$ is the largest eigenvalues of $L$.
Then, the structure-aware node features $X^{{\mathcal G}}\in {\mathbb R}^{N\times d^{{\mathcal G}}}$ can be obtained with $K$ order chebyshev-polynomial approximation graph convolution as $$X^{{\mathcal G}}_{:,j}=\sum\limits_{i=1}^{d}\sum\limits_{k=0}^K \theta_{ij,k}T_k({\mathcal {\widetilde L}})X_{:,i} \quad 0 \leq j \leq d_{{\mathcal G}}$$ where $X^{\mathcal G}_{:,i}$ is the $i$-th channel of node features and $\theta_{ij,k}$ is the learned parameters.
Note that we naturally use the physical connectivity and distances among sensors to construct graph, so that we can explicitly explore the road topology information through graph convolution.
### Dynamical Graph Convolution Layer
Previous GCNs-based models such as [@yu2018spatio] [@li2018dcrnn_traffic] can only model fixed spatial dependencies. To capture the hidden spatial dependencies evolving time, here, we propose a novel dynamical graph convolution network to dynamically calculate spatial dependencies in learned latent high-dimensional subspaces. Specifically, we learn linear mappings to project input features of each node to latent high-dimensional subspaces and then the spatial dependencies are dynamically computed by a self-attention mechanism between the projected features. Through such mechanism, we can efficiently model the dynamical spatial dependencies according to the changing input graph signals. Notably, the learned dependencies are directed and we can also learn multiple linear mappings to model spatial dependencies in various representation subspaces as to reveal more hidden spatial dependencies influenced by different relationships factors.
Here, we adopt the self-attention to dynamically model the spatial dependencies among sensors. Self-attention mechanism is widely used in computer vision and natural language processing tasks. It proves powerful to model the relationships among individual items. In GAT, self-attention mechanism is also adopted to calculate the weights among nodes, however, their graph topology are predefined, thus they only model the edges weights dynamically. In traffic networks, there are some hidden spatial dependencies that are not fully reflected by the road topology, thus the graph should also be dynamical constructed as well as the edge weights. The self-attention mechanism we adopted in this paper is shown in Fig.4.
Formally, the input features (e.g traffic condition together with spatial-temporal position information) are first projected into latent high dimensional subspaces with learnable mappings which are realized with feed-forward neural networks. Basically, for single-head attention model which model one relationship pattern, three subspaces are obtained for each node, namely, query subspace $Q^{\mathcal S}\in {\mathbb R}^{N\times d^{\mathcal S}_q}$, key subspace $K^{\mathcal S}\in {\mathbb R}^{N\times d^{\mathcal S}_k}$ and value subspace $V^{\mathcal S}\in {\mathbb R}^{N\times d^{\mathcal S}_v}$. And the latent subspace learning process can be formulated as $$\begin{array}{lcr}
Q^{\mathcal S}=X^{{\mathcal S}}W^{\mathcal S}_q\\
K^{\mathcal S}=X^{{\mathcal S}}W^{\mathcal S}_k\\
V^{\mathcal S}=X^{{\mathcal S}}W^{\mathcal S}_v
\end{array}$$ where $W^{\mathcal S}_q, W^{\mathcal S}_k, W^{\mathcal S}_v$ are the weight matrices for $Q^{\mathcal S}, K^{\mathcal S}, V^{\mathcal S}$, respectively.
After obtaining the three latent high dimensional subspaces, dynamical spatial dependencies $S^{\mathcal S}\in {\mathbb R^{N\times N}}$ are further calculated by dot-product as $$S^{\mathcal S}=\text{softmax}(\frac{Q^{\mathcal S}(K^{\mathcal S})^T}{\sqrt{d^{\mathcal S}_k}})$$ where $S^{\mathcal S}$ is the learned dynamical dependencies matrix among nodes. Then, new node features $M^s\in{\mathbb R^{N\times d^{\mathcal S}_v}}$ are further updated with $$M^s=S^{\mathcal S}V_{\mathcal S}$$ Note that we adopt dot product to calculate dependencies between nodes as it is much fast and space-efficient in practice. The softmax is then applied to normalize the dependencies. Scaling by $\sqrt{d^{\mathcal S}_k}$ is to avoid the softmax from reaching the saturation when gradients are extremely small. Multiple dependencies can be learned with multi-heads attention mechanism by learning multiple subspace pairs, thereby revealing different hidden spatial dependencies from various latent subspaces.
To further improve the prediction ability of learned node features, a shared three-layer feed-forward neural network with nonlinear activation is applied on each node to explore the interactions among features channels and update the node features as $Y^s\in {\mathbb R}^{N \times d^s_o}$. $$U^{\mathcal S}=\text{ReLu}(\text{ReLu}(M^{\mathcal S}W^{\mathcal S}_0)W^{\mathcal S}_1)W^{\mathcal S}_2$$ where $W^{\mathcal S}_0, W^{\mathcal S}_1, W^{\mathcal S}_2$ are the weight matrices, and $U^{\mathcal S}=X^{\mathcal S}+M^{\mathcal S}$ is the residual connection for stable training.
To model more complicated spatial dependencies, we can stack several dynamical graph convolution layers for deep model to improve the model capacity.
### Gate mechanism for features fusion
There are two kind of spatial features, one is obtained with fixed spatial dependencies while the other is calculated with dynamical dependencies. To fuse the two kind of features, a gate mechanism is adopted. We first learn the gate $g$ as $$g=\text{Sigmoid}(f_s(U^{\mathcal S})+f_g(X^{\mathcal G})+b)$$ where $f_s$ and $f_g$ are fully connected layer and $b$ is the bias term. The final output of spatial transformer is $$Y^{\mathcal S}=gU^{\mathcal S}+(1-g)X^{\mathcal G}$$
### General Dynamical Graph Neural Networks
As we have mentioned above, previous graph convolutional networks include spectral-based and spatial-based models, all rely on predefined graph structure, and can not adapt to input graph signal. Here, we demonstrate the entire spatial transformer can be formulated as a general message passing dynamical graph neural network. Formally, we denote $x_v$ as the input features of node $v$. The whole spatial feature learning process can be rewritten as $$m_v=\sum_{w\in \mathcal V }F(x_v, x_w)$$ $$y_v=G(m_v ,x_v)$$ where $\mathcal V$ is the set of nodes in the traffic network, $F$ is the message-passing function that computes the spatial dependencies and passes message $m$ among nodes (Eq.4$-$Eq5), $G$ is the shared position-wise feed forward network to update the node features as $y$ (Eq.6).
Then, the whole spatial transformer can be summarized as $$Y^{\mathcal S}={\mathcal S}(X, A)$$
In summary, the proposed spatial transformer has the following properties. 1) Dynamical hidden spatial dependencies are calculated in the learned high-dimensional latent subspaces, and prior graph structure information can be incorporated via residual structures. 2)The learned hidden spatial dependencies are not restricted to local nodes but can be globally, and multiple spatial dependencies can be explored in different latent relationship spaces. 3)It is fully feed-forward, thus computations can be performed in parallel in a fast and scalable fashion, in other words, we can easily scale our model to large-scale traffic forecasting with proper distribution computation scheme.
Temporal Transformers
---------------------
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We also develop a temporal transformer to efficiently and effectively capture long-range temporal dependencies [@NIPS2017_7181] over time. Compared with RNNs and the variants, it can not only capture long-range dependencies but also allow parallel calculations so that it can be easily scaled to long sequences. The structures are illustrated in the bottom right of Fig.3.
### Self-attention Layer for Long-range Modeling
Similar to temporal position adopted in spatial transformer, we first concatenate the learned spatial features in each time step with temporal embedding which is also initialized with one-hot encode and updated during training as ${X^{\mathcal T}=[Y^{\mathcal S}, D^{\mathcal T'}]}$.
And then self-attention mechanism is also adopted to model temporal dependencies. The input to the temporal transformer is a temporal sequence $X^{\mathcal T}\in {\mathbb R}^{M\times d^{\mathcal T}}$ with a slide window of length $M$ and $d^{\mathcal T}$ channels. Similar to spatial transformer, temporal dependencies are dynamically computed in high-dimensional subspaces. The three subspaces, the queries $Q^ {\mathcal T}\in {\mathbb R}^{M\times d^{\mathcal T}_q}$, the keys $K^{\mathcal T}\in {\mathbb R}^{M\times d^{\mathcal T}_k}$, and the values $V^{\mathcal T}\in {\mathbb R}^{M\times d^{\mathcal T}_v}$, can be computed as $$\begin{array}{lcr}
Q^ {\mathcal T}=X^{{\mathcal T}}W^{\mathcal T}_q\\
K^{\mathcal T}=X^{\mathcal T}W^{\mathcal T}_k\\
V^{\mathcal T}=X^{\mathcal T}W^{\mathcal T}_v
\end{array}$$ where $W_q\in {\mathbb R}^{d^{\mathcal T}\times d^{\mathcal T}_q}$,$W_k\in {\mathbb R}^{d^{\mathcal T}\times d^{\mathcal T}_k}$ and $W_v\in {\mathbb R}^{d^{\mathcal T}\times d^{\mathcal T}_v}$ are the learned liner mappings.
Unlike RNNs-based models which temporal dependencies are limited to previous time steps, we learn bidirectional temporal dependencies with scaled dot-product function as $$S^{\mathcal T}=\text{softmax}(\frac{Q^{\mathcal T}(K^{\mathcal T})^T}{\sqrt{d^{\mathcal T}_k}})$$ Then, the temporal features are obtained by aggregating the values $V^{\mathcal T}$ weighted by the bidirectional temporal dependencies. $$M^{\mathcal T}=S^{\mathcal T}V^{\mathcal T}$$ The temporal dependencies modeling process is illustrated in Fig.4. To explore the interactions among latent features, they are further processed with a shared three-layer feed-forward neural network $$Y^{\mathcal T}=\text{ReLu}(\text{ReLu}(U^{\mathcal T}W^{\mathcal T}_0)W^{\mathcal T}_1)W^{\mathcal T}_2$$ where $U^{\mathcal T}=M^{\mathcal T}+X^{\mathcal T}$ is the residual connection for stable training.
As Eq. 11, the temporal transformer can be formulated as $$Y^{\mathcal T}={\mathcal T}(X^{\mathcal T})$$
It is worth noting that the in each temporal transformer layer, each time step can attend to all other time steps, thus long-range bidirectional temporal can be efficiently captured in each temporal layer. Furthermore, temporal transformer can be easily scaled to long sequence by increasing the slide window length $M$ without much sacrifice in computation efficiency. While RNNs-based models can not deal with long sequences due to the gradients vanishing/exploding. As for convolution-based model, for different sequences length, more layers are need and the filters windows and number of layers require explicitly designed to ensure that the whole sequence are covered to capture long-range dependencies.
Model Structures
----------------
### Spatial-temporal Blocks
The future traffic flow conditions of one location are determined by the traffic conditions of its neighbor locations, and the time when the influence will happen as well as some sudden changes (e.g traffic accidents, weather condition s). Thus to make accurate predictions, the spatial and temporal dependencies of the traffic networks should be jointly modeled, so we integrate the proposed spatial and temporal transformer as a spatial-temporal transformer block, and residual connections are adopted among them as well. The structure of spatial-temporal block is illustrated in Fig. 3.
The input to the $l$-th spatial-temporal block is a 3D tensor $X^l\in {\mathbb R}^{M\times N\times d^l}$ with $M$ time steps of observations from $N$ nodes. Let ${\mathcal T}$ and $\mathcal S$ be the spatial and temporal transformers as aforementioned. Then the output $X^{l+1}\in {\mathbb R}^{M\times N\times d^{l+1}}$ can be formulated as $$X^{l+1}=X^l+{\mathcal S}(X^l,A)+{\mathcal T}(X^l+{\mathcal S}(X^l,A))$$ For simplicity, we denote $\mathcal S$ as spatial transformer and $\mathcal T$ as temporal transformer. Multiple spatial-temporal blocks can be stacked to improve the model capacity according tasks at hand.
### Prediction Layer
In the prediction layer, final time steps spatial-temporal features from the last spatial-temporal block are feed as input. And two stacked classical convolutional layers are then adopted to make multisteps predictions.
Formally, the input to the prediction layer is a 2D tensor $X^{\mathcal {ST}}\in {\mathbb R}^{N\times d^{{\mathcal {ST}}}}$, which is the last time step spatial-temporal features extracted by stacked spatial-temporal blocks. The final multi-steps prediction is $$Y=\text{Conv}(\text{Conv}(X^{\mathcal {ST}}))$$ where $Y\in {\mathbb R}^{N\times T}$. Mean absolute loss are then adopted to train the model which can be formulated as $$L=\|Y-Y^{gt}\|_1$$
Experiments
===========
In this section, we evaluate the proposed model by conducting extensive experiments on two real-world datasets, PeMSD7(M) and PEMS-BAY. In particular, we will demonstrate the state-of-the-art performances especially in making long-term predictions. To validate effectiveness of the proposed spatial and temporal transformers as well as verify some arguments we have proposed above, extensive ablation studies will be further conducted. Specifically, we will first show that direct prediction can alleviate the error propagation problem encounted in auto-regressive prediction, and then we will validate the effectiveness of dynamical spatial dependencies as well as the proposed spatial transformer. Next, we demonstrate that long-range temporal dependencies are important for accurate predictions, thus should be well utilized and the proposed temporal is efficient and effective in model long-range temporal dependencies. Finally, we analyze the influence of different model configurations as well the model complexity.
Dataset and Data preprocessing
------------------------------
**PEMS-BAY** [@li2018dcrnn_traffic] contains $6$ months traffic information collected from $325$ sensors in the Bay Area of California, starting from Jan $1$st 2017 through May $31$th 2017. **PeMSD7(M)** [@yu2018spatio] collects traffic information from $228$ monitoring stations in the California state highway system during the weekdays from May through June of 2012. Traffic speeds are aggregated every five minutes, and normalized with $Z$-Score as inputs.
The road topology information is represented by a graph adjacent matrix. The graph of PeMS-BAY dataset is prior designed as a directed graph to differentiate the influence of different directions, thus forward and backward diffusion graph convolutions can be adopted to model the directed spatial dependencies [@li2018dcrnn_traffic]. However, it is difficult to metric the influence of direction with manually, resulting in difficulty in constructing the directed graph. In this paper, we use self-attention mechanism to model the directed spatial dependencies in a data-driven manner, bypassing the heavy burden on differentiating the influence of directionality. In our model, a simple undirected graph adjacent matrix which can reveal the distance and connectivity among sensors are required. For PeMS-bay dataset, the undirected graph is constructed by adopting the maximum weight on the two directed edges between each pair of nodes as undirected edge, and use the undirected graph to represent the road topology information. For the PeMSD7 dataset, the adjacent matrix is already symmetric based on the road distances between sensors, thus can be directed adopted.
Experiment Settings
-------------------
All experiments are conducted on a NVIDIA 1080 Ti GPU. The proposed model is trained with the mean absolute error loss using the RMSprop optimizer for $50$ epochs with a batch size of $50$. The initial leaning rate is set to $10^{-3}$, and it decays at a rate of $0.7$ every five epochs. We independently perform the same experiments on each dataset five times, and report the average results in Table 1. To demonstrate the performance of the proposed model, we compare with the results reported in [@li2018dcrnn_traffic][@yu2018spatio], where the current $12$ observations ($60$ minutes) are used to predict the traffic conditions in the next $15, 30, 45$ minutes on PeMSD7(M) and $15,30,60$ minutes on PEMS-BAY, respectively. For GraphWaveNet, we trained the public code released by the author on PeMSD7(M) dataset and the best performance is reported.
\[table1\]
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Evaluation Metrics and Baselines
--------------------------------
Three widely used metrics are adopted for evaluation – Mean Absolute Errors (MAE), Mean Absolute Percentage Errors (MAPE), and Root Mean Squared Errors (RMSE).
We mainly compared with STGCN, DCRNN and GraphWaveNet. STGCN [@yu2018spatio] and DRCNN [@li2018dcrnn_traffic] are the two most representative deep learning methods for traffic forecasting, while GraphWaveNet is the the latest model which has achieved state-of-the-art performance. In addition, the other methods we will compare with include 1) Historical Average (HA); 2) Linear Support Vector Regression (LSVR); 3) Auto-Regressive Integrated Moving Average (ARIMA); 4) Feed-forward Neural Network (FNN); 5) Fully-Connected LSTM (FC-LSTM) are reported in STGCN [@yu2018spatio] and DRCNN [@li2018dcrnn_traffic].
The proposed model is denoted as STTN in the following sections. For PeMSD7(M) dataset, only one spatial-termporal block are adopted. Two hidden layers and single attention are adopted for spatial and temproal transformer and the feature channels are all set as 64. For PeMS-BAY dataset which are much larger than PeMSD7(M) both termporally and spatially, three spatial-temporal transformer blocks are stacked to model deep spatial and temporal dependencies. For each spatial and temporal transformer in each block, one hidden layer and single attention are adopted, and the feature channels are still set as 64. In our model, residual structures are adopted for stable learning and fast convergence.
Experiments Results
-------------------
The results are shown in Table I. We can observe that 1) For PeMSD7(M) dataset, the proposed model achieves the state-of-the-art performance and we outperform STGCN and DCRNN by a large margin. The gaps are increased with prediction time gets longer. In comparsion with GraphWaveNet, we both take advantages of long-range temporal dependencies and GraphWaveNet models hidden fixed spatial dependencies while we are able to model dynamical spatial dependencies. We both consistently outperform other models by a large margin, demonstrating the effectiveness of long-range temporal dependencies and hidden spatial dependencies. It can also be observed that we perform much better than GraphWaveNet for long-term predictions while similar in short-term predictions. It is quiet reasonable as the spatial dependencies as well as temporal dependencies can be viewed as stationary in short term prediction, GraphWaveNet adopts convolutional kernels with small receptive fields perform quiet well in capturing short-range dependencies.
2\) For PEMS-BaY dataset, we achieve comparable performance with state-of-the-art (GraphWaveNet) while performing much better than STGCN and DCRNN. For GraphWaveNet and DCRNN, bidirectional diffusion graph convolution are performed on explicitly designed directed matrix to consider the influence of directionality, so that they outperform STGCN by a large margin. However the influence of directionality is complicated and hard to measure. Instead, we use self-attention mechanism to capture the dynamical directed spatial dependencies in a data-driven manner and we achieve similar results with simple undirected adjacent matrix, demonstrating the effectiveness of the dynamical spatial dependencies modeling. Furthermore, compared with STGCN which also adopted three pairs of spatial and temporal units, we consistently outperform it with different prediction range. It can also validate the effectiveness of our model. To better demonstrate the superiority of our model in capturing changing spatial-temporal depedencies, we further illustrate the one-day prediction results (averaged along saptial dimension) on test dataset with different prediction time for STTN, GraphWaveNet, STGCN and DCRNN. The results are shown in Fig.5. As we can observed that compared with STGCN and DCRNN, STTN and GraphWaveNet perform much better in changing area (e.g. $[60,84]$). For STGCN and DCRNN, there are considerately large time-shift in the prediction results curve especially, and the shifts grow with the prediction steps, contributing to large prediction error in sharp changing area. Compare STTN with GraphWaveNet, we are able to capture continously sharp changes.(e.g. \[84,192\]) while GraphWaveNet cannot. We believe it is due to our superiority in capturing dynamical spatial dependencies as well as long-range temporal dependencies.
Ablation Studies
----------------
In this section, we conduct extensive experiments for through analysis of the proposed model and verify the arguments we have proposed above. Since PeMSD7(M) dataset is much more challenging than PEMS-BAY while smaller both temporally as well as spatially, e.g the standard deviation of collected speeds is much larger than that of PeMS-bay, so we take PeMS-D7 dataset for example and all the experiments below are performed on PeMSD7(M) dataset. To better demonstrate the influence of different factors, in this section, for our model, we use only one spatial-temporal block with feature dimension 64 for each spatial and temporal transformer.
### Direct multi-step prediction is better than auto-regressive prediction
\[table2\]
Most previous models make predictions in a auto-regressive fashion, in which the predicted results are adopted to make next-step prediction until all predictions are obtained. Since predictions are error-prone, thus it is inevitable to propagate errors for next predictions, resulting in poor performance in long-term predictions. To alleviate this problem, [@li2018dcrnn_traffic] DCRNN proposed a sampling scheme. We argue that it is better to directly make long-term predictions with true observations instead of the error-prone prediction results. To verify this argument, we perform experiments on both STGCN and our one-block model, and both of them make predictions directly as well as auto-regressively. The results are shown in Table II. Form Table II, we can easily observe that, by direct make predictions from previous true observations, the performance especially in long-term prediction is much significantly improved compared with that of auto-regressive predictions. Furthermore, the errors increase much slower, which firmly validates our argument.
### Effectiveness of Spatial Transformer in modeling dynamic spatial dependencies
\[table3\]
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![(a) Illustration of traffic predictions for short term prediction (5 min) with fixed and dynamical spatial dependencies. (b) Illustration of traffic predictions for long term prediction (60 min) with fixed and dynamical spatial dependencies.[]{data-label="fig5"}](spatial_short_v2.pdf "fig:"){width="1\columnwidth" height="0.6\columnwidth"}
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![(a) Illustration of traffic predictions for short term prediction (5 min) with fixed and dynamical spatial dependencies. (b) Illustration of traffic predictions for long term prediction (60 min) with fixed and dynamical spatial dependencies.[]{data-label="fig5"}](spatial_long_v2.pdf "fig:"){width="1\columnwidth" height="0.6\columnwidth"}
(b)
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Dynamic spatial dependencies play an important role in effective traffic flow prediction. In this experiment, we validate that dynamical spatial dependencies is necessary for accurate long-term predictions as well as the effectiveness of the proposed spatial transformer in modeling dynamical spatial dependencies. To better demonstrate the influence of dynamical spatial dependencies, here, we adopt a model which has only one graph convolution layer and one convolution-base sequence modeling module (GLU layer) as our baseline, and then we replace the graph convolution layer with the proposed spatial transformer as a comparison denoted as STTN-S(a,h) in table III, where $a $ and $h$ denote the number of attention heads and hidden layers, respectively. Note that the only difference between the two models is that the baseline can only model fixed spatial dependencies that can not evolving with time while we model dynamical spatial dependencies adapt to real-time traffic-conditions. For spatial transformer, we change the number of attention heads as well as the number of hidden layers (e.g $(a,h)$) to make a through analysis of the proposed spatial transformer.
The results are shown in Table III. From Table III, we can observed that STTN-S (1,1) outperforms baseline by a large margin in each time steps, and the performance gap becomes larger as the prediction time gets longer. This demonstrates the effectiveness of dynamic spatial dependencies in long-term predictions. We further illustrate averaged one-day prediction results on test dataset with short and long-term prediction (5 min and 60 min) for Baseline and STTN-S (1,1). The results are shown in fig 6. We can observe that, for short-term predictions, the performance of dynamical and fixed spatial dependencies are similar. As the spatial dependencies can be viewed as stationary in short-term prediction, and the assumptions are usually made for traditional short term prediction models. However, for long term predictions, the spatial dependencies changes considerately. The proposed model are able to model dynamical spatial dependencies evolving with time, resulting in better performance in sharly changing area (e.g.\[48,84\] in Fig.6 (b)). Thus it validates our argument that the dynamical spatial dependencies are important for long-term predictions.
Compared with other attention-based dynamical spatial models such as GraphWaveNet [@pan2019urban] which limited their dynamical spatial dependencies in $k$-nearest neighbor nodes, we can capture long-range spatial dependencies beyond local nodes, thus can better to capture sudden changes, resultsing our excellent performance in sharp changing area. To validate this claim, we masked the dynamical dependencies matrix to restrict the learned spatial dependencies to local nodes. The model is denoted as STTN-S(local), shown in table III. As we can see that the performance are largely degraded with this constraint. We further compare the learned spatial dependencies in Figure 7. We can see that we both learn dynamical spatial dependencies evolving with time, but we are not limited to local nodes. On the other hand, from (b) and (c), we can observe the spatial dependencies in the most recent time steps are more than the earliest time steps. It is reasonable as the distances among sensors are quite small.
[cc]{}\
(a)\
![We take first 50 sensors for example. (a) Illustration of adjacent matrix and only local nodes are kept. (b)(c) Illustrations of dynamical spatial dependencies for STTN-S(1,1) on time step 1 and 12. (d)(e) Illustrations of dynamcial spatial dependencies for STTN-S(local) on time step 1 and 12[]{data-label="fig7"}](spatial_50_1.pdf "fig:"){width="0.55\columnwidth" height="0.45\columnwidth"}&![We take first 50 sensors for example. (a) Illustration of adjacent matrix and only local nodes are kept. (b)(c) Illustrations of dynamical spatial dependencies for STTN-S(1,1) on time step 1 and 12. (d)(e) Illustrations of dynamcial spatial dependencies for STTN-S(local) on time step 1 and 12[]{data-label="fig7"}](spatial_50_12.pdf "fig:"){width="0.55\columnwidth" height="0.45\columnwidth"}\
(b)&(c)\
![We take first 50 sensors for example. (a) Illustration of adjacent matrix and only local nodes are kept. (b)(c) Illustrations of dynamical spatial dependencies for STTN-S(1,1) on time step 1 and 12. (d)(e) Illustrations of dynamcial spatial dependencies for STTN-S(local) on time step 1 and 12[]{data-label="fig7"}](masked_spatial_50_1.pdf "fig:"){width="0.55\columnwidth" height="0.45\columnwidth"}&![We take first 50 sensors for example. (a) Illustration of adjacent matrix and only local nodes are kept. (b)(c) Illustrations of dynamical spatial dependencies for STTN-S(1,1) on time step 1 and 12. (d)(e) Illustrations of dynamcial spatial dependencies for STTN-S(local) on time step 1 and 12[]{data-label="fig7"}](masked_spatial_50_12.pdf "fig:"){width="0.55\columnwidth" height="0.45\columnwidth"}\
(d)&(e)\
We further change the attention heads and hidden layers in spatial transformer, the results are also shown in Table III. By increasing attention heads, the performance is continuously improved in all time steps. This is because multi-heads attention can model spatial dependencies in different relationship space, thus more hidden dependenceies pattern can be further utilized. The increase of hidden layers does small benefit to the performance. As PeMSD7(M) dataset are relatively small, thus one hidden layer is able to model the spatial dependencies.
### Effectiveness of Temporal Transformer in modeling long-range dependencies.
In this subsection, we first validate the effectiveness of long-range temporal dependencies in accurate traffic flow prediction and then demonstrate the effectiveness of the proposed temporal transformer in capture long-range dependencies, thus contributing to better prediction performance.
To demonstrate the influence of long-range temporal dependencies, here, we adopt the model with one graph convolution layer and one GLU layer as our baseline (the same as last subsection). By adopting convolution-based sequence model (GLU layer) to model temporal dependencies, we can easily control the receptive fields, in other words, the temporal dependencies range the model are utilized in the experiments. In practice, we change the convolution kernel size from $3$ to $6, 9$ and $12$. With the convolution kernel size increases, the temporal dependencies utilized are increasing and all the input sequence are covered when kernel size is $12$. The results are shown in upper part of Table IV, As we can seen that, with longer temporal dependencies are utilized, the long-term prediction performance is continuously increasing, which validates that long-range temporal dependencies benefits accurate prediction results for long term predictions.
\[table4\]
To validate the effectiveness of the proposed temporal transformer, we replace the GLU layer with our temporal transformer and we further change the number of hidden layers and attention heads to better understand temporal transformer. As shown in Table VI, STTN-T (1,1) performs similar with baseline in the short-term prediction, while the performance are much better in long term prediction, which demonstrates the effectiveness of the proposed temporal transformer in capturing long-range temporal dependencies. We further illustrate some temporal transformer attention matrix during test as in Fig 8.
As we can see that for different node, the temporal attention weight are different and in some node, earliest time-steps are utilized with long-range dependencies for predicitons.
![Illustration of temporal attention matrix of the first nine sensors for the first prediction on test dataset[]{data-label="fig6"}](Temporal_attention.pdf "fig:"){width="1\columnwidth" height="0.7\columnwidth"}\
Model Configuration analysis
----------------------------
STTN enables flexible model configurations and we can adjust it according to tasks at hand, such as the number of spatial-temporal blocks, the number of hidden layers, the attention dimensions, the number of attention heads as well as the hidden units dimensions. To better understand the STTN as well as the influence of different parameters settings, in this subsection, we present the results of STTN with different parameter settings.
First, we analyze the influence of different number of spatial-temporal blocks. The results are shown in the first three rows of Table V. We can observe that by stacking multiple spatial-temporal blocks to model deep spatial-temporal dependencies jointly, the performance is increasing. We can further see that two blocks are enough to capture the complicate dependencies in PeMSD7(M) dataset, so that this is no gain by increasing another block.
Then, we investigate the influence of feature channels. In STTN, the feature channel denote the dimension of subsection where dependencies are dynamically computed, thus the dimension of output spatial and temporal features. The results are shown in the fourth and fifth rows which demonstrate that higher dimension of attention subspace can result in better performance. It is reasonable as more information can be explored with higher dimension.
Next, we analyze the influence of a single spatial or temporal transformer capacity to the final prediction performance. Increasing the capacity of temporal transformer, in other words, adding the hidden layers, benefits a little to the performance, while it is contributes to large improvement to increase the capacity of spatial transformer, especially in long-term prediction, validating the importance of dynamical spatial dependencies in long-term prediction. And increasing the capacity of spatial and temporal transformer jointly will further improve the model.
As multi-attention can be adopted in STTN, and in real-world traffic networks, the relationships among nodes may fall into different relationship space. For example, the relationship among sensors can be formed as the similarity of their traffic flow pattern, or just the influence among them. Thus, multi-heads attention may be helpful to explore more hidden relationship patterns. As shown in Table V, by adopting multi-heads attention, the performance are improved, and the gain of more spatial relationship patterns does more benefits to the model, which is conform to our prior knowledge.
Finally, we explore the importance of positional embedding in STTN. We first remove the positional embedding in spatial and temporal transformer, respectively, and then remove all the embedding. The results are reported. We can concluded that
\[table7\]
Computational Complexity
------------------------
We further compare the computational costs among DCRNN, STGCN and STTN. All the experiments are conducted on the same GPU server, and we report the average training time of one epoch. The results are shown in Table 4. STGCN adopts fully convolutional structures so that it is fastest, and DRCNN uses the recurrent structures, which are very time consuming. Furthermore, as DCRNN adopts multiple step prediction errors to train the model, the training time will proportionally increase with the prediction time gets longer. In contrast, the proposed model has a competitive efficiency compared with DCRNN, and it is more scalable to making long-term predictions without increasing computation complexity.
\[table6\]
Conclusion
==========
In this paper, we propose a novel paradigm of spatial temporal transformer architecture to improve the long-term traffic forecasting. It can dynamically model various scales of spatial dependencies as well as capture long-range temporal dependencies. Experiment results on two real-world datasets demonstrate the superior performance of the proposed model in long-term forecasting. Furthermore, the proposed spatial transformer can be generalized into dynamical graph features learning in various applications, left for future research.
|
---
author:
- |
Radovan Dermíšek\
Department of Physics, Indiana University, Bloomington, IN 47405, USA
- |
John F. Gunion\
Department of Physics, University of California, Davis, CA 95616, USA\
and\
Theory Group, CERN, CH-1211, Geneva 23, Switzerland
title: 'Direct production of a light CP-odd Higgs boson at the Tevatron and LHC'
---
Introduction
============
Many motivations for the existence of a light CP-odd Higgs boson, $a$, have emerged in a variety of contexts in recent years. Of particular interest is the $\ma<2m_B$ region, for which a light Higgs, $h$, with SM-like $WW$, $ZZ$ and fermionic couplings can have mass below the nominal LEP limit of $\mh>114\gev$ by virtue of $h\to aa\to 4\tau$ decays being dominant [@Dermisek:2005ar; @Dermisek:2005gg; @Dermisek:2006wr; @Dermisek:2007yt] (see also [@Chang:2005ht; @Chang:2008cw]). For $\mh\lsim 105\gev$, the Higgs provides perfect agreement with the rather compelling precision electroweak constraints, and for $\br(h\to aa)\gsim 0.75$ also provides an explanation for the $\sim 2.3\sigma$ excess observed at LEP in $\epem \to Z b\anti b$ in the region $M_{b\anti b}\sim
100\gev$ if $\mh\sim 100\gev$. This is sometimes referred to as the “ideal” Higgs scenario. More generally, superstring modeling suggests the possibility of many light $a$’s, at least some of which couple to $\mupmum$, $\tauptaum$ and $b\anti b$. Further, it is not excluded that a light $a$ with $\ma>8\gev$ and enhanced $ab\anti b$ coupling could be responsible for the deviation of the measured muon anomalous magnetic moment $a_\mu$ from the SM prediction [@Gunion:2008dg]. Below, we will show that a light $a$ with the required $ab\anti b$ and $a\mupmum$ couplings would have been seen in existing Tevatron data for the $\mupmum$ final state at low $\mmumu$. More generally, current muon pair Tevatron data places significant limits on a light $a$. These will be further strengthened with increased Tevatron integrated luminosity and by $\mupmum$ data obtained at the LHC.
The possibilities for discovery of an $a$ and limits on the $a$ are phrased in terms of the $a\mu^-\mu^+$, $a\tau^-\tau^+$, $ab\anti b$ and $at\anti t$ couplings defined via \_[aff]{}i C\_[aff]{}[ig\_2m\_f2m\_W]{}f \_5 f a. \[cabbdef\] In this paper, we assume a Higgs model in which $\camumu=\catautau=\cabb$, as typified by a two-Higgs-doublet model (2HDM) of either type-I or type-II, or more generally if the lepton and down-type quark masses are generated by the same combination of Higgs fields. However, one should keep in mind that there are models in which $r=(\camumu=\catautau) /\cabb\gg 1$ — such models include those in which the muon and tau masses are generated by different Higgs fields than the $b$ mass. In a 2HDM of type-II and in the MSSM, $\camumu=\catautau=\cabb=\tanb$ (where $\tanb=h_u/h_d$ is the ratio of the vacuum expectation values for the doublets giving mass to up-type quarks vs. down-type quarks) and $\catt=\cotb$. These results are modified in the NMSSM (see, [@Ellis:1988er] and [@hhg]). [^1] In the NMSSM, both $\catt$ and $\cabb=\camumu=\catautau$ are multiplied by a factor $\cta$, where $\cta$ is defined by a=a\_[MSSM]{}+a\_S, where $a$ is the lightest of the 2 CP-odd scalars in the model (sometimes labeled as $\ai$). Above, $a_{MSSM}$ is the CP-odd (doublet) scalar in the MSSM sector of the NMSSM and $a_S$ is the additional CP-odd singlet scalar of the NMSSM. In terms of $\cta$, $\camumu=\catautau=\cabb=\cta\tanb$ and $\catt=\cta\cotb$. Quite small values of $\cta$ are natural when $\ma$ is small as a result of being close to the $U(1)_R$ limit of the model. In the most general Higgs model, $\camumu$, $\catautau$, $\cabb$ and $\catt$ will be more complicated functions of the vevs of the Higgs fields and the structure of the Yukawa couplings. In this paper, we assume $\camumu=\catautau=\cabb$ and $\cabb/\catt=\tan^2\beta$.
One should keep in mind, however, the fact that the above are tree-level couplings and that the $b\anti b a_{MSSM}$ coupling is especially sensitive to radiative corrections from SUSY particle loops that can be large when $\tanb$ is large [@Hall:1993gn; @Carena:1994bv; @Pierce:1996zz]. These are typically characterized by the quantity $\Delta_b$ which is crudely of order ${\mu\tanb\over 16\pi^2M_{SUSY}}$. The correction to the coupling then takes the form of $1/(1+\Delta_b)$. Since $\mu$ can have either sign, $\cabb$ can be either enhanced or suppressed relative to equality with $\catautau $ (the corrections to which are much smaller) and $\camumu$ (the corrections to which are negligible).
In the past, probes of a light $a$ have mainly relied on production of a primary particle ( an Upsilon) which then decays to a lighter $a$ with the emission of a known SM particle ( a photon). Such probes are strictly limited to a maximum accessible $\ma$ by simple kinematics. The only exceptions to this statement have been probes based on $\epem\to b\anti b a$ followed by $a\to\tauptaum$ or $a\to
b\anti b$, with LEP providing the strongest (but still rather weak) limits on the $a$ based on this type of radiative production process.
In contrast, hadron colliders potentially have a large reach in $\ma$ as a result of the fact that an $a$ can be produced via $gg\to a$. The $gga$ coupling derives from quark triangle loops. This process, plus higher order corrections thereto, leads to a large cross section for the $a$ due to the large $gg$ “luminosity” at small gluon momentum fractions, provided the $aq\anti q$ coupling deriving from the doublet component of the $a$ is significant. This large cross section will typically lead to a significant number of $gg\to a \to
\mupmum$ events even though $\br(a\to\mupmum)$ is not very large (and in fact is quite small for $\ma>2\mtau$). Further, since the $a$ is a very narrow resonance, all $a$ events will typically fall into a single bin of size given by the $\mmumu$ mass resolution of the experiment, typically below $100\mev$. This implies very controllable background levels, mostly deriving from heavy flavor production ($b\anti b$ and $c\anti c$), once isolation and promptness cuts have been imposed on the muons.
The Higgs doublet component of the $a$ can be suppressed when the $a$ mixes with one or more SM singlet CP-odd field, the $a_S$ of the NMSSM as made explicit above. However, what is important for limits is $\cabb$. In the NMSSM context, the scenarios allowing a light scalar Higgs to escape LEP limits by virtue of $h\to aa$ decays with $\ma<2m_B$ are such that $\cta$ is only small when $\tanb$ is large; for example, preferred scenarios for $\tanb=10$ are such that $\cta\sim 0.1$, implying $\cabb\sim 1$. As a result, the Tevatron and LHC provide very significant probes of such a light $a$ despite its being only 1% doublet (at the probability level). In addition, there are models beyond the MSSM [@Dermisek:2008id; @Dermisek:2008sd], including scenarios within the NMSSM [@Dermisek:2008uu], in which the doublet component of the $a$ can be quite substantial. (For a review, see also [@Dermisek:2009si].) In such models there are typically several Higgs scalars (including charged Higgs bosons) with masses near $100\gev$ that have escaped discovery because of decays involving the light $a$. This type of scenario requires that $\tanb$ be small ($1\lsim \tanb\lsim 3$). When the doublet component of the $a$ is substantial $\cabb=\cta\tanb$ will have magnitude $\geq 1$ and hadron colliders will almost certainly discover or exclude the $a$. In the case of the NMSSM, there is a portion of the preferred parameter region for low $\tanb$ in which precisely this kind of scenario arises. However, in the NMSSM at low $\tanb$ there is a second part of the preferred parameter region in which there are many light Higgs bosons but the $a$ is mainly singlet. In this latter case, $|\cabb|$ will be relatively small and direct searches for the light $a$ will be more difficult than in NMSSM models with larger $\tanb$.
The organization of the paper is as follows. In Sec. \[currentlimits\], we review some basic facts about a light $a$ and limits on $\cabb=\camumu=\catautau$ coming from non-hadron-collider data. In Sec. \[tevatron\], we discuss the additional limits that can be placed on the couplings of the $a$ implied by existing Tevatron analyses and data and extrapolate these existing results to $L=10\fbi$ data sets. In Sec. \[lhc\], we analyze prospects for discovering, or at least further improving limits on the couplings of, a light $a$ using early LHC data. Sec. \[conclusions\] summarizes our conclusions and provides a few additional comments.
Phenomenology and limits for a light CP-odd $a$ {#currentlimits}
===============================================
![$\br(a\to \mupmum)$ is plotted as a function of $\ma$ for a variety of $\tanb$ values. $\br(a\to\mupmum)$ is independent of $\cta$ at tree-level. []{data-label="bramumu"}](bramumu.ps){width="65.00000%"}
One key ingredient in understanding current limits and future prospects is the branching ratio for $a\to \mupmum$ decays. This branching ratio (which is independent of $\cta$ at tree-level due to the absence of tree-level $a\to VV$ couplings and similar) is plotted in Fig. \[bramumu\]. Note that $\br(a\to\mupmum)$ changes very little with increasing $\tanb$ at any given $\ma$ once $\tanb\gsim 2$.
![We plot results from [@Gunion:2008dg] for $\ctamax$ in the NMSSM (where $\cabb=\cta\tanb$) as a function of $\ma$ for $\ma>2\mtau$. The different curves correspond to $\tanb=1$ (upper curve), $3$, $10$, $32$ and $50$ (lowest curve). The region between $\sim 2\mtau$ and $\sim 8\gev$ is strongly constrained by CLEO III data [@:2008hs] on $\upsi\to \gam \tauptaum$ decays. The plotted limits do not include the BaBar $\upsiii\to \gam \tauptaum$ and $\upsiii\to \gam\mupmum$ limits that became available after the analysis of [@Gunion:2008dg].[]{data-label="ctamaxvsma"}](ctamaxvsma_rbt_magt2mtau.ps){width="65.00000%"}
Limits on $|\cabb|=|\cta|\tanb$ were analyzed in [@Gunion:2008dg] (see also [@Domingo:2008rr]), based on data available at the time. The analysis of [@Gunion:2008dg] employed limits from $\Upsilon\to \gam a$ decays, the importance of which was emphasized in [@Dermisek:2006py] (especially within the NMSSM context), as well as from $\epem\to b\anti b a$ production at LEP. The analysis of [@Gunion:2008dg] was done prior to the very recently released BaBar $\Upsilon(nS)\to \gam a$ results [@Aubert:2009ck; @Aubert:2009cp]. Without including the $\upsiii$ BaBar data, limits in the $8\gev<\ma<2m_B$ range (especially, $\mupsi<\ma<2m_B$) are quite weak and suffer from uncertainty regarding $\eta_b-a$ mixing. An update employing the $\upsiii$ data will be performed in a separate paper. In the present paper, the limits implicit in Tevatron data are compared to the limits obtained in [@Gunion:2008dg]. We will also briefly summarize how this comparison will change after inclusion of the $\upsiii$ BaBar results.
Focusing on the NMSSM, we note that it is always possible to choose $\cta$ so that the limits on $\cabb$ as a function of $\tanb$ are satisfied. The maximum allowed value of $|\cta|$, $\ctamax$, as a function of $\ma$ for various $\tanb$ values as obtained in [@Gunion:2008dg] is plotted in Fig. \[ctamaxvsma\]. Constraints are strongest for $\ma\lsim 7\gev$ for which Upsilon limits are strong, and deteriorate rapidly above that.
Turning to the 2HDM(II), where $\cabb=\tanb$, we note that any point for which $\ctamax$ is smaller than $1$ corresponds to an $\ma$ and $\tanb$ choice that is not consistent with the experimental limits. Disallowed regions emerge for $\ma\lsim 10\gev$ at higher $\tanb$. A disallowed region also arises over a limited $\ma$ range starting from $\ma>12\gev$ when $\tanb\gsim 18$, the larger the value of $\tanb$ the larger the interval. For example, for $\tanb=50$ the 2HDM(II) is not consistent for $\ma<10\gev$ nor for $12\lsim \ma\lsim 37\gev$. In contrast, for $\tanb=10$ the 2HDM(II) model is only inconsistent for $\ma\lsim 9\gev$.
Before proceeding, we note that constraints from precision electroweak data are easily satisfied for a light $a$ in both the 2HDM(II) and NMSSM cases (see [@Gunion:2008dg] for more discussion). We also wish to make note of the regions of interest for obtaining a new physics contribution, $\Delta a_\mu$, of order $\Delta a_\mu\sim
27.5\times 10^{-10}$ (the current discrepancy between observation and the SM prediction). These can be roughly described as follows. In the 2HDM(II) context, such $\delta a_\mu$ requires a rather precisely fixed value of $\tanb\sim 30-32$ and $\ma\sim 9.9-12\gev$. In the NMSSM context, the strong constraints from Upsilon physics imply that significant contributions to $a_\mu$ are not possible until $\ma$ exceeds roughly $9.2\gev$. The maximal $\delta a_\mu$ can exceed $\Delta a_\mu\equiv 27.5\times 10^{-10}$ for $9.9\gev\lsim \ma\lsim
12\gev$ if $\tanb\geq 32$, with an almost precise match to this value for $\tanb=32$. For $\tanb=50$, one can match $\Delta a_\mu$ by using a value of $\cta$ below $\ctamax$. (The fact that matching is possible for $9.9\gev\lsim \ma\lsim 2m_B$ is particularly interesting in the context of the ideal Higgs scenario.) Further, the maximal $\delta
a_\mu$ is in the $7-20\times 10^{-10}$ range for $12\gev < \ma\lsim
48\gev$ for $\tanb=32$ and for $12\gev < \ma\lsim 70\gev$ for $\tanb=50$.
At this point, it is worth discussing in more depth the “ideal” $\mh\sim 100\gev$, $\ma\lsim 2m_B$, $\br(h\to aa)>0.75$ Higgs scenario as discussed in [@Dermisek:2005ar; @Dermisek:2005gg; @Dermisek:2006wr; @Dermisek:2007yt]. These references examined the degree to which obtaining the observed value of $m_Z$ requires very precisely tuned values of the GUT scale parameters of the MSSM and NMSSM. One finds that in any supersymmetric model this finetuning is always minimized for GUT scale parameters that yield a SM-like $h$ with $\mh\leq 100-105\gev$, something that is only consistent with LEP data if the $h$ has unexpected decays that reduce the $h\to b\anti b$ branching ratio while not contributing to $h\to b\anti b b\anti b$ (also strongly constrained by LEP data). A Higgs sector with a light $a$ for which $\br(h\to aa)>0.75$ and with $\ma$ small enough that $a$ decays to $B\anti B$ final states are disallowed (i.e. $\ma < 10.56\gev$) provides a very natural possibility for allowing minimal finetuning. The NMSSM provides one possible example. As a useful benchmark, in the context of the NMSSM the $\tanb=10$ scenarios that yield the required $\ma<2m_B$ and $\br(h\to aa)>0.75$ are ones with $0.35\gsim |\cabb|$ ($|\cta|\gsim
0.035$). The lower limit arises from the fact that $\br(h\to aa)$ falls below the $0.75$ level needed for the ideal Higgs scenario if $|\cta|$ is too small. From Fig. \[ctamaxvsma\] we see that such $|\cta|$ values are not yet excluded for any $\ma>2\mtau$. This range becomes more restricted if, in addition, one requires small finetuning of the $\alam$ and $\akap$ soft-SUSY-breaking NMSSM parameters that determine the properties of the $a$ — such finetuning is characterized by a parameter we call $G$, defined in [@Dermisek:2006wr]. At $\tanb=10$, $0.6\lsim |\cabb|\lsim 1.2$ ($0.06\lsim |\cta|\lsim 0.12$) is required if $G<20$ is imposed as well as requiring $\ma<2m_B$ and $\br(h\to aa)>0.75$. For $\tanb\lsim
2$, the means for escaping the LEP constraints on the light scalar Higgses are a bit more complex since two $\sim 100\gev$ Higgses can share the $ZZ$-Higgs couplings squared, but there is always a lower limit on $|\cta|$ for which such escape is possible. In Table \[ctatable\], we tabulate more precisely the values of $\cta$ for various $\tanb$ values that: a) have $\ma<2m_B$ and large enough $\br(h\to aa)$ to escape LEP limits on the $h$, with no constraint on $G$; and b) have small $\akap,\alam$ finetuning measure $G<20$ as well as $\ma<2m_B$ and large enough $\br(h\to aa)$.
[|c|c|c|]{} $\tanb$ & $\cta$ ranges & $\cta$ ranges, $G<20$ required1.7 & $<-0.3$ or $>0.1$ & $[-0.6,-0.5]$ or $\sim +0.1$ 2 & $<-0.3$ or $>0.1$ & $[-0.7,-0.5]$ 3 & $<-0.06$ & $[-0.35,-0.08]$ 10 & $<-0.06$ or $>0.035$ & $[-0.12,-0.08]$ or $[0.06,0.08]$ 50 & $<-0.04$ or $>0.04$ & $[-0.06,-0.04]$ or $\sim +0.04$
We can summarize the implications of this table as follows. First, comparing to the existing limits on $|\cta|$ as plotted in Fig. \[ctamaxvsma\], we see that only ideal Higgs scenarios (ones with $\mh<105\gev$ and $\br(h\to aa)$ large enough to escape LEP limits) with $\tanb>30$ and $\ma\lsim 8\gev$ are excluded. Ideal Higgs scenarios with $\tanb<10$ are fairly far from being excluded. If we wish to eliminate ideal Higgs scenarios then: for $1.7\lsim\tanb\lsim
2$,[^2] we must exclude $|\cabb|\gsim 0.17$; for $\tanb=3$, we must exclude $|\cabb|\gsim
0.18$; for $\tanb=10$, we must exclude $|\cabb|\gsim 0.35$ and for $\tanb=50$, we must exclude $|\cabb|\gsim 2$. If we only wish to exclude such scenarios that also have $G<20$, then the required $|\cabb|$ levels for $\tanb=1.7,2,3,10,50$ are $0.17, 1, 0.24, 0.6,2$, respectively. As we shall see, completely probing even the latter levels for all $\ma<2m_B$ will be challenging, but hadron colliders may ultimately play a leading role. Indeed, those scenarios with $G<20$ typically have $\ma$ values above $7.5\gev$ and most often above $\mupsiii$. Of course, the many scenarios with larger $|\cta|$ than the values listed above will be correspondingly easier to exclude or verify.
Finally, we comment on the implications of the recent ALEPH results [@cranmer] which place a limit on \^2()\[()\]\^2 as a function of $\mhi$ and $\mai$.[^3] (Above, we use the more precise notation $\hi$ and $\ai$ for the lightest CP-even and CP-odd Higgs bosons of the NMSSM.) According to the ALEPH analysis, to have $\mhi\lsim 100\gev$, $\xi^2_1\lsim 0.52$ ($0.42$) is required if $\mai\sim 10\gev$ ($4\gev$). These limits rise rapidly with increasing $\mhi$ — for $\mhi=105\gev$ (the rough upper limit on $\mhi$ such that electroweak finetuning remains quite small and precision electroweak constraints are fully satisfied) the ALEPH analysis requires $\xi^2\lsim 0.85$ ($\lsim 0.7$) at $\mai\sim
10\gev$ ($4\gev$). These limits are such that the easily viable NMSSM scenarios are ones: i) with $\mai$ below but fairly close to $2m_B$, which is, in any case, strongly preferred by minimizing the light-$\ai$ finetuning measure $G$; and/or ii) with $\tanb$ relatively small ($\lsim 2$). These are also the scenarios for which Upsilon constraints are either weak or absent. Details will be provided in a forthcoming paper [@upsilonupdate]. Here, we present a simple summary. In particular, we note the following: a) all $\tanb\leq 2$ cases provide $\mhi\leq 100\gev$ scenarios that escape the ALEPH limits; b) there are a few $G<20$, $\tanb=3$ scenarios with $\mhi$ as large as $98\gev$ and $99\gev$ and with $\xi^2$ essentially equal to the ALEPH limits of $\xi^2\leq 0.42$ and $\xi^2\leq 0.45$ applicable at these respective $\mhi$ values; c) $\tanb=10$ ideal scenarios easily allow for $\mhi\sim 100-105\gev$ (because the tree-level Higgs mass is larger at $\tanb=10 $ than at $\tanb=3$) and at $\mai\lsim 2m_B$ many $\mhi\gsim 100\gev$ points have $\xi^2<0.5$ in the fixed-$\mu$ scan and a few of the full-scan points have $\xi^2<0.6$ for $\mhi\sim 105\gev$, both of which are below the $\mai=10\gev$ ALEPH upper limits on $\xi^2$ of 0.52 at $\mhi\sim 100\gev$ and $0.85$ at $\mhi=105\gev$; d) at $\tanb=50$ there are some $G<20$ points with $\mhi\sim 100\gev$ and $\mai\lsim 2m_B$ having $\xi^2$ below the $0.52$ ALEPH limit. Finally, we note that for the entire range of Higgs masses studied the ALEPH limits were actually $\sim 2\sigma$ stronger than expected. Thus, it is not completely unreasonable to consider the possibility that the weaker expected limits should be employed. These weaker limits for example allow $\xi^2$ as large as $0.52$ at $\mhi\sim
95\gev$ and $0.9$ for $\mhi\sim 100\gev$. These weaker limits allow ample room for the majority of the $\mai\lsim 2m_B$ ideal Higgs scenarios.
The role of the Tevatron {#tevatron}
========================
Potentially, the Tevatron can probe precisely the $\ma$ range close to and above the $\Upsilon(nS)$ masses which cannot be probed in $\Upsilon(nS)$ decays. Some relevant analyzes have been performed looking for a very narrow resonance, denoted $\eps$, that is produced in the same way as the $\upsi$. The published/preprinted results are those of [@Apollinari:2005fy] and [@Aaltonen:2009rq] from the CDF experiment. The latter results employ data corresponding to $L=630\pbi$ and exclude the potential $\eps$ peak at $\mmumu\sim
7.2\gev$ present in the $L=110\pbi$ data of the first paper. The analysis was only performed for the region from $6.3\gev\leq \mmumu
\leq 9\gev$. The reason for not performing the analysis at lower $\mmumu$ is that the acceptance of the $\mupmum$ pair relative to that of the $\upsi$ (used as a normalizing cross section) would be highly mass dependent. This is due to the fact that CDF is only able to see $\mupmum$ pairs with $p_T>5\gev$ and for $\mmumu<6.3\gev$ the fraction of pairs that fail this cut becomes highly mass dependent. No reason for not analyzing the region of $\mmumu>9\gev$ is given, although it is in this region that the $\Upsilon(1S,2S,3S)$ peaks are present.
Our goal here is to use the above $\eps$ analyses to place limits on a CP-odd $a$. This is possible under certain assumptions detailed below. In particular, we will place limits on $\cabb$.
![The total cross section for $a$ production at the Tevatron is plotted vs $\ma$ for $\tanb=1,2,3,10$ (lowest to highest point sets). For each $\ma$ and $\tanb$ value, the lower (higher) point is the cross section without (with) resolvable parton final state contributions. []{data-label="totsigs"}](higlucomb_loop_alltb_tevatron.ps){width="65.00000%"}
The dominant production mechanism for a light $a$ at a hadron collider is different than that for the $\eps$ (assumed to be the same or very similar in kinematic shape for the $\upsi$). The production mechanisms for the $\upsi$ remain uncertain. It has recently been claimed [@Artoisenet:2008fc] that an NNLO version of the leading order (LO) calculation can reproduce the Tevatron results for direct production of the $\upsi$ at larger $p_T$ (roughly $p_T>5\gev$). The diagrams employed begin with the LO $\calo(\alpha_S^3)$ process $gg\to
b\anti b g$ with the $b\anti b$ pair turning into the $\upsi$ with probability determined by $|R_{\upsi}(0)|^2$, leading to an $\upsi+g$ final state. At NLO, the $\alpha_S^4$ diagrams include virtual correction diagrams that also lead to the $\upsi+g$ final state and several diagrams containing an extra quark or gluon in the final state ($\upsi+2g$ and $\upsi+b\anti b$). Several $\calo(\alpha_S^5)$ diagrams leading to $\upsi+3j$ final states (especially $\upsi+3g$) are argued to be of importance at larger $p_T$ and are also included. Resummation [@Berger:2004cc] is necessary to get the low $p_T$ portion of the cross section. From CDF and D0 data, the direct $\upsi$ production cross section is measured to be about 50% of the total. Indirect contributions coming from, for example, $gg\to \chi_b$ followed by $\chi_b\to \upsi \gamma$ make up the remaining 50% of the total $\upsi$ production rate. In contrast, $a$, being a spin-0 resonance, will be dominantly produced via $gg\to a$ through the quark-loop induced $gga$ coupling. In addition, there are large QCD corrections to the one-loop-induced cross section. These are of two basic types: a) virtual corrections and soft gluon corrections; b) corrections containing an extra resolvable gluon or quark in the final state (the dominant diagram is $gg\to ag$) in close proximity to the $a$. The total cross sections predicted by HIGLU [@Spira:1996if] are plotted as a function of $\ma$ for $\cabb=1/\catt=\tanb=1,2,3,10$ in Fig. \[totsigs\] with and without the resolvable parton final state QCD corrections. The HIGLU results agree well with a private program for this process. We note that the cross sections do not scale precisely as $\tan^2\beta$ at large $\tanb$ (as naively predicted by dominance of the $b$-quark loop diagram for the $gg\to a$ coupling at high $\tanb$) due to the virtual corrections. In any case, very substantial cross sections are predicted. In the NMSSM context, at any given $\tanb$ value one should reduce the plotted result for that $\tanb$ by a factor of $(\cta)^2$.
Among the cuts employed in the CDF analysis there is an isolation requirement whereby events are only included if both muons have less than $4\gev$ scalar summed $p_T$ in a cone of size $\Delta R=0.4$ about the muon. The impact of the isolation requirement was studied for the $\upsi$ and it was found that this isolation requirement was 99.8% efficient for the $\upsi$ despite the fact that $\upsi$’s are produced along with one or more extra particles in the final state. Thus, in our analysis for the $a$ we will make the assumption that the components of the $a$ cross section coming from final states containing an extra $q$ or $g$ are not significantly affected by the isolation cut and thus we will employ the full QCD-corrected $a$ cross section. In addition, in the analysis of [@Aaltonen:2009rq] only events for which the $\mupmum$ pair resides in the $|y|<1$ region are retained. Thus, what we actually employ are the cross sections (a)\_[|y|<1]{}=.[d(a)dy]{}|\_[y=0]{}2, which is an excellent approximation given that the cross section is essentially flat in $y$ over this region. At the Tevatron, the ratio varies from roughly $0.12$ at $\ma\sim 2\gev$ to $0.19$ at $\ma\sim 12\gev$ with very weak dependence on $\tanb$. At $\ma=\mupsi$ the ratio is $\sim 0.15$.
In [@Aaltonen:2009rq], what is given are limits on the ratio for production of a very narrow resonance, the $\eps$, relative to that for the $\upsi$ R=[()()()()]{} \[rdef\] under the assumption that the same mechanism is responsible for $\eps$ production as is responsible for $\upsi$ production. As stated earlier, since the $a$ can be produced directly via $gg\to a$ whereas the $\upsi$ cannot, an interpretation for the $a$ of the limits given for the generic $\eps$ requires actually knowing what the $\upsi$ cross section is. It also requires an assumption regarding the efficiency for the $a$ of acceptance and isolation requirements relative to those employed for the $\eps$.
Our analysis is the following. In Ref. [@Apollinari:2005fy], it is stated that the cross section for $\upsi$ production in the $|y|<0.6$ region at $\rts=1.8\tev$ was measured to be $34,600\pb$. In contrast, the cuts of Ref. [@Apollinari:2005fy] accept $\upsi$ events with $|y|<1$. In Ref. [@Apollinari:2005fy], it is stated that the efficiency for $\upsi$ detection (due to geometric and kinematic acceptance cuts as well as trigger and reconstruction efficiencies, but before imposing the isolation requirement noted above) is $0.066$. From [@Abe:1995an], we infer that this acceptance times efficiency factor is one that applies at any fixed value of $y$ that is relatively central and after integrating over accepted $p_T$’s. Then, using $\br(\upsi\to\mupmum)=0.0248$ and the integrated luminosity of $L=110\pbi$ one then predicts 34600([2 1.2]{})1100.0248 0.066=10383 \[estimate\] events where the parenthetical fraction corrects for the increased $\dely$ acceptance compared to that used in measuring the $\upsi$ cross section. This compares favorably to the 9838 number of events that were observed before including the isolation cuts and promptness cuts of Table 1 in Ref. [@Apollinari:2005fy]. A cross check on the cross section is to note that the $\left.{d\sigma(\upsi)\over dy}\right|_{y=0}\times \br(\upsi\to \mupmum) \sim 753\pb$ value measured in [@Abe:1995an] is comparable to the estimate based on Eq. (\[estimate\]) of .[d()dy]{}|\_[y=0]{}=[()\_[|y|<0.6]{}=34600=1.2]{}0.0248\~715. \[xsecest\] Because the earlier paper [@Abe:1995an] may have employed slightly different procedures, efficiencies and so forth, we use the value of Eq. (\[xsecest\]) at $\sqrt s=1.8\tev$.
Moving to the higher energy of $\rts=1.96\tev$, it is stated in [@Aaltonen:2009rq] that the $|y|<0.6$ $\upsi$ cross section increases relative to $\rts=1.8\tev$ by about 10%, implying .[d() dy]{}|\_[y=0]{}(1.96)() \~787 . \[dsigdybr1.96\] This is the value we shall employ. As another cross check, we note that a 10% increase in the total cross section would yield about $38,330\pb$ at $\rts=1.96\tev$. At $\rts=1.96\tev$, [@Aaltonen:2009rq] states that 52,700 $\upsi\to\mupmum$ events are observed using the same cuts as in Ref. [@Apollinari:2005fy] (that imply that only an $\upsi$ with $|y|<1$ will be accepted) [*and*]{} after imposing the isolation and promptness criteria detailed in Ref. [@Apollinari:2005fy]. The latter imply an additional efficiency factor of $0.921$ relative to the $0.066$ efficiency referenced earlier. Multiplying these two efficiencies yields a net efficiency of $\sim 0.061$. With the 10% cross section increase and accounting for the increased luminosity of $L=630\pbi$, the $0.061$ net efficiency implies an expected event number of 60,244. Although this is not in perfect agreement with the 52,700 events actually observed, we will use the result of Eq. (\[dsigdybr1.96\]) below.
Relative to the $\upsi$ efficiency, purely geometric effects alter the efficiency for $\eps$ production and we assume that the same geometric changes apply to the $a$. The formula of [@Apollinari:2005fy] is: efficiency() = efficiency()\[efficiency\] We will employ this same relative efficiency for the $a$ as a function of $\ma$ using as well $efficiency(\upsi)=0.061$ as obtained above.
The most precise limits on $\cabb$ are obtained using the ratio $R$ defined in Eq. (\[rdef\]). We recall from Refs. [@Apollinari:2005fy; @Aaltonen:2009rq] that the limits on $R$ are obtained by performing a smooth fit to the event distribution and looking for fluctuations about this smooth fit. The limits on the $\eps$ (or the $a$ in our case) are then obtained by placing small Gaussians at each possible $\ma$ value and placing limits using the observed fluctuations about the smooth fit. In the mass region for which CDF has performed this analysis, $6.3\gev\lsim \ma\lsim 9\gev$, it is very convenient to simply directly employ their results. As stated, we assume that the $a$ efficiencies are the same as for the $\eps$, in which case we can compute the ratio $R$ as R \[rcomp\] where $\left.{d\sigma(a)\over dy}\right|_{y=0}$ is computed using HIGLU, $\br(a\to \mupmum)$ is taken from Fig. \[bramumu\] and $\left.{d\sigma(\upsi)\over dy}\right|_{y=0}\brupsimumu$ is as given in Eq. (\[dsigdybr1.96\]). Note that the exact values of the efficiencies, Eq. (\[efficiency\]), are not important using this procedure so long as the efficiency for the $a$ is the same as for the $\eps$.
![We plot the 90% CL limits on the ratio $R$ for $a$ production at the Tevatron as a function of $\ma$ compared to NMSSM predictions using HIGLU for the following cases: $(\tanb=1,\cta=1)$ (red $+$’s), $(\tanb=2,\cta=1)$ (blue diamonds), $(\tanb=3,\cta=1)$ (green $\times$’s) and $(\tanb=10,\cta=0.1)$ (yellow squares). []{data-label="90percentCL"}](tevatron90percent_630pbi.ps){width="62.00000%"}
With these assumptions and inputs we can then predict the ratio $R$ for the case of $\eps=a$ and compare to the 90% CL upper limits of [@Aaltonen:2009rq] based on $L=630\pbi$ of analyzed CDF data. This comparison appears in Fig. \[90percentCL\] for a number of $\cabb=\cta\tanb$ choices. We observe that the predicted $R$ depends almost entirely on $|\cabb|$, with extremely little dependence on $\tanb$ separately for the $\tanb\geq 1$, $\ma\geq 4\gev$ parameter region on which we focus. The corresponding bin-by-bin limits on $|\cabb|$ obtained by interpolation appear in Fig. \[90percentCL2lums\]. In the 2HDM(II), they are limits on $\tanb=\cabb$. In the NMSSM, these are limits on $\cabb=\cta\tanb$. In both cases, the interpolations are only accurate for $\tanb\geq1$ and $\ma\geq 4\gev$. From Fig. \[90percentCL2lums\], we find that the limits based on the existing $L=630\pbi$ analysis roughly exclude $|\cabb|>3$ for $6.8\lsim\ma\leq 9\gev$ and $|\cabb|>2$ for $8.2\lsim \ma\leq 9\gev$, but do not exclude $\cabb=1$ for any of the $\ma$ values in the analysis range. In the 2HDM(II) case the $\cabb=\tanb$ limits from the Tevatron are stronger than those from Upsilon decays and LEP data, as summarized earlier, for $8\gev\lsim \ma \lsim 9\gev$. In the NMSSM case the limits on $|\cta|$ from the Tevatron data are the stronger in much the same mass range, as we detail shortly. In Fig. \[90percentCL2lums\] we also plot the statistically extrapolated limits that would result by increasing the data sample to $L=10\fbi$. (Presumably, a real analysis of a high luminosity data set would do even better.) Since the $a$ signal cross section varies roughly as $(\cabb)^2$, even this large luminosity increase leads to limits that are improved by only a factor of a bit more than two. Nonetheless, one approaches the $|\cabb|\sim 1$ level of interest in the NMSSM at $\ma=9\gev$.
![We plot the $90\%$ CL upper limits on $|\cabb|$ obtained using the results for the ratio $R$ of Fig. \[90percentCL\]. The $10\fbi$ results are obtained by statistical extrapolation of the $630\fbi$ results. In the context of the 2HDM(II), $\cabb=\tanb$. In the context of the NMSSM, $\cabb=\cta\tanb$. In both cases, limits were derived assuming $\tanb\geq 1$. []{data-label="90percentCL2lums"}](tanblim_tevatron_2lums.ps){width="62.00000%"}
![We plot the $90\%$ CL $\ctamax$ values in the NMSSM context obtained from the results of Fig. \[90percentCL\] for the $630\pbi$ CDF data set, in comparison to the $\ctamax$ values plotted in Fig. \[ctamaxvsma\]. For clarity, the plot is limited to the $\ma$ region over which the Tevatron data are relevant. The curve types are as in Fig. \[ctamaxvsma\]. The Fig. \[ctamaxvsma\] results for a given curve type are those for which $\ctamax$ starts at lower values at low $\ma$ rising to higher values at higher $\ma$. The new CDF limits are those that begin near $\ma\sim 6.3\gev$ and terminate at $\ma\sim 9\gev$ and that fall (with fluctuations) as $\ma$ increases.[]{data-label="ctamaxtevatron"}](ctamaxvsma_rbt_withtevatron.ps){width="65.00000%"}
Focusing now on the NMSSM, we compute the upper limit on $\cta$, $\ctamax$, obtained by appropriate interpolation of the results of Fig. \[90percentCL\]. We again emphasize that although $\br(a\to
\mupmum)$ is $\tanb$ dependent as shown in Fig. \[bramumu\], it is nonetheless the case that for $\tanb\geq 1$ and $\ma\geq 4\gev$ the limits on $\cabb=\cta\tanb$ are almost independent of $\tanb$ at fixed $\cabb$, as found in Fig. \[90percentCL\] (compare the two $\cta\tanb=1$ cases — $\tanb=\cta=1$ vs. $\tanb=10$, $\cta=0.1$). This can be understood as follows. At low $\tanb$, although $\br(a\to\mupmum)$ is suppressed, contributions to the $gga$ coupling from loops involving the top quark are substantial relative to loops involving the bottom quark. In comparison, at large $\tanb$ one finds that $\br(a\to\mupmum)$ is maximal but top-quark loops are relatively suppressed compared to bottom quark loops. These two effects very nearly cancel one another leaving the net $a$ cross section unchanged at fixed $\cabb=\cta\tanb$. As a result, it is easy to extract the $\ctamax$ values for different $\tanb$ values directly from the plotted limit on $|\cabb|$ shown in Fig. \[90percentCL2lums\].
The resulting $\ctamax$ limits are shown in Fig. \[ctamaxtevatron\] in comparison to the upper limits plotted earlier in Fig. \[ctamaxvsma\]. The figure focuses on the $6\gev\lsim\ma\lsim
9\gev$ region for which we have extracted the Tevatron limits using $R$. What we observe is that the $630\pbi$ 90% CL limits become the strongest for $\ma\gsim 8.3\gev$. In a forthcoming paper, we will analyze the impact of BaBar data for $\upsiii\to\gam \tauptaum$ decays on the $6\gev\lsim\ma\lsim 10\gev$ region. Our preliminary results suggest that the Tevatron limits plotted above and the $\upsiii$ limits are very similar at $\ma\sim 9\gev$, with $\upsiii$ limits being superior for lower $\ma$.
Given the above, the great value of extending the Tevatron analysis above $\mmumu=9\gev$ is apparent. A full analysis of existing and future data all the way out to $\ma\sim 12\gev$ is needed. First, it might strongly constrain the properties of any light $a$ with $\ma\lsim 2m_B$ that would allow for the ideal Higgs scenario. Second, it might completely eliminate the possibility that a light $a$ could provide a major contribution to $a_\mu$. At the moment, the Tevatron limits on $\cabb$ shown rule out a significant contribution to $\Delta
a_\mu$ from an $a$ with $\ma<9\gev$, while in the range $9\gev\lsim\ma\lsim 2m_B$ these limits leave open the possibility of $\Delta a_\mu$ arising from diagrams involving a CP-odd $a$. Only the Tevatron and/or LHC can probe the region of $\ma$ above the Upsilon masses.
![We plot the upwards fluctuation in the number of events in a given bin corresponding to a $1.646\sigma$ (90% CL) excess as predicted using the $L=630\pbi$ event numbers of [@Aaltonen:2009rq]. These limits are compared to the predicted number of events for an $a$ resonance centered on that bin spread out by the experimental resolution. The same $(\tanb,\cta)$ values as in Fig. \[90percentCL\] are considered. []{data-label="90percentCLnevt630pbi"}](tevatron90percentCLnevt_630pbi.ps){width="65.00000%"}
Absent a full analysis by CDF of limits on $R$ in the region $\mmumu>9\gev$ and given that this region is of great interest, we wish to make some estimates of limits on $|\cabb|$ based on the event number plots of Ref. [@Aaltonen:2009rq]. We have employed the following procedure. First, for this analysis, we must know the efficiency for detecting the $a$. For our estimates we use $efficiency(\ma)$ from Eq. (\[efficiency\]) and $efficiency(\upsi)=0.061$ (as motivated earlier below Eq. (\[dsigdybr1.96\])) to predict the number of $a$ events as a function of $\ma$. Second, we wish to determine how many of the total number of $a$ events fall into a $50\mev$ bin centered on $\ma$. To do so, we need to know the resolution as a function of $\ma$. In [@Aaltonen:2009rq], it is stated that the resolution, $\sigma_r$ varies from $32\mev$ to $50\mev$ in going from $\ma=6.3\gev$ to $\ma=9\gev$, with a value of $52\mev$ at $\mupsi$. We use a simple linear interpolation for other values of $\ma$, but do not allow $\sigma_r$ to fall below $25\mev$ at low $\ma$. The fraction of $a$ events distributed as a Gaussian of width $\sigma_r$ that fall into a $50\mev$ bin (which should be thought of as a bin of half width $25\mev$) that is centered on $\ma$ is given by f(m)=Erf([25 \_r(m)]{}) where $\sigma_r(m)$ is in $\mev$. In Fig. \[90percentCLnevt630pbi\], we plot the $1.646\sigma$, 90% CL, fluctuation number for each of the CDF $50\mev$ bins compared to the predicted number of $a$ events that would fall into that bin. We do this for the same selection of $(\tanb,\cta)$ values as employed in Fig. \[90percentCL\]. One observes that for $\ma\sim 6\gev$ ($\ma\sim 9\gev$) 90% CL sensitivity is anticipated for $\cabb\gsim
3$ ($\cabb\gsim 2$). This anticipates in an average sense the more precise (and more fluctuating) results based on the $R$ analysis found in Fig. \[90percentCL2lums\].
Sensitivity to the $a$ in the $S/\sqrt B$ sense can actually be improved by taking a bin size that properly matches $\sigma_r$. If the background is flat then the optimal bin size is $2\sqrt 2
\sigma_r$ which retains a fraction $Erf(1)=0.843$ of the total $a$ signal and yields $B=2\sqrt 2 \sigma_r {d \sigma_B\over d\mmumu}$. Following this procedure we can then use interpolation to extract the $|\cabb|$ value such that $S/\sqrt B$ is $1.646$. The resulting values of $|\cabb|$ which correspond to this 90 CL fluctuation in the $\Delta \mmumu=2\sqrt 2 \sigma_r$ acceptance window are plotted in Fig. \[tanblim\_tevatron\_2lums\_evts\]. As anticipated above, this event counting method turns out to give a good average representation of the results obtained using $R$ (which analysis was based on bin-by-bin fits of the fluctuations about a smooth curve) at the 90% CL. Thus, despite relatively small $S/B$ levels (typically of order 0.02 in each of two neighboring bins for a $1.646\sigma$ net fluctuation), our estimates for expectations for $\ma>9\gev$ (using the approach of assuming there were no $1.646\sigma$ fluctuations in the absolute $L=630\pbi$ event numbers in acceptance windows of size $\Delta \mmumu$) should give a good idea of the limits that are implicit in current data.
As an aside, we note that even though the shape of the $\Upsilon(nS)$ resonances (which are also very narrow) will also be determined by $\sigma_r$, one can learn if there is an excess in the $\mupmum$ final state by also looking at the $\Upsilon(nS)$ resonance in the $\epem$ final state to which the $a$ will not contribute. Assuming lepton universality, the $\Upsilon(nS)$ contribution in the $\mupmum$ final state can be subtracted from the $\mupmum$ spectrum, after which any residual excess from the presence of an $a$ would become apparent. Statistical errors resulting from this subtraction will be roughly a factor of $\sqrt 2$ larger than employed above. However, this procedure does rely on a precise understanding of efficiencies, resolutions and the like for electrons. Another technique that could be considered is comparing one Upsilon resonance to another. If the relative normalization between the two Upsilon resonances can be sufficiently precisely predicted, including both theoretical and experimental uncertainties, then an $a\to \mupmum$ signal hiding under one of the Upsilons could become apparent.
![We plot approximate limits on $|\cabb|$ as a function of $\ma$ estimated assuming that bins centered on $\ma$ and encompassing an $\ma$ range of $2\sqrt{2}\times \sigma_r$, where $\sigma_r$ is the experimental $\mmumu$ resolution at the given $\ma$, do not have a $90\%$ CL fluctuation relative to the number of events observed and plotted over this bin range in [@Aaltonen:2009rq] for $L=630\pbi$. The $L=10\fbi$ histogram corresponds to simply scaling the predicted $a$ event rate and the $90\%$ CL fluctuations to the increased luminosity. []{data-label="tanblim_tevatron_2lums_evts"}](tanblim_tevatron_2lums_evts_90percentCL.ps){width="65.00000%"}
![We plot the upwards fluctuation in the number of events in a given bin corresponding to a $1.646\sigma$ (90% CL) excess as predicted using the $L=630\pbi$ event numbers of [@Aaltonen:2009rq] scaled up to $L=10\fbi$. These limits are compared to the predicted number of events for $L=10\fbi$ for an $a$ resonance centered on that bin spread out by the experimental resolution. The same $(\tanb,\cta)$ values as in Fig. \[90percentCL\] are considered. []{data-label="90percentCLnevt10fbi"}](tevatron90percentCLnevt_10fbi.ps){width="65.00000%"}
Both CDF and D0 will continue to accumulate data far in excess of $L=630\pbi$. Thus, it is useful to extrapolate to higher luminosity using the observed number of events for $L=630\pbi$ plotted in [@Aaltonen:2009rq]. We rescale the observed number of events in each bin to $L=10\fbi$ and compute the $90\%$ CL fluctuation upper limit in each bin as $1.646\times\sqrt{ N_{evt}(bin)}$. In Fig. \[90percentCLnevt10fbi\], we plot these extrapolated $1.646\sigma$ fluctuation levels in each bin compared to the predicted number of $a$ events in each bin for the same selection of $(\tanb,\cta)$ values as employed in Fig. \[90percentCL\]. The extracted limits on $|\cabb|$ are plotted in Fig. \[tanblim\_tevatron\_2lums\_evts\]. We see that the eventual Tevatron limits from just one experiment could easily be superior to those currently available from Upsilon decays, in particular probing the $\cabb=\cta\tanb\lsim 1$ coupling level, of particular interest for the “ideal” Higgs scenario, all the way up to $\ma=2m_B$, except in the vicinity of the Upsilon peaks.
Some further comments are the following. First, we emphasize that the $1.646\sigma$ approximate procedure leads to the expectation that quite important limits are possible for $L=10\fbi$ in the $9\gev\lsim \ma\lsim 2m_B$ region of great interest in the NMSSM version of the ideal Higgs scenario for which $\ma\gsim 8$ and $|\cabb|\sim 0.2 - 2$ is a strongly preferred parameter region. If this scenario were to be nature’s choice, there is a decent chance of observing an $a$ using the dimuon spectrum analysis and the ultimate Tevatron data set.
Second, we again note that even the $L=630\pbi$ estimated limits of Fig. \[tanblim\_tevatron\_2lums\_evts\] for $9\gev\leq \ma\leq 12\gev$ would rule out the enhanced $|\cabb|$ values of order $30$ needed for $a$-exchange graphs to explain the $a_\mu$ discrepancy for $\ma$ in this mass region, which is the only relatively low mass region for which other current constraints are sufficiently weak that the $a$ might provide the observed discrepancy. It is thus quite important for CDF (and D0) to perform the needed analysis using the $R$ or similar technique and determine whether or not the rough limits we obtained above using event numbers are approximately correct.
LHC Prospects {#lhc}
=============
The basic question is whether the LHC will be able to improve over the Tevatron $L=10\fbi$ projected results and limits. The total cross sections at the LHC appear in Figs. \[totsigslhc\] and \[totsigslhc10\] for $\rts=14\tev$ and $\rts=10\tev$ respectively; they are plotted analogously to those for the Tevatron appearing in Fig. \[totsigs\]. Recall that these cross sections are those appropriate in the 2HDM(II) context. We see that, relative to the Tevatron, the $\rts=14\tev$ cross sections are about a factor of $3-7$ higher, the smaller (larger) ratio applying at small (large) $\ma$. Relative to the $\rts=14\tev$ cross section, the $\rts=10\tev$ cross sections are roughly a factor of $\sim 1.2$ smaller at $\ma=2\gev$ and a factor $\sim 1.34$ smaller at $\ma=10\gev$, more or less independent of the $\tanb$ value. It now appears that perhaps as much as a year will be spent running at $\rts=7\tev$. Thus, we also plot in Fig. \[totsigslhc7\] the cross sections for this latter energy. At $\ma=2\gev$, the $a$ cross section at $\rts=10\tev$ is a factor of about 1.15 larger than the $\rts=7\tev$ cross section; the factor rises to $\sim 1.35$ for $\ma=10\gev$. The modest decrease of the $a$ cross section with decreasing energy is a result of the fact that the $gg$ luminosity at low $\ma$ varies slowly with $\rts$. This is one of the reasons why searches for a light $a$ are very appropriate in early LHC running.
In going to the NMSSM, one takes these results for any given $\tanb$ choice and then multiplies by $(\cta)^2$, the square of the overlap fraction of the $a$ with the 2HDM component.
![The total cross section for $a$ production at the LHC for $\rts=14\tev$ is plotted vs $\ma$ for $\tanb=1,2,3,10$ (lowest to highest point sets). For each $\ma$ and $\tanb$ value, the lower (higher) point is the cross section without (with) resolvable parton final state contributions. []{data-label="totsigslhc"}](higlucomb_loop_alltb_lhc.ps){width="65.00000%"}
![The total cross section for $a$ production at the LHC for $\rts=10\tev$ is plotted vs $\ma$ for $\tanb=1,2,3,10$ (lowest to highest point sets). For each $\ma$ and $\tanb$ value, the lower (higher) point is the cross section without (with) resolvable parton final state contributions. []{data-label="totsigslhc10"}](higlucomb_loop_alltb_lhc10.ps){width="65.00000%"}
![The total cross section for $a$ production at the LHC for $\rts=7\tev$ is plotted vs $\ma$ for $\tanb=1,2,3,10$ (lowest to highest point sets). For each $\ma$ and $\tanb$ value, the lower (higher) point is the cross section without (with) resolvable parton final state contributions. []{data-label="totsigslhc7"}](higlucomb_loop_alltb_lhc7.ps){width="65.00000%"}
As an example of how limits obtained at the LHC in early running will compare to the Tevatron limits, let us consider the case of $\tanb=10$ and $\cta=0.1$, for which $\cabb=1$. As shown in Fig. \[tanblim\_tevatron\_2lums\_evts\], even with $L=10\fbi$ the Tevatron is not fully able to probe at the $90\%$ CL the predicted relatively small $a$ event levels except at $\ma$ values close to $2m_B$ but outside the $\Upsilon(nS)$ peaks. In more detail, for the above parameter choices, the predicted number of $a\to \mupmum$ events for $|y|\leq 1$ in a $\Delta \mmumu=2\sqrt 2 \sigma_r$ bin centered on $\ma$ is 436, 615 and 475 at $\ma=8\gev$, $\mupsi$ and $10.5\gev$, respectively, where the event numbers quoted incorporate the $Erf(1)=0.8427$ reduction factor associated with keeping only events in an interval of size $\Delta \mmumu=2\sqrt 2 \sigma_r$. The actual $\Delta \mmumu=2\sqrt{2}\sigma_r$ values are $43\mev$, $52\mev$ and $57\mev$ at $8\gev$, $\mupsi$ and $10.5\gev$, respectively. As regards the background, we take the 50 MeV bin event numbers in the CDF plot of the number of events in each bin and rescale to the $\Delta \mmumu$ interval sizes at the above $\mmumu=\ma$ choices. This gives us a background event number $N_{\Delta \mmumu}$ at each $\ma$. The $1\sigma$ fluctuations in these background event numbers, $\sqrt{N_{\Delta \mmumu}}$, are 468, 945 and 285, respectively. The statistical significances of the $a$ signals are then $\sim 0.93\sigma$, $\sim 0.65\sigma$ and $\sim 1.67\sigma$, respectively. Only the latter is (slightly) above the $1.646\sigma$ level corresponding to 90% CL. However, to repeat, this high $\ma\lsim 2m_B$ region is particularly favored in the model context. But, to reach $5\sigma$ at $\ma=10.5\gev$ would require about nine times as much integrated luminosity, $L\sim 90\fbi$ and $5\sigma$ at $\ma=\mupsi$ would require $L\sim 590\fbi$.
Projections for the LHC have been made public by ATLAS. In Fig. 1 of [@priceatlas], one finds a plot of $d\sigma/d\mmumu$ coming from $b\anti b$ production, Drell-Yan production and $\upsi$ production. The dimuon Drell-Yan contribution is negligible compared to that from $b\anti b$ production even after the latter is reduced by muon isolation requirements. We ignore the Drell-Yan contribution in all subsequent discussions.
In generating the $b\anti b$ and $\upsi$ cross sections, only events with $p_T$ cuts requiring one muon with $p_T>6\gev$ and a 2nd muon with $p_T>4\gev$, both with $|\eta|<2.4$, were retained. A recent Monte Carlo study [@student] finds that these events constitute 20% of the total inclusive cross section. The fraction of these events that survive after further requirements related to triggering, reconstruction and the final analysis selection cuts is 50%. Thus, the net efficiency for the $\upsi$ events plotted in Fig. 1 of [@priceatlas] is $\sim 0.5\times 0.2=0.1$. Therefore, we will write $\effatlas=0.1 r$ for the fraction of inclusive $a$ events that will be retained, where $r\sim 1$ for the cuts and triggering strategies studied so far, but $r>1$ is probably achievable if these are optimized for the CP-odd $a$.
Returning to Fig. 1 of [@priceatlas], we observe a $b\anti
b$-induced dimuon cross section level for $\rts=14\tev$ of order $d\sigma/d\mmumu\sim 50-90\pb/100\mev$ in the $\mmumu\in[8\gev,2m_B]$ interval when outside the Upsilon peak region. This is the dimuon cross section from $b\anti b$ heavy flavor production only. The author of [@priceatlas] estimates [@dpprivatecom] that one should at most double this cross section to account for $c\anti c$ production and other contributions. We will make estimates based on multiplying the $b\anti b$-induced dimuon cross section by a factor of two. To this, we add the $\upsi$ cross section as plotted in Fig. 1. The net resulting spectrum constitutes the background to the $a$ signal that we discuss shortly.
As in the CDF case, we will use a bin size of $\Delta_{\mmumu}=2\sqrt
2\sigma_r$ (which optimizes $S/\sqrt B$ for a flat background) for comparing the $a$ signal to the above stipulated background. As for resolutions, it is stated in [@priceatlas] that the resolution at the $J/\psi$ is around $54\mev$ while that at the $\upsi$ is close to $170\mev$. We use a linear interpolation for other values of $\mmumu$. Assuming $L=10\pbi$ of integrated luminosity, the background event numbers $N_{\Delta {\mmumu}}$ in the intervals of size $\Delta {\mmumu}=2\sqrt 2\sigma_r$ are $4055$ at $\ma=8\gev$, $50968$ at $\ma=\mupsi$ and $9620$ at $\ma=10.5\gev$. We take the square root to determine the $1\sigma$ fluctuation level.
We now consider the $a\to\mupmum$ signal rates. From Fig. \[totsigslhc\], we see that at $\tanb=10$ the total $a$ cross section ranges from about $4.2\times 10^5\pb(\cta)^2\sim 4200\pb$ at $\ma=8\gev$ to $\sim 8500\pb$ at $\ma\lsim 2m_B$ for $\rts=14\tev$. The cross section for $a\to \mupmum$ assuming $\tanb=10$ and $\cta=0.1$ will then range from $4200-8500\pb \times (\br(a\to
\mupmum)\sim 0.003)\sim 12-25\pb$. As discussed above, we will write the total $a$ efficiency in the form $\effatlas=0.1 \times r$. Multiplying the above cross section by $\effatlas$ and by the $Erf(1)=0.8427$ acceptance factor for the ideal interval being employed and using $L=10\pbi$ (as employed above in computing the number of background events), we obtain $a$ event numbers of $10\times r$, $18.5\times r$ and $21\times r$ at $\ma=8\gev$, $\mupsi$ and $10.5\gev$, respectively. The statistical significances of the $a$ peaks for $L=10\pbi$ are then $r\times$ the $r=1$ results of $0.16
\sigma$, $0.08 \sigma$ and $0.22 \sigma$, respectively.
Of course, we currently expect that substantial early running will mostly take place at $\sqrt s=7\tev$ and $\sqrt s=10\tev$. As noted earlier, lower $\rts$ implies a somewhat smaller $a$ cross section in the $[8\gev,2m_B]$ mass interval on which we are focusing. Roughly, relative to $\rts=14\tev$, the $a$ cross section decreases by a factor of $\sim 1.3$ at $\rts=10\tev$ and a factor of $\sim 1.7$ at $\rts=7\tev$ in this mass interval. Since the backgrounds are also basically $gg$ fusion induced, we presume that these same factors will apply to them. At $\rts=10\tev$ ($\rts=7\tev$) this then will reduce the statistical significances given above by a factor of $1/\sqrt{1.3}$ ($1/\sqrt{1.7}$). The statistical significances at $\ma=8\gev$, $\mupsi$ and $10.5\gev$ are, respectively, then $0.14\sigma$, $0.07\sigma$, $0.19\sigma$ at $10\tev$ and $0.12\sigma$, $0.06\sigma$, $0.17\sigma$ at $7\tev$, all to be multiplied by $r$.
[|c|c|c|c|]{} Case & $\ma=8\gev$ &$\ma=\mupsi$ & $\ma\lsim 2m_B$ ATLAS LHC7 & $17/r^2$ & $63/r^2$ & $9/r^2$ ATLAS LHC10 & $13/r^2$ & $48/r^2$ & $7/r^2$ ATLAS LHC14 & $10/r^2$ & $37/r^2$ & $5.4/r^2$
Given the above results, we can tabulate the integrated luminosity $L$ needed to achieve a $5\sigma$ significance at each of the three energies. The results appear in Table \[comparisontable2\]. The required $L$’s away from the Upsilon resonance may be achieved after a year or two of LHC operation. The sensitivity of the required luminosities to $r$ shows the importance of firmly establishing the precise efficiencies for background and signal. We look forward to continued and detailed work by the ATLAS collaboration in this area. Of course, we must not forget that the required $L$’s are very sensitive to $\tanb$, $\cta$ and $\br(a\to \mupmum)$; very roughly for $\tanb\neq 10$, $\cta\neq 0.1$ and/or $\br(a\to\mupmum)\neq 0.003$ the tabulated luminosities need to be multiplied by ([0.003(a)]{})\^2([0.1 ]{})\^4([10]{})\^[3.2-3.6]{}, where the $3.2$ applies for $\ma\sim 8\gev$ and the $3.6$ applies for $\ma\lsim 2m_B$. Depending upon the precise value of $\ma$ and $\tanb$, in the $\ma$ mass range under discussion Fig. \[bramumu\] shows that $\br(a\to \mupmum)$ can range from a low of $0.0023$ at $\tanb=1.5$ and $\ma\lsim 2m_B$ to a high of $0.0033$ for $\tanb\geq
3$ and $\ma=8\gev$. The minimum values of $\cta$, with and without placing a maximum on the light-$a$ finetuning measure $G$, were detailed in Table \[ctatable\].
Studies by CMS analogous to the ATLAS studies discussed above are under way [@bortprivatecom].
Conclusions
===========
In this paper we have shown that a dedicated analysis of the dimuon spectrum at the Tevatron and LHC at low masses, $\mmumu\lsim
2m_B$, will provide very important constraints on models containing a light CP-odd Higgs boson. We employed the published $L=630\pbi$ CDF analysis of the dimuon spectrum between $\sim 6.3\gev$ and $9\gev$ by CDF and found that constraints on the $b\anti b a$ coupling $\cabb$ become competitive with those from $\Upsilon(nS)\to \gam a$ decays for $8.5\gev\lsim \ma\lsim 9\gev$, and will be superior for larger data sets. In addition, only hadron colliders have the kinematic reach to constrain $|\cabb|$ in the important region $\mupsiii\lsim\ma\lsim
2m_B$. In particular, for $L=10\fbi$, the Tevatron will provide significant constraints on the $|\cabb|\gsim 1$ portion of the $8\gev\lsim \ma\lsim 2m_B$ mass region that would allow an NMSSM ideal Higgs scenario with an $\mh\sim 100-105\gev$ CP-even $h$ decaying primarily via $h\to aa \to \tauptaum\tauptaum$ to be possible with neither electroweak finetuning nor “light-$a$” finetuning. It is also very noteworthy that our rough estimates of the limits that CDF could place on $|\cabb|$ using the $L=630\fbi$ event rates in the $9\leq\ma\leq 12\gev$ region are such that the observed $a_\mu$ discrepancy could not be explained by a light $a$.
For the LHC, we have obtained rough estimates of what will be possible using information available from the ATLAS collaboration, in particular regarding the efficiency (for triggering, tracking, $p_T$ cuts, etc.) for retaining $a\to \mupmum$ events. We find that it will be possible to obtain a $5\sigma$ signal for a light $a$ with $\tanb\sim 10$ and $\cta\sim 0.1$ throughout the entire range $8\gev\lsim \ma\lsim 2m_B$ away from the Upsilon peaks for $L\sim
13\fbi$ at $\rts=10\tev$ or $L\sim 17\fbi$ at $\rts=7\tev$. For example, at $\ma=10.5\gev$, only $L\sim 7\fbi$ at $\rts=10\tev$ or $L\sim 9\fbi$ at $\rts=7\tev$ is required to achieve a $5\sigma$ signal for such an $a$.
Of course, not all acceptable NMSSM models have $|\cabb|$ as large as $\sim 1$. As an extreme example, at $\tanb=1.7$, $\cta\sim 0.1$ is possible for small light-$a$ finetuning (corresponding to $\cabb\sim
0.17$). In this case, the $a$ cross section at $\ma\lsim 2m_B$ is about a factor of 18 smaller than at $\tanb=10$ and $\cta\sim 0.1$. Using statistical extrapolation this suggests that as much as $324$ times more luminosity would be needed to achieve the same statistical significances as above. However, one should keep in mind that it may in the end be possible to obtain net efficiencies for the $a$ at ATLAS and CMS in excess of the current ATLAS estimate of $10\%$. Indeed, early CMS studies suggest that net efficiencies might be as high as $30\%$ [@bortprivatecom]. Since the needed $L$ scales inversely with the square of the efficiency, assuming $\rts=14\tev$ and $r=3$ one finds that a $5\sigma$ signal could be achieved for $\tanb=1.7$ and $\cta\sim 0.1$ with $L\sim 195\fbi$, an integrated luminosity that should be achieved in the not too distant future, although background levels might be larger at the higher instantaneous luminosities needed to achieve such large total $L$.
Overall, this kind of search is quite important given that there are many models in which light $a$’s are present that have significant, even if not enhanced, couplings to gluons via quark loops and that would have reasonable $a\to \mupmum$ branching ratio. Searching for such $a$’s and constraining their possible masses and couplings is an important general goal and both Tevatron and LHC data will be of great value.
We thank D. Price, D. Bortoletto, H. Evans, Z. Gecse, C. Mariotti and M. Pellicioni for helpful conversations regarding the ATLAS and CMS analyses, respectively. We particularly want to thank Y. Yang for performing the Monte Carlo study needed to estimate the net ATLAS efficiency. JFG was supported by U.S. DOE grant No. DE-FG03-91ER40674, the National Science Foundation under Grant No. PHY05-51164 while at KITP, the Aspen Center for Physics and, during the completion of the work, as a Scientific Associate at CERN.
[99]{}
R. Dermisek and J. F. Gunion, Phys. Rev. Lett. [**95**]{}, 041801 (2005) \[arXiv:hep-ph/0502105\]. R. Dermisek and J. F. Gunion, Phys. Rev. D [**73**]{}, 111701 (2006) \[arXiv:hep-ph/0510322\]. R. Dermisek and J. F. Gunion, Phys. Rev. D [**75**]{}, 075019 (2007) \[arXiv:hep-ph/0611142\]. R. Dermisek and J. F. Gunion, Phys. Rev. D [**76**]{}, 095006 (2007) \[arXiv:0705.4387 \[hep-ph\]\]. S. Chang, P. J. Fox and N. Weiner, JHEP [**0608**]{}, 068 (2006) \[arXiv:hep-ph/0511250\].
S. Chang, R. Dermisek, J. F. Gunion and N. Weiner, arXiv:0801.4554 \[hep-ph\]. J. F. Gunion, JHEP [**0908**]{}, 032 (2009) \[arXiv:0808.2509 \[hep-ph\]\].
J. R. Ellis, J. F. Gunion, H. E. Haber, L. Roszkowski and F. Zwirner, Phys. Rev. D [**39**]{}, 844 (1989). John F. Gunion, Howard E. Haber, Gordon Kane, Sally Dawson. 1990. Series: Frontiers in Physics, 80; QCD161:G78
U. Ellwanger, J. F. Gunion and C. Hugonie, JHEP [**0502**]{}, 066 (2005) \[arXiv:hep-ph/0406215\].
U. Ellwanger and C. Hugonie, Comput. Phys. Commun. [**175**]{}, 290 (2006) \[arXiv:hep-ph/0508022\]. L. J. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D [**50**]{}, 7048 (1994) \[arXiv:hep-ph/9306309\]. M. S. Carena, M. Olechowski, S. Pokorski and C. E. M. Wagner, Nucl. Phys. B [**426**]{}, 269 (1994) \[arXiv:hep-ph/9402253\]. D. M. Pierce, J. A. Bagger, K. T. Matchev and R. j. Zhang, Nucl. Phys. B [**491**]{}, 3 (1997) \[arXiv:hep-ph/9606211\]. R. Dermisek, arXiv:0806.0847 \[hep-ph\]. R. Dermisek, AIP Conf. Proc. [**1078**]{}, 226 (2009) \[arXiv:0809.3545 \[hep-ph\]\].
R. Dermisek and J. F. Gunion, Phys. Rev. D [**79**]{}, 055014 (2009) \[arXiv:0811.3537 \[hep-ph\]\].
R. Dermisek, Mod. Phys. Lett. A [**24**]{}, 1631 (2009) \[arXiv:0907.0297 \[hep-ph\]\].
R. Dermisek, J. F. Gunion and B. McElrath, Phys. Rev. D [**76**]{}, 051105 (2007) \[arXiv:hep-ph/0612031\]. B. Aubert [*et al.*]{} \[BABAR Collaboration\], arXiv:0906.2219 \[hep-ex\]. B. Aubert [*et al.*]{} \[BABAR Collaboration\], arXiv:0905.4539 \[hep-ex\].
W. Love [*et al.*]{} \[CLEO Collaboration\], Phys. Rev. Lett. [**101**]{}, 151802 (2008) \[arXiv:0807.1427 \[hep-ex\]\]. F. Domingo, U. Ellwanger, E. Fullana, C. Hugonie and M. A. Sanchis-Lozano, JHEP [**0901**]{}, 061 (2009) \[arXiv:0810.4736 \[hep-ph\]\].
The ALEPH Collaboration, arXiv:1003.0705 \[hep-ex\].
R. Dermisek and J. F. Gunion, arXiv:1002.1971 \[hep-ph\].
G. Apollinari [*et al.*]{}, Phys. Rev. D [**72**]{}, 092003 (2005) \[arXiv:hep-ex/0507044\].
T. Aaltonen [*et al.*]{} \[CDF Collaboration\], Eur. Phys. J. C [**62**]{}, 319 (2009) \[arXiv:0903.2060 \[hep-ex\]\].
P. Artoisenet, J. M. Campbell, J. P. Lansberg, F. Maltoni and F. Tramontano, Phys. Rev. Lett. [**101**]{}, 152001 (2008) \[arXiv:0806.3282 \[hep-ph\]\]. E. L. Berger, J. w. Qiu and Y. l. Wang, Phys. Rev. D [**71**]{}, 034007 (2005) \[arXiv:hep-ph/0404158\]. M. Spira, Nucl. Instrum. Meth. A [**389**]{}, 357 (1997) \[arXiv:hep-ph/9610350\]. arXiv:hep-ph/9510347. F. Abe [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett. [**75**]{}, 4358 (1995). D. D. Price, arXiv:0808.3367 \[hep-ex\].
Based on result obtained by Yi Yang under the direction of H. Evans.
D. Price, private communication.
D. Bortoletto and Z. Gecse, private communication.
[^1]: A convenient program for exploring the NMSSM Higgs sector is NMHDECAY [@Ellwanger:2004xm; @Ellwanger:2005dv].
[^2]: Scenarios with $\tanb$ much below $1.7$ are problematical since it is difficult to retain perturbativity for Yukawa couplings all the way up to the unification scale.
[^3]: In this arXiv version of this paper, we have access to final results and so we have updated this paragraph relative to the published version of this paper.
|
---
abstract: |
The fractional Poisson process has recently attracted experts from several fields of study. Its natural generalization of the ordinary Poisson process made the model more appealing for real-world applications. In this paper, we generalized the standard and fractional Poisson processes through the waiting time distribution, and showed their relations to an integral operator with a generalized Mittag–Leffler function in the kernel. The waiting times of the proposed renewal processes have the generalized Mittag–Leffler and stretched-squashed Mittag–Leffler distributions. Note that the generalizations naturally provide greater flexibility in modeling real-life renewal processes. Algorithms to simulate sample paths and to estimate the model parameters are derived. Note also that these procedures are necessary to make these models more usable in practice. State probabilities and other qualitative or quantitative features of the models are also discussed.
*Keywords:* Fractional Poisson process, generalized Mittag–Leffler distribution, renewal processes, Prabhakar operator.
author:
- |
$\text{Dexter O. Cahoy}_1$, $\text{Federico Polito}_2$\
(1) – Department of Mathematics and Statistics\
College of Engineering and Science, Louisiana Tech University, USA\
Tel: +1 318 257 3529, fax: +1 318 257 2182\
Email address: dcahoy@latech.edu\
(2) – Department of Mathematics, University of Torino, Italy\
Tel: +39 011 6702937, fax: +39 011 6702878\
Email address: federico.polito@unito.it\
(Corresponding author)
bibliography:
- 'cahpol6.bib'
nocite: '[@*]'
title: 'Renewal processes based on generalized Mittag–Leffler waiting times'
---
Introduction
============
The fractional Poisson process [@laskin; @laskin2; @sib; @beg; @beg2; @cuw; @mainardi; @mainardi2; @sibatov; @meer; @scalas; @scalas2; @scalas3] gained popularity in many areas of research as it naturally generalizes the standard or classical Poisson process. Recall that the inter-event time density function of the fractional Poisson process $N^\nu(t)$, $t \ge 0$, $\nu \in (0,1]$, was originally derived in @ras00 (known to date) and has the following integral form: $$\label{2e6}
f^\nu(t)=\frac{1}{t}\int\limits_0^\infty
e^{-x}\phi_\nu(\lambda t/x) \, \mathrm dx, \qquad \nu \in (0,1], \: t > 0, \lambda > 0,$$ where $$\begin{aligned}
\phi_\nu(\xi)=\frac{\sin(\nu\pi)}{\pi[\xi^\nu+\xi^{-\nu}+2\cos(\nu\pi)]}.
\end{aligned}$$ The preceding density function suggests that the tail distribution of the waiting time is of the form $$\label{2e5}
\Pr \, (T^\nu>t)= E_\nu(-\lambda t^\nu),$$ where $$\label{MLfunction}
E_\beta(z)=\sum\limits_{n=0}^\infty\frac{z^n}{\Gamma(\beta n+1)},
\qquad \: z \in \mathbb{C}, \: \beta \in \mathbb{C}, \Re (\beta)> 0,$$ is the Mittag–Leffler function. Note that the Mittag–Leffler density has been widely used to describe distributions appearing in anomalous diffusion, finance and economics, transport of charge carriers in semiconductors, and light propagation through random media (see, e.g., [@uz99; @psw05]). In view of equations and , the interarrival time density for the fractional Poisson process directly follows as $$\label{2e16}
f^\nu(t)=\lambda t^{\nu-1} E_{\nu,\,\nu}(-\lambda t^{\nu}), \qquad t > 0,$$ where $$\begin{aligned}
E_{\beta,\gamma}(z) = \sum_{r=0}^{\infty}\frac{z^{r}}{\Gamma
(\beta r + \gamma)}, \qquad z \in \mathbb{C}, \: \beta,\gamma \in \mathbb{C}, \: \Re(\beta)> 0
\end{aligned}$$ is the two-parameter Mittag–Leffler function. The $q$th fractional moment [@cap10] of the random interarrival time is $$\mathbb{E}\left[T^\nu\right]^q =\frac{\pi \Gamma (1 + q)}{\lambda^q \Gamma ( q / \nu )\sin ( \pi q/ \nu)
\Gamma (1- q )}, \qquad 0< q < \nu.$$ In addition, the above information automatically gives the probability density function $$f_m^\nu(t)= \lambda^m \frac{ t^{\nu m-1}}{(m-1)!}E_{\nu,\nu}^{(m-1)}\big(- \lambda t^{\nu} \big),
\label{2e20}$$ of the $m$-th arrival time because its Laplace transform, $$\begin{aligned}
\textsf{L} \big\lbrace f_m^\nu(t) \big\rbrace (s) =
\int_0^\infty e^{-s t} f_m^\nu(t) \, \mathrm dt = \frac{\lambda^m}{(\lambda + s^\nu)^{m}},
\end{aligned}$$ where $ E_{\nu,\nu}^{(k)} (-\lambda t^\nu)$ is the $k$th derivative of $E_{\nu,\nu} (z)$ evaluated at $z = -\lambda t^\nu$. As $\nu \to 1$, the above distribution converges to the classical Erlang distribution.
In another approach to the study the fractional Poisson process, @laskin used the fractional Kolmogorov–Feller-type differential equation system to characterize the one-dimensional state probability distributions as (see @laskin [formula (25)] and @beg2 [formula (2.5)]) $$\begin{aligned}
\label{2e14}
p_k^\nu(t) = \Pr \{ N^\nu(t) = k \} = \frac{(\lambda
t^\nu)^{k}}{k!}\sum_{r=0}^{\infty}\frac{(r+k)!}{r!}\frac{(-\lambda
t^{\nu})^r}{\Gamma (\nu (r+k) +1)}, \qquad
k \ge 0, \: t \ge 0.
\end{aligned}$$ One can also show [@laskin] that the moment generating function (MGF) of the fractional Poisson process is $$\begin{aligned}
M_\nu(s,t) & = E_\nu \left[ \lambda(e^{-s}-1)t^\nu \right]
= \sum_{r=0}^{\infty}\frac{\left[\lambda t^\nu\left(e^{-s}-1 \right)\right]^r}{\Gamma ( \nu r+1)},
\end{aligned}$$ which permits calculation (see Table \[t1\]) of the moments. A summary of the characteristics of the classical and fractional Poisson processes is shown in Table \[t1\] below.
\[t1\]
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Poisson process $(\nu=1)$ Fractional Poisson Process $(\nu < 1)$
---------------- ------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------
$\Pr(T^\nu>t)$ $e^{-\lambda t}$ $E_{\nu}(-\lambda t^{\nu} )$
$f^\nu (t)$ $ \lambda e^{-\lambda t} $ $\lambda t^{\nu-1} E_{\nu,\,\nu}(-\lambda t^{\nu})$
$p_k^\nu(t)$ $\frac{(\lambda t)^{k}}{k!}e^{-\lambda t}$ $\frac{(\lambda t^\nu)^{k}}
{k!}\sum_{r=0}^{\infty}\frac{(r+k)!}{r!}\frac{(-\lambda t^{\nu})^r}{\Gamma (\nu (r+k) +1)}$
Mean $\lambda t$ $\lambda t^{\nu}/\Gamma (\nu + 1)$
Variance $\lambda t$ $\frac{\lambda t^\nu}{\Gamma(\nu+1)} + (\lambda t^\nu)^2
\left[ \frac{1}{\nu\Gamma(2\nu)} - \frac{1}{\Gamma^2(\nu+1)} \right]$
$k$th moment $ (-1)^k \frac{\partial^k}{\partial s^k} \exp\left[ \lambda(e^{-s}-1)t\right]\big|_{s=0}$ $
\left( -1\right)^k\frac{\partial^k}{\partial s^k} E_\nu\left[\lambda (e^{-s}-1)t^\nu\right]\big|_{s=0}$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: *Properties of fractional Poisson process compared with those of the standard Poisson process.*
In this paper, we generalize the standard and fractional Poisson processes through their waiting time distributions. In particular, we propose two renewal processes that have waiting times that are generalized Mittag–Leffler and stretched-squashed Mittag–Leffler distributed. These generalizations naturally provide more flexibility in capturing real-world renewal processes. Algorithms to simulate sample paths and estimate the model parameters are derived and tested. State probabilities and other qualitative or quantitative features of the models are also discussed.
The rest of the paper is organized as follows. In Section \[00sec\], a renewal process with generalized Mittag–Leffler distributed waiting times is presented. Procedures to generate sample paths and to estimate parameters are also derived. In Section \[00anothersec\], another generalization based on stretching and squashing the Mittag–Leffler distributed inter-event times is developed. Methods to simulate sample trajectories and to estimate parameters are also showcased. More discussions are provided in Section \[summary\]. Finally, computational test results are shown in the appendix.
Generalization I {#00sec}
================
We consider the generalized Mittag–Leffler distribution (see e.g. @pil) built from the generalized Mittag–Leffler function [@saigo; @prab]. Let $T^{\nu,\delta}$ be a generalized Mittag–Leffler distributed random variable. Then the probability density function is $$\begin{aligned}
\label{00fu}
f^{\nu,\delta}(t) = \lambda^\delta t^{\delta \nu -1} E_{\nu, \delta \nu}^\delta (-\lambda t^\nu),
\qquad t>0, \: \lambda > 0, \: \nu \in (0, 1], \: \delta \in \mathbb{R},
\end{aligned}$$ where $$\begin{aligned}
E_{\beta,\gamma}^\xi(z) = \sum_{r=0}^\infty \frac{(\xi)_r}{r!\Gamma(\beta r+\gamma)} z^r,
\quad \beta,\gamma,\xi, z \in \mathbb{C}, \: \Re(\beta)>0
\end{aligned}$$ is the generalized Mittag–Leffler function (see Figure \[f1\]). The Pochhammer symbol $(\xi)_r$ can be written also as $(\xi)_r = \xi(\xi+1)\dots (\xi+r-1)$, $\xi \ne 0$. When $\delta \nu < 1$ the function has an asymptote at $t=0$, while in the particular case $\delta \nu = 1$ $$\begin{aligned}
\left. f^{\nu,\delta}(t) \right|_{t=0} = \left. \lambda^{1/\nu}
E_{\nu,1}^{1/\nu}(-\lambda t^\nu) \right|_{t=0} = \lambda^{1/\nu}.
\end{aligned}$$ The Laplace transform of reads $$\begin{aligned}
\textsf{L} \big\lbrace \lambda^\delta t^{\delta\nu-1} E_{\nu,\delta\nu}^\delta (-\lambda t^\nu) \big\rbrace
(s)= \frac{\lambda^\delta}{(s^\nu+\lambda)^\delta}
\end{aligned}$$ (see @mathai, formula (2.3.24), page 95). Below are the plots of the generalized Mittag–Leffler densities.
![\[f1\]The generalized Mittag–Leffler density plots (see ) for parameter values $( \delta, \lambda )=( 0.5, 1)$ (top) and $(\delta, \lambda )=(2, 1)$ (bottom) where $\nu$ goes from 0.2 to 1 with step size of 0.2.](plot1a.pdf "fig:") ![\[f1\]The generalized Mittag–Leffler density plots (see ) for parameter values $( \delta, \lambda )=( 0.5, 1)$ (top) and $(\delta, \lambda )=(2, 1)$ (bottom) where $\nu$ goes from 0.2 to 1 with step size of 0.2.](plot1b.pdf "fig:")
Now, consider $m$ i.i.d. random waiting times $\mathcal{T}_i$, $i=1,\dots, m$ of a renewal point process, here denoted as $N^{\nu,\delta}(t)$, $t \ge 0$, which are distributed as in . Furthermore, denote $T_m^{\nu,\delta} = \mathcal{T}_1+\dots +\mathcal{T}_m$ as the waiting time of the $m$th renewal event and $T_0^{\nu,\delta} = 0$. Then $$\begin{aligned}
\mathbb{E} e^{-s T_m^{\nu,\delta}} = \left( \mathbb{E}e^{-s\mathcal{T}_i} \right)^m
= \frac{\lambda^{\delta m}}{(s^\nu+\lambda)^{\delta m}}.
\end{aligned}$$ By formula (2.3.24) of @mathai we have $$\begin{aligned}
\Pr \{ T_m^{\nu,\delta} \in \mathrm dt \} / \mathrm dt = \lambda^{\delta m} t^{\nu \delta m-1}
E_{\nu,\nu\delta m}^{\delta m} (-\lambda t^\nu), \qquad t > 0, \: \lambda > 0.
\end{aligned}$$ It is rather immediate now to obtain the state probabilities $p_k^{\nu,\delta}(t) = \Pr \{ N^{\nu,\delta}(t) = k \}$, $k \ge 0$ because $$\begin{aligned}
\label{00state}
& \int_0^\infty e^{-st} \Pr \{ N^{\nu,\delta}(t) = k \} \, \mathrm dt \\
& = \int_0^\infty e^{-st} \left( \Pr\{ T_k^{\nu,\delta}<t \} -
\Pr \{ T_{k+1}^{\nu,\delta}<t \} \right) \mathrm dt \notag \\
& = \int_0^\infty e^{-st} \left\{ \int_0^t \Pr\{ T_k^{\nu,\delta} \in \mathrm dy \}
- \int_0^t \Pr\{ T_{k+1}^{\nu,\delta} \in \mathrm dy \} \right\} \, \mathrm dt \notag \\
& = \int_0^\infty \Pr\{ T_k^{\nu,\delta} \in \mathrm dy \} \int_y^\infty e^{-st} \mathrm dt
- \int_0^\infty \Pr\{ T_{k+1}^{\nu,\delta} \in \mathrm dy \} \int_y^\infty e^{-st} \mathrm dt \notag \\
& = s^{-1} \left[ \int_0^\infty e^{-sy} \Pr \{ T_k^{\nu,\delta} \in \mathrm dy \} - \int_0^\infty e^{-sy} \Pr
\{ T_{k+1}^{\nu,\delta} \in \mathrm dy \} \right] \notag \\
& = s^{-1} \left( \frac{\lambda^{\delta k}}{(s^\nu+\lambda)^{\delta k}}
-\frac{\lambda^{\delta(k+1)}}{(s^\nu+\lambda)^{\delta(k+1)}} \right) \notag, \qquad k \ge 0.
\end{aligned}$$ Inverting the preceding Laplace transform we readily arrive at $$\begin{aligned}
\label{001}
p_k^{\nu,\delta}(t) =
\lambda^{\delta k} t^{\nu\delta k} E_{\nu,\nu\delta k+1}^{\delta k} (-\lambda t^\nu)
-\lambda^{\delta (k+1)} t^{\nu\delta(k+1)} E_{\nu,\nu\delta(k+1)
+1}^{\delta(k+1)} (-\lambda t^\nu), \qquad k \ge 0.
\end{aligned}$$ When $(0)_r=0$, $r \in \mathbb{N}$, $(0)_0=1$, and $k=0$, equation becomes $$\begin{aligned}
p_0^{\nu,\delta}(t) = 1-\lambda^\delta t^{\nu\delta} E_{\nu,\nu\delta+1}^\delta (-\lambda t^\nu).
\end{aligned}$$ Clearly, $$\begin{aligned}
\label{40p}
p_k^{\nu,\delta}(0) = \Pr \{ N^{\nu,\delta}(0) = k\} =
\begin{cases}
1, & k = 0, \\
0, & k \ge 1.
\end{cases}
\end{aligned}$$
The state probabilities $p_k^{\nu,\delta}(t) = \Pr \{ N^{\nu,\delta}(t) = k \}$, $k \ge 1$, satisfy the convolution-type Volterra equation of the first kind $$\begin{aligned}
\label{00hat}
p_k^{\nu,\delta}(t) = \lambda^\delta \int_0^t (t-w)^{\nu\delta -1}
E_{\nu,\nu\delta}^\delta \left(-\lambda(t-w)^\nu \right)
p_{k-1}^{\nu,\delta}(w) \, \mathrm dw.
\end{aligned}$$
We start by rewriting by means of the following well-known relation (see e.g. @sax, formula (11.7), page 17): $$\begin{aligned}
\label{01for}
\int_0^x (x-t)^{\beta-1} E_{\alpha,\beta}^\gamma [a(x-t)^\alpha] t^{\nu-1}
E_{\alpha,\nu}^\sigma (at^\alpha) \mathrm dt = x^{\beta + \nu-1} E_{\alpha, \beta+\nu}^{\gamma+\sigma}
(ax^\alpha),
\end{aligned}$$ where $\alpha,\beta,\gamma,a,\nu,\sigma \in \mathbb{C}$, and $\Re(\alpha)>0$, $\Re(\beta)>0$, $\Re(\gamma)>0$, $\Re(\nu)>0$, $\Re(\sigma)>0$. Then $$\begin{aligned}
p_k^{\nu,\delta} (t)
= {} & \lambda^{\delta k} t^{\nu\delta k} E_{\nu,\nu\delta k +1}^{\delta k}(-\lambda t^\nu)
- \lambda^{\delta(k+1)} t^{\nu\delta(k+1)} E_{\nu,\nu\delta(k+1)+1}^{\delta(k+1)}(-\lambda t^\nu) \\
= {} & \lambda^{\delta k} \int_0^t (t-s)^{\nu\delta (k-1)} E_{\nu,\nu\delta(k-1)+1}^{\delta(k-1)}
\left[ -\lambda (t-s)^\nu \right] s^{\nu\delta -1} E_{\nu,\nu\delta}^\delta (-\lambda s^\nu)
\, \mathrm ds
\notag \\
& - \lambda^{\delta(k+1)} \int_0^t (t-s)^{\nu\delta k} E_{\nu,\nu\delta k +1}^{\delta k}
\left[ -\lambda(t-s)^\nu \right] s^{\nu\delta -1}
E_{\nu,\nu\delta}^\delta (-\lambda s^\nu) \, \mathrm ds \notag \\
& \!\!\!\!\!\!\!\!\!\!\!\!\!\! \overset{w=t-s}{=} \lambda^\delta \int_0^t (t-w)^{\nu\delta -1}
E_{\nu,\nu\delta}^\delta \left[ -\lambda(t-w)^\nu \right]
\lambda^{\delta(k-1)} w^{\nu\delta(k-1)} E_{\nu,\nu\delta(k-1)+1}^{\delta(k-1)}
(-\lambda w^\nu) \, \mathrm dw \notag \\
& - \lambda^\delta \int_0^t (t-w)^{\nu\delta -1} E_{\nu,\nu\delta}^\delta \left[
-\lambda(t-w)^\nu \right]
\lambda^{\delta k} w^{\nu\delta k} E_{\nu,\nu\delta k+1}^{\delta k} (-\lambda w^\nu)
\, \mathrm dw \notag \\
= {} & \lambda^\delta \int_0^t (t-w)^{\nu\delta -1}
E_{\nu,\nu\delta}^\delta \left[ -\lambda (t-w)^\nu \right] \notag \\
& \times \left[ \lambda^{\delta(k-1)} w^{\nu\delta(k-1)} E_{\nu,\nu\delta(k-1)+1}^{\delta(k-1)}
- \lambda^{\delta k} w^{\nu\delta k} E_{\nu,\nu\delta k +1}^{\delta k}
(-\lambda w^\nu) \right] \mathrm dw \notag \\
= {} & \lambda^\delta \int_0^t (t-w)^{\nu\delta -1}
E_{\nu,\nu\delta}^\delta \left[ (-\lambda(t-w)^\nu) \right]
p_{k-1}^{\nu,\delta}(w) \, \mathrm dw. \notag
\end{aligned}$$
Result can also be conveniently expressed by means of the Prabhakar operator [@prab], defined as $$\begin{aligned}
\left( \bm{\mathrm E}^\gamma_{\rho,\mu,\omega;a+} \phi \right)(x) =
\int_a^x (x-y)^{\mu-1} E_{\rho,\mu}^\gamma
\left(\omega(x-y)^\rho\right) \phi(y) \, \mathrm dy, \qquad x>a, \: \rho,\mu,\gamma \in \mathbb{C}, \:
\Re (\rho), \Re(\mu) >0,
\end{aligned}$$ which is a generalization of the Riemann–Liouville fractional integral of $\phi (x)$. Therefore, we obtain $$\begin{aligned}
\label{arancione}
p_k^{\nu,\delta}(t)
= \lambda^\delta \left( \bm{\mathrm E}_{\nu,\nu\delta,-\lambda;0+}^\delta \,
p_{k-1}^{\nu,\delta} \right) (t).
\end{aligned}$$
When $\delta = 1$ equation clearly reduces to $$\begin{aligned}
p_k^\nu(t) = \lambda \int_0^t (t-w)^{\nu-1} E_{\nu,\nu}(-\lambda(t-w)^\nu)
p_{k-1}^\nu(w) \, \mathrm dw.
\end{aligned}$$ We now check that the state probabilities $p_k^\nu(t)$ of the fractional Poisson process $N^\nu(t)$, $t \ge 0$ (see e.g. @beg2) satisfy the above integral equation. By recalling that $$\begin{aligned}
p_k^\nu(t) = (\lambda t^\nu)^k E_{\nu,\nu k+1}^{k+1} (-\lambda t^\nu), \qquad k \ge 0, \:
t \ge 0,
\end{aligned}$$ we can write $$\begin{aligned}
p_k^\nu(t) = {} & \lambda \int_0^t (t-w)^{\nu-1} E_{\nu,\nu}(-\lambda(t-w)^\nu)
\, p_{k-1}^\nu \, \mathrm dw \\
= {} & \lambda \int_0^t (t-w)^{\nu-1} E_{\nu,\nu}(-\lambda(t-w)^\nu)
\, (\lambda w)^{k-1} E_{\nu,\nu(k-1)+1}(-\lambda w^\nu) \, \mathrm dw \notag \\
& \hspace{-.76cm} \overset{\text{by \eqref{01for}}}{=}
(\lambda t^\nu)^k E_{\nu,\nu k+1}^{k+1} (-\lambda t^\nu) \notag.
\end{aligned}$$
Notice also that for $\nu=\delta=1$ (classical case), equation reduces to $$\begin{aligned}
\label{00eq}
p_k(t) = \lambda \int_0^t e^{-\lambda(t-w)}
p_{k-1}(w) \, \mathrm dw,
\end{aligned}$$ where $p_k(t)$, $k \ge 1$, $t \ge 0$, are the state probabilities of a homogeneous Poisson process $N(t)$, $t \ge 0$. Equation is easily solvable and the solution reads $$\begin{aligned}
p_{k-1}(t) = \lambda^{-1} \frac{\mathrm d}{\mathrm dt} p_k(t) +
p_k(t), \qquad k \ge 1.
\end{aligned}$$ Finally we note that the above equation is clearly the difference-differential equation governing the state probabilities of a homogeneous Poisson process.
\[teio1\] The state probabilities $p_k^{\nu,\delta}(t) = \text{Pr} \{ N^{\nu,\delta}(t) = k \}$, $k \ge 0$, $t \ge 0$, satisfy the equations $$\begin{aligned}
\label{arancione1}
\frac{\mathrm d^{\nu \delta +\theta}}{\mathrm dt^{\nu\delta+\theta}}
\left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta} \, p_k^{\nu,\delta} \right) (t)
= \lambda^\delta p_{k-1}^{\nu,\delta}(t)
+\delta_{k,0} \left( t^{-\nu\delta} E_{\nu,1-\nu\delta}^{-\delta}(-\lambda t^\nu) -\lambda^\delta \right),
\end{aligned}$$ for any $\theta \in \mathbb{C}$, $\Re (\theta)>0$, where $\delta_{k,0}$ is the Kronecker’s delta and where the operator $\frac{\mathrm d^{\nu \delta +\theta}}{\mathrm dt^{\nu\delta+\theta}}$ is the Riemann–Liouville fractional derivative of order $\nu\delta+\theta$.
We start by considering $k\ge 1$. Applying the the operator $\bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta}$ to equation , we obtain $$\begin{aligned}
& \left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta} \, p_k^{\nu,\delta}
\right) (t) = \lambda^\delta
\left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta}
\left( \bm{\mathrm E}_{\nu,\nu\delta,-\lambda;0+}^\delta \, p_{k-1}^{\nu,\delta}
\right) \right) (t) \\
& \Leftrightarrow \quad
\left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta} \, p_k^{\nu,\delta} \right) (t) =
\lambda^\delta J^{\nu\delta+\theta}_{t,0+} p_{k-1}^{\nu,\delta}(t), \notag
\end{aligned}$$ where $J^{\nu\delta+\theta}_{t,0+}$ is the Riemann–Liouville fractional integral operator. By recalling that the Riemann–Liouville fractional derivative is the left inverse operator of the Riemann–Liouville fractional integral (see e.g. @Diethelm, Theorem 2.14, page 30), we readily arrive at the claimed result. For $k=0$, it is sufficient to show that $$\begin{aligned}
\vspace{-1cm} \frac{\mathrm d^{\nu \delta +\theta}}{\mathrm dt^{\nu\delta+\theta}} &
\left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta} \, p_0^{\nu,\delta} \right) (t) \\
= {} & \frac{\mathrm d^{\nu \delta +\theta}}{\mathrm dt^{\nu\delta+\theta}}
\left( \int_0^t (t-y)^{\theta-1} E_{\nu,\theta}^{-\delta} \left[ -\lambda(t-y)^\nu \right]
p_0^{\nu,\delta}(y) \, \mathrm dy \right) \notag \\
= {} & \frac{\mathrm d^{\nu \delta +\theta}}{\mathrm dt^{\nu\delta+\theta}}
\left( \int_0^t (t-y)^{\theta-1} E_{\nu,\theta}^{-\delta} \left[ -\lambda(t-y)^\nu \right]
\, \mathrm dy \right. \notag \\
& \left. - \lambda^\delta \int_0^t (t-y)^{\theta-1} E_{\nu,\theta}^{-\delta}
\left[ -\lambda (t-y)^\nu \right] y^{\nu\delta} E_{\nu,\nu\delta+1}^\delta (-\lambda y^\nu)
\, \mathrm dy \right) \notag \\
= {} & \frac{\mathrm d^{\nu \delta +\theta}}{\mathrm dt^{\nu\delta+\theta}}
\left( t^\theta E_{\nu,\theta+1}^{-\delta} (-\lambda t^\nu)
-\lambda^\delta t^{\nu\delta+\theta} E_{\nu,\nu\delta+\theta+1}^0 (-\lambda t^\nu) \right) \notag \\
= {} & \frac{\mathrm d^{\nu \delta +\theta}}{\mathrm dt^{\nu\delta+\theta}}
\left( t^\theta E_{\nu,\theta+1}^{-\delta} (-\lambda t^\nu)
-\lambda^\delta \frac{t^{\nu\delta+\theta}}{\Gamma(\nu\delta +\theta+1)} \right) \notag \\
= {} & t^{-\nu\delta} E_{\nu,1-\nu\delta}^{-\delta} (-\lambda t^\nu) -\lambda^\delta. \notag
\end{aligned}$$
For more information on the inverse operator appearing in , the reader can consult @saigo, Section 6.
When $\delta=1$ and $k\ge 1$, equation can be written as $$\begin{aligned}
p_{k-1}^\nu (t) & = \lambda^{-1} \frac{\mathrm d^{\nu+\theta}}{\mathrm d t^{\nu+\theta}}
\int_0^t (t-w)^{\theta-1} E_{\nu,\theta}^{-1} (-\lambda(t-w)^\nu) p_k^\nu(t) \, \mathrm dw \\
& = \lambda^{-1} \frac{\mathrm d^{\nu+\theta}}{\mathrm d t^{\nu+\theta}}
\int_0^t (t-w)^{\theta-1} \left[ \frac{1}{\Gamma(\theta)} + \lambda(t-w)^\nu \frac{1}{\Gamma(\nu+\theta)}
\right] p_k^\nu(w) \, \mathrm dw \notag \\
& = \lambda^{-1} \frac{\mathrm d^\nu}{\mathrm dt^\nu} \frac{\mathrm d^\theta}{\mathrm dt^\theta}
\frac{1}{\Gamma(\theta)} \int_0^t (t-w)^{\theta-1} p_k^\nu(w) \, \mathrm dw
+ \frac{\mathrm d^{\nu+\theta}}{\mathrm dt^{\nu+\theta}} \frac{1}{\Gamma(\nu+\theta)} \int_0^t
(t-w)^{\nu+\theta-1} p_k^\nu(w) \, \mathrm dw \notag,
\end{aligned}$$ while, for $k=0$, and considering that $$\begin{aligned}
t^{-\nu} E_{\nu,1-\nu}^{-1} (-\lambda t^\nu) -\lambda & = t^{-\nu} \left( \frac{1}{\Gamma(1-\nu)}
+\lambda t^\nu \right) -\lambda
= \frac{t^{-\nu}}{\Gamma(1-\nu)},
\end{aligned}$$ we have $$\begin{aligned}
\frac{t^{-\nu}}{\Gamma(1-\nu)} & = \frac{\mathrm d^{\nu+\theta}}{\mathrm d t^{\nu+\theta}}
\int_0^t (t-w)^{\theta-1} E_{\nu,\theta}^{-1} (-\lambda(t-w)^\nu) \, p_0^\nu(t) \, \mathrm dw \\
& = \frac{\mathrm d^{\nu+\theta}}{\mathrm d t^{\nu+\theta}}
\int_0^t (t-w)^{\theta-1} \left[ \frac{1}{\Gamma(\theta)} + \lambda(t-w)^\nu \frac{1}{\Gamma(\nu+\theta)}
\right] p_0^\nu(w) \, \mathrm dw \notag \\
& = \frac{\mathrm d^\nu}{\mathrm dt^\nu} \frac{\mathrm d^\theta}{\mathrm dt^\theta}
\frac{1}{\Gamma(\theta)} \int_0^t (t-w)^{\theta-1} p_0^\nu(w) \, \mathrm dw
+ \lambda \frac{\mathrm d^{\nu+\theta}}{\mathrm dt^{\nu+\theta}} \frac{1}{\Gamma(\nu+\theta)} \int_0^t
(t-w)^{\nu+\theta-1} p_0^\nu(w) \, \mathrm dw \notag.
\end{aligned}$$ Hence, we retrieve the fractional difference-differential equations governing the state probabilities of a fractional Poisson process [@laskin]: $$\begin{aligned}
\label{pejo}
\frac{\mathrm d^\nu}{\mathrm dt^\nu} p_k^\nu(t)
= -\lambda p_k^\nu(t) + \lambda p_{k-1}^\nu(t) + \delta_{k,0}
\frac{t^{-\nu}}{\Gamma(1-\nu)}, \qquad k \ge 0.
\end{aligned}$$
From equation we can easily arrive at the following partial differential equation for the probability generating function $\mathcal{G}_{\nu,\delta}(u,t) = \sum_{k=0}^\infty u^k
p_k^{\nu,\delta}(t)$. $$\begin{aligned}
\label{12gen}
\frac{\partial^{\nu\delta+\theta}}{\partial t^{\nu\delta+\theta}}
\left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta} \mathcal{G}_{\nu,\delta}(u,\cdot) \right) (t)
= \lambda^\delta u \, \mathcal{G}_{\nu,\delta}(u,t)
+ t^{-\nu\delta} E_{\nu,1-\nu\delta}^{-\delta} (-\lambda t^\nu) -\lambda^\delta.
\end{aligned}$$ From the above equation and by recalling the formula $\frac{\partial}{\partial u} \mathcal{G}_{\nu,\delta}(u,t)|_{u=1} = \mathbb{E} \,
N^{\nu,\delta}(t)$, it is now immediate to derive the differential equation involving the mean value as $$\begin{aligned}
\label{12mean}
\frac{\mathrm d^{\nu\delta+\theta}}{\mathrm d t^{\nu \delta + \theta}}
\left( \bm{\mathrm E}_{\nu,\delta,-\lambda;0+}^{-\delta} \mathbb{E}\, N^{\nu,\delta}(\cdot) \right) (t)
= \lambda^\delta \left( 1+\mathbb{E} \, N^{\nu,\delta}(t)\right).
\end{aligned}$$ Observe that equations and reduce to the corresponding equations in the pure fractional case when $\delta=1$ (see @laskin [formula (22)] for the differential equation involving the probability generating function). For the fractional Poisson process $N^\nu(t)$, $t \ge 0$ ($\delta= 1$), equation becomes $$\begin{aligned}
\label{30water}
\frac{\mathrm d^{\nu+\theta}}{\mathrm d t^{\nu+\theta}} & \left( \bm{\mathrm{E}}_{\nu,\theta,-\lambda;0+}^{-1}
\mathbb{E} \, N^\nu(\cdot) \right)(t) = \lambda + \lambda \mathbb{E} \, N^\nu(t) \\
\Leftrightarrow {} \qquad & \frac{\mathrm d^{\nu+\theta}}{\mathrm d t^{\nu+\theta}}
\int_0^t (t-y)^{\theta-1} E_{\nu,\theta}^{-1} [-\lambda (t-y)^\nu] \, \mathbb{E} \, N^\nu(t) \, \mathrm dy
= \lambda + \lambda \mathbb{E} \, N^\nu(t) \notag \\
\Leftrightarrow {} \qquad & \frac{\mathrm d^\nu}{\mathrm d t^\nu} \frac{\mathrm d^\theta}{\mathrm dt^\theta}
\frac{1}{\Gamma(\theta)} \int_0^t (t-y)^{\theta-1} \mathbb{E} \, N^\nu(y) \, \mathrm dy \notag \\
& + \lambda \frac{\mathrm d^{\nu+\theta}}{\mathrm dt^{\nu+\theta}} \frac{1}{\Gamma(\nu+\theta)}
\int_0^t (t-y)^{\nu+\theta-1} \mathbb{E}\, N^\nu(y)\, \mathrm dy
= \lambda + \lambda \mathbb{E} \, N^\nu(t) \notag \\
\Leftrightarrow {} \qquad & \frac{\mathrm d^\nu}{\mathrm dt^\nu} \mathbb{E} \, N^\nu(t)
= \lambda, \notag
\end{aligned}$$ with $\mathbb{E} \, N^\nu(0) = 0$ and considering that the second step is justified by the semigroup property of the Riemann–Liouville fractional derivative (see @Diethelm [Theorem 2.2, page 14]). The solution to is well-known and reads [@laskin formula (26)] $$\begin{aligned}
\label{meanpure}
\mathbb{E}\, N^\nu(t) = \frac{\lambda t^\nu}{\Gamma(\nu+1)}, \qquad \nu \in (0,1] \: t \ge 0.
\end{aligned}$$
The following theorem derives the mean value of the process $N^{\nu,\delta}(t)$, $t \ge 0$.
Let $\nu \in (0,1]$, $\delta \in \mathbb{C}$, $\theta \in \mathbb{C}$, $\Re (\theta)>0$. The solution to $$\begin{aligned}
\label{30mean}
\begin{cases}
\frac{\mathrm d^{\nu\delta+\theta}}{\mathrm d t^{\nu \delta + \theta}}
\left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta} \mathbb{E}\, N^{\nu,\delta}
(\cdot) \right) (t)
= \lambda^\delta \left( 1+\mathbb{E} \, N^{\nu,\delta}(t)\right), \\
\left[ \frac{\mathrm d^{\nu\delta +\theta -k-1}}{\mathrm dt^{\nu\delta +\theta -k-1}}
\left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta}
\mathbb{E} \, N^{\nu,\delta} (\cdot) \right) (t) \right]_{t \to 0} =0,
\qquad \forall k= 0, \dots, n-1,
\: n-1 \le \Re (\nu\delta+\theta) < n,
\end{cases}
\end{aligned}$$ reads $$\begin{aligned}
\label{30meanres}
\mathbb{E} \, N^{\nu,\delta}(t) = \sum_{r=0}^\infty \lambda^{\delta(r+1)} t^{\nu\delta (r+1)}
E_{\nu,\nu\delta(r+1)+1}^{\delta(r+1)}(-\lambda t^\nu).
\end{aligned}$$
We start by taking the Laplace transform of , obtaining $$\begin{aligned}
\label{30mouse}
& \int_0^\infty e^{-st}
\frac{\mathrm d^{\nu\delta+\theta}}{\mathrm d t^{\nu \delta + \theta}}
\left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-\delta} \mathbb{E}\, N^{\nu,\delta}
(\cdot) \right) (t)
\, \mathrm dt = \frac{\lambda^\delta}{s} + \lambda^\delta \int_0^\infty e^{-st} \mathbb{E} \,
N^{\nu,\delta}(t) \, \mathrm dt \\
& \Leftrightarrow \quad s^{\nu\delta+\theta} \int_0^\infty e^{-st} \int_0^t (t-y)^{\theta-1}
E_{\nu,\theta}^{-\delta} \left[ -\lambda (t-y)^\nu \right]
\mathbb{E}\, N^{\nu,\delta} (y) \, \mathrm dy \, \mathrm dt
= \frac{\lambda^\delta}{s} + \lambda^\delta \int_0^\infty e^{-st} \mathbb{E} \,
N^{\nu,\delta}(t) \, \mathrm dt \notag \\
& \Leftrightarrow \quad s^{\nu\delta+\theta}
\int_0^\infty \mathbb{E}\, N^{\nu,\delta}(y) \, \mathrm dy
\int_{y}^\infty e^{-st} (t-y)^{\theta-1}
E_{\nu,\theta}^{-\delta} \left[ -\lambda (t-y)^\nu \right]\, \mathrm dt
= \frac{\lambda^\delta}{s} + \lambda^\delta \int_0^\infty e^{-st} \mathbb{E} \,
N^{\nu,\delta}(t) \, \mathrm dt \notag \\
& \Leftrightarrow \quad s^{\nu\delta+\theta}
\int_0^\infty \mathbb{E}\, N^{\nu,\delta}(y) \, \mathrm dy
\int_0^\infty e^{-s(z+y)} z^{\theta-1} E_{\nu,\theta}^{-\delta} (-\lambda z^\nu)\, \mathrm dz
= \frac{\lambda^\delta}{s} + \lambda^\delta \int_0^\infty e^{-st} \mathbb{E} \,
N^{\nu,\delta}(t) \, \mathrm dt \notag \\
& \Leftrightarrow \quad s^{\nu\delta+\theta}
\int_0^\infty e^{-sy} \mathbb{E}\, N^{\nu,\delta}(y) \, \mathrm dy
\int_0^\infty e^{-sz} z^{\theta-1} E_{\nu,\theta}^{-\delta} \, \mathrm dz
= \frac{\lambda^\delta}{s} + \lambda^\delta \int_0^\infty e^{-st} \mathbb{E} \,
N^{\nu,\delta}(t) \, \mathrm dt \notag \\
& \Leftrightarrow \quad s^{\nu\delta +\theta}
\textsf L \left\{ \mathbb{E}\, N^{\nu,\delta} (t) \right\} (s) s^{-\theta}
(1+\lambda s^{-\nu})^\delta = s^{-1} \lambda^\delta +\lambda^\delta
\textsf{L} \left\{ \mathbb{E}\, N^{\nu,\delta} (t) \right\} (s) \notag \\
& \Leftrightarrow \quad \textsf{L} \left\{ \mathbb{E}\, N^{\nu,\delta} (t) \right\} (s)
= \frac{\lambda^\delta}{s \left[ (s^\nu+\lambda)^\delta -\lambda^\delta \right]}. \notag
\end{aligned}$$ Notice that the first step in is justified by the formula for the Laplace transform of the Riemann–Liouville fractional derivative and by applying the initial conditions. Before inverting the Laplace transform, it can be shown that $$\begin{aligned}
\label{30acer}
\textsf{L} \left\{ \mathbb{E} \, N^{\nu,\delta} (t) \right\} (s)
& = \frac{\lambda^\delta}{s(s^\nu+\lambda)^\delta
\left( 1-\frac{\lambda^\delta}{(s^\nu+\lambda)^\delta} \right)} \\
& = \frac{\lambda^\delta}{s(s^\nu+\lambda)^\delta}
\sum_{r=0}^\infty \left[ \frac{\lambda^\delta}{(s^\nu+\lambda)^\delta} \right]^r
\notag \\
& = \frac{1}{s} \sum_{r=0}^\infty \frac{\lambda^{\delta(r+1)}}{(s^\nu+\lambda)^{\delta(r+1)}}.
\notag
\end{aligned}$$ Thus, the mean value is now easily found by inverting term by term: $$\begin{aligned}
\mathbb{E}\, N^{\nu,\delta} (t) & = \sum_{r=0}^\infty \int_0^t \lambda^{\delta(r+1)}
y^{\nu\delta(r+1)-1} E_{\nu,\nu\delta(r+1)}^{\delta(r+1)} (-\lambda y^\nu) \, \mathrm dy \\
& = \sum_{r=0}^\infty \lambda^{\delta(r+1)} t^{\nu\delta (r+1)}
E_{\nu,\nu\delta(r+1)+1}^{\delta(r+1)}(-\lambda t^\nu). \notag
\end{aligned}$$
For $\delta = 1$, the mean value reduces to that of the pure fractional case . Indeed, $$\begin{aligned}
\mathbb{E} \, N^\nu(t) & = \sum_{r=0}^\infty \lambda^{r+1} t^{\nu(r+1)} E_{\nu,\nu(r+1)+1}^{r+1}
(-\lambda t^\nu),
\end{aligned}$$ and passing now to the Laplace transform we get $$\begin{aligned}
\textsf{L} \left\{ \mathbb{E}\, N^\nu (t) \right\} (s)
= \sum_{r=0}^\infty \lambda^{r+1} s^{-\nu(r+1)} \left( 1+\lambda s^{-\nu} \right)^{-(r+1)}
= \lambda / s^\nu,
\end{aligned}$$ which immediately leads to . When $\delta=1$, reduces to $$\begin{aligned}
& \left[ \frac{\mathrm d^{\nu +\theta -k-1}}{\mathrm dt^{\nu +\theta -k-1}}
\left( \bm{\mathrm E}_{\nu,\theta,-\lambda;0+}^{-1}
\mathbb{E} \, N^\nu (\cdot) \right) (t) \right]_{t \to 0} = 0 \\
& \Leftrightarrow \quad \left[ \frac{\mathrm d^{\nu +\theta -k-1}}{\mathrm dt^{\nu +\theta -k-1}}
\int_0^t (t-y)^{\theta-1} E_{\nu,\theta}^{-\delta} [-\lambda(t-y)^\nu] \mathbb{E} \, N^\nu
(y) \, \mathrm dy \right]_{t \to 0} = 0 \notag \\
& \Leftrightarrow \quad \left[ \frac{\mathrm d^{\nu +\theta -k-1}}{\mathrm dt^{\nu +\theta -k-1}}
\int_0^t (t-y)^{\theta-1} \left[ \frac{1}{\Gamma(\theta)}
+\lambda (t-y)^\nu \frac{1}{\Gamma(1+\theta)} \right] \mathbb{E} \, N^\nu
(y) \, \mathrm dy \right]_{t \to 0} = 0 \notag \\
& \Leftrightarrow \quad\left[ \frac{\mathrm d^{\nu -k-1}}{\mathrm dt^{\nu -k-1}}
\mathbb{E} \, N^\nu(t) + \lambda \mathbb{E} \, N^\nu(t) \right]_{t \to 0} = 0, \notag
\end{aligned}$$ for each $k= 0, \dots, n-1$, $n-1 \le \Re (\nu+\theta) < n$. By recalling $\mathbb{E}\,
\mathcal{N}(t) = \sum_{r=0}^\infty r p_r^\nu(t)$ and equation for $\delta=1$ we obtain $$\begin{aligned}
\left[ \frac{\mathrm d^{\nu-1}}{\mathrm dt^{\nu-1}}
\mathbb{E} \, N^\nu(t) \right]_{t \to 0} = 0,
\end{aligned}$$ where we considered only $k=0$ and therefore only one initial condition is used.
Path simulation and parameter estimation
----------------------------------------
It is straightforward to generate a sample trajectory of generalization I by noting that the generalized Mittag–leffler random variable $T^{\nu,\delta}$ (see, e.g., @pil) is a mixture of gamma densities, i.e., $$T^{\nu,\delta} \stackrel{d}{=} U^{1/\nu} V_\nu,$$ where $U$ is gamma distributed with density function $$f_U(u)=\frac{\lambda^\delta}{\Gamma (\delta)} u^{\delta-1} e^{-\lambda u}, \quad u>0,$$ and $V_\nu$ is strictly positive-stable distributed with $\exp ( - s^\nu)$ as the Laplace transform of the corresponding density function. Note that the $q$th fractional moment of the inter-event time can be easily shown as $$\mathbb{E}\left[ T^{\nu,\delta} \right]^q =\frac{\pi \Gamma (q/ \nu +
\delta)}{\lambda^{q / \nu} \Gamma ( q / \nu )\sin ( \pi q/ \nu) \Gamma (1- q )}, \qquad 0< q < \nu.$$ Typically, generating $T^{\nu,\delta}$ and adding one (corresponding to a single jump or event) each time gives a sample trajectory.
Given $m$ jumps (corresponding to $m$ renewal times), we propose method-of-moments estimators for the parameters $\nu, \delta,$ and $\lambda$ to make the preceding generalization usable in practice. Getting the logarithm of $T^{\nu,\delta}$ we have $$T' \stackrel{d}{=} \frac{1}{\nu} U' + V_\nu',$$ where $T'=\ln (T^{\nu,\delta})$, $U'=\ln (U)$, and $V_\nu'=\ln (V_\nu)$. Following @cuw we get the estimating equations: $$\mu_{T'} = \mathbb{E} \left( T' \right) = \eta \bigg( \frac{1}{\nu} -1\bigg) +
\frac{\psi ( \delta )-\ln (\lambda)}{\nu },$$ $$\sigma_{T'}^2= \frac{\pi^2}{6}\bigg( \frac{1}{\nu^2} -1\bigg) + \frac{1}{\nu^2} \psi^{(1)} (\delta),$$ $$\mu_3= \mathbb{E} \left( T'- \mu_{T'} \right)^3 =
\frac{ \psi^{(2)} (\delta) -2\left(\nu^3 -1\right)\zeta (3)}{\nu^3},$$ $\eta \approx 0.57721$ is the Euler’s constant, and $\zeta (3)$ is the Riemann Zeta function evaluated at 3. Using the equations of the variance and the third central moment above, we can solve for the estimates $\hat{\delta}$ and $\hat{\nu}$ using $\hat{\mu}_3$ and $\hat{\sigma}_{T'}^2$. Plugging $\hat{\nu}$ and $\hat{\delta}$ into the mean equation above, we obtain the estimate of $\lambda$ as $$\hat{\lambda}=\exp \left(- \left[\hat{\nu} \left( \hat{\mu}_{T^{'}} -\eta (1/\hat{\nu}-1) \right)
-\psi (\hat{\delta} ) \right] \right).$$
Furthermore, we tested the above procedure using the following estimate of the digamma function: $$\psi(\tau) = \log (\tau) - 1/(2\tau) - 1/(12\tau^2) + 1/(120\tau^4) - 1/(252\tau^6) + O(1/\tau^8).$$ We then calculated the bias and the root-mean-square-error (RMSE) based on the 1000 generated data samples for different parameter values. Table \[t2\] in the appendix generally indicated positive results for the proposed method.
Generalization II {#00anothersec}
=================
Recall that a random variable $X$ is Mittag–Leffler-distributed with parameters $\lambda>0$ and $\nu \in (0,1]$ if it has probability density function $$\begin{aligned}
\label{tonno}
f_X(x) = \lambda x^{\nu-1} E_{\nu,\nu}(-\lambda x^\nu), \qquad x \in \mathbb{R}^+,
\end{aligned}$$ where $$\begin{aligned}
E_{\alpha,\beta}(x) = \sum_{r=0}^\infty \frac{x^r}{\Gamma(\alpha r + \beta)}, \qquad x \in \mathbb{R},
\end{aligned}$$ is the Mittag–Leffler function. Note that $\text{Pr} \{ X>x \} = E_{\nu,1}(-\lambda x^\nu)$.
Let $Y=1/X$. Then the random variable $Y$ has the inverse Mittag–Leffler distribution, that is, $$\begin{aligned}
\label{aa}
\Pr \{ Y< y \} = \text{Pr} \{ 1/X < y \} = \text{Pr} \{ X > 1/y \} = E_{\nu,1}(-\lambda
y^{-\nu}).
\end{aligned}$$ Hence, the corresponding probability density function is $$\begin{aligned}
\label{a}
\frac{\mathrm d}{\mathrm dy} \Pr \{ Y<y \} & = \frac{\lambda}{\nu} E_{\nu,\nu} (-\lambda y^{-\nu})
\nu y^{-\nu-1} \\
& = \lambda y^{-\nu-1} E_{\nu,\nu} (-\lambda y^{-\nu}), \qquad y \in \mathbb{R}^+. \notag
\end{aligned}$$ When $\nu=1$, formula is the probability density function of an inverse exponential random variable, that is, $$\begin{aligned}
f(y) = \frac{\lambda}{y^2} e^{-\frac{\lambda}{y}}, \qquad y \in \mathbb{R}^+.
\end{aligned}$$
We now give a single definition for both the Mittag–Leffler and the inverse Mittag–Leffler distributions. Note that the probability density function in equation can be written as $$\begin{aligned}
\label{b}
f_Y(y) = \lambda y^{\gamma -1} E_{\nu,\nu} (-\lambda y^\gamma), \qquad y \in \mathbb{R}^+, \: \nu
\in (0,1],
\end{aligned}$$ where $\gamma = \pm \nu$. By freeing the parameter $\gamma$ in formula , we arrive at the probability density $$\begin{aligned}
\label{c}
f_\Xi(\xi) = \frac{|\gamma|}{\nu} \lambda \xi^{\gamma-1} E_{\nu,\nu}(-\lambda \xi^\gamma),
\qquad \xi \in \mathbb{R}^+, \: \nu\ \in (0,1], \: \gamma \in \mathbb{R} \backslash \{0\}
\end{aligned}$$ (see Figure \[afig3\]). Observe that $\int_0^\infty f_\Xi(\xi)\, \mathrm d \xi =1$ as $$\begin{aligned}
\label{ciao}
\frac{|\gamma|}{\nu} \lambda \int_0^\infty \xi^{\gamma -1} E_{\nu,\nu} (-\lambda \xi^\gamma)\,
\mathrm d\xi
& \overset{(\xi=z^{\nu/\gamma})}{=} \frac{|\gamma|}{\nu} \lambda
\int_0^\infty \frac{\nu}{|\gamma|} z^{\nu-\nu/\gamma} E_{\nu,\nu}(-\lambda z^\nu)
\, z^{\nu/\gamma-1} \mathrm dz \\
& \quad = \int_0^\infty \lambda z^{\nu-1} E_{\nu,\nu} (-\lambda z^\nu)\, \mathrm dz = 1. \notag
\end{aligned}$$ Note that in the second-to-last line of , $\text{sgn} (\gamma)$ is used to stabilize the domain of the integral.
The Laplace transform $\mathbb{E} e^{-s \xi}$ can be shown as $$\begin{aligned}
\label{acqua}
\mathbb{E} e^{-s \xi} & = \lambda \frac{|\gamma|}{\nu} \int_0^\infty e^{-s \xi} \xi^{\gamma -1}
E_{\nu,\nu} (- \lambda \xi^\gamma)\, \mathrm d \xi \\
& = \lambda s^{-\gamma} \frac{|\gamma|}{\nu} \: {}_2 \psi_1 \left[ -\lambda s^{-\nu}
\left|
\begin{array}{l}
(1,1),(\gamma,\nu) \\
(\nu,\nu)
\end{array}
\right. \right] \notag \\
& = \lambda s^{-\gamma} \frac{|\gamma|}{\nu} \sum_{r=0}^\infty \left( -\lambda s^{-\nu} \right)^r
\frac{\Gamma (r \nu + \gamma)}{\Gamma(r \nu + \nu)}, \notag
\end{aligned}$$ where we use formula (2.2.22) of @mathai. When $\gamma = \nu \in (0,1]$ we obtain $$\begin{aligned}
\mathbb{E} e^{-s \xi} = \lambda s^{-\nu} \sum_{r=0}^\infty \left( -\lambda s^{-\nu} \right)^r
= \frac{\lambda}{s^\nu + \lambda},
\end{aligned}$$ as in @beg, formula (4.15).
From the above discussion it is clear that $\Xi = X^{\nu/\gamma}$, $\nu \in (0,1]$, $\gamma \in \mathbb{R}\backslash \{ 0 \}$, where $X$ is the Mittag–Leffler distribution with probability density function , therefore the waiting times $\Xi$ of a newly constructed renewal process are simple time-stretching or time-squashing of the original Mittag–Leffler distributed waiting times $X$. Below are density plots of $\Xi = X^{\nu/\gamma}$ where $X$ has the generalized Mittag–Leffler distribution given in .
![\[afig3\]The stretched-squashed Mittag–Leffler density plots (see ) for parameter values $(\nu, \delta, \lambda, \gamma )=((0.1, 0.5, 1), 2, 1, -0.5) $ (top), $(\nu, \delta, \lambda, \gamma )=((0.1, 0.5, 1), 2, 1, 1) $ (middle), and $(\nu, \delta, \lambda, \gamma )=((0.1, 0.5, 1), 2, 1, 5) $ (bottom).](plot2a.pdf "fig:")\
![\[afig3\]The stretched-squashed Mittag–Leffler density plots (see ) for parameter values $(\nu, \delta, \lambda, \gamma )=((0.1, 0.5, 1), 2, 1, -0.5) $ (top), $(\nu, \delta, \lambda, \gamma )=((0.1, 0.5, 1), 2, 1, 1) $ (middle), and $(\nu, \delta, \lambda, \gamma )=((0.1, 0.5, 1), 2, 1, 5) $ (bottom).](plot2b.pdf "fig:")\
![\[afig3\]The stretched-squashed Mittag–Leffler density plots (see ) for parameter values $(\nu, \delta, \lambda, \gamma )=((0.1, 0.5, 1), 2, 1, -0.5) $ (top), $(\nu, \delta, \lambda, \gamma )=((0.1, 0.5, 1), 2, 1, 1) $ (middle), and $(\nu, \delta, \lambda, \gamma )=((0.1, 0.5, 1), 2, 1, 5) $ (bottom).](plot2c.pdf "fig:")
Now, consider $m$ i.i.d. random inter-event imes $\Xi_1, \dots, \Xi_m$ of a counting process $\mathcal{N} (t)$, $t\ge 0$, which are distributed according to . If $W_m = \Xi_1 + \dots + \Xi_m$ is the waiting time till the $m$th event, we have $$\begin{aligned}
\mathbb{E} e^{-s W_m} = s^{-m \gamma} \left( \lambda \frac{|\gamma|}{\nu} \right)^m
{}_2 \psi_1^m \left[ -\lambda s^{-\nu}
\left|
\begin{array}{l}
(1,1),(\gamma,\nu) \\
(\nu,\nu)
\end{array}
\right. \right].
\end{aligned}$$ To determine the state probabilities $q_k^\nu(t) = \Pr \{ \mathcal{N}(t) = k \}$, $k \ge 0$, we write $$\begin{aligned}
\int_0^\infty e^{-s t} q_k^\nu(t) \, \mathrm dt {} = &
\int_0^\infty e^{-s t} \left( \Pr \{ W_k<t \} - \Pr \{ W_{k+1}<t \} \right) \mathrm dt \\
= {} & s^{-k\gamma-1}\left( \lambda \frac{|\gamma|}{\nu} \right)^k
{}_2 \psi_1^k \left[ -\lambda s^{-\nu} \left|
\begin{array}{l}
(1,1),(\gamma,\nu) \\
(\nu,\nu)
\end{array}
\right. \right] \notag \\
& - s^{-(k+1)\gamma-1}\left( \lambda \frac{|\gamma|}{\nu} \right)^{k+1} \! \! \!
{}_2 \psi_1^{k+1} \left[ -\lambda s^{-\nu} \left|
\begin{array}{l}
(1,1),(\gamma,\nu) \\
(\nu,\nu)
\end{array}
\right. \right]. \notag
\end{aligned}$$
Path generation and parameter estimation
----------------------------------------
Simulating a sample path of generalization II directly follows from generalization I. Note that the $\Xi$’s can be generated using the algorithm of @cuw. It is also straighforward to show that the $q$th fractional moment of the random inter-event time is $$\mathbb{E}\mathcal{T}^q =\frac{\pi \Gamma (q \nu / \gamma + 1)}{\lambda^{q \nu / \gamma}
\Gamma ( q / \nu )\sin ( \pi q/ \nu) \Gamma (1- q )}, \qquad 0< q < \nu.$$
Given $m$ renewal times, we propose a formal procedure to estimate the parameters $\nu$, $\gamma$, and $\lambda$ of generalization II. Let $\Xi' = \ln (\Xi)$ and $X' = \ln(X)$. Following @cuw, we can deduce that $$\label{e1}
\mu_{\Xi'} = \mathbb{E} \left( \Xi' \right) = \frac{\nu}{\gamma} \left( \frac{-\ln (\lambda) }{\nu}
- \eta \right),$$ $$\label{e2}
\sigma_{\Xi'}^2= \left( \frac{\nu}{\gamma} \right)^2\left[ \pi^2 \left( \frac{1}{3\nu^2} -\frac{1}{6}
\right) \right],$$ and $$\mu_3= \mathbb{E} \left( \Xi'- \mu_{\Xi'} \right)^3 = -2 \zeta (3) \left( \frac{\nu}{\gamma} \right)^3.$$ Using the estimating equations above, we can eliminate $\gamma$ and solve for $\nu$ by getting the $2/3$ root of the third central moment and dividing it by the variance. Thus, we obtain $$\hat{\nu}= \sqrt{ \frac{c \pi^2}{3\left[ \left(2 \zeta (3) \right)^{2/3} + \frac{c \pi^2}{6}\right]} },$$ where $c=( \hat{\mu}_3^{2/3} )/ \hat{\sigma}_{\Xi'}^2$. Substituting $\hat{\nu}$ to the variance equation , we get $$\hat{\gamma}= \sqrt{ \left( \frac{\hat{\nu}}{\hat{\sigma}_{\Xi'}^2} \right)^2\left[ \pi^2 \left(
\frac{1}{3 \hat{\nu}^2} -\frac{1}{6} \right) \right] }.$$ Finally, plugging $\hat{\nu}$ and $\hat{\gamma}$ into the mean equation above, we have $$\hat{\lambda}= \exp \left[ - \left( \hat{\mu}_{\Xi'} \hat{\gamma} + \eta \hat{\nu} \right)\right].$$
We also tested the above explicit forms of the estimators by calculating the bias and the root-mean-square-error (RMSE) based on the 1000 generated data samples for different parameter and total jump size values. Overall, Table \[t3\] in the appendix showed favorable results for the proposed procedure.
Concluding remarks {#summary}
==================
We proposed two generalizations of the standard and the fractional Poisson processes through their renewal time distributions which naturally provided greater flexibility in modeling real-life renewal processes. Statistical properties such as the state probabilities and process moments were derived. Algorithms to simulate trajectories and to estimate model parameters were also developed. Generally, tests provided additional merits to the proposed procedures.
Although some work have already been done, there are still a few things that need to be pursued. For instance, the complete analysis of the counting process related to the renewal process that has stretched-squashed generalized Mittag–Leffler distributed waiting times would be a worthy pursuit. Also, the development of estimators using likelihood approaches would be of interest as well.
Appendix
========
[cc|ccc|ccc]{} & &\
$\nu$ & $Est$ & $m=100$ & $1000$ & $10000$ & $m=100$ & $1000$ & $10000$\
& $\hat{\nu}$ & 0.018 & 0.001 & 0.000 & 0.184 & 0.049 & 0.013\
& $\hat{\delta}$ & 0.082 & 0.011 & 0.002 & 0.253 & 0.074 & 0.023\
& $\hat{\lambda}$ & 0.207 & 0.027 & 0.003 & 0.525 & 0.151 & 0.048\
& $\hat{\nu}$ & 0.026 & 0.003 & 0.000 & 0.210 & 0.068 & 0.016\
& $\hat{\delta}$ & 0.065 & 0.010 & 0.000 & 0.205 & 0.074 & 0.023\
& $\hat{\lambda}$ & 0.174 & 0.026 & 0.001 & 0.442 & 0.151 & 0.047\
& $\hat{\nu}$ & 0.024 & 0.001 & 0.000 & 0.254 & 0.056 & 0.018\
& $\hat{\delta}$ & 0.069 & 0.009 & 0.000 & 0.207 & 0.067 & 0.023\
& $\hat{\lambda}$ & 0.176 & 0.022 & 0.001 & 0.434 & 0.135 & 0.046\
& $\hat{\nu}$ & 0.005 & 0.003 & 0.000 & 0.264 & 0.077 & 0.020\
& $\hat{\delta}$ & 0.080 & 0.008 & 0.001 & 0.199 & 0.072 & 0.023\
& $\hat{\lambda}$ & 0.202 & 0.022 & 0.004 & 0.429 & 0.147 & 0.047\
& $\hat{\nu}$ & 0.022 & 0.001 & 0.000 & 0.349 & 0.079 & 0.021\
& $\hat{\delta}$ & 0.070 & 0.009 & 0.002 & 0.187 & 0.067 & 0.022\
& $\hat{\lambda}$ & 0.184 & 0.024 & 0.004 & 0.405 & 0.135 & 0.044\
[cc|ccc|ccc]{} & &\
$\nu$ & $Est$ & $m=100$ & $1000$ & $10000$ & $m=100$ & $1000$ & $10000$\
& $\hat{\nu}$ & 0.217 & 0.079 & -0.022 & 0.280 & 0.176 & 0.124\
& $\hat{\gamma}$ & -0.038 & -0.016 & 0.001 & 0.072 & 0.034 & 0.016\
& $\hat{\lambda}$ & -0.041 & -0.014 & 0.006 & 0.090 & 0.044 & 0.030\
& $\hat{\nu}$ & 0.136 & -0.021 &0.000 & 0.222 & 0.153 & 0.100\
& $\hat{\gamma}$ & -0.026 & 0.018 & 0.051 & 0.067 & 0.032 & 0.016\
& $\hat{\lambda}$ & -0.017 & 0.005 & 0.001 & 0.085 & 0.041 & 0.023\
& $\hat{\nu}$ & 0.052 &-0.022 &-0.008 & 0.188 & 0.149 & 0.060\
& $\hat{\gamma}$ &-0.015 & 0.001 & 0.001 & 0.071 & 0.034 & 0.013\
& $\hat{\lambda}$ &-0.003 & 0.007 & 0.002 & 0.086 & 0.040 & 0.014\
& $\hat{\nu}$ &-0.013 &-0.030 &-0.003 & 0.172 & 0.129 & 0.034\
& $\hat{\gamma}$ & 0.004 & 0.006 & 0.001 & 0.073 & 0.035 & 0.011\
& $\hat{\lambda}$ & 0.007 & 0.008 & 0.000 & 0.086 & 0.035 & 0.009\
& $\hat{\nu}$ &-0.043 &-0.004 & 0.000 & 0.131 & 0.044 & 0.013\
& $\hat{\gamma}$ & 0.021 & 0.002 & 0.000 & 0.083 & 0.028 & 0.008\
& $\hat{\lambda}$ & 0.014 & 0.002 & 0.000 & 0.073 & 0.022 & 0.007\
|
---
abstract: 'We derive a simple formula for the fluctuations of the time average $\overline{x}(t)$ around the thermal mean $\langle x \rangle_{{\rm eq}}$ for overdamped Brownian motion in a binding potential $U(x)$. Using a backward Fokker-Planck equation, introduced by Szabo, Schulten, and Schulten in the context of reaction kinetics, we show that for ergodic processes these finite measurement time fluctuations are determined by the Boltzmann measure. For the widely applicable logarithmic potential, ergodicity is broken. We quantify the large non-ergodic fluctuations and show how they are related to a super-aging correlation function.'
author:
- 'A. Dechant'
- 'E. Lutz'
- 'D. A. Kessler'
- 'E. Barkai'
title: Fluctuations of time averages for Langevin dynamics in a binding force field
---
Current technology permits tracking of trajectories of individual molecules with exquisite precision. The motion of a Brownian particle in a binding potential field $U(x)$ is used to model many such physical, biological and chemical processes. From statistical mechanics, we know that if the process is ergodic, and if the measurement time $t \to \infty$, then the time average $\overline{x}(t) =\int_0 ^t x(t') {\rm d} t'/t$ is equal to the corresponding ensemble average $\langle x \rangle_{{\rm eq}}$. In experiment the measurement time might be long, but it is always finite. Hence it is natural to ask what the fluctuations of $\overline{x}$ are. Such an analysis sheds light on deviations from the thermal equilibrium average due to finite time measurement, a general theme which has attracted much interest in the context of fluctuation theorems [@Fluc]. The Boltzmann measure, due to ergodicity, yields equilibrium properties of thermal systems. Surprisingly, we find that for Langevin dynamics, the Boltzmann measure also determines the deviations from ergodicity.
As we will show, for binding fields $U(x)$ where the Fokker-Planck (FP) operator exhibits a discrete eigenspectrum, the fluctuations of the time average $\overline{x}$ become small as time increases, as expected from ordinary ergodic statistical mechanics. For this type of field, ergodicity is related to the work of Szabo, Schulten, and Schulten [@Szabo] on the seemingly unrelated problem of reaction kinetics (see details below). A more interesting case is that of a logarithmic binding field [@Lutz] $U(x) \sim U_0 \ln(|x|)$ when $|x| \to \infty$, since for such a potential the fluctuations of $\overline{x}$ are not small even in the long time limit. Here the Boltzmann measure exhibits power law tails, $P^{\rm eq} (x) \propto
|x|^{ - U_0/(k_B T)}$. Starting at the origin, the particle during its evolution tends to sample larger and larger values of $|x|$ as illustrated in Fig. \[fig1\]. Large fluctuations in the amplitude of $x(t)$ cause the time average of this special process to remain random even in the long time limit. In what follows, we calculate the magnitude of these fluctuations and show how they are related to a super-aging correlation function. Importantly, such logarithmic potentials model many physical systems, ranging from optical lattices [@Zoller], charges in vicinity of a long charged polymer [@Manning], DNA dynamics [@Fogedby], membrane induced forces [@Farago], a nano-particle in a trap [@Adam], to long ranged interacting models [@Chavanis]. At the end of this Letter we discuss the connection between our theory and a recent experiment [@Sagi].
[*Model and observable.*]{} Brownian dynamics in a force field $f(x)= - {\rm d} U(x)/ {\rm d} x$ obeys the equation [@Risken] $${{\rm d} x \over {\rm d} t } = - { f(x) \over \gamma} + \eta(t).
\label{eq01}$$ Here $\gamma$ is the friction constant, $\eta(t)$ is Gaussian white noise obeying the fluctuation dissipation relation $\langle \eta(t) \eta(t')\rangle = 2 D \delta(t-t')$, and $D=k_B T/ \gamma$ according to the Einstein relation. From the trajectory $x(t)$ we construct the time average $\overline{x}(t) = \int_0 ^t x(t') {\rm d} t' /t$. For a binding potential, in the long time limit, $x$ obeys the equilibrium Boltzmann distribution: $$P^{{\rm eq}} (x) = { \exp\left[ - {U(x) \over k_B T} \right] \over Z}; \quad Z=\int_{-\infty} ^\infty e^{-\frac{U(x)}{k_B T}} {\rm d} x
\label{eq02}$$ where $Z$ is the normalizing partition function [*which is assumed to be finite*]{}. We consider symmetric potentials $U(x) = U(-x)$ and then the ensemble average in equilibrium $\langle x \rangle_{{\rm eq}}=
\int_{-\infty} ^\infty x P^{{\rm eq}} (x) {\rm d} x =0 $. If the process is ergodic then in the long time limit $\overline{x} \to \langle x \rangle_{{\rm eq}}=0$. If $\lim_{t \to \infty} \langle \overline{x}^2(t) \rangle \neq 0$ the process is non-ergodic, where $\langle \cdots \rangle$ stands for an ensemble mean. In the second part of our work we show that not all binding potentials satisfy the ergodic hypothesis.
[*Szabo-Schulten-Schulten equation yields the fluctuations of the time average.*]{} The variance of the time average is given by $$\langle \overline{x}^2(t) \rangle = {1 \over t^2} \int_0 ^t {\rm d } t_2 \int_0 ^t {\rm d} t_1 \langle x(t_2) x(t_1) \rangle
\label{eq03}$$ where $\langle x(t_2) x(t_1) \rangle$ is the correlation function. For the Markovian process under investigation, and for a particle starting at the origin at time $t=0$ we have [@Risken]
$$\langle \overline{x}^2(t) \rangle = {2 \over t^2} \int_0 ^t {\rm d } t_2 \int_0 ^{t_2 } {\rm d} t_1 \int_{-\infty} ^\infty \int_{-\infty} ^\infty x_2 x_1 P\left(x_2, t_2| x_1 , t_1\right) P\left( x_1, t_1 | 0, 0\right){\rm d} x_1 {\rm d} x_2
\label{eq04}$$
where $P(x_2,t_2|x_1,t_1)$ is the conditional probability density to find the particle on $x_2$ at time $t_2$ once it is located at $x_1$ at time $t_1$. In the limit of long times, the major contribution to the integration over $t_1$ comes from long times; hence one replaces $P(x_1,t_1|0,0)$ with $P^{{\rm eq}} (x_1)$. To proceed, it is useful to define $$\xi(x_1) = \int_0 ^\infty E(x_1,\tau){\rm d} \tau
\label{eq05}$$ where $E(x_1,\tau)$ is the averaged position of a particle at a time $\tau$ after it starts at $x_1$. Two cases are of interest; the first is when $\xi(x_1)$ is finite, the other when it diverges. We shall start with the former case which is clearly relevant to potential fields where the FP eigenspectrum [@Risken] has a finite energy gap to the ground state, since then the relaxation of $E(x_1,\tau)$ is exponential. From Eq. (\[eq04\]) it follows that in the long time limit $$\langle \overline{x}^2(t) \rangle \sim {2 \over t} \int_{-\infty}
^\infty x_1 \xi(x_1) P^{{\rm eq}} (x_1){\rm d} x_1 .
\label{eq06}$$ As is well known the backward FP equation [@Risken] $$\begin{array}{l}
L^{\dagger} _{{\rm FP}} P(x_2, \tau|x_1,0) = {\partial \over \partial \tau} P(x_2,\tau|x_1,0), \\
\ \ \\
L^{\dagger} _{{\rm FP}} = D \left[ {\partial^2 \over \partial (x_1)^2 } + { f(x_1) \over k_B T} {\partial \over \partial x_1} \right]
\end{array}
\label{eq07}$$ governs the dynamics where $L^{\dagger} _{{\rm FP}}$ is the adjoint FP operator and $P(x_2,0|x_1,0)=\delta(x_2 - x_1)$. By definition $E(x_1,\tau)=\int_{-\infty} ^\infty x_2 P(x_2,\tau|x_1,0){\rm d} x_2 $ which implies $$L^{\dagger} _{{\rm FP}} E(x_1,\tau) =
{\partial \over \partial \tau} E(x_1,\tau)
\label{eq11}$$ with $E(x_1,0) = x_1$. Using Eq. (\[eq05\]), we find $$L^{\dagger} _{{\rm FP}} \xi(x_1) = - x_1
\label{eq08}$$ with $\xi(0)=0$. Eq. (\[eq08\]) was obtained previously in [@Szabo] in the context of reaction kinetics. Eqs. (\[eq04\]-\[eq08\]) are so general that they could be extended to arbitrary Markovian processes. The latter equations thus serve as a starting point for the investigation of fluctuations of time averages for a wide class of systems.
[*Fluctuations of time averages determined from Boltzmann statistics.*]{} Eq. (\[eq08\]) is easy to solve, and upon using Eq. (\[eq06\]) we find the general formula $$\langle \overline{x}^2 \rangle \sim { 2 \over D t} \int_{-\infty} ^\infty {e^{U(x) / (k_B T)} \over Z} {\rm d} x \left[ \int_{x} ^\infty x' e^{ - U(x')/ (k_B T)} {\rm d} x' \right]^2 .
\label{eq09}$$ As is well known, Boltzmann statistics can be used to determine the time average of ergodic processes: $\overline{x} \to \langle x \rangle$ in the long time limit. Eq. (\[eq09\]) shows that also the finite time fluctuations of $\overline{x}$ are determined by the Boltzmann distribution. Surprisingly, Eq. (\[eq09\]) shows that the difficult task of finding the entire eigenspectrum of the FP operator is not required. Eq. (\[eq09\]) is easily generalized to dimensions greater than one, and to non-thermal processes whose equilibrium density is non-Boltzmannian. As expected from ergodicity, the magnitude of the fluctuations decays to zero with time, provided that the integrals in Eq. (\[eq09\]) converge. For example, for the harmonic potential $U(x) = m \omega^2 x^2 /2$ we get $\langle \overline{x}^2 \rangle \sim 2 (k_B T)^2 / [ D ( m \omega^2)^2 t]$. An interesting case where the integrals diverge is the logarithmic potential $U(x) \sim U_0 \ln(|x|) $ for $|x| \to \infty$ and $U_0/(k_B T) < 5$. This leads to a non-ergodic behavior which we now investigate.
[*Logarithmic potential.*]{} We will first find the two-point correlation function $\langle x(t_2) x(t_1) \rangle$ for a general logarithmic potential which satisfies $U(x) \sim U_0 \ln(|x/a|)$, e.g. $U(x) = 0.5 U_0 \ln[1+ (x/a)^2]$. We will then use (\[eq03\]) to obtain the fluctuations of the time average showing that for high enough temperature the fluctuations increase with time. For this potential, for $1<U_0/(k_B T)<5$, due to the slow convergence of the tail of the distribution to $P^{\rm eq}$, and the slow power-law decay of $E(x_1,\tau)$ (which we shall shortly demonstrate), rendering $\xi(x_1)$ infinite for $U_0/(k_B T)<2$, one must consider the full time dependent problem instead of the time independent Eq. (\[eq08\]) and $P^{{\rm eq}}(x)$. Generally the correlation function is given by $$\langle x(t_2) x(t_1 ) \rangle= \int_{-\infty} ^\infty x_1 E(x_1, t_2 - t_1) P(x_1, t_1|0,0){\rm d} x_1 .
\label{eq10}$$ To solve this problem we used two approaches; the first is based on an eigenfunction expansion of the solution of the FP equation [@Andreas]. Such a calculation is lengthy and hence we adopt here a scaling approach. As seen from Eq. (\[eq10\]) the key quantity to calculate is the ensemble mean $E(x_1,\tau)$ using Eq. (\[eq11\]). Due to the homogenous character of the large $x$ Fokker-Planck operator, it is natural to adopt a scaling ansatz: $$E(x_1,\tau)\sim \tau^\alpha g\left( { x_1 \over \tau^\beta} \right)
\label{eq13}$$ where $\alpha$ and $\beta$ are scaling exponents. Since for short time $E(x_1,\tau)\simeq x_1$ we have $g(y) \simeq y$ for large $y$ and $\alpha=\beta$. Inserting Eq. (\[eq13\]) in Eq. (\[eq11\]) we find to leading order $$\tau^{-\beta} D \left( g''- \tilde{U}_0 {g'\over y}\right) = \tau^{\beta- 1} \beta \left( g - y g'\right),
\label{eq14}$$ where $\tilde{U}_0 = U_0 /(k_B T)$ is a key dimensionless parameter. To achieve a $t$-independent equation, we must have $\beta=1/2$, typical of Brownian motion. Then, $$g(y) = c_1 y^{1+ \tilde{U}_0} e^{ - {y^2 \over 4 D}} M\left({3 \over 2} , {3 + \tilde{U}_0 \over 2} , {y^2 \over 4 D} \right)
\label{eq15}$$ where $M(a,b,x)$ \[also denoted $_1 F_1 (a;b;x)$\] is the Kummer $M$ function [@Abr] and we rejected a second solution in terms of the Kummer $U$ function since it does not satisfy the boundary condition $E(x_1,\tau) \to 0$ when $\tau \to \infty$ (i.e., relaxation to equilibrium). The constant $c_1$ is found by matching the solution in the $y\to \infty$ limit which corresponds to short times. Using $M(a,b,x) \sim \exp(x) \Gamma(b) x^{a-b}/\Gamma(a)$ and $g(y) \sim y $ we find $c_1 = \{\Gamma(3/2) / \Gamma[(3 + \tilde{U}_0 )/2]\}(4 D)^{- \tilde{U}_0 /2}$. In particular, for long times, $E(x_1,\tau) \sim \tau^{-\tilde{U}_0/2}$, so as we claimed, $\xi(x_1)$ diverges for $\tilde{U}_0<2$.
[*Steady state cannot be used to obtain the correlation function.*]{} To complete the calculation, we must have $P(x_1,t_1|0,0)$ which was recently obtained [@KesslerPRL]. The equilibrium PDF, since it decays as a power law $P^{{\rm eq}} (x) \propto |x|^{- \tilde{U}_0} $ would give, for $1<\tilde{U}_0<3$, $\langle x(t_2) x(t_1) \rangle = \infty$ for $t_1=t_2$. This is an unphysical behavior: at finite time one cannot have an infinite value for the correlation function, since the particle cannot travel faster than diffusion permits. Specifically in the limit of long $t_1$ we have [@KesslerPRL] $$P(x_1,t_1| 0 ,0 ) \sim P^{{\rm eq }} (x_1) {\Gamma( { 1 + \tilde{U}_0 \over 2} , {(x_1)^2 \over 4 D t_1} ) \over \Gamma( { 1 + \tilde{U}_0 \over 2} )}.
\label{eq16}$$ Since $\Gamma(a,0)=\Gamma(a)$ as $t \to \infty$, thermal equilibrium is reached. Nevertheless, for the calculation of correlation functions one must take into account the finite time correction which is represented by the ratio of $\Gamma$ functions.
[*Aging correlation function.*]{} Inserting Eqs. (\[eq15\],\[eq16\]) in Eq. (\[eq10\]) we find the non-stationary correlation function for the temperature range $1 < \tilde{U}_0 < 3$: $$\langle x(t_2) x(t_1) \rangle \sim \langle x^2 (t_1) \rangle f_{\tilde{U}_0}\left( { t_2 - t_1 \over t_1} \right)
\label{eq18}$$ where $$\begin{array}{c}
f_{\tilde{U}_0} (s) =
{ \sqrt{ \pi} \left(3 - \tilde{U}_0\right) \over 2 \Gamma \left( { 3 + \tilde{U} _0 \over 2} \right) } s^{ {3 - \tilde{U}_0 \over 2}} \times \\
\int_0 ^\infty {\rm d} y y^2 e^{- y^2} M\left( { 3 \over 2} , {3 + \tilde{U}_0 \over 2} , y^2 \right) \Gamma \left( { \tilde{U}_0 + 1 \over 2}, y^2 s \right).
\end{array}
\label{eq19}$$ The behavior in Eq. (\[eq18\]) is very different than the stationary case where the correlation function is a function of the time difference $t_2 - t_1$. In this temperature regime the equilibrium mean square displacement diverges, $\langle x^2 \rangle_{{\rm eq}}= \infty$, while the time dependent solution Eq. (\[eq16\]) gives [@KesslerPRL] $ \langle x^2 (t_1) \rangle = a^2 c_2 (4 D t_1/ a^2)^{ (3 - \tilde{U}_0 )/2}$, $c_2=2 (a/Z) [\Gamma(1/2+\tilde{U}_0/2)(3 - \tilde{U}_0)]^{-1}$. We find $f_{\tilde{U}_0} (0) = 1$ which implies that $C(t_1,t_1) = \langle x^2(t_1) \rangle$ as it should. In the opposite limit $t_2 \gg t_1$, we obtain $$\langle x(t_2) x(t_1) \rangle \sim c_3 \langle x^2 (t_1) \rangle \left( { t_2 \over t_1} \right)^{ - \tilde{U}_0 \over 2}
\label{eq20}$$ with $c_3=(3/2-\tilde{U}_0/2)\sqrt{\pi}\Gamma(2+\tilde{U}_0/2)/3 \Gamma(3/2+\tilde{U}_0/2)$. In Fig. \[fig2\] we compare our analytical Eq. (\[eq19\]) with Langevin simulations showing excellent agreement for various measurement times.
As mentioned we assume that the partition function function $Z$ is finite and hence the steady state $P^{{\rm eq}}(x)$ is normalizable. This excludes the well known Bessel process [@Bray] which can be mapped onto $U(x)= U_0 \ln|x|$ with its singularity at the origin. It is important to emphasize that $\langle x(t_2) x(t_1) \rangle \sim 1/Z$ depends on the shape of the potential [*in the whole space*]{} through $Z$. Hence for the calculation of the correlation function the regularity of the potential on the origin is vital. Interestingly this is not the case for all observables; e.g., $E(x_1,\tau)$ Eqs. (\[eq13\],\[eq15\]) is $Z$ independent and hence related to the Bessel process [@Bray].
[*Ergodicity of the dynamics*]{} is classified in four domains which are controlled by temperature.\
[**(a)**]{} The most interesting case is the regime $1<\tilde{U}_0 <3$. As we showed, a normalized steady state exists and from symmetry $\langle x \rangle_{{\rm eq}} = 0$. If we naively assume ergodicity $\overline{x} \to \langle x \rangle_{{\rm eq}}=0$ and $\lim_{t \to \infty} \langle \overline{x}^2(t) \rangle=0$. Rather, from Eqs. (\[eq03\],\[eq18\]) we find [@remark] $$\langle \overline{x}^2(t) \rangle \sim { 2 \langle x^2(t) \rangle \over t^2} \int_0 ^t {\rm d} t_1 \int_{t_1} ^t {\rm d} t_2 \left( { t_1 \over t } \right)^{{ 3 - \tilde{U}_0 \over 2} } f_{\tilde{U}_0 } \left( { t_2 - t_1 \over t_1} \right).
\label{eq21}$$ Changing variables to $s=t_2/t_1-1,w=t/t_1-1$, we find $$\langle \overline{x}^2(t) \rangle \sim c_4 \langle x^2(t) \rangle \propto t^{3 - \tilde{U}_0 \over 2}
\label{eq22}$$ where $c_4=4\int_0 ^\infty {\rm d} w (1 + w)^{(\tilde{U}_0 - 7)/ 2} (7-\tilde{U}_0)^{-1} f_{\tilde{U}_0} (w)$. We see that the fluctuations grow with time, hence ergodicity is broken. We find that $c_4 \approx 0.2397$ for $\tilde{U}_0 = 1$ and that it decreases monotonically to $c_4 = 0$ at $\tilde{U}_0 = 3$.\
[**(b)**]{} For lower temperature, $3 < \tilde{U}_0< 5$, the integrals in Eq. (\[eq09\]) still diverge, and $\langle \overline{x}^2(t) \rangle$ decays as $t^{(3 - \tilde{U}_0) / 2} $; indicating an anomalously slow approach to ergodicity.\
[**(c)**]{} For $\tilde{U}_0 >5$ the temperature is low enough that Eq. (\[eq09\]) is now valid. For $U(x) = 0.5 U_0 \ln[1+ (x/a)^2]$ we find $$\langle \overline{x}^2 (t)\rangle \sim { 2 (\tilde{U}_0 -4) \over
(\tilde{U}_0 - 2) (\tilde{U}_0 - 3) } { a^4 \over (\tilde{U}_0 - 5) D t}
\label{eq23}$$ which diverges when $\tilde{U}_0 \to 5$.\
[**(d)**]{} Finally, for very high temperatures $\tilde{U}_0 < 1$, the equilibrium state Eq. (\[eq02\]) is not defined as the partition function $Z$ diverges. Here $\langle \overline{x}^2(t) \rangle \propto t$, exactly the diffusive behavior of a free particle, $U_0=0$ [@Andreas].\
These four different behaviors are confirmed via numerical simulations presented in Fig. \[fig3\], which illustrates convergence on reasonable computer time scales. A summary of the scaling regimes is presented in Table 1.
$\langle \overline{x}^2(t) \rangle$ $\langle x^2(t) \rangle$
------------------------ ----------------------------------------- ------------------------------
$\tilde{U}_0 < 1$ $t$ $t$
$1 <\tilde{U}_0 < 3\ $ $t^{(3 - \tilde{U}_0)/2}$ $t^{(3 - \tilde{U}_0)/2}$
$3 <\tilde{U}_0 < 5$ $t^{(3 - \tilde{U}_0)/2}$ $t^0$
$5 <\tilde{U}_0 $ $ t^{-1}$ $t^0$
: Scaling behavior of $\langle \overline{x}^2 (t) \rangle$ and $\langle x^2 (t) \rangle$ for various values of $\tilde{U}_0 = U_0/(k_B T)$. []{data-label="Tab1"}
[*Relation with experiment.*]{} After the submission of this manuscript, an experiment on anomalous diffusion of ultra-cold atoms which employs the well known Sisyphus cooling scheme was reported [@Sagi]. In the semi-classical approximation, the atomic velocity distribution follows Fokker-Planck dynamics in an asymptotically logarithmic potential [@Zoller; @KesslerPRL; @Renzoni]. Our work provides the theoretical mean-square displacement in this experiment by identifying our position $x$ with the velocity $v$ of the atoms. The measured atomic position is $x(t) = \int_0 ^t v(t) {\rm d} t$ and hence $x(t)/t$ corresponds to the time averaged velocity. The PDF of the atoms in the experiment has been described with a Lévy distribution, with a divergent variance. However, our results show that the mean square displacement is finite for any finite measurement time. These seemingly contradicting findings are related to the well known dilemma whether Lévy flights are at all physical, since they predict diverging mean square displacement, which must be tamed [@Zoller; @Klafter]. We speculate that the Lévy distribution found in the experiment describes the center part of the packet, which eventually is cut off to give a finite mean square displacement. Furthermore, using our results one can estimate the time in which the atoms remain within a finite domain, which is of course crucial for experiments. Experimentally one may also control the depth of the optical potential, here modeled with $\tilde{U}_0$ and hence explore the nontrivial dependence of our results on this parameter. We will elaborate on these interesting points in a longer publication.
[*Discussion.*]{} Aging correlation functions and ergodicity breaking typically describe glassy dynamics [@Fisher; @PNAS] (and Ref. therein). Our work shows that aging and ergodicity breaking can be found also for simple Markovian dynamics, without the need to introduce heavy-tailed waiting times into the kinetic scheme, nor disorder or many-body physics. The aging correlation function (\[eq18\]) has a signature very different than most previous work. The prefactor $\langle x^2 (t_1) \rangle$ grows with time, and hence we call it super-aging. This is in contrast to normal aging where the correlation function is of the form $C(t_2,t_1) = \langle x^2 \rangle_{{\rm eq}}f(t_2/t_1)$ with a finite equilibrium value $\langle x^2 \rangle_{{\rm eq}}$. A similar non-normal aging behavior, albeit with a logarithmic time dependence, has been found in Sinai’s model of diffusion in a random environment [@Fisher]. Unlike previous scenarios to ergodicity breaking, the amplitude of the stochastic process $x(t)$ in our work increases with time, since the particle explores more and more of the tails of the equilibrium PDF as time goes on. Thus rare events where the amplitude $x(t)$ of the Markovian process attains a large value are responsible for the non-ergodic behavior. This is clearly related to the power law tail of the equilibrium steady state $P^{{\rm eq}} (x) \propto |x|^{- \tilde{U}_0}$. More importantly, physical systems with fat tailed equilibrium states are common and hence this type of ergodicity breaking may find broad applications.
[**Acknowledgement**]{} This work was supported by the Israel Science Foundation, the Emmy Noether Program of the DFG (contract No LU1382/1-1) and the cluster of excellence Nanosystems Initiative Munich.
[99]{}
C. Jarzynski, [*Annu. Rev. Condens. Matter Phys.*]{} [**2**]{}, 329 (2011).
A. Szabo, K. Schulten, and Z. Schulten, [*J. Chem. Phys.*]{} [**72**]{}, 4350 (1980).
E. Lutz, [*Phys. Rev. Lett.*]{} [**93**]{}, 190602 (2004).
S. Marksteiner, K. Ellinger, and P. Zoller, [*Phys. Rev. A*]{} [**53**]{}, 3409 (1996).
G. S. Manning, [*J. of Chemical Physics*]{} [**51**]{}, 924 (1969).
H. C. Fogedby, and R. Metzler, [*Phys. Rev. Lett.*]{} [**98**]{}, 070601 (2007).
O. Farago, [*Phys. Rev. E*]{} [**81**]{}, 050902 (2010).
A. E. Cohen, [*Phys. Rev. Lett.*]{} [**94**]{}, 118102 (2005).
P. H. Chavanis and R. Mannella, [*Eur. Phys. J. B*]{} [**78**]{}, 139 (2010).
Y. Sagi, M. Brook, I. Almog, and N. Davidson arXiv:1109.1503v1 \[quant-ph\] (2011).
H. Risken, [*The Fokker Planck Equation*]{} Springer 1996 (Berlin).
A. J. Bray, [*Phys. Rev. E*]{} [**62**]{}, 103 (2000).
A. Dechant, Diploma thesis, University of Augsburg (2011).
M. Abramowitz and I. A. Stegun, [*Handbook of Mathematical Functions*]{} Dover 1972 (New York)
D.A. Kessler and E. Barkai, [*Phys. Rev. Lett.*]{} [**105**]{}, 120602 (2010).
This is related to the observation that the FP eigenspectrum for the logarithmic potential has no gap. A. Dechant, E. Lutz, E. Barkai, D. A. Kessler [*J. Stat. Phys.*]{} [**145**]{}, 1524 (2011).
P. Douglas, S. Bergamini, and F. Renzoni [*Phys. Rev. Lett.*]{} [**96**]{}, 110601 (2006).
J. Klafter, M. F. Shlesinger, and G. Zumofen [*Physics Today*]{} [**49**]{} 33 (1996).
D. S. Fisher, P. Le Doussal, and C. Monthus, [*Phys. Rev. Lett.*]{}[**80**]{}, 3539 (1998). P. Le Doussal, C. Monthus and D. S. Fisher [*Phys. Rev. E*]{} [**59**]{} 4795 (1999). See Eq. (159) therein.
J. P. Bouchaud, [*J. Phys. I*]{} France [**2**]{}, 1705 (1992). S. Burov, R. Metzler, E. Barkai, [*PNAS*]{} [**107**]{}, 13228 (2010).
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---
abstract: 'We review the interior structure and evolution of Jupiter, Saturn, Uranus and Neptune, and giant exoplanets with particular emphasis on constraining their global composition. Compared to the first edition of this review, we provide a new discussion of the atmospheric compositions of the solar system giant planets, we discuss the discovery of oscillations of Jupiter and Saturn, the significant improvements in our understanding of the behavior of material at high pressures and the consequences for interior and evolution models. We place the giant planets in our Solar System in context with the trends seen for exoplanets.'
address:
- 'Laboratoire Lagrange, Université de Nice-Sophia Antipolis, Observatoire de la Côte d’Azur, CNRS, CP 34229, 06304 NICE Cedex 04, France'
- 'LESIA, Observatoire de Paris, CNRS FRE 2461, 5 pl. J. Janssen, 92195 Meudon Cedex, France'
author:
- Tristan Guillot
- Daniel Gautier
bibliography:
- 'geophys\_guillot.bib'
title: Giant Planets
---
Giant planets, exoplanets, Jupiter, Saturn, Uranus, Neptune, planet formation
Introduction
============
In our solar system, four planets stand out for their sheer mass and size. Jupiter, Saturn, Uranus, and Neptune indeed qualify as “giant planets” because they are larger than any terrestrial planet and much more massive than all other objects in the solar system, except the Sun, put together (Figure \[fig:inventory\]). Because of their gravitational might, they have played a key role in the formation of the solar system, tossing around many objects in the system, preventing the formation of a planet in what is now the asteroid belt, and directly leading to the formation of the Kuiper Belt and Oort Cloud. They also retain some of the gas (in particular hydrogen and helium) that was present when the Sun and its planets formed and are thus key witnesses in the search for our origins.
Because of a massive envelope mostly made of hydrogen helium, these planets are [*fluid*]{}, with no solid or liquid surface. In terms of structure and composition, they lie in between stars (gaseous and mostly made of hydrogen and helium) and smaller terrestrial planets (solid and liquid and mostly made of heavy elements), with Jupiter and Saturn being closer to the former and Uranus and Neptune to the latter (see fig. \[fig:inventory\]).
The discovery of many extrasolar planets of masses from a few thousands down to a few Earth masses and the possibility to characterize them by the measurement of their mass and size prompts a more general definition of giant planets. For this review, we will adopt the following: “a giant planet is a planet mostly made of hydrogen and helium and too light to ignite deuterium fusion.” This is purposely relatively vague – depending on whether the inventory is performed by mass or by atom or molecule, Uranus and Neptune may be included or left out of the category. Note that Uranus and Neptune are indeed relatively different in structure than Jupiter and Saturn and are generally referred to as “ice giants”, due to an interior structure that is consistent with the presence of mostly “ices” (a mixture formed from the condensation in the protoplanetary disk of low- refractivity materials such as H$_2$O, CH$_4$ and NH$_3$, and brought to the high-pressure conditions of planetary interiors – see below).
Globally, this definition encompasses a class of objects that have similar properties (in particular, a low viscosity and a non-negligible compressibility) and inherited part of their material directly from the same reservoir as their parent star. These objects can thus be largely studied with the same tools, and their formation is linked to that of their parent star and the fate of the circumstellar gaseous disk present around the young star.
We will hereafter present some of the key data concerning giant planets in the solar system and outside. We will then present the theoretical basis for the study of their structure and evolution. On this basis, the constraints on their composition will be discussed and analyzed in terms of consequences for planet formation models.
Observations and global properties
==================================
Visual appearances
------------------
In spite of its smallness, the sample of four giant planets in our solar system exhibits a large variety of appearances, shapes, colors, variability, etc. As shown in Figure \[fig:visual\], all four giant planets are flattened by rotation and exhibit a more or less clear zonal wind pattern, but the color of their visible atmosphere is very different (this is due mostly to minor species in the high planetary atmosphere), their clouds have different compositions (ammonia for Jupiter and Saturn, methane for Uranus and Neptune) and depths, and their global meteorology (number of vortexes, long-lived anticyclones such as Jupiter’s Great Red Spot, presence of planetary-scale storms, convective activity) is different from one planet to the next.
We can presently only wonder about what is in store for us with extrasolar giant planets since we cannot image and resolve them. But with orbital distances from as close as 0.01 AU to 100AU and more, a variety of masses, sizes, and parent stars, we should expect to be surprised!
Gravity fields {#sec:gravity}
--------------
The mass of our giant planets can be obtained with great accuracy from the observation of the motions of their natural satellites: 317.834, 95.161, 14.538 and 17.148 times the mass of the Earth ($1\mea =5.97369\times 10^{27}\g$) for Jupiter, Saturn, Uranus and Neptune, respectively. More precise measurements of their gravity field can be obtained through the analysis of the trajectories of a spacecraft during flyby, especially when they come close to the planet and preferably in a near-polar orbit. The gravitational field thus measured departs from a purely spherical function due to the planets’ rapid rotation. The measurements are generally expressed by expanding the components of the gravity field in Legendre polynomials $P_i$ of progressively higher orders: V\_[ext]{}(r,)=-{ 1-\_[i=1]{}\^( )\^i J\_i P\_i() }, where $V_{\rm ext}(r,\theta)$ is the gravity field evaluated outside the planet at a distance $r$ and colatitude $\theta$, $R_{\rm eq}$ is the equatorial radius, and $J_i$ are the gravitational moments. Because the giant planets are very close to hydrostatic equilibrium the coefficients of even order are the only ones that are not negligible. We will see how these gravitational moments, as listed in table \[tab:moments\], help us constrain the planets’ interior density profiles.
[l r@[.]{}l r@[.]{}l r@[.]{}l r@[.]{}l]{} & & & &\
$M\p{-26}$ \[kg\] & 18&986112(15) & 5&68463036(16) & 0&8683205(34) & 1&0243547861(15)\
$R_{\rm eq}\p{-7}$ \[m\] & 7&1492(4) & 6&0268(4) & 2&5559(4) & 2&4766(15)\
$R_{\rm pol}\p{-7}$ \[m\] & 6&6854(10) & 5&4364(10) & 2&4973(20) & 2&4342(30)\
$\overline{R}\p{-7}$ \[m\] & 6&9894(6) & 5&8210(6) & 2&5364(10) & 2&4625(20)\
$\overline{\rho}\p{-3}$ \[$\rm kg\,m^{-3}$\] & 1&3275(4) & 0&6880(2) & 1&2704(15) & 1&6377(40)\
$R_{\rm ref}\p{-7}$ \[m\] & 7&1398 & 6&0330 & 2&5559 & 2&5225\
$J_2\p{2}$ & 1&4736(1) & 1&629071(27) & 0&35160(32) & 0&34084(45)\
$J_4\p{4}$ & $-5$&87(5) & $-9$&358(28) & $-0$&354(41) & $-0$&334(29)\
$J_6\p{4}$ & 0&31(20) & 0&861(96) & &\
$P_{\omega}\p{-4}$ \[s\] & 3&57297(41) & 3&83624(47) ? & 6&206(4) & 5&800(20)\
$q$ & 0&08923(5) & 0&15491(10) & 0&02951(5) & 0&02609(23)\
$C/M\req^2$ & 0&258 & 0&220 & 0&230 & 0&241\
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Table \[tab:moments\] also indicates the radii obtained with the greatest accuracy by radio-occultation experiments. An important consequence obtained is the fact that these planets have low densities, from $0.688\gcc$ for Saturn to $1.64\gcc$ for Neptune, to be compared with densities of 3.9 to 5.5$\gcc$ for the terrestrial planets in the solar system. Considering the compression that strongly increases with mass, one is led naturally to the conclusion that these planets contain an important proportion of light materials including hydrogen and helium. It also implies that Uranus and Neptune which are less massive must contain a relatively larger proportion of heavy elements than Jupiter and Saturn. This may lead to a sub-classification between the hydrogen-helium giant planets Jupiter and Saturn, and the “ice giants” or “sub giants” Uranus and Neptune.
The planets are also relatively fast rotators, with periods of $\sim
10$ hours for Jupiter and Saturn, and $\sim 17$ hours for Uranus and Neptune. The fact that this fast rotation visibly affects the figure (shape) of these planets is seen by the significant difference between the polar and equatorial radii. It also leads to gravitational moments that differ significantly from a null value. However, it is important to stress that there is no unique rotation frame for these fluid planets: atmospheric zonal winds imply that different latitudes rotate at different velocities (see § \[sec:dynamics\]), and the magnetic field provides another rotation period. Because the latter is tied to the deeper levels of the planet, it is believed to be more relevant when interpreting the gravitational moments. The rotation periods listed in Table \[tab:moments\] hence correspond to that of the magnetic field. The case of Saturn is complex and to be discussed in the next section.
Magnetic fields {#sec:magnetic fields}
---------------
As the Earth, the Sun and Mercury, our four giant planets possess their own magnetic fields. These magnetic fields $\bf B$ may be expressed in form of a development in spherical harmonics of the scalar potential $W$, such that ${\bf B}=-\bfnabla W$: $$W=a\sum_{n=1}^{\infty} \left(\frac{a}{r} \right)^{n+1}
\sum_{m=0}^n \left\{g_n^m \cos(m\phi)+h_n^m \sin(m\phi)\right\}
P_n^m(\cos\theta).
\label{eq:W}$$ $r$ is the distance to the planet’s center, $a$ its radius, $\theta$ the colatitude, $\phi$ the longitude and $P_n^m$ the associated Legendre polynomials. The coefficients $g_n^m$ and $h_n^m$ are the magnetic moments that characterize the field. They are expressed in magnetic field units.
One can show that the first coefficients of relation (for $n=0$ and $n=1$) correspond to the potential of a magnetic dipole such that $W={\bf M\cdot r}/r^3$ of moment: M=a\^3 {(g\_1\^0)\^2 + (g\_1\^1)\^2 + (h\_1\^1)\^2}\^[1/2]{}.
As shown by the Voyager 2 measurements, Jupiter and Saturn have magnetic fields of essentially dipolar nature, of axis close to the rotation axis ($g_1^0$ is much larger than the other harmonics); Uranus and Neptune have magnetic fields that are intrinsically much more complex. To provide an idea of the intensity of the magnetic fields, the value of the dipolar moments for the four planets are $4.27\,\rm Gauss\,R_J^3$, $0.21\rm\,Gauss\,R_S^3$, $0.23\rm\,Gauss\,R_U^3$, $0.133\rm\,Gauss\,R_N^3$, respectively [@1982Natur.298...44C; @1983JGR....88.8771A; @1986Sci...233...85N; @1989Sci...246.1473N see also chapter by Connerney].
A true surprise from [*Voyager*]{} that has been confirmed by the [*Cassini-Huygens*]{} mission is that Saturn’s magnetic field is axisymetric [*to the limit of the measurement accuracy*]{}: Saturn’s magnetic and rotation axes are perfectly aligned [e.g., @RussellDougherty2010]. Voyager measurements indicated nevertheless a clear signature in the radio signal at $\rm 10^h 39^m 22.4^s$ believed to be a consequence of the rotation of the magnetic field. New measurements of a slower spin period of $\rm 10^h 47^m 6^s$ by Cassini [@2005Sci...307.1255G; @2006Natur.441...62G] have shown that the kilometric radiation was not directly tied to the period of the magnetic field but resulted from a complex interplay between the spin of the planetary magnetic field and the solar wind [e.g. @2005JGRA..11012203C]. New periods have been proposed: $\rm 10^h 32^m 35^s$ based on a minimization of the zonal differential rotation [@AndersonSchubert2007] and $\rm 10^h 33^m 13^s$ based on the latitudinal distribution of potential vorticity [@Read+2009]. The problem still stands out.
These magnetic fields must be generated in the conductive parts of the interiors, i.e., in metallic hydrogen at radii which are about 80% and 60% of the planetary radius for Jupiter and Saturn respectively (see sections \[sec:EOS\], \[sec:JupSat\] and e.g., @StanleyGlatzmaier2010). Recent models that consistently join the slowly convecting metallic interior with the non-conducting outer molecular envelope dominated by zonal flows result in a mainly dipolar magnetic field similar to the observations and further show that Jupiter’s stronger magnetic field and Saturn’s broader equatorial jet (see next section) can be interpreted as resulting from the deeper location of the transition region in Saturn [@HeimpelGomezPerez2011]. These explanations remain largely qualitative rather than quantitative and are further complicated by a necessary overforcing of the simulations [see @Showman+2011 and next section]. The question of why Saturn’s magnetic field is much more axisymmetric than Jupiter’s remains. Dipolar fields are obtained relatively naturally through the forcing of zonal jets extending down to the conducting region [@Guervilly+2012] but why Jupiter differs is unexplained. A possibility is that both fields are non-axisymmetric at deep levels but that Saturn’s is filtered by a more extended helium sedimentation region [@1983RPPh...46..555S], but in practice, a realistic solution yielding Saturn’s measured field has not been found [@StanleyGlatzmaier2010 and references therein].
Within Uranus and Neptune, the magnetic field is believed to be generated within a layer in which water is in an ionic phase, below about 80% of their total radius [see @Redmer+2011 and section \[sec:UraNep\] hereafter]. Their complex, multipolar magnetic fields has been thought to be a consequence of a strong stratification and of a dynamo generated in a thin shell [@StanleyBloxham2004]. However, this point of view is now challenged by new simulations that generate both planets’ magnetic fields and zonal wind structures through a thick shell dynamo [@Soderlund+2013]. Further work however must involve realistic variations of the interior density and conductivity.
Atmospheric dynamics: winds and weather {#sec:dynamics}
---------------------------------------
The atmospheres of all giant planets are evidently complex and turbulent in nature. This can, for example, be seen from the mean zonal winds (inferred from cloud tracking), which are very rapidly varying functions of the latitude [see e.g., @Ingersoll+95]: while some of the regions rotate at the same speed as the interior magnetic field (in the so-called “system III” reference frame), most of the atmospheres do not. Jupiter and Saturn both have superrotating equators ($+100$ and $+400\m\sec^{-1}$ in system III, for Jupiter and Saturn, respectively), Uranus and Neptune have subrotating equators, and superrotating high latitude jets. Neptune, which receives the smallest amount of energy from the Sun has the largest peak-to-peak latitudinal variations in wind velocity: about $600\m\sec^{-1}$. It can be noted that, contrary to the case of the strongly irradiated planets to be discussed later, the winds of Jupiter, Saturn, Uranus and Neptune, are significantly slower than the planet itself under its own spin (from 12.2km$\sec^{-1}$ for Jupiter to 2.6km$\sec^{-1}$ for Neptune, at the equator).
It is not yet clear whether the observed winds are driven from the bottom or from the top. The first possibility is that surface winds are related to motions in the planets’ interiors, which, according to the Taylor-Proudman theorem, should be confined by the rapid rotation to the plane perpendicular to the axis of rotation [@Busse78]. This is now backed by simulations in the anelastic limit (i.e., accounting for compressibility) which show that the outcome strongly depends on the density stratification in the interior. A small density contrast (as expected in Jupiter and Saturn) leads to equatorial superrotation whereas for a large one (as expected for Uranus and Neptune), the equatorial jet tends to subrotate [@Glatzmaier+2009; @Gastine+2013]. However, the application of these numerical results to the true conditions prevailing in the giant planets requires an extrapolation over at least 6 orders of magnitude [@Showman+2011]. The second possibility (not exclusive) is that winds are driven from the top by the injection of turbulence at the cloud level, which can also lead to the correct winds for the four giant planets [@LianShowman2010; @LiuSchneider2011].
Information on the gravity field of the planets can be used to constrain the interior rotation profile, as in the case of Uranus and Neptune whose observed jets appear to only extend to the outer 0.15% and 0.20% of the mass (corresponding to pressures of 2 and 4kbar) for Uranus and Neptune, respectively [@Kaspi+2013]. The method is promising with the perspective of the Juno measurements at Jupiter [@Liu+2013].
Our giant planets also exhibit planetary-scale to small-scale storms with very different temporal variations. For example, Jupiter’s great red spot is a 12000km-diameter anticyclone found to have lasted for at least 300 years [e.g. @SimonMiller+02]. Storms developing over the entire planet have even been observed on Saturn [@SanchezLavega+96]. Uranus and Neptune’s storm system has been shown to have been significantly altered since the Voyager era [@Rages+02; @Hammel+05; @dePater+2011]. On Jupiter, small-scale storms related to cumulus-type cloud systems have been observed [e.g., @Gierasch+00; @HSG02], and lightning strikes have been monitored by Galileo [e.g., @Little+99]. These represent only a small arbitrary subset of the work concerning the complex atmospheres of these planets.
It is tempting to extrapolate these observations to the objects outside our Solar System as well. However, two features governing the weather in these are not necessarily present for exoplanets [e.g., @Guillot99b]: their rapid rotation, and the presence of abundant condensing species and in particular one, water, whose latent heat can fuel powerful storms. But as we will see briefly in section \[sec:exoplanets\] theoretical models for exoplanets are now complemented by measurements of wind speeds and of global temperature contrasts, offering the perspective of a global approach to planetary weather and atmospheric dynamics.
Energy balance and atmospheric temperature profiles
---------------------------------------------------
Jupiter, Saturn and Neptune are observed to emit more energy than they receive from the Sun (see Table \[tab:flux\]). The case of Uranus is less clear. Its intrinsic heat flux $F_{\rm int}$ is significantly smaller than that of the other giant planets. With this caveat, all four giant planets can be said to emit more energy than they receive from the Sun. @Hubbard68 showed in the case of Jupiter that this can be explained simply by the progressive contraction and cooling of the planets.
[l r@[$\pm$]{}l r@[$\pm$]{}l r@[$\pm$]{}l r@[$\pm$]{}l]{} & & & &\
Absorbed power \[$10^{16}$ Js$^{-1}$\] & 50.14&2.48 & 11.14&0.50 & 0.526&0.037 & 0.204&0.019\
Emitted power \[$10^{16}$ Js$^{-1}$\] & 83.65&0.84 & 19.77&0.32 & 0.560&0.011 & 0.534&0.029\
Intrinsic power \[$10^{16}$ Js$^{-1}$\]& 33.5& 2.6 & 8.63&0.60 & & 0.330& 0.035\
Intrinsic flux \[Js$^{-1}$m$^{-2}$\] & 5.44& 0.43 & 2.01& 0.14 & & 0.433& 0.046\
Bond albedo \[\] & 0.343&0.032 & 0.342&0.030 & 0.300&0.049 & 0.290&0.067\
Effective temperature \[K\] & 124.4& 0.3 & 95.0& 0.4 & 59.1& 0.3 & 59.3& 0.8\
1-bar temperature \[K\] & 165&5 & 135&5 & 76&2 & 72&2\
\
A crucial consequence of the presence of an intrinsic heat flux is that it requires high internal temperatures ($\sim 10,000\K$ or more), and that consequently the giant planets are [*fluid*]{} (not solid) ([@Hubbard68]; see also [@Hubbard+95]). Another consequence is that they are essentially convective, and that their interior temperature profile are close to [*adiabats*]{}. We will come back to this in more detail.
The deep atmospheres (more accurately tropospheres) of the four giant planets are indeed observed to be close to adiabats, a result first obtained by spectroscopic models [@Trafton67], then verified by radio-occultation experiments by the Voyager spacecrafts, and by the [*in situ*]{} measurement from the Galileo probe (fig. \[fig:atm\_temp\]). The temperature profiles show a temperature minimum, in a region near 0.2| called the tropopause. At higher altitudes, in the stratosphere, the temperature gradient is negative (increasing with decreasing pressure). In the regions that we will be mostly concerned with, in the troposphere and in the deeper interior, the temperature always increases with depth. It can be noticed that the slope of the temperature profile in fig. \[fig:atm\_temp\] becomes almost constant when the atmosphere becomes convective, at pressures of a fraction of a bar, in the four giant planets.
It should be noted that the 1 bar temperatures listed in table \[tab:flux\] and the profiles shown in fig. \[fig:atm\_temp\] are retrieved from radio-occultation measurements using a helium to hydrogen ratio which, at least in the case of Jupiter and Saturn, was shown to be incorrect. The new values of $Y$ are found to lead to increased temperatures by $\sim 5\K$ in Jupiter and $\sim 10\K$ in Saturn [see @Guillot99a]. However, the Galileo probe found a 1 bar temperature of $166\K$ [@Seiff+98], and generally a good agreement with the Voyager radio-occultation profile with the wrong He/H$_2$ value.
When studied at low spatial resolution, it is found that all four giant planets, in spite of their inhomogeneous appearances, have a rather uniform brightness temperature, with pole-to-equator latitudinal variations limited to a few kelvins [e.g., @Ingersoll+95]. However, in the case of Jupiter, some small regions are known to be very different from the average of the planet. This is the case of hot spots, which cover about 1% of the surface of the planet at any given time, but contribute to most of the emitted flux at 5 microns, due to their dryness (absence of water vapor) and their temperature brightness which can, at this wavelength, peak to 260.
Atmospheric compositions {#sec:compositions}
------------------------
In fluid planets, the distinction between the atmosphere and the interior is not obvious. We name “atmosphere” the part of the planet which can directly exchange radiation with the exterior environment. This is also the part which is accessible by remote sensing. It is important to note that the continuity between the atmosphere and the interior does not guarantee that compositions measured in the atmosphere can be extrapolated to the deep interior, even in a fully convective environment: Processes such as phase separations [e.g., @Salpeter73; @SS77a; @FH03], phase transitions [e.g., @1989oeps.book..539H], chemical reactions [e.g., @FL94] and cloud formation [e.g. @Rossow1978] can occur and decouple the surface and interior compositions. Furthermore, imperfect mixing may also occur, depending on the initial conditions [e.g., @1985Icar...62....4S].
The conventional wisdom is however that these processes are limited to certain species (e.g. helium) or that they have a relatively small impact on the global abundances, so that the hydrogen-helium envelopes may be considered relatively uniform, from the perspective of the global abundance in heavy elements. An important caveat is that measurements must probe deeper than the condensation altitude for any volatile (e.g. ammonia, water, etc.). We first discuss measurements made in the atmosphere before inferring interior compositions from interior and evolution models.
### Hydrogen and helium {#sec:hhe}
The most important components of the atmospheres of our giant planets are also among the most difficult to detect: H$_2$ and He have a zero dipolar moment and hence absorb very inefficiently visible and infrared light. Absorption in the infrared becomes important only at high pressures as a result of collision-induced absorption [e.g., @1997AA...324..185B]. On the other hand, lines due to electronic transitions correspond to very high altitudes in the atmosphere, and bear little information on the structure of the deeper levels. The only robust result concerning the abundance of helium in a giant planet is by [*in situ*]{} measurement by the Galileo probe in the atmosphere of Jupiter [@1998JGR...10322815V]. The helium mole fraction (i.e., number of helium atoms over the total number of species in a given volume) is $q_{\rm He}=0.1359\pm 0.0027$. The helium mass mixing ratio $Y$ (i.e., mass of helium atoms over total mass) is constrained by its ratio over hydrogen, $X$: $Y/(X+Y)=0.238\pm 0.05$. This ratio is by coincidence that found in the Sun’s atmosphere, but because of helium sedimentation in the Sun’s radiative zone, it was larger in the protosolar nebula: $Y_{\rm proto}=0.275\pm 0.01$ and $(X+Y)_{\rm
proto}\approx 0.98$ [e.g., @1995RvMP...67..781B]. Less helium is therefore found in the atmosphere of Jupiter than inferred to be present when the planet formed. We will discuss the consequences of this measurement later: let us mention that the explanation invokes helium settling due to a phase separation in the interiors of massive and cold giant planets.
Helium is also found to be depleted compared to the protosolar value in Saturn’s atmosphere. However, in this case the analysis is complicated by the fact that Voyager radio occultations combined with the far-IR sounding (to separate effects of helium from that of temperature) led to a wrong value for Jupiter when compared to the Galileo probe data and hence are suspect for the other planets. The current adopted value from IR data only is now $Y=0.18-0.25$ [@ConrathGautier2000], in agreement with values predicted by interior and evolution models [@Guillot99b; @Hubbard+99].
Finally, as shown in table \[tab:comp\] hereafter, Uranus and Neptune are found to have near-protosolar helium mixing ratios, but with considerable uncertainty.
### Heavy elements {#sec:heavies}
[lllllll]{} & [**Element**]{} & [**Carrier**]{} & [**Abundance ratio/H**]{}$^\dagger$ & [**Protosun**]{}$^a$ & $\frac{\mbox{\bf Planet}}{\mbox{\bf Protosun}}$ & [**Method**]{}\
\
& He/H & He & $(7.85\pm 0.18)\times 10^{-2}$ & $9.69\times 10^{-2}$ & $0.810\pm 0.019$ & Galileo/GPMS $^b$\
& C/H & CH$_4$ & $(1.185\pm 0.019)\times 10^{-3}$ & $2.75\times 10^{-4}$ & $4.31\pm 0.07$ & Galileo/GPMS$^c$\
& N/H & NH$_3$ & $(3.3\pm 1.3)\times 10^{-4}$ & $8.19\times 10^{-5}$ & $4.05\pm 1.55$ &Galileo/GPMS$^c$\
& O/H & H$_2$O$^\star$ & $(1.49^{+0.98}_{-0.68})\times 10^{-4}$ & $6.06\times 10^{-4}$ & $0.25^{+0.16}_{-0.11}$ & Galileo/GPMS@19 bar$^c$\
& S/H & H$_2$S & $(4.5\pm 1.1)\times 10^{-5}$ & $1.55\times 10^{-5}$ & $2.88\pm 0.68$ & Galileo/GPMS$^c$\
& Ne/H & Ne & $(1.20\pm 0.12)\times 10^{-5}$ & $1.18\times 10^{-4}$ & $0.10\pm 0.01$ & Galileo/GPMS$^d$\
& Ar/H & Ar & $(9.10\pm 1.80)\times 10^{-6}$ & $3.58\times 10^{-6}$ & $2.54\pm 0.50$ & Galileo/GPMS$^d$\
& Kr/H & Kr & $(4.65\pm 0.85)\times 10^{-9}$ & $2.15\times 10^{-9}$ & $2.16\pm 0.40$ & Galileo/GPMS$^d$\
& Xe/H & Xe & $(4.45\pm 0.85)\times 10^{-10}$ & $2.11\times 10^{-10}$ & $2.11\pm 0.40$ & Galileo/GPMS$^d$\
& P/H & PH$_3$$^\star$ & $(1.11\pm 0.06)\times 10^{-6}$ & $3.20\times 10^{-7}$ & $3.45\pm 0.18$ & Cassini/CIRS$^e$\
& Ge/H & GeH$_4$$^\star$ & $(4.1\pm 1.2)\times 10^{-10}$ & $4.44\times 10^{-9}$ & $0.09\pm 0.03$ & Voyager/IRIS$^f$\
& As/H & AsH$_3$$^\star$ & $(1.3\pm 0.6)\times 10^{-10}$ & $2.36\times 10^{-10}$ & $0.54\pm 0.27$ & Ground/IR$^g$\
\
\
& He/H & He & $(6.75\pm 1.25)\times 10^{-2}$ & $9.69\times 10^{-2}$ & $0.70\pm 0.13$ & Voyager/IRIS$^h$\
& C/H & CH$_4$ & $(2.67\pm 0.11)\times 10^{-3}$ & $2.75\times 10^{-4}$ & $9.72\pm 0.41$ & Cassini/CIRS$^i$\
& N/H & NH$_3$$^\star$ & $(2.27\pm 0.57)\times 10^{-4}$ & $8.19\times 10^{-5}$ & $2.77\pm 0.69$ & Cassini/VIMS$^j$\
& S/H & H$_2$S & $(1.25\pm 0.17)\times 10^{-4}$ & $1.55\times 10^{-5}$ & $8.08\pm 1.10$ & Ground/radio$^k$\
& P/H & PH$_3$$^\star$ & $(4.65\pm 0.32)\times 10^{-6}$ & $3.20\times 10^{-7}$ & $14.5\pm 1.0$ & Cassini/CIRS$^e$\
& & & $(1.76\pm 0.17)\times 10^{-6}$ & $3.20\times 10^{-7}$ & $5.49\pm 0.53$ & Cassini/VIMS$^i$\
& & & $(4.0^{+1.7}_{-1.1})\times 10^{-6}$ & $3.20\times 10^{-7}$ & $12.4^{+5.3}_{-3.5}$ &Ground/IR$^l$\
& Ge/H & GeH$_4$$^\star$ & $(2.3\pm 2.3)\times 10^{-10}$ & $4.44\times 10^{-9}$ & $0.05\pm 0.05$ & Ground/IR$^l$\
& As/H & AsH$_3$$^\star$ & $(1.25\pm 0.17)\times 10^{-9}$ & $2.36\times 10^{-10}$ & $5.33\pm 0.73$ & Cassini/VIMS$^i$\
& & & $(1.71\pm 0.57)\times 10^{-9}$ & $2.36\times 10^{-10}$ & $7.3\pm 2.4$ & Ground/IR$^l$\
\
& He/H & He & $(9.0\pm 2.0)\times 10^{-2}$ & $9.69\times 10^{-2}$ & $0.93\pm 0.20$ & Voyager/IRIS+occult$^m$\
& C/H & CH$_4$$^\star$ & $(2.36\pm 0.30)\times 10^{-2}$ & $2.75\times 10^{-4}$ & $85.9\pm 10.7$ & Hubble/STIS$^n$\
& S/H & H$_2$S$^\star$ & $(3.2\pm 1.6)\times 10^{-4}$ & $1.55\times 10^{-5}$ & $21.0\pm 10.5$ & Ground/radio$^o$\
\
\
& He/H & He & $(1.17\pm 0.20)\times 10^{-1}$ & $9.69\times 10^{-2}$ & $1.21\pm 0.20$ & Voyager/IRIS+occult$^p$\
& C/H & CH$_4$$^\star$ & $(1.85\pm 0.43)\times 10^{-2}$ & $2.75\times 10^{-4}$ & $67.5\pm 15.8$ & Ground/IR$^q$\
& & & $(2.47\pm 0.62)\times 10^{-2}$ & $2.75\times 10^{-4}$ & $89.9\pm 22.5$ & Hubble/STIS$^r$\
& S/H & H$_2$S$^\star$ & $(3.2\pm 1.6)\times 10^{-4}$ & $1.55\times 10^{-5}$ & $21.0\pm 10.5$ & Ground/radio$^o$\
$^\star$: Species which condense or are in chemical disequilibrium, i.e., with vertical/horizontal variations of their concentration. The global elemental abundances are estimated from the maximum measured mixing ratio, but like in the case of H$_2$O in Jupiter (believed to correspond to the measurement in a dry downdraft), they may only be lower limits to the bulk abundance.\
$^\dagger$: Abundance ratios $r$ are measured with respect to atomic hydrogen. In these atmospheres dominated by molecular hydrogen and helium, mole fractions $f$ are found by $f=2r/(1+r_{\rm He})$ where $r_{\rm He}$ is the He/H abundance ratio.\
$^a$: protosolar abundances from [@Lodders+2009]; $^b$: [@VonZahn+1998]; $^c$: [@Wong+04]; $^d$: [@Atreya+2003]; $^e$: [@Fletcher+2009b]; $^f$: [@Kunde+1982]; $^g$: [@Noll+1990]; $^h$: [@ConrathGautier2000]; $^i$: [@Fletcher+2009a]; $^j$: [@Fletcher+2011]; $^k$: [@BS89]; $^l$: [@NollLarson1991]; $^m$: [@Conrath+1987]; $^n$: [@Sromovsky+2011]; $^o$: [@dePater+1991]; $^p$: [@Conrath+1991]; $^q$: [@BainesSmith1990]; $^r$: [@KarkoschkaTomasko2011].
The abundance of elements other than hydrogen and helium (that we will call hereafter “heavy elements”) bears crucial information for the understanding of the processes that led to the formation of these planets. Table \[tab:comp\] summarizes the present situation after [*in situ*]{} measurements in Jupiter by the Galileo probe, as well as spectroscopic measurements from spacecraft and from the ground for the other planets.
The elemental abundances in the giant planets’ atmospheres are most usefully compared to those in the Sun since they all originated from the protosolar disk. The solar abundances have seen very significant revisions in the past decade because it has been realized that convective motions in the Sun’s atmosphere affect spectral lines more extensively than was previously thought. It is not yet clear at this date whether the solar abundances have converged. Furthermore, as discussed for helium, heavy elements gradually settle towards the Sun’s interior so that a proper reference for the giant planets is not the solar atmosphere today, but its value 4.5 billion years ago which is model-dependent. Table \[tab:comp\] provides the values obtained for the protosun by [@Lodders+2009]. These are used as reference without accounting for their uncertainties.
The most abundant heavy elements in the envelopes of our four giant planets are O (presumably) and C, N and S. It is possible to model the chemistry of gases in the tropospheres from the top of the convective zone down to the 2000 K temperature level [@FL94]. Models conclude that, whatever the initial composition in these elements of planetesimals which collapsed with hydrogen onto Jupiter and Saturn cores during the last phase of the planetary formation, C in the upper tropospheres of giant planets is mainly in the form of gaseous CH$_4$, N in the form of NH$_3$, S in the form of H$_2$S, and O in the form of H$_2$O. All these gases but methane in Jupiter and Saturn condense in the upper troposphere and vaporize at deeper levels when the temperature increases. Noble gases do not condense even at the tropopauses of Uranus and Neptune, the coldest regions in these atmospheres.
Jupiter is the planet which has been best characterized thanks to the measurements of the Galileo atmospheric probe which precisely measured the abundances of He, Ne, Ar, Kr, Xe, CH$_4$, NH$_3$, H$_2$S, and H$_2$O down to pressures around 22bars. As helium, neon was found to be depleted compared to the protosolar value, in line with theoretical predictions that this atom would fall in with the helium droplets [@RS95; @WilsonMilitzer2010 and section \[sec:others\]]. C, N, and S were found to be supersolar by a factor 2.5 to 4.5 [@Wong+04], which was not unexpected because condensation of nebula gases results in enriching icy grains and planetesimals. The surprise came from Ar, Kr, Xe, which were expected to be solar because they are difficult to condense, but turned out to be supersolar by a factor $\sim 2$ [@Owen+99; @Wong+04].
H$_2$O is difficult to measure in all four giant planets because of its condensation relatively deep. It was hoped that the Galileo probe would provide a measurement of its deep abundance, but the probe fell into one of Jupiter’s 5-micron hot spots, now believed to be a dry region mostly governed by downwelling motions [e.g., @SI98]. As a result, and although the probe provided measurements down to 22 bars, well below water’s canonical 5 bar cloud base, it is believed that this measurement of a water abundance equal to a fraction of the solar value is only a lower limit. An indirect determination comes from the measurement of the disequilibrium species CO which has to be transported fast from the deep levels where H$_2$O and CH$_4$ tend to form more CO (and H$_2$). This predicts a mostly solar to slightly supersolar (by a factor 2) abundance of O in Jupiter [@VisscherMoses2011] and much larger enrichments in Neptune [@LF94]. This however depends crucially on the reaction network and somewhat on assumptions on mixing, both of which are not well known. The abundance of oxygen, the most abundant element in the Universe after hydrogen and helium and a crucial planetary building block is essentially unknown for what concerns our four giant planets.
Three other species can help us probe the bulk elemental abundance inside Jupiter, although with larger difficulties perhaps because they are not necessarily in chemical equilibrium at the levels where they are detected and their measured abundances are thus not necessarily representative of their bulk abundance: these are PH$_3$, GeH$_4$ and AsH$_3$. All three where detected remotely rather than in situ. The first one is clearly supersolar in Jupiter, with an enrichment in between that measured for C and S. GeH$_4$ is clearly subsolar, but this is not surprising because of condensation into solid Ge and GeS [@FL94]. The same chemical models would predict that the measured abundance of AsH$_3$ should be close to its bulk abundance. The measured abundance therefore could be interpreted as a subsolar bulk abundance of As, but with considerable uncertainty.
Table \[tab:comp\] shows that Saturn’s atmosphere is more enriched in heavy elements than Jupiter. Unfortunately, unlike Jupiter, no in situ measurement has been performed in this planet and we can only rely on remote sensing. But we can confidently assess that the abundances of C, S, P and As are significantly higher than in Jupiter. Saturn has about twice more C (as CH$_4$) and S (as H$_2$S) than Jupiter, for a given mass of atmosphere. The situation for PH$_3$ is unclear, both because it is highly variable both vertically and latitudinally: the enrichment could be only slightly more than in Jupiter to more than 4 times that value [@Fletcher+2009b; @Fletcher+2011]. Note that in the presence of horizontal variability (as for this molecule) table \[tab:comp\] indicates the maximum abundances measured - which should be closer to the bulk abundance, except if there exist mechanisms to preferentially trap certain species. The enrichment in N (as NH$_3$) is smaller than in Jupiter. This may be due to its condensation deeper as NH$_4$SH [@GJO78], although it does not explain why Jupiter and Saturn would be that different in that respect. The enrichment in As (as AsH$_4$) is considerably larger than in Jupiter, which is also a mystery [@Fletcher+2009a]. At least, GeH$_4$ appears to be of equally low abundance in both planets, but this is probably more related to its condensation than to its bulk abundance.
Finally, Uranus and Neptune provide all signs of a significant enrichment in heavy elements, even though very few elements have been detected. Methane is the most important one, although the fact that it condenses in these planets complicates the interpretation of the spectroscopic measurements. Large-scale variations with latitude are observed, in particular in Uranus [@Sromovsky+2011], but less so in Neptune [@KarkoschkaTomasko2011]. However, the deep abundances provided in table \[tab:comp\] are very similar for both planets, with a $\sim 90$ times solar enrichment. This is much higher than the $\sim 30$ times solar enrichments discussed in past reviews [e.g., @Gautier+95] for two reasons: one is a decrease of the protosolar abundance itself. The other is the fact that it relied on spectroscopic measurements probing higher atmospheric levels affected by methane condensation [see @KarkoschkaTomasko2011].
The other key species detected in Uranus and Neptune thanks to ground-based radio observations is H$_2$S, which points to a $10$ to $30$ enrichment in sulfur [@dePater+1991]. Because this element condenses at even greater depths than methane, the bulk abundance of S in these planets may be larger if the global circulation is indeed important down to the deep levels probed by the radio waves. The measurement is however a difficult one, with other potential absorbers affecting the smooth microwave spectra yielding degenerate solutions.
Overall, the global picture that can be drawn is that of an increase of the abundance of heavy elements compared to the solar value with increasing distance to the Sun, from Jupiter which shows a $\sim2$ to 4 enrichment, Saturn a $3$ to $10$ one, and Uranus and Neptune which are enriched by a factor $\sim 90$ in carbon and by at least 10 to 30 in sulfur. In spite of their different atmospheric dynamics, and with the present accuracy of the measurements, the two ice giants have very similar abundances.
Isotopic ratios
---------------
[llllll]{} & [**Isotope**]{} & [**Isotopic ratio**]{} & [**Protosun**]{}$^a$ & [**Planet/Protosun**]{} & [**Comments**]{}\
\
& D/H & $(2.25\pm 0.35)\times 10^{-5}$&$1.94\times 10^{-5}$&$1.16\pm0.18$& ISO/SWS$^q$\
&$^3$He/$^4$He&$(1.66\pm 0.06)\times 10^{-4}$&$1.66\times 10^{-4}$&$1.00\pm0.03$ &Galileo/GPMS$^d$\
&$^{13}$C/$^{12}$C&$(1.08\pm 0.05)\times 10^{-2}$&$1.12\times 10^{-2}$&$1.04\pm 0.05$&Galileo/GPMS$^d$\
&$^{15}$N/$^{14}$N&$(2.30\pm 0.30)\times 10^{-3}$&$2.27\times 10^{-3}$&$0.99\pm0.13$&Galileo/GPMS$^d$\
&$^{22}$Ne/$^{20}$Ne&$(7.7\pm 1.2)\times 10^{-2}$&$7.35\times 10^{-2}$&$0.96\pm 0.15$ &Galileo/GPMS$^d$\
&$^{38}$Ar/$^{36}$Ar&$(1.79\pm 0.08)\times 10^{-1}$&$1.82\times 10^{-1}$&$1.02\pm 0.05$&Galileo/GPMS$^d$\
&$^{128}$Xe/Xe&$(1.80\pm 0.20)\times 10^{-2}$&$2.23\times 10^{-2}$&$1.24\pm 0.14$&Galileo/GPMS$^d$\
&$^{129}$Xe/Xe&$(2.85\pm 0.21)\times 10^{-1}$&$2.75\times 10^{-1}$&$0.96\pm 0.07$&Galileo/GPMS$^d$\
&$^{130}$Xe/Xe&$(3.80\pm 0.50)\times 10^{-2}$&$4.38\times 10^{-2}$&$1.15\pm 0.15$&Galileo/GPMS$^d$\
&$^{131}$Xe/Xe&$(2.03\pm 0.18)\times 10^{-1}$&$2.18\times 10^{-1}$&$1.07\pm 0.10$&Galileo/GPMS$^d$\
&$^{132}$Xe/Xe&$(2.90\pm 0.20)\times 10^{-1}$&$2.64\times 10^{-1}$&$0.91\pm 0.06$&Galileo/GPMS$^d$\
&$^{134}$Xe/Xe&$(9.10\pm 0.70)\times 10^{-2}$&$9.66\times 10^{-2}$&$1.06\pm 0.08$&Galileo/GPMS$^d$\
\
&D/H&$(1.60\pm 0.20)\times 10^{-5}$&$1.94\times 10^{-5}$&$0.83\pm 0.11$&Cassini/CIRS$^h$\
& &$(1.70^{+0.75}_{-0.45})\times 10^{-5}$&$1.94\times 10^{-5}$&$0.88^{+0.39}_{-0.23}$& ISO/SWS$^q$\
&$^{13}$C/$^{12}$C&$(1.09\pm 0.10)\times 10^{-2}$&$1.12\times 10^{-2}$&$1.03\pm 0.09$&Cassini/CIRS$^h$\
\
&D/H&$(4.40\pm 0.40)\times 10^{-5}$&$1.94\times 10^{-5}$&$2.27\pm 0.21$&Herschel/PACS$^r$\
\
&D/H&$(4.10\pm 0.40)\times 10^{-5}$&$1.94\times 10^{-5}$&$2.11\pm 0.21$&Herschel/PACS$^r$\
$^a$: protosolar abundances from [@Lodders+2009], except $^{15}$N/$^{14}$N which is corrected by [@Marty+2011]; $^d$: [@Atreya+2003]; $^h$: [@Fletcher+2009a]; $^q$: [@Lellouch+2001]; $^r$: [@Feuchtgruber+2013].
The measurement of isotopic ratios in planetary atmospheres is a powerful tool to understand their origin. Table \[tab:isotopes\] provides the ensemble of isotopic ratios measured in our giant planets, and a comparison to their values in the Sun.
Of course, the Galileo probe and its onboard mass spectrometer have provided us a strikingly clear picture of Jupiter’s atmosphere: it is directly formed from the same material as our Sun, with isotopic ratios which are, to the accuracy of the measurements, indistinguishable (i.e., within 2 sigma) from the solar values and for elements as diverse as D, He, C, N, Ne, Ar and Xe with as many as 6 isotopes measured. This was expected because indeed Jupiter’s composition is globally similar to that of the Sun, but given the fact that the abundances of elements are far from being Sun-like, it is perhaps surprising to find such a good match! By extension, this applies to Saturn although only the deuterium to hydrogen and $^{13}$C/$^{12}$C isotopic ratios could be measured by remote spectroscopic observations. This confirms that the atmospheres and envelopes of Jupiter and Saturn originated from the same material that formed the Sun and that mechanisms leading to isotopic fractionation (e.g., atmospheric evaporation) were of limited importance.
In the case of Uranus and Neptune, only the deuterium to hydrogen ratio was measured, from the ground in the infrared [@Irwin+2014] and most precisely by recent far infrared spectroscopy from Herschel [@Feuchtgruber+2013]. Interestingly, it is about twice larger than the protosolar value, and a factor 2 to 6 times smaller than the D/H value in comets. Given our present knowledge of the interiors of Uranus and Neptune, @Feuchtgruber+2013 conclude that either these planets contain much more rocks than expected or that the ices in their interior have not been fully mixed.
Moons and rings
---------------
A discussion of our giant planets motivated by the opportunity to extrapolate the results to objects outside our solar system would be incomplete without mentioning the moons and rings that these planets all possess (see chapters by Breuer & Moore, by Peale & Canup and by Hussmann et al.). First, the satellites/moons can be distinguished from their orbital characteristics as regular or irregular. The first ones have generally circular, prograde orbits. The latter tend to have eccentric, extended, and/or retrograde orbits.
These satellites are numerous: After the Voyager era, Jupiter was known to possess 16 satellites, Saturn to have 18, Uranus 20 and Neptune 8. Recent extensive observation programs have seen the number of satellites increase considerably, with a growing list of satellites presently reaching 62, 56, 27 and 13 for Jupiter, Saturn, Uranus and Neptune, respectively. All of the new satellites discovered since Voyager are classified as irregular.
The presence of regular and irregular satellites is due in part to the history of planet formation. It is believed that the regular satellites have mostly been formed in the protoplanetary subnebulae that surrounded the giant planets (at least Jupiter and Saturn) at the time when they accreted their envelopes. On the other hand, the irregular satellites are thought to have been captured by the planet. This is, for example, believed to be the case of Neptune’s largest moon, Triton, which has a retrograde orbit.
A few satellites stand out by having relatively large masses: it is the case of Jupiter’s Io, Europa, Ganymede and Callisto, of Saturn’s Titan, and of Neptune’s Triton. Ganymede is the most massive of them, being about twice the mass of our Moon. However, compared to the mass of the central planet, these moons and satellites have very small weights: $10^{-4}$ and less for Jupiter, $1/4000$ for Saturn, $1/25000$ for Uranus and $1/4500$ for Neptune. All these satellites orbit relatively closely to their planets. The farthest one, Callisto revolves around Jupiter in about 16 Earth days.
The four giant planets also have rings, whose material is probably constantly resupplied from their satellites. The ring of Saturn stands out as the only one directly visible with binoculars. In this particular case, its enormous area allows it to reflect a sizable fraction of the stellar flux arriving at Saturn, and makes this particular ring as bright as the planet itself. The occurrence of such rings would make the detection of extrasolar planets slightly easier, but it is yet unclear how frequent they can be, and how close to the stars rings can survive both the increased radiation and tidal forces.
Seismology
----------
The best way to directly probe planetary (or stellar) interiors is through seismology, i.e., by measuring the spectrum of waves propagating through the interior. Our knowledge of the interior structure of the Earth, the Sun and even other stars is largely due to the ability to detect the oscillations of these objects. Because Jupiter (and by extent the other giant planets in our Solar System) are similar in composition and density to the Sun, the possibility to detect pressure waves in their atmosphere has been proposed in the 1970’s. In particular, [@VZL76] showed that waves with periods smaller than about 10 minutes would be trapped and reflected downwards in Jupiter’s and Saturn’s atmospheres, creating the possibility for resonant waves similar to those observed in the Sun to exist. On the theoretical side, the possibility of how these waves may be excited has however remained problematic [e.g., @BS87].
After two decades of promising but slow progress, the case for the existence of detectable free oscillations of giant planets has recently taken a new turn. First, using ground-based Doppler imaging of Jupiter, [@Gaulme+11] detected an oscillation pattern with frequencies between 0.8 and 2 mHz and a characteristic spacing of the peak of $155.3\pm 2.2\,\mu$Hz, in agreement with theoretical models. Separately, [@HedmanNicholson2013] confirmed that waves observed by the Cassini spacecraft in Saturn’s rings cannot be caused by satellites and therefore must result from oscillations in Saturn, as had been proposed by [@MarleyPorco1993].
Further observations are required to better characterize the oscillations of these planets and start using them as probes of the planetary interiors [e.g., @Jackiewicz+2012]. This is very promising however and should lead to a revolution in our understanding of the giant planets.
Exoplanets {#sec:exoplanets}
----------
Huge progress has been made in the field of extrasolar planets since the detection of the first giant planet orbiting a solar-type star by @MQ95. As shown in figure \[fig:exo\], more than a thousand planets are known at the time of this review, and importantly, more than four hundred planets that transit their star at each orbital revolution have been identified [@Wright+11; @Schneider+11]. These transiting planets are especially interesting because of the possibility to measure both their mass and size and thus obtain constraints on their global composition.
In spite of their particular location just a few stellar radii away from their stars, the transiting giant planets that have been discovered bear some resemblance with their Solar System cousins in the sense that they are also mostly made of hydrogen and helium [e.g., @Burrows+00; @Guillot05; @Baraffe+05]. They are, however, much hotter due to the intense irradiation that they receive.
Although obtaining direct information on these planets represents a great observational challenge, several key steps have been accomplished: Atomic sodium, predicted to be detectable [@SS00], has indeed been detected by transit spectroscopy[@CBNG02] early on, and a tentative abundance measured in planet HD209458b: According to @Sing+2009, it appears to be oversolar by a factor $\sim 2$ at pressures deeper than about $\sim 3\,$mbar and undersolar above that level [see also @Vidal-Madjar+2011]. Hydrodynamically escaping species (including hydrogen, oxygen, carbon, nitrogen and heavier ions) have also been detected around the brightest hot Jupiters [e.g., @VidalMadjar+03; @Fossati+10; @Bourrier+13]. A theoretical study of the atmospheric dynamics of hot Jupiters [@SG02] predicted strong day-night temperature variations (up to 100’s K), fast km/s zonal jets and a displacement of the hottest point west of the substellar point. These have been confirmed by observations of transiting and non-transiting planets in the infrared [@Harrington+06; @Knutson+07] and by doppler-imaging of planetary CO lines [@Snellen+10].
Unfortunately, the list of chemical species thought to have been detected [see @SD10 for a review] has dwindled in recent years due to the realization that instrumental effects could mimic spectral signatures [e.g., @Desert+2009; @Gibson+2011; @Crouzet+2012], due to new observations with a better instrument [e.g., @Deming+2013] and generally because of the unexpected prevalence of hazes in these close-in exoplanets [e.g., @Sing+2009; @Pont+13]. After examination, claims of a high C/O ratio in some of these atmospheres also appear to be highly uncertain [@Crossfield+2012].
In any case, in spite of the hiccups, there is obviously a big potential for growth in this young field, and the comparison between fine observations made for giant planets in our Solar System and the more crude, but also more statistically significant data obtained for planets around other stars promise to be extremely fruitful to better understand these objects.
The calculation of interior and evolution models
================================================
Basic equations {#sec:basic}
---------------
The structure and evolution of a giant planet is governed by the following hydrostatic, thermodynamic, mass conservation and energy conservation equations: $$\begin{aligned}
{{\partial P\over\partial r}}&=&-\rho g \label{eq:dpdr}\\
{\partial T\over\partial r}&=&{\partial P\over \partial r}{T\over
P}\nabla_T. \label{eq:dtdr}\\
{\partial m\over\partial r}&=&4\pi r^2\rho. \label{eq:dmdr}\\
{\partial L\over\partial r}&=&4\pi r^2\rho \left(\dot{\epsilon}-
T{\partial S\over \partial t}\right),\label{eq:dldr}\end{aligned}$$ where $P$ is the pressure, $\rho$ the density, and $g=Gm/r^2$ the gravity ($m$ is the mass, $r$ the radius and $G$ the gravitational constant). The temperature gradient $\nabla_T\equiv(d\ln T/d\ln P)$ depends on the process by which the internal heat is transported. $L$ is the intrinsic luminosity, $t$ the time, $S$ the specific entropy (per unit mass), and $\dot{\epsilon}$ accounts for the sources of energy due e.g., to radioactivity or more importantly nuclear reactions. Generally it is a good approximation to assume $\dot{\epsilon}\sim 0$ for objects less massive than $\sim 13\mjup$, i.e., too cold to even burn deuterium (but we will see that in certain conditions this term may be useful, even for low mass planets).
The boundary condition at the center is trivial: $r=0$; ($m=0$, $L=0$). The external boundary condition is more difficult to obtain because it depends on how energy is transported in the atmosphere. One possibility is to use the Eddington approximation, and to write [e.g., @Chandrasekhar39]: $r=R$; ($T_0=\teff$, $P_0=2/3\,g/\kappa$), where $\teff$ is the effective temperature (defined by $L=4\pi R\sigma\teff^4$, with $\sigma$ being the Stephan-Boltzmann constant), and $\kappa$ is a mean opacity. Note for example that in the case of Jupiter $\teff=124$K, $g=26\rm\,m\,s^{-2}$ and $\kappa\approx 5\times 10^{-3}
(P/1\rm\,bar)\,m^2\,kg^{-1}$. This implies $P_0\approx 0.2$bar (20,000Pa), which, given the simplicity of the calculation, is surprisingly close to the location of Jupiter’s real tropopause where $T\approx 110$K. Actually, the properties of the opacities of important absorbing chemical species like water and their pressure dependence imply that photospheres around 0.1 bar should be common [@RobinsonCatling2014].
However, the Eddington boundary condition should not be used in the case of irradiated atmospheres because it does not properly account for both the incoming flux (mostly at visible wavelengths for planets around solar-type stars) and the intrinsic flux (in the infrared). The fact that opacities differ at these wavelengths yields the possibility of thermal inversions (higher visible than infrared opacities) or a greenhouse effect (lower visible than infrared opacities) and thus a hotter interior, something that cannot be captured without accounting for the different fluxes. Analytical solutions of the radiative transfer problem exist in the semi-grey case (two opacities for the visible and infrared, respectively) [@Hansen2008; @Guillot2010], and can even be extended to include non-grey effects [@ParmentierGuillot2014]. Numerical solutions in the non-irradiated, solar-composition case are provided by @Saumon+96, and for the irradiation levels and compositions relevant for the solar system giant planets by @Fortney+2011. In that case, a grid is used to relate the atmospheric temperature and pressure at a given level to the radius $R$, intrinsic luminosity $L$ and incoming stellar luminosity $\linc$: $r=R$; ($T_0=T_0(R,L,\linc)$, $P_0=P_0(R,L,\linc)$). $P_0$ is chosen to satisfy the condition that the corresponding optical depth at that level should be much larger than unity.
High pressure physics & equations of state {#sec:EOS}
------------------------------------------
### Hydrogen
In terms of pressures and temperatures, the interiors of giant planets lie in a region for which accurate equations of state (EOS) are extremely difficult to calculate. This is because both molecules, atoms, and ions can all coexist, in a fluid that is partially degenerate (free electrons have energies that are determined both by quantum and thermal effects) and partially coupled (Coulomb interactions between ions are not dominant but must be taken into account). The presence of many elements and their possible interactions further complicate matters. For lack of space, this section will mostly focus on hydrogen whose EOS has seen the most important developments in recent years. A phase diagram of hydrogen (fig. \[fig:phase\_diag\]) illustrates some of the important phenomena that occur in giant planets.
The photospheres of giant planets are generally relatively cold (50 to 3000K) and at low pressure (0.1 to 10bar, or $10^4$ to $10^6$Pa), so that hydrogen is in molecular form and the perfect gas conditions apply. As one goes deeper into the interior hydrogen and helium progressively become fluid. (The perfect gas relation tends to underestimate the pressure by 10% or more when the density becomes larger than about $0.02\gcc$ ($P\wig{>} 1\,$kbar in the case of Jupiter)).
Characteristic interior pressures are considerably larger however: as implied by Eqs. \[eq:dpdr\] and \[eq:dmdr\], $P_{\rm c}\approx
GM^2/R^4$, of the order of 10-100Mbar for Jupiter and Saturn. As shown in fig. \[fig:phase\_diag\], all the central pressures and temperatures of giant planets and brown dwarfs (from Uranus to CoRoT-15b) lie in a regime of high pressures and temperatures lower than the corresponding Fermi temperature $T_{\rm F}$, implying that electrons are degenerate: their pressure is mostly a function of the density of the material. In Jupiter and Saturn, the degeneracy parameter $\theta=T/T_{\rm F}$ is always close to $0.03$. Even for the warmer CoRoT-15b (a $\sim 60\rm\,M_{Jup}$ brown dwarf discovered in transit in front of its parent star – see @Bouchy+2011), $\theta\approx 0.2$. This implies that for these objects, the thermal component is small so that the energy of electrons in the interior is expected to be only slightly larger than their non-relativistic, fully degenerate limit: $u_{\rm e}\ge
3/5\,kT_{\rm F} =15.6\left(\rho/\mu_{\rm e}\right)^{2/3}\ \rm eV$, where $k$ is Boltzmann’s constant, $\mu_{\rm e}$ is the number of nucleons per electron and $\rho$ is the density in $\rm
g\,cm^{-3}$. For pure hydrogen, when the density reaches $\sim
0.8\gcc$, the average energy of electrons becomes larger than hydrogen’s ionization potential, even at zero temperature: hydrogen pressure-ionizes and becomes metallic. This molecular to metallic transition occurs near Mbar pressures, but exactly how this happens is a result of the complex interplay of thermal, Coulomb and degeneracy effects.
Recent laboratory measurements on fluid deuterium have been able to reach extremely high pressures up to 20Mbar [@Mochalov+2012]. Beyond that experimental feat, most of the progress of the decade in the domain has been the improvement in our understanding of the hydrogen metallization region at pressures of a fraction to a few Mbars, both from an experimental and numerical point of view [see the very complete review by @McMahon+2012]. Already in the 1990’s, gas-gun experiments had been able to measure a rise in the conductivity of molecular hydrogen up to $T\sim 3000\K$, $P\sim 1.4\,$Mbar, a sign that metallization had been reached [@WMN96]. A very sharp transition, probably discontinuous, was then later measured by isentropic convergent explosive shock experiments [@Fortov+2007] at pressures between 1.5 and 2.5Mbar but uncertain temperatures below 4000K[see @McMahon+2012]. New experiments at higher temperatures using laser compression, directly [@Sano+2011] and from precompressed targets [@Loubeyre+2012], confirmed that this transition from a weakly conducting molecular fluid to a metal-like hydrogen fluid occurs around pressures near 1Mbar and temperatures as high as 15,000K, but that it is continuous at these temperatures. In parallel, [*ab initio*]{} calculations of the behavior of fluid hydrogen in this thermodynamical regime predicted the existence of a first order liquid-liquid phase transition (the so-called PPT for [*Plasma Phase Transition*]{}) with a critical point near $T\sim 1500-2000\,$K and $P\sim 2.2\,$Mbar [@Morales+2010; @Morales+2013a]. This discontinuous transition at low temperatures merges into a continuous transition at temperatures above the critical point, in good agreement with the experimental data. The PPT is indicated by a thick almost vertical line in fig. \[fig:phase\_diag\]. It is prolonged by a dashed line indicating the location of the continuous transition from molecular to metallic hydrogen that extends up to the region of thermal dissociation and ionization of hydrogen.
The controversy that had arisen between laser-induced shock compression [@daSilva+97; @Collins+98] and pulsed-power shock compression [@Knudson+04] regarding the maximum compression of deuterium along the principal shock Hugoniot has now been resolved in favor of the latter thanks to new experiments [@Boriskov+2005; @Hicks+2009] and the realization that the equation of state of quartz used to calibrate the laser-induced shock experiments was incorrect [@KnudsonDesjarlais2009]. Similarly, the existence of a PPT of hydrogen at high temperatures in a regime crossing the adiabats of Jupiter and Saturn [@SCvH95] have now been shown to be a spurious effect resulting from the different treatment of molecules, atom and ions within the so-called chemical picture [@Chabrier+2007]. Both laboratory experiments and independent models based on first-principles [@Militzer+01; @Desjarlais03; @BMG04; @Vorberger+2007; @French+2012] now agree and show that the transition from molecular to metallic hydrogen should occur continuously in all known giant planets.
Progress has also been made on the issue of the solidification of hydrogen [see @McMahon+2012 and references therein]. This has led to the confirmation that the interiors of the hydrogen-helium giant planets and brown dwarfs are [*fluid*]{} whatever their age, a result expected since the pioneering study by @Hubbard68. Of course, because of their initial gravitational energy, these objects are warm enough to avoid the critical point for the liquid gas transition in hydrogen and helium, at very low temperatures, but they also lie comfortably above the solidification lines for hydrogen and helium. (An [*isolated*]{} Jupiter should begin partial solidification only after at least $\sim
10^3\,$Ga of evolution.) They are considered to be fluid because at the high pressures and relatively modest temperatures in their interiors, Coulomb interactions between ions play an important role in the EOS and yield a behavior that is more reminiscent of that of a liquid than that of a gas, contrary to what is the case in e.g., solar-like stars. For Uranus and Neptune, the situation is actually more complex because at large pressures they are not expected to contain a significant amount of hydrogen (see next section).
As fig. \[fig:phase\_diag\] highlights, while some highly irradiated planets and brown dwarfs like CoRoT-15b have temperature profiles that get close to the hydrogen thermal dissociation line, most of them are well within the molecular hydrogen regime at low-pressures and in the metallic, degenerate regime at high pressures. Stars like our Sun lie in a higher temperature regime for which the EOS is dominated by thermal effects and electrons are essentially non-degenerate.
### Other elements and mixtures {#sec:others}
Hydrogen is of course a key element, but it is not sufficient to describe the structure of all giant planets. A description of the high-pressure behavior of other elements would go beyond the scope of the present review. We only sketch a few important results here.
In order to obtain tractable equations of state in the entire domain of pressure and temperature spanned by the planets during their evolution, one has to consider simplifications, among which the first one is to consider that an element (e.g., hydrogen) dominates, and that others can be considered as a perturbation. This is done for the hydrogen-helium mixture for the now classical EOS by @SCvH95, and now with more up-to-date EOSs [@Caillabet+2011; @Nettelmann+2012; @MilitzerHubbard2013]. The addition of other elements can be done through the so-called additive volume rule which is generally a good approximation given other sources of uncertainty (@Vorberger+2007; see also @ChabrierAshcroft1990).
Equations of state for elements other than hydrogen and helium in the parameter range relevant for giant planetary interiors have traditionally been difficult to obtain, and are often not easily shared. Beyond an extrapolation from the classical ANEOS and Sesame tables [see @SG04 and references therein], new results have become available. In particular, ab-initio simulations revealed that when compressing water or ammonia along an isentrope from conditions relevant to the atmospheres of Uranus and Neptune, they transition from a molecular to a ionic fluid, then to a superionic fluid and finally to a plasma [@Cavazzoni+1999]. The superionic fluid corresponds to a state in which protons move relatively freely among a lattice of oxygen atoms. An equation of state for water has been calculated from first-principles by @French+2009 and is found to be in good agreement with experiments [@Knudson+2012]. A similar equation of state for ammonia is presented by @Bethkenhagen+2013. In models of Uranus and Neptune, for water, the ionic transition occurs near 0.1Mbar and the superionic transition near 1Mbar [@Redmer+2011].
In some cases however, generally at low enough temperatures for a given pressure range, mixtures cannot remain homogeneous. This further complicates the calculation of equations of state and has important physical consequences for the planetary structure: the two components having different molecular weights, they tend to be separated by gravity so that the heavier component settles down under the lighter one. This is the case of the hydrogen and helium mixture for which it was proposed that such a separation would occur in Saturn already in the 1970s [@Salpeter73; @SS77b], but for which realistic calculations have only become possible in the past decade or so [see @Morales+2009; @Lorenzen+2011; @Morales+2013b]. Figure \[fig:hhe\] shows the comparison between two of these calculations based on first-principles simulations. According to these calculations, Saturn’s interior is in the phase separation region below 1Mbar but whether Jupiter is too depends on which calculation is considered.
The separation of other mixtures have also been calculated and is relevant to understand the initial formation and subsequent possible erosion of the cores of giant planets [see @GSHS04]. First-principles calculations of the water in metallic hydrogen (at pressures above 10Mbar) predict a critical phase separation temperature of less than 4000K [@WilsonMilitzer2010] implying that water is completely soluble in hydrogen in the interiors of Jupiter, Saturn and generally gas giants. This is also the case of iron, with a critical temperature of around 2000K [@Wahl+2013]. Finally, mixing rocks (specifically MgO) and metallic hydrogen is also relatively easy, even though the critical temperature for the same pressure range is higher, of order 10,000K or less [@WilsonMilitzer2012].
Heat transport
--------------
Giant planets possess hot interiors, implying that a relatively large amount of energy has to be transported from the deep regions of the planets to their surface. This can either be done by radiation, conduction, or, if these processes are not sufficient, by convection. Convection is generally ensured by the rapid rise of the opacity with increasing pressure and temperature. At pressures of a bar or more and relatively low temperatures (less than 1000K), the three dominant sources of opacities are water, methane and collision-induced absorption by hydrogen molecules.
However, in the intermediate temperature range between $\sim 1200$ and $1500\K$, the Rosseland opacity due to the hydrogen and helium absorption behaves differently: the absorption at any given wavelength increases with density, but because the temperature also rises, the photons are emitted at shorter wavelengths, where the monochromatic absorption is smaller. As a consequence, the opacity can decrease. This was shown by Guillot et al. (1994) to potentially lead to the presence of a deep radiative zone in the interiors of Jupiter, Saturn and Uranus.
This problem must however be reanalyzed in the light of observations and analyses of brown dwarfs. Their spectra show unexpectedly wide sodium and potassium absorption lines (see Burrows, Marley & Sharp 2000), in spectral regions where hydrogen, helium, water, methane and ammonia are relatively transparent. It thus appears that the added contribution of these elements (if they are indeed present) would wipe out any radiative region at these levels [@GSHS04].
At temperatures above $1500\sim 2000\K$ two important sources of opacity appear: (i) the rising number of electrons greatly enhances the absorption of H$_2^-$ and H$^-$; (ii) TiO, a very strong absorber at visible wavelengths is freed by the vaporization of CaTiO$_3$. Again, the opacity rises rapidly which ensures a convective transport of the heat. Still deeper, conduction by free electrons becomes more efficient, but the densities are found not to be high enough for this process to be significant, except perhaps near the central core [see @Hubbard68; @SS77a].
While our giant planets seem to possess globally convective interiors, strongly irradiated extrasolar planets must develop a radiative zone just beneath the levels where most of the stellar irradiation is absorbed. Depending on the irradiation and characteristics of the planet, this zone may extend down to kbar levels, the deeper levels being convective. In this case, a careful determination of the opacities is necessary (but generally not possible) as these control the cooling and contraction of the deeper interior [see @Freedman+2008 for a discussion of opacities and tables for substellar atmospheres and interiors].
The contraction and cooling histories of giant planets {#sec:virial}
------------------------------------------------------
The interiors of giant planets are expected to evolve with time from a high entropy, high $\theta$ value, hot initial state to a low entropy, low $\theta$, cold degenerate state. The essential underlying physics can be derived from the well-known virial theorem and the energy conservation which link the planet’s internal energy $E_{\rm i}$, gravitational energy $E_{\rm g}$ and luminosity through: $$\begin{aligned}
\xi E_{\rm i} + E_{\rm g} &=&0,\label{eq:virial_E}\\
L &=& -{\xi-1\over \xi}{dE_{\rm g}\over dt},\label{eq:virial_L}\end{aligned}$$ where $\xi=\int_0^M 3(P/\rho)dm / \int_0^M u dm\approx <\!3P/\rho
u\!>$, the brackets indicating averaging, and $u$ is the specific internal energy. For a diatomic perfect gas, $\xi=3.2$; for fully-degenerate non-relativistic electrons, $\xi=2$.
Thus, for a giant planet or brown dwarf beginning its life mostly as a perfect H$_2$ gas, two third of the energy gained by contraction is radiated away, one third being used to increase $E_{\rm i}$. The internal energy being proportional to the temperature, the effect is to heat up the planet. This represents the slightly counter-intuitive but well known effect that a star or giant planet initially heats up while radiating a significant luminosity [e.g., @KW94].
Let us now move further in the evolution, when the contraction has proceeded to a point where the electrons have become degenerate. For simplicity, we will ignore Coulomb interactions and exchange terms, and assume that the internal energy can be written as $E_{\rm
i}=E_{\rm el}+E_{\rm ion}$, and that furthermore $E_{\rm el}\gg E_{\rm
ion}$ ($\theta$ is small). Because $\xi\approx 2$, we know that half of the gravitational potential energy is radiated away and half of it goes into internal energy. The problem is to decide how this energy is split into an electronic and an ionic part. The gravitational energy changes with some average value of the interior density as $E_{\rm g}\propto 1/R \propto \rho^{1/3}$. The energy of the degenerate electrons is essentially the Fermi energy: $E_{\rm
el}\propto \rho^{2/3}$. Therefore, $\dot{E}_{\rm g}\approx 2(E_{\rm
g}/ E_{\rm el})\dot{E}_{\rm el}$. Using the virial theorem and specifically eq. (\[eq:virial\_E\]) we get that $\dot{E}_{\rm g}\approx -\dot{E}_{\rm el}$. The luminosity is by definition $L=-(\dot{E}_{\rm g}+\dot{E}_{\rm i})$ and therefore $$L \approx -\dot{E}_{\rm ion} \propto -\dot{T}.$$ In this limit, the gravitational energy lost is entirely absorbed by the increase in pressure of the degenerate electrons and the observed luminosity is due to the thermal cooling of the ions [@Guillot05].
Several simplifications limit the applicability of this result (that would be valid in the white dwarf regime). In particular, the Coulomb and exchange terms in the EOS introduce negative contributions that cannot be neglected. However, the approach is useful to grasp how the evolution proceeds: in its very early stages, the planet is very compressible. It follows a standard Kelvin-Helmholtz contraction. When degeneracy sets in, the compressibility becomes much smaller ($\alpha T\sim 0.1$, where $\alpha$ is the coefficient of thermal expansion), and the planet gets its luminosity mostly from the thermal cooling of the ions. The luminosity can be written in terms of a modified Kelvin-Helmholtz formula: $$L\approx \eta {GM^2\over R\tau},
\label{eq:lapprox}$$ where $\tau$ is the age, and $\eta$ is a factor that hides most of the complex physics. In the approximation that Coulomb and exchange terms can be neglected, $\eta\approx\theta/(\theta +1)$. The poor compressibility of giant planets in their mature evolution stages imply that $\eta\ll 1$ ($\eta\sim 0.03$ for Jupiter): the luminosity is not obtained from the entire gravitational potential, but from the much more limited reservoir constituted by the thermal internal energy. Equation (\[eq:lapprox\]) shows that to first order, $\log
L\propto -\log\tau$: very little time is spent at high luminosity values. In other words, the problem is (in most cases) weakly sensitive to initial conditions. However, it is to be noticed that with progress in our capability to detect very young objects, i.e., planets and brown dwarfs of only a few million years of age, the problem of the initial conditions does become important [@Marley+06]. Interestingly, at these early stages, their luminosity appears to strongly depend on their core mass [@Mordasini2013].
Figure \[fig:L vs t\] shows more generally how giant planets, but also brown dwarfs and small stars see their luminosities evolve as a function of time. The $1/\tau$ slope is globally conserved, with some variations for brown dwarfs during the transient epoch of deuterium burning, and of course for stars, when they begin burning efficiently their hydrogen and settle on the main sequence: in that case, the tendency of the star to contract under the action of gravity is exactly balanced by thermonuclear hydrogen fusion.
Mass-radius relation
--------------------
The relation between mass and radius has very fundamental astrophysical applications. Most importantly it allows one to infer the gross composition of an object from a measurement of its mass and radius. This is especially relevant in the context of the discovery of extrasolar planets with both radial velocimetry and the transit method, as the two techniques yield relatively accurate determination of $M$ and $R$, these determinations being often limited by the uncertainty on the stellar parameters themselves.
Figure \[fig:mass\_rad\] shows mass-radius relations for compact degenerate objects from giant planets to brown dwarfs and low-mass stars. The right-hand side of the diagram shows a rapid increase of the radius with mass in the stellar regime which is directly due to the onset of stable thermonuclear reactions. In this regime, observations and theoretical models agree [see however @Ribas06 for a more detailed discussion]. The left-hand side of the diagram is obviously more complex, and this can be understood by the fact that planets have much larger variations in composition than stars, and because external factors such as the amount of irradiation they receive do affect their contraction in a significant manner.
Let us first concentrate on isolated or nearly-isolated gaseous planets. The black curves have a local maximum near $4\mjup$: at small masses, the compression is small so that the radius increases with mass. At large masses, degeneracy sets in and the radius decreases with mass.
This can be understood on the basis of polytropic models based on the assumption that $P=K\rho^{1+1/n}$, where $K$ and $n$ are constants. Because of degeneracy, a planet of large mass will tend to have $n\rightarrow 1.5$, while a planet of smaller mass will be less compressible ($n\rightarrow 0$). Indeed, it can be shown that in their inner 70 to 80% in radius isolated solar composition planets of 10, 1 and $0.1\mjup$ have $n=1.3$, 1.0 and 0.6, respectively. From polytropic equations [e.g., @Chandrasekhar39]: $$R\propto K^{n\over 3-n} M^{1-n\over 3-n}.
\label{eq:m-r-k}$$ Assuming that $K$ is independant of mass, one gets $R\propto
M^{0.16}$, $M^{0}$, and $M^{-0.18}$ for $M=10$, 1 and $0.1\mjup$, respectively, in relatively good agreement with (the small discrepancies are due to the fact that the intrinsic luminosity and hence $K$ depend on the mass considered).
Figure \[fig:mass\_rad\] shows already that the planets in our Solar System are not made of pure hydrogen and helium and require an additional fraction of heavy elements in their interior, either in the form of a core, or distributed in the envelope (dotted line).
For extrasolar planets, the situation is complicated by the fact that the intense irradiation that they receive plays a major role in their evolution. The present sample is already quite diverse, with equilibrium temperature (defined as the effective temperature corresponding to the stellar flux received by the planet) ranging from 1000 to 2500K. Their compositions are also quite variable, with some planets having large masses of heavy elements [@Sato+05; @Guillot+06]. The orange and yellow curves in fig. \[fig:mass\_rad\] show theoretical results for equilibrium temperatures of 1000 and 2000K, respectively. Two extreme models have been plotted: assuming a purely solar composition planet (top curve), and assuming the presence of a $100\mea$ central core (bottom curve). In each case, an additional energy source proportional to 0.5% of the incoming luminosity was also assumed (see discussion in § \[sec:irradiated\] hereafter).
The increase in radius for decreasing planetary mass for irradiated, solar-composition planets with little or no core can be understood using the polytropic relation (eq. \[eq:m-r-k\]), but accounting for variations of $K$ as defined by the atmospheric boundary condition. Using the Eddington approximation, assuming $\kappa\propto
P$ and a perfect gas relation in the atmosphere, one can show that $K\propto (M/R^2)^{-1/2n}$ and that therefore $R\propto M^{1/2-n\over
2-n}$. With $n=1$, one finds $R\propto M^{-1/2}$. Strongly irradiated hydrogen-helium planets of small masses are hence expected to have the largest radii which qualitatively explain the positions of the extrasolar planets in . Note that this estimate implicitly assumes that $n$ is constant throughout the planet. The real situation is more complex because of the growth of a deep radiative region in most irradiated planets, and because of structural changes between the degenerate interior and the perfect gas atmosphere [@Guillot05].
In the case of the presence of a fixed mass of heavy elements, the trend is inverse because of the increase of mean molecular mass (or equivalently core/envelope mass) with decreasing total mass. Thus, small planets with a core are much more tightly bound and less subject to evaporation than those that have no core.
Rotation and the figures of planets {#sec:rotation}
-----------------------------------
The mass and radius of a planet informs us on its global composition. Because planets are also rotating, one is allowed to obtain more information on their deep interior structure. The hydrostatic equation becomes more complex however: $${\bfnab P\over \rho}=\bfnab\left(G\int\!\!\!\int\!\!\!\int
{\rho(\bfr')\over |\bfr - \bfr'|}d^3\bfr'\right) - \bfOm\times(\bfOm\times\bfr),
\label{eq:full_hydrostat}$$ where $\bfOm$ is the rotation vector. The resolution of eq. (\[eq:full\_hydrostat\]) is a complex problem. It can however be somewhat simplified by assuming that $|\bfOm|\equiv\omega$ is such that the centrifugal force can be derived from a potential. The hydrostatic equilibrium then writes $\nabla P = \rho \nabla U$, and the [*figure*]{} of the rotating planet is then defined by the $U=constant$ level surface.
One can show [e.g., @ZT78] that the hydrostatic equation of a fluid planet can then be written in terms of the mean radius $\rbar$ (the radius of a sphere containing the same volume as that enclosed by the considered equipotential surface): $${1\over \rho}{{\partial P\over\partial \rbar}}=-{Gm\over \rbar^2}+{2\over 3}\omega^2
\rbar + {GM\over \overline{R}^3} \rbar\varphi_\omega,$$ where $M$ and $\overline{R}$ are the total mass and mean radius of the planet, and $\varphi_\omega$ is a slowly varying function of $\rbar$. (In the case of Jupiter, $\varphi_\omega$ varies from about $2\times 10^{-3}$ at the center to $4\times 10^{-3}$ at the surface.) Equations (\[eq:dtdr\]-\[eq:dldr\]) remain the same with the hypothesis that the level surfaces for the pressure, temperature, and luminosity are equipotentials. The significance of rotation is measured by the ratio of the centrifugal acceleration to the gravity: $$q={\omega^2 \req^3\over GM}.
$$
As discussed in section \[sec:gravity\], in some cases, the external gravity field of a planet can be accurately measured in the form of gravitational moments $J_{k}$ (with zero odd moments for a planet in hydrostatic equilibrium) that measure the departure from spherical symmetry. Together with the mass, this provides a constraint on the interior density profile (see [@ZT74] -see also chapters by Van Hoolst and Sohl & Schubert): $$\begin{aligned}
M&=&\int\!\!\!\int\!\!\!\int \rho(r,\theta) d^3\tau, \\
J_{2i} &=& -{1\over M R_{\rm eq}^{2i}}\int\!\!\!\int\!\!\!\int \rho(r,\theta) r^{2i}
P_{2i}(\cos\theta) d^3\tau,\end{aligned}$$ where $d\tau$ is a volume element and the integrals are performed over the entire volume of the planet.
Figure \[fig:contrib\] shows how the different layers inside a planet contribute to the mass and the gravitational moments. The figure applies to Jupiter, but would remain relatively similar for other planets. Note however that in the case of Uranus and Neptune, the core is a sizable fraction of the total planet and contributes both to $J_2$ and $J_4$. Measured gravitational moments thus provide information on the external levels of a planet. It is only indirectly, through the constraints on the outer envelope that the presence of a central core can be infered. As a consequence, it is impossible to determine this core’s state (liquid or solid), structure (differentiated, partially mixed with the envelope) and composition (rock, ice, helium...) from the gravity field data.
The Juno [@Bolton2010] and Cassini Solstice [@Spilker2012] missions are expected to yield considerable improvements in our determination of the gravity fields of Jupiter and Saturn, respectively. Because the theory of figures is limited by its expansion in terms of the rotation parameter $q$ and because purely barotropic solutions are possible only in the limit of solid-body rotation and of pure rotation on cylinders, these high precision measurements will require new approaches to include rotation in planetary models [see @Hubbard1999; @Hubbard2013]. Separately, these measurements will enable new constraints such as the determination of the planets’ angular momentum through the measurement of the relativistic Lense-Thirring effect [@Iorio2010] and the determination of their moment of inertia [@Helled2011; @Helled+2011b]. But probably the most important prospects lie in the possibility to couple measurements on gravity field, magnetic fields and wind speeds with combined tri-dimensional magnetohydrodynamical models (see sections \[sec:magnetic fields\] and \[sec:dynamics\]).
For planets outside the solar system, although measuring their gravitational potential is presently beyond reach, an indirect measurement of the planets’ Love number $k_2$ may be possible in systems of planets in which one is locked into a so-called fixed-point eccentricity. So far, one such transiting system is known, HAT-P-13 [@Batygin+2009b; @Mardling2010], because it contains a transiting planet, HAT-P-13b and a companion, HAT-P-13c whose minimum mass $M\sin i$ and orbital eccentricity are known [@Bakos+2009]. The value of $k_2$ constrains the interior structure in a way that is very similar to $J_2$ and has been used to obtain first constraints on the interior structure of this planet [@Kramm+2012].
Interior structures and evolutions
==================================
Jupiter and Saturn {#sec:JupSat}
------------------
As illustrated by fig. \[fig:intjupsat\], the simplest interior models of Jupiter and Saturn matching all observational constraints assume the presence of three main layers: (i) an outer hydrogen-helium envelope, whose global composition is that of the deep atmosphere; (ii) an inner hydrogen-helium envelope, enriched in helium because the whole planet has to fit the H/He protosolar value; (iii) a central dense core. Because the planets are believed to be mostly convective, these regions are expected to be globally homogeneous. (Many interesting thermochemical transformations take place in the deep atmosphere, but they are of little concern to us).
The transition from a helium-poor upper envelope to a helium-rich lower envelope is thought to take place through the formation of helium-rich droplets that fall deeper into the planet due to their larger density. These droplets form when the temperature-pressure profiles enter the separation region for the initial helium abundance, as shown in section \[sec:others\]. Three-layer models implicitly make the hypothesis that this region is adiabatic (this is justified only if convection is not inhibited by the formation of helium droplets, as discussed by [@SS77b]) and narrow. Figure \[fig:hhe\] shows that this zone may be extended, especially in present-day Saturn [see also @FH03].
As discussed by @SS77b [@Stevenson82] the planets would start from an initially hot and homogeneous state and would start entering the phase separation region progressively, leading to a depletion of helium in the outer region and its increase in the deeper interior. According to the calculations by [@Lorenzen+2011] and [@Morales+2013b] the separation would first occur at a pressure between 1 and 2 Mbar and the inhomogeneous region would grow inward but not so much towards lower pressures because of the higher solubility of helium in molecular hydrogen. According to the simulations, although needed to explain the abundances in Jupiter’s atmosphere (section \[sec:compositions\]), it is not yet clear that the process has begun in this planet. Fully consistent calculations that account both for the constraints on the planets’ gravitational moments and their atmospheric composition should become possible, especially with the additional information brought by the Juno [@Bolton2010] and Cassini Solstice [@Spilker2012] space missions.
In the absence of these calculations, adiabatic three-layer models can be used as a useful guidance to a necessarily hypothetical ensemble of allowed structures and compositions of Jupiter and Saturn. A relatively extensive exploration of the parameter space has been performed by several authors [@SG04; @FortneyNettelmann2010; @Nettelmann+2012; @Nettelmann+2013b; @HelledGuillot2013]. The calculations account for a transition from a helium-poor outer envelope to a helium-rich inner envelope. The abundance of heavy elements may or may not be held constant across this transition. Many sources of uncertainties are taken into account however; among them, the most significant are on the equations of state of hydrogen and helium, the uncertain values of $J_4$ and $J_6$, the presence of differential rotation deep inside the planet, the location of the helium-poor to helium-rich region, and the uncertain helium to hydrogen protosolar ratio.
Their results indicate that Jupiter’s core is smaller than $\sim
10\mea$, and that its global composition is pretty much unknown (between 10 to 42$\mea$ of heavy elements in total). The models indicate that Jupiter is enriched compared to the solar value by a factor 1.5 to 8 times the solar value. This enrichment is compatible with a global uniform enrichment of all species near the atmospheric Galileo values, but allows many other possibilities as well.
Other models of Jupiter based on an ab-initio equation of state by [@Militzer+2008] led to a solution with a large core mass and a very small enrichment in heavy elements in the envelope incompatible with either the Galileo probe measurements or the protosolar helium abundance. A significant update in the EOS is presented by @MilitzerHubbard2013. This updated EOS yields a warmer interior than the 2008 models and should therefore lead to a smaller core mass and larger amount of heavy elements in the envelope, in line with the other results. At the same time, the @MilitzerHubbard2013 EOS is much more accurate than the range of EOSs used by @SG04 and it differs slightly from the other ab-initio EOS used by @Nettelmann+2012 and @Nettelmann+2013b. New constraints should therefore be derived on the basis of those new data.
In the case of Saturn, the solutions depend less on the hydrogen EOS because the Mbar pressure region is comparatively smaller. The total amount of heavy elements present in the planet can therefore be estimated with a better accuracy than for Jupiter, and is between $16$ and $30\mea$ [@Nettelmann+2013b; @HelledGuillot2013]. The uncertainty on the core mass is found to be larger than for Jupiter because a heavy-element rich inner envelope can mimic the gravitational signature of the core. As a result only an upper limit on the core mass of $20\mea$ is derived.
Concerning the [*evolutions*]{} of Jupiter and Saturn, the three main sources of uncertainty are, by order of importance: (1) the magnitude of the helium separation; (2) the EOS; (3) the atmospheric boundary conditions. Figure \[fig:jup-sat-evolution\] shows an ensemble of possibilities that attempts to bracket the minimum and maximum cooling. In all these quasi-adiabatic cases, helium sedimentation is needed to explain Saturn’s present luminosity [see @Salpeter73; @SS77b; @Hubbard77; @Hubbard+99; @FH03]. In the case of Jupiter, the sedimentation of helium that appears to be necessary to explain the low atmospheric helium abundance poses a problem for evolution models because it appears to generally prolong its evolution beyond 4.55Ga, the age of the Solar System [@Fortney+2011; @Nettelmann+2012]. However, different solutions are possible, including improvements of the EOS and atmospheric boundary conditions, or even the possible progressive erosion of the central core that would yield a lower luminosity of Jupiter at a given age [@GSHS04].
A different set of solutions appears when one considers the possibility that the envelopes of Jupiter and Saturn are [*not*]{} homogeneous and adiabatic on large scales, but are instead of variable composition, with heavier elements at the bottom. In that case, convection can be strongly inhibited which requires the temperature gradient to be larger in order to transport the same intrinsic luminosity. In the case of giant planets, this process, known as semiconvection or diffusive convection leads to the formation of a non-static staircase structure with diffusive interfaces with abrupt variations of temperature and composition and a homogeneous adiabatic structure inbetween [@Stevenson1985; @Rosenblum+2011]. By assuming that this structure is maintained over the entire envelopes of Jupiter and Saturn, @LeconteChabrier2012 derive much warmer interior structures with also 30% to 60% more heavy elements than in conventional models. However, the assumption that this non-homogenous composition is maintained in the entire envelopes is ad hoc. In reality, one may expect semi-convection to be confined to a much smaller region and thus have a more limited effect. The problem is open however and requires further study. Its understanding is also critical for deciding whether Jupiter’s core can erode into its envelope [see @GSHS04; @WilsonMilitzer2012].
As discussed in sections \[sec:magnetic fields\] and \[sec:dynamics\], classical interior models of Jupiter and Saturn based on the assumption of a homogeneous structure in the molecular envelope and an increase of the conductivity mainly due to hydrogen metallization do yield solutions for their magnetic fields and atmospheric zonal winds that globally match the observations. However, important “details” such as why Saturn’s magnetic field is axisymmetric and Jupiter is not remain unexplained. Coupling interior and dynamical models in order to fit both Jupiter and Saturn’s gravity and magnetic fields as well as their observed zonal winds (see sections \[sec:magnetic fields\] and \[sec:dynamics\]) should bring a more global understanding of the planetary structures.
Uranus and Neptune {#sec:UraNep}
------------------
Although the two planets are relatively similar, table \[tab:comp\] already shows that Neptune’s larger mean density compared to Uranus has to be due to a slightly different composition: either more heavy elements compared to hydrogen and helium, or a larger rock/ice ratio. The gravitational moments impose that the density profiles lie close to that of “ices” (a mixture initially composed of e.g., H$_2$O, CH$_4$ and NH$_3$, but which rapidly becomes a ionic fluid of uncertain chemical composition in the planetary interior), except in the outermost layers, which have a density closer to that of hydrogen and helium [@MGP95; @PPM00]. As illustrated in , three-layer models of Uranus and Neptune consisting of a central “rocks” core (magnesium-silicate and iron material), an ice layer and a hydrogen-helium gas envelope have been calculated [@PHS91; @Hubbard+95; @FortneyNettelmann2010; @Helled+2011; @Nettelmann+2013a].
According to the models of [@Nettelmann+2013a], Uranus contains a minimum of $1.8$ to $2.2\mea$ of hydrogen and helium and Neptune $2.7$ to $3.3\mea$. The global ice to rock ratio that is derived is very high (19 to 36) in Uranus, while Neptune has a wide range of solutions from $3.6$ to $14$. These values are much larger than the canonical ice to rock ratio of 2 to 3 for the protosun that accounts for the abundances of all elements condensing at low temperatures (“ices”) versus that of more refractory elements (“rocks”). The fact that either planet would have accreted much less rocks than ices is puzzling and unexplained by formation models. It is probably an artefact from assuming ices being confined to the envelope and rocks to the core.
The evolution of the two planets also remains a mystery. While Neptune’s present luminosity may be explained by the adiabatic cooling of the planet over the age of the Solar System, this is not the case of Uranus’s very low luminosity [@PWM95; @Fortney+2011; @Nettelmann+2013a]. This could be explained by the presence of a strongly inhibiting compositional gradient decoupling an inner region which would remain hot and an outer envelope which would cool progressively[@PWM95]. Such regions could also be present in Neptune but considerably deeper. Unfortunately, this qualitative explanation cannot be tied to the inferred interior structures. Apart from the latest models by [@Nettelmann+2013a], the models of Uranus and Neptune are too similar (and so are their magnetic fields – see section \[sec:magnetic fields\]) to explain why Uranus would have such a small intrinsic heat flux and not Neptune.
In fact, it is likely that all present models of Uranus and Neptune are inadequate because of the assumption of an adiabatic temperature structure across interfaces with different compositions. Instead, diffusive-convection should occur and lead to strongly superadiabatic temperature gradient [e.g., @Rosenblum+2011]. As in the case of Jupiter and Saturn [see @LeconteChabrier2012], this would lead to higher temperatures in the interior and very different constraints on the interior composition. The amount of rocks required to fit the mean density and gravitational moments would certainly rise, potentially solving the ice to rock ratio problem. The evolution of the planets would be very different as the present-day luminosity would be mostly governed by the leak of heat from the hot interior by diffusion at the interfaces.
Irradiated giant planets {#sec:irradiated}
------------------------
### Interior structure and dynamics
The physics that governs the calculation of interior structure and evolution models of giant planets described in the previous sections can be applied in principle to any gaseous exoplanet and brown dwarf. We focus the discussion on the ones that orbit extremely close to their star because of the possibility to directly characterise them and measure their mass, radius and in some cases even the properties of their atmosphere. Two planets are proxies for this new class of objects: the first extrasolar giant planet discovered, 51Pegb, with an orbital period of $P=4.23$ days, and the first [*transiting*]{} extrasolar giant planet, HD209458b, with $P=3.52$ days. Following widespread usage, we call these planets “hot Jupiters” (a.k.a “Pegasids” since these two archetypes have been discovered in the Pegasus constellation).
With such a short orbital period, these planets are for most of them subject to an irradiation from their central star that is so intense that the absorbed stellar energy flux can be about $\sim 10^4$ times larger than their intrinsic flux. The atmosphere is thus prevented from cooling, with the consequence that a radiative zone develops and governs the cooling and contraction of the interior [@Guillot+96]. Typically, for a planet like HD209458b, this radiative zone extends to kbar levels, $T\sim 4000\K$, and is located in the outer 5% in radius ($0.3\%$ in mass) [@GS02].
Problems in modeling the evolution of hot Jupiters arise because of the uncertain outer boundary condition. The intense stellar flux implies that the atmospheric temperature profile is extremely dependent upon the opacity sources considered. Depending on the chosen composition, the opacity data used, the assumed presence of clouds, the geometry considered, resulting temperatures in the deep atmosphere can differ by up to $\sim 600\K$ [@SS00; @Goukenleuque+00; @BHA01; @SBH03; @IBG05; @Fortney+06]. Furthermore, as illustrated by , the strong irradiation and expected synchronization of the planets’ spin implies that strong inhomogeneities should exist in the atmosphere with in particular strong ($\sim 500$K) day-night and equator-to-pole differences in effective temperatures [@SG02; @IBG05; @CS05; @BHA05].
Figure \[fig:parmentier\] illustrates the expected structure for the atmosphere of HD209458b from a modern, tri-dimensional global circulation model coupled with a one-dimensional radiative transfer algorithm [@Parmentier+2013]. All the caveats concerning these overforced simulations discussed in section \[sec:dynamics\] of course also apply and add to the uncertainties stemming from the poorly known chemical composition. For example, the particular simulation of fig. \[fig:parmentier\] assumes the presence of TiO in the atmosphere, which yields very high temperatures at low pressures on the day side of the planet. It is not clear that this molecule is present or has condensed at deeper levels [see also @Spiegel+2009]. Aside from that, the eastward equatorial circulation and the strong equator to pole gradient now appear to be a robust feature of these simulations [e.g., @SG02; @RauscherMenou2013; @Parmentier+2013]. As seen in fig. \[fig:parmentier\], the equatorial jet redistributes heat between the day side and the night side relatively efficiently at large pressures and on the equator, but this is not the case at the poles, and at low pressures, in line with the observational constraints (see section \[sec:exoplanets\]).
These strong temperature variations must influence at some point the cooling and contraction histories of hot Jupiters. When opacities variations are not included, they result in more loss of the intrinsic heat and a faster contraction than when assuming that the stellar irradiation is homogeneously redistributed across the planetary surface [@GS02; @Guillot2010; @Budaj+2012; @SpiegelBurrows2013]. However, given other uncertainties (e.g., on the chemical composition and opacities to be used), this has been neglected in planetary evolution models thus far.
### Thermal evolution and inferred compositions
We have seen in fig. \[fig:mass\_rad\] that the measured masses and radii of transiting planets can be globally explained in the framework of an evolution model including the strong stellar irradiation and the presence of a variable mass of heavy elements, either in the form of a central core, or spread in the planet interior. However, when analyzing the situation for each planet, it appears that several planets are too large to be reproduced by standard models, i.e., models using the most up-to-date equations of state, opacities, atmospheric boundary conditions and assuming that the planetary luminosity governing its cooling is taken solely from the lost gravitational potential energy (see section \[sec:virial\]).
Figure \[fig:ev-hd209458b\] illustrates the situation for the particular case of HD209458b: unless using an unrealistically hot atmosphere, or arbitrarily increasing the internal opacity, or decreasing the helium content, one cannot reproduce the observed radius which is 10 to 20% larger than calculated using standard models [@BLM01; @BLL03; @GS02; @Baraffe+03]. The fact that the measured radius corresponds to a low-pressure ($\sim$mbar) level while the calculated radius corresponds to a level near 1bar is not negligible [@BSH03] but too small to account for the difference. This is problematic because while it is easy to invoke the presence of a massive core to explain the small size of a planet, a large size such as that of HD209458b requires an additional energy source, or significant modifications in the data/physics involved.
The discovery of many transiting hot Jupiters has shown that this phenomenon is widespread, with at least a third of them being oversized compared to predictions from the standard evolution of a solar-composition planet with no core [@Guillot+06; @Guillot08; @Laughlin+11]. Numerous explanations have been put forward to explain this large size. The first ones, invoking tidal dissipation of eccentricity [@BLM01] or inclination [@WinnHolman2005] imply that orbital energy is transfered to the planet. These are generally too short-lived [e.g., @Leconte+10] or of a low probability of occurrence [@Levrard+07]. The second ones posit that part of the irradiation energy is transferred into kinetic energy and is then dissipated deeper into the planet. This is the case of weather-noise [@SG02], ohmic dissipation [@BS10], thermal tides [@ArrasSocrates10] and turbulent burial [@YoudinMitchell10] models. These mechanisms appear quite promising as they are long-lived and generally require only a small fraction of order 1% or less of the irradiation luminosity to be transported and dissipated at deeper levels to explain the observed planets [@GS02]. Finally, a third class of models is based on a reduced cooling, either through an ad hoc increase of opacities [@Burrows+07] or inefficient heat transport due to semi-convection [@CB07]. Validating these models is becoming possible thanks to a large number of planets allowing statistical tests [see @Laughlin+11], but will require further work.
In any case, the fact that a large number of planets are oversized lends weight to a mechanism that would apply to each planet. Masses of heavy elements can then be derived by imposing that all planets should be fitted by the same model with the same hypotheses. This can be done by inverting the results of , as proposed by @Guillot+06. On the basis of this hypothesis, figure \[fig:correlation\] shows that some of the hot Jupiters contain a large fraction of heavy elements in their interior and that this fraction is correlated with the metallicity of the parent star. The large fraction of heavy elements is inferred both for relatively small giant planets (i.e., Saturn mass) and for planets which are several times the mass of Jupiter. A typical archetype of the first is HD149026b which must contain around $70\mea$ of heavy elements, a conclusion that is hard to escape because of the low total mass and high irradiation of the planet [see @Ikoma+06; @Fortney+06]. For the latter, CoRoT-20b appears to push the models to the limit, with predicted masses of heavy elements in excess of $400\mea$ [@Deleuil+12].
The correlation between mass of heavy elements in hot Jupiters and stellar metallicity first obtained by [@Guillot+06] appears to stand the trial of time [@Burrows+07; @Guillot08; @Laughlin+11; @Moutou+2013], and remains valid when applied to planets with low irradiation levels which do not require additional physics to explain their large sizes [@MF11]. It is important to realize that simply accreting slightly more metal-rich gas with the composition of the parent star would lead to a much smaller increase of a few percent at most. This correlation requires an efficient mechanism to collect solids in the protoplanetary disk and bring them into the hot Jupiters, something that is just beginning to be included into planet formation models [@Mordasini+12].
Another intriguing possibility concerning hot Jupiters is that of a sustained mass loss due to the high irradiation dose that the planets receive. Indeed, this effect was predicted [@BL95; @Guillot+96; @Lammer+03] and detected [@VidalMadjar+03; @Fossati+10; @Bourrier+13]. While its magnitude is still uncertain, it appears to have sculpted the population of planets in very close orbits around their star [e.g., @Lopez+12].
The harvest of the Kepler and CoRoT missions opens the possibility to extend these studies to smaller planets. These objects are especially interesting but pose difficult problems in terms of structure because depending on their formation history, precise composition and location, they may be fluid, solid, or they may even possess a global liquid ocean [see @Kuchner03; @Leger+04].
Implications for planetary formation models
===========================================
The giant planets in our Solar System have in common a large mass of hydrogen and helium, but they are obviously quite different in their appearances, compositions and internal structures. Although studies cannot be conducted with the same level of details, we can safely conclude that extrasolar planets show a greater variety of compositions and structures, and imagine that their appearances differ even more significantly.
A parallel study of the structures of our giant planets and of giant planets orbiting around other stars should provide us with key information regarding planet formation in the next decade or so. But, already, some conclusions, some of them robust, others still tentative, can be drawn:
[*Giant planets formed in circumstellar disks, before these were completely dissipated:*]{}\
This is a relatively obvious consequence of the fact that giant planets are mostly made of hydrogen and helium: these elements had to be acquired when they were still present in the disk. Because the observed lifetime of gaseous circumstellar disks is of the order of a few million years, this implies that these planets formed (i.e., acquired most of their final masses) in a few million years also, quite faster than terrestrial planets in the Solar System.
[*Giant planets migrated:*]{}\
The observed orbital distribution of extrasolar planets and the presence of planets extremely close to their star is generally taken a strong evidence for an inward migration of planets, and various mechanisms have been proposed for that [see @IL04a; @AMBW05; @MA05 ...etc.]. Separately, it was shown that several properties of our Solar System can be explained if Jupiter, Saturn, Uranus and Neptune ended the early formation phase in the presence of a disk with quasi-circular orbit, and with Saturn, Uranus and Neptune significantly closer to the Sun than they are now, and that these three planets subsequently migrated outward [@TGML05].
[*Accretion played a key role for giant planet formation:*]{}\
Although formation by direct gas instability still remains a possibility [e.g., @HelledBodenheimer2011; @Boley+2011], several indications point towards a formation of giant planets that is dominated by accretion of heavy elements: First, Jupiter, Saturn, Uranus and Neptune are all significantly enriched in heavy elements compared to the Sun. This feature can be reproduced by core-accretion models, for Jupiter and Saturn at least [@AMBW05]. Second, the probability to find a giant planet around a solar-type star (with stellar type F, G or K) is a strongly rising function of stellar metallicity [@Gonzalez98; @SIM04; @FV05], a property that is also well-reproduced by standard core accretion models [@IL04b; @AMBW05]. Third, the large masses of heavy elements inferred in some transiting extrasolar planets as well as the apparent correlation between mass of heavy elements in the planet and stellar metallicity [@Guillot+06; @Burrows+07; @Guillot08; @Laughlin+11] is a strong indication that accretion was possible and that it was furthermore efficient. [*Giant planets were enriched in heavy elements by core accretion, planetesimal delivery and/or formation in an enriched protoplanetary disk:*]{}\
The giant planets in our Solar System are unambigously enriched in heavy elements compared to the Sun, both globally, and when considering their atmosphere. This may also be the case of extrasolar planets, although the evidence is still tenuous. The accretion of a central core can explain part of the global enrichment, but not that of the atmosphere. The accretion of planetesimals may be a possible solution but in the case of Jupiter at least the rapid drop in accretion efficiency as the planet reaches appreciable masses ($\sim 100\mea$ or so) implies that such an enrichment would have originally concerned only very deep layers, and would require a relatively efficient upper mixing of these elements, and possibly an erosion of the central core [@GSHS04; @WilsonMilitzer2012].
Although not unambiguously explained, the fact that Jupiter is also enriched in noble gases compared to the Sun is a key observation to understand some of the processes occuring in the early Solar System. Indeed, noble gases are trapped into solids only at very low temperatures, and this tells us either that most of the solids that formed Jupiter were formed at very low temperature to be able to trap gases such as argon, probably as clathrates [@GHML01; @HGL04; @Mousis+2012], or that the planet formed in an enriched disk as it was being evaporated [@GH06]. The fact that in Jupiter, argon, krypton and xenon have a comparable enrichment over the solar value within the error bars [see @Lodders2008 and table \[tab:comp\]] slightly favors the latter explanation.
Future prospects
================
We have shown that the compositions and structures of giant planets remain very uncertain. This is an important problem when attempting to understand and constrain the formation of planets, and the origins of the Solar System. However, the parallel study of giant planets in our Solar System by space missions such as Galileo and Cassini, and of extrasolar planets by both ground based and space programs has led to rapid improvements in the field, with in particular a precise determination of the composition of Jupiter’s troposphere, and constraints on the compositions of a dozen of extrasolar planets.
Improvements on our knowledge of the giant planets requires a variety of efforts. Fortunately, nearly all of these are addressed at least partially by adequate projects in the next few years. The efforts that are necessary thus include (but are not limited to):
- Continue progresses on EOSs in order to obtain reliable results that can be used for a wide range of temperatures and pressures in the astrophysical context. This should be done with the help of laboratory experiments, for instance by using powerful lasers such as the NIF in the USA and the MégaJoule laser in France. Extensive numerical calculations should be performed as well, in particular with mixtures of elements.
- Calculate phase diagrams for a variety of mixtures, in particular involving superionic water, rocks, iron (and hydrogen). The hydrogen-helium phase diagram should also be refined because it is critical to understand the evolution and structure of Jupiter and Saturn.
- Have a better yardstick to measure solar and protosolar compositions. This has not been fully addressed by the Genesis mission and may require another mission and/or progresses in modeling the Sun’s composition.
- Improve the measurement of Jupiter’s gravity and magnetic fields, and determine the abundance of water in the deep atmosphere. This will be done by the Juno mission [@Bolton2010] which is to arrive at Jupiter in 2016, and whose polar orbit skimming a mere 5000km above the cloud tops should enable exquisite measurements of these quantities.
- Measure with high precision Saturn’s gravity field. Saturn’s gravitational moments have already been improved, but an important increase in accuracy can be obtained as part of the Cassini Solstice mission [@Spilker2012], while the spacecraft plunges onto the planet. This should lead to better constraints, and possibly a determination of whether the interior of Saturn rotates as a solid body.
- Pursue the discovery of transiting extrasolar planets including some with longer orbital periods and around bright stars. A large number of these objects will enable detailed statistical studies which will be key in understanding this population of objects.
- Develop consistent models for the formation, evolution, present structure and magnetic field of Uranus and Neptune, in order to understand ice giants as a class of planets.
- It would be highly desirable to send a probe similar to the Galileo probe into Saturn’s atmosphere [e.g., @Marty+2009]. The comparison of the abundance of noble gases would discriminate between different models of the enrichment of the giant planets, and the additional measurement of key isotopic ratio would provide further tests to understand our origins.
- In the long term, a mission to the ice giants Uranus or Neptune would bring new views of these fascinating planets and help to complete our knowledge of the outer solar system.
Clearly, there is a lot of work on the road, but the prospects for a much improved knowledge of giant planets and their formation are bright.
Acknowledgements {#acknowledgements .unnumbered}
================
The manuscript improved significantly thanks to the insightful reviews of Nadine Nettelmann and another reviewer. The authors also wish to thank Vivien Parmentier, Leigh Fletcher, Didier Saumon, Emmanuel Lellouch, Paul Loubeyre, Ravit Helled, Bill Hubbard, Erich Karkoschka, Imke De Pater and Miguel Morales for very useful comments and suggestions.
|
---
abstract: |
The continuous increase in performance requirements, for both scientific computation and industry, motivates the need of a powerful computing infrastructure. The Grid appeared as a solution for inexpensive execution of heavy applications in a parallel and distributed manner. It allows combining resources independently of their physical location and architecture to form a global resource pool available to all grid users. However, grid environments are highly unstable and unpredictable. Adaptability is a crucial issue in this context, in order to guarantee an appropriate quality of service to users.
Migration is a technique frequently used for achieving adaptation. The objective of this report is to survey the problem of strong migration in heterogeneous environments like the grids’, the related implementation issues and the current solutions.
author:
- 'Anolan Milanés, Noemi Rodriguez and Bruno Schulze'
bibliography:
- 'survey.bib'
title: Heterogeneous Strong Computation Migration
---
Introduction
============
*Computation Migration* can be defined as the transfer of a computation from one host to another during execution. This includes encapsulating and transmiting the computation state (namely, data, code and execution state) and restoring it at the destination machine. Migration can be done transparently, so the programmer has no control over the migration process, or the system may provide some way to control it.
Discussions about advantages and disadvantages of migration can be found in [@Eskicioglu90; @LO99; @CHK94]. Although techniques for process migration have been studied for several years [@MDPW+00; @Nuttall94], it has never been extensively adopted. As discussed in [@MDPW+00] and [@CHK94], this may be due to several factors, such as performance penalties when compared to alternative solutions, security issues and sociological factors. (While sociological factors can be overridden by guaranteeing higher execution priorities or a pre-defined degree of resources occupation for the machine owner, they still represent a limitation for the adoption of this technique, in particular because security is a real issue.) The study of process migration was initially strongly motivated by load-balancing concerns in distributed systems. In this context, migrating a process makes sense only when its remaining execution time is much larger than migration time. However, it is in general very hard to predict the remaining execution time for a running program. This, combined with the relatively high performance costs generally incurred by migration mechanisms, may have contributed strongly to the decrease of interest in the technique.
However, the current availability of high speed networks implies in lower penalties for the technique. Besides, there are other motivations for migration, in the context of which performance may not be the main issue. For instance, a computation may be migrated to a node where specific resources, data, or services are available. Migration may also be used to fulfill requirements for fault tolerance and uninterruptible services.
The emergence of new scenarios, such as those of mobile devices and grid computing, gives rise to new interest in migration [@LSM00]. Grid environments are usually characterized by concurrent execution, domain autonomy, resource heterogeneity and high failure probability, which imply in unpredictable resource utilization. This motivates the use of techniques that provide adaptation, reliability and maintenance. In this scenario, the study of migration seems to acquire new relevance. Furthermore, one important direction of current work in grid computing is that of [*opportunistic computing*]{}, in which resources are made available for remote users only when local users do not require them. Migration is an important mechanism to evict remotely started computations when the machine owner returns [@TTL05]. For this reason, we believe it is worthwhile to regain insight into the area by surveying the migration techniques that have been proposed. The problem of migration involves several issues. In this survey we concentrate on the question of [*how*]{} to move, that is, how to implement the transfer of an executing computation. This question has very different answers depending on whether we consider that migration will occur among machines with the same architecture and operating system (homogeneous migration) or among different platforms (heterogeneous migration). As we have mentioned, grid environments are naturally heterogeneous, and because such environments are our motivation, we concentrate, in this study, on heterogeneous migration.
This survey is organized as follows. Section 2 provides background on computation migration, intending to clarify the terms usually employed in the literature, describing various classifications and the problems related to this technique. Section 3 surveys work done in the area of strong heterogeneous migration. Finally, we conclude in Section 4 comparing some of the presented systems and discussing the approaches they take for the implementation of this technique.
Preliminaries
=============
Computation migration consists in moving the execution of a computation from one node to another while preserving its state. The composition of the state depends on the context. For instance, in Unix processes it includes the address space (heap contents, stack, global variables), the execution state (processor state) and the environment information (or resource information, that is, information about open files, but also about messages). As another example, the state of a thread running on the Java Virtual Machine (JVM) consists of [@LY99] the method area (the set of Java classes that includes a Java method currently being executed by the thread), the object heap (objects accessible from the thread’s execution stack), and the Java stack, organized in blocks called frames.
In general, migration can be initiated from inside the process (*proactive* or *subjective migration*), or from outside (*reactive* or *objective migration*). The latter case is usually found in load balancing facilities, allowing the controller engine to command the movement of a computation in response to changes on the environment (new resources appeared, performance deterioration was detected, etc). Migration can be made at process level, usually called process migration, or at thread level (thread migration). Migration granularity can be finer: a single object or a set of objects may move together. Emerald [@JLHB88], for instance, is a distributed language and system designed for the support of object mobility. *Mobility* is a term commonly used for referring to migration of objects.
The term mobility can also be found in literature with other meanings. In this report we will use it as a synonym of migration. Mobile computation, or computation mobility, is different from *mobile computing* in that the latter has to do with physical mobility (related to physical devices), whereas the first refers to virtual (logic) mobility [@cardelli99]. Another term that has been used in literature referring to migration is Dynamic Software Migration [@ABBC+88] or just Dynamic Migration [@Shub90].
It is also important to note the difference between *Computation Migration* and *Code Migration*. While code migration involves sending code to some location, computation migration requires the computation state to be transferred as well. That is, computation migration requires support for — but is not equal to — code migration. [*Mobile agents*]{}, moreover, require support for computation migration and also for transferring authority to act on the owner’s behalf.
Mobile agents are code-containing objects that may be transmitted between communicating participants in a distributed system [@Knabe95]. While process migration is typically initiated from outside the process, mobile agents can determine the moment and destination of the migration (although some agent platforms allow migration to be also initiated also from outside). A mobile agent system is the infrastructure that implements a mobile agent paradigm. The agent server, a protected agent execution environment, is responsible for executing agent code and provides primitive operations to programmers. When an agent requests to be transported to another host, the agent server deactivates the agent, saves its state and sends it to the remote agent server, which restores and reactivates it. Garbage collection is insured by forcing the return of the agent to the creator server after termination. Agents are frequently implemented using interpreted languages, because of their features of platform independence and dynamic code loading. Most agent systems have been implemented over the Java Virtual Machine (JVM).
The methods used for implementing migration of processes, threads, or agents are quite similar. However, terms used to describe them are often different. In this survey, we will employ the term [*computation migration*]{} to refer indistinctively to the movement of any kind of computation (processes or threads) from a source machine to a target machine, specifying the kind only if necessary. The moved computation itself will be referred to as computation, executing unit (EUs), as in [@FPV98] or migration unit, as in [@Shub90].
Fugetta et al. proposed in [@FPV98] a conceptual framework for understanding code mobility that has been extensively cited in the literature. The authors assign a slightly broader meaning to mobility than the one used in the present work: for them, mobility can be achieved either through migration or through [*remote cloning*]{} mechanisms. Remote cloning basically creates a copy of an Executing Unit at the destination. Unlike the case of migration, the original Executing Unit is not destroyed, or detached.
Figure \[Fig:classificationMobility\] shows the classification of mobility mechanisms proposed in the work of Fugetta. In this classification, mobility can be either [*weak*]{} or [*strong*]{}. Weak migration is the simplest and in consequence, most implemented form of migration. In this case, only the code segment is transferred, optionally with some initialization data. An example of weak migration in agent systems is provided by Aglets [@LO98]. After migration, an Aglet will always resume execution from the beginning of the program. On the other hand, in strong migration, the execution state is also transferred, allowing execution to restart at exactly the instruction at which it was interrupted at the source host. NOMADS [@SBBG+00] and D’Agents [@GCKP+02] are examples of agent systems with strong migration. Strong migration can be better for programs performing intensive computations and/or long executions, but it is harder to implement. Because weak migration is extensively implemented, it makes sense to use it as a basis for providing support for strong migration.
In order to achieve total migration transparency, that is, for the effects of the movement to be hidden from the user and the application, the references to objects and resources must also be transferred (open files, etc.). Some authors classify the case in which this is handled as a third type of mobility called [*full migration*]{} [@BD01], but most regard it as a special case of strong migration. The term *transparent migration* has also been used in literature as a synonym of strong migration, whether or not the environment is restored at the target machine [@Funfrocken98; @TRVC+00]. Because of the complexity of this issue, implementations typically impose restrictions over the reconstruction of the original environment at the target machine. Another problem with transparency is that, while hiding the information about the movement of the computation from the programmer certainly reduces programming complexity, it also disallows the control of errors (maybe caused by network problems or latency) and possible optimizations based on location.
![Classification of mobility mechanisms [@FPV98][]{data-label="Fig:classificationMobility"}](classificationMobility){width="\textwidth"}
The data space management mechanisms shown in Figure \[Fig:classificationMobility\] address this problem of relocating resources and reconfigurating bindings. The resources characteristics and the way they are bound to the EU restrict the methods that can be used by the data space management mechanisms in each case. For example, a huge database can be considered as a non-transferable resource, which eliminates copy as a possibility. Data space management is orthogonal to the mechanisms that suports the mobility of code and execution state.
To reduce the initial costs of migration, some of the computation state can be transferred on demand instead of at migration time. This technique is called [*lazy evaluation*]{}. [*Residual dependency*]{} is related to the references the migrated process leaves in the node from where it comes or where it was created, typically as a consequence of using lazy evaluation techniques or for providing transparency in communications, by redirecting communications through the previously established links to the migrated process [@MDPW+00].
Migration can be done at [*user*]{} or [*kernel*]{} level. Kernel level migration modifies the operating system kernel, which allows accessing directly the whole state of the process but is complex to implement and makes the mechanism dependent of the operating system. Implementing migration as a user-level mechanism allows it to be installed with no modification to the OS kernel. Besides, typically, user-level implementations are simpler than their kernel-level counterparts, because of the higher level at which state capture and serialization are done. On the other hand, operations for manipulating process state are generally not freely available at user-level, thus implying in limitations for this approach, and often higher performance penalties as well.
Given that the motivation of our work is migration in the Grid and similar environments, which requires independence of the underlying architecture and the Operating System, in this report we will not focus on migration at kernel level, although techniques for process migration in operating systems have been employed in higher level migration. Methods for implementing kernel-level migration have extensively studied elsewhere [@MDPW+00; @Smith88].
The concept of handling the state of a process is related to the notion of [*reflection*]{} in programming languages, which is the capacity of a program to get and modify information about its own state at runtime. Reflective mechanisms allow an executing program to access its computation state and to have knowledge about its structure, and also to adapt its behavior as a consequence. The implementations of migration can take advantage of reflection techniques. As an example, Proactive [@BCHV00] uses reflection to choose the method to be executed by the agent at arrival. X-Klaim [@BD01] also uses reflection, in this case to capture the agent code to be sent to the remote site. The problem of migration implies in taking policy decisions to solve the questions of *where* to move, *which* (EU) to move and *when*, and also *how* to implement a mechanism to effectively migrate the process. In this report we examine different techniques for the implementation of mechanisms for strong heterogeneous migration. Homogeneous migration assumes that data representation, machine registers, heap, stack, data segments and machine’s data instruction sets are the same. Except otherwise specified, all reported proposals refer to strong migration in heterogeneous systems.
In fact, when implementing a migration mechanism one must consider two layers. The inner layer implements state capture, serialization, deserialization, and restoration. The outer layer manages the problems of mobility at a linguistic level, which includes the management of locations, etc. Due to the extent of the problem, this paper studies only the first of these layers.
Heterogeneous Strong Computation Migration
==========================================
Migration in homogeneous environments implies in suspending the executing computation, encapsulating its state, transferring the code and the state information, and restarting the computation at the destination using the transmitted information. When the source and destination platforms are different, the problem of migration gets more complicated, because of the need for translation of the state of the computation to a format that may be understood at the destination machine. This is called *heterogeneous migration* and is the typical case in grids, where no homogeneity assumptions can be usually made. Because of the inherent complexity of heterogeneous migration, a convenient approach is to provide only homogeneous migration even in systems running on heterogeneous platforms. An example is the widely deployed Condor system [@TTL05]. Migration in Condor is strong but not heterogeneous: the destination node is chosen among those with the same platform as the source machine.
For the better understanding of the problem of heterogeneous strong computation migration, we will subdivide the problem of how to migrate into the problems of the capture/restoration of the state and that of representing the state to be transmitted. The capturing and restoring problem involves manipulating information about the internal (local) structure of the execution state. The problem of transmissible representations [@Knabe97] is related to determining an appropriate representation for this information when transferring it between machines.
Heterogeneous strong state capture/restore
------------------------------------------
A basic mechanism for capturing execution state and restoring it later is to use the memory image of the computation. This works up to a point in homogeneous enviroments, but heterogeneous migration introduces specific requirements, because of the need of appropriate translations. It is no longer possible to transfer memory dumps to the destination with no further modification. The main issues arise due to differences in instruction sets and data representations, which will, for instance, invalidate a Program Counter value from one platform to another. Even solving a simple representation problem such as endianness involves knowing the type and size of the data to be read.
Data can be translated at the origin, either to a specified architecture or to an architecture-independent representation, or alternatively can be translated at the destination. In this last case, information about data types must be transmitted along with the data itself.
The problem of capturing the structure of data is related to the programming language’s type system. Most compiled languages keep no runtime information about data types. Besides, some type mechanisms, such as C’s unions, limit the possibility of obtaining type information. Possible solutions to this issue are to restrict unsafe features of the language or to modify compilers to deal with these features [@SH98]. Both can result in non-standard language behaviour.
Much work has been devoted to migrating Java threads [@Funfrocken98; @IKKW02; @TRVC+00; @BHKP+04]. Java restricts the internal and native information made available by the virtual machine [@IKKW02]. The state of Java threads is internal to the JVM: there is no standard API allowing access to it. Moreover, the state of the stack is also non-portable. The stack is implemented in most JVMs as a native data structure (a C structure) [@BHKP+04]. This makes the stack information dependent of the underlying architecture. A translation step is required to represent the stack state in a platform-independent format (a Java Object) during the marshalling or serialization process, and the reverse is true for unmarshalling or deserialization. This implies in translating the values of local variables and operands to Java values, which requires access to the type of the values, but Java does not offer this information at runtime. Runtime type information is embedded in the bytecode of the methods that push the data on the stack. Techniques commonly used to overcome those problems either have drawbacks on serialization performance or on portability.
Restoring the computation state consists in creating a new process or thread, reconstructing the execution state from the transferred state, and restarting execution. A service must be available to managed the required actions at the destination. Depending on the implementation platform, the mechanism to restore the stack, the local variables and the current instruction pointer in every frame can be more or less complicated. The execution must resume from the point at which it was suspended at the origin. But not every language allows jumps to specific points in the code, and even when they do, translation issues may require the definition of logical points marking the next command to be executed. Java, for instance, facilitates the execution of the received code via dynamic class loading, but there is no service allowing to restart the execution from the last executed instruction. The state can be restored by the transferred program itself, by detecting at the beginning of execution that it must reconstruct its state from a predefined data source, or by an external service that will restore the whole execution state and then initiate the program. Migration based on threads involves the additional issue of synchronization.
Performance penalties due to migration can either be distributed over the execution and the migration procedure itself, or be concentrated on this last step. The latter is better for programs where migration is not frequent.
A source code or bytecode instruction may be composed of various machine code instructions, thus, when migration is initiated at machine code level, it is necessary to define the points at which migration is allowed, to avoid inconsistencies. The fact that architecture heterogeneity may cause the program counter location to be different at the destination can also be solved by placing logical points, acting as labels, to reinitiate the execution at the right instruction, given that the migration will only hapen at those points. There is a third application for the placement of logical points in the code, which is to check for migration requests in the cases in which the system allows for objective migration. Logical points can be found in literature under different names, like *poll points[@FCG97], bus stops[@JLHB88], preemption points [@SH98]* or *safe points [@AF02]*, and their use was reported in many of the implementations we studied. The number and location of those points is a compromise between the performance overhead if they are frequently inserted, and the delay in responding to a migration request when they are very sparsely distributed.
The problem of migration is closely related to that of computation persistency [@Bouchenak01]. Indeed, the persistence of a computation can be seen as the problem of moving it to the same location, or otherwise, computation mobility can be achieved through the restauration of a computation persisted in a different host. Several techniques for capturing and restoring state are based on checkpointing facilities [@VD03; @AF02]. Checkpointing an application is the act of saving the application state in persistent storage in a form from which it can be restarted later. It is mainly used in fault tolerance to avoid the need for restarting from the beginning a process formerly running in a faulty host.
Heterogeneous transmissible representations
-------------------------------------------
After the state is captured, it must be prepared for transmission and then transferred to its destination. The code and execution state must be transferred in a format, or representation, that can be understood by a possibly different architecture.
The application code may be either transmitted along with the execution state or obtained, on demand, by the destination host (for instance, by download from a code base). The transmissible representation of the code can be either an architecture-independent representation (to be compiled or interpreted), or the machine code for the target architecture.
Recompiling source code at the destination machine guarantees portability to any platform. Also, the execution performance of the compiled program will be better than in interpreted schemes. On the other hand, there is a delay in restarting due to the recompilation process.
The use of interpreted languages is a valid alternative due to their features of dynamic adaptation and portability. Interpretation allows assuming a homogeneous execution environment, supposing that there is an interpreter for every available platform. The program to be migrated can be expressed in a platform-independent form, as well as the state data, when captured at this level. While the implementation of migration using interpreted languages seems to be straightforward (migration would consist on the implementation of a mechanism to transmit state data and code), common interpreters lack support for execution state capture/restore.
Besides limitations imposed by runtime environments, there are performance losses inherent to the interpretation procedure. Some interpreted languages, to improve performance, allow parts of the program to be coded in a compiled language. This is called *dual programming model* [@URI02], or *interleaving* [@AF02]. While it can help on the performance point of view, it also contributes to the loss of the platform-independence offered by the interpreted approach and makes the capture/restoration of the state information more difficult, as part of it will not be available from the Virtual Machine. This kind of application is not usually supported in migration systems.
If machine code is to be transmitted, it is necessary to generate as many versions of the compiled programs as the number of platforms that will be supported. At migration time, the appropriate program will be selected, if it is not already available at the destination. This implies in generating a new pre-compiled program for every new supported platform, and also in the availability of storage space. The Tui system [@SH98] is an example of the approach based on the migration of native code. It was built to provide a migration mechanism of Ansi-C programs for four architectures within the Unix environment. Capture and recovery is carried out with full knowledge of the destination platform, and also of the data types and variables used within the program. The programs are compiled for each of the four machine types supported by Tui, producing four different binaries. The compiler detects and avoids migration-unsafe features, such as *Unions*, to allow the extraction of typing information. When the process is selected for migration, a program is called to checkpoint the process to an intermediate representation that will be sent to the target machine. This program uses the type information generated by the compiler to extract correctly the data from the executable file. On the destination, another program takes the transmitted representation and creates a new process. After reconstructing all the execution state, the process is restarted from the point at which it was checkpointed. The system specifies points in the code where the migration is allowable, called preemption points.
On the other hand, [@TH91] describes an implementation of heterogeneous process migration based on recompilation. In this case, migration involves transmitting a machine-independent program that, when started at the destination, reconstructs the process’ state and then continues the normal execution of the process. This approach has the advantage that it hides the details of data translation in the compilers of each machine, but it has the drawback of the increase in the time caused by the recompilation and relinking of the program. Migration consists of the following steps: suspending the process to be migrated, translating the machine-dependent state data to a machine independent representation, creating a machine-independent program that represents the process state, transferring that program to the target machine, compiling and linking the transferred program, destroying the source program, and loading and running the final program on the target machine.
The Extended Facile [@Knabe95; @Knabe97] system, although supporting only weak migration, takes a hybrid approach to marshalling which is worth mentioning. Extended Facile is an extension of Facile, a strongly typed functional programming language based on Standard ML with support for concurrency and distribution. Extended Facile supports both architecture-independent and machine code representations, and the joint transmission of several representations, allowing the programmer to choose the representation best suited for its agent. It allows the program, for instance, to choose a machine code implementation when the destination host has the same architecture as the origin.
Classification of heterogeneous strong migration techniques
===========================================================
Methods for the implementation of strong heterogeneous migration are basically similar in different environments. They consist in a mechanism for capturing the execution state information and saving it in stationary or transient storage, and next, transmitting the saved status and restarting the saved computation at the remote location.
Regarding their approach to transmissible representations of code, they can be divided between those which use an architecture-independent representation and those which use machine code. Architecture-independent representations can use either interpretation or recompilation.
On the other hand, a general classification of the methods used to implement state capture and restoration is not quite clear. Bouchenak [@BHKP+04] proposed a classification for approaches to capturing the execution state of Java threads. We believe this classification can be applied in the more general problem of heterogeneous state capture.
The classification originally proposed by Bouchenak identifies an application-level approach and a JVM-level approach, which we generalize to the following:
1. User program pre-processing: Consists of introducing fragments of code (automatically or not) into the user program, in order to make it capable of auto-saving/restoring its status. This implies in runtime and space overhead caused by the inserted code.
2. Platform modification: Consists of modifying the underlying platform or virtual machine to make it provide the required data and functionalities necessary to achieve migration.
In Section \[other\], we will mention another approach, based on languages which provide some built-in facilities for state capture.
Across the following subsections, we survey systems that were implemented according to each of these approaches.
Code pre-processing, or application-level approach
--------------------------------------------------
In this approach, the user program is modified by inserting fragments of code that allow the program to save its execution state and restart the computation by itself. Such modifications can be done to source or compiled code. In the case of interpreted languages, the compiled code would be the pre-compiled code (hereafter called bytecode).
Ferrari et al. [@FCG97], in one of the earliest works discussing heterogeneous state capture/restoration, propose a mechanism called *Process Introspection*. This consists on pre-processing the application source code (written in Ansi-C) to incorporate autonomous checkpoint and restart facilities. The target application domain are scientific applications, whose high performance requirements would not allow the use of interpreted languages for generating platform-independent checkpoints. The implementation consists of a library (the Process Introspection Library, PIL), and a source-code generator called APrIL (Automatic application of the PRocess Introspection Library), which automates the implementation of state capture and recovery. PIL provides a mechanism for describing, saving and restoring data values, and an event-based mechanism for coordinating the capture/restore activities. It can be automatically applied by APrIL to incorporate capture and restore functions for platform-independent modules, that is, those written in a high-level language, that are type safe, and do not rely on the underlying features of a particular platform for correctness [@FCG97]. The transformation consists in adding prologues to every function, which include calls to PIL to register all the local variables or parameter addresses found in the function body in the local variable table. The compiler inserts [*poll points*]{} followed by code which checks migration requests. For the cases when automatic transformation is not possible, the user can make use of the library directly.
The states traversed by a migrated process are:
$Normal Execution \rightarrow State Capture \rightarrow State Recovery \rightarrow Normal Execution$
The capture of stack state is made through a [*native subroutine return*]{} mechanism (see Figure \[Fig:StateCapture\]), which consists in saving the state of the current procedure (including the logical location of the poll point at which the current frame was saved and the local variables and parameters of the function) and returning, and is executed recursively until the base routine is reached.
![Native subroutine return mechanism: State Capture[]{data-label="Fig:StateCapture"}](state_capture){width="\textwidth"}
For restart, every subroutine recovers the data for its local frame and jumps to the poll point at which the stack of the current stack frame was captured, as shown in Figure \[Fig:StateRecovery\]. For this mechanism to work, there must be a poll point in the program after each subroutine call. Since this mechanism is specified at a platform-independent level, the captured state is also valid in any platform, assuming the associated data is stored in a universally recognizable format for masking issues such as those of data representation.
![Native subroutine return mechanism: State Recovery[]{data-label="Fig:StateRecovery"}](state_recovery.eps){width="70.00000%"}
Fünfrocken proposes a similar approach [@Funfrocken98] for saving and restoring the state of Java programs, using exception handling. A compiler inserts try-catch statements in the source code to save the state of execution of each thread in different Java backup objects. On restart, all the threads that were active at the time of the capture are restarted by creating new threads which will be initialized with the contents of their respective saved counterparts. Synchronization issues emerge here, due to the impredictability of the situation of each thread at the moment of state capture. The proposal addresses only subjective migration, and thus the points at which migration is to occur are explicit in the code. The solution was to provide the programmer with a method [*allowGo*]{} to signal that the calling thread is ready to save its state. State saving only occurs if all threads have called this method or have initiated state saving.
The serialization of Java threads in the Brakes framework [@TRVC+00] is achieved through the instrumentation of bytecode. The technique employed for thread stack capture is very similar to the *native subroutine return* mechanism proposed by [@FCG97]. In the [*state capture*]{} state, every method saves the current frame in the Context object and returns to the previous method recursively until the end of the stack is reached. Given that the instruction counter is not available in Java, a counter was created to represent the “last performed instruction” (LPI). On recovery, the thread restores the first frame from the context stack, removes it from the context and moves the program counter to the saved LPI (using the goto instruction, available only at bytecode level), which could correspond to a method being called at the moment of the checkpoint. In this case, the method will be called and it will recover its frame from the context, in the same way as the previous method did before. State verification is done by code inserted automatically by the bytecode transformer at the beginning of every invoked method.
The MAG grid middleware proposal [@FS05], an extension of the Integrade middleware for the execution of Bag-of-Tasks applications, is built over the Brakes framework [@TRVC+00] and the JADE framework. Its goal was to provide for strong migration, thus portability was a major requirement. MAG mobile agents can migrate strongly, objectively and subjectively: the Brakes framework was modified to allow the migration to be initiated by an external entity. The programmer can also insert new points where the checkpoint may occur. The published implementation didn’t include the migration of external dependencies or multi-threaded applications.
In JavaGo [@SMY99], strong mobility is provided by source code transformations. An exception mechanism is employed for capturing the execution state: exceptions are thrown recursively until the whole call stack is saved as a chain of State objects. Because Java does not include a [*goto*]{} statement, these transformations must resort to switch-case statements. This requires some pre-processings (including splitting expressions with side effects) and unfolding techniques. An [*undock*]{} statement marks the part of the code that will be migrated. At restart, unfolding techniques must be also used for loops.
MobileScope [@MPOY04] is a programming language which supports both weak and strong migration. Strong migration is supported by the integration with the JavaGo framework. MobileScope extended JavaGo to support triggering the migration externally. One important aspect of MobileScope is the provision of full mobility, by means of channel mobility, which allows modifying the resource bindings at runtime.
Bettini and De Nicola [@BD01] propose a technique for implementing strong mobility through a weak mechanism, adaptable to any language with support for transmitting data and code. The authors exploit this technique in the implementation of the X-Klaim programming language.
In Aglets [@LO98], the event model allows the programmer to save the state before migration. This has been exploited in MobiGrid [@BG04] which focuses on long sequential applications and opportunistic computation. In MobiGrid, the support for strong migration is achieved by extending Aglets to allow the programmer to save the state of execution periodically, allowing it to be restarted later. Drawbacks of this method are the required programmer effort and the fact that the execution flow is split into several callback procedures, making it hard for the compiler to perform optimizations. The fact that the mobility model in Aglets, and, in general, in weak mobile systems, forces the programmer to manage the objects to be transferred during migration affects the transparency of the process.
Bytecode instrumentation offers better portability than the Virtual Machine modifications techniques and better performance that the source interpreted code approach [@BHKP+04]. A drawback originated by the bytecode-based implementation is the need of code maintenance. There is no guarantee of backward bytecode compatibility in Virtual Machine upgrades, so it may be necessary to modify the implementation after JVM upgrades. Working over the bytecode (for the case of implementations over Virtual Machines), instead of over the source code, has the advantage of the wider instruction set and also a better performance. Also, for pre-processing, the source code must be available, which is not the case with libraries and legacy code. A problem common to both approaches is that the execution state cannot be captured in all situations [@BHKP+04]. In general, shortcomings of code pre-processing are that it implies in changes to program flow and in some time and space penalties at runtime, caused by the inserted code.
Modifying/extending the Runtime System
--------------------------------------
Another alternative for introducing support for state capture and restoration is to modify or extend existing platforms. In this case, there is no need to modify the user code eliminating the requirements of code availability and lowering the execution and space overhead. On the other hand, portability is affected.
Agbaria and Friedman [@AF02] propose a transparent mechanism for checkpoint/restart in heterogeneous environments focusing on fault tolerance. Their approach is to checkpoint the application state at virtual machine level. The implementation was done over the OCaml virtual machine (OCVM). Since the systems operates at VM level, the checkpoint is only allowed at points where the state of the application is consistent, that is, between instructions or during an instruction which does not modify the state of the system. Whith that in mind, the interpreter checks for a flag signalling a checkpoint request before fetching a new instruction. State capture is based on the tagging of data types and the garbage collection features of OCaml. File-descriptor checkpointing is based on the OCaml support for I/O interception. A multi-threaded consistent checkpoint is achieved by taking the checkpoint only after stopping all threads.
Agbaria and Friedman’s work assumes that failures are rare, and that, on that account, it is preferrable to penalize restart, maintaining checkpointing overhead minimal. Thus, data is saved in its native representation and translated, if necessary, when the application is restarted. In order to avoid blocking the application during checkpoint, a new process is forked to save the state and then exits.
D’Agents [@GCKP+02] (formerly called Agent Tcl) is a mobile agent system with support for subjective strong migration. Initially implemented in Tcl, it currently allows mobile agents to be written in Tcl, Java, and Scheme, and supports strong mobility in Tcl and Java with “significant” modifications to the respective interpreters. Migration in D’Agents is accomplished with the [*agent\_jmp*]{} command. [*agent\_jmp*]{} captures the internal state of the Tcl script and transfers it to the destination machine, where the execution continues from the next command. The D’Agents server is multi-threaded: every agent runs in a separate thread.
Bouchenak et al. [@BHKP+04] present a solution for Java thread serialization/deserialization which intends to build thread mobility or persistence while avoiding the performance overhead incurred by previous approaches. This is achieved through type inference and dynamic de-optimization techniques. The Java Virtual Machine is extended to capture the state of a Java thread as an object and to initialize a thread with a particular state. The Java compiler was not modified. Thread serialization does not handle problems of object sharing between threads, distribution, synchronization, or the management of object dependencies.
The authors compare two prototypes, based on Interpreter-based serialization (ITS) and on capture time-based thread serialization (CTS). The first was based on the modification of the interpreter to capture the data type every time a bytecode instruction saved data on the stack. For CTS, the type inference was done by analyzing the bytecode only at thread serialization time. The work concluded that given that the serialization of the interpreter (ITS) is not compliant with JIT compilation, it would not be a realistic solution. It also introduces an execution performance overhead. On the other hand, CTS avoids any execution performance overhead, but the cost is transferred to the serialization latency and thus it is not advisable for applications with a high serialization frequency, such as mobile agents. Application-level Java thread serialization would probably be the best solution for those cases.
Illman and others [@IKKW02] propose the use of the Java Platform debugging interface to achieve transparent migration in the context of the CIA project (Collaboration and Coordination Infrastructure for Personal Agents), which deals with the development of an infrastructure for software agents. The Java debugging architecture provides access to runtime information like stack frames, local variables and the program counter. It is possible to stop and resume execution, execute single bytecode instructions and set/unset breakpoints. A problem in this proposal is how to reestablish the program counter after the whole state has been transferred. Given that the Java debugging architecture does not include this feature, the solutions proposed are the modification of the JVM, or else, the instrumentation of the byte code, and take us back to the previous approaches.
Migration with Language Support {#other}
-------------------------------
Functional languages facilitate the manipulation of functions, allowing their transfer to remote hosts for execution. Heterogeneous strong migration can be implemented in those systems by means of continuations, if those are offered as transmissible structures. Tarau and Dahl[@TD01], for instance, use this strategy to implement strong heterogeneous migration in BinProlog.
Other authors have dealt with the problem of migration using functional languages. An implementation for strong mobility over mHaskell [@DTL03] is reported in [@DTL06]. It is based on weak mobility, higher-order channels, and first-class continuations, without the need for changes to the run-time system or built-in support for continuations. Unlike other proposals, the implementation is based on [*Monads*]{}, available in Haskell and in other purely functional languages.
Systems like ARA, NOMADS and Telescript implement support for strong migration by creating new platforms that offer all the required information to execute the procedure. A comparison among those systems is presented in [@GCKP+02].
Discussion and Conclusions
==========================
Cardelli in Mobile Computation [@cardelli97], commenting the fact that traditional languages and traditional compilers are not well suited for network computing, asserts that languages that are not portable on-line will be abandoned because they don’t provide mobility. But what support should a run-time engine provide for heterogeneous strong migration?
We have identified two layers for the implementation of migration mechanisms. The inner layer implements state capture, serialization, and restoration. The outer layer manages the problems of mobility at a linguistic level. At the inner layer, we believe the following items should be considered.
- A mechanism to capture and serialize the state of the execution (including data types, program counter);
- A mechanism for transferring a computation (data, code and state of execution);
- Support for deserializing and restoring the computation (it means, a way to transform back the captured values from the independent representation, and also a way to restart execution from the point where it stopped);
- The performance offered by these mechanism must satisfy the application goals.
Generalizing from the surveyed work, the mechanisms for capture and restore can be grouped in two major categories, according to the support provided by the programming language. In the case when there is no language support for state capture/restoration, the problem can be solved in two ways, which are based on:
1. User program pre-processing: Consists in introducing fragments of code (automatically or not) into the user program in order to make it capable of auto-saving/restoring its status. This implies in runtime and space overhead caused by the injected code.
2. RTS modification (or extension): Consists in modifying or extending the underlying platform or virtual machine, seeking to provide the required data and functionalities necessary to achieve migration. This usually results in better performance but implies in less portability and maintanability.
Language support for state capture/restoration can be expressed either by means of mechanisms such as first-class continuations, typically offered by functional languages, or by explicit primitives for state manipulation.
The use of continuations has the advantage of enabling strong migration to be implemented using weak mobility. Nevertheless, continuations are almost exclusively supported by functional languages, which have well known performance limitations. In general, features commonly present in the so called dynamic programming languages like *introspection*, *continuations*, and the possibility of modifying programs at runtime make them interesting vehicles for the implementation of migration. On the other hand, languages and systems that do not offer enough support for state capture and restoration force the implementor to choose between preprocessing techniques or modifying the platform. This implies in a compromise between portability and performance.
The issue of transferring the computation involves the transmitting the code and captured state in a way that is understandable at the target node. The code can be transferred either as an architecture-independent representation (to be compiled or interpreted), or as machine code for the target architecture. The transmission of the captured data seems to be well handled by current solutions such as Java’s RMI.
Table \[TAB:generalization\] shows the classification we propose for the case of the surveyed works.
--------------------------------------------------------------- -- --
**Name & **Capture/Recovery Method & **Code Representation\
Tui & RTS modification & Native Code\
Recompilation & RTS modification & Source Code\
Process Introspection & Program Transformation & Native Code\
JavaGo & Program pre-processing & Interpreted\
Brakes & Program pre-processing & Interpreted\
Bouchenak & RTS modification & Interpreted\
CIA project & Debugger Platform & Interpreted\
D’Agents & RTS modification & Interpreted\
DTL06 & Language Supported & Interpreted\
******
--------------------------------------------------------------- -- --
: Evaluation according to classification[]{data-label="TAB:generalization"}
Performance is still a issue in the implementation of migration. However, as we mentioned in the introduction, current motivations for migration, such as evicting processes in opportunistic computing systems, make us evaluate migration performance from a new perspective.
Is migration a good idea? It certainly is, for a number of applications and in specific conditions. Is strong migration a good idea? Actually, given the costs implied in the mechanisms for adapting it to a heterogeneous environment, it seems that a combination between weak and strong migration (if possible using weak migration) would be the best answer for systems requiring to use this technology. It will depend strongly on the focus of the application: if it is a long lasting and high performance application it will require a fault tolerance service, thus the need for checkpointing, and high migration performance. It is probably not the case for an agent sniffing for information on the internet. Further work must be done to evaluate the advantages of the implementation of migration for the current potential applications. New solutions for security are currently being investigated, like the possibility of code signing mobile agents. Error management issues should also be better studied.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been partially supported by CNPq Brasil and PCI/LNCC.
|
---
abstract: 'With the goal of recovering high-quality image content from its degraded version, image restoration enjoys numerous applications, such as in surveillance, computational photography, medical imaging, and remote sensing. Recently, convolutional neural networks (CNNs) have achieved dramatic improvements over conventional approaches for image restoration task. Existing CNN-based methods typically operate either on full-resolution or on progressively low-resolution representations. In the former case, spatially precise but contextually less robust results are achieved, while in the latter case, semantically reliable but spatially less accurate outputs are generated. In this paper, we present a novel architecture with the collective goals of maintaining spatially-precise high-resolution representations through the entire network, and receiving strong contextual information from the low-resolution representations. The core of our approach is a multi-scale residual block containing several key elements: (a) parallel multi-resolution convolution streams for extracting multi-scale features, (b) information exchange across the multi-resolution streams, (c) spatial and channel attention mechanisms for capturing contextual information, and (d) attention based multi-scale feature aggregation. In the nutshell, our approach learns an enriched set of features that combines contextual information from multiple scales, while simultaneously preserving the high-resolution spatial details. Extensive experiments on five real image benchmark datasets demonstrate that our method, named as MIRNet, achieves state-of-the-art results for a variety of image processing tasks, including image denoising, super-resolution and image enhancement.'
author:
- Syed Waqas Zamir
- Aditya Arora
- Salman Khan
- Munawar Hayat
- Fahad Shahbaz Khan
- 'Ming-Hsuan Yang'
- Ling Shao
bibliography:
- 'MIRNet.bib'
title: Learning Enriched Features for Real Image Restoration and Enhancement
---
Introduction
============
Image content is exponentially growing due to the ubiquitous presence of cameras on various devices. During image acquisition, degradations of different severities are often introduced. It is either because of the physical limitations of cameras, or due to inappropriate lighting conditions. For instance, smart phone cameras come with narrow aperture, and have small sensors with limited dynamic range. Consequently, they frequently generate noisy and low-contrast images. Similarly, images captured under the unsuitable lighting are either too dark, or too bright. The art of recovering the original clean image from its corrupted measurements is studied under the image restoration task. It is an ill-posed inverse problem, due to the existence of many possible solutions.
Recently, deep learning models have made significant advancements for image restoration and enhancement, as they can learn strong (generalizable) priors from large-scale datasets. Existing CNNs typically follow one of the two architecture designs: 1) an encoder-decoder, or 2) high-resolution (single-scale) feature processing. The encoder-decoder models [@ronneberger2015u; @kupyn2019deblurgan; @chen2018; @zhang2019kindling] first progressively map the input to a low-resolution representation, and then apply a gradual reverse mapping to the original resolution. Although these approaches learn a broad context by spatial-resolution reduction, on the downside, the fine spatial details are lost, making it extremely hard to recover them in the later stages. On the other side, the high-resolution (single-scale) networks [@dong2015image; @DnCNN; @zhang2020residual; @ignatov2017dslr] do not employ any downsampling operation, and thereby produce images with spatially more accurate details. However, these networks are less effective in encoding contextual information due to their limited receptive field. Image restoration is a position-sensitive procedure, where pixel-to-pixel correspondence from the input image to the output image is needed. Therefore, it is important to remove only the undesired degraded image content, while carefully preserving the desired fine spatial details (such as true edges and texture). Such functionality for segregating the degraded content from the true signal can be better incorporated into CNNs with the help of large context, *e.g.*, by enlarging the receptive field. Towards this goal, we develop a new *multi-scale* approach that maintains the original high-resolution features along the network hierarchy, thus minimizing the loss of precise spatial details. Simultaneously, our model encodes multi-scale context by using *parallel convolution streams* that process features at lower spatial resolutions. The multi-resolution parallel branches operate in a manner that is complementary to the main high-resolution branch, thereby providing us more precise and contextually enriched feature representations.
The main difference between our method and existing multi-scale image processing approaches is the way we aggregate contextual information. First, the existing methods [@tao2018scale; @nah2017; @gu2019self] process each scale in isolation, and exchange information only in a top-down manner. In contrast, we progressively fuse information across all the scales at each resolution-level, allowing both top-down and bottom-up information exchange. Simultaneously, both fine-to-coarse and coarse-to-fine knowledge exchange is laterally performed on each stream by a new *selective kernel* fusion mechanism. Different from existing methods that employ a simple concatenation or averaging of features coming from multi-resolution branches, our fusion approach dynamically selects the useful set of kernels from each branch representations using a self-attention approach. More importantly, the proposed fusion block combines features with varying receptive fields, while preserving their distinctive complementary characteristics.
Our main contributions in this work include:
- A novel feature extraction model that obtains a complementary set of features across multiple spatial scales, while maintaining the original high-resolution features to preserve precise spatial details.
- A regularly repeated mechanism for information exchange, where the features across multi-resolution branches are progressively fused together for improved representation learning.
- A new approach to fuse multi-scale features using a selective kernel network that dynamically combines variable receptive fields and faithfully preserves the original feature information at each spatial resolution.
- A recursive residual design that progressively breaks down the input signal in order to simplify the overall learning process, and allows the construction of very deep networks.
- Comprehensive experiments are performed on five real image benchmark datasets for different image processing tasks including, image denoising, super-resolution and image enhancement. Our method achieves state-of-the-results on *all* five datasets. Furthermore, we extensively evaluate our approach on practical challenges, such as generalization ability across datasets.
Related Work
============
With the rapidly growing image media content, there is a pressing need to develop effective image restoration and enhancement algorithms. In this paper, we propose a new approach capable of performing image denoising, super-resolution and image enhancement. Different from existing works for these problems, our approach processes features at the original resolution in order to preserve spatial details, while effectively fuses contextual information from multiple parallel branches. Next, we briefly describe the representative methods for each of the studied problems. **Image denoising.** Classic denoising methods are mainly based on modifying transform coefficients [@yaroslavsky1996local; @donoho1995noising; @simoncelli1996noise] or averaging neighborhood pixels [@smith1997susan; @tomasi1998bilateral; @perona1990scale; @rudin1992nonlinear]. Although the classical methods perform well, the self-similarity [@efros1999texture] based algorithms, *e.g.*, NLM [@NLM] and BM3D [@BM3D], demonstrate promising denoising performance. Numerous patch-based algorithms that exploit redundancy (self-similarity) in images are later developed [@dong2012nonlocal; @WNNM; @mairal2009non; @hedjam2009markovian]. Recently, deep learning-based approaches [@MLP; @RIDNet; @Brooks2019; @Gharbi2016; @CBDNet; @N3Net; @DnCNN; @FFDNetPlus] make significant advances in image denoising, yielding favorable results than those of the hand-crafted methods.
**Super-resolution (SR).** Prior to the deep-learning era, numerous SR algorithms have been proposed based on the sampling theory [@keys1981cubic; @irani1991improving], edge-guided interpolation [@allebach1996edge; @li2001new; @zhang2006edge], natural image priors [@kim2010single; @xiong2010robust], patch-exemplars [@chang2004super; @freedman2011image] and sparse representations [@yang2010image; @yang2008image]. Currently, deep-learning techniques are actively being explored, as they provide dramatically improved results over conventional algorithms. The data-driven SR approaches differ according to their architecture designs [@wang2019deep; @anwar2019deep]. Early methods [@dong2014learning; @dong2015image] take a low-resolution (LR) image as input and learn to directly generate its high-resolution (HR) version. In contrast to directly producing a latent HR image, recent SR networks [@VDSR; @tai2017memnet; @tai2017image; @hui2018fast] employ the residual learning framework [@He2016] to learn the high-frequency image detail, which is later added to the input LR image to produce the final super-resolved result. Other networks designed to perform SR include recursive learning [@kim2016deeply; @han2018image; @ahn2018fast], progressive reconstruction [@wang2015deep; @Lai2017], dense connections [@tong2017image; @wang2018esrgan; @zhang2020residual], attention mechanisms [@RCAN; @dai2019second; @zhang2019residual], multi-branch learning [@Lai2017; @EDSR; @dahl2017pixel; @li2018multi], and generative adversarial models [@wang2018esrgan; @park2018srfeat; @sajjadi2017enhancenet; @SRResNet]. **Image enhancement.** Oftentimes, cameras provide images that are less vivid and lack contrast. A number of factors contribute to the low quality of images, including unsuitable lighting conditions and physical limitations of camera devices. For image enhancement, histogram equalization is the most commonly used approach. However, it frequently produces under-enhanced or over-enhanced images. Motivated by the Retinex theory [@land1977retinex], several enhancement algorithms mimicking human vision have been proposed in the literature [@bertalmio2007; @palma2008perceptually; @jobson1997multiscale; @rizzi2004retinex]. Recently, CNNs have been successfully applied to general, as well as low-light, image enhancement problems. Notable works employ Retinex-inspired networks [@Shen2017; @wei2018deep; @zhang2019kindling], encoder-decoder networks [@chen2018encoder; @Lore2017; @ren2019low], and generative adversarial networks [@chen2018deep; @ignatov2018wespe; @deng2018aesthetic].
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Framework of the proposed network MIRNet that learns enriched feature representations for image restoration and enhancement. MIRNet is based on a recursive residual design. In the core of MIRNet is the multi-scale residual block (MRB) whose main branch is dedicated to maintaining spatially-precise high-resolution representations through the entire network and the complimentary set of parallel branches provide better contextualized features. It also allows information exchange across parallel streams via selective kernel feature fusion (SKFF) in order to consolidate the high-resolution features with the help of low-resolution features, and vice versa.[]{data-label="fig:framework"}](Images/framework.png "fig:"){width="\textwidth"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Proposed Method
===============
In this section, we first present an overview of the proposed MIRNet for image restoration and enhancement, illustrated in Fig. \[fig:framework\]. We then provide details of the *multi-scale residual block*, which is the fundamental building block of our method, containing several key elements: **(a)** parallel multi-resolution convolution streams for extracting (fine-to-coarse) semantically-richer and (coarse-to-fine) spatially-precise feature representations, **(b)** information exchange across multi-resolution streams, **(c)** attention-based aggregation of features arriving from multiple streams, **(d)** dual-attention units to capture contextual information in both spatial and channel dimensions, and **(e)** residual resizing modules to perform downsampling and upsampling operations.
**Overall Pipeline.** Given an image $\mathbf{I} \in \mathbb{R}^{H\times W \times 3}$, the network first applies a convolutional layer to extract low-level features $\mathbf{X_0} \in \mathbb{R}^{H\times W \times C}$. Next, the feature maps $\mathbf{X_0}$ pass through $N$ number of recursive residual groups (RRGs), yielding deep features $\mathbf{X_d} \in \mathbb{R}^{H\times W \times C}$. We note that each RRG contains several multi-scale residual blocks, which is described in Section \[sec:msrb\]. Next, we apply a convolution layer to deep features $\mathbf{X_d}$ and obtain a residual image $\mathbf{R} \in \mathbb{R}^{H\times W \times 3}$. Finally, the restored image is obtained as $\mathbf{\hat{I}} = \mathbf{I} + \mathbf{R}$. We optimize the proposed network using the Charbonnier loss [@charbonnier1994]:
$$\label{Eq:loss}
\mathcal{L}(\mathbf{\hat{I}},\mathbf{I}^*) = \sqrt{ {\|\mathbf{\hat{I}}-\mathbf{I}^*\|}^2 + {\varepsilon}^2 },$$
where $\mathbf{I}^*$ denotes the ground-truth image, and $\varepsilon$ is a constant which we empirically set to $10^{-3}$ for all the experiments.
Multi-scale Residual Block (MRB) {#sec:msrb}
--------------------------------
In order to encode context, existing CNNs [@ronneberger2015u; @newell2016stacked; @noh2015learning; @xiao2018simple; @badrinarayanan2017segnet; @peng2016recurrent] typically employ the following architecture design: **(a)** the receptive field of neurons is fixed in *each* layer/stage, **(b)** the spatial size of feature maps is *gradually* reduced to generate a semantically strong low-resolution representation, and **(c)** a high-resolution representation is *gradually* recovered from the low-resolution representation. However, it is well-understood in vision science that in the primate visual cortex, the sizes of the local receptive fields of neurons in the same region are different [@hubel1962receptive; @riesenhuber1999hierarchical; @serre2007robust; @hung2005fast]. Therefore, such a mechanism of collecting multi-scale spatial information in the same layer needs to be incorporated in CNNs [@huang2017multi; @hrnet; @fourure2017residual; @Szegedy2015]. In this paper, we propose the multi-scale residual block (MRB), as shown in Fig. \[fig:framework\]. It is capable of generating a spatially-precise output by maintaining high-resolution representations, while receiving rich contextual information from low-resolutions. The MRB consists of multiple (three in this paper) fully-convolutional streams connected in parallel. It allows information exchange across parallel streams in order to consolidate the high-resolution features with the help of low-resolution features, and vice versa. Next, we describe individual components of MRB.
**Selective kernel feature fusion (SKFF).** One fundamental property of neurons present in the visual cortex is to be able to change their receptive fields according to the stimulus [@li2019selective]. This mechanism of adaptively adjusting receptive fields can be incorporated in CNNs by using multi-scale feature generation (in the same layer) followed by feature aggregation and selection. The most commonly used approaches for feature aggregation include simple concatenation or summation. However, these choices provide limited expressive power to the network, as reported in [@li2019selective]. In MRB, we introduce a nonlinear procedure for fusing features coming from multiple resolutions using a self-attention mechanism. Motivated by [@li2019selective], we call it selective kernel feature fusion (SKFF).
The SKFF module performs dynamic adjustment of receptive fields via two operations –[*Fuse* and *Select*, as illustrated in Fig. \[fig:skff\]]{}. The *fuse* operator generates global feature descriptors by combining the information from multi-resolution streams. The *select* operator uses these descriptors to recalibrate the feature maps (of different streams) followed by their aggregation. Next, we provide details of both operators for the three-stream case, but one can easily extend it to more streams. **(1) Fuse:** SKFF receives inputs from three parallel convolution streams carrying different scales of information. We first combine these multi-scale features using an element-wise sum as: $\mathbf{L = L_1 + L_2 + L_3}$. We then apply global average pooling (GAP) across the spatial dimension of $\mathbf{L} \in \mathbb{R}^{H\times W \times C}$ to compute channel-wise statistics $\mathbf{s} \in \mathbb{R}^{1\times 1 \times C}$. Next, we apply a channel-downscaling convolution layer to generate a compact feature representation $\mathbf{z} \in \mathbb{R}^{1\times 1 \times r}$, where $r=\frac{C}{8}$ for all our experiments. Finally, the feature vector $\mathbf{z}$ passes through three parallel channel-upscaling convolution layers (one for each resolution stream) and provides us with three feature descriptors $\mathbf{v_1}, \mathbf{v_2}$ and $\mathbf{v_3}$, each with dimensions $1\times1\times C$. **(2) Select:** this operator applies the softmax function to $\mathbf{v_1}, \mathbf{v_2}$ and $\mathbf{v_3}$, yielding attention activations $\mathbf{s_1}, \mathbf{s_2}$ and $\mathbf{s_3}$ that we use to adaptively recalibrate multi-scale feature maps $\mathbf{L_1}, \mathbf{L_2}$ and $\mathbf{L_3}$, respectively. The overall process of feature recalibration and aggregation is defined as: $\mathbf{U = s_1 \cdot L_1 + s_2\cdot L_2 + s_3 \cdot L_3}$. Note that the SKFF uses $\sim6\times$ fewer parameters than aggregation with concatenation but generates more favorable results (an ablation study is provided in experiments section).
**Dual attention unit (DAU).** While the SKFF block fuses information across multi-resolution branches, we also need a mechanism to share information within a feature tensor, both along the spatial and the channel dimensions. Motivated by the advances of recent low-level vision methods [@RCAN; @RIDNet; @dai2019second; @zhang2019residual] based on the attention mechanisms [@hu2018squeeze; @wang2018non], we propose the dual attention unit (DAU) to extract features in the convolutional streams. The schematic of DAU is shown in Fig. \[fig:dau\]. The DAU suppresses less useful features and only allows more informative ones to pass further. This feature recalibration is achieved by using channel attention [@hu2018squeeze] and spatial attention [@woo2018cbam] mechanisms. **(1) Channel attention (CA)** branch exploits the inter-channel relationships of the convolutional feature maps by applying *squeeze* and *excitation* operations [@hu2018squeeze]. Given a feature map $\mathbf{M} \in \mathbb{R}^{H\times W \times C}$, the squeeze operation applies global average pooling across spatial dimensions to encode global context, thus yielding a feature descriptor $\mathbf{d} \in \mathbb{R}^{1\times 1 \times C}$. The excitation operator passes $\mathbf{d}$ through two convolutional layers followed by the sigmoid gating and generates activations $\mathbf{\hat{d}} \in \mathbb{R}^{1\times 1 \times C}$. Finally, the output of CA branch is obtained by rescaling $\mathbf{M}$ with the activations $\mathbf{\hat{d}}$. **(2) Spatial attention (SA)** branch is designed to exploit the inter-spatial dependencies of convolutional features. The goal of SA is to generate a spatial attention map and use it to recalibrate the incoming features $\mathbf{M}$. To generate the spatial attention map, the SA branch first independently applies global average pooling and max pooling operations on features $\mathbf{M}$ along the channel dimensions and concatenates the outputs to form a feature map $\mathbf{f} \in \mathbb{R}^{H\times W \times 2}$. The map $\mathbf{f}$ is passed through a convolution and sigmoid activation to obtain the spatial attention map $\mathbf{\hat{f}} \in \mathbb{R}^{H\times W \times 1}$, which we then use to rescale $\mathbf{M}$.
**Residual resizing modules.** The proposed framework employs a recursive residual design (with skip connections) to ease the flow of information during the learning process. In order to maintain the residual nature of our architecture, we introduce residual resizing modules to perform downsampling (Fig. \[fig:downsample\]) and upsampling (Fig. \[fig:upsample\]) operations. In MRB, the size of feature maps remains constant along convolution streams. On the other hand, across streams the feature map size changes depending on the input resolution index $i$ and the output resolution index $j$. If $i<j$, the input feature tensor is downsampled, and if $i>j$, the feature map is upsampled. To perform $2\times$ downsampling (halving the spatial dimension and doubling the channel dimension), we apply the module in Fig. \[fig:downsample\] only once. For $4\times$ downsampling, the module is applied twice, consecutively. Similarly, one can perform $2\times$ and $4\times$ upsampling by applying the module in Fig. \[fig:upsample\] once and twice, respectively. Note in Fig. \[fig:downsample\], we integrate anti-aliasing downsampling [@zhang2019making] to improve the shift-equivariance of our network.
[0.49]{} {width="\textwidth"}
[0.49]{} {width="\textwidth"}
Experiments
===========
In this section, we perform qualitative and quantitative assessment of the results produced by our MIRNet and compare it with the previous best methods. Next, we describe the datasets, and then provide the implementation details. Finally, we report results for **(a)** image denoising, **(b)** super-resolution and **(c)** image enhancement on five real image datasets. The source code and trained models will be released publicly[^1].
Real Image Datasets
-------------------
**Image denoising.** **(1) DND [@dnd]** consists of $50$ images captured with four consumer cameras. Since the images are of very high-resolution, the dataset providers extract $20$ crops of size $512\times512$ from each image, yielding $1000$ patches in total. All these patches are used for testing (as DND does not contain training or validation sets). The ground-truth noise-free images are not released publicly, therefore the image quality scores in terms of PSNR and SSIM can only be obtained through an online server [@dndwebsite]. **(2) SIDD [@sidd]** is particularly collected with smartphone cameras. Due to the small sensor and high-resolution, the noise levels in smartphone images are much higher than those of DSLRs. SIDD contains $320$ image pairs for training and $1280$ for validation. **Super-resolution.** **(1) RealSR [@RealSR]** contains real-world LR-HR image pairs of the same scene captured by adjusting the focal-length of the cameras. RealSR have both indoor and outdoor images taken with two cameras. The number of training image pairs for scale factors $\times2$, $\times3$ and $\times4$ are $183$, $234$ and $178$, respectively. For each scale factor, $30$ test images are also provided in RealSR.
**Image enhancement.** **(1) LoL [@wei2018deep]** is created for low-light image enhancement problem. It provides 485 images for training and 15 for testing. Each image pair in LoL consists of a low-light input image and its corresponding well-exposed reference image. **(2) MIT-Adobe FiveK [@mit_fivek]** contains $5000$ images of various indoor and outdoor scenes captured with several DSLR cameras in different lighting conditions. The tonal attributes of all images are manually adjusted by five different trained photographers (labelled as experts A to E). Same as in [@hu2018exposure; @park2018distort; @wang2019underexposed], we also consider the enhanced images of expert C as the ground-truth. Moreover, the first 4500 images are used for training and the last 500 for testing.
Implementation Details
----------------------
The proposed architecture is end-to-end trainable and requires no pre-training of sub-modules. We train three different networks for three different restoration tasks. The training parameters, common to all experiments, are the following. We use 3 RRGs, each of which further contains $2$ MRBs. The MRB consists of $3$ parallel streams with channel dimensions of $64, 128, 256$ at resolutions $1, \frac{1}{2}, \frac{1}{4}$, respectively. Each stream has $2$ DAUs. The models are trained with the Adam optimizer ($\beta_1 = 0.9$, and $\beta_2=0.999$) for $7\times10^5$ iterations. The initial learning rate is set to $2\times10^{-4}$. We employ the cosine annealing strategy [@loshchilov2016sgdr] to steadily decrease the learning rate from initial value to $10^{-6}$ during training. We extract patches of size $128\times128$ from training images. The batch size is set to $16$ and, for data augmentation, we perform horizontal and vertical flips.
Image Denoising
---------------
In this section, we demonstrate the effectiveness of the proposed MIRNet for image denoising. We train our network only on the training set of the SIDD [@sidd] and directly evaluate it on the test images of both SIDD and DND [@dnd] datasets. Quantitative comparisons in terms of PSNR and SSIM metrics are summarized in Table \[table:sidd\] and Table \[table:dnd\] for SIDD and DND, respectively. Both tables show that our MIRNet performs favourably against the data-driven, as well as conventional, denoising algorithms. Specifically, when compared to the recent best method VDN [@VDN], our algorithm demonstrates a performance gain of $0.44$ dB on SIDD and $0.50$ dB on DND. Furthermore, it is worth noting that CBDNet [@CBDNet] and RIDNet [@RIDNet] use additional training data, yet our method provides significantly better results. For instance, our method achieves $8.94$ dB improvement over CBDNet [@CBDNet] on the SIDD dataset and $1.82$ dB on DND.
In Fig. \[fig:dnd example\] and Fig. \[fig:sidd example\], we present visual comparisons of our results with those of other competing algorithms. It can be seen that our MIRNet is effective in removing real noise and produces perceptually-pleasing and sharp images. Moreover, it is capable of maintaining the spatial smoothness of the homogeneous regions without introducing artifacts. In contrast, most of the other methods either yield over-smooth images and thus sacrifice structural content and fine textural details, or produce images with chroma artifacts and blotchy texture.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB/SSID_0324_noisy "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB/SSID_0324_CBD "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB/SSID_0324_RIDNet "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB/SSID_0324_VDN "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB/SSID_0324_Ours_MSRNet "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB/SSID_0324_gt "fig:"){width=".16\textwidth"}
18.25 dB 28.84 dB 35.57 dB 36.39 dB **36.97 dB**
![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB1/SSID_3828_noisy "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB1/SSID_3828_CBD "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB1/SSID_3828_RIDNet "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB1/SSID_3828_VDN "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB1/SSID_3828_Ours_MSRNet "fig:"){width=".16\textwidth"} ![Denoising examples from SIDD [@sidd]. Our method effectively removes real noise from challenging images, while better recovering structural content and fine texture.[]{data-label="fig:sidd example"}](Images/Denoising/SIDD/RGB1/SSID_3828_gt "fig:"){width=".16\textwidth"}
18.16 dB 20.36 dB 29.83 dB 30.31 dB **31.36 dB**
Noisy CBDNet [@CBDNet] RIDNet [@RIDNet] VDN [@VDN] MIRNet (Ours) Reference
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
**Generalization capability.** The DND and SIDD datasets are acquired with different sets of cameras having different noise characteristics. Since the DND benchmark does not provide training data, setting a new state-of-the-art on DND with our SIDD trained network indicates the good generalization capability of our approach.
Super-Resolution (SR)
---------------------
We compare our MIRNet against the state-of-the-art SR algorithms (VDSR [@VDSR], SRResNet [@SRResNet], RCAN [@RCAN], LP-KPN [@RealSR]) on the testing images of the RealSR [@RealSR] for upscaling factors of $\times2$, $\times3$ and $\times4$. Note that all the benchmarked algorithms are trained on the RealSR [@RealSR] dataset for fair comparison. In the experiments, we also include bicubic interpolation [@keys1981cubic], which is the most commonly used method for generating super-resolved images. Here, we compute the PSNR and SSIM scores using the Y channel (in YCbCr color space), as it is a common practice in the SR literature [@RCAN; @RealSR; @wang2019deep; @anwar2019deep]. The results in Table \[table:realSR\] show that the bicubic interpolation provides the least accurate results, thereby indicating its low suitability for dealing with real images. Moreover, the same table shows that the recent method LP-KPN [@RealSR] provides marginal improvement of only $\sim0.04$ dB over the previous best method RCAN [@RCAN]. In contrast, our method significantly advances state-of-the-art and consistently yields better image quality scores than other approaches for all three scaling factors. Particularly, compared to LP-KPN [@RealSR], our method provides performance gains of $0.45$ dB, $0.74$ dB, and $0.22$ dB for scaling factors $\times2$, $\times3$ and $\times4$, respectively. The trend is similar for the SSIM metric as well.
Visual comparisons in Fig. \[fig:sr example\] show that our MIRNet recovers content structures effectively. In contrast, VDSR [@VDSR], SRResNet [@SRResNet] and RCAN [@RCAN] reproduce results with noticeable artifacts. Furthermore, LP-KPN [@RealSR] is not able to preserve structures (see near the right edge of the crop). Several more examples are provided in Fig. \[fig:sr crop examples\] to further compare the image reproduction quality of our method against the previous best method [@RealSR]. It can be seen that LP-KPN [@RealSR] has a tendency to over-enhance the contrast (cols. 1, 3, 4) and in turn causes loss of details near dark and high-light areas. In contrast, the proposed MIRNet successfully reconstructs structural patterns and edges (col. 2) and produces images that are natural (cols. 1, 4) and have better color reproduction (col. 5).
**Cross-camera generalization.** The RealSR [@RealSR] dataset consists of images taken with Canon and Nikon cameras at three scaling factors. To test the cross-camera generalizability of our method, we train the network on the training images of one camera and directly evaluate it on the test set of the other camera. Table \[table:realSR generalization\] demonstrates the generalization of competing methods for four possible cases: (a) training and testing on Canon, (b) training on Canon, testing on Nikon, (c) training and testing on Nikon, and (d) training on Nikon, testing on Canon. It can be seen that, for all scales, LP-KPN [@RealSR] and RCAN [@RCAN] shows comparable performance. In contrast, our MIRNet exhibits more promising generalization.
Image Enhancement
-----------------
In this section, we demonstrate the effectiveness of our algorithm by evaluating it for the image enhancement task. We report PSNR/SSIM values of our method and several other techniques in Table \[table:lol\] and Table \[table:fivek\] for the LoL [@wei2018deep] and MIT-Adobe FiveK [@mit_fivek] datasets, respectively. It can be seen that our MIRNet achieves significant improvements over previous approaches. Notably, when compared to the recent best methods, MIRNet obtains $3.27$ dB performance gain over KinD [@zhang2019kindling] on the LoL dataset and $0.69$ dB improvement over DeepUPE [@wang2019underexposed] on the Adobe-Fivek dataset.
We show visual results in Fig. \[Fig:qual\_lol\] and Fig. \[Fig:qual\_fivek\]. Compared to other techniques, our method generates enhanced images that are natural and vivid in appearance and have better global and local contrast.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Visual comparison of low-light enhancement approaches on the LoL dataset [@wei2018deep]. Our method reproduces image that is visually closer to the ground-truth in terms of brightness and global contrast.[]{data-label="Fig:qual_lol"}](Images/Enhancement/lol_images/lol_input "fig:"){width="24.40000%"} ![Visual comparison of low-light enhancement approaches on the LoL dataset [@wei2018deep]. Our method reproduces image that is visually closer to the ground-truth in terms of brightness and global contrast.[]{data-label="Fig:qual_lol"}](Images/Enhancement/lol_images/lol_lime "fig:"){width="24.40000%"} ![Visual comparison of low-light enhancement approaches on the LoL dataset [@wei2018deep]. Our method reproduces image that is visually closer to the ground-truth in terms of brightness and global contrast.[]{data-label="Fig:qual_lol"}](Images/Enhancement/lol_images/crm "fig:"){width="24.40000%"} ![Visual comparison of low-light enhancement approaches on the LoL dataset [@wei2018deep]. Our method reproduces image that is visually closer to the ground-truth in terms of brightness and global contrast.[]{data-label="Fig:qual_lol"}](Images/Enhancement/lol_images/lol_retx "fig:"){width="24.40000%"}
Input image LIME [@guo2016lime] CRM [@ying2017bio] Retinex-Net [@wei2018deep]
![Visual comparison of low-light enhancement approaches on the LoL dataset [@wei2018deep]. Our method reproduces image that is visually closer to the ground-truth in terms of brightness and global contrast.[]{data-label="Fig:qual_lol"}](Images/Enhancement/lol_images/srie "fig:"){width="24.40000%"} ![Visual comparison of low-light enhancement approaches on the LoL dataset [@wei2018deep]. Our method reproduces image that is visually closer to the ground-truth in terms of brightness and global contrast.[]{data-label="Fig:qual_lol"}](Images/Enhancement/lol_images/lol_kind "fig:"){width="24.40000%"} ![Visual comparison of low-light enhancement approaches on the LoL dataset [@wei2018deep]. Our method reproduces image that is visually closer to the ground-truth in terms of brightness and global contrast.[]{data-label="Fig:qual_lol"}](Images/Enhancement/lol_images/lol_ours "fig:"){width="24.40000%"} ![Visual comparison of low-light enhancement approaches on the LoL dataset [@wei2018deep]. Our method reproduces image that is visually closer to the ground-truth in terms of brightness and global contrast.[]{data-label="Fig:qual_lol"}](Images/Enhancement/lol_images/lol_gt "fig:"){width="24.40000%"}
SRIE [@fu2016weighted] KinD [@zhang2019kindling] MIRNet (Ours) Ground-truth
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Visual results of image enhancement on the MIT-Adobe FiveK [@mit_fivek] dataset. Compared to the state-of-the-art, our MIRNet makes better color and contrast adjustments and produces image that is vivid, natural and pleasant in appearance. ](Images/Enhancement/fivek/input "fig:"){width="32.40000%"} ![Visual results of image enhancement on the MIT-Adobe FiveK [@mit_fivek] dataset. Compared to the state-of-the-art, our MIRNet makes better color and contrast adjustments and produces image that is vivid, natural and pleasant in appearance. ](Images/Enhancement/fivek/hdrnet "fig:"){width="32.40000%"} ![Visual results of image enhancement on the MIT-Adobe FiveK [@mit_fivek] dataset. Compared to the state-of-the-art, our MIRNet makes better color and contrast adjustments and produces image that is vivid, natural and pleasant in appearance. ](Images/Enhancement/fivek/dpe "fig:"){width="32.40000%"}
Input image HDRNet [@Gharbi2017] DPE [@chen2018deep]
![Visual results of image enhancement on the MIT-Adobe FiveK [@mit_fivek] dataset. Compared to the state-of-the-art, our MIRNet makes better color and contrast adjustments and produces image that is vivid, natural and pleasant in appearance. ](Images/Enhancement/fivek/deepupe "fig:"){width="32.40000%"} ![Visual results of image enhancement on the MIT-Adobe FiveK [@mit_fivek] dataset. Compared to the state-of-the-art, our MIRNet makes better color and contrast adjustments and produces image that is vivid, natural and pleasant in appearance. ](Images/Enhancement/fivek/ours "fig:"){width="32.40000%"} ![Visual results of image enhancement on the MIT-Adobe FiveK [@mit_fivek] dataset. Compared to the state-of-the-art, our MIRNet makes better color and contrast adjustments and produces image that is vivid, natural and pleasant in appearance. ](Images/Enhancement/fivek/gt "fig:"){width="32.40000%"}
DeepUPE [@wei2018deep] MIRNet (Ours) Ground-truth
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\[Fig:qual\_fivek\]
Ablation Studies
================
In this section we study the impact of each of our architectural components and design choices on the final performance. All the ablation experiments are performed for the super-resolution task with $\times3$ scale factor. Table \[table:ablation main\] shows that removing skip connections causes the largest performance drop. Without skip connections, the network finds it difficult to converge and yields high training errors, and consequently low PSNR. Furthermore, we note that the information exchange among parallel convolution streams via SKFF is helpful and leads to an improved performance. Similarly, DAU also makes positive influence to the final image quality. Next, we analyze the feature aggregation strategy in Table \[table:ablation aggregation\]. It shows that the proposed SKFF generates favorable results compared to summation and concatenation. Moreover, it can be seen that our SKFF uses $\sim6\times$ fewer parameters than concatenation. Finally, in Table \[table: ablation MRB\] we study how the number of convolutional streams and columns (DAU blocks) of MRB affect the image restoration quality. We note that increasing the number of streams provides significant improvements, thereby justifying the importance of multi-scale features processing. Moreover, increasing the number of columns yields better scores, thus indicating the significance of information exchange among parallel streams for feature consolidation. Additional ablation studies and qualitative results are provided in the supplementary material.
Concluding Remarks
==================
Conventional image restoration and enhancement pipelines either stick to the full resolution features along the network hierarchy or use an encoder-decoder architecture. The first approach helps retain precise spatial details, while the latter one provides better contextualized representations. However, these methods can satisfy only one of the above two requirements, although real-world image restoration tasks demand a combination of both conditioned on the given input sample. In this work, we propose a novel architecture whose main branch is dedicated to full-resolution processing and the complementary set of parallel branches provides better contextualized features. We propose novel mechanisms to learn relationships between features within each branch as well as across multi-scale branches. Our feature fusion strategy ensures that the receptive field can be dynamically adapted without sacrificing the original feature details. Consistent achievement of state-of-the-art results on five datasets for three image restoration and enhancement tasks corroborates the effectiveness of our approach.
[^1]: <https://github.com/swz30/MIRNet>
|
---
abstract: 'Wireless sensor networks are often deployed in public or otherwise untrusted and even hostile environments, which prompts a number of security issues. Although security is a necessity in other types of networks, it is much more so in sensor networks due to the resource-constraint, susceptibility to physical capture, and wireless nature. In this work we emphasize two security issues: (1) secure communication infrastructure and (2) secure nodes scheduling algorithm. Due to resource constraints, specific strategies are often necessary to preserve the network’s lifetime and its quality of service. For instance, to reduce communication costs nodes can go to sleep mode periodically (nodes scheduling). These strategies must be proven as secure, but protocols used to guarantee this security must be compatible with the resource preservation requirement. To achieve this goal, secure communications in such networks will be defined, together with the notions of secure scheduling. Finally, some of these security properties will be evaluated in concrete case studies.'
author:
-
bibliography:
- 'mabase.bib'
title: 'A Security Framework for Wireless Sensor Networks: Theory and Practice'
---
Wireless Sensor Networks; Security; Secure Scheduling; Indistinguability; Nonmalleability.
Introduction
============
In the last few years, wireless sensor networks (WSN) have gained increasing attention from both the research community and actual users. As sensor nodes are generally battery-powered devices, the critical aspects to face concern how to reduce the energy consumption of nodes, so that the network lifetime can be extended to reasonable times. Therefore, energy conservation is a key issue in the design of systems based on wireless sensor networks. In the literature, we can find different techniques to extend the sensor network lifetime [@87]. For example, energy efficient protocols are aimed at minimizing the energy consumption during network activities. However, a large amount of energy is consumed by node components (CPU, radio, etc.) even if they are idle. Power management schemes are thus used for switching off node components that are not temporarily needed [@47; @46]. Other techniques suitable to reduce the energy consumption of sensors is data acquisition (i.e. sampling or transmitting) reduction as data fusion and aggregation [@88; @89].
On the other hand, sensor networks are often deployed in unattended even hostile environments, thus leaving these networks vulnerable to passive and active attacks by the adversary. The communication between sensor nodes can be eavesdropped by the adversary and can forge the data. Sensor nodes should be resilient to these attacks. Therefore, One of the major challenges in such networks is how to provide connection between sensors and the base station and how to exchange the data while maintaining the security requirements and taking into consideration their limited resources. In this paper we emphasize two security issues:
[**Secure communication infrastructure:**]{} In wireless sensor networks a sensor node generally senses the data and sends to its neighbor nodes or to the sink. Stationary adversaries equipped with powerful computers and communication devices may access whole WSN from a remote location. For instance, an intrusion detection system detects the different type of attacks and sends the report to base station. It uses all nodes or some special nodes to detect these types of attacks. These nodes co-operate each other to take the decision and finally send the report to the base station. It requires lots of communication between the nodes. If adversary can trap the message exchanging between the nodes then he can easily tamper the messages and send the false information to the other nodes. Secure communication is a necessary condition in order to make the network smooth so that nodes can send data or exchange the message securely. In our paper, we provide the definition of a communication system for WSNs, and define some of the required security properties dedicated to sensor networks.
[**Secure scheduling:**]{} The main objective of a secure scheduling is to prolong the whole network lifetime while fulfilling the surveillance application needs. In other words, a common approach is to define a subset of the deployed nodes to be active while the other nodes can sleep. In this paper, we present a novel scheduling algorithm where only a subset of nodes contribute significantly to detect intruders and prevent malicious attacker to predict the behavior of the network prior to intrusion. We present a random scheduling to solve this issue, by guaranteeing an uniform coverage while preventing attackers to predict the list of awaken nodes.
The remainder of this research work is organized as follows. In the next section we provide a general presentation for security in WSN. A rigorous formalism for secure communications in wireless sensor networks is presented in \[sec:secure comm\], in which the notions of communication systems, indistinguability, nonmalleability, and message detection resistance are formalized rigorously in the WSN framework. In Section \[sc:Secure Scheduling\], the notion of secure scheduling is defined and applied on a given example. This research work ends by a conclusion section.
Security in WSN: General presentation
=====================================
Wireless nature of communication, lack of infrastructure and uncontrolled environment improve capabilities of adversaries in WSN. Stationary adversaries equipped with powerful computers and communication devices may access whole WSN from a remote location. They can gain mobility by using powerful laptops, batteries and antennas, and move around or within the WSN. In this section, we consider a WSN where nodes communicate together by sending data publicly. These *transmitted data* contain a *message* whose confidentiality must be preserved. For instance, transmitted data is the cryptogram of a message, modulated in an electromagnetic radiation, or the message is dissimulated into the electromagnetic radiation by using a spread spectrum information hiding technique.
Wireless communication helps adversaries to perform variety of attacks. A secure communication can be used to provide the following general security goals:
[**One-wayness (OW)**]{}, the adversary who sees transmitted data is not able to compute the corresponding message.
[**Indistinguability (IND)**]{}, observing transmitted data, the adversary learns nothing about the contained message.
[**Non-malleability (NM)**]{}, the adversary, observing data for a message $m$, cannot derive another data for a meaningful message $m'$ related to $m$.
The OW and IND goals relate to the confidentiality of messages sending through the WDN. The IND goal is, however, much more difficult to achieve than the one-wayness. Non-malleability guarantees that any attempt to manipulate the observed data to obtain a valid data will be unsuccessful (with a high probability).
The power of a polynomial attacker (with polynomial computing resources) very much depends on his/her knowledge about the system used to transform *information* in *data*. The weakest attacker is an outsider who knows the public embedding algorithm together with other public information about the setup of the system. The strongest attacker seems to be an insider (he/she is inside the network) who can access the extraction device (recovering information from data) in regular interval. The access to the extraction key is not possible as the extraction device is assumed to be tamperproof.
An *extraction oracle* is a formalism that mimics an attacker’s access to the extraction device. The attacker can experiment with it providing *data* and collecting corresponding *information* from the oracle (the attacker cannot access to the decryption key). In general, the public-key WSN can be subjected to the following attacks (ordered in increasing strength):
[**Chosen information attack (CIA)**]{} The attacker knows the embedding algorithm and the public elements including the public key (the embedding oracle is publicly accessible).
[**Nonadaptative chosen data attack (CDA1)**]{} The attacker has access to the extraction oracle before he sees a data that he wishes to manipulate.
[**Adaptative chosen data attack (CDA2)**]{} The attacker has access to the extraction oracle before and after he observes a data $s$ that he wishes to manipulate (assuming that he is not allowed to query the oracle about the data $s$).
The security level that a public-key WSN achieves can be specified by the pair (goal, attack), where the goal can be either OW, IND, or NM, and the attack can be either CIA, CDA1, or CDA2. For example, the level (NM,CIA) assigned to a public-key network says that the system is nonmalleable under the chosen message attack. There are two sequences of trivial implications
- (NM,CDA2) $\Rightarrow$ (NM, CDA1) $\Rightarrow$ (NM,CIA),
- (IND,CDA2) $\Rightarrow$ (IND, CDA1) $\Rightarrow$ (IND,CIA),
which are true because the amount of information available to the attacker in CIA, CDA1, and CDA2 grows. Figure \[fig:rel\] shows the interrelation among different security notions. Consequently, we can identify the hierarchy of security levels. The top level is occupied by $(NM,CDA2)$ and $(IND,CDA2)$. The bottom level contains $(IND,CIA)$ only as the weakest level of security. If we are after the strongest security level, its enough to prove that our network attains the $(IND,CDA2)$ level of security.
$$\begin{array}{ccccc}
(NM,CDA2) & \longrightarrow & (NM, CDA1) & \longrightarrow & (NM,CIA)\\
\updownarrow & & \downarrow & & \downarrow\\
(IND,CDA2) & \longrightarrow & (IND, CDA1) & \longrightarrow & (IND,CIA)
\end{array}$$
Rigorous Formalism for Secure Communications in WSNs {#sec:secure comm}
====================================================
In this section, we present a new principles formalism for secure communication in wireless sensor networks.
Communication System in a WSN
-----------------------------
\[def:secure system\] Let $\mathcal{S}, \mathcal{M}$, and $\mathcal{K}=\{0,1\}^\ell$ three sets of words on $\{0,1\}$ called respectively the sets of transmission supports, of messages, and of keys (of size $\ell$).
A *communication system* on $(\mathcal{S}, \mathcal{M}, \mathcal{K})$ is a tuple $(\mathcal{I},\mathcal{E}, inv)$ such that:
- $\mathcal{I}:\mathcal{S} \times \mathcal{M} \times \mathcal{K} \longrightarrow \mathcal{S}$, $(s,m,k) \longmapsto \mathcal{I}(s,m,k)=s'$, is the *insertion function*, which put the message $m$ into the support of transmission $s$ according to the key $k$, leading to the transmitted data $s'$.
- $\mathcal{E}:\mathcal{S} \times \mathcal{K} \longrightarrow \mathcal{M}$, $(s,k) \longmapsto \mathcal{E}(s,k) = m'$, defined as the *extraction function*, which extract a message $m'$ from a transmitted data $s$, depending on a key $k$.
- $inv:\mathcal{K} \longrightarrow \mathcal{K}$, s.t. $\forall k \in
\mathcal{K}, \forall (s,m)\in \mathcal{S}\times\mathcal{M},
\mathcal{E}(\mathcal{I}(s,m,k),inv(k))=m$, which is the function that can “invert” the effects of the key $k$, producing the message $m$ that has been embedded into $s$ using $k$.
- $\mathcal{I}$ and $\mathcal{E}$ can be computed in polynomial time, and $\mathcal{I}$ is a probabilistic algorithm (the same values inputted twice produce two different transmitted data).
$k$ is called the embedding key and $k'=inv(k)$ the extraction key. If $\forall k \in \mathcal{K}, k=inv(k)$, the communication system through the WSN is said *symmetric* (private-key), otherwise it is *asymmetric* (public-key).
Indistinguability
-----------------
Suppose that the adversary has two messages $m_1,m_2$ and a transmitted data $s$ in his/her possession. He/she knows that $s$ contains either $m_1$ or $m_2$. Our intention is to define the fact that, having all these materials, the key, and the insertion function (we take place into the (IND,CIA) context), he cannot determine with a non negligible probability the message that has been embedded into $s$.
The difficulty of the challenge comes, for a large extend, from the fact that the insertion algorithm $\mathcal{I}$ is a probabilistic one, which is a common-sense assumption usually required in cryptography.
An Indistinguability I-adversary is a couple $(\mathsf{A}_1,\mathsf{A}_2)$ of nonuniform algorithms, each with access to an oracle $\mathcal{O}$.
For a public communication system in WSN $(\mathcal{I},\mathcal{E}, inv)$ on $(\mathcal{S}, \mathcal{M}, \{0,1\}^\ell)$, define the advantage of an I-adversary $\mathsf{A}$ by
$$Adv_{\mathsf{A}}^{I-\mathcal{O}} = Pr\left[
\begin{array}{c} k \xleftarrow{\$} \{0,1\}^\ell \\ (m_0,m_1,s) \leftarrow \mathsf{A}_1(k) \\ b \leftarrow \{0,1\} \\ \alpha = \mathcal{I}(s,m_b,k)\end{array}:\begin{array}{c} \mathsf{A}_2(k,s,m_1,m_2,\alpha) = b\end{array}\right]$$
We define the insecurity of $S=(\mathcal{I},\mathcal{E}, inv)$ with respect to the Indistinguability as
$$InSec_S^{I-\mathcal{O}}(t) = \max_\mathsf{A}\left\{Adv_\mathsf{A}^{I-\mathcal{O}}\right\}$$ where the maximum is taken over all adversaries $\mathsf{A}$ with total running time $t$.
We distinguish three kinds of oracles:
- The Non-adaptative oracle, denoted $\mathcal{NA}$, where $A_1$ and $A_2$ can only access to the elements of the communication system.
- The Adaptative oracle, denoted $\mathcal{AD}1$, where $A_1$ has access the communication system and to an oracle that can in a constant time provide a message $m^\prime$ from any transmitted data $\mathcal{I}(M^\prime,m^\prime,k^\prime)$, without knowing neither $M^\prime$ nor $k^\prime$ nor $inv(k^\prime)$. In this context, $A_2$ has no access to this oracle.
- The Strong adaptative oracle, denoted $\mathcal{AD}2$, where $A_1$ has access to the communication system and to an oracle that can in a constant time provide a message $m^\prime$ from any transmitted data $\mathcal{I}(M^\prime,m^\prime,k^\prime)$, without knowing neither $M$ nor $k^\prime$ nor $inv(k^\prime)$. In this context, $A_2$ has also access to this oracle but for the message $\mathcal{I}(M,m_b,k)$.
Relation Based Non-malleability
-------------------------------
In some scenarios malicious nodes can integrate the WSN, hoping by doing so to communicate false information to the other nodes. We naturally suppose that communications are secured. The problem can be formulated as follows: is it possible for the attacker to take benefits from his/her observations, in order to forge transmitted data either by embedding erroneous messages, or sending data that appear to be similar with what a node is supposed to produce.
As wireless sensor networks have usually a dynamical architecture, the (dis)appearance of nodes is not necessarily suspect. Authentication protocols can be deployed into the WSN, but in some cases such authentication is irrelevant, because of its energy consumption, communication cost, or rigidity. We focus in this section on the possibility to propose a secured communication scheme in WSN that prevents an attacker to forge such malicious transmitted data. Such non-malleability property can be formulated as follows.
A Relation Based NM-adversary is a nonuniform algorithm $\mathsf{A}$ having access to an oracle $\mathcal{O}$.
For a public communication system $(\mathcal{I},\mathcal{E}, inv)$ on $(\mathcal{S}, \mathcal{M}, \{0,1\}^\ell)$, define the advantage of a NM-adversary $\mathsf{A}$ by
$$~~~~~~Adv_{\mathsf{A}}^{NM-\mathcal{O}}(m) = Pr\left[
\begin{array}{c}
s \leftarrow \mathcal{S} \\
k \xleftarrow{\$} \{0,1\}^\ell \\
s' \leftarrow \mathsf{A}(\mathcal{I}(s,m,k)) \\
m' \leftarrow \mathcal{E}(s',k)
\end{array}
:\begin{array}{c} m' \in R(m) \end{array}\right]$$
where $R:\mathcal{M} \longrightarrow \mathcal{P}(\mathcal{M})$ is a function that map any message $m$ to a subset of $\mathcal{M}$ containing messages related to $m$ (for a given property). For instance, if we suppose that an attacker has inserted or corrupted some nodes in a network that measures temperature, he can make these nodes send wrong temperatures values fixed [*a priori*]{}. We can now define the insecurity of $S=(\mathcal{I},\mathcal{E}, inv)$ with respect to the Relation Based Non-malleability as $$InSec_S^{NM-\mathcal{O}}(t) = \max_\mathsf{A}\left\{\max_{m \in \mathcal{M}}\left\{Adv_\mathsf{A}^{NM-\mathcal{O}}(m)\right\}\right\}$$ where the maximum is taken over all adversaries $\mathsf{A}$ with total running time $t$. Similar kinds of oracles than previously can be defined in that context.
Message Detection Resistance
----------------------------
We now address the particular case where transmitted data can contain or not an embedded message. For security reasons, it is sometimes required that an attacker cannot determine when information are transmitted through the network. For instance, in a video surveillance context, suppose that an attacker can determine when an intrusion is detected, or when something considered as suspicious is forwarded through the nodes to the sink. Then he/she can use this knowledge to deduce what kind of behavior is suspicious for the network, adapting so his/her attacks. Decoys are often proposed to make such attacks impossible: transmitted data do not always contain information, some of the communications are only realized to mislead the attacker. The quantity and frequency of these decoys must naturally respect the energy consumption requirement, and a compromise must be found on the message/decoy rate to face such attacks while preserving the WSN lifetime. However, such an approach supposes that the attacker is unable to make the distinction between decoys and meaningful communications. Such a supposition leads to the following definition.
A Detection Resistance DR-adversary is a couple $(\mathsf{A}_1,\mathsf{A}_2)$ of nonuniform algorithms, each with access to an oracle $\mathcal{O}$.
For a public communication system $(\mathcal{I},\mathcal{E}, inv)$ on $(\mathcal{S}, \mathcal{M}, \{0,1\}^\ell)$, define the advantage of a DR-adversary $\mathsf{A}$ by
$$\begin{small}Adv_{\mathsf{A}}^{DR-\mathcal{O}} = Pr\left[
\begin{array}{c} M_0, M_1 \leftarrow \mathcal{S} \\ k \xleftarrow{\$} \{0,1\}^\ell \\ m \leftarrow \mathsf{A}_1(k) \\ b \leftarrow \{0,1\} \\ \alpha = \left\{M_b, \mathcal{I}(M_{\overline{b}},m,k))\right\}\end{array}:\begin{array}{c} \mathsf{A}_2(m,k,\alpha) = M_b\end{array}\right]\end{small}$$
where the set defining $\alpha$ is a non-ordered one.
We define the insecurity of $S=(\mathcal{I},\mathcal{E}, inv)$ with respect to the Message Detection Resistance as $$InSec_S^{DR-\mathcal{O}}(t) = \max_\mathsf{A}\left\{Adv_\mathsf{A}^{DR-\mathcal{O}}\right\}$$ where the maximum is taken over all adversaries $\mathsf{A}$ with total running time $t$. Similar kinds of oracles than previously can be defined in that context.
Secure Scheduling {#sc:Secure Scheduling}
=================
Motivations
-----------
A common way to enlarge lifetime of a wireless sensor network is to consider that not all of the nodes have to be awakened: a subset of well-chosen nodes participates temporarily to the task devoted to the network [@91], whereas the other nodes sleep in order to preserve their batteries. Obviously, the scheduling process determining the nodes that have to be awakened at each time must be defined carefully, both for guaranteeing a certain level of quality in the assigned task and to preserve the network capability over time. Problems that are of importance in that approach are often related to coverage, ratio of awaken vs sleeping nodes, efficient transmission of wake up orders, and capability for the partial network to satisfy, with a sufficient quality, the objectives it has been designed for. Existing surveillance applications works focus on finding an efficient deployment pattern so that the average overlapping area of each sensor is bounded. The authors in [@94] analyze new deployment strategies for satisfying some given coverage probability requirements with directional sensing models. A model of directed communications is introduced to ensure and repair the network connectivity. Based on a rotatable directional sensing model, the authors in [@95] present a method to deterministically estimate the amount of directional nodes for a given coverage rate. A sensing connected sub-graph accompanied with a convex hull method is introduced to model a directional sensor network into several parts in a distributed manner. With adjustable sensing directions, the coverage algorithm tries to minimize the overlapping sensing area of directional sensors only with local topology information. Lastly, in [@93], the authors present a distributed algorithm that ensures both coverage of the deployment area and network connectivity, by providing multiple cover sets to manage Field of View redundancies and reduce objects disambiguation.
All the above algorithms depend on the geographical location information of sensor nodes. These algorithms aim to provide a complete-coverage network so that any point in the target area would be covered by at least one sensor node. However, this strategy is not as energy-efficient as what we expect because of the following two reasons. Firstly, the energy cost and system complexity involved in obtaining geometric information may compromise the effect of those algorithms. Secondly, sensor nodes located at the edge of the area of interest must be always in an active state as long as the region is required to be completely covered. These nodes will die after some time and their coverage area will be left without surveillance. Thus, the network coverage area will shrink gradually from outside to inside. This condition is unacceptable in surveillance applications and (intelligent) intrusion detection, because the major goal here is to detect intruders as they cross a border or as they penetrate a protected area. In case of hostile environments, security play an important role in the written of the scheduling program. Indeed an attacker, observing the manner nodes are waken up, should not be able to determine the scheduling program. For instance, in a video surveillance context, if the attacker is able to determine at some time the list of the sleeping nodes, then he can possibly achieve an intrusion without being detected [@bgmp11:ip].
Obviously, a random scheduling can solve the issues raised above, by guaranteeing a uniform coverage while preventing attackers to predict the list of awaken nodes. However, this approach needs random generators into nodes, which cannot be obtained by deterministic algorithms embedded into the network. Even if truly random generators (TRG) can be approximated by physical devices, they need a certain quantity of resources, suppose that the environment under observation has a sufficient variability of a given set of physical properties (to produce the physical noise source required in that TRG), and are less flexible or adaptable on demand than pseudorandom number generators (PRNGs). Furthermore, neither their randomness nor their security can be mathematically proven: these generators can be biased or wrongly designed. Being able to guarantee a certain level of security in scheduling leads to the notion of *secure scheduling* proposed below.
Secure Scheduling in Wireless Sensor Networks
---------------------------------------------
Two kinds of scheduling processes can be defined: each node can embed its own program, determining when it has to sleep (local approach), or the sink or some specific nodes can be responsible of the scheduling process, sending sleep or wake up orders to the nodes that have to change their states (global approach).
We consider that a deterministic scheduling algorithm is a function $S: \{0,1\}^n \rightarrow \{0,1\}^M$, where $M > n$. This definition can be understood as follows:
- The value inputted in $S$ is the secret key launching the scheduling process. It can be shown as the seed of a PRNG.
- In case of a local approach, the binary sequence produced by this function corresponds to the moments where the node must be awaken: if the $k-$th term of this sequence is 0, then the node can go to sleep mode between $t_k$ and $t_{k+1}$.
- In case of a global approach, the binary sequence returned by $S$ can be divided by blocs, such that each bloc contains the id of the node to which an order of state change will be send.
Loosely speaking, $S$ is called a secure scheduling if it maps uniformly distributed input (the secret key or seed of the scheduling process) into an output which is computationally indistinguishable from uniform. The precise definition is given below.
A $T$-time algorithm $\mathcal{D} : \{0, 1\}^M \longrightarrow {0, 1}$ is said to be a $(T,\varepsilon)$-distinguisher for $S$ if $$\left| Pr[\mathcal{D}(S(\mathfrak{U}_2^n )) = 1] - Pr[\mathcal{D}(\mathfrak{U}_2^M) = 1] \right| \geqslant \varepsilon.$$
where $\mathfrak{U}_2$ is the uniform distribution on $\{0,1\}$.
Algorithm $S$ is called a $(T,\varepsilon )$-secure scheduling if no $(T,\varepsilon)$-distinguisher exists for $S$.
Adapting the proofs of [@Yao82; @Goldreich86], it is possible to show that a $(T,\varepsilon)-$distinguisher exists if and only if a $T$-time algorithm can, knowing the first $l$ bits of a scheduling $s$, predict the $(l + 1)-$st bit of $s$ with probability significantly greater than $0.5$. This comes from the fact that a PRNG passes the next-bit test if and only if it passes all polynomial-time statistical tests [@Yao82; @Goldreich86].
An important question is what level of security $(T,\varepsilon)$ suffices for practical applications in scheduled wireless sensor networks. Unfortunately, the level of security is often chosen arbitrarily. It is reasonable to require that a scheduling process is secure for all pairs $(T,\varepsilon)$ such that the time-success ratio $T/\varepsilon$ is below a certain bound. In the next section we present an illustration of this notion.
Practical Study
---------------
Suppose that a wireless sensor node has been scheduled by a Blum-Blum-Shub BBS pseudorandom generator. This generator produces bits $y_0, y_1, \hdots$, and the node is awaken during the time interval $[t_i;t_{i+1}[$ if and only if $y_i=1$.
Let us recall that the Blum Blum Shum generator [@Blum:1985:EPP:19478.19501] (usually denoted by BBS) is defined by the following process:
1. Generate two large secret random and distinct primes $p$ and $q$, each congruent to 3 modulo 4, and compute $N = pq$.
2. Select a random and secret seed $s \in \llbracket 1, N - 1 \rrbracket$ such that $gcd(s, N) = 1$, and compute $x_0 = s^2 (\textrm{mod } N)$.
3. For $1 \leqslant i \leqslant l$ do the following:
1. $x_i = x_{i-1}^2 (\textrm{mod } N)$.
2. $y_i =$ the least significant bit of $x_i$.
4. The output sequence is $y_1 , y_2 , \hdots, y_l$.
Suppose now that the network will work during $M=100$ time units, and that during this period, an attacker can realize $10^{12}$ clock cycles. We thus wonder whether, during the network’s lifetime, the attacker can distinguish this sequence from truly random one, with a probability greater than $\varepsilon = 0.2$. We consider that $N$ has 900 bits.
The scheduling process is the BBS generator, which is cryptographically secure. More precisely, it is $(T,\varepsilon)-$secure: no $(T,\varepsilon)-$distinguishing attack can be successfully realized on this PRNG, if [@Fischlin] $$T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)$$ where $M$ is the length of the output ($M=100$ in our example), and $$L(N)=2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right)$$ is the number of clock cycles to factor a $N-$bit integer.
A direct numerical application shows that this attacker cannot achieve its $(10^{12},0.2)$ distinguishing attack in that context.
Results study
-------------
This section presents simulation results on comparing our approach to the standard C++ [rand()]{}-based approach with random intrusions. We use the OMNET++ simulation environment and the next node selection will either use our approach or the C++ [rand()]{} function ([rand() % $2^n$]{}) to produce a random number between 0 and $2^n$. For these set of simulations, $128$ sensor nodes are randomly deployed in a $75m * 75m$ area. Figure \[graph-activenode\] shows the percentage of active nodes. Both our approach and the standard [rand()]{} function have similar behavior: the percentage of active nodes progressively decreases due to battery shortage.
![Percentage of active nodes.[]{data-label="graph-activenode"}](graph-activenode.png){width="\linewidth"}
Another result we want to show is the energy consumption distribution. We recorded every 10s the energy level of each sensor node in the field and computed the mean and the standard deviation. Figure \[graph-stddev\] shows the evolution of the standard deviation during the network lifetime. We can see that our approach selection provides a slightly better distribution of activity than the standard [rand()]{} function.
![Evolution of the energy consumption’s standard deviation.[]{data-label="graph-stddev"}](graph-stddev){width="\linewidth"}
Conclusion
==========
In this document, a rigorous framework for security in wireless sensor networks has been formalized. The definition of a communication system in WSNs has been introduced, and security properties (indistinguability, nonmalleability, and message detection resistance) have been formalized in that context. Furthermore, the definitions of secure scheduling, specific to such networks, have been given too. With this theoretical framework, it has been possible to evaluate the security of a scheduling scheme based on the BBS cryptographically secure PRNG.
|
---
abstract: 'The Shannon capacity of a graph is an important graph invariant in information theory that is extremely difficult to compute. The Lovász number, which is based on semidefinite programming relaxation, is a well-known upper bound for the Shannon capacity. To improve this upper bound, previous researches tried to generalize the Lovász number using the ideas from the sum-of-squares optimization. In this paper, we consider the possibility of developing general conic programming upper bounds for the Shannon capacity, which include the previous attempts as special cases, and show that it is impossible to find better upper bounds for the Shannon capacity along this way.'
author:
- |
Yingjie Bi and Ao Tang\
Cornell University, Ithaca, NY, 14850
bibliography:
- 'SOS.bib'
title: On Upper Bounding Shannon Capacity of Graph Through Generalized Conic Programming
---
Introduction
============
The *Shannon capacity of a graph* is a graph invariant originated from computing the maximum achievable rate to transmit information with zero possibility of error through a noisy channel [@Shannon1956]. To state the definition of Shannon capacity, we need the following notions in graph theory: For an undirected graph $G$, let $V(G)$ and $E(G)$ be its vertex set and edge set, respectively. Let $\alpha(G)$ be the independence number (aka stability number) of $G$, i.e., the size of the maximum independent set in $G$. For two vertices $i,j \in V(G)$, the notation $i \sim_G j$ means either $i=j$ or $(i,j) \in E(G)$. The *strong product* $G \boxtimes H$ of two graphs $G$ and $H$ is a graph such that
- its vertex set $V(G \boxtimes H)$ is the Cartesian product $V(G) \times V(H)$ and
- $(i,j) \sim_{G \boxtimes H} (k,l)$ if and only if $i \sim_G k$ and $j \sim_H l$.
The Shannon capacity $\Theta(G)$ of graph $G$ is defined by $$\Theta(G)=\sup_k\sqrt[k]{\alpha(G^k)},$$ where $G^k$ is the strong product of $G$ with itself for $k$ times.
The Shannon capacity is unknown for most graphs, including certain simple cases such as odd cycles $C_{2n+1}$ when $n \geq 3$. By definition, for any positive integer $k$, $\sqrt[k]{\alpha(G^k)}$ provides a direct lower bound for the Shannon capacity $\Theta(G)$, although it is still hard to calculate due to the NP-hardness of maximum independent set problem and the exponential growth of the size of $G^k$. Finding a good upper bound for $\Theta(G)$ is even more difficult. One well-known upper bound is the Lovász number $\vartheta(G)$ proposed in [@Lovasz1979], which can be efficiently computed by solving a semidefinite program (SDP). The most famous application of Lovász number is the establishment of the Shannon capacity for the pentagon graph $C_5$: $$\sqrt 5=\sqrt{\alpha(C_5^2)} \leq \Theta(C_5) \leq \vartheta(C_5)=\sqrt 5.$$ However, for 7-cycle $C_7$, $\vartheta(C_7) \approx 3.3177$, while the best known lower bound [@PS2019] at the time of writing is $$\Theta(C_7) \geq \sqrt[5]{\alpha(C_7^5)} \geq \sqrt[5]{367} \approx 3.2578.$$ Determining the exact value for the Shannon capacity $\Theta(C_7)$ remains an open problem.
One interesting direction is to look for a tighter upper bound for the Shannon capacity than the Lovász number. Since the definition of the Shannon capacity is closely related to the independence number, and in fact the Lovász number itself can be derived from approximating the independence number of a graph, it is tempting to find better upper bounds for the Shannon capacity by using tighter approximations for the independence number. The major challenge here is to ensure that the new approximation is still an upper bound for the Shannon capacity. In Section \[sec:conic\], we will look at general conic programming approximation for the independence number, which is a natural generalization of the SDP-based Lovász number. Next, in Section \[sec:product\], we will propose a condition called the *product property* over the cones appeared in the above approximate optimization problem. This property guarantees that the optimal value of the approximation is an upper bound for the Shannon capacity. Surprisingly, in Section \[sec:optimal\] it is shown that the semidefinite cone used by the Lovász number is the largest cone with such a property, thus ruling out the possibility of improving the estimation of the Shannon capacity along this way.
Conic Programming for the Independence Number {#sec:conic}
=============================================
In this section, we will first formulate the maximum independent set problem as a copositive program. If the semidefinite cone is used as an inner approximation for the copositive cone in this program, the obtained objective value is exactly the Lovász number. As a generalization, we consider all the possible cones that are subsets of the copositive cone, and the corresponding conic programs will be the candidates to generate better upper bounds for the Shannon capacity. Before we start, we summarize the common notations used in the paper below:
- ${\mathbb{R}}_+^n$ is the set of $n \times 1$ nonnegative column vectors.
- $J_n$ is the $n \times n$ matrix of all ones.
- $\mathcal S_n$ is the cone of $n \times n$ symmetric matrices.
- $\mathcal P_n$ is the cone of $n \times n$ positive semidefinite matrices.
- $\mathcal N_n$ is the cone of $n \times n$ nonnegative symmetric matrices.
- $\mathcal C_n$ is the cone of $n \times n$ copositive matrices, i.e., all symmetric matrices $Q \in \mathcal S_n$ such that $x^TQx \geq 0$ for any $x \in {\mathbb{R}}_+^n$.
Our starting point is the Motzkin-Straus theorem, which gives the exact value of the independence number of a graph:
\[thm:motzkin\] If $A$ is the adjacency matrix of a graph $G$ with $n$ vertices, then the independence number of $G$ is given by $$\frac{1}{\alpha(G)}=\min_{x \in {\mathbb{R}}_+^n,\sum_ix_i=1}x^T(I+A)x.$$
In [@KP2002], the optimization problem in Theorem \[thm:motzkin\] is converted into the following equivalent form: $$\label{eq:copold}
\begin{split}
\alpha(G)=\min \quad & \lambda \\
\operatorname*{s.t.}\quad & \lambda(I+A)-J_n \in \mathcal C_n.
\end{split}$$ In order to make the above problem closer to the formulation for the Lovász number, we are going to further rewrite it as follows: $$\label{eq:cop}
\begin{split}
\min \quad & \lambda \\
\operatorname*{s.t.}\quad & Y-J_n \in \mathcal C_n, \\
& Y_{ii}=\lambda, \quad \forall i=1,\dots,n, \\
& Y_{ij}=0, \quad \forall i \not\sim_G j, \\
& Y \in \mathcal S_n.
\end{split}$$ Since problem can be viewed as problem with the additional constraint $Y=\lambda(I+A)$, problem is a relaxation of the original problem . To show that these two problems are indeed equivalent, the following property of copositive matrices will be useful:
\[lem:copmat\] Assume $Q$ is a copositive matrix whose diagonal entries are all equal to $\mu$. $R$ is another symmetric matrix of the same size. If for each entry of $R$ either $R_{ij}=Q_{ij}$ or $R_{ij}=\mu$, then $R$ is also copositive.
We only need to consider the case in which $R=Q$ except for some off-diagonal entry $R_{st}=\mu$ (and also $R_{ts}=\mu$), since the general result can be obtained by repeating the same argument for each difference between $R$ and $Q$. For any $x \in {\mathbb{R}}_+^n$ with $\sum_ix_i=1$, $$\label{eq:bilinear}
x^TRx=\mu x_s^2+\mu x_t^2+2\mu x_sx_t+\sum_{\substack{(i,j) \neq (s,s),\\(s,t),(t,s),(t,t)}}Q_{ij}x_ix_j.$$ Fix $x_i$, $i \neq s,t$, as constants and regard $x^TRx$ as a function of $x_s$ by replacing $$x_t=1-x_s-\sum_{i \neq s,t}x_i.$$ Then the first part of $$\mu x_s^2+\mu x_t^2+2\mu x_sx_t=\mu(x_s+x_t)^2=\mu\left(1-\sum_{i \neq s,t}x_i\right)^2$$ becomes a constant. Since the remaining terms in are all linear functions of $x_s$, $x^TRx$ is also linear as a function of $x_s$ and thus must achieve the minimum when $x_s=0$ or $x_s=1$. However, in both cases, $x^TRx=x^TQx \geq 0$, which implies that $R$ is also copositive.
Now we can prove that the problems and have the same optimal value. Consider an arbitrary feasible solution $(\lambda,Y)$ to problem . Let $$Q=Y-J_n, \quad R=\lambda(I+A)-J_n.$$ All the diagonal entries of $Q$ are $\lambda-1$. By Lemma \[lem:copmat\], the matrix $R$ is also copositive and thus $\lambda \geq \alpha(G)$ by . On the other hand, the solution $\lambda^*=\alpha(G)$, $Y^*=\alpha(G)(I+A)$ is feasible to , so it must be optimal.
The copositive cone constraint in makes the problem hard to solve. If we substitute the copositive cone $\mathcal C_n$ in with the semidefinite cone $\mathcal P_n$, the optimal value for the modified problem is exactly the Lovász number[^1], which will be denoted as $\vartheta(G)$. Since $\mathcal P_n \subseteq \mathcal C_n$, we immediately get $\alpha(G) \leq \vartheta(G)$. Naturally, to find a tighter bound for the Shannon capacity $\Theta(G)$, we can replace the copositive cone $\mathcal C_n$ in by some cone between $\mathcal C_n$ and $\mathcal P_n$, which may lead to some problem whose optimal value is potentially between the Shannon capacity $\Theta(G)$ and the Lovász number $\vartheta(G)$.
The above discussion illuminates us to construct more general approximations for the independence number $\alpha(G)$ by introducing an arbitrary cone $\mathcal A_n \subseteq \mathcal C_n$ into the problem $$\label{eq:conic}
\begin{split}
\min \quad & \lambda \\
\operatorname*{s.t.}\quad & Y-J_n \in \mathcal A_n, \\
& Y_{ii}=\lambda, \quad \forall i=1,\dots,n, \\
& Y_{ij}=0, \quad \forall i \not\sim_G j, \\
& Y \in \mathcal S_n.
\end{split}$$
In the case when the cone $\mathcal A_n$ is chosen to be the semidefinite cone $\mathcal P_n$, the above problem gives the Lovász number $\vartheta(G)$. To provide some other examples of $\mathcal A_n$, one can approximate the copositive cone $\mathcal C_n$ based on sum-of-squares programming [@KP2002; @DR2010]. Note that the copositivity of a matrix $Q \in \mathcal S_n$ is equivalent to the condition $$\label{eq:sospoly}
p_Q(x)=\sum_{i,j}Q_{ij}x_i^2x_j^2 \geq 0, \quad \forall x \in {\mathbb{R}}^n.$$ Like determining copositivity, it is NP-hard to decide whether the polynomial $p_Q(x)$ is nonnegative or not. However, if $p_Q(x)$ can be written as a sum of squares, i.e., $$p_Q(x)=\sum_kg_k^2(x),$$ where $g_k(x)$ are arbitrary polynomials of $x \in {\mathbb{R}}^n$, then clearly $p_Q(x)$ is nonnegative. All symmetric matrices $Q \in \mathcal S_n$ whose corresponding polynomial $p_Q(x)$ given by is a sum of squares constitute a cone, which will be denoted as $\mathcal C_n^{(0)}$ in the following. By the above discussion, $\mathcal C_n^{(0)} \subseteq \mathcal C_n$, and furthermore it is tractable to determine whether a matrix $Q$ is in the cone $\mathcal C_n^{(0)}$ through SDP. In fact, $\mathcal C_n^{(0)}$ has a simple characterization [@Parrilo2000]: $$\mathcal C_n^{(0)}=\mathcal P_n+\mathcal N_n.$$ In other words, the polynomial $p_Q(x)$ is a sum of squares if and only if the matrix $Q$ is a sum of a positive semidefinite matrix and a nonnegative symmetric matrix.
For any graph $G$, the optimal value of problem , in which $\mathcal A_n=\mathcal C_n^{(0)}$, is called $\vartheta'(G)$, the Schrijver $\vartheta'$-function [@Schrijver1979]. Since $$\mathcal P_n \subseteq \mathcal C_n^{(0)} \subseteq \mathcal C_n,$$ we have $$\alpha(G) \leq \vartheta'(G) \leq \vartheta(G).$$ Moreover, there exists some graph for which the second inequality is strict (see [@Schrijver1979]). Given these properties, $\vartheta'(G)$ seems to be a good candidate for upper bounding the Shannon capacity.
More generally, we can find even better approximations for the copositive cone $\mathcal C_n$ by using higher order sum-of-squares polynomials. For each nonnegative integer $r$, define $\mathcal C_n^{(r)}$ to be the set of all symmetric matrices $Q \in \mathcal S_n$ such that $$\left(\sum_ix_i^2\right)^rp_Q(x)$$ is a sum of squares. Then $\mathcal C_n^{(r)}$ is a cone, and $$\mathcal P_n \subseteq \mathcal C_n^{(0)} \subseteq \mathcal C_n^{(1)} \subseteq \dots \subseteq \mathcal C_n^{(r)} \subseteq \dots \subseteq \mathcal C_n.$$ Similar to the Schrijver $\vartheta'$-function, we denote the optimal value of the corresponding problem as $\vartheta^{(r)}(G)$.
For higher-order sum-of-squares cones $\mathcal C_n^{(r)}$ where $r>0$, although $\vartheta^{(r)}(G)$ is a tighter upper bound for the independence number $\alpha(G)$ than $\vartheta(G)$ or $\vartheta'(G)$, it is too tight to be an upper bound for the Shannon capacity $\Theta(G)$. For instance, for the pentagon graph $C_5$, if $r>0$, $$\alpha(C_5)=\vartheta^{(r)}(C_5)=2<\Theta(C_5)=\vartheta(C_5)=\vartheta'(C_5)=\sqrt 5.$$ Therefore, to obtain an upper bound for the Shannon capacity from cones $\mathcal C_n^{(r)}$, we have to add extra constraints in the problem to restrict these cones. Whatever the exact form of constraints is, we can still analyze the restricted problem as a special case of as long as these constraints define a cone.
In the next, we will assume $\mathcal A_n$ in the above problem to be an arbitrary cone satisfying $\mathcal A_n \subseteq \mathcal C_n$, and the optimal value will be called $f(G)$. To ensure that $f(G)$ is still an upper bound for the Shannon capacity $\Theta(G)$, we will look at the key property of the semidefinite cone $\mathcal P_n$ used by the Lovász number $\vartheta(G)$ that guarantees $\Theta(G) \leq \vartheta(G)$ and then try to enforce the same property on the cone $\mathcal A_n$ in .
Product Property and Upper Bounds for the Shannon Capacity {#sec:product}
==========================================================
One fundamental property[^2] of the Lovász number is $$\label{eq:lovaszproduct}
\vartheta(G \boxtimes H) \leq \vartheta(G)\vartheta(H)$$ for any graphs $G$ and $H$, which immediately implies that $$\sqrt[k]{\alpha(G^k)} \leq \sqrt[k]{\vartheta(G^k)} \leq \vartheta(G)$$ for all positive integers $k$, and thus $$\Theta(G)=\sup_k\sqrt[k]{\alpha(G^k)} \leq \vartheta(G).$$
The above argument can also be applied to the graph function $f(G)$ defined as the optimal value of . Since $\alpha(G) \leq f(G)$, as long as $f(G)$ satisfies the similar inequality $$\label{eq:product}
f(G \boxtimes H) \leq f(G)f(H),$$ $f(G)$ will also be an upper bound for the Shannon capacity $\Theta(G)$. To find out what leads to the inequality , we need to generalize the proof for the property of the Lovász number, which itself is a special case of the general product rules in semidefinite programming [@MS2007].
Consider two graphs $G$ of $n$ vertices and $H$ of $m$ vertices. Assume $(\lambda',Y')$ and $(\lambda'',Y'')$ are the optimal solutions to the problem for graph $G$ and $H$, respectively. Let $Y=Y' \otimes Y''$, i.e., the Kronecker product of $Y'$ and $Y''$, which is an $nm \times nm$ matrix given by $$Y=\begin{pmatrix}
Y'_{11}Y'' & \cdots & Y'_{1n}Y'' \\
\vdots & \ddots & \vdots \\
Y'_{n1}Y'' & \cdots & Y'_{nn}Y''
\end{pmatrix}.$$ If we index the rows of $Y$ by pairs $(i,j)$ and the columns by pairs $(k,l)$, the above definition can be rewritten as $$Y_{(i,j)(k,l)}=Y'_{ik}Y''_{jl}.$$ When $(i,j) \not\sim_{G \boxtimes H} (k,l)$, by definition either $i \not\sim_G k$ or $j \not\sim_H l$, which implies either $Y'_{ik}=0$ or $Y''_{jl}=0$ and thus $Y_{(i,j)(k,l)}=0$. Since all the diagonal entries of $Y$ equal to $\lambda'\lambda''$, if we can show that $Y-J_{nm} \in \mathcal A_{nm}$, $(\lambda'\lambda'',Y)$ will be a feasible solution to the problem for the product graph $G \boxtimes H$. In this case, we have $$f(G \boxtimes H) \leq \lambda'\lambda''=f(G)f(H).$$
Let $$Q=Y'-J_n, \quad R=Y''-J_m.$$ Then $Q \in \mathcal A_n$, $R \in \mathcal A_m$. The only missing part that remains to be shown is $$Y-J_{nm}=Y' \otimes Y''-J_{nm}=(Q+J_n) \otimes (R+J_m)-J_{nm} \in \mathcal A_{nm},$$ which will be encapsulated into the following definition:
Given two symmetric matrices $Q \in \mathcal S_n$, $R \in \mathcal S_m$, define $$Q \odot R=(Q+J_n) \otimes (R+J_m)-J_{nm}.$$ A sequence of cones $\mathcal A_n \subseteq \mathcal S_n$ is said to have the *product property* if for any matrices $Q \in \mathcal A_n$, $R \in \mathcal A_m$, we have $Q \odot R \in \mathcal A_{nm}$.
Based on this definition, the above argument can be summarized as follows:
\[thm:bound\] If the cones $\mathcal A_n$ in problem satisfy $\mathcal A_n \subseteq \mathcal C_n$ and the product property, then $\Theta(G) \leq f(G)$ for any graph $G$.
As an example, we check that the product property holds for semidefinite cones $\mathcal P_n$ in the Lovász number. Assume matrices $Q \in \mathcal P_n$, $R \in \mathcal P_m$. Then the matrix $$Q \odot R=(Q+J_n) \otimes (R+J_m)-J_{nm}=Q \otimes R+Q \otimes J_m+J_n \otimes R$$ is also positive semidefinite, because the Kronecker product of two positive semidefinite matrices is still positive semidefinite. Therefore, Theorem \[thm:bound\] implies that the Lovász number $\vartheta(G) \geq \Theta(G)$.
The product property is a sufficient condition for the functional inequality and further for being an upper bound for the Shannon capacity. However, neither the product property nor the inequality is necessary for being the upper bound. In any case, from the proof of Theorem \[thm:bound\], one can see that the product property is the most natural condition to guarantee $\Theta(G) \leq f(G)$. In the next, we will study the product property holds for what choice of cones $\mathcal A_n$.
Optimality of the Lovász Number {#sec:optimal}
===============================
In the previous section, we have stated the product property, the condition for our new function $f(G)$ to be an upper bound for the Shannon capacity. At the same time, we do not want $f(G)$ to be much larger than the Lovász number $\vartheta(G)$ for the same graph $G$. Note that the Lovász number satisfies the following sandwich inequality: $$\alpha(G) \leq \vartheta(G) \leq \chi(\bar G),$$ where $\chi(\bar G)$ is the chromatic number for the complement graph of $G$. Choose $G=\bar K_2$, the edgeless graph of two vertices, then $$2=\alpha(\bar K_2) \leq \vartheta(\bar K_2) \leq \chi(K_2)=2.$$ If the new function $f(G)$ satisfies the similar sandwich inequality, we must have $f(\bar K_2)=2$, which means that the matrix $$\Lambda=\begin{pmatrix}
1 & -1 \\
-1 & 1
\end{pmatrix} \in \mathcal A_2.$$
We want to find a sequence of cones $\mathcal A_n$ satisfying all the above desired conditions. However, it turns out that the only possible cones $\mathcal A_n$ must be subsets of the corresponding semidefinite cones $\mathcal P_n$, and consequently the obtained upper bound $f(G)$ would be at least the Lovász number.
\[thm:optimal\] Suppose a sequence of cones $\mathcal A_n$ satisfies the following properties:
1. $\mathcal A_n \subseteq \mathcal C_n$ for all $n$.
2. The matrix $\Lambda \in \mathcal A_2$.
3. The sequence $\mathcal A_n$ has the product property.
Then we must have $\mathcal A_n \subseteq \mathcal P_n$ for all $n$.
We prove by contradiction. Suppose there is a matrix $A \in \mathcal A_n$ that is not positive semidefinite and $v \in {\mathbb{R}}^n$ is a vector such that $v^TAv<0$.
The first step is to construct a matrix $B \in \mathcal A_m$ with $m=2n$ and a vector $w \in {\mathbb{R}}^m$ satisfying $$w^TBw<0, \quad \sum_iw_i=0.$$ For any $k>0$, let $B=\Gamma \odot (kA)$, then by the cone property and the product property, $$B=\begin{pmatrix}
2kA+J_n & -J_n \\
-J_n & 2kA+J_n
\end{pmatrix} \in \mathcal A_m.$$ If we let $$w=\begin{pmatrix}
v \\
-v
\end{pmatrix},$$ then $$w^TBw=4kv^TAv+4v^TJ_nv.$$ In the above argument, we can choose $k$ with $$k>-\frac{v^TJ_nv}{v^TAv},$$ and now the matrix $B$ and the vector $w$ will have all the desired properties.
Next, we are going to construct another matrix $C \in \mathcal A_{2m}$ which is not copositive. Define $$x=\max(w,0), \quad y=\max(-w,0).$$ Then $x,y \geq 0$ and $w=x-y$. For any $k>1$, by the product property again, the matrix $$C=(k\Gamma) \odot B=\begin{pmatrix}
(k+1)B+kJ_m & -(k-1)B-kJ_m \\
-(k-1)B-kJ_m & (k+1)B+kJ_m
\end{pmatrix} \in \mathcal A_{2m}.$$ Consider $$\begin{gathered}
\begin{pmatrix}
x^T & y^T
\end{pmatrix}C\begin{pmatrix}
x \\
y
\end{pmatrix}=(k+1)(x^TBx+y^TBy)-2(k-1)x^TBy \\
+k(x^TJ_mx+y^TJ_my)-2kx^TJ_my,\end{gathered}$$ in which the second part $$k(x^TJ_mx+y^TJ_my)-2kx^TJ_my=k(x-y)^TJ_m(x-y)=k\left(\sum_iw_i\right)^2=0.$$ On the other hand, for sufficiently large $k$, $$w^TBw=(x-y)^TB(x-y)=x^TBx+y^TBy-2x^TBy<0$$ implies $$x^TBx+y^TBy<2\frac{k-1}{k+1}x^TBy.$$ In this case, $$\begin{pmatrix}
x^T & y^T
\end{pmatrix}C\begin{pmatrix}
x \\
y
\end{pmatrix}<0.$$ Now we have exhibited a matrix $C \in \mathcal A_{2m}$ and $C$ is not copositive, which is a contradiction.
Theorem \[thm:optimal\] tells us that either the cones do not have the product property or the resulting function $f(G) \geq \vartheta(G)$. In other words, it is impossible to derive an upper bound for the Shannon capacity that is better than the Lovász number by enforcing the product property on cones $\mathcal A_n$.
For the Schrijver $\vartheta'$-function, the corresponding cones $\mathcal C_n^{(0)}$ satisfy the first and second condition of Theorem \[thm:optimal\] but not the conclusion $\mathcal C_n^{(0)} \subseteq \mathcal P_n$. Therefore, by Theorem \[thm:optimal\], the cones $\mathcal C_n^{(0)}$ do not have the product property. Although not having the product property for $\mathcal C_n^{(0)}$ does not directly imply that the Schrijver $\vartheta'$-function is not an upper bound for the Shannon capacity, it strongly suggests such negative result. In fact, it is quite difficult to disprove that the Schrijver $\vartheta'$-function is an upper bound, because at least for graphs $G$ of moderate size the two values $\vartheta'(G)$ and $\vartheta(G)$ are very close to each other. In order to prove that $\vartheta'(G)$ is not an upper bound, we have to find some sufficiently large $k$ and show that $$\vartheta'(G)<\sqrt[k]{\alpha(G^k)} \leq \vartheta(G),$$ which is extremely hard even if $G$ contains only a few vertices. We believe that the Schrijver $\vartheta'$-function is not an upper bound for the Shannon capacity due to its lack of the product property, but whether this is actually true or not remains open.
[^1]: The common definition of the Lovász number that appears in the literature is the dual form of ours.
[^2]: In fact, the equality holds in , but the reverse direction is not relevant for our purpose.
|
---
abstract: 'This paper proposes and analyzes a novel multi-agent opinion dynamics model in which agents have access to actions which are quantized version of the opinions of their neighbors. The model produces different behaviors observed in social networks such as disensus, clustering, oscillations, opinion propagation, . The main results of the paper provides the characterization of preservation and diffusion of actions under general communication topologies. A complete analysis allowing the opinion forecasting is given in the particular cases of complete and ring communication graphs. Numerical examples illustrate the main features of this model.'
author:
- 'N. R. Chowdhury, I.-C. Morărescu, S. Martin, S. Srikant [^1]'
bibliography:
- 'diss.bib'
title: ' Continuous opinions and discrete actions in social networks: a multi-agent system approach[^2] '
---
Multiagent systems; .
Introduction {#sec:intro}
============
The analysis of social networks received an increasing interest during the last decade. This is certainly related with the increasing use of Facebook, Twitter, LinkedIn and other on-line platforms allowing to get information about social networks. A way to model opinion dynamics in social networks is by the intermediate of multi-agent systems. The existing models can be split in two main classes: those considering that opinions can evolve in a discrete set and those considering a continuous set of values that can be taken by each agent. The models in first class come from statistical physics and the most employed are the Ising [@IsingThesis1924], voter [@CliffordSudbury1973] and Sznajd [@Sznajd2000] models. When the opinions are not constrained to , we can find in the literature two popular models: the Deffuant [@Deffuant2000] and the Hegselmann-Krause [@krause2002] models. They are usually known as bounded confidence models since they depend on one parameter characterizing the fact that one agent takes into account the opinion of another only if their opinions are close enough. In order to more accurately describe the opinion dynamics and to recover more realistic behaviors, a mix of continuous opinion with discrete actions (CODA) was proposed in [@Martins2008]. This model reflects the fact that even if we often face binary actions, the opinion behind evolves in a continuous space of values. For instance we may think that car A is 70% more appropriate for our use than car B. However, our action will be 100% buy car A. Moreover, our neighbors only see our action without any access to our opinion. Similar idea was employed in [@CeragioliFrascaECC2015] where it is studied the emergence of consensus under quantized all-to-all communication. In this paper the authors assume constant interaction weights and quantized information on the opinion of all the other individuals belonging to a given social network.
Whatever is the model employed to describe the opinion dynamics, many studies focus on the emergence of consensus in social networks [@GalamMoscovici1991; @Axelrod1997; @Deffuant2000; @Fortunato2005]. Nevertheless, this behavior is rarely observed in real large social networks.
From mathematical view-point the model proposed in this paper is close to the one in [@CeragioliFrascaECC2015] with the difference that we are considering state-dependent interaction weights instead of constant ones. Beside the heterogeneity introduced in the model by the state-dependent interaction weights we are also dealing with more general interaction topologies and we are not trying to characterize consensus. Instead we highlight that extremist individuals are less influenceable, that several equilibria can be reached and one can also have oscillatory behaviors in the network. From behavioral view-point our model is close to the one in [@Martins2008] . The difference here is that we are able to analytically study this model and go beyond simulations and theirs interpretation.
The main contributions of this work are as follows:
- [**we propose a consensus-like dynamics**]{} that approaches the dynamics described in [@Martins2008]. This CODA model is given by a quantized consensus system with state-dependent interaction weights.
- [**we describe the possible equilibria**]{} of the proposed model and [**depict the main properties**]{} characterizing the opinions dynamics. Precisely, we provide a criterion to detect stabilization of the actions of a group of agents and to predict the propagation of this action throughout the network. Our criterion depends on the initial conditions and interaction topology only.
- [**we completely analyze**]{} some particular interaction topologies such as: the complete graph and the ring graph.
The rest of the paper is organized as follows. Section \[problem\_formulation\] formulates the problem and illustrates that our model is close to the one proposed in [@Martins2008]. In Section \[COCAmodel\] we show when the quantization effect is removed [[*i.e.* ]{}]{}in the context of continuous opinions with continuous actions (COCA), that consensus is always achieved provided that the interaction graph is connected. The main features of our CODA model under general interaction topologies are derived in Section \[CODAanalysis\]. Precisely we characterize the preservation and the propagation of actions inside a group and outside it, respectively. Some particular interaction topologies such as: the complete graph and the ring graph are analyzed in \[Particular-Top\].
Problem formulation and preliminaries {#problem_formulation}
=====================================
Model description
-----------------
In this work we consider a network of $n$ individuals/agents denoted by $\V=\{1,\ldots, n\}$. The interaction topology between agents is described by a fixed graph $\G=(\V,\E)$ that can be directed or not. Let us also denote by $N_i$ the set of agents that influence $i$ according to the graph $\G$ ([[*i.e.* ]{}]{}$j\in N_i \Leftrightarrow (j,i)\in\E$) and by $n_i$ the cardinality of $N_i$. Initially, agent $ i\in\V $ has a given opinion $ p_i(0)=p_i^0 \in[0,1]$ and this opinion evolves according to a discrete-time protocol. Let $p_i(k)$ be the opinion of agent $i\in\V$ at time $k$. Assume that $\forall i\in\V, \ p_i(0)\neq\frac{1}{2}$ and introduce the action value $q_i(k)\in\{0,1\}$ as a quantized version of $p_i(k)$ defined by:
$$q_i(k)= \left\{
\begin{array}{l}
0 \text{ if }\left( p_i(k)<\frac{1}{2}\right),\\
0 \text{ if }\left( p_i(k)=\frac{1}{2}~\text{and}~p_i(k-1)<\frac{1}{2}\right),\\
1 \text{ otherwise.}
\end{array}\right.$$
Two distinct situations are considered in the following.\
[**COCA model:**]{} each agent has access to the opinion of its neighbors. In this case, our model simply writes as a consensus protocol with state-dependent interaction weights. Precisely, the opinion of an agent $i\in\V$ updates according to the following rule $$p_i(k+1)=p_i(k)+\frac{p_i(k)(1-p_i(k))}{n_i}\sum_{j\in N_i} (p_j(k)-p_i(k)). \label{eq:mn1}$$
Denoting by $p(\cdot)=(p_1(\cdot),\ldots,p_n(\cdot))$ the vector that collects the opinions of all agents, the collective opinion dynamics is given by: $$p(k+1)=\left( I_n+A(p)\right) p(k). \label{eq:mn2_vector}$$
We assume that $p_i^0$ belongs to $(0,1)$ which is a normalized version of $\R$. Doing so, the matrix $I_n+A(p)$ is row stochastic and for all $k\in\N$ one has $p(k)\in(0,1)$.
[**CODA model:**]{} provides the main model under study in this paper. This model assumes that each individual has access only to the action of its neighbors. The opinion of agent $i\in\V$ in this case updates according to the following rule: $$\begin{aligned}
p_i(k+1)=p_i(k)+\frac{p_i(k)(1-p_i(k))}{n_i}\sum_{j\in N_i} (q_j(k)-p_i(k)). \label{eq:mn2}\end{aligned}$$
For an agent $i\in\V$, both COCA and CODA models propose interaction weights depending only on the opinion $p_i$.
We can emphasize a natural partition of $\V$ in two subsets $N^-(k)=\{i\in\V\mid q_i(k)=0\}$ and $N^+(k)=\{i\in\V\mid q_i(k)=1\}$. The main objective of this paper is to study how these sets evolve in time and what is the behavior of the opinions $p_i(k)$ inside these sets.\
Throughout the paper we denote by $n^-(k)$ and $n^+(k)$ the cardinality of $N^-(k)$ and $N^+(k)$, respectively. Similarly, for an agent $i$ we denote by $N_i^-(k)=N_i\cap N^-(k)$ and $N_i^+(k)=N_i\cap N^+(k)$ and by $n_i^-(k)$ and $n_i^+(k)$ the cardinalities of these sets.
Related model {#sec:martins}
-------------
The CODA model was inspired by Martins’ model [@Martins2008] which was formulated in terms of the following bayesian update. Let us denote by $\tilde{p}_i(k)$ the opinion of agent $i\in\V$ at time $k\in\N$ when using Martins’ model [@Martins2008]. The updates of this opinion follows the rule described below. When agent $i$ is influenced by agent $j$, $$\label{eq:MARTINS_CODA}
\frac{\tilde{p}_i(k+1)}{1-\tilde{p}_i(k+1)} =
\frac{\tilde{p}_i(k)}{1-\tilde{p}_i(k)} \cdot
\frac{\alpha}{1-\alpha} \text{ if } q_j(k) = 1,
$$ and where $\alpha$ is replaced by $1-\alpha$ if $q_j(k) = 0$. The constant $\alpha \ge 0.5$ is a model parameter linked to the amplitude of opinion change as a result of interactions. This parameter does not appear in our model. The study [@Martins2008] is based on numerical experiments and do not present a theoretical analysis. The simulations found in [@Martins2008] where obtained with $\alpha = 0.7$. The simulations appear to be qualitatively close to the ones resulting from our update (see Section \[sec:numerical\_simulations\]). To understand this fact, one can show that for $\alpha = 2/3 \approx 0.7$, and $q_j(k) = 1$, update and are equivalent for small $p_i(k)$ values. Similarly, if $q_j(k)=0$ the two updates are equivalent for $p_i(k)$ values close to $1$. For other $p_i(k)$ values, $p_i(k+1)$ still remains close to $\tilde{p}_i(k+1)$. These facts are illustrated in Figure \[fig:CODA\_vs\_Martins\]. As a consequence, our CODA model can be seen as a consensus-type version of Martins’ model. One advantage of our model is to explicitely explains why extremist agents with opinion close to $0$ or $1$ hardly change their actions. This is due to the weight $p_i (1-p_i)$ in update .
![Comparison of the CODA model and the model by Martins [@Martins2008]. The two top curves display the update and when the influencing agent has action $q_j = 1$. The two bottom curves display the update and when the influencing agent has action $q_j = 0$. Discontinuous red lines correspond to update while strait blue lines correspond to update . []{data-label="fig:CODA_vs_Martins"}](original_CODA_versus_consensus_CODA_3.pdf)
Analysis of the COCA model {#COCAmodel}
==========================
It is noteworthy that an opinion stays constant according to dynamics if it is initialized at the value $0$ or $1$. Therefore, by the following assumption we exclude these extreme cases from the analysis.
\[COCA1\] There exists a *strictly* positive constants $ \epsilon\in\left(0,\frac{1}{2}\right)$ such that for all $ i\in\mathcal{V} $, one has $p_i^0\in[\epsilon,1-\epsilon]$.
For a subset of agents, $ \V'\subseteq \V $ and for all $ k\in \N $, let us define $m_{\V'}(k):=\min\limits_{j\in\V'}p_j(k) $ and $ M_{\V'}(k):=\max\limits_{j\in\V'}p_j(k)$. The following result can be easily proven by induction.
\[COCA2\] Let $\mathcal{V}'\subseteq \mathcal{V}$ such that $\V'$ is isolated [[*i.e.* ]{}]{}for all $ i\in \mathcal{V}' $, for all $ j\in \mathcal{V}\setminus \mathcal{V}' $, $ (j,i)\notin \mathcal{E} $. Then,
1. $M_{\mathcal{V}'}(k+1)\leq M_{\mathcal{V}'}(k)$
2. $m_{\mathcal{V}'}(k+1)\geq m_{\mathcal{V}'}(k)$
Let us assume for the rest of this section that the graph is strongly connected.
Applying Lemma \[COCA2\] with $\V'=\V$ one obtains that sequences $m_{\V}(k)$ and $M_{\V}(k)$ are both monotonic and bounded, thus convergent. The interaction weight from the agent $j\in N_i$ to the agent $i$ at time $k$ is defined by $a_{ij}(k):=\frac{p_i(k)\big(1-p_i(k)\big)}{n_i}$. Taking into account that $p_i^0\in[\epsilon,1-\epsilon]$, the behavior of the function $x\mapsto x(1-x)$ on $[0,1]$ and Lemma \[COCA2\], one deduces that $a_{ij}(k)\in[\frac{\epsilon(1-\epsilon)}{n_i},\frac{1}{4n_i}]$ for all $k\in\N$ and $j\in N_i$. Moreover, for all $i\in\V$ one has $1\leq n_i\leq n-1$ yielding $$\label{aij}a_{ij}(k)\in\left[\frac{\epsilon(1-\epsilon)}{n-1},\frac{1}{4}\right], \quad \forall i\in\V,\ \forall j\in N_i.$$
Since $a_{ii}=1-\sum_{j\in N_i} a_{ij}$, straightforward computation shows that $$\label{aii}a_{ii}(k)\in\left[\frac{3}{4},1-\epsilon(1-\epsilon)\right], \quad \forall i\in\V.$$
Equation and shows that Assumption 1 in [@blondel2005] holds true. Moreover, in this paper we are dealing with a fixed (strongly) connected graph meaning that Assumption 2 and 3 in [@blondel2005] also hold. Therefore the following result is a consequence of Theorems 1 and Lemma 1 in [@blondel2005].
As far as the graph $\G$ is (strongly) connected and Assumption \[COCA1\] is satisfied, the update rule guarantees asymptotic consensus of all opinions. Furthermore, the convergence rate to consensus satisfies $$M_{\V}(k+1)-m_{\V}(k+1)\leq \beta(k) \Big(M_{\V}(k)-m_{\V}(k)\Big)$$ with $\beta(k)\leq\beta(\epsilon):=\Big(1-\min\{\frac{3}{4},\frac{ \epsilon(1-\epsilon)}{n-1}\}\Big)$.
Analysis of the CODA model {#CODAanalysis}
==========================
For the remaining part of this work we consider the update rule . In this section we derive the set of possible equilibrium points and we give the conditions guaranteeing that the action $q_i$ of the agent $i\in\V$ is preserved/changed over time. Throughout the rest of the paper we assume that $\forall i\in\V, \ p_i(0)\in(0,1)$.
Characterization of equilibria
------------------------------
Let us define the following finite set of rational numbers: $$\label{S}
\S=\left\{\frac{k}{m} \ \Big|\ k,m\in{{\textcolor{black}{\N}}}, k\leq m\leq n-1 \right\}\subset \Q .$$ The main result of this section states that the possible equilibrium points of the opinions belong to $\S$. Let us first introduce an instrumental result. For the rest of the paper we use the notation $r_i(k):=\displaystyle\frac{n_i^+(k)}{n_i}$.
\[lemma:CODA\] Let $i\in\V$, $p_i(0)\in(0,1)$. Then, , one of the following relations holds $$\label{eq1}
p_i(k)< p_i(k+1)< r_i(k),$$ $$\label{eq2}
p_i(k)> p_i(k+1)> r_i(k),$$ $\forall k\in\N,\ p_i(k)\in(0,1)$.
The discrete dynamics can be rewritten by replacing $q_j(k)$ with $0$ or $1$ respectively. At time $k$ the agent $i$ has $n_i^-(k)$ neighbors having the action equals 0 and $n_i^+(k)$ with action equal 1. Then rewrites as: $$\label{eq:model_rewritten_with_ni_plus}
\begin{array}{ll}
p_i(k+1)&=p_i(k)+\frac{p_i(k)(1-p_i(k))}{n_i}( n_i^+(k) -n_ip_i(k))\\
&=p_i(k)+p_i(k)(1-p_i(k))\left( r_i(k) -p_i(k)\right),
\end{array}$$ where we have used the fact that $n_i^+(k)+n_i^-(k)$ is a constant equal to $n_i$. if $p_i(k)<\displaystyle r_i(k)$ then $p_i(k+1)> p_i(k)$ and reversely if $p_i(k)>\displaystyle r_i(k)$ then $p_i(k+1)< p_i(k)$. Moreover, $$p_i(k+1)- r_i(k)=\left(p_i(k)- r_i(k)\right)(1-p_i(k)(1-p_i(k)))$$ yielding $p_i(k+1)<\displaystyle r_i(k)$ if $p_i(k)<\displaystyle r_i(k)$\
and $p_i(k+1)>\displaystyle r_i(k)$ if $p_i(k)>\displaystyle r_i(k)$. In these cases, either or holds
\[CODAeq\]
If the sequence $\big(n_i^+(k)\big)_{k\geq0}$ is stationary, one gets that $n_i^+(k)$ is constant for $k$ bigger than or equal to a fixed integer $k^*$. Let us denote by ${{\textcolor{black}{\rho}}}_i^*$ the value of $ r_i(k)$ for $k\geq k^*$. In the first case $\big(p_i(k)\big)_{k\geq0}$ is increasing and upper-bounded, in the second $\big(p_i(k)\big)_{k\geq0}$ is decreasing and lower-bounded. Thus $\big(p_i(k)\big)_{k\geq0}$ converges.
Preservation of actions
-----------------------
In this subsection we investigate the conditions ensuring that the action $q_i$ does not change over time. More precisely, we provide a criterion to detect when a group of agents sharing the same action will preserve it for all time. For the sake of simplicity we focus on $q_i(0)=0$ but using similar arguments the same results can be obtained for $q_i(0)=1$.
\[lemma:actionpreserv\] Let $i\in\V$, if $n_i^-(k)\geq n_i^+(k)$ and $q_{i}(k)=0$ then $q_{i}(k+1)=0$.
Following the proof of Proposition \[CODAeq\] one obtains that either , .\
If holds then $p_{i}(k+1)\leq\displaystyle r_i(k)$. Since $n_i^-(k)\geq n_i^+(k)$ one deduces that $p_{i}(k+1)\leq\displaystyle\frac{1}{2}$ and since $q_{i}(k)=0$ one obtains that $q_{i}(k+1)=0$.\
If holds, then $p_{i}(k+1)\leq p_i(k)$. Since $q_{i}(k)=0$ one deduces that $p_i(k)\leq\displaystyle\frac{1}{2}$. Therefore, $p_{i}(k+1)\leq\displaystyle\frac{1}{2}$ meaning again that $q_{i}(k+1)=0$.
Throughout the paper we denote by $|A|$ the cardinality of a set $A$.
\[Polarized-Cluster\] We say a subset of agents $A\subseteq\V$ is a *robust polarized cluster* if the following hold:
- $\forall i,j\in A,\ q_i(0)=q_{j}(0)$;\
- $\forall i\in A,\ |N_i\cap A|\geq |N_i\setminus A|$.
Notice that, in this section we do not make any assumption on the connectivity of the interaction graph. This means in particular that it may happen to have $n_i=0$ for some agents belonging to $A$ above.
The next result explains why the word *robust* appears in the previous definition.
\[prop:actionpreserv\] If $A$ is a robust polarized cluster with $q_i(0)=0$ for a certain $i\in A$ then
- $\forall i\in A,\ \forall k\in\N, \ q_i(k)=0$;\
- $\forall i\in A, \ \displaystyle\lim_{k\rightarrow\infty}p_i(k)\leq\displaystyle\frac{1}{2}$.
The proof will be done by induction. Let us remark first that following the first item of Definition \[Polarized-Cluster\] we have $q_i(0)=0, \forall i\in A$. Let us assume that for a fixed $k^*$ and $\forall i\in A$ one has $q_i(k^*)=0$. Let us also recall that the interaction graph under consideration is fixed. Therefore we still have that $|N_i\cap A|\geq |N_i\setminus A|$. Moreover, $|N_i\cap A|\subseteq N_i^-(k^*)$ implying that $n_i^-(k^*)\geq n_i^+(k^*), \forall i\in A$. Thus, Lemma \[lemma:actionpreserv\] yields that $ \forall i\in A$ one has $q_i(k^*+1)=0$ and the proof ends.
Change and diffusion of actions
-------------------------------
The goal of this subsection is twofold. First to provide conditions at a given time $k\in\N$ guaranteeing that the action of a fixed agent $i\in\V$ will change at time $k+1$. Secondly, we analyze the propagation/diffusion of the action of a robust polarized cluster inside the overall network. Due to the symmetry of reasonings we continue to focus only on one case $q_i(0)=0$ or $q_i(0)=1$.
\[prop:changeopaction\] Let $i\in\V$ and $k\in\N$ such that $p_{i}(k)>\displaystyle\frac{1}{2}$ and $n_i^-(k)>n_i^+(k)$. Let $\epsilon(n_i^+(k),n_i) \in \left(0,\displaystyle\frac{1}{2}\right)$ be the unique positive real solution of the equation: $$\label{eps-x def}
x^3-\left(\frac{1}{2}- r_i(k)\right)x^2-\frac{3}{4}x+\left(\frac{1}{2}- r_i(k)\right)\frac{1}{4}=0.$$ Then $p_i(k+1)\leq\displaystyle\frac{1}{2}$ if and only if $p_i(k)<\displaystyle\frac{1}{2}+\epsilon(n_i^+(k),n_i)$.
Let us introduce $x:=p_i(k)-\displaystyle\frac{1}{2}\in\left(0,\displaystyle\frac{1}{2}\right]$. Recall that rewrites as: $$\begin{aligned}
p_i(k+1)=p_i(k)+p_i(k)(1-p_i(k))\left( r_i(k) -p_i(k)\right).\end{aligned}$$ Consequently, $p_i(k+1)\leq\displaystyle\frac{1}{2}$ if and only if $$p_i(k)+p_i(k)(1-p_i(k))\left( r_i(k) -p_i(k)\right)\leq\frac{1}{2},$$ which rewrites in term of $x$ as $$x^3+\left(\frac{1}{2}- r_i(k)\right)x^2+\frac{3}{4}x-\left(\frac{1}{2}- r_i(k)\right)\frac{1}{4}\leq0.$$ the inequality above is satisfied for $x=0$ but it does not hold for $x=\displaystyle\frac{1}{2}$. Therefore, by continuity, has a solution into $\left(0,\displaystyle\frac{1}{2}\right)$. Moreover, by using the first derivative it is straightforward that the function $$x\mapsto x^3+\left(\frac{1}{2}- r_i(k)\right)x^2+\frac{3}{4}x-\frac{1}{4}\left(\frac{1}{2}- r_i(k)\right)$$ is strictly increasing for $x\geq0$. Thus has an unique positive solution Denoting it by $\epsilon(n_i^+(k))$ one deduces that $p_i(k+1)\leq\displaystyle\frac{1}{2}$ if and only if $p_i(k)\in\left[0,\displaystyle\frac{1}{2}+\epsilon(n_i^+(k),n_i) \right)$ provided that $n_i^-(k)>n_i^+(k)$.
The previous result states that an agent will change its action when it is influenced by more opposite actions, only if its opinion is sufficiently close to the boundary between the actions. The notion of sufficiently closed depends on the proportion of opposite actions that influence the agent and is exactly quantified by $\epsilon(n_i^+(k),n_i)$.
\[lemma:actionchange\] Then it exists $k^*\geq T_i$ such that $q_{i}(k)=0,\ \forall k\geq k^*$.
To obtain a contradiction, let us suppose that $p_i(k)\geq\displaystyle\frac{1}{2},\ \forall k\geq T_i$. Since $n_i^-(k)> n_i^+(k)$, $ \forall k\geq T_i $ we get that $\displaystyle r_i(k)<\displaystyle\frac{1}{2}, \forall k\geq T_i$. We recall that $$p_i(k+1)=p_i(k)+p_i(k)(1-p_i(k))\left( r_i(k)-p_i(k)\right).$$ Therefore, the sequence $\big(p_i(k)\big)_{k\geq T_i}$ is strictly decreasing and lower-bounded by $\displaystyle\frac{1}{2}$. Consequently $\big(p_i(k)\big)_{k\in\N}$ converges and its limit is bigger than or equal to $\displaystyle\frac{1}{2}$ .\
From Proposition \[CODAeq\] the limit of $\big(p_i(k)\big)_{k\in\N}$ is $\displaystyle\frac{\displaystyle\lim_{k\rightarrow\infty}n_i^+(k)}{n_i}$. On the other hand $n_i^-(k)> n_i^+(k)$, $ \forall k\geq T_i$ Therefore $\displaystyle\frac{\displaystyle\lim_{k\rightarrow\infty}n_i^+(k)}{n_i}<\displaystyle\frac{1}{2}$ contradicting the fact that $\big(p_i(k)\big)_{k\in\N}$ converges toward a value bigger than or equal to $\displaystyle\frac{1}{2}$.\
The previous reasoning shows that it exists $k^*\geq T_i$ such that $q_{i}(k^*)=0$. To get $q_{i}(k)=0,\ \forall k\geq k^*$ it is sufficient to recursively apply Lemma \[lemma:actionpreserv\].
The next result characterizes the diffusion of the action of a robust polarized cluster over the network.
\[prop:actionchange\] Let us consider the sets $A_1, A_2,\ldots, A_d$ such that
- $A_1$ is a robust polarized set with $q_i(0)=0$ for a certain $i\in A_1$ (and thus $\forall i\in A_1,\ q_i(0)=0$);\
- $\forall h\in\{1,\ldots,d-1\}$ and $\forall i\in A_{h+1}$ one has $$|N_i\cap A_h|> |N_i\setminus A_h|.$$
Consequently, $\forall i\in \displaystyle\bigcup_{h=1}^d A_h$ one has $\displaystyle\lim_{k\rightarrow\infty}p_i(k)\leq\displaystyle\frac{1}{2}$.
we proceed recursively. Assume that the proposition holds for $h\in\{1,\ldots, f\}$ with $f<d$ and show that it holds for $h=f+1$. We know that $\forall i\in A_{f+1}$ one has $$|N_i\cap A_f|> |N_i\setminus A_f|.$$ Moreover $N_i\cap A_f\subseteq N_i^-(k),\ \forall k\geq T_f$ and $\forall i\in A_{f+1}$. Therefore we can apply Lemma \[lemma:actionchange\] for any $ i\in A_{f+1}$. Choosing $T_{f+1}=\displaystyle\max_{i\in A_{f+1}}\ T_i$ one obtains that the proposition holds for $h=f+1$.
The last part of the statement is a simple consequence of the fact that $\forall i\in \displaystyle\bigcup_{h=1}^d A_h$ one has $q_i(k)=0, \forall k\geq T$ where $T=\displaystyle\max_{h\in \{2,\ldots,d\}}\ T_h$.
Some particular network topologies {#Particular-Top}
==================================
Complete graph
--------------
In this subsection we use previous results to completely characterize the opinion dynamics when the interactions are described by the complete graph.
\[complete\] If $n^-(0)>n^+(0)$ then $\forall i\in\V$ the limit behavior of the opinion is given by $\displaystyle\lim_{k\rightarrow\infty}p_i(k)=0$. Reversely, $n^+(0)>n^-(0)$ implies $\displaystyle\lim_{k\rightarrow\infty}p_i(k)=1$.
If $n^-(0)>n^+(0)$, due to all-to-all connections one has that $N^-(0)$ is a robust polarized cluster. Moreover, $\forall i\in N^+(0)$ one has that $|N_i\cap N^-(0)|> |N_i\setminus N^-(0)|.$ Thus, applying Proposition \[prop:actionchange\] with $A_1=N^-(0)$ and $A_2=N^+(0)$ we obtain that it exists a value $T\in\N$ such that $q_i(k)=0, \ \forall i\in\V, \forall k\geq T$. Following Proposition \[CODAeq\] this yields that $$\displaystyle\lim_{k\rightarrow\infty}p_i(k)=0.$$ The case $n^+(0)>n^-(0)$ is proven in a similar way and as we have done throughout the paper we omit the details.
Let us consider now the case $n^-(0)=n^+(0)$. It is noteworthy that in this case $n=|\V|$ is even and
- $\forall i\in N^+(0)$ one has $$|N_i\cap N^-(0)|=\frac{n}{2}>\frac{n}{2}-1= |N_i\setminus N^-(0)|.$$
- $\forall i\in N^-(0)$ one has $$|N_i\cap N^+(0)|=\frac{n}{2}>\frac{n}{2}-1= |N_i\setminus N^+(0)|.$$
If the initial conditions are not symmetric w.r.t. $\displaystyle\frac{1}{2}$ an agent will cross from $N^+$ to $N^-$ (or reversely) and we recover the situation treated in Proposition \[complete\]. Therefore, to finish the analysis we give the following result that deals with initial conditions symmetrically displayed w.r.t. $\displaystyle\frac{1}{2}$. This case emphasizes an interesting oscillatory behavior of the opinions.
\[completesym\] Assume that $n^+(0)=n^-(0)$ and moreover $\forall i\in\{1,\ldots,\frac{n}{2}\}$ there exist $\eta_i(0)\in\left(0,\displaystyle\frac{1}{2}\right)$ such that $$\label{init-cond complete-graph}
p_i(0)=\frac{1}{2}-\eta_i(0) \ \mbox{ and } p_{\frac{n}{2}+i}(0)=\frac{1}{2}+\eta_i(0).$$ Then $n^+(k)=n^-(k),\ \forall k\in\N$ and $\exists k^*\in\N$ such that $$\begin{split}&|p_j(k)-\frac{1}{2}|\leq \epsilon^* \text{ and } \left(p_j(k)-\frac{1}{2}\right)\left(p_j(k+1)-\frac{1}{2}\right)<0,
\end{split}$$ where $\epsilon^*<\frac{1}{6(n-1)}$ is the unique positive solution of the equation $$x^3+\frac{1}{2(n-1)}x^2+\frac{3}{4}x-\frac{1}{8(n-1)}=0.$$
The result above states that all the agents in the network will finish by oscillating around $\frac{1}{2}$ in a $2\epsilon^*$ strip.
For all $i\in\{1,\ldots,\frac{n}{2}\}$ let us denote $\bar{i}=\frac{n}{2}+i$ and observe that due to all-to-all communications one has $$\frac{n_i^+(0)}{n_i}=\frac{1}{2}+\frac{1}{2(n-1)} \mbox{ and } \frac{n_{\bar{i}}^+(0)}{n_{\bar{i}}}=\frac{1}{2}-\frac{1}{2(n-1)}.$$ For all the couples $(i,\bar{i})$ the following holds $$\begin{aligned}
p_i(1)&=p_i(0)+p_i(0)(1-p_i(0))\left(\frac{n_i^+(0)}{n_i}-p_i(0)\right)\\
&=p_i(0)+(\frac{1}{4}-\eta_i(0)^2)\left(\frac{1}{2(n-1)}+\eta_i(0)\right)\\
p_{\bar{i}}(1)&=p_{\bar{i}}(0)+p_{\bar{i}}(0)(1-p_{\bar{i}}(0))\left(\frac{n_{\bar{i}}^+(0)}{n_{\bar{i}}}-p_{\bar{i}}(0)\right)\\
&=p_{\bar{i}}(0)-(\frac{1}{4}-\eta_i(0)^2)\left(\frac{1}{2(n-1)}+\eta_i(0)\right).\end{aligned}$$ Therefore, $p_i(1)$ and $p_{\bar{i}}(1)$ remain symmetric w.r.t. $\displaystyle\frac{1}{2}$. Inductively one obtains that $p_i(k)$ and $p_{\bar{i}}(k)$ remain symmetric w.r.t. $\displaystyle\frac{1}{2}$. In other words if an agent $i\in\{1,\ldots,\frac{n}{2}\}$ changes its action, the agent $\bar{i}$ will also change its action and the changing will be in the opposite sense. Consequently, $n^+(k)=n^-(k),\ \forall k\in\N$.\
Let us show now that $\exists k^*\in\N$ such that $$|p_j(k)-\frac{1}{2}|\leq \epsilon^*,\ \forall k\geq k^*, \forall j\in\V.$$ Notice that for all $i\in\{1,\ldots,\frac{n}{2}\}$ one has $p_i(0)<\frac{1}{2}$, $n_i^+(0)>n_i^-(0)$, $p_{\bar{i}}(0)>\frac{1}{2}$ and $n_{\bar{i}}^-(0)>n_{\bar{i}}^+(0)$. Thus, from the proof of Lemma \[lemma:actionchange\] $\exists k_i^*\in\N$ such that $p_{\bar{i}}(k_i^*+1)<\frac{1}{2}$ and from the reasoning above $p_i(k_i^*+1)>\frac{1}{2}$. From Proposition \[prop:changeopaction\] this change of actions happens if and only if $p_{\bar{i}}(k_i^*)\in\left(\frac{1}{2},\frac{1}{2}+\epsilon^*\right)$ with $\epsilon^*=\epsilon(\frac{n}{2}-1,n-1) $ the unique positive solution of $$x^3+\frac{1}{2(n-1)}x^2+\frac{3}{4}x-\frac{1}{8(n-1)}=0.$$ For $x=\frac{1}{6(n-1)}$ the expression above is positive and for $x=\frac{1}{8(n-1)}$ the expression is negative yielding that $\epsilon^* \in\left(\frac{1}{8(n-1)},\frac{1}{6(n-1)} \right)$. Let $x=\frac{1}{2}-p_i(k^*)\leq\epsilon^*$ meaning that $$\begin{aligned}
&x^3+\frac{1}{2(n-1)}x^2+\frac{3}{4}x-\frac{1}{8(n-1)}\leq0.\end{aligned}$$ One notes that $p_i(k^*+1)<\frac{1}{2}+\epsilon^*$ is equivalent to $$x^3+\frac{1}{2(n-1)}x^2+\frac{3}{4}x+\epsilon^*-\frac{1}{8(n-1)}>0,$$ which is always true since $x\geq0$ and $\epsilon^*-\frac{1}{8(n-1)}>0$. Therefore, once $p_i$ enters in the tube of radius $\epsilon^*$ around $\frac{1}{2}$ it never escapes and moreover Proposition \[prop:changeopaction\] applies at each iteration implying $$\left(p_j(k)-\frac{1}{2}\right)\left(p_j(k+1)-\frac{1}{2}\right)<0,
\ \forall k\geq k^*, \forall j\in\V.$$
Ring graph
----------
Throughout this section we consider the particular configuration in which the interactions are described by an undirected graph in which each vertex has exactly two neighbors. In the following we identify agent $n+1$ as agent $1$ and agent $0$ as agent $n$. For a precise representation of the graph we assume that $\forall i\in\{1,\ldots,n\}$ one has $N_i=\{i-1,i+1\}$.
\[prop:ring\] Under the ring graph topology the opinions dynamics leads to the following properties:
- the set $\S$ defined in reduces to $\left\{0,\displaystyle\frac{1}{2},1\right\}$;
- if $\exists i\in\V$ such that $q_i(0)=q_{i+1}(0)=0$ then $\{i,i+1\}$ is a robust polarized cluster ([[*i.e.* ]{}]{}$q_i(k)=q_{i+1}(k),\ \forall k\in\N$);
- if $\forall i\in\V$ one has $q_i(0)=1-q_{i+1}(0)$ then
- either the initial opinions are not symmetric w.r.t. $\displaystyle\frac{1}{2}$ and agents will change actions asynchronously leading to robust polarized sets $\{i-1,i,i+1\}$.
- or $p_i(0)\in\left\{\frac{1}{2}-\sigma,\frac{1}{2}+\sigma\right\},\forall i\in\V$ and agents will change actions synchronously preserving $n^-(k)=n^+(k),\forall k\in\N$. Moreover, for $\sigma$ solving $$8\sigma^3+8\sigma^2+14\sigma-1=0$$ one has $p_i(k)\in\left\{\frac{1}{2}-\sigma,\frac{1}{2}+\sigma\right\},\forall i\in\V,\forall k\in\N$
The first item is a consequence of Proposition \[CODAeq\]. The second item follows from Proposition \[prop:actionpreserv\]. The third item follows the ideas in Proposition \[completesym\]. Imposing $p_i(k)=\frac{1}{2}+\sigma$, $p_i(k+1)= \frac{1}{2}-\sigma$ and using the ring graph topology we obtain: $$\begin{aligned}
\frac{1}{2}-\sigma=\frac{1}{2}+\sigma-\left(\frac{1}{2}+\sigma\right)^2\left(\frac{1}{2}-\sigma\right)\end{aligned}$$ or equivalently $$\begin{aligned}
8\sigma^3+8\sigma^2+14\sigma-1=0.\end{aligned}$$
Numerical illustrations {#sec:numerical_simulations}
=======================
Influencial minority {#sec:simu_minority}
--------------------
We illustrate that a well connected minority can *convince* a majority of agents located in the periphery of the interaction network. Proposition \[prop:actionchange\] can be used to predict the phenomenon given the topology of the social network and the initial actions only. Figure \[fig:diffusion\_of\_action\] illustrates this fact. From Proposition \[prop:actionchange\], taking $A_1 = \{1,2,3,4\}$, $A_2 = \{5,6,7,8\}$, $A_3 = \{9\}$ and $A_4 = \{10\}$, we predict that all agents will converge to a state with action $\lim q_i = 0$. We see in Figure \[fig:diffusion\_of\_action\]-B that initially agents $9$ and $3$ tend to approach $1/2$ since they have neighbors equally distributed over $1/2$. This behavior changes when agent $8$ passes the $1/2$ threshold. Moreover, the action diffusion propagates to agent $10$ even though it originally had no neighbor with $q_i(0) = 0$. The decrease of agent $10$ towards $0$ only starts when agent $9$ passes the $1/2$ threshold.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------
**(A)** **(B)**
![Illustration of the action diffusion process described in Proposition \[prop:actionchange\]. Agents $1,2,3$ and $4$ start with $q_i(0) = 0$ and form a robust polarized cluster (Definition \[Polarized-Cluster\]), while agents $5,6,7,8,9$ and $10$ start with $q_i(0) = 1$. All agents converge to a state with action $\lim q_i = 0$. []{data-label="fig:diffusion_of_action"}](toy_example_diffusion.pdf "fig:")
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CODA on a square lattice {#sec:simu_lattice}
------------------------
We illustrate our results when the topology of interactions is a square lattice. First, we use a $6\times 6$ lattice (see Figure \[fig:6-lattice\]). As illustrated in Figure \[fig:6-lattice\]-B, for this type of structure, the smallest robust clusters are formed by $2\times 2$ squares. As expected , the robust clusters keep their inital actions and patches of same actions form around the robust clusters. Notice also that the values of convergence lie in set $\{0,1/4,1/3,1/2,2/3,3/4,1\}$ as predicted by Proposition \[CODAeq\].
**(A)** **(B)** **(C)**\
**(D)**\
Patches of agents with same actions are also observed for bigger lattice (see Figure \[fig:50-lattice\] for an instance final actions in a $50\times 50$ lattice ). This is in accordance with the patterns found in [@Martins2008].
Oscillatory dynamics on a ring graph {#sec:simu_complete_graph}
------------------------------------
The following simulation displays the oscillation of agents’opinion around $1/2$ when the interaction graph is complete and when the initial opinions are symmetrically distributed around $1/2$ (see Proposition \[completesym\]). Figure \[fig:100-complete-graph\] shows an instance of this phenomenon for a system of $100$ agents.
![Trajectories of $ 100 $ agents with the complete interaction graph and initial opinions distributed symmetrically around $1/2$.[]{data-label="fig:100-complete-graph"}](coda_random_initial-eps-converted-to.pdf)
Conclusions
===========
In this paper we have introduced a novel opinion dynamics model in which agents have access to actions which are quantized version of the opinions of their neighbors. The model reproduces different behaviors observed in social networks such as dissensus, clustering, oscillations, opinion propagation. The main results of the paper provides the characterization of preservation and diffusion of action under general communication topologies. A complete analysis of the opinions behavior is given in the particular cases of complete and ring communication graphs. Numerical examples illustrate the main features of this model.
[^1]: I.-C. Morărescu and S. Martin are with Université de Lorraine, CRAN, UMR 7039 and CNRS, CRAN, UMR 7039, 2 Avenue de la Forêt de Haye, Vandœuvre-lès-Nancy, France. e-mails: [constantin.morarescu@univ-lorraine.fr, samuel.martin@univ-lorraine.fr]{}
[^2]: The work of I.-C. Morărescu, S. Martin was funded by the ANR project COMPACS - “Computation Aware Control Systems”, ANR-13-BS03-004.
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---
abstract: 'Two novel computing models based on an infinite tessellation of space-time are introduced. They consist of recursively coupled primitive building blocks. The first model is a scale-invariant generalization of cellular automata, whereas the second one utilizes self-similar Petri nets. Both models are capable of hypercomputations and can, for instance, “solve” the halting problem for Turing machines. These two models are closely related, as they exhibit a step-by-step equivalence for finite computations. On the other hand, they differ greatly for computations that involve an infinite number of building blocks: the first one shows indeterministic behavior whereas the second one halts. Both models are capable of challenging our understanding of computability, causality, and space-time.'
author:
- Martin Schaller
- Karl Svozil
title: 'Scale-Invariant Cellular Automata and Self-Similar Petri Nets'
---
Introduction
============
Every physically relevant computational model must be mapped into physical space-time and [*vice versa*]{} [@landauer-89; @maxwell-demon; @bennett-73]. In this line of thought, Von Neumann’s self-reproducing Cellular Automata [@v-neumann-66] have been envisioned by Zuse [@zuse-69] and other researchers [@fredkin; @toffoli-margolus-90; @wolfram-2002; @Margenstern:jucs_5_9:a_polynomial_solution_for] as “calculating space;” i.e., as a locally connected grid of finite automata [@hopcroft] capable of universal algorithmic tasks, in which intrinsic [@svozil-94] observers are embedded [@toffoli:79]. This model is conceptually discreet and noncontinuous and resolves the eleatic “arrow” antinomy [@zeno; @ki-57; @gruenbaum:68; @salmon-01] against motion in discrete space by introducing the concept of information about the state of motion in between time steps.
Alas, there is no direct physical evidence supporting the assumption of a tessellation of configuration space or time. Given enough energy, and without the possible bound at the Planck length of about $10^{-35}$m, physical configuration space seems to be potentially infinitely divisible. Indeed, infinite divisibility of space-time has been utilized for proposals of a kind of “Zeno oracle” [@weyl:49], a progressively accelerated Turing machine [@gruenbaum:74; @Davies01; @ord-2006] capable of hypercomputation [@Davis-2004; @Doria-2006; @Davis-2006]. Such accelerated Turing machines have also been discussed in the relativistic context [@DBLP:conf/mcu/Durand-Lose04; @Nemeti2006118]. In general, a physical model capable of hypercomputation by some sort of “Zeno squeezing” has to cope with two seemingly contradictory features: on the one hand, its infinite capacities could be seen as an obstacle of evolution and therefore require a careful analysis of the principal possibility of motion in finite space and time [*via*]{} an infinity of cycles or stages. On the other hand, the same infinite capacities could be perceived as an advantage, which might yield algorithms beyond the Turing bound of universal computation, thus extending the Church-Turing thesis.
The models presented in this article unify the connectional clarity of von Neumann’s Cellular Automaton model with the requirement of infinite divisibility of cell space. Informally speaking, the scale-invariant cellular automata presented “contain” a multitude of “spatially” and “temporally” ever decreasing copies of themselves, thereby using different time scales at different layers of cells. The cells at different levels are also capable to communicate, i.e., exchange information, with these copies, resulting in ever smaller and faster cycling cells. The second model is based on Petri nets which can enlarge themselves.
The advantage over existing models of accelerated Turing machines — which are just Turing machines with a geometrically progression of accelerated time cycles — resides in the fact that the underlying computational medium is embedded into its environment in a uniform and homogeneous way. In these new models, the entire universe, and not just specially localized parts therein, is uniformly capable of the same computational capacities. This uniformity of the computational environment could be perceived as one further step towards the formalization of continuous physical systems [@CIEChapter2007] in algorithmic terms. In this respects, the models seem to be closely related to classical continuum models, which are at least in principle capable of unlimited divisibility and information flows at arbitrary small space and time dimensions. At present however, for all practical purposes, there are finite bounds on divisibility and information flow.
To obtain a taste of some of the issues encountered in formalizing this approach, note that an infinite sequence of ever smaller and faster cycling cells leads to the following situation. Informally speaking, let a [*self-similar cellular automaton*]{} be a variant of a one-dimensional elementary cellular automaton, such that each cell is updated twice as often as its left neighbor. The cells of a self-similar cellular automaton can be enumerated as $\ldots, c_{-2}, c_{-1}, c_{0}, c_{1}, c_{2}, \ldots$. Starting at time 0 and choosing an appropriate time unit, cell $c_i$ is updated at times $1 / 2^{i}, 2 /2^{i}, 3 / 2^{i}, \ldots$. Remarkably, this definition leads to indeterminism. To see this, let $s(i, t)$ be the state of cell $i$ at time $t$. Now, the state $s(0, 1)$ depends on $s(1, 1/2)$, which itself depends on $s(2, 1/2^2)$ and so on, leading to an infinite regress. In general, in analogy to Thomson’s paradox [@salmon-01; @1011191], this results in an undefined or at least nonunique and thus indeterministic behavior of the automaton.
This fact relates to the following variant of Zeno’s paradox of a runner, according to which the runner cannot even get started [@salmon-01]. He must first run to the half way point, but before that he must run half way to the half way point and so on indefinitely. Whereas Zeno’s runner can find rescue in the limit of convergent real sequences, there is no such relieve for the discrete systems considered.
Later on, two restrictions on self-similar automata (build from scale-invariant cellular automata) are presented, which are sufficient conditions for deterministic behavior, at least for finite computations. Furthermore, a similar model based on a variant of Petri nets will be introduced, that avoids indeterminism and halts in the infinite limit, thereby coming close to the spirit of Zeno’s paradox.
The article is organized as follows. Section \[sec-tm\] defines the Turing machine model used in the remainder of the article, and introduces two hypercomputing models: the accelerated and the right-accelerated Turing machine. In section \[chap:sica\] self-similar as well as scale-invariant cellular automata are presented. Section \[chap:hypercomputer\] is devoted to the construction of a hypercomputer based on self-similar cellular automata. There is a strong resemblance between this construction and the right-accelerated Turing machine, as defined in section \[sec-tm\]. A new computing model, the self-similar Petri net is introduced in section \[chap:petri\]. This model features a step-to-step equivalence to self-similar cellular automata for finite computations, but halts in the infinite case. The same construction as in section \[chap:hypercomputer\] is used to demonstrate that self-similar Petri nets are capable of hypercomputation. The final section contains some concluding remarks and gives some directions for future research.
Turing machines and accelerated Turing machines {#sec-tm}
===============================================
The Turing machine is, beside other formal systems that are computationally equivalent, the most powerful model of classical computing [@rogers1; @odi:89; @odi:99]. We use the following model of a Turing machine [@hopcroft].
Formally, a [*Turing machine*]{} is a tuple $M = (Q, \Sigma, \Gamma, \delta, q_0, B, F)$, where $Q$ is the finite set of states, $\Gamma$ is the finite set of tape symbols, $\Sigma \subset \Gamma$ is the set of input symbols, $q_0 \in Q$ is the start state, $B \in \Gamma \backslash \Sigma$ is the blank, and $F \subset Q$ is the set of final states. The next move function or transition function $\delta$ is a mapping from $Q \times \Gamma$ to $Q \times \Gamma \times \{L, R\}$, which may be undefined for some arguments.
The Turing machine $M$ works on a tape divided into cells that has a leftmost cell but is infinite to the right. Let $\delta(q, a) = (p, b, D)$. One step (or move) of $M$ in state $q$ and the head of $M$ positioned over input symbol $a$ consists of the following actions: scanning input symbol $a$, replacing symbol $a$ by $b$, entering state $p$ and moving the head one cell either to the left ($D=L$) or to the right ($D=R$). In the beginning, $M$ starts in state $q_0$ with a tape that is initialized with an input word $w \in \Sigma^*$, starting at the leftmost cell, all other cells blank, and the head of $M$ positioned over the first symbol of $w$. We need sometimes the function $\delta$ split up into three separate functions: $\delta(q,a) = (\delta_Q(q,a), \delta_\Gamma(q,a), \delta_D(q,a))$.
The configuration of a Turing machine $M$ is denoted by a string of the form $\alpha_1 q \alpha_2$, where $q \in Q$ and $\alpha_1, \alpha_2 \in \Gamma^*$. Here $q$ is the current state of $M$, $\alpha_1$ is the tape content to the left, and $\alpha_2$ the tape content to the right of the head including the symbol that is scanned next. Leading and trailing blanks will be omitted, except the head has moved to the left or to the right of the non-blank content. Let $\alpha_1 q \alpha_2$ and $\alpha_1^\prime p \alpha_2^\prime$ be two configurations of $M$. The relation $\alpha_1 q \alpha_2 \vdash_M \alpha_1^\prime p \alpha_2^\prime$ states that $M$ with configuration $\alpha_1 q \alpha_2$ changes in one step to the configuration $\alpha_1^\prime p \alpha_2^\prime$. The relation $\vdash_M^*$ denotes the reflexive and transitive closure of $\vdash_M$.
The original model of a Turing machine as introduced by Alan Turing contained no statement about the time in which a step of the Turing machine has to be performed. In classical computation, a “yes/no”-problem is therefore decidable if, for each problem instance, the answer is obtained in a finite number of steps. Choosing an appropriate time scheduling, the Turing machine can perform infinitely many steps in finite time, which transcends classical computing, thereby leading to the following two hypercomputing models. The concept of an accelerated Turing machine was independently proposed by Bertrand Russell, Ralph Blake, Hermann Weyl and others (see Refs. [@ord-2006; @potgieter-06]).
An accelerated Turing machine is a Turing machine which performs the $n$-th step of a calculation in $1/2^n$ units of time.
The first step is performed in time 1, and each subsequent step in half of the time before. Since $1 + 1/2 + 1/4 + 1/8 + \ldots = 2$, the accelerated Turing machine can perform infinitely many steps in finite time. The accelerated Turing machine is a hypercomputer, since it can, for example, solve the halting problem, see e.g., Ref. [@ord-2006]. If the output operations are not carefully chosen, the state of a cell becomes indeterminate, leading to a variation of Thomson’s lamp paradox. The open question of the physical dynamics in the limit reduces the physical plausibility of the model. The following model of a hypercomputing Turing machine has a different time scheduling, thereby avoiding some of the paradoxes that might arise from the previous one.
Let the cells of the tape be numbered from the left to the right $c_0, c_1, c_2, \ldots$. A right-accelerated Turing machine is a Turing machine that takes $1/2^n$ units of time to perform a step that moves the head from cell $c_n$ to one of its neighbor cells.
\[th-right-acc-tm\] There exists a right-accelerated Turing machine that is a hypercomputer.
Let $M_U$ be a universal Turing machine. We construct a Turing machine $\overline{M}_U$ that alternates between simulating one step of $M_U$ and shifting over the tape content one cell to the right. We give a sketch of the construction, Ref. [@hopcroft] contains a detailed description of the used techniques. The tape of $\overline{M}_U$ contains one additional track that is used to mark the cell that is read next by the simulated $M_U$. The finite control of $\overline{M}_U$ is able to store simultaneously the state of the head of $M_U$ as well as a tape symbol of $M_U$. We assume that the input of $M_U$ is surrounded by two special tape symbols, say $\$$. At the start of a cycle, the head of $\overline{M}_U$ is initially positioned over the left delimiter $\$$. $\overline{M}_U$ scans the tape to the right, till it encounters a flag in the additional track that marks the head position of $M_U$. Accessing the stored state of $M_U$, $\overline{M}_U$ simulates one step of $M_U$ thereby marking either the left or the right neighbor cell as the cell that has to be visited next in the simulation of $M_U$. If necessary, a blank is inserted left to the right delimiter $\$$, thereby extending the simulated tape of $M_U$. Afterwards the head of $\overline{M}_U$ moves to the right delimiter $\$$ to start the shift over that is performed from the right to the left. $\overline{M}_U$ repeatedly stores the symbols read in its finite control and prints them to the cell to the right. After the shift over, the head of $M_U$ is positioned over the left delimiter $\$$ which finishes one cycle.
We now give an upper bound of the cycle time. Let $n$ be the number of cells, from the first $\$$ to the second one. Without loss of generality we assume that $c_0$ contains the left $\$$. $\overline{M}_U$ scans from the left to the right and simulates one step of $M_U$ which might require to go an additional step to the left. If cell $c_1$ is to be read next, the head of $M_U$ cannot move to the right, otherwise it would fall off the tape of $M_U$. Therefore the worst case occurs if the cell $c_2$ is marked as cell that $M_U$ has to be read next. In this case we obtain $1 + 1/2 + 1/4 + 1/2 + 1/4 + 1/8 + \ldots + 1/2^{n-1} < 3$. The head of $\overline{M}_U$ is now either over cell $c_{n-1}$, or over cell $c_n$ if a insertion was performed. The shift over visits each cell $c_i, 1 \leq i < n$ three times, and $c_0$ two times. Therefore the following upper bound of the time of the shift over holds: $3(1 + 1/2 + 1/4 + \ldots 1/2^n) < 6$. We conclude that if the cycle started initally in cell $c_n$ it took less than time $9/2^n$. If $M_U$ halts on its input, $\overline{M}_U$ finishes the simulation in a time less than $9(1 + 1/2 + 1/4 + \ldots) = 18$. $\overline{M}_U$ therefore solves the halting problem of Turing machines. We remark that if $M_U$ does not halt, the head of $\overline{M}_U$ vanishes in infinity, leaving a blank tape behind.
A right-accelerated Turing machine is, in contrast to the accelerated one, in control over the acceleration. This can be used to transfer the result of a computation back to slower cells. The construction of an infinite machine, as proposed by Davies [@Davies01], comes close to the model of a right-accelerated Turing machine, and his reasoning shows that a right-accelerated Turing machine could be build within a continuous Newtonian universe.
Self-similar and scale-invariant cellular automata {#chap:sica}
==================================================
Basic definitions
-----------------
*Cellular automata* are dynamical systems in which space and time are discreet. The states of cells in a regular lattice are updated synchronously according to a local deterministic interaction rule. The rule gives the new state of each cell as a function of the old states of some “nearby” neighbor cells. Each cell obeys the same rule, and has a finite (usually small) number of states. For a more comprehensive introduction to cellular automata, we refer to Refs. [@v-neumann-66; @wolfram-86; @gutowitz; @ilachinski01; @wolfram-2002].
A [*scale-invariant cellular automaton*]{} operates like an ordinary [*cellular automaton*]{} on a cellular space, consisting of a regular arrangement of cells, whereby each cell can hold a value from a finite set of states. Whereas the cellular space of a cellular automaton consists of a regular one- or higher dimensional lattice, a scale-invariant cellular automaton operates on a cellular space of recursively nested lattices which can be embedded in some Euclidean space as well.
The time behavior of a scale-invariant cellular automaton differs from the time behavior of a cellular automaton: Cells in the same lattice synchronously change their state [@Morelli_Zanette], but as cells are getting smaller in deeper nested lattices, the time steps between state changes in the same lattice are assumed to [*decrease*]{} and approach zero in the limit. Thereby, a finite speed of signal propagation between adjacent cells is always maintained. The scale-invariant cellular automaton model gains its computing capabilities by introducing a local rule that allows for interaction between adjacent lattices [@BoFeng_MengDing]. We will introduce the scale-invariant cellular automaton model for the one-dimensional case, the extension to higher dimensions [@Brunnet_Chate] is straightforward.
A scale-invariant cellular automaton, like a cellular automaton, is defined by a cellular space, a topology that defines the neighborhood of a cell, a finite set of states a cell can be in, a time model that determines when a cell is updated, and a local rule that maps states of neighborhood cells to a state. We first define the cellular space of a scale-invariant cellular automaton. To this end, we make use of standard interval arithmetic. For a scalar $\lambda \in \mathbb{R}$ and a (half-open) interval $[x,y) \subset \mathbb{R}$ set: $\lambda + [x,y) = [\lambda + x, \lambda + y)$ and $\lambda [x,y) = [\lambda x, \lambda y)$. We denote the unit interval $[0,1)$ by $\mathbbm{1}$.
The cellular space $\mathcal{C}$, the set of all cells of the scale-invariant cellular automaton, is the set $\mathcal{C} = \{ 2^k (i + \mathbbm{1}) | i, k \in \mathbb Z\}$. The neighborhood of a cell $c$ is determined by the following operators $\mathit{op}: \mathcal{C} \rightarrow \mathcal{C}$. For a cell $c = 2^k (i + \mathbbm{1})$ in $\mathcal{C}$ let $c_{\leftarrow} = 2^k (i - 1 + \mathbbm{1})$ be the left neighbor, $c_{\rightarrow} = 2^k (i + 1 + \mathbbm{1})$ the right neighbor, $c_{\uparrow} = 2^{k + 1} (\lfloor \frac{i}{2} \rfloor + \mathbbm{1})$ the parent, $c_{\swarrow} = 2^{k-1}(2i + \mathbbm{1})$ the left child, and $c_{\searrow} = 2^{k-1}(2i + 1 + \mathbbm{1})$ the right child of $c$. The predicate $\mathit{left}(c)$ is true if and only if the cell $c$ is the left child of its parent.
The cellular space $\mathcal{C}$ is the union of all lattices $L_k=\{2^k (i + \mathbbm{1})| i \in \mathbb Z\}$, where $k$ is an integer. This topology is depicted in Fig. \[fig:1-dim-interaction\]. For notational convenience, we introduce a further operator, this time from $\mathcal{C}$ to $\mathcal{C} \times \mathcal{C}$, that maps a cell to its both child cells: $c_{\downarrow} = (c_{\swarrow}, c_{\searrow})$. We remark that according to the last definition for each cell either $\mathit{left}(c)$ or $\neg \mathit{left}(c)$ is true. Later on, we will consider scale-invariant cellular automata where not each cell has a parent cell. If $c = 2^k (i + \mathbbm{1})$ is such a cell, we set by convention $\mathit{left}(c) = 1$ if $i \mod 2 = 0$, otherwise $\mathit{left}(c) = 0$.
All cells in lattice $L_k$ are updated synchronously at time instances $2^k i$ where $i$ is an integer. The time interval between two cell updates in lattice $L_k$ is again a half-open interval $2^k (i + \mathbbm{1})$ and the cycle time, that is the time between two updates of the cell, is therefore $2^k$. A simple consequence of this time model is that child cells cycle twice as fast and the parent cell cycle half as fast as the cell itself.
The time scale $\mathcal{T}$ is the set of all possible time intervals, which is in the one-dimensional case equal to the set $\mathcal{C}$: $\mathcal{T} = \{ 2^k (i + \mathbbm{1}) | i, k \in \mathbb Z\}$. The temporal dependencies of a cell are expressed by the following time operators $\mathit{op}: \mathcal{T} \rightarrow \mathcal{T}$. For a time inverval $t = 2^k (i + \mathbbm{1})$ let $t_\leftarrow = 2^k (i - 1 + \mathbbm{1})$, $t_\uparrow = 2^{k + 1} (\lfloor \frac{i-1}{2} \rfloor + \mathbbm{1})$, $t_\swarrow = 2^{k-1} (2i - 2 + \mathbbm{1})$, and $t_\searrow = 2^{k-1} (2i - 1 + \mathbbm{1})$. The predicate $\mathit{coupled}(t)$ is true if and only if the state change of a cell at the beginning of $t$ occurs simultaneously with the state change of its parent cell.
The usage of time intervals instead of time instances, has the advantage that a time interval uniquely identifies the lattice where the update occurs. Fig. \[fig:timeops\] depicts the temporal dependencies of a cell: to the left it shows a coupled state change, to the right an uncoupled one. We remark that we denoted space and time operators by the same symbols, even if their mapping is different. In applying these operators, we take in the remainder of this paper care, that the context of the operator is always clearly defined.
At any time, each cell is in one state from a finite state set $Z$. The cell state in a given time interval is described by the state function $s(c,t)$, which maps cells and time intervals to the state set. The space-time scale $\mathcal{S}$ of the scale-invariant cellular automaton describes the set of allowed pairs of cells and time intervals: $\mathcal{S} = \{(c,t)| c \in \mathcal{C}, t \in \mathcal{T} \mbox{ and } |c| =|t|\}$. Then, the state function $s$ can be expressed as a mapping $s: \mathcal{S} \rightarrow Z$. The local rule describes the evolution of the state function.
For a cell $c$ and a time interval $t$, where $(c, t)$ is in $\mathcal{S}$, the evolution of the state is given by the local rule $f$ of the scale-invariant cellular automaton $$\label{eq:local-rule}
\small
s(c,t) = f(
s(c_\uparrow, t_\uparrow),
s(c_\leftarrow, t_\leftarrow), s(c, t_\leftarrow), s(c_\rightarrow, t_\leftarrow),
s(c_\downarrow, t_\swarrow), s(c_\downarrow, t_\searrow), \mathit{left}(c), \mathit{coupled}(t)
)$$
In accordance with the definition, the expanded form of a expression of the kind $s(c_\downarrow, t_\swarrow)$ is $(s(c_\swarrow, t_\swarrow), s(c_\searrow, t_\swarrow))$. The local rule $f$ is a mapping from $Z^8 \times \{0,1\}^2$ to $Z$. Beside the dependencies on the states of the neighbor cells, the new state of the cell further depends on whether the cell is the left or the right child of its parent cell and whether the state change is coupled or uncoupled to the state change of its parent cell. Formally, a scale-invariant cellular automaton $A$ is denoted by the tuple $A = (Z, f)$. There are some simplifications of the local rule possible, if one allows for a larger state set. For instance, the values of the predicates $\mathit{left}$ and $\mathit{coupled}$ could be stored as substate in the initial configuration. If the local rule accordingly updates the value of $\mathit{coupled}$, the dependencies on the boolean predicates could be dropped from the local rule.
As noted in the introduction the application of the local rule in its general form might lead to indeterministic behavior. The next subsection introduces two restrictions of the general model that avoid indeterminism at least for finite computations. A special case of the local rule is a rule of the form $f(s(c_\leftarrow, t_\leftarrow), s(c, t_\leftarrow), s(c_\rightarrow, t_\leftarrow))$, which is the constituting rule of a one-dimensional 3-neighborhood cellular automaton. In this case, the scale-invariant cellular automaton splits up in a sequence of infinitely many nonconnected cellular automata. This shows that the scale-invariant cellular automaton model is truly an extension of the cellular automaton model and allows us to view a scale-invariant cellular automaton as an infinite sequence of interconnected cellular automata.
We now examine the signal speed that is required to communicate state changes between neighbor cells. To this end, we select the middle point of a cell as the source and the target of a signal that propagates the state change of a cell to one of its neighbor cells. A simple consideration shows that the most restricting cases are the paths from the space time points $(c_\leftarrow, t_\leftarrow)$, $(c_\uparrow, t_\uparrow)$, $(c_\swarrow, t_\searrow)$ to $(c,t)$ if not $\mathit{coupled}(t)$. The simple calculation delivers the results $1,1$, and $\frac{1}{2}$, respectively, hence a signal speed of 1 is sufficient to deliver the updates in the given timeframe. A more general examination takes also the processing time of a cell into account. If a cell in $L_k$ takes time $2^k p$ to process their inputs and if we assume a finite signal speed of $v$, the cycle time of a cell in $L_k$ must be at least $2^k (p + v)$. In sum, as long as the processing time is proportional to the diameter of a cell, we can always find a scaling factor $t \rightarrow \lambda t$, such that the scale-invariant cellular automaton has cycle times that conform to the time scale $\mathcal{T}$.
Self-similar cellular automata and indeterminism
------------------------------------------------
The construction of a hypercomputer in section \[chap:hypercomputer\] uses a simplified version of a scale-invariant cellular automaton, which we call a Self-similar Cellular Automaton.
A [*self-similar cellular automaton*]{} has the cellular space $\mathcal{C} = \{2^k \mathbbm{1} | k \in \mathbb Z\}$, the time scale $\mathcal{T} = \{2^k (i + \mathbbm{1})| i, k \in \mathbb{Z}\}$, and the finite state set $Z$. The space-time scale of a self-similar cellular automaton is the set $\mathcal{S} = \{(c,t)| c \in \mathcal{C}, t \in \mathcal{T} \mbox{ and } |c| = |t|\}$. The self-similar cellular automaton has the following local rule: for all $(c, t) \in \mathcal{S}$ $$s(c,t) =
f(
s(c_\uparrow, t_\uparrow),
s(c, t_\leftarrow),
s(c_\swarrow, t_\swarrow),
s(c_\swarrow, t_\searrow),
\mathit{coupled}(t)
)$$
The local rule $f$ is a mapping from $Z^4 \times \{0,1\}$ to $Z$. Formally, a self-similar cellular automaton $A$ is denoted by a tuple $A = (Z, f)$. By restricting the local rule of a scale-invariant cellular automaton, a self-similar cellular automaton can also be constructed from a scale-invariant cellular automaton. Consider a scale-invariant cellular automaton whose local rule does not depend on the cell neighbors $c_\leftarrow$, $c_\rightarrow$, and $c_\searrow$. Then, the resulting scale-invariant cellular automaton contains the self-similar cellular automaton as subautomaton.
We introduce the following notation for self-similar cellular automata. We index a cell $[0, 2^k)$ by the integer $-k$, that is a cell with index $k$ has a cyle time of $2^{-k}$. We call the cell $k - 1$ the upper neighbor and the cell $k + 1$ the lower neighbor of cell $k$. Time instances can be conveniently expressed as a binary number. If not stated otherwise, we use the cycle time of cell 0 as time unit.
We noted already in the introduction that the evolution of a scale-invariant cellular automaton might lead to indeterministic behavior. We offer two solutions, one based on a special quiescent state, the other one based on a dynamically growing lattice.
A state $q$ in the state set $Z$ is called a quiescent state with regard to the short-circuit evaluation, if $f(q, q, ?, ?, ?) = q$, where the question mark sign “$?$” either represents an arbitrary state or a boolean value, depending on its position. Whenever a cell is in state $q$, the cell does not access its lower neighbor.
The cell remains as long in the quiescent state as long as the upper neighbor is in the quiescent state, too. This modus of operandi corresponds to the short-circuit evaluation of logical expressions in programming languages like C or Java. If the self-similar cellular automaton starts in an initial configuration of the form $z_0 z_1 \ldots z_n q q q\ldots$ at cell $0$, the infinite regress is interrupted, since cell $n+2$ evolves to $q$ without being dependent on cell $n+3$.
Let $q$ be a state in the state set $Z$, called the quiescent state. A dynamically growing self-similar cellular automaton initially starts with the finite set of cells $0, \ldots, n$ and the following boundary condition. Whenever cell $0$ or the cell with the highest index $k$ is evolved, the state of the missing neighbor cell is assumed to be $q$. The self-similar cellular automaton dynamically appends cells to the lower end when needed: whenever the cell with the highest index $k$ enters a state that is different from the quiescent state, a new cell $k + 1$ is appended, initialized with state $q$, and connected to the cell $k$. To be more specific: If $k$ is the highest index, and cell $k$ evolves at time $2^{-k} i$ to state $z \neq q$, a new cell $k + 1$ in state $q$ is appended. The cell performs its first transition at time $2^{-k}(i + 1/2)$, assuming state $q$ for its missing lower neighbor cell.
We note that the same technique could also be applied to append upper cells to the self-similar cellular automaton, although in the remainder of this paper we only deal with self-similar cellular automata which are growing to the bottom. Both enhancements ensure a deterministic evaluation either for a configuration where only a finite number of cells is in a nonquiescent state or for a finite number of cells.
A configuration of a self-similar cellular automaton $A$ is called finite if only a finite number of cells is different from the quiescent state. Let $C$ be a finite configuration and $C^\prime$ the next configuration in the evolution that is different to $C$. $C^\prime$ is again finite. We denote this relationship by $C \vdash_{A} C^\prime$. The relation $\vdash_{A}^*$ is again the reflexive and transitive closure of $\vdash_{A}$.
A self-similar cellular automaton as a scale-invariant cellular automaton cannot halt by definition and runs forever without stopping. The closest analogue to the Turing machine halting occurs, when the configuration stays constant during evolution. Such a configuration that does not change anymore is called final.
Constructing a hypercomputer {#chap:hypercomputer}
============================
In this section, we shall construct an accelerated Turing machine based on a self-similar cellular automaton. A self-similar cellular automaton which simulates the Turing machine $\overline{M}_U$ specified in the proof of Theorem \[th-right-acc-tm\] in a step-by-step manner is a hypercomputer, since the resulting Turing machine is a right-accelerated one. We give an alternative construction, where the shift over to the right is directly embedded in the local rule of the self-similar cellular automaton. The self-similar cellular automaton will simultaneously simulate the Turing machine and shift the tape content down to faster cycling cells. The advantages of this construction are the smaller state set as well as a resulting faster simulation.
Specification
-------------
Let $M = (Q, \Sigma, \Gamma, \delta, q_0, B, F)$ be an arbitrary Turing machine. We construct a self-similar cellular automaton $A_M = (Z, f)$ that simulates $M$ as follows. First, we simplify the local rule by dropping the dependency on $t_\swarrow$, obtaining $$s(c,t) =
f(
s(c_\uparrow, t_\uparrow),
s(c, t_\leftarrow),
s(c_\swarrow, t_\searrow),
\mathit{coupled}(t)
).$$ The state set $Z$ of $A_M$ is given by $$Z = \Gamma \cup (\Gamma \times \{\rightarrow\}) \cup (Q \times \Gamma)
\cup (Q \times \Gamma \times \{\rightarrow\}) \cup
\{\Box, \blacktriangleleft, \lhd, \overrightarrow{\lhd}, \rhd, \rhd_B, \rhd_\blacktriangleleft\}.$$ We write $\overrightarrow{a}$ for an element $(a, \rightarrow)$ in $\Gamma \times \{\rightarrow\}$, $\langle q,a \rangle$ for an element $(q, a)$ in $Q \times \Gamma$, and $\overrightarrow{\langle q,a \rangle}$ for an element $(q, a, \rightarrow)$ in $Q \times \Gamma \times \{\rightarrow\}$. To simulate $M$ on the input $w=a_1 \ldots a_n$ in $\Sigma^*$, $n \geq 1$, $A_M$ is initialized with the sequence $\overrightarrow{\lhd} \langle q_0,a_1 \rangle a_2 a_3\ldots a_n\rhd $ starting at cell 0, all other cells shall be in the quiescent state $\Box$. If $w=a_1$, $A_M$ is initialized with the sequence $\overrightarrow{\lhd} \langle q_0,a_1 \rangle B\rhd $, and if $w=\epsilon$, the empty word, $A_M$ is initialized with the sequence $\overrightarrow{\lhd} \langle q_0,B \rangle B\rhd $. We denote the initial configuration by $C_0$, or by $C_0(w)$ if we want to emphasize the dependency on the input word $w$. The computation is started at time 0, i.e. the first state change of cell $k$ occurs at time $2^{-k}$.
The elements $\langle q, a \rangle$ and $\overrightarrow{\langle q, a \rangle}$ act as head of the Turing machine including the input symbol of the Turing machine that is scanned next. To accelerate the Turing machine, we have to shift down the tape content to faster cycling cells of the self-similar cellular automaton, thereby taking care that the symbols that represent the non-blank content of the Turing machine tape are kept together. We achieve this by sending a pulse, which is just a symbol from a subset of the state set, from the left delimiter $\lhd$ to the right delimiter $\rhd$ and back. Each zigzag of the pulse moves the tape content one cell downwards and triggers at least one move of the Turing machine. Furthermore a blank is inserted to the right of the simulated head if necessary. The pulse that goes down is represented by exactly one element of the form $\overrightarrow{\lhd}, \overrightarrow{a}, \overrightarrow{\langle q,a \rangle}, \rhd_B$, or $\rhd_\blacktriangleleft$, the upgoing pulse is represented by the element $\blacktriangleleft$.
The specification of the values for the local rule $f$ for all possible arguments is tedious, therefore we use the following approach. A coupled transition of two neighbor cells can perform a simultaneous state change of the two cells. If the state changes of these two neighbor cells is independent of their other neighbors, we can specify the state changes as a transformation of a state pair into another one. Let $z_1, z_2, z_1^\prime, z_2^\prime$ be elements in $Z$. We call a mapping of the form $z_1 \: z_2 \mapsto z_1^\prime \: z_2^\prime$ a block transformation. The block transformation $z_1 \: z_2 \mapsto z_1^\prime \: z_2^\prime$ defines a function mapping of the form $
f(x, z_1, z_2, 0) = f(x, z_1, z_2, 1) = z_1^\prime
$ and $
f(z_1, z_2,y, 1) = z_2^\prime
$ for all $x, y$ in $Z$. Furthermore, we will also allow block transformations that might be ambiguous for certain configurations. Consider the block transformations $z_1 \: z_2 \mapsto z_1^\prime \: z_2^\prime$ and $z_2 \: z_3 \mapsto z_2^{\prime\prime} \: z_3^\prime$ that might lead to an ambiguity for a configuration that contains $z_1z_2z_3$. Instead of resolving these ambiguities in a formal way, we will restrict our consideration to configurations that are unambiguous.
The evolution of the self-similar cellular automaton $A_M$ is governed by the following block transformations:
1. *Pulse moves downwards.* Set $$\overrightarrow{\lhd} \: \langle q, a \rangle \mapsto \lhd \:
\overrightarrow{\langle q, a \rangle};
\label{tr:start-state}$$ $$\overrightarrow{a} \: b \mapsto a \: \overrightarrow{b};
\label{tr:down}$$ $$\overrightarrow{\lhd} \:a \mapsto \lhd \: \overrightarrow{a}.
\label{tr:start}$$ If $\delta(q,a) = (p,c,R)$ set $$\overrightarrow{b} \: \langle q, a \rangle \mapsto b \:
\overrightarrow{\langle q, a \rangle};
\label{tr:down-to-head}$$ $$\overrightarrow{\langle q,a \rangle} \: b \mapsto c \:
\overrightarrow{\langle p, b \rangle};
\label{tr:right-2}$$ $$\overrightarrow{\langle q,a \rangle} \: \rhd \mapsto \langle q,a \rangle \:
\rhd_B.
\label{tr:down-state-right-delimiter-blank}$$ If $\delta(q,a) = (p,c,L)$ set $$\overrightarrow{b} \: \langle q, a \rangle \mapsto \langle p, b \rangle \:
\overrightarrow{c};
\label{tr:left-1}$$ $$\overrightarrow{\langle q,a \rangle} \: b \mapsto \langle q,a \rangle \:
\overrightarrow{b};
\label{tr:left-no-move}$$ $$\overrightarrow{\langle q,a \rangle} \: \rhd \mapsto \langle q,a \rangle \:
\rhd_\blacktriangleleft.
\label{tr:down-state-right-delimiter}$$ Set $$\overrightarrow{a} \: \rhd \mapsto a \: \rhd_\blacktriangleleft;
\label{tr:down-a-rhd}$$ $$\rhd_B \: \Box \mapsto B \: \rhd_\blacktriangleleft;
\label{tr:new-blank}$$ $$\rhd_\blacktriangleleft \: \Box \mapsto \blacktriangleleft \: \rhd.
\label{tr:reflection-right}$$
2. *Pulse moves upwards*. Set $$a \: \blacktriangleleft \mapsto \blacktriangleleft \: a;
\label{tr:up}$$ $$\langle q,a \rangle \: \blacktriangleleft \mapsto \blacktriangleleft \:
\langle q,a \rangle;
\label{tr:up-state}$$ $$\lhd \: \blacktriangleleft \mapsto \Box \: \overrightarrow{\lhd}.
\label{tr:up-lhd}$$
If to a certain cell no block transformation is applicable the cell shall remain in its previous state. Furthermore, we assume a short-circuit evaluation with regard to the quiescent state: $f(\Box, \Box, ?, ?) = \Box$, whereby the lower neighbor cell is not accessed.
Example
-------
------- ------------- ------------- ------------- ------------- -------------
State 0 1 $X$ $Y$ $B$
$q_0$ $(q_1,X,R)$ — — $(q_3,Y,R)$ —
$q_1$ $(q_1,0,R)$ $(q_2,Y,L)$ — $(q_1,Y,R)$ —
$q_2$ $(q_2,0,L)$ — $(q_0,X,R)$ $(q_2,Y,L)$ —
$q_3$ — — — $(q_3,Y,R)$ $(q_4,B,R)$
$q_4$ — — — — —
------- ------------- ------------- ------------- ------------- -------------
We illustrate the working of $A_M$ by a simple example. Let $L$ be the formal language consisting of strings with $n$ 0’s, followed by $n$ 1’s: $L = \{0^n1^n | n \geq 1\}$. A Turing machine that accepts this language is given by $M = (\{q_0, q_1, q_2, q_3, q_4\}, \{0,1\}, \{0,1,X,Y,B\}, \delta, q_0, B, \{q_4\})$ [@hopcroft] with the transition function depicted in Fig. \[fig:example-delta\]. Note that $L$ is a context-free language, but $M$ will serve for demonstration purposes. The computation of $M$ on input $01$ is given below: $$q_001 \vdash Xq_11 \vdash q_2XY \vdash Xq_0Y \vdash XYq_3 \vdash XYBq_4.$$ Fig. \[fig:example-hyper-sca-2\] depicts the computation of $A_M$ on the Turing machine input 01. The first column of the table specifies the time in binary base. $A_M$ performs 4 complete pulse zigzags and enters a final configuration in the fifth one after the Turing machine simulation has reached the final state $q_4$. Fig. \[fig:evolution\] depicts the space-time diagram of the computation. It shows the position of the left and right delimiter (gray) and the position of the pulse (black).
Proof
-----
We split the proof that $A_M$ is a hypercomputer into several steps. We first show that the block transformations are well-defined and the pulse is preserved during evolution. Afterwards we will prove that $A_M$ simulates $M$ correctly and we will show that $A_M$ represents an accelerating Turing machine.
Let $D = \{\overrightarrow{\lhd}, \rhd_B, \rhd_\blacktriangleleft, \overrightarrow{a}, \overrightarrow{\langle q,a \rangle} \}$ be the set of elements that represent the downgoing pulse, $U = \{\blacktriangleleft\}$ be the singleton that contains the upgoing pulse, $P = D \cup U$, and $R = Z \backslash P$ the remaining elements. The following lemma states that the block transformations are unambiguous for the set of configurations we consider and that the pulse is preserved during evolution.
If the finite configuration $C$ contains exactly one element of $P$ then the application of the block transformations \[tr:start-state\] – \[tr:up-lhd\] is unambiguous and at most one block transformation is applicable. If a configuration $C^\prime$ with $C \vdash_{A_M} C^\prime$ exists, then $C^\prime$ contains exactly one element of $P$ as well.
Note that the domains of all block transformations are pairwise disjoint. This ensures that for all pairs $z_1z_2$ in $Z \times Z$ at most one block transformation is applicable. Block transformations \[tr:start-state\] – \[tr:new-blank\] are all subsets or elements of $(D \times R) \times (R \times D)$, block transformation \[tr:reflection-right\] is element of $(D \times R) \times (U \times R)$, block transformations \[tr:up\] and \[tr:up-state\] are subsets of $(R \times U) \times (U \times R)$, and finally block transformation \[tr:up-lhd\] is element of $(R \times U) \times (R \times D)$. Since the domain is either a subset of $D \times R$ or $R \times U$ the block transformations are unambiguous if $C$ contains at most one element of $P$. A configuration $C^\prime$ with $C \vdash_{A_M} C^\prime$ must be the result of the application of exactly one block transformation. Since each block transformation preserves the pulse, $C^\prime$ contains one pulse if and only if $C$ contains one.
We introduce a mapping $\gamma$ that aims to decode a self-similar cellular automaton configuration into a Turing machine configuration. Let $C$ be a finite configuration. Then $\gamma(C)$ is the string in $(\Gamma \cup Q)^{*}$ that is formed of $C$ as following:
1. All elements in $\{\Box, \blacktriangleleft, \lhd, \overrightarrow{\lhd}, \rhd, \rhd_B, \rhd_\blacktriangleleft\}$ are omitted.
2. All elements of the form $\overrightarrow{a}$ are replaced by $a$ and all elements of the form $\langle q,a \rangle$ or $\overrightarrow{\langle q,a \rangle}$ are replaced by the two symbols $q$ and $a$.
3. All other elements of the form $a$ are added as they are.
4. Leading or trailing blanks of the resulting string are omitted.
The following lemma states that $A_M$ correctly simulates $M$.
Let $c_1$, $c_2$ be configurations of $M$. If $c_1 \vdash_M^* c_2$, then there exist two finite configurations $C_1$, $C_2$ of $A_M$ such that $\gamma(C_1) = c_1$, $\gamma(C_2) = c_2$, and $C_1 \vdash_{A_M}^* C_2$. Especially if the initial configuration $C_0$ of $A_M$ satisfies $\gamma(C_0) = c_1$, then there exists a finite configuration $C_2$ of $A_M$, such that $\gamma(C_2) = c_2$ and $C_0 \vdash_{A_M}^* C_2$.
If $c_1$ has the form $a_1 \ldots a_n q$ we consider without loss of generality $a_1 \ldots a_n q B$. Therefore let $c_1 = a_1 \ldots a_{i-1} q a_i \ldots a_n$. If $i < n$ or $i = n$ and $\delta_D(q, a_n) = L$ we choose $C_1 = \overrightarrow{\lhd} a_1 \ldots a_{i-1} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd$. If $i = n$ and $\delta_D(q, a_n) = R$ we insert an additional blank: $C_1 = \overrightarrow{\lhd} a_1 \ldots a_{n-1} \langle q,a_n \rangle B \rhd$. In any case $\gamma(C_1)=c_1$ holds. We show the correctness of the simulation by calculating a complete zigzag of the pulse for the start configuration: $\overrightarrow{\lhd} a_1 \ldots a_{i-1} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd$. The number of the block transformation that is applied, is written above the derivation symbol. We split the zigzag up into three phases.
1. Pulse moves down from the left delimiter to the left neighbor cell of the simulated head.
For $i > 1$ we obtain $$\begin{array}{l}
\overrightarrow{\lhd} a_1 \ldots a_{i-1} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd
\stackrel{(\ref{tr:start})}{\vdash_{A_M}}
\lhd \overrightarrow{a_1} \ldots a_{i-1} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd
\stackrel{(\ref{tr:down})}{\vdash_{A_M}} \\
\lhd a_1 \overrightarrow{a_2} \ldots a_{i-1} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd
\stackrel{(\ref{tr:down})}{\vdash_{A_M}} \ldots \stackrel{(\ref{tr:down})}{\vdash_{A_M}}
\lhd a_1 \ldots \overrightarrow{a_{i-1}} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd.
\label{der:start}
\end{array}$$
If $i = 1$ the pulse piggybacked by the left delimiter $\overrightarrow{\lhd}$ is already in the left neighbor cell of the head and this phase is omitted.
2. Downgoing pulse passes the head.
If in the beginning of the zigzag the head was to the right of the left delimiter then $$\overrightarrow{\lhd} \langle q,a_1 \rangle a_{2} \ldots a_n \rhd
\stackrel{(\ref{tr:start-state})}{\vdash_{A_M}}
\lhd \overrightarrow{\langle q,a_1 \rangle} a_{2} \ldots a_n \rhd.$$ If $\delta_D(q,a_1)=L$ no further block transformation is applicable and the configuration is final. The case $\delta_D(q,a_1)=R$ will be handled later on. We now continue the derivation \[der:start\]. If $\delta(q,a_i) = (p, b, L)$ then $$\lhd a_1 \ldots \overrightarrow{a_{i-1}} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd
\stackrel{(\ref{tr:left-1})}{\vdash_{A_M}}
\lhd a_1 \ldots \langle p,a_{i-1} \rangle \overrightarrow{b} a_{i+1} \ldots a_n \rhd.
\label{der:tm-step-left}$$
If $\delta(q,a_i) = (p, b, R)$ then $$\lhd a_1 \ldots \overrightarrow{a_{i-1}} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd
\stackrel{(\ref{tr:down-to-head})}{\vdash_{A_M}}
\lhd a_1 \ldots a_{i-1} \overrightarrow{\langle q,a_i \rangle} a_{i+1} \ldots a_n \rhd.$$ We distinguish two cases: $i < n$ and $i = n$. If $i < n$ then $$\lhd a_1 \ldots a_{i-1} \overrightarrow{\langle q,a_i \rangle} a_{i+1} \ldots a_n \rhd
\stackrel{(\ref{tr:right-2})}{\vdash_{A_M}}
\lhd a_1 \ldots a_{i-1} b \overrightarrow{\langle p,a_{i+1} \rangle} a_{i+2} \ldots a_n \rhd.
\label{pulse-passed}$$ If the next steps of $M$ are moving the head again to the right, block transformation \[tr:right-2\] will repeatedly applied, till the head changes its direction or till the head is left of the right delimiter $\rhd$. If the Turing machine $M$ changes its direction before the right delimiter is reached, we obtain $$\lhd a_1 \ldots a_{i-1} b_1 \ldots b_j \overrightarrow{\langle r ,a_{k} \rangle} a_{k+1} \ldots a_n \rhd
\stackrel{(\ref{tr:left-no-move})}{\vdash_{A_M}}
\lhd a_1 \ldots a_{i-1} b_1 \ldots b_j \langle r ,a_{k} \rangle \overrightarrow{a_{k+1}} \ldots a_n \rhd$$ or if the direction change happens just before the right delimiter then $$\lhd a_1 \ldots a_{i-1} b_1 \ldots b_j \overrightarrow{\langle r ,a_{n} \rangle} \rhd
\stackrel{(\ref{tr:down-state-right-delimiter})}{\vdash_{A_M}}
\lhd a_1 \ldots a_{i-1} b_1 \ldots b_j \langle r ,a_{n} \rangle \rhd_\blacktriangleleft.
\label{der:right}$$ If $i=n$ or if the right-moving head hits the right delimiter the derivation has the following form $$\lhd a_1 \ldots a_{n-1} \overrightarrow{\langle q,a_n \rangle} \rhd
\stackrel{(\ref{tr:down-state-right-delimiter-blank})}{\vdash_{A_M}}
\lhd a_1 \ldots a_{n-1} \langle q,a_n \rangle \rhd_B
\stackrel{(\ref{tr:new-blank})}{\vdash_{A_M}}
\lhd a_1 \ldots a_{n-1} \langle q,a_n \rangle B \rhd_\blacktriangleleft,
\label{der:new-blank}$$ which inserts a blank to the right of the simulated head.
3. Downgoing pulse is reflected and moves up.
We proceed from configurations of the form $\lhd c_1 \ldots c_{i-1} \langle p,c_{i} \rangle \overrightarrow{c_{i+1}} \ldots c_n \rhd$. Then $$\begin{array}{l}
\lhd c_1 \ldots c_{i-1} \langle p,c_{i} \rangle \overrightarrow{c_{i+1}} \ldots c_n \rhd
\stackrel{(\ref{tr:down})}{\vdash_{A_M}} \ldots \stackrel{(\ref{tr:down})}{\vdash_{A_M}}
\lhd c_1 \ldots c_{i-1} \langle p,c_{i} \rangle c_{i+1} \ldots \overrightarrow{c_n} \rhd
\stackrel{(\ref{tr:down-a-rhd})}{\vdash_{A_M}} \\
\lhd c_1 \ldots c_{i-1} \langle p,c_{i} \rangle c_{i+1} \ldots c_n \rhd_\blacktriangleleft
\stackrel{(\ref{tr:reflection-right})}{\vdash_{A_M}}
\lhd c_1 \ldots c_{i-1} \langle p,c_{i} \rangle c_{i+1} \ldots c_n \blacktriangleleft \rhd
\stackrel{(\ref{tr:up})}{\vdash_{A_M}} \ldots \stackrel{(\ref{tr:up})}{\vdash_{A_M}} \\
\lhd c_1 \ldots c_{i-1} \langle p,c_{i} \rangle \blacktriangleleft c_{i+1} \ldots c_n \rhd
\stackrel{(\ref{tr:up-state})}{\vdash_{A_M}}
\lhd c_1 \ldots c_{i-1} \blacktriangleleft \langle p,c_{i} \rangle c_{i+1} \ldots c_n \rhd
\stackrel{(\ref{tr:up})}{\vdash_{A_M}} \ldots \stackrel{(\ref{tr:up})}{\vdash_{A_M}} \\
\lhd \blacktriangleleft c_1 \ldots c_{i-1} \langle p,c_{i} \rangle c_{i+1} \ldots c_n \rhd
\stackrel{(\ref{tr:up-lhd})}{\vdash_{A_M}}
\overrightarrow{\lhd} c_1 \ldots c_{i-1} \langle p,c_{i} \rangle c_{i+1} \ldots c_n \rhd,
\end{array}
\label{der:up}$$ which finishes the zigzag. Note that the continuation of derivations \[der:right\] and \[der:new-blank\] is handled by the later part of derivation \[der:up\]. We also remark that the zigzag has shifted the whole configuration one cell downwards.
All block transformations except transformations \[tr:right-2\] and \[tr:left-1\] keep the $\gamma$-value of the configuration unchanged. Block transformations \[tr:right-2\] and \[tr:left-1\] correctly simulate one step in the calculation of the Turing machine $M$: if $C \stackrel{(\ref{tr:right-2}) or (\ref{tr:left-1})}{\vdash_{A_M}} C^\prime$, $\gamma(C)=c$, and $\gamma(C^\prime)=c^\prime$ then $c \vdash_M c^\prime$. Let $C_1^\prime$ be the resulting configuration of the zigzag. We conclude that $\gamma(C_1) \vdash_M^* \gamma(C_1^\prime)$ holds. We have chosen $C_1$ in such a way that at least one step of $M$ is performed, if $M$ does not halt, either by block transformation \[tr:right-2\] or \[tr:left-1\]. If $M$ does not halt the configuration after the zigzag is again of the form $\overrightarrow{\lhd} a_1 \ldots a_{i-1} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd$. The case $i = n$ and $\delta_D(q,a_n) = R$ is excluded by derivation \[der:new-blank\], which inserts a blank to the right of the head, if $\delta_D(q,a_n) = R$. This means that $C_1^\prime$ has the same form as $C_1$ and that any subsequent zigzag will perform at least one step of $M$ as well if $M$ does not halt.
In summary, we conclude that $A_M$ reaches after a finite number of zigzags a configuration $C_2$ such that $\gamma(C_2) = c_2$. On the other hand, if $M$ halts, $A_M$ enters a final configuration since derivations \[der:tm-step-left\] or \[pulse-passed\] are not applicable anymore and the pulse cannot cross the simulated head. Since we have chosen $C_0$ to be of the same form as $C_1$ in the beginning of the proof, the addendum of the lemma regarding the initial configuration is true.
Next, the time behavior of the self-similar cellular automaton $A_M$ will be investigated.
Let $C=\overrightarrow{\lhd} a_1 \ldots a_{i-1} \langle q,a_i \rangle a_{i+1} \ldots a_n \rhd$ be a finite configuration of $A_M$ that starts in cell $k$. If $M$ does not halt, the zigzag of the pulse takes 3 cycles of cell $k$ and $A_M$ is afterwards in a finite configuration $C^\prime=\overrightarrow{\lhd} b_1 \ldots b_{j-1} \langle p,b_j \rangle b_{j+1} \ldots b_m \rhd$ that starts in cell $k + 1$.
Without loss of generality, we assume that the finite configuration starts in cell 0. We follow the zigzag of the pulse, thereby tracking all times, compare with Fig. \[fig:example-hyper-sca-2\] and Fig. \[fig:evolution\]. The pulse reaches at time 1 cell 1, and at time $\sum_{i=0}^1 2^{-i}$ cell 2. In general, the downgoing pulse reaches cell $r$ in time $\sum_{i=0}^{r-1} 2^{-i}$. At time $\sum_{i=0}^{n+1} 2^{-i}$ the cell $n+2$ changes to $\rhd_\blacktriangleleft$ which marks the reversal of direction of the pulse. The next configuration change ($\rhd_\blacktriangleleft \Box \mapsto \blacktriangleleft \rhd$) occurs at $\sum_{i=0}^{n+1} 2^{-i} + 2^{-(n+1)} = 2$. The pulse $\blacktriangleleft$ reaches cell $n+1$ in time $2 + 2^{-(n+1)}$ and in general cell $r$ in time $2 + 2^{-r}$. The final configuration change of the zigzag ($\lhd \blacktriangleleft \mapsto \Box \overrightarrow{\lhd}$) that marks also the beginning of a new pulse zigzag occurs synchronously in cell 0 and cell 1 at time 3. We remark that the overall time of the pulse zigzag remains unchanged if the simulated head inserts a blank between the two delimiters.
\[th-rca\] If $M$ halts on $w$ and $A_M$ is initialized with $C_0(w)$ then $A_M$ enters a final configuration in a time less than 6 cycles of cell 0, containing the result of the calculation between the left and right delimiter. If $M$ does not halt, $A_M$ enters after 6 cycles of cell 0 the final configuration that consists of an infinite string of the quiescent element: $\Box^\infty$.
$A_M$ needs 3 cycles of cell 0 to perform the first zigzag of the pulse. After the 3 cycles the configuration is shifted one cell downwards, starting now in cell 1. The next zigzag takes 3 cycles of cell 1 which are 3/2 cyles of cell 0, and so on. Each zigzags performs at least one step of the Turing machine $M$, if $M$ does not halt. We conclude that if $M$ halts, $A$ enters a final configuration in a time less than $\sum_{i=0}^\infty 3/2^{i} = 6$ cycles of cell 0. If $M$ does not halt, the zigzag disappears in infinity after 6 cycles of cell 0 leaving a trail of $\Box$’s behind.
If $M$ is a universal Turing machine, we immediately obtain the following result, which proves that $A_M$ is a hypercomputer for certain Turing machines $M$.
Let $M_U$ be a universal Turing machine. Then $A_{M_U}$ solves the halting problem for Turing machines.
Initialize $A_{M_U}$ with an encoded Turing machine $M$ and an input word $w$. Then $A_M$ enters a final configuration with the result of $M$ on $w$ in less than 6 cycles of cell 0 if and only if $M$ halts.
In the current form of Turing machine simulation the operator has to scan a potentially unlimited number of cells to determine whether $M$ has halted or not, which limits its practical value. If $M$ has halted, we would like to propagate at least this fact back to the upper cells. The following obvious strategy fails in a subtle way. Add a rule to $A_M$ that whenever $\langle q, a \rangle$ has no next move, replaces it by the new symbol $H$. Add the rule $f(?, ?, H, ?) = H$ to $A_M$ that propagates $H$ upwards to cell $0$. The propagation upwards is only possible if we change also the block transformation \[tr:up-lhd\] to $\lhd \blacktriangleleft \mapsto \Diamond \overrightarrow{\lhd}$, thereby introducing a new symbol $\Diamond$ that is not subject of the short-circuit evaluation. The last point, even if necessary, causes the strategy to fail, since if $A_M$ does not halt, $A_M$ is after 6 cycles in the configuration $\Diamond^\infty$ that leads to indeterministic behavior of $A_M$. This is in so far problematic, since we can not be sure whether a state $H$ in cell $0$ is really the outcome of a halting Turing machine or the result of indeterministic behavior. Instead of enhancing the self-similar cellular automaton model, we will introduce in the next section a computing model that is computational equivalent for finite computations, but avoids indeterminism for infinite computations.
Self-similar Petri nets {#chap:petri}
=======================
The evolution of a cellular automaton as well as the evolution of a self-similar cellular automaton depends on an extrinsic clock representing a global time that triggers the state changes. Since a self-similar cellular automaton cannot halt, a self-similar cellular automaton is forced to perform a state change, even if no state with a causal relationship to the previous one exists, leading to indeterministic behavior, as described in the introduction. In this section, we present a model based on Petri nets, the self-similar Petri nets, with a close resemblance to self-similar cellular automata. Even though Petri nets in general are not deterministic, there exist subclasses that are. As will be shown below, self-similar Petri Nets are deterministic. They are also capable of hypercomputing, but compared to self-similar cellular automata, their behavior differ in the limit. Whereas a self-similar cellular automaton features indeterministic behavior, the self-similar Petri net halts.
Petri nets
----------
C.A. Petri introduced Petri nets in the 1960s to study asynchronous computing systems. They are now widely used to describe and study information processing systems that are characterized as being concurrent, asynchronous, distributed, parallel, nondeterministic, and/or stochastic. It is interesting to note that very early, and clearly ahead of its time, Petri investigated the connections between physical and computational processes, see e.g., Ref. [@petri-82]. In what follows, we give a brief introduction to Petri nets to define the terminology. For a more comprehensive treatment we refer to the literature; e.g., to Ref. [@Murata89].
A Petri net is a directed, weighted, bipartite graph consisting of two kinds of nodes, called places and transitions. The weight $w(p,t)$ is the weight of the arc from place $p$ to transition $t$, $w(t,p)$ is the weight of the arc from transition $t$ to place $p$. A marking assigns to place $p$ a nonnegative integer $k$, we say that $p$ is marked with $k$ tokens. If a place $p$ is connected with a transition $t$ by an arc that goes from $p$ to $t$, $p$ is an input place of $t$, if the arc goes from $t$ to $p$, $p$ is an output place. A Petri net is changed according to the following transition (firing) rule:
1. a transition $t$ may fire if each input place $p$ of $t$ is marked with at least $w(p,t)$ tokens, and
2. a firing of an enabled transition $t$ removes $w(p,t)$ tokens from each input place $p$ of $t$, and adds $w(t,p)$ tokens to each output place $p$ of $t$.
Formally, a Petri net $N$ is a tuple $N=(P,T,F,W,M_0)$ where $P$ is the set of places, $T$ is the set of transitions, $F \subseteq (P \times T) \cup (T \times P)$ is the set of arcs, $W: F \rightarrow \mathbb{N}$ is the weight function, and $M_0: P \rightarrow \mathbb{N}$ is the initial marking.
In graphical representation, places are drawn as circles and transitions as boxes. If a place is input place of more than one transition, the Petri net becomes in general indeterministic, since a token in this place might enable more than one transition, but only one can actually fire and consume the token. The subclass of Petri nets given in the following definition avoids these conflicts and is therefore deterministic. In a standard Petri net, tokens are indistinguishable. If the Petri net model is extended so that the tokens can hold values, the Petri net is called a colored Petri net.
A marked graph is a Petri Net such that each place has exactly one input transition and exactly one output transition. A colored Petri net is a Petri net where each token has a value.
Self-similarity
---------------
It is well-known that cellular automata can be modeled as colored Petri Nets. To do this, each cell of the cellular automaton is replaced by a transition and a place for each neighbor. The neighbor transitions send their states as token values to their output places, which are the input places of the transition under consideration. The transition consumes the tokens, calculates the new state, and send its state back to its neighbors. A similar construction can be done for self-similar cellular automata, leading to the class of self-similar Petri nets.
![\[petri\] Underlying graph of a self-similar Petri net.](2008-sica-RcaPetri.eps)
A [*self-similar Petri net*]{} is a colored Petri net with some extensions. A self-similar Petri net has the underlying graph partitioned into cells that is depicted in Fig. \[petri\]. We denote the transition of cell $n$ by $t(n)$, the place to the left of the transition by $p_l(n)$, the place to the right of the transition by $p_r(n)$ and the central place, in the figure the place above the transition, by $p_c(n)$. Let $Z$ be a finite set, the state set, $q \in Z$ be the quiescent state, and $f$ be a (partial) function $Z^4 \times \{0,1\} \rightarrow Z$. The set $V=Z \cup (\{0,1\} \times Z)$ is the value set of the tokens. Tokens are added to a place and consumed from the place according to a first-in first-out order. Initially, the self-similar Petri net starts with a finite number of cells $0, 1, \ldots, n$, and is allowed to grow to the right. The notation $p \leftarrow z$ defines the following action: create a token with value $z$ and add it to place $p$. The firing rule for a transition in cell $n$ of a self-similar Petri net extends the firing rule of a standard Petri net in the following way:
1. If the transition $t(n)$ is enabled, the transition removes token $\mathit{Tk}_l$ from place $p_l(n)$, token $\mathit{Tk}_c$ from $p_c(n)$ and tokens $\mathit{Tk}_{r1}, \mathit{Tk}_{r2}$ from $p_r(n)$. The value of token $\mathit{Tk}_{l}$ shall be of the form $(\mathit{coupled}, z_l)$ in $V=\{0,1\} \times Z$, the other token values $z_c, z_{r1}$ and $z_{r2}$ shall be in $Z$. If the tokens do not conform, the behavior of the transition is undefined.
2. The transition calculates $z = f(z_l, z_c, z_{r1}, z_{r2}, \mathit{coupled})$.
3. *(Left boundary cell)* If $n = 0$ then $p_l(0) \leftarrow (\neg coupled, q)$, $p_c(0) \leftarrow z$, $p_l(1) \leftarrow (0, z)$, $p_l(1) \leftarrow (1,z)$.
4. *(Inner cell)* If $n > 0$ and $n$ is not the highest index, then: $p_r(n-1) \leftarrow z$, $p_c(n) \leftarrow z$, $p_l(n+1) \leftarrow (0, z)$, $p_l(n+1) \leftarrow (1, z)$.
5. *(Right boundary cell)* If $n$ is the highest index then:
1. *(Quiescent state)* \[firing-rule-quiescent\] If $z = q$ then $p_r(n-1) \leftarrow q$, $p_c(n) \leftarrow q$, $p_r(n) \leftarrow q$, $p_r(n) \leftarrow q$
2. *(New cell allocation)* If $z \neq q$ then a new cell $n + 1$ is created and connected to cell $n$. Furthermore: $p_r(n-1) \leftarrow z$, $p_c(n) \leftarrow z$, $p_r(n) \leftarrow q$, $p_l(n+1) \leftarrow (0, z)$, $p_l(n+1) \leftarrow (1, z)$, $p_c(n + 1) \leftarrow q$, $p_r(n + 1) \leftarrow q$, $p_r(n + 1) \leftarrow q$.
Formally, we denote the self-similar Petri net by a tuple $N = (Z, f)$.
A self-similar Petri net is a marked graph and therefore deterministic. The initial markup is chosen in such a way that initially only the rightmost transition is enabled.
(Initial markup) Let $a_0 a_1 \ldots a_m$ be an input word in $Z^{m+1}$ and let $N$ be a self-similar Petri net with $n$ cells, whereby $n > m + 1$. The initial markup of the Petri net is as follows:
- $p_l(0) \leftarrow (0, q)$, ($p_l(i) \leftarrow (0, a_{i-1})$, $p_l(i) \leftarrow (1, a_{i-1})$) for $0 < i \leq m + 1$, ($p_l(i) \leftarrow (0, q)$, $p_l(i) \leftarrow (1, q)$) for $i > m + 1$
- $p_c(i) \leftarrow a_i$ for $i \leq m$, $p_c(i) \leftarrow q$ for $i > m$,
- $p_r(i) \leftarrow a_{i+1}$ for $i < m$, $p_r(i) \leftarrow q$ for $i \geq m$, and $p_r(n) \leftarrow q$.
Note that the place $p_r(n)$ is initialized with two tokens. We identify the state of a cell with the value of its $p_c$-token. If $p_c$ is empty, because the transition is in the process of firing, the state shall be the value of the last consumed token of $p_c$.
![\[token-flow\] Token flow in a self-similar Petri net.](2008-sica-TokenFlow.eps)
Fig. \[token-flow\] depicts the token flow of a self-similar Petri net consisting of $4$ cells under the assumption that the self-similar Petri net does not grow. Tokens that are created and consumed by the same cell are not shown. The numbers indicate whether the firing is uncoupled (0) or coupled (1). The only transition that is enabled in the begin is $t(3)$, since $p_r(3)$ was initialized with 2 tokens. The firing of $t(3)$ bootstraps the self-similar Petri net by adding a second token to $p_r(2)$, thereby enabling $t(2)$, and so on, until all transitions have fired, and the token flow enters periodic behavior.
Comparison of self-similar cellular automata and self-similar Petri nets
------------------------------------------------------------------------
We now compare self-similar Petri nets with self-similar cellular automata. We call a computation finite, if it involves either only a finite number of state updates of a self-similar cellular automaton, or a finite number of transition firings of a self-similar Petri net, respectively.
\[lemma:comp-equivalence\] For finite computations, a dynamically growing self-similar cellular automaton $A = (Z, f)$ and a self-similar Petri net $N = (Z, f)$ are computationally equivalent on a step-by-step basis if the start with the same number of cells and the same initial configuration.
Let $N$ be a self-similar Petri net which has $n$ cells initially. For the sake of the proof consider an enhanced self-similar Petri net $N^\prime$ that is able to timestamp its token. A token $\mathit{Tk}$ of $N^\prime$ does not hold only a value, but also a time interval. We refer to the time interval of $\mathit{Tk}$ by $\mathit{Tk}.t$ and to the value of $\mathit{Tk}$ by $\mathit{Tk}.v$. We remark that the timestamps serve only to compare the computations of a self-similar cellular automaton and a self-similar Petri net and do not imply any time behavior of the self-similar Petri net. The firing rule of $N^\prime$ works as for $N$, but has an additional pre- and postprocessing step:
- *(Preprocessing)* Let $\mathit{Tk}_c$, $\mathit{Tk}_l$, $\mathit{Tk}_{r1}$, and $\mathit{Tk}_{r2}$ be the consumed token, where the alphabetical subscript denotes the input place and the numerical subscript the order in which the tokens were consumed. Calculate $t = (\mathit{Tk}_c.t)_\rightarrow$, where $\rightarrow$ is the inverse time operator of $\leftarrow$. If $\mathit{Tk}_{r1}.t \neq t_\swarrow$ or $\mathit{Tk}_{r2}.t \neq t_\searrow$ or $\mathit{Tk}_{l}.t \neq t_\uparrow$ the firing fails and the transition becomes permanently disabled.
- *(Postprocessing)* For each created token $\mathit{Tk}$, set $\mathit{Tk}.t = t$.
The initial marking must set the $t$-field, otherwise the first transitions will fail. For the initial tokens in cell $k$, set $\mathit{Tk}_l.t = 2^{-k+1} \mathbbm{1}$ for both tokens in place $p_l$, $\mathit{Tk}_c.t = 2^{-k} \mathbbm{1}$, and $\mathit{Tk}_d.t = 2^{-k-1} \mathbbm{1}$. Set $\mathit{Tk}_d.t = 2^{-n-1} (1 + \mathbbm{1})$ for the second token in $p_r(n)$. The firings of cell $k$ add tokens with timestamps $2^{-k} \mathbbm{1}, 2^{-k} (2 + \mathbbm{1}), 2^{-k} (3 + \mathbbm{1}) \ldots$ to the output place $p_c(k)$. If transition $t(k)$ does not fail, the state function for the arguments $c=2^{-k} \mathbbm{1}$ and $t=2^{-k} (i + \mathbbm{1})$ is well-defined: $s^\prime(c,t) = z$ if cell $k$ has produced or was initialized in place $p_r$ with a token $\mathit{Tk}$ with $\mathit{Tk}.t = t$ and $\mathit{Tk}.v = z$. Let $s(c,t)$ be the state function of the scale-invariant cellular automaton $A$. Due to the initialization, the two state functions are defined for the first $n$ cells and first time intervals $2^{-k} \mathbbm{1}$. Assume that the values of $s$ and $s^\prime$ differ for some argument or that their domains are different. Consider the first time interval $t_1$ where the difference occurs: $s(c,t_1) \neq s^\prime(c,t_1)$, or exactly one of $s(c,t_1)$ or $s^\prime(c,t_1)$ is undefined. If there is more than one time interval choose an arbitrary one of these. Since $t_1$ was the first time interval where the state functions differ, we know that $s(c_\uparrow, {t_1}_\uparrow) = s^\prime(c_\uparrow, {t_1}_\uparrow)$, $s(c, {t_1}_\leftarrow) = s^\prime(c, {t_1}_\leftarrow)$, $s(c_\swarrow, {t_1}_\swarrow) = s^\prime(c_\swarrow, {t_1}_\swarrow)$, and $s(c_\swarrow, {t_1}_\searrow)= s^\prime(c_\swarrow, {t_1}_\searrow)$. We handle the case that the values of the state functions are different or that $s^\prime$ is undefined for $(c,t_1)$ whereas $s$ is. The other case ($s^\prime$ defined, but not $s$) can be handled analogously. If $c = 2^{-k} \mathbbm{1}$, we conclude that tokens with timestamps ${t_1}_\uparrow$, ${t_1}_\leftarrow$, ${t_1}_\swarrow$, ${t_1}_\searrow$ were sent to cell $k$, and no other tokens were sent afterwards to cell $k$, since the timestamps are created in chronological order. Hence, the precondition of the firing rule is satisfied and we conclude that $s(c,t_1) = s^\prime(c,t_1)$, which contradicts our assumption. The allocation of new cells introduces some technicalities, but the overall strategy of going back in time and concluding that the conditions for a state change or cell allocation were the same in both models works here also. We complete the proof, by the simple observation that $N$ and $N^\prime$ perform the same computation.
The proof can be simplified using the following more abstract argumentation. A comparison of Fig. \[token-flow\] with Fig. \[fig:timeops\] shows that each computation step has in both models the same causal dependencies. Since both computers use the same rule to calculate the value of a cell, respectively the value of a token, we conclude that the causal nets [@Levin81] of both computations are the same for a finite computation, and therefore both computers yield the same output, in case the computation is finite.
Timed self-similar Petri nets that hypercompute
-----------------------------------------------
A large number of different approaches to introducing time concepts to Petri nets have been proposed since the first extensions in the mid 1970s. We do not delve into the depths of the different models, but instead, define a very simple time schedule for the class of self-similar Petri nets.
A timed self-similar Petri net is a self-similar Petri net that fires as soon as the transition is enabled and where a firing of an enabled transition $t(k)$ takes the time $2^{-k}$. In the beginning of the firing, the tokens are removed from the input places, and at the end of the firing the produced tokes of the firing are simultaneously entered into the output places.
This time model can be satisfied if the cells of the timed self-similar Petri net are arranged as the cells of a self-similar cellular automaton. Under the assumption of a constant token speed, a firing time that is proportional to the cell length, and an appropriate unit of time we yield again cycle times of $2^{-k}$.
We now come back to the simulation of Turing machines and construct a hypercomputing timed self-similar Petri net, analogous to the hypercomputing self-similar cellular automaton in section \[chap:hypercomputer\]. Let $M = (Q, \Sigma, \Gamma, \delta, q_0, B, F)$ be an arbitrary Turing machine. Let $Z$ be the state set that we used in the simulation of a Turing machine by a self-similar cellular automaton, and let $f$ the local rule that is defined by the block transformations \[tr:start-state\] - \[tr:up-lhd\], without the short-circuit evaluation. By Lemma \[lemma:comp-equivalence\] we know that the timed self-similar Petri net $N_M = (Z, f)$ simulates $M$ correctly for a finite number of Turing machine steps. Hence, if $M$ halts on input $w$, $N_M$ enters a final configuration in less than 6 cyles of cell 0. We examine now the case that $M$ does not halt. A pivotal difference between a self-similar cellular automaton and a self-similar Petri net is the ability of the latter one to halt on a computation. This happens if all transitions of the self-similar Petri net are disabled.
\[lemma:apn-halting\] Let $M = (Q, \Sigma, \Gamma, \delta, q_0, B, F)$ be an arbitrary Turing machine and $w$ an input word in $\Sigma^*$. If $M$ does not halt on $w$, the timed self-similar Petri net $N_M$ halts on $C_0(w)$ after 6 cycles of cell 0.
As long as the number of cells is finite, the boundary condition \[firing-rule-quiescent\] of the firing rule adds by each firing two tokens to the $p_r$-place of the rightmost cell that successively enable all other transitions as well. This holds no longer for the infinite case. Let $M$ be a Turing machine, and $w$ an input word, such that $M$ does not halt on $w$. We consider again the travel of the pulse zigzags down to infinity for the timed self-similar Petri net $N_M$ with initial configuration $C_0(w)$, thereby tracking the marking of the $p_r$-places for times after the zigzag has passed by. The first states of cell $0$ are $\overrightarrow{\lhd}$, $\lhd$, $\lhd$, and $\Box$, including the initial one. The state $\Box$ is the result of the firing at time 3, exhausting thereby the tokens in place $p_r(0)$. At time 3 the left delimiter ($\overrightarrow{\lhd}$) of the pulse zigzag is now in cell 1. Cell 1 runs from time 3 on through the same state sequence $\overrightarrow{\lhd}$, $\lhd$, $\lhd$, and $\Box$, thereby adding in summary 4 tokens to $p_r(0)$. After creating the token with value $\Box$, $p_r(1)$ is empty as well. We conclude that after the zigzag has passed by a cell, the lower cell sends in summary 4 tokens to the upper cell, till the zigzag has left the lower cell as well. For each cell $k$ these four tokens in $p_r(k)$ enable two firings of cell $k$ thereby adding two tokens to $p_r(k-1)$. These two tokens of $p_r(k-1)$ enable again one firing of cell $k-1$ thereby adding one token to $p_r(k-2)$. We conclude that each cell fires 3 times after the zigzag has passed by and that the final marking of each $p_r$ is one. Hence, no $p_r$ has the necessary two tokens that enable the transition, therefore all transitions are disabled and $N_M$ halts at time 6.
Since $N_M$ halts for nonhalting Turing machines, there are no longer any obstacles that prevent the construction of the proposed propagation of the halting state back to upper cells. We replace block transformation \[tr:start-state\] with the following two and add one new.
If $\delta(q,a) = (p,c,R)$ set $$\overrightarrow{\lhd} \: \langle q, a \rangle \mapsto \lhd \:
\overrightarrow{\langle q, a \rangle}.
\label{tr:start-state2}$$
If $\delta(q,a) = (p,c,L)$ or $\delta(q,a)$ is not defined set $$\overrightarrow{\lhd} \: \langle q, a \rangle \mapsto \lhd \: H.
\label{tr:H}$$
If $\delta(q,a)$ is not defined set $$\overrightarrow{b} \: \langle q, a \rangle \mapsto b \: H.
\label{tr:down-to-head2}$$
The following definition propagates the state $H$ up to cell $0$: $$f(?, ?, H, ?) = H.
\label{eq:H-up}$$ We denote the resulting timed self-similar Petri net by $\overline{N}_M$. The following theorem makes use of the apparently paradoxical fact, that $\overline{N}_M$ halts if and only if the simulated Turing machine does not halt.
Let $M_U$ be a universal Turing machine. Then $\overline{N}_{M_U}$ solves the halting problem for Turing machines.
Consider a Turing machine $M$ and an input word $w$. Initialize $\overline{N}_{M_U}$ with $C_0(\langle M, w \rangle )$ where $\langle M, w \rangle$ is the encoding of $M$ and $w$. If $M$ does not halt on $w$, $\overline{N}_{M_U}$ halts at time 6 by Lemma \[lemma:apn-halting\]. If $M$ halts on $w$, then one cell of $\overline{N}_{M_U}$ enters the state $H$ by block transformation \[tr:H\] or \[tr:down-to-head2\] according to Theorem \[th-rca\] and Lemma \[lemma:comp-equivalence\] and taking the changes in $f$ into account. The mapping \[eq:H-up\] propagates $H$ up to cell 0. An easy calculation shows that cell 0 is in state $H$, in time 7 or less.
We have proven that $\overline{N}_{M_U}$ is indeed a hypercomputer without the deficiencies of the scale-invariant cellular automaton-based hypercomputer. We end this section with two remarks. The timed self-similar Petri net $N_M$ sends a flag back to the upper cells, if the simulated Turing machine halts. Strictly speaking, this is not necessary, if the operator is able to recognize whether the timed self-similar Petri net has halted or not. On the other hand, a similar construction is essential, if the operator is interested in the final tape content of the simulated Turing machine. Transferring the whole tape content of the simulated Turing machine upwards, could be achieved by implementing a second pulse that performs an upwards-moving zigzag. The construction is even simpler as the described one, since the tape content of the Turing machine becomes static as soon as the Turing machine halts.
The halting problem of Turing machines is not the only problem that can be solved by self-similar cellular automata, scale-invariant cellular automata, or timed self-similar Petri nets, but is unsolvable for Turing machines. A discussion of other problems unsolvable by Turing machines and of techniques to solve them within infinite computing machines, can be found in Davies [@Davies01].
Summary
=======
We have presented two new computing models that implement the potential infinite divisibility of physical configuration space. These models are purely information theoretic and do not take into account kinetic and other effects. With these provisos, it is possible, at least in principle, to use the potential infinite divisibility of space-time to perform hypercomputation, thereby extending the algorithmic domain to hitherto unsolvable decision problems.
Both models are composed of elementary computation primitives. The two models are closely related but are very different ontologically. A cellular automaton depends on an [*extrinsic*]{} time requiring an [*external*]{} clock and a rigid synchronization of its computing cells, whereas a Petri net implements a causal relationship leading to an [*intrinsic*]{} concept of time.
Scale-invariant cellular automata as well as self-similar Petri nets are built in the same way from their primitive building blocks. Each unit is recursively coupled with a sized-down copy of itself, potentially leading to an infinite sequence of ever decreasing units. Their close resemblance leads to a step-by-step equivalence of finite computations, yet their ontological difference yields different behaviors for the for the case that the computation involves an infinite number of units: a scale-invariant cellular automaton exhibits indeterministic behavior, whereas a self-similar Petri net halts. Two supertasks which operate identically in the finite case but differ in their limit is a puzzling observation which might question our present understanding of supertasks. This may be considered an analogy to a theorem [@Specker49] in recursive analysis about the existence of recursive monotone bounded sequences of rational numbers whose limit is not a computable number.
One striking feature of both models is their scale-invariance. The computational behavior of these models is therefore the first example for what might be called scale-invariant or self-similar computing, which might be characterized by the property that any computational space-time pattern can be arbitrary squeezed to finer and finer regions of space and time.
Although the basic definitions have been given, and elementary properties of these new models have been explored, a great number of questions remain open for future research. The construction of a hypercomputer was a first demonstration of the extraordinary computational capabilities of these models. Further investigations are necessary to determine their limits, and to relate them with the emerging field of hypercomputation [@2002-cal-pav; @ord-2002; @Davis-2004; @Doria-2006; @Davis-2006; @potgieter-06; @1011191]. Another line of research would be the investigation of their phenomenological properties, analogous to the statistical mechanics of cellular automata [@wolfram83; @wolfram-2002].
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url \#1[[\#1]{}]{}urlprefix
R. Landauer, “Computation, Measurement, Communication and Energy Dissipation,” in [*Selected Topics in Signal Processing*]{}, S. Haykin, ed. (Prentice Hall, Englewood Cliffs, NJ, 1989), p. 18.
H. S. Leff and A. F. Rex, [*Maxwell’s Demon*]{} (Princeton University Press, Princeton, 1990).
C. H. Bennett, “Logical Reversibility of Computation,” IBM Journal of Research and Development [**17**]{}, 525–532 (1973), reprinted in [@maxwell-demon pp. 197-204].
J. von Neumann, [*Theory of Self-Reproducing Automata*]{} (University of Illinois Press, Urbana, 1966), a. W. Burks, editor.
K. Zuse, [*[R]{}echnender [R]{}aum*]{} (Friedrich Vieweg & Sohn, Braunschweig, 1969), [E]{}nglish translation [@zuse-70].
E. Fredkin, “An informational process based on reversible universal cellular automata,” Physica [**D45**]{}, 254–270 (1990). http://dx.doi.org/10.1016/0167-2789(90)90186-S
T. Toffoli and N. Margolus, “Invertible cellular automata: A review,” Physica D [**45**]{}, 229–253 (1990). http://dx.doi.org/10.1016/0167-2789(90)90185-R
S. Wolfram, [*A New Kind of Science*]{} (Wolfram Media, Inc., Champaign, IL, 2002).
M. Margenstern and K. Morita, “A Polynomial Solution for 3-SAT in the Space of Cellular Automata in the Hyperbolic Plane,” Journal of Universal Computer Science [**5**]{}, 563–573 (1999). http://www.jucs.org/jucs\_5\_9/a\_polynomial\_solution\_for
J. E. Hopcroft and J. D. Ullman, [*Introduction to Automata Theory, Languages, and Computation*]{} (Addison-Wesley, Reading, MA, 1979).
K. Svozil, “Extrinsic-intrinsic concept and complementarity,” in [ *Inside versus Outside*]{}, H. Atmanspacker and G. J. Dalenoort, eds. (Springer-Verlag, Heidelberg, 1994), pp. 273–288.
T. Toffoli, “The role of the observer in uniform systems,” in [ *Applied General Systems Research, Recent Developments and Trends*]{}, G. J. Klir, ed. (Plenum Press, New York, London, 1978), pp. 395–400.
H. D. P. Lee, [*Zeno of Elea*]{} (Cambridge University Press, Cambridge, 1936).
G. S. Kirk and J. E. Raven, [*The Presocratic Philosophers*]{} (Cambridge University Press, Cambridge, 1957).
A. Gr[ü]{}nbaum, [*Modern Science and Zeno’s paradoxes*]{} (Allen and Unwin, London, 1968), 2nd edn.
W. C. Salmon, [*Zeno’s Paradoxes*]{} (Hackett Publishing Company, 1970, 2001).
H. Weyl, [*Philosophy of Mathematics and Natural Science*]{} (Princeton University Press, Princeton, 1949).
A. Gr[ü]{}nbaum, [*Philosophical problems of space and time (Boston Studies in the Philosophy of Science, vol. 12)*]{} (D. Reidel, Dordrecht/Boston, 1974), 2nd edn.
E. B. Davies, “Building Infinite Machines,” The British Journal for the Philosophy of Science [**52**]{}, 671–682 (2001). http://dx.doi.org/10.1093/bjps/52.4.671
T. Ord, “The many forms of hypercomputation,” Applied Mathematics and Computation [**178**]{}, 143–153 (2006). http://dx.doi.org/10.1016/j.amc.2005.09.076
M. Davis, “The myth of hypercomputation,” in [*Alan Turing: Life and Legacy of a Great Thinker*]{}, C. Teuscher, ed. (Springer, Berlin, 2004), pp. 195–212.
F. A. Doria and J. F. Costa, “Introduction to the special issue on hypercomputation,” Applied Mathematics and Computation [**178**]{}, 1–3 (2006). http://dx.doi.org/10.1016/j.amc.2005.09.065
M. Davis, “Why there is no such discipline as hypercomputation,” Applied Mathematics and Computation [**178**]{}, 4–7 (2006). http://dx.doi.org/10.1016/j.amc.2005.09.066
J. Durand-Lose, “Abstract Geometrical Computation for Black Hole Computation,” in [*Machines, Computations, and Universality, 4th International Conference, MCU 2004, Saint Petersburg, Russia, September 21-24, 2004, Revised Selected Papers*]{}, M. Margenstern, ed. (Springer, 2005), pp. 176–187. http://dx.doi.org/10.1007/b106980
I. N[é]{}meti and G. D[á]{}vid, “Relativistic computers and the Turing barrier,” Applied Mathematics and Computation [**178**]{}, 118–142 (2006), special Issue on Hypercomputation. http://dx.doi.org/10.1016/j.amc.2005.09.075
O. Bournez and M. L. Campagnolo, “A Survey on Continuous Time Computations,” in [*New Computational Paradigms. Changing Conceptions of What is Computable*]{}, S. Cooper, B. L[ö]{}we, and A. Sorbi, eds. (Springer Verlag, New York, 2008), pp. 383–423.
O. Shagrir, “Super-tasks, accelerating [T]{}uring machines and uncomputability,” Theoretical Computer Science [**317**]{}, 105–114 (2004). http://dx.doi.org/10.1016/j.tcs.2003.12.007
H. [Rogers, Jr.]{}, [*Theory of Recursive Functions and Effective Computability*]{} (MacGraw-Hill, New York, 1967).
P. Odifreddi, [*Classical Recursion Theory, Vol. 1*]{} (North-Holland, Amsterdam, 1989).
P. Odifreddi, [*Classical Recursion Theory, Vol. 2*]{} (North-Holland, Amsterdam, 1999).
P. H. Potgieter, “Zeno machines and hypercomputation,” Theoretical Computer Scienc [**358**]{}, 23–33 (2006). http://dx.doi.org/10.1016/j.tcs.2005.11.040
S. Wolfram, [*Theory and Application of Cellular Automata*]{} (World Scientific, Singapore, 1986).
H. Gutowitz, “Cellular Automata: Theory and Experiment,” Physica [ **D45**]{}, 3–483 (1990), previous CA conference proceedings in [*International Journal of Theoretical Physics*]{} [**21**]{}, 1982; as well as in [*Physica*]{}, [**D10**]{}, 1984 and in [**Complex Systems**]{} [**2**]{}, 1988.
A. Ilachinski, [*Cellular Automata: A Discrete Universe*]{} (World Scientific Publishing Co., Inc., River Edge, NJ, USA, 2001).
L. G. Morelli and D. H. Zanette, “Synchronization of stochastically coupled cellular automata,” Physical Review E [**58**]{}, R8–R11 (1998). http://dx.doi.org/10.1103/PhysRevE.58.R8
B. Feng and M. Ding, “Block-analyzing method in cellular automata,” Physical Review E [**52**]{}, 3566–3569 (1995). http://dx.doi.org/10.1103/PhysRevE.52.3566
L. G. Brunnet and H. Chat[é]{}, “Cellular automata on high-dimensional hypercubes,” Physical Review E [**69**]{}, 057201 (2004). http://dx.doi.org/10.1103/PhysRevE.69.057201
C. A. Petri, “State-transition structures in physics and in computation,” International Journal of Theoretical Physics [**21**]{}, 979–992 (1982). http://dx.doi.org/10.1007/BF02084163
T. Murata, “Petri nets: Properties, analysis and applications,” Proceedings of the IEEE [**77**]{}, 541–580 (1989). http://dx.doi.org/10.1109/5.24143
L. A. Levin, “Causal Nets or What is a Deterministic Computation,” Information and Control [**51**]{}, 1–19 (1981).
E. Specker, “Nicht konstruktiv beweisbare [S]{}ätze der [A]{}nalysis,” The Journal of Smbolic Logic [**14**]{}, 145–158 (1949), reprinted in [@specker-ges pp. 35–48]; [E]{}nglish translation: [*Theorems of Analysis which cannot be proven constructively*]{}.
C. S. Calude and B. Pavlov, “Coins, Quantum Measurements, and [T]{}uring’s Barrier,” Quantum Information Processing [**1**]{}, 107–127 (2002). http://arxiv.org/abs/quant-ph/0112087
T. Ord, “Hypercomputation: computing more than the Turing machine,” (2002). http://arxiv.org/abs/math/0209332
S. Wolfram, “Statistical Mechanics of Cellular Automata,” Reviews of Modern Physics [**55**]{}, 601–644 (1983).
K. Zuse, [*Calculating Space. MIT Technical Translation AZT-70-164-GEMIT*]{} (MIT (Proj. MAC), Cambridge, MA, 1970).
E. Specker, [*Selecta*]{} (Birkh[ä]{}user Verlag, Basel, 1990).
|
---
abstract: 'We present extensive multi-wavelength observations of the extremely rapidly declining Type Ic supernova, SN2005ek. Reaching a peak magnitude of $M_R = -17.3$ and decaying by $\sim 3$ mag in the first 15 days post-maximum, SN2005ek is among the fastest Type I supernovae observed to date. The spectra of SN2005ek closely resemble those of normal SN Ic, but with an accelerated evolution. There is evidence for the onset of nebular features at only nine days post-maximum. Spectroscopic modeling reveals an ejecta mass of $\sim 0.3$ M$_\odot$ that is dominated by oxygen ($\sim 80$%), while the pseudo-bolometric light curve is consistent with an explosion powered by $\sim 0.03$ M$_\odot$ of radioactive $^{56}$Ni. Although previous rapidly evolving events (e.g., SN1885A, SN1939B, SN2002bj, SN2010X) were hypothesized to be produced by the detonation of a helium shell on a white dwarf, oxygen-dominated ejecta are difficult to reconcile with this proposed mechanism. We find that the properties of SN2005ek are consistent with either the edge-lit double detonation of a low-mass white dwarf or the iron-core collapse of a massive star, stripped by binary interaction. However, if we assume that the strong spectroscopic similarity of SN2005ek to other SN Ic is an indication of a similar progenitor channel, then a white-dwarf progenitor becomes very improbable. SN2005ek may be one of the lowest mass stripped-envelope core-collapse explosions ever observed. We find that the rate of such rapidly declining Type I events is at least 1–3% of the normal SN Ia rate.'
author:
- 'M. R. Drout, A. M. Soderberg, P. A. Mazzali, J. T. Parrent, R. Margutti, D. Milisavljevic, N. E. Sanders, R. Chornock, R. J. Foley, R. P. Kirshner, A. V. Filippenko, W. Li, P. J. Brown, S. B. Cenko, S. Chakraborti, P. Challis, A. Friedman, M. Ganeshalingam, M. Hicken, C. Jensen, M. Modjaz, H. B. Perets, J. M. Silverman, D. S. Wong'
title: The Fast and Furious Decay of the Peculiar Type Ic Supernova 2005ek
---
Introduction
============
The advent of dedicated supernova (SN) searches has dramatically increased the rate at which unusual transients are discovered. In particular, high-cadence surveys have uncovered a diverse set of rapidly evolving events which reach SN luminosities (absolute magnitude between $-20$ and $-15$) but have observed properties that challenge the parameter space easily explained by traditional SN models (e.g., the collapse of the core of a massive star, or the thermonuclear disruption of a white dwarf).
The plethora of objects that have been referred to as “rapidly evolving” include both Type I (hydrogen poor) and Type II (hydrogen rich) events (see @Filippenko1997 for a review of traditional SN classifications). Although the main physical process leading to optical emission varies among supernovae, in all cases the characteristic timescale offers insight into the amount of participating material. Rapid evolution typically implies lower masses. For supernovae powered by hydrogen recombination (e.g., Types IIP, IIL) and radioactive decay (e.g., Types Ia, Ib, Ic), rapid timescales indicate a low hydrogen envelope mass and a short photon diffusion timescale, respectively. For supernovae powered by interaction with external gas (e.g., Type IIn), a rapid decline implies a steep decrease in circumstellar medium (CSM) density, and a short overall timescale implies a small radius over which this material is located.
![\[fig:host\] *Top:* R-band Palomar 60-inch (P60) image of SN2005ek, on the outskirts of UGC2526. The SN location is marked by red crosshairs. *Bottom:* P60 template image of the region around SN2005ek, taken on 2007 Aug. 26. ](Figure1.ps){width="\columnwidth"}
Among the Type I events labeled as rapidly evolving are the SN 1991bg-like SN Ia [@Filippenko92; @Leibundgut93], the “calcium-rich” transients for which SN2005E is the prototype [@Perets2010; @Kasliwal2012; @Valenti2013], and some members of the recently defined Type Iax supernovae [@Foley2013a]. They earn the title “rapidly evolving” because they decay by 1–2 mag in the first 15 days post-maximum. Many of these objects possess peak luminosities lower than those of normal SN I ($\gtrsim -15$ mag) and are thought to be powered by radioactive decay. Although their host galaxies are diverse, members of the first two classes above have exploded in elliptical galaxies. In addition, several luminous (M $\lesssim -19$ mag) transients have been observed that decay on similar timescales, but show narrow hydrogen and/or helium emission lines in their spectra, indicating that they are at least partially powered by interaction with a dense CSM. These include the Type IIn SN PTF09uj [@Ofek2010] and the Type Ibn SN1999cq [@Matheson2000].
However, the record for the most rapidly declining SN observed thus far does not belong to any of these objects. SN2002bj [@Poznanski2010] and SN2010X [@Kasliwal2010] easily outstrip them, declining by $\gtrsim 3$ mag in the first 15 days post-maximum. The ejecta masses inferred for these two events are very small ($\lesssim 0.3$ M$_\odot$), but their peak luminosities are within the typical range for SN Ib/Ic ($-19 \lesssim$ M $\lesssim -17$; @Drout2011). These two facts, coupled with the lack of hydrogen in their spectra (SN2002bj is a SN Ib, SN2010X a SN Ic), have led several authors to hypothesize that they were produced by the detonation of a helium shell on a white dwarf (a “.Ia” supernova; @Woosley1986 [@Chevalier1988; @Bildsten2007; @Shen2010; @Waldman2011; @Sim2012]).
Two potential other members of this class include SN1885A and SN1939B [see, e.g., @Perets2011; @Chevalier1988; @deVaucouleurs1985; @Leibundgut1991]. While both SN2002bj and SN2010X were found in star-forming galaxies, SN1939B exploded in an elliptical, a fact suggestive of an old progenitor system. However, while the post-maximum decline rates of these objects are similar, even the well-studied events show differences in their other properties. SN2002bj was $\sim 1.5$ mag brighter, significantly bluer, and exhibited lower expansion velocities than SN2010X. It has not yet been established whether all (or any) of these extremely rapidly declining objects belong to the same class of events.
Here we present the discovery and panchromatic follow-up observations of SN2005ek, another very rapidly declining and hydrogen-free event that closely resembles SN2010X. In §\[sec:Obs\] we present our extensive multi-wavelength observations, while in §\[sec:lc\], §\[sec:SpecModel\], §\[sec:explosion\], and §\[sec:HostProps\] we respectively describe the photometric and spectroscopic properties, explosion parameters, and host-galaxy environment of SN2005ek. Section \[sec:rates\] examines the rates of such transients. Finally, in §\[sec:theories\], we discuss progenitor channels that could lead to such a rapidly evolving explosion.
Observations {#sec:Obs}
============
Discovery
---------
SN2005ek was discovered by the Lick Observatory Supernova Search (LOSS) using the Katzman Automatic Imaging Telescope [KAIT; @Filippenko01] on 2005 Sep. 24.53 (UT dates are used throughout this paper) with an unfiltered (clear) $m \approx 17.5$ mag. The object was not detected in previous KAIT images on Sep. 18.51 to a limit of $m \approx 19$ mag, while subsequent imaging on Sep. 25.37 revealed that the transient had brightened to $m \approx 17.3$ mag [@i8604]. SN2005ek is located in the outskirts of its host galaxy, UGC 2526, with distance $D = 66.6 \pm 4.7$ Mpc[^1] and morphology Sb.
{width="90.00000%"}
@wps+05 obtained a spectrum of SN2005ek on Sep. 26 with the Shane 3-m reflector (plus Kast spectrograph) at Lick Observatory and reported that SN2005ek was a “young supernova, probably of Type Ic.” After this spectroscopic identification, we promptly initiated a panchromatic follow-up program spanning the radio, infrared (IR), optical, ultraviolet (UV), and X-ray bands.
Palomar 60-inch Imaging {#sec:p60}
-----------------------
We obtained nightly multi-band images of SN2005ek with the robotic Palomar 60-inch telescope (P60; @Cenko2006) beginning on Sep. 26.3 and spanning through Oct. 15.2. Each epoch consisted of 4–10 120 s frames in filters $B$, $V$, $R$, and $I$. All P60 images were reduced with IRAF[^2] using a custom real-time reduction pipeline [@Cenko2006]. Nightly images were combined using standard IRAF tools. Images of the transient and host galaxy constructed from P60 data are shown in Figure \[fig:host\].
For the P60 $V$ and $R$ bands we obtained template images of the region surrounding SN2005ek on 2007 Aug. 16 (bottom panel of Fig. \[fig:host\]), after the SN had faded from view. We subtracted the host-galaxy emission present in the template image using a common point-spread function (PSF) method and then performed aperture photometry on the resulting difference images. For the $B$ and $I$ bands, no suitable template images were obtained, and we therefore performed PSF photometry on our stacked images directly. A comparison of these two methods with our $V$- and $R$-band data revealed that the resulting photometry was consistent within the measured uncertainties.
In all cases, we measured the relative magnitude of the SN with respect to five field stars within the full $13' \times 13'$ P60 field of view. Absolute calibration was performed based on Sloan Digital Sky Survey (SDSS) photometry of the field stars [@Ahn2012], converted to the $BVRI$ system using the relations from @Smith2002. Our resulting P60 photometry is listed in Table \[tab:PhotomP60\] and shown as filled circles in the main panel of Figure \[fig:Photom\]. These data reveal that the light curve reaches maximum at $m_R \approx 17.4$ mag only $\sim 9$ days after the KAIT nondetection and subsequently decays very rapidly in all bands.
Palomar 200-inch Imaging
------------------------
On 2005 Oct. 11 and Dec. 5 we imaged SN2005ek with the Large Format Camera (LFC) mounted on the Palomar 200-inch (5 m) telescope in the $g'$, $r'$, and $i'$ bands (120 s expsoures). Image processing and PSF photometry were performed using standard packages within IRAF. We performed our absolute calibration using SDSS photometry of field stars, in the same manner as described above. Our resulting photometry is listed in Table \[tab:Photom\] and supplements the P60 data in the main panel of Figure \[fig:Photom\]. On our final epoch, the transient was only detected in the $i'$ band at $\sim 21.5$ mag.
Lick Observatory Imaging
------------------------
The LOSS unfiltered images of SN2005ek were reanalyzed for this work in the manner described by @Li2003. These include the discovery images (Sep. 24.5 and 25.4) as well as an additional detection on Sep. 29.5 (orange circles with asterisks in Fig. \[fig:Photom\]; also see Table \[tab:Photom\]). These detections indicate that SN2005ek was discovered on the rise, while our P60 observations, beginning on Sep. 26.3, show a continued rise for at most one day before a very rapid decline.
FLWO Imaging
------------
We supplement our photometry with the $JHK$ data presented by Modjaz ([-@Modjaz2007]; violet circles in Fig. \[fig:Photom\]). These data were obtained on the PAIRITEL telescope at the Fred Lawrence Whipple Observatory (FLWO) and are well sampled over the same time period as the P60 observations described above. In addition, @Modjaz2007 obtained an $r'$-band detection with the FLWO 1.2 m telescope on 2005 Nov. 5 (open orange circle; Fig. \[fig:Photom\]).
Swift UVOT Imaging {#sec:uvot}
-------------------
*Swift*-UVOT [@Roming05] observations of SN2005ek were triggered beginning on 2005 Sep. 29. Seven epochs were obtained in the $uvw2$, $uvm2$, $uvw1$, $u$, $b$, and/or $v$ filters over a time period of 20 days. The data were analyzed following the prescriptions of [@Brown09] and photometry is based on the UVOT photometric system of [@Poole08] with the sensitivity corrections and revised UV zeropoints of @Breeveld2011. All data are listed in Table \[tab:PhotomUVOT\]. The $uvw2$-, $uvm2$-, $uvw1$-, and $u$-band data are shown in the inset of Figure \[tab:Photom\], while the $b$- and $v$-band data are plotted as squares in the main panel. The flux from SN2005ek appears to fall off in the blue, with only upper limits obtained in the UV bands. The spectral energy distribution (SED) of SN2005ek will be examined in more detail in §\[PB\_LC\].
Optical Spectroscopy {#sec:05ekSpec}
---------------------
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Between 2005 Sep. 17 and 2005 Oct. 8, we obtained eight low-resolution spectra of SN2005ek from the FAST spectrograph [@Fabricant1998] on the FLWO 60-inch Tillinghast telescope, the Kast double spectrograph [@Miller93] on the 3-m Shane reflector at Lick Observatory, the DEIMOS spectrograph [@Faber03] mounted on the Keck-II 10-m telescope, and the 9.2-m Hobby-Eberly Telescope (HET) at McDonald Observatory. Technical details for all of our spectroscopic observations are summarized in Table \[tab:Spectra\], and the epochs on which they were obtained are marked by dashed vertical lines in Figure \[fig:Photom\].
The spectra were reduced in the manner described by @Matheson2008, @Blondin2012, and @Silverman2012. Standard IRAF routines were used to subtract the overscan region and flatfield the two-dimensional CCD frames using a combined and normalized flatfield image. One-dimensional spectra were extracted and wavelength calibrated using comparison lamps obtained immediately following each exposure. The FAST, Kast and DEIMOS spectra were flux calibrated utilizing a set of custom IDL routines which fit spectrophotometric standards to the data. In addition, these routines apply a small shift to the wavelength calibration after cross-correlating night-sky lines with a template night-sky spectrum, apply a heliocentric correction, and use the spectrophotometric standards to remove telluric absorption features from the SN spectra (see, e.g., @Matheson2000). The HET spectrum was flux calibrated in IRAF and no telluric correction was made.
All eight spectra are displayed in the top panel of Figure \[fig:SpecEvol\], where the FAST and Kast spectra have been smoothed with windows of 11 Å and 7 Å, respectively. A thorough analysis of the spectroscopic features and evolution of SN2005ek will be performed in §\[sec:SpecModel\]. The spectra closely resemble those of the rapidly declining SN2010X (bottom-left panel of Fig. \[fig:SpecEvol\]).
We obtained two spectra of the host galaxy, UGC 2526, with the Blue Channel Spectrograph [@Schmidt1989] on the 6.5-m MMT on 2011 Feb. 22 and 23. The first observation was positioned on the explosion site, while the second was on the galactic nucleus and aligned along the major axis of the galaxy. No strong nebular emission lines were detected in the explosion-site spectrum. Reduction, extraction, flux calibration, and telluric correction were performed in the manner described above, and the final spectrum centered on the galaxy nucleus is shown in the bottom-right panel of Figure \[fig:SpecEvol\]. Weak, narrow H$\alpha$ and \[N II\] emission lines are evident, along with a red continuum.
Radio Observations {#sec:vla}
-------------------
{width="67.00000%"}
We observed SN2005ek with the Very Large Array (VLA) on 2005 Sep. 29 under our Target-of-Opportunity program to study the nonthermal properties of local Type Ib/Ic supernovae[^3] . At 8.46 GHz we did not detect a coincident radio source, and we place an upper limit of $F_{\nu} \lesssim 128~\mu$Jy ($2\sigma$) on the flux density. At a distance of $\sim 67$ Mpc, this corresponds to a radio luminosity of $L_{\nu}\lesssim 7 \times 10^{26}~\rm ergs~s^{-1}~Hz^{-1}$. This value is a factor of $\sim 7$ above the peak radio luminosity observed for the subluminous radio supernova SN2007gr [@Soderberg2010].
X-ray Observations {#sec:X-ray}
-------------------
SN2005ek was also observed with the X-ray Telescope [XRT; @Burrows05]) onboard *Swift* [@Gehrels04] beginning on 2005 Sep. 28. The data were analyzed using the latest version of the HEASOFT package available at the time of writing (v. 6.13) and corresponding calibration files. Standard filtering and screening criteria were applied. All XRT data were coadded, resulting in a final 13 ks map spanning 19 days (median time of arrival = 7.44 days). No X-ray source is detected coincident with SN2005ek with a $3\sigma$ upper limit of $F_{x}<1.3\times 10^{13}~\rm{ergs\,s^{-1}\,cm^{-2}}$ (unabsorbed, 0.3–10 keV energy band). The Galactic neutral hydrogen column density in the direction of the SN is $1.0\times10^{21}~\rm{cm^{-2}}$ [@Kalberla05]. At $D \approx 67$ Mpc this yields a $3\sigma$ limit on the luminosity of $L_{\nu}\lesssim 6.9 \times 10^{40}~\rm ergs~s^{-1}~Hz^{-1}$. This value lies above the peak luminosity level for all but the most X-ray loud SN I at a similar time after explosion (e.g., SN1998bw; @Pian2000) .
Light-Curve Properties {#sec:lc}
=======================
Reddening {#dred}
----------
Reddening due to the Milky Way in the direction of UGC 2526 has a value of $E(B-V) = 0.210$ mag, according to the infrared dust maps of @Schlegel1998. In order to estimate the host-galaxy contribution to the total reddening, we examine our spectra for evidence of narrow Na I D lines, which have been shown to correlate with extinction due to dust [@Turatto2003; @Poznanski2012]. It is only in our highest signal-to-noise ratio (S/N) spectra (DEIMOS, +9 days) that we see weak Na I D absorption at the redshift of UGC 2526[^4] with an equivalent width EW$_{\mathrm{Na~I~D}} \sim 0.31$ Å. Using the empirical relation of @Poznanski2012, this implies $E(B-V)_{\mathrm{host}} \approx 0.03$ mag. Given the low level of this inferred effect, combined with the uncertainties in the Na I D relation (e.g., @Poznanski2011), we incorporate this value into our error budget for $E(B-V)_{\mathrm{tot}}$. Throughout this paper we adopt an $R_V = A_V/E(B-V) = 3.1$ Milky Way extinction curve with a total reddening value of $E(B-V) = 0.210^{+0.036}_{-0.006}$ mag.
Optical Light-Curve Evolution {#lcs}
-----------------------------
{width="90.00000%"}
In the left four panels of Figure \[fig:AppPan\] we display our $BVRI$ light curves, normalized to peak magnitude and epoch along with the $BVRI$ light curves for other SN I which have previously been referred to as fast or rapidly evolving (black circles). These include the rapid Type Ic SN1994I [@Richmond1996], the “calcium-rich” Type Ib SN2005E [@Perets2010], the SN1991bg-like Type Ia SN1998de [@Modjaz2001], and the extremely low-luminosity SN2008ha [@Foley2009; @Foley2010; @Valenti2009]. Also shown are SN2010X (@Kasliwal2010; colored stars), SN2002bj (@Poznanski2010; colored squares), SN1885A (@deVaucouleurs1985 [@Perets2011]; plus signs), and SN1939B (@Leibundgut1991 [@Perets2011]; asterisks).
From Figure \[fig:AppPan\] it is clear that SN2005ek is an outlier even among rapid Type I supernovae, decaying by $\gtrsim 3$ mag in 15 days and showing an unusually linear decline immediately post-maximum. However, our final $r'$ and $i'$ detections do show evidence for a change in slope around 20–30 days post-maximum. We can place an upper limit of $\sim 0.029$ mag day$^{-1}$ on the late-time $i'$-band slope of SN2005ek by comparing the two P200 $i'$ detections. Both the timing of this transition and the late-time slope are comparable to those of the other rapid SN I plotted in Figure \[fig:AppPan\], although SN2005ek decays by 1–2 mag more before settling onto this late-time tail.
Basic properties for the $BVRIJHK$ bands are given in Table \[tab:PhotomProps\]. We find the $R$-band peak epoch by fitting a low-order polynomial to the P60 $R$-band light curve supplemented with the Lick unfiltered photometry (which most closely mimics and is calibrated to the $R$ band; @Li2003). This yields a peak epoch (MJD) of 53639.9 $\pm$ 0.3 day. Unless otherwise noted, all phases throughout this paper are in reference to $R$-band maximum. After correcting for distance and reddening we derive peak absolute $BVRI$ magnitudes ranging from $-16.72 \pm 0.15$ ($B$ band) to $-17.38 \pm 0.15$ ($I$ band). This places SN2005ek at a peak optical magnitude very similar to SN2010X and $\sim 1.5$ mag below SN2002bj, SN1885A, and SN1939B. In the right panel of Figure \[fig:AppPan\] we compare the absolute $R$-band light curves of SN2005ek, SN2010X, and SN2002bj.
In order to quantify the rapid decline of SN2005ek we calculate the time over which the magnitude declines by a factor of $e^{-1}$ ($\tau_e$), the number of magnitudes the light curve declines in the first 15 days past maximum ($\Delta m_{15}$), and the linear decline rate in magnitudes per day. The first two quantities are calculated by linearly interpolating our data, and are measured with respect to the observed peak magnitude/date. The decline rates are estimated from linear least-square fits to the data between +2 and +16 days. Uncertainties for all properties listed in Table \[tab:PhotomProps\] were estimated using a Monte Carlo technique to produce and analyze 1000 realizations of our data[^5].
Our best constraint on the rise time of SN2005ek comes from the nondetection in a LOSS search image obtained on Sep. 18.5 (only $\sim 9$ days before the observed $R$-band maximum). Despite this relatively short time frame, the upper limit of $\sim 19$ mag only moderately constrains the explosion epoch (Figure \[fig:Photom\]). We can infer that SN2005ek rose slightly faster than its initial decline.
One of the most distinctive features of SN Ia is the tight correlation between light-curve peak magnitude and decay rate. This is in stark contrast to SN Ib/Ic, which have been shown to fill a large portion of this parameter space [@Drout2011]. In Figure \[fig:DeltaM15\] we plot peak absolute magnitude versus $\Delta m_{15}$ for SN2005ek and other supernovae of Type I.
![\[fig:Colors\] $B-V$, $V-R$, and $R-I$ color evolution for SN2005ek (stars) and other rapidly evolving SN (circles). See text for details.](Figure6.ps){width="\columnwidth"}
The left panel of Figure \[fig:DeltaM15\] displays these values as measured in the $B$ band, which allows for a comparison with the well-studied @Phillips93 relation for normal SN Ia (@Phillips1999; grey shaded region and solid black line) and the steeper relation found by @Taubenberger2008 (dark blue points and dashed black line) for “fast” SN Ia, as well as SN1939B. Also shown in this panel are the literature sample of SN Ib/Ic from @Drout2011 (red points), the SN Iax sample from @Foley2013a (light blue points), and several other peculiar Type I events. The right panel, measured in the $R$ band, allows for a comparison to the full sample of SN Ib/Ic from @Drout2011 as well as SN2010X (for which $r$ was the only well-sampled band obtained). We also include SN1885A which was observed in a photographic band which most closely resembles modern day V-band. SN2005ek falls well outside the phase space covered by normal SN Ib/Ic and is inconsistent with a simple extrapolation of either of the two SN Ia scaling relations.
Color Evolution {#Color}
----------------
In Figure \[fig:Colors\] we plot the $B-V$, $V-R$, and $R-I$ colors for SN2005ek, along with the color evolution for other rapidly evolving events. Also displayed is the Lira relation [@Phillips1999] which describes the remarkably similar $B-V$ color evolution for SN Ia between 30 and 90 days past $V$-band maximum (dashed line; top panel). In the middle panel we also show the SN Ib/Ic template color curve from @Drout2011 (grey shaded region). @Drout2011 demonstrated that dereddened SN Ib/Ic show a very similar $V-R$ color evolution with a minimum dispersion at $\sim 10$ days post-maximum.
In the first $\sim 20$ days post-maximum, the $B-V$ and $V-R$ colors of SN2005ek appear consistent with those of other rapidly evolving SN I: they exhibit a steady reddening with time and tentative evidence for the onset of a plateau between $\sim 10$ and 15 days. Although the $R-I$ colors of SN2005ek also show a steady reddening, they do so at a much steeper rate than the other events displayed in Figure \[fig:Colors\]. Because this unusual evolution is present only in $R-I$, it cannot be fully explained by a rapid cooling of the SN ejecta, but must be caused, in part, by a strong spectroscopic feature. Indeed, from Figure \[fig:SpecEvol\] we see that by +9 days the emission component of the near-IR triplet (which falls solidly inside the $I$ band) has grown substantially.
Although the colors of SN2005ek are broadly consistent with those of other SN I, they vary substantially from those of SN2002bj. SN2002bj appears much bluer than any other object in $B-V$ and actually shows very little color evolution in either $V-R$ or $R-I$ until the final epoch, when it drastically reddens.
Spectral Energy Distribution {#SED}
----------------------------
![\[fig:SED\] UV through IR spectral energy distribution of SN2005ek at three days post-maximum. Upper limits are indicated as triangles and bandpass shapes are shown in the lower panel. The best-fitting 7000 K blackbody is shown as a red line.](Figure7.ps){width="\columnwidth"}
In Figure \[fig:SED\] we plot the UV-optical-IR spectral energy distribution (SED) of SN2005ek from Sep. 30 (3 days post-maximum). Like many SN I near maximum brightness, the SED of SN2005ek peaks in the optical.
Blackbody fits to various portions of the SED yield temperatures clustered around 7000 K (Fig. \[fig:SED\], red curve). This likely represents a lower limit on the true temperature due to the strong UV line blanketing produced by iron-peak elements in the spectra of SN I (see, e.g., @Mazzali1993 [@Bongard2008]). Detailed modeling of the photospheric spectra reveals an ionization temperature of 9000–10,000 K near maximum brightness (§\[sec:SpecModel\]).
Pseudo-Bolometric Light Curve {#PB_LC}
-----------------------------
![\[fig:PseudoB\] Pseudo-bolometric light curve for SN2005ek (red stars) along with those of several other rapidly evolving supernovae (circles). The red circles show our constraints on the late-time pseudo-bolometric luminosity of SN2005ek based on single-band detections. The hatched region is meant to guide the eye.](Figure8.ps){width="\columnwidth"}
To construct a pseudo-bolometric light curve we sum our observed $BVRIJHK$ data by means of a trapezoidal interpolation and attach a blackbody tail with a temperature and radius found by fitting a Planck function to the data. For epochs where we do not possess $JHK$ data we add a factor to our summed $BVRI$ data such that the total IR contribution ranges from $\sim 20$% near maximum to $\sim 40$% at the end of our observations [@Valenti2007]. We do not include a UV correction as our [*Swift*]{} UV observations contain only upper limits. Using this method, we find a peak bolometric luminosity of $(1.2 \pm 0.2) \times 10^{42}$ ergs s$^{-1}$ and a total radiated energy between $-1$ and +16 days of $(8.2 \pm 0.3) \times 10^{47}$ ergs.
We also use the $r'$-band detection at +38 days and the $i'$-band detection at +68 days to place constraints on the late-time pseudo-bolometric evolution of SN2005ek. First, we sum the observed flux at each epoch over the width of the appropriate filter to yield estimates for the minimum bolometric luminosity at +38 and +68 days. Second, by comparing the luminosity contained in our P200 $r'i'$-band observations at +15 days to our inferred bolometric luminosity at that epoch, we find that at +15 days the $r'$ and $i'$ bands contained $\sim 15$% and $\sim 14$% of the bolometric luminosity, respectively. We use this fact to estimate a maximum pseudo-bolometric luminosity at +38 and +68 days under the assumption that the $r'$ and $i'$ contributions to the total luminosity will continue to increase. The true bolometric luminosity likely lies closer to the higher of these two constraints.
In Figure \[fig:PseudoB\] we plot our pseudo-bolometric light curve of SN2005ek (red stars) along with the pseudo-bolometric light curves of SN1994I [@Richmond1996], SN2008ha [@Moriya2010], SN2002bj [@Poznanski2010], SN2010X [@Kasliwal2010], SN2005E, and SN1998de. These last two were constructed in the manner described above from the photometry of @Perets2010 and @Modjaz2001. The bolometric curve of SN2010X was constructed by computing $\nu F_{\nu}$ in the $r$ band. A similar analysis of SN2005ek yields a pseudo-bolometric curve which is broadly consistent with our analysis above, but declines slightly more rapidly. The hatched region in Figure \[fig:PseudoB\] represents our constraints on the late-time pseudo-bolometric evolution of SN2005ek.
Spectroscopic Properties {#sec:SpecModel}
=========================
The spectra of SN2005ek shown in Figure \[fig:SpecEvol\] show considerable evolution over a short time period, and most closely resemble those of normal SN Ic. In Figure \[fig:Icspec\] we compare the maximum-light and transitional[^6] spectra of SN2005ek to a set of SN Ic (SN2010X, SN1994I, SN2007gr, and SN2004aw). SN2005ek reaches the transitional phase much faster than the other events but the spectroscopic similarities at both epochs are clear. Of the events displayed in Figure \[fig:Icspec\], only SN2010X decays on a timescale similar to that of SN2005ek. The others possess $\Delta m_{15,R}$ values ranging from $\sim 1.4$ (SN1994I) to $\sim 0.4$ (SN2004aw).
In this section, we examine the spectroscopic properties and evolution of SN2005ek utilizing two modeling techniques. Initial line identifications and estimates of photospheric velocities were made with the spectral synthesis code SYN++ [@Thomas2011][^7]. This analysis offers information about the ions present in a particular ionization state and spectral range, allowing one to cover a large parameter space with minimal time and computational resources. In addition, we model a subset of the spectra using a one-dimensional Monte Carlo radiative transport code developed for SN outflows [@Mazzali1993; @Lucy1999; @Mazzali2000], which allows quantitative assessments of ion abundances to be made. Using the results of both techniques, we compare several distinctive spectroscopic features of SN2005ek to those of other SN I and comment on the consequences for explosive nucleosynthesis.
SYN++ Evolution and Photospheric Velocities
-------------------------------------------
SYN++ is a parameterized spectral synthesis code which allows empirical fitting of SN spectra without [*a priori*]{} assumptions of the ejecta’s density and composition structure. It operates under the assumption of spherical symmetry, homologous expansion (radius proportional to velocity), a sharp photosphere, and a pseudo-blackbody continuum level. Line formation is due to pure resonant scattering (treated using the Sobolev approximation) and Boltzmann statistics are utilized to determine the relative line strengths for a given ion. For more details, see @Branch2002, and @Thomas2011.
In Figure \[fig:Jerod2\] we show three representative SYN++ fits, covering the evolution of SN2005ek between $-1$ and +9 days. Major spectroscopic features are labeled. In all cases, the excitation temperature was set to 10,000 K and we chose an exponential form for the optical depth profile. The photospheric velocity used in these fits ranges from $\sim 9000$ km s$^{-1}$ (day $-1$) to $\sim 7000$ km s$^{-1}$ (day +9).
![\[fig:Icspec\] Comparison of SN2005ek (red) and other well-studied SN Ic (black; SN1994I, SN2004aw, SN2007gr, and SN2010X). Strong similarities are seen. *top:* Near maximum light. The regions around $\lambda$6582 and $\lambda$7774 are shaded. *bottom:* Transitional spectra. The region around $\lambda$7774 and $\lambda$9095 are shaded.](Figure9a.eps "fig:"){width="\columnwidth"} ![\[fig:Icspec\] Comparison of SN2005ek (red) and other well-studied SN Ic (black; SN1994I, SN2004aw, SN2007gr, and SN2010X). Strong similarities are seen. *top:* Near maximum light. The regions around $\lambda$6582 and $\lambda$7774 are shaded. *bottom:* Transitional spectra. The region around $\lambda$7774 and $\lambda$9095 are shaded.](Figure9b.eps "fig:"){width="\columnwidth"}
The near-maximum-light spectra of SN2005ek can be modeled with a combination of , , , , , , and some at 8000–9000 km s$^{-1}$. S II is also included, although evidence for it is very weak. Between the $-1$ day and maximum-light models was added to describe the feature near 5700 Å. In Figure \[fig:Jerod1\] we present the individual ion components of the maximum-light model. Also displayed (blue, lower panel) is a model constructed from the same set of ions for the $-4$ day spectrum of SN2010X. This highlights the similarities between the spectra of SN2005ek and SN2010X and demonstrates that our fitting scheme is equally applicable to both events.
Despite growth in the emission component of the near-IR triplet, the +9 day spectrum of SN2005ek still shows a partial environment of resonant-line scattering, indicating it can be approximately modeled with SYN++. By this epoch, the and features found near maximum light have already faded. A majority of the features can be attributed to and , along with , , and a decent fit to $\lambda$9095 (consistent with the presence of at earlier epochs) at $\sim 7000$ km s$^{-1}$. SYN++ fits to the intermediate epochs are consistent with the decrease in velocity, increase in prominence of and features, and decrease in prominence of the and features between the maximum-light and +9 day spectra.
Abundance Modeling {#sec:Paolo}
------------------
![\[fig:Jerod2\] SYN++ model fits to the $-1$ day, maximum-light, and $+9$ day spectra of SN2005ek (red lines). Observed spectra are shown in black. Major spectroscopic features are labeled.](Figure10_lowres.eps){width="\columnwidth"}
To obtain quantitative estimates of the elemental abundances present in the ejecta of SN2005ek, we also model the $-1$ day, maximum-light, and +9 day spectra with the one-dimensional Monte Carlo radiation transport code described by @Mazzali1993, @Lucy1999, and @Mazzali2000. The code assumes spherical symmetry and that radiation is emitted as a blackbody at a lower boundary (a pseudo-photosphere). The SN ejecta are defined by a run of density vs. velocity (an “explosion model” ) and a depth-dependent set of abundances. Energy packets are allowed to interact with the ejecta gas via excitation processes and electron scattering. The state of the gas is computed according to that of the radiation using a lambda iteration and adopting the modified nebular approximation [@Mazzali1993], while the emerging spectrum is computed by formally solving the transfer equation in a final step [@Lucy1999; @Mazzali2000]. This method has been successfully applied to both SN Ia (e.g., @Mazzali2008) and SN Ib/Ic (e.g., @Sauer2006).
In order to model the rapidly evolving spectra of SN2005ek we first had to establish a reasonable explosion model. The fast light curve and moderate velocities indicate a small ejecta mass. We experimented with different possibilities, and found that a model with mass $\sim 0.3$ M$_\odot$ and $E_K \approx 2.5 \times 10^{50}$ ergs provides a reasonable match to the evolution of the spectra. The distribution of density with velocity resembles a scaled-down W7 model [@Nomoto1984].
Using this model, and assuming a rise time of 14 days, we were able to reproduce the spectroscopic evolution of SN2005ek (Fig. \[fig:Paolo\]). As the photosphere recedes inward, each model epoch constrains a larger amount of the ejected material. With spectral coverage out to +9 days we are able to probe the outer $\sim 0.2$ M$_{\odot}$ ($\sim 66$%) of the ejecta.
![\[fig:Jerod1\] *top:* SYN++ model for the maximum-light spectrum of SN2005ek, separated by ion. *bottom:* Full model (red) along with a similar model constructed for the $-4$ day spectrum of SN2010X (blue). Observed spectra are shown in black.](Figure11_lowres.eps){width="\columnwidth"}
The results are qualitatively consistent with the SYN++ line identifications given above. However, abundances do not necessarily correlate with line strength. Notably, the strength of the $\lambda$7774 line requires a high oxygen abundance, so that the composition is dominated by oxygen ($\sim 80$%). Fe is present and is responsible for the absorption near 5000Å, while Ti and Cr are also important for shaping the spectrum near 4000Å, but small abundances are sufficient for this as well as for Ca, despite the strength of the near-IR triplet absorption. Smaller fractions of Mg, C, and Si are also present; Mg II lines leave a strong imprint in the earlier spectra, especially in the red (features at 8900Å and 7600Å, the latter in a blend with the stronger O I line), but also in the blue (4481Å, which is the stronger contributor to the absorption near 4000Å).
The composition shows little or no variation between the three epochs. The ejecta are dominated by oxygen ($\sim 86$%), with intermediate mass elements such as C, Mg, Si, and limited amounts of S and Ca contributing $\sim 13.5$% and iron-peak elements (Fe, Ni, Cr, Co, Ti) contributing only $\sim 0.5$%.
Notable Spectroscopic Features
------------------------------
The ions utilized in the models above are typical of SN I. However, several features warrant further discussion in the context of SN2005ek.
### Carbon Features
The $\lambda$6582 feature near 6400 Å is noticeably prominent in the near-maximum-light spectra of SN2005ek. Its strength is comparable to the $\lambda$6355 feature. This is due in part to the relative weakness of the feature, which, in the one-dimensional Monte Carlo models presented above, is caused by a combination of the low overall Si abundance ($\sim 2$%) and the relatively low-density environment. However, the strength of the feature is unusual in its own right, and it is still present several days post-maximum (Fig. \[fig:SpecEvol\]).
![\[fig:Paolo\] One-dimensional Monte Carlo radiation transport models for the $-1$ day (top panel), maximum-light (middle panel), and $+9$ day (lower panel) spectra of SN2005ek. Observed spectra are shown in black, models in red. Regions showing excess with respect to the $+9$ day model, which may represent the onset of nebular features, are shaded (grey).](Figure12.ps){width="\columnwidth"}
In SN Ia, carbon features trace the distribution of unburned material from the carbon-oxygen (C-O) white dwarf. features are observed in $\sim$30% of SN Ia although they are rarely either this strong or after maximum light [@Howell2006; @Thomas2007; @Parrent2011; @Silverman2012b; @Folatelli2012]. In general, the detection of carbon is also considered to be unusual in SN Ic. However, all the spectra shown in Figure \[fig:Icspec\] also show evidence for a notch in this region (highlighted in grey), exemplifying the spectroscopic similarity of these objects. This feature was specifically identified as C II in SN2007gr [@Valenti2008] and SN2004aw [@Taubenberger2006], while it is more debated in SN1994I (@Wheeler1994) where features are broader and, hence, more blended. In the bottom panel of Figure \[fig:Icspec\], we also highlight the region around the feature we identify as $\lambda$9095. A similar notch is seen in SN2007gr. Note that despite the strength of the feature, the inferred carbon to oxygen abundance ratio is still small ($\sim$0.02).
### Iron-Peak Features
In the $-1$ day spectrum of SN2005ek the broad feature between 4600 Å and 5200 Å, attributed mainly to , is noticeably weaker than in the pre-maximum-light spectra of other SN Ic (top panel of Fig. \[fig:Icspec\]). The depth of the feature does increase in later epochs (Fig. \[fig:SpecEvol\] and Fig. \[fig:fastspec\]). Attempting to replicate this feature with other iron-peak elements produces less satisfactory model spectra.
The presence of a strong feature around 4400 Å is one of the distinctive spectroscopic features of the rapidly-declining SN1991bg-like SN Ia. Although is also identified in the models above, the lack of a strong absorption line (another distinctive feature of SN1991bg-like SN) distinguishes SN2005ek from these events.
### Helium Features
None of the models presented above include helium. However, although its presence is not required, our data do not completely rule out its presence. Due to the similarity in wavelength of ($\lambda$5875, $\lambda$6678) to $\lambda$5890 and $\lambda$6582, some ambiguity between ions can result (e.g., @Kasliwal2010). In our case, the lack of features between 5200 Å and 5800 Å in the $-1$ day spectrum (see Fig. \[fig:Icspec\]) would require any such helium features to have increased in strength after maximum light. This would not be inconsistent with our abundance models, where we required an increase in sodium abundance to model the feature in the +9 day spectrum. However, is difficult to excite [@Mazzali1998; @Dessart2012], especially in relatively cool radiation fields like those of SN2005ek, and we have no direct evidence for helium emission. We find that the lack of helium in our fits does not require the presence of aluminum as implied by @Kasliwal2010. Rather, a majority of the SN features can be reproduced with a combination of and .
### Nebular Features
The +9 day spectrum of SN2005ek appears to be transitional between the photospheric and nebular phases, with the Ca II near-IR feature significantly influenced by net emission. In addition, there are slight excesses with respect to both the SYN++ and Monte Carlo models near 5500 Å, 6300 Å, and 7300 Å (shaded regions, bottom panel, Fig. \[fig:Paolo\]) which may be due to the emergence of \[O I\] $\lambda$5577, \[O I\] $\lambda \lambda$6300, 6364, and \[Ca II\] $\lambda\lambda$7291, 7324 in emission. This may represent the earliest onset of nebular features in a SN I observed to date.
Comparison to SN2002bj
----------------------
The decline rate of SN2005ek is most closely matched by SN2002bj and SN2010X and in Figure \[fig:fastspec\] we compare the spectra of these three objects. While SN2010X possesses very similar spectroscopic features at both early and late times (with moderately higher photospheric velocities), the spectroscopic similarities between SN2005ek and SN2002bj are less clear.
![\[fig:fastspec\] Comparison of SN2005ek, SN2010X, and SN2002bj spectra. *top:* “Early”-time spectra. *bottom:* “Late”-time spectra. No spectra from a similar phase are available for SN2002bj.](Figure13.eps){width="\columnwidth"}
@Poznanski2010 model SN2002bj with SYNOW, finding that a majority of features can be fit with intermediate-mass elements (C, Si, S) and helium at the relatively low velocity of $\sim 4000$ km s$^{-1}$. Notably absent from their fit were any iron-peak elements. This is in stark contrast to the maximum-light spectra of SN2005ek where both iron and titanium play a significant role, and sulfur is not required. Indeed, the first spectrum of SN2002bj (top, Fig. \[fig:fastspec\]) shows a significantly bluer continuum than any of the spectra obtained for either SN2005ek or SN2010X. This spectrum was obtained seven days post-maximum. By +11 days (when a second, lower quality, spectrum was obtained), SN2002bj appears to show a distinctly different morphology in the blue (Fig. \[fig:fastspec\]). The continuum is more depressed in the region between 3000 Å and 5500 Å, with features that appear similar to those attributed to iron-peak elements in SN2005ek. This spectroscopic evolution is clearly distinct from that of SN2005ek and SN2010X.
Consequences for Nucleosynthesis {#sec:nuc}
--------------------------------
The ejecta of SN2005ek are dominated by oxygen. Unlike the “calcium-rich” objects, such as SN2005E whose ejecta was $\sim 50$% calcium [@Perets2010], this does not necessarily imply an unusual nucleosynthetic channel. Oxygen-dominated ejecta are common in models of SN Ic which are due to core collapse in a stripped C-O star. In that situation, the large oxygen abundance is due to a combination of the initial composition along with partial carbon burning. In contrast, producing oxygen-dominated ejecta via primarily helium burning (the mechanism invoked in the “.Ia” scenario) is not straightforward. @Perets2010 use a one-zone model to examine the explosive nucleosynthetic outputs of He, C, and O mixtures at several temperatures. None of their helium-dominated trials yield oxygen-dominated ejecta.
Power Source and Explosion Parameters {#sec:explosion}
=====================================
The optical light curves of normal SN Ia and SN Ib/Ic are powered mainly by the radioactive decay of $^{56}$Ni. For SN2005ek, the change in light-curve slope between +20 and +40 days (Figure \[fig:PseudoB\]) points to radioactive decay as a possible power source. However, the initial decay timescale as well as the almost linear post-maximum decline are unusual. In this section we examine whether a $^{56}$Ni power source is consistent with the observations of SN2005ek described above.
For the purposes of simplified analytic models, the bolometric light curves of SN Ib/Ic are usually divided into two regimes as follows. (a) The *photospheric phase*, when the optical depth is high and the shape of the light curve is dependent both on the rate of energy deposition and the photon diffusion time scale [@Arnett1982]. In normal SN Ib/Ic, the effects of radiative transfer result in an initial post-maximum decline rate which is *slower* than $^{56}$Ni $\rightarrow$ $^{56}$Co decay [@Drout2011]. (b) The *nebular phase*, when the optical depth has decreased and the SN luminosity is determined by the instantaneous rate of energy deposition. Normal SN Ib/Ic enter this stage at a late epoch ($\gtrsim 60$ days;@Valenti2007) when the dominant energy source is expected to be $^{56}$Co $\rightarrow$ $^{56}$Fe decay. The late-time slope of SN Ib/Ic light curves are well matched by this decay rate when the effects of incomplete gamma-ray trapping are included [@Clocchiatti1997; @Valenti2007].
The time when a SN will transition to the second of these two phases is determined in large part by the total ejecta mass and kinetic energy of the explosion. In SN2005ek, both the small inferred ejecta mass ($\sim 0.3$ M$_\odot$) and the early onset of nebular spectroscopic features indicate that the assumption of optically thick ejecta may break down within a few days of maximum light, making the models of @Arnett1982 inapplicable. Further, we note that the early portion of the pseudo-bolometric light curve appears linear, and decays at a rate of 0.15 mag day$^{-1}$, comparable to the 0.12 mag day$^{-1}$ given by $^{56}$Ni $\rightarrow$ $^{56}$Co decay. In Figure \[fig:Nickel\] we again plot the pseudo-bolometric light curve of SN2005ek. Also shown are lines which describe the decay rate of $^{56}$Ni $\rightarrow$ $^{56}$Co and $^{56}$Co $\rightarrow$ $^{56}$Fe.
With this early decay rate as motivation, we construct a model for the entire post-maximum pseudo-bolometric light curve of SN2005ek based on the instantaneous rate of energy deposition from the $^{56}$Ni $\rightarrow$ $^{56}$Co $\rightarrow$ $^{56}$Fe decay chain. The model is similar to the nebular phase model of @Valenti2007, although we allow for incomplete trapping of the gamma-rays produced from $^{56}$Ni $\rightarrow$ $^{56}$Co decay. One effect of incomplete trapping at this early phase is that the light curve should decay by a larger number of magnitudes before settling onto the $^{56}$Co tail. This prediction is in good agreement with our comparison of SN2005ek to other SN I in §\[lcs\] (see Fig. \[fig:AppPan\]).
Under these assumptions, the luminosity of the SN can be modeled as (@Valenti2007, @Sutherland1984, @Cappellaro1997; we use the notation of @Valenti2007)
$L (t) = S^{\rm Ni}(\gamma) +S^{\rm Co}(\gamma)+S^{\rm Co}_{e^+}(\gamma)+S^{\rm Co}_{e^+}({\rm KE})$,
where the four terms describe the energy due to gamma-rays from nickel decay, gamma-rays from cobalt decay, gamma-rays from the annihilation of positrons created in cobalt decay, and the kinetic energy of positrons created in cobalt decay, respectively. These are given by
$S^{\rm Ni}(\gamma) = \mathrm{M}_{\mathrm{Ni}} \epsilon_{\mathrm{Ni}} e^{-t/\tau_{\mathrm{Ni}}} (1 - e^{-F/t^2})$\
$S^{\rm Co}(\gamma) = 0.81 \times \mathcal{E}_{\mathrm{Co}} (1 - e^{-(F/t)^2})$\
$S_{e^+}^{\rm Co}(\gamma) = 0.164 \times \mathcal{E}_{\mathrm{Co}} (1 - e^{-(F/t)^2}) (1 - e^{-(G/t)^2})$\
$S_{e^+}^{\rm Co}({\rm KE}) = 0.036 \times \mathcal{E}_{\mathrm{Co}} (1 - e^{-(G/t)^2})$,
where
$\mathcal{E}_{\mathrm{Co}} = \mathrm{M}_{\mathrm{Ni}} \epsilon_{\mathrm{Co}}( e^{-t/\tau_{\mathrm{Co}}}-e^{-t/\tau_{\mathrm{Ni}}})$.
![\[fig:Nickel\] Radioactive models for the pseudo-bolometric light curve of SN2005ek. Black lines show the decay rates for $^{56}$Ni, $^{56}$Co, $^{48}$Cr, and $^{48}$V, assuming full trapping of gamma-rays. The gold curve shows the best-fit model described in the text, assuming $\sim 0.03$ M$_\odot$ of $^{56}$Ni, a luminosity which tracks the instantaneous energy input, and incomplete gamma-ray trapping.](Figure14.ps){width="\columnwidth"}
The incomplete trapping of gamma-rays and positrons is incorporated with the terms $(1 - e^{-(F/t)^2})$ and $(1 - e^{-(G/t)^2})$. $F$ and $G$ are constants such that the gamma-ray and positron optical depths decrease by a factor proportional to $t^{-2}$ as expected for an explosion in homologous expansion [@Clocchiatti1997] and are functions of the total ejecta mass, kinetic energy, and density distribution of the ejecta [@Clocchiatti1997]. Using this model with $M_{\rm Ni} = 0.03$ M$_\odot$, $F = 12.8$ days, and G $\approx 16.1F$ [@Valenti2007], we find the gold curve shown in Figure \[fig:Nickel\] which matches the early decay rate of SN2005ek and is also consistent with our late-time constraints. Adopting the parameterization of @Valenti2007 where $F \approx 32~M_{\rm ej,\odot}/\sqrt{E_{K,51}}$, this value of $F$ implies a $M_{\rm ej,\odot}/\sqrt{E_{K,51}} \approx 0.4$. Using the observed photospheric velocity near maximum ($\sim 8500$ km s$^{-1}$), we can break the degeneracy between $M_{\rm ej}$ and $E_K$ to yield explosion parameters of $M_{\rm ej} \approx 0.7$ M$_\odot$ and $E_K \approx 5.2 \times 10^{50}$ ergs. These values are a factor of two larger than those used in our spectroscopic modeling ($M_{\rm ej} = 0.3$ M$_\odot$ and $E_K = 2.5 \times 10^{50}$ ergs; §\[sec:Paolo\]), but given the number of assumptions required to extract explosion parameters from this simplified analytic model the two are relatively consistent. We adopt conservative estimates of the explosion parameters to be $M_{\rm ej} = 0.3$–0.7 M$_\odot$ and $E_K = 2.5$–5.2 $\times 10^{50}$ ergs.
It has been suggested [e.g. @Shen2010] that other radioactive decay chains such as $^{48}$Cr $\rightarrow$ $^{48}$V $\rightarrow$ $^{48}$Ti, which possess shorter decay times than $^{56}$Ni $\rightarrow$ $^{56}$Co $\rightarrow$ $^{56}$Fe, may contribute to the luminosity of rapidly evolving events. In Figure \[fig:Nickel\] we also include lines that represent the decay rates of $^{48}$Cr $\rightarrow$ $^{48}$V and $^{48}$V $\rightarrow$ $^{48}$Ti. The rapid $^{48}$Cr decay time ($\tau_{\rm Cr} = 1.3$ days) implies that by a few days post-explosion the power input should already be dominated by $^{48}$V $\rightarrow$ $^{48}$Ti decay. Although photon diffusion likely plays a role at very early times, it is difficult to reconcile this power source with the change in light-curve slope observed between +20 and +40 days. In addition, in order to fit both of our late-time luminosity constraints with $^{48}$V decay, we would require nearly full gamma-ray trapping ($\sim 0.05$ mag day$^{-1}$, five times steeper than $^{56}$Co), which is inconsistent with our low derived ejecta mass. Thus, although we cannot completely rule out some (especially early) contributions from other radioactive decay chains, we find that our observations are consistent with SN2005ek being powered by the radioactive decay of $\sim 0.03$ M$_\odot$ of $^{56}$Ni. We summarize this and our other inferred explosion parameters in Table \[tab:ExpPara\].
Host Galaxy: UGC 2526 {#sec:HostProps}
=====================
Global Properties
-----------------
SN2005ek exploded in the outskirts of UGC 2526, an edge-on spiral galaxy of morphology Sb. In Figure \[fig:hostsed\] we show the SED of UGC 2526, which was compiled from the SDSS [@Ahn2012] and IRAS [@MD2005] catalogs and supplemented with upper limits from our radio observations described in §\[sec:Obs\]. Also shown is an Sb model template from the SWIRE database [@Silva1998]. Using this template, we derive a star-formation rate (SFR) of $\sim 2$–5 M$_\odot$ yr$^{-1}$ [@Yun2002; @Kennicutt1998]. SN2010X, SN2002bj and SN1885A also exploded in star-forming galaxies.
UGC2526 has a low radio luminosity when compared to the Sb template which provides a best fit at other wavelengths. According to @Chakraborti2012 this can be explained if electrons responsible for producing the radio synchrotron emission undergo significant inverse-Compton losses. Synchrotron and inverse-Compton losses are proportional to the energy density in magnetic fields and seed photons, respectively. The energy density of a characteristic Milky-Way-like magnetic field of $\sim 5~\mu$G is $\sim 10^{-12}$ ergs cm$^{-3}$. We estimate that the bolometric luminosity of the host galaxy contributes an energy density of $\sim 3 \times 10^{-12}$ ergs cm$^{-3}$ which could lead to significant inverse-Compton losses.
![\[fig:hostsed\] SED for UGC2526 (circles) and the best-fit Sb galaxy model (grey line).](Figure15.ps){width="\columnwidth"}
We measure the metallicity of UGC2526 using our host-galaxy spectrum centered on the galaxy nucleus (§\[sec:Obs\]). H$\alpha$ and \[N II\] line fluxes were measured using the Markov Chain Monte Carlo (MCMC) technique described by @Sanders2012a and, using the relations of @Pettini2004, we find the metallicity of the UGC 2526 to be 12 $+\log{\rm (O/H)}_{\rm PP04N2} = 8.79 \pm 0.06$. This value is approximately solar[^8]. This metallicity measurement may be affected by absorption from the underlying stellar population, although H$\alpha$ should be relatively less affected than H$\beta$, resulting in a slight overestimation of the actual metallicity. For typical SN host galaxies the correction factor is on the order of 0.05 dex and in extreme cases $\sim 0.2$ dex. Additionally, the relatively high \[N II\] contribution suggests that the galaxy may be weakly active [probably a LINER; e.g., @HFS97].
Explosion-Site Properties
-------------------------
The explosion site of SN2005ek is offset nearly 30 kpc ($1.5'$) in projection from the center of UGC 2526, which possesses a major diameter of $D_{25} \approx 69$ kpc. This places SN2005ek at the extreme high end of the distribution of offsets seen for all SN subtypes [@Prieto2008; @Kelly2011], which is especially notable because SN Ib/Ic typically exhibit smaller offsets than SN II and SN Ia. This large offset, coupled with the observation of metallicity gradients in many spiral galaxies [@Zaritsky1994], implies that the explosion-site metallicity is likely lower than the value we measured in the galaxy nucleus. In contrast, SN2010X, SN2002bj, and SN1885A all exploded with low projected offsets [@Kasliwal2010; @Poznanski2010; @Perets2011].
The lack of nebular emission lines in the explosion-site spectrum allows us to place a strict limit on the amount of star formation within the $1''$ slit ($\sim 0.3$ kpc at the distance of UGC 2526). We measure a $3\sigma$ upper limit on the H$\alpha$ line flux of $3.3\times 10^{37}$ ergs s$^{-1}$, which corresponds to an upper limit on the local SFR of $2.6 \times 10^{-4}$ M$_\odot$ yr$^{-1}$ (Eq. 2, @Kennicutt1998). While this value is an order of magnitude below the mean H$\alpha$ flux measured for H II regions associated with core-collapse supernovae in the sample of @Crowther2012, negligible H$\alpha$ flux at the explosion site of a core-collapse supernova is not unprecedented. Only 21 of 39 supernovae in the @Crowther2012 sample show evidence for an associated H II region.
However, a much higher fraction of SN Ib/Ic are associated with H II regions compared to SN II ($70 \pm 26$% versus $38 \pm 11$%; @Crowther2012). This trend may reflect the fact that massive stars with $M \lesssim 12$ M$_\odot$ (which are expected to explode as SN II) have lifetimes longer than the roughly 20 Myr lifetimes of giant regions. Thus, the lack of observed emission makes the explosion site of SN2005ek more comparable to those of SN II and is consistent with a progenitor older than $\sim 20$ Myr.
Rates {#sec:rates}
=====
SN2005ek was discovered as part of the LOSS survey, which achieved a high level of completeness. From the LOSS data, @Li2011 constructed volume-limited samples of Type Ia and core-collapse supernovae out to distances of 80 Mpc and 60 Mpc, respectively, which were used to derive relative rates for various SN subtypes. Although SN2005ek was not included in this original analysis[^9], we can use the LOSS data to obtain a rough estimate for the relative rate of SN2005ek-like transients. SN2002bj was also discovered by the LOSS survey, and we derive two sets of rates below, one of which assumes SN2002bj and SN2005ek are members of the same class of objects.
The LOSS volume-limited SN Ia sample contains 74 objects and is 99% complete to a distance of 80 Mpc. In order to estimate the incompleteness correction for SN2005ek, we examine the correction factors for SN2002dk and SN2002jm, two SN1991bg-like objects with peak magnitudes similar to that of SN2005ek. @Li2011 find that both objects are $\sim 97$% complete to a distance of 80 Mpc. This correction factor is based on a combination of peak magnitude and light-curve shape, and should therefore be taken as an upper limit for the completeness factor of the rapidly declining SN2005ek. If the completeness factor of SN2005ek lies between 50% and 100%, we may estimate that the rate of such transients is $\sim 1$–2% of the SN Ia rate. If we include SN2002bj in the same category of objects as SN2005ek, this rate rises to $\sim 2$–3% of the SN Ia rate. These rates should be taken as lower limits, and are consistent with those estimated by @Poznanski2010 and @Perets2011. It should also be noted that Poisson errors in this small number regime are large.
Possible Progenitor Channels {#sec:theories}
============================
The observations and analysis presented above allow us to examine several possible progenitor models for SN2005ek. SN2005ek shows a very rapid post-maximum decline, a peak luminosity of $\sim 10^{42}$ ergs s$^{-1}$, colors which redden with time, and photospheric velocities which evolve from $\sim 9000$ km s$^{-1}$ near maximum to $\sim 7000$ km s$^{-1}$ at +9 days. Spectroscopic modeling reveals a small ejecta mass (0.3–0.7 M$_\odot$) which is predominantly oxygen ($\sim 85$%), with smaller amounts of other intermediate-mass elements (Mg, C, Si) and an explosion kinetic energy of (2.5–5.0) $\times 10^{50}$ ergs. The pseudo-bolometric light curve is consistent with an explosion powered by $\sim 0.03$ M$_\odot$ of $^{56}$Ni, assuming a non-negligible fraction of the gamma-rays escape at early times. This assumption is consistent with our low inferred ejecta mass and the emergence of nebular features at only 9 days post maximum light. Finally, both the large offset and low level of H$\alpha$ emission from the explosion site of SN2005ek are consistent with a progenitor older than $\sim 20$ Myr.
A robust progenitor model should be able to reproduce all of the above properties. In addition, if one accepts that the spectroscopic and compositional similarities between SN2005ek and other normal SN Ic imply that they should have a common class of progenitors, the model should be capable of producing a variety of observed decline rates and ejecta masses. Possible progenitor channels can be divided into two classes: those involving a white dwarf (WD) or neutron star (NS) and those involving a massive star. We examine both below.
Degenerate Objects {#sec:degen}
------------------
Explosion models involving degenerate objects make attractive models for faint, rapidly evolving transients, as they naturally predict small ejecta masses. In addition, the old stellar environment of SN2005ek is consistent with a progenitor system containing at least one degenerate object. Here we discuss several specific scenarios in the context of SN2005ek: the accretion-induced collapse of a WD, a WD-NS or NS-NS merger, and the detonation of a helium shell on a low-mass WD.
### Accretion-Induced Collapse (AIC)
Under certain circumstances, when an accreting WD nears the Chandrasekhar mass, electron capture may occur in its core, causing it to collapse to a NS rather than undergo a thermonuclear explosion [@Nomoto1991]. Modern simulations suggest that the subsequent bounce and neutrino-driven wind can lead to the ejection of a small amount of material, producing a weak, rapidly evolving transient powered by radioactive decay (e.g., @Metzger2009 [@Darbha2010; @Fryer1999; @Fryer2009]). The observational properties of these AIC transients should vary if the AIC is caused by the merger of two WDs rather than the collapse of a single degenerate object.
The single-degenerate case is unlikely to produce a transient similar to SN2005ek. The simulations of @Dessart2006 predict ejecta masses and explosion energies of $\sim 10^{-2}$ M$_\odot$ and $\sim 10^{49}$ ergs, respectively, an order of magnitude below those inferred for SN2005ek. This is evident in the far-left panel of Figure \[fig:theories\], where the theoretical light curves of Darbha et al. (2010; black lines) are significantly faster and fainter than those of SN2005ek (red stars). In addition, the predicted velocities are on the order of $0.1c$, and a majority of the ejecta is likely processed to nuclear statistical equilibrium, implying a dearth of intermediate-mass elements [@Darbha2010; @Metzger2009; @Fryer1999].
An AIC due to the merger of two WDs may be “enshrouded” [@Metzger2009] by $\sim 0.1$ M$_\odot$ of unburned material left in a remnant disk [@Yoon2007]. This material will be shock heated by the ensuing explosion, potentially synthesizing intermediate-mass elements, as well as slowing the initially rapid ejecta velocity. However, models of @Fryer2009 suggest that this heating may lead to a transient which peaks in the UV bands, inconsistent with our observations of SN2005ek.
### WD-NS/NS-NS Merger
{width="98.00000%"}
Both NS-NS and NS-WD mergers have also been theorized to produce faint optical transients. In the former case, r-process nucleosynthesis is thought to occur during the ejection of neutron-rich tidal tails, yielding a rapidly evolving transient with peak luminosities between $10^{41}$ and $10^{42}$ ergs s$^{-1}$ (@Metzger2010 [@Roberts2011]; solid line, center left panel of Figure \[fig:theories\]). However, similar to the single-degenerate AIC scenario described above, the ejecta masses are lower ($\sim 10^{-2}$ M$_\odot$), ejecta velocities are higher ($\sim 0.1c$) and the nucleosynthetic yields are inconsistent (mainly r-process elements) with our observations of SN2005ek. In addition, using improved r-process opacities @Barnes2013 find that these transients may be fainter, longer lived, and significantly redder than previously hypothesized.
In contrast, the tidal disruption of a WD by a NS or BH yields a set of explosion parameters at least broadly consistent with those observed for SN2005ek. By examining the evolution of and nucleosynthesis within the accretion disk formed during the disruption @Metzger2012 produce a set of models with ejecta masses between 0.3 and 1.0 M$_\odot$, ejecta velocities between 1000 and 5000 km s$^{-1}$, synthesized Nickel masses between $10^{-3}$ and $10^{-2}$ M$_\odot$, and peak luminosity between $10^{39}$ and $10^{41.5}$ ergs s$^{-1}$. The disruption of a larger WD yields a transient with a larger ejecta mass, expansion velocity, and nickel mass. Two example light curves are shown as dashed lines in the middle-left panel of Figure \[fig:theories\]. Although the curves fall slightly below our observations (a consequence of the slightly lower inferred nickel mass) the shape is reproduced. In addition, because the outer layers of the disk do not burn to nuclear statistical equilibrium, the final ejecta composition is at least qualitatively consistent with our results for SN2005ek (mainly O, C, Si, Mg, Fe, and S), although the exact compositional fractions may not be reproduced. In particular, in order to synthesize enough $^{56}$Ni to power SN2005ek, the current models would require a total ejecta mass greater than 1.0 M$_\odot$.
### Helium-Shell Detonation
Finally, we examine the detonation of a helium shell on the surface of a WD, a model for which numerous theoretical light curves and spectra have been produced [e.g., @Woosley1986; @Shen2010; @Fryer2009; @Waldman2011; @Sim2012] and which has been invoked to explain a number of unusual recent transients (e.g., SN2005E, @Perets2010; SN2002bj, @Poznanski2010; SN2010X, @Kasliwal2010; SN1885A @Chevalier1988). The term “.Ia,” which is often associated with this explosion mechanism, was used by @Bildsten2007 to describe the specific case where the detonation occurs after mass transfer within an AM CVn binary system. In this case, the predicted helium-shell mass at the time of detonation is relatively small ($\lesssim 0.1$ M$_\odot$), which leads to a faint ($-15 > M_R > -18$ mag) and rapidly evolving transient. In the center-right panel of Figure \[fig:theories\], we show two “.Ia” model light curves from @Shen2010. SN2005ek falls comfortably between the two.
However, despite the similarity in light-curve morphology, the “.Ia” model in its basic form fails to reproduce the inferred abundances of SN2005ek. The detonation of a predominately helium shell should yield ejecta dominated by calcium, iron-peak elements (especially titanium), and unburned helium, with a notable absence of other intermediate-mass elements [@Shen2010; @Perets2010]. As discussed in §\[sec:nuc\], it is not straightforward to produce a high oxygen abundance from helium burning, and although calcium and titanium features are present in the spectra of both SN2005ek and SN2010X, these do not correspond to high abundances.
One possible reconciliation of these issues is if the helium-shell detonation triggers a second detonation within the C-O WD. Such “double-detonation” scenarios in solar-mass WDs have been thoroughly investigated as a possible explosion mechanism for normal SN Ia. However, @Sim2012 extend this analysis to low-mass WDs, considering both core-compression and edge-lit secondary detonations. In the former case significant amounts of iron-peak elements are synthesized and a much more slowly evolving light curve is produced. However, in the latter case the main modification to the observable parameters is the production of additional intermediate-mass elements. Depending on the precise abundances synthesized and amount of unburned C-O material ejected, it is possible that this scenario can explain most of the observational properties of SN2005ek. In the far-right panel of Figure \[fig:theories\] we show two edge-lit models from @Sim2012, demonstrating that they are capable of reproducing the morphology of SN2005ek.
Massive Stars
-------------
In our above discussion, we found that both the edge-lit double detonation of a low mass WD and the tidal disruption of a WD by a NS could potentially explain the observed properties of SN2005ek. However, as discussed in §\[sec:SpecModel\], SN2005ek shows remarkable spectroscopic similarity to a number of normal SN Ic. If one assumes a single set of progenitors for these objects, and accepts their inferred ejecta masses (between 1 and 10 M$_\odot$; @Mazzali2009 [@Mazzali2013]), it is difficult to reconcile the data with a WD progenitor. We therefore now examine the possibility that the progenitor of SN2005ek was a massive star.
The inferred ejecta mass and abundances provide some constraints on the pre-explosion mass and envelope density structure of any massive-star progenitor for SN2005ek. With typical core-collapse SN producing a compact remnant having a mass $\gtrsim 1.3$ M$_\odot$ [@Fryer2001], the ejecta of SN2005ek would have only constituted a small fraction of the pre-explosion progenitor mass. At the same time, the presence of a small but non-negligible amount of Ni, Si, and Mg implies that a portion of the ejecta was nuclearly processed during the explosion. In this context, we discuss three potential massive-star explosion mechanisms for SN2005ek: an iron core-collapse SN, an electron-capture SN, and a fallback SN.
### Iron Core-Collapse SN {#sec:iron}
Stars with masses $\gtrsim 11$ M$_\odot$ are expected to proceed through silicon burning before undergoing an iron-core collapse at the ends of their lives. While some amount of fallback is expected in all core-collapse SN [@Fryer1999; @Woosley2002], for stars with initial masses $\lesssim 20$ M$_\odot$ the final remnant mass will be dominated by the mass of the iron core at the time of collapse (i.e., approximately the Chandrasekhar mass; we will examine the case of more significant fallback in §\[sec:fall\]). In this case, we would infer a pre-explosion mass for SN2005ek of $\sim 2$ M$_\odot$. This may additionally apply for stars with initial masses around $\sim 50$ M$_\odot$ in the case of strong Wolf-Rayet winds (@Woosley2002; see their Fig. 16).
The environment of SN2005ek makes the latter situation (a high-mass progenitor with exceptionally strong Wolf-Rayet winds) less likely. With a large offset from the center of its host (see §\[sec:HostProps\]) and the metallicity gradients common in spiral galaxies [@Zaritsky1994], it is likely that SN2005ek exploded in a low-metallicity environment. In this case, mass loss from massive stars should be reduced rather than enhanced [@Vink2001]. In addition, such a high-mass progenitor would necessarily be short lived, and the lack of H$\alpha$ flux at the explosion site is more consistent with a lower-mass progenitor ($\lesssim 12$ M$_\odot$; §\[sec:HostProps\]).
For a lower-mass progenitor, it would likely be necessary to invoke binary stripping as stars of this initial mass are not expected to strip their hydrogen envelopes via winds. This situation would be similar to that described by @Nomoto1994 for SN1994I. In that model, a star having an initial mass of $\sim 15$ M$_\odot$ was stripped via binary interaction, yielding a C-O star of $\sim 2$ M$_\odot$ which underwent core collapse. Recall that the spectra of SN1994I closely resembled those of SN2005ek (Fig. \[fig:Icspec\]) and its decline rate was intermediate between SN2005ek and the bulk of other SN Ib/Ic (Fig. \[fig:DeltaM15\]). The ejecta mass @Nomoto1994 derived for SN1994I was 0.88 M$_\odot$, slightly larger than that inferred for SN2005ek. Taken at face value, it would be necessary to invoke either a star having a smaller initial mass (only slightly above the lower limit for iron-core collapse, consistent with the host environment described above) or more extreme stripping to reproduce SN2005ek.
In either case, we would expect the amount of radioactive material produced to be on the low side for SN Ib/Ic (due to the small amount of material at sufficient densities) and the composition to be dominated by oxygen, along with carbon and neon. Both predictions are well matched by our observations of SN2005ek. The ejecta abundances we derive for SN2005ek are very similar to those found for SN2007gr [@Mazzali2010] and SN1994I [@Sauer2006]. If SN2005ek is due to the iron-core collapse of a massive star, it exhibits one of the lowest kinetic energies (2.5–5.2 $\times 10^{50}$ ergs) and most extreme ratios of ejecta mass to remnant mass ($M_{\rm ej}/M_{\rm remnant} < 1.0$) ever observed. However, the inferred ratio of kinetic energy to ejecta mass is similar to that of other SN Ic ($E_{K,51}/M_{\rm ej,\odot} \approx 1$).
Detailed modeling would be necessary to determine if the low kinetic energy inferred for SN2005ek is consistent with what one would expect from the iron-core collapse of a stripped low-mass progenitor. @Fryer2001 argue that although stripping the hydrogen envelope from a massive star should not significantly impact the resulting explosion energy, the same may not be true if the stripping extends into the C-O core (as is the case for SN Ic). In deriving the distribution of remnant masses, @Fryer2001 assume an explosion energy which is proportional to the C-O core mass, an assumption which is qualitatively consistent with the low kinetic energy inferred for SN2005ek.
### Electron-Capture SN
If the progenitor of SN2005ek was a relatively low-mass star, it is possible that the ensuing explosion was due to a (stripped) electron-capture SN rather than a traditional iron-core-collapse event. Stars within a narrow mass range ($\sim 8$–10 M$_\odot$; @Nomoto1987 [@Woosley1980; @Iben1997; @Kitaura2006]) are expected to undergo electron capture in their O-Ne-Mg core, decreasing pressure support, and causing the core to collapse to a NS [@Nomoto1987]. @Kitaura2006 have shown that the explosion proceeds in a similar manner to that of an iron-core-collapse SN (e.g., a delayed explosion driven by neutrinos), although a lower-energy explosion is produced and the density structure of the overlying material differs. The explosion energy found in the models of @Kitaura2006 is $\sim 10^{50}$ ergs, comparable to the kinetic energy inferred for SN2005ek. However, the envelopes surrounding O-Ne-Mg cores are expected to be relatively diffuse. This leads to an explosion which synthesizes only very small amounts of $^{56}$Ni ($\sim 10^{-3}$; @Kitaura2006 [@Bethe1985]) and ejecta which show few signs of nuclear processing. On both these points, our observations of SN2005ek are more consistent with a low-mass iron-core-collapse event than with an electron-capture SN.
### Fallback SN {#sec:fall}
Alternatively, it is possible that the low ejecta mass of SN2005ek is not caused by a low pre-explosion mass, but by a significant amount of fallback onto the proto-NS. Qualitatively, this is expected to occur when the binding energy of the outer envelope is high, although the actual fallback criteria are complex, depending on the evolution of the shock velocity within the envelope as well as parameters such as metallicity and rotation. It has been theorized that significant fallback should occur for stars with masses $\gtrsim 30$ M$_\odot$ [@Fryer2001; @MacFadyen2001]. Although the successful ejecta in such explosions can possess low kinetic energies [@MacFadyen2001], events of this sort lacking a hydrogen envelope are expected to be quite faint, as a majority of the radioactive $^{56}$Ni is synthesized in the inner portions of the ejecta (which fall back onto the proto-NS). The peak luminosity of SN2005ek would likely require a non-negligible amount of mixing prior to fallback. Additionally, the environment-based arguments against a very massive progenitor for SN2005ek (§\[sec:iron\]) still hold.
Applicability of Conclusions to Other Rapidly Declining Events
--------------------------------------------------------------
In the discussion above we found that SN2005ek could potentially be explained by either the edge-lit double detonation of a low mass white dwarf, the tidal disruption of a WD by a NS, or the iron core-collapse of stripped massive star, with the latter option preferred if one takes the strong spectroscopic similarity of SN2005ek to normal SN Ic to be an indication of a similar progenitor channel. However, we note that these conclusions may not broadly apply to all of the rapidly declining SN I in the literature to date. While SN2005ek, SN2010X, SN2002bj, SN1885A, and SN1939B all exhibit similar post-maximum decline rates and have low inferred ejecta masses ($\lesssim$ 0.3 M$_\odot$), only SN2010X possess a similar peak luminosity and spectroscopic evolution to SN2005ek. SN2002bj, SN1939B, SN1885 are all significantly ($\gtrsim$ 1.3 mag) more luminous. Coupled with their fast decline rate it is unclear whether a similar modeling scheme evoked in §\[sec:explosion\] would produce a self-consistent solution for these events. SN2002bj also shows significantly bluer colors, slower expansion velocities, and a distinct spectroscopic evolution near maximum. In addition while we argue that the explosion site of SN2005ek does not rule out the possibility of a massive star progenitor (by analogy with with explosion sites of many SN IIP), SN1939B exploded in an elliptical galaxy and SN1885A (which exploded in the bulge of M31) shows no signs of a NS in its remnant [see, @Perets2011]. In short, while the conclusions above likely also apply to the rapidly-evolving SN2010X, their validity with respect to SN2002bj, SN1885A, and SN1939B is less clear. More detailed modeling will be required to distinguish various possibilities. Such modeling is currently underway for SN2010X (Kleiser et al. *in prep.*).
Summary and Conclusions
=======================
We have presented the discovery and extensive multi-wavelength observations of the rapidly evolving Type I SN2005ek. Here we summarize our main conclusions.
- Reaching a peak of $M_R = -17.3$ mag and declining by $\sim 3$ mag in the first 15 days post-maximum, SN2005ek is one of the fastest declining SN I known thus far.
- Late-time photometric detections show a shallower decay timescale which is similar to the late-time evolution of other SN I.
- The spectra of SN2005ek closely resemble those of other SN Ic in both morphology and velocity. However, SN2005ek enters the optically thin phase at a much earlier epoch. We present evidence for the onset of nebular spectroscopic features at only 9 days post-maximum.
- The bolometric light curve of SN2005ek peaks at $\sim 10^{41}$ ergs s$^{-1}$. Its evolution is consistent with an explosion powered by $\sim 0.03$ M$_\odot$ of $^{56}$Ni, with incomplete gamma-ray trapping at early times.
- We estimate the ejecta mass and kinetic energy of SN2005ek to be 0.3–0.7 M$_\odot$ and 2.5–5.2 $\times$ 10$^{50}$ ergs, respectively.
- SN2005ek exploded in a star forming galaxy, but with a large projected offset in an area lacking strong H$\alpha$ emission.
- The ejecta of SN2005ek are dominated by oxygen ($\sim 86$%). Other intermediate-mass elements (C, Mg, Si, S, Ca) account for $\sim 13.5$% of the ejecta, while iron-peak elements make up only 0.5%. These oxygen-dominated ejecta are inconsistent with the helium-shell detonation model (“.Ia”) which has previously been invoked to explain such rapidly declining events.
- Many of the observed properties of SN2005ek could be explained by either the edge-lit double detonation of a low-mass WD or the tidal disruption of a WD by a NS. However, if we assume that the strong spectroscopic similarities between SN2005ek and other normal SN Ic (with a wide range of decline timescales and inferred ejecta masses) to be an indication of a similar progenitor channel, a WD progenitor becomes very unlikely.
- Our observations and modeling of SN2005ek are also consistent with the iron-core collapse of a low-mass star (12–15 M$_\odot$), stripped by binary interaction. In particular, the abundances derived are very similar to those found for other SN Ic. In this case, SN2005ek may possess the most extreme ratio of ejecta mass to remnant mass observed for a core-collapse SN to date. The ratio of kinetic energy to ejecta mass is similar to that of other SN Ic.
- The rate of such rapidly declining SN I is at least 1–3% of the normal SN Ia rate.
- Based on their strong photometric and spectroscopic similarities, our conclusions likely also apply to the rapidly evolving Type I SN2010X. However, despite their similar decline rates, there are several important observational differences between SN2005ek and SN2002bj, SN1939B, and SN1885A. More-detailed analysis will be required to determine if these three (more luminous) objects belong to the same class of explosions.
We thank L. Bildsten, K. Shen, and T. Janka for helpful discussions. We are grateful to the staffs at the numerous observatories where we gathered data. We kindly acknowledge Sung Park, Katsuki Shimasaki, and Tom Matheson for assisting in the acquisition of some of the observations presented here. We acknowledge useful conversations at a Sky House workshop.
MRD is supported in part by the NSF through a Graduate Research Fellowship. Support for this work was provided by the David and Lucile Packard Foundation Fellowship for Science and Engineering awarded to AMS. JMS is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1302771. ASF acknowledges support from the US National Science Foundation (NSF) under grant SES 1056580.
This work was supported in part by the National Science Foundation under Grant No. PHYS-1066293 and the hospitality of the Aspen Center for Physics. We also acknowledge the hospitality of the Kavli Institute for Theoretical Physics and partial support by the National Science Foundation under Grant No. NSF PHY11-25915.
The W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA; the observatory was made possible by the generous financial support of the W. M. Keck Foundation. Observations reported here were obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. The Hobby-Eberly Telescope (HET), is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen. The HET is named in honor of its principal benefactors, William P. Hobby and Robert E. Eberly. The Very Large Array is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. PAIRITEL is operated by the Smithsonian Astrophysical Observatory (SAO) and was made possible by a grant from the Harvard University Milton Fund, the camera loan from the University of Virginia, and the continued support of the SAO and UC Berkeley. We thank M. Skrutskie for his continued support of the PAIRITEL project.
The supernova research of A.V.F.’s group at U.C. Berkeley is supported by Gary & Cynthia Bengier, the Richard & Rhoda Goldman Fund, the Christopher R. Redlich Fund, the TABASGO Foundation, and NSF grant AST-1211916. KAIT and its ongoing work were made possible by donations from Sun Microsystems, Inc., the Hewlett-Packard Company, AutoScope Corporation, Lick Observatory, the NSF, the University of California, the Sylvia & Jim Katzman Foundation, and the TABASGO Foundation.
[121]{} natexlab\#1[\#1]{}
, C. P., [Alexandroff]{}, R., [Allende Prieto]{}, C., [et al.]{} 2012, , 203, 21
Arnett, W. D. 1982, ApJ, 253, 785
, J., & [Kasen]{}, D. 2013, ArXiv e-prints
, H. A., & [Wilson]{}, J. R. 1985, , 295, 14
Bildsten, L., Shen, K. J., Weinberg, N. N., & Nelemans, G. 2007, ApJ, 662, L95
Blondin, S., Matheson, T., Kirshner, R. P., [et al.]{} 2012, AJ, 143, 126
Bongard, S., Baron, E., Smadja, G., Branch, D., & Hauschildt, P. H. 2008, ApJ, 687, 456
Branch, D., Benetti, S., Kasen, D., [et al.]{} 2002, ApJ, 566, 1005
, A. A., [Landsman]{}, W., [Holland]{}, S. T., [et al.]{} 2011, in American Institute of Physics Conference Series, Vol. 1358, American Institute of Physics Conference Series, ed. J. E. [McEnery]{}, J. L. [Racusin]{}, & N. [Gehrels]{}, 373–376
, P. J., [Holland]{}, S. T., [Immler]{}, S., [et al.]{} 2009, , 137, 4517
, D. N., [Hill]{}, J. E., [Nousek]{}, J. A., [et al.]{} 2005, , 120, 165
, E., [Mazzali]{}, P. A., [Benetti]{}, S., [et al.]{} 1997, , 328, 203
Cenko, S. B., Fox, D. B., Moon, D.-s., [et al.]{} 2006, PASP, 118, 1396
Chakraborti, S., Yadav, N., Cardamone, C., & Ray, A. 2012, ApJ, 746, L6
, R. A., & [Plait]{}, P. C. 1988, , 331, L109
Clocchiatti, A., & Wheeler, J. C. 1997, ApJ, 491, 375
Crowther, P. a. 2012, MNRAS, 1943, 1927
Darbha, S., Metzger, B. D., Quataert, E., [et al.]{} 2010, MNRAS, 409, 846
, G., & [Corwin]{}, Jr., H. G. 1985, , 295, 287
, L., [Burrows]{}, A., [Ott]{}, C. D., [et al.]{} 2006, , 644, 1063
, L., [Hillier]{}, D. J., [Li]{}, C., & [Woosley]{}, S. 2012, , 424, 2139
Drout, M. R., Soderberg, A. M., Gal-Yam, A., [et al.]{} 2011, ApJ, 741, 97
, S. M., [Phillips]{}, A. C., [Kibrick]{}, R. I., [et al.]{} 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. M. [Iye]{} & A. F. M. [Moorwood]{}, 1657–1669
Fabricant, D., Cheimets, P., Caldwell, N., & Geary, J. 1998, PASP, 110, 79
Filippenko, A. V. 1997, ARA&A
, A. V., [Li]{}, W. D., [Treffers]{}, R. R., & [Modjaz]{}, M. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 246, IAU Colloq. 183: Small Telescope Astronomy on Global Scales, ed. B. [Paczynski]{}, W.-P. [Chen]{}, & C. [Lemme]{}, 121
, A. V., [Richmond]{}, M. W., [Branch]{}, D., [et al.]{} 1992, , 104, 1543
, G., [Phillips]{}, M. M., [Morrell]{}, N., [et al.]{} 2012, , 745, 74
Foley, R. J., Brown, P. J., Rest, A., [et al.]{} 2010, ApJ, 708, L61
Foley, R. J., Chornock, R., Filippenko, A. V., [et al.]{} 2009, AJ, 138, 376
Foley, R. J., Challis, P. J., Chornock, R., [et al.]{} 2013, ApJ, 767, 57
Fryer, C. 1999, ApJ, 522, 413
Fryer, C. L., & Kalogera, V. 2001, ApJ, 554, 548
Fryer, C. L., Brown, P. J., Bufano, F., [et al.]{} 2009, ApJ, 707, 193
, N., [Chincarini]{}, G., [Giommi]{}, P., [et al.]{} 2004, , 611, 1005
, L. C., [Filippenko]{}, A. V., & [Sargent]{}, W. L. W. 1997, , 112, 315
, D. A., [Sullivan]{}, M., [Nugent]{}, P. E., [et al.]{} 2006, , 443, 308
, Jr., I., [Ritossa]{}, C., & [Garcia-Berro]{}, E. 1997, , 489, 772
, P. M. W., [Burton]{}, W. B., [Hartmann]{}, D., [et al.]{} 2005, , 440, 775
Kasliwal, M. M., Kulkarni, S. R., Gal-Yam, A., [et al.]{} 2010, ApJ, 723, L98
—. 2012, ApJ, 755, 161
Kelly, P. L., & Kirshner, R. P. 2012, ApJ, 759, 107
Kennicutt, R. C. 1998, ARA&A, 36, 189
, H., & [Li]{}, W. 2005, Central Bureau Electronic Telegrams, 232, 1
Kitaura, F. S., Janka, H.-T., & Hillebrandt, W. 2006, A&A, 450, 345
, B., [Tammann]{}, G. A., [Cadonau]{}, R., & [Cerrito]{}, D. 1991, , 89, 537
, B., [Kirshner]{}, R. P., [Phillips]{}, M. M., [et al.]{} 1993, , 105, 301
Li, W., Filippenko, A. V., Chornock, R., & Jha, S. 2003, PASP, 115, 844
Li, W., Leaman, J., Chornock, R., [et al.]{} 2011, MNRAS, 412, 1441
Lucy, L. 1999, A&A, 220, 211
MacFadyen, A., Woosley, S., & Heger, A. 2001, ApJ, 20, 410
Matheson, T., Filippenko, A. V., Chornock, R., Leonard, D. C., & Li, W. 2000, AJ, 119, 2303
Matheson, T., Kirshner, R. P., Challis, P., [et al.]{} 2008, AJ, 135, 1598
Mazzali, P., & Lucy, L. 1993, A&A, 279, 447
Mazzali, P. A. 2000, A&A, 363, 705
, P. A., [Deng]{}, J., [Hamuy]{}, M., & [Nomoto]{}, K. 2009, , 703, 1624
, P. A., & [Lucy]{}, L. B. 1998, , 295, 428
Mazzali, P. a., Maurer, I., Valenti, S., Kotak, R., & Hunter, D. 2010, MNRAS, 408, 87
Mazzali, P. a., Sauer, D. N., Pastorello, A., Benetti, S., & Hillebrandt, W. 2008, MNRAS, 386, 1897
, P. A., [Walker]{}, E. S., [Pian]{}, E., [et al.]{} 2013,
Metzger, B. D. 2012, MNRAS, 419, 827
Metzger, B. D., Piro, a. L., & Quataert, E. 2009, MNRAS, 396, 1659
Metzger, B. D., Martínez-Pinedo, G., Darbha, S., [et al.]{} 2010, MNRAS, 406, 2650
, J. S., & [Stone]{}, R. P. S. 1993, Lick Observatory Technical Reports, Vol. 66 (Santa Cruz, CA: Lick Obs.)
, M.-A., & [Lagache]{}, G. 2005, , 157, 302
, M. 2007, PhD thesis, Harvard University
Modjaz, M., Li, W., Filippenko, A. V., [et al.]{} 2001, PASP, 113, 308
Moriya, T., Tominaga, N., Tanaka, M., [et al.]{} 2010, ApJ, 719, 1445
, J. R., [Huchra]{}, J. P., [Freedman]{}, W. L., [et al.]{} 2000, , 529, 786
, K. 1987, , 322, 206
, K., & [Kondo]{}, Y. 1991, , 367, L19
Nomoto, K., Thielemann, F., & Yokoi, K. 1984, ApJ, 286, 644
Nomoto, K., Yamaoka, H., Pols, O. R., [et al.]{} 1994, Nature, 371, 227
Ofek, E. O., Rabinak, I., Neill, J. D., [et al.]{} 2010, ApJ, 724, 1396
Parrent, J. T., Thomas, R. C., Fesen, R. a., [et al.]{} 2011, ApJ, 732, 30
Perets, H. B., Badenes, C., Arcavi, I., Simon, J. D., & Gal-yam, A. 2011, ApJ, 730, 89
Perets, H. B., Gal-Yam, A., Mazzali, P. a., [et al.]{} 2010, Nature, 465, 322
Pettini, M., & Pagel, B. E. J. 2004, MNRAS, 348, L59
, M. M. 1993, , 413, L105
Phillips, M. M., Lira, P., Suntzeff, N. B., [et al.]{} 1999, AJ, 118, 1766
, E., [Amati]{}, L., [Antonelli]{}, L. A., [et al.]{} 2000, , 536, 778
, T. S., [Breeveld]{}, A. A., [Page]{}, M. J., [et al.]{} 2008, , 383, 627
Poznanski, D., Ganeshalingam, M., Silverman, J. M., & Filippenko, A. V. 2011, MNRAS, 415, L81
, D., [Prochaska]{}, J. X., & [Bloom]{}, J. S. 2012, , 426, 1465
Poznanski, D., Chornock, R., Nugent, P. E., [et al.]{} 2010, Science, 327, 58
Prieto, J. L., Stanek, K. Z., & Beacom, J. F. 2008, ApJ, 673, 999
Richmond, M. W., van Dyk, S. D., Ho, W., [et al.]{} 1996, AJ, 111, 327
, L. F., [Kasen]{}, D., [Lee]{}, W. H., & [Ramirez-Ruiz]{}, E. 2011, , 736, L21
, P. W. A., [Kennedy]{}, T. E., [Mason]{}, K. O., [et al.]{} 2005, , 120, 95
Sanders, N. E., Soderberg, a. M., Levesque, E. M., [et al.]{} 2012, ApJ, 758, 132
Sauer, D. N., Mazzali, P. a., Deng, J., [et al.]{} 2006, MNRAS, 369, 1939
Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525
, G. D., [Weymann]{}, R. J., & [Foltz]{}, C. B. 1989, , 101, 713
Shen, K. J., Kasen, D., Weinberg, N. N., Bildsten, L., & Scannapieco, E. 2010, ApJ, 715, 767
Silva, L., Granato, G. L., Bressan, A., & Danese, L. 1998, ApJ, 509, 103
, J. M., & [Filippenko]{}, A. V. 2012, , 425, 1917
Silverman, J. M., Foley, R. J., Filippenko, A. V., [et al.]{} 2012, MNRAS, 425, 1789
Sim, S. a., Fink, M., Kromer, M., [et al.]{} 2012, MNRAS, 420, 3003
Smith, J. A., Tucker, D. L., Kent, S., [et al.]{} 2002, AJ, 123, 2121
, A. M. 2007, in American Institute of Physics Conference Series, Vol. 937, Supernova 1987A: 20 Years After: Supernovae and Gamma-Ray Bursters, ed. S. [Immler]{}, K. [Weiler]{}, & R. [McCray]{}, 492–499
Soderberg, a. M., Brunthaler, a., Nakar, E., Chevalier, R. a., & Bietenholz, M. F. 2010, ApJ, 725, 922
, P. G., & [Wheeler]{}, J. C. 1984, , 280, 282
Taubenberger, S., Pastorello, A., Mazzali, P. a., [et al.]{} 2006, MNRAS, 371, 1459
Taubenberger, S., Hachinger, S., Pignata, G., [et al.]{} 2008, MNRAS, 385, 75
Thomas, R. C., Nugent, P. E., & Meza, J. C. 2011, PASP, 123, 237
, R. C., [Aldering]{}, G., [Antilogus]{}, P., [et al.]{} 2007, , 654, L53
Turatto, M., Benetti, S., Cappellaro, E., Astronomico, O., & Osservatorio, V. 2003, in From Twilight to Highlight: The Physics of Supernovae, ed. W. Hillebrandt & B. Leibundgut (Springer-Verlag), 200
Valenti, S., Benetti, S., Cappellaro, E., [et al.]{} 2007, MNRAS, 383, 1485
Valenti, S., Elias-Rosa, N., Taubenberger, S., [et al.]{} 2008, ApJ, 673, L155
Valenti, S., Pastorello, a., Cappellaro, E., [et al.]{} 2009, Nature, 459, 674
, S., [Yuan]{}, F., [Taubenberger]{}, S., [et al.]{} 2013, ArXiv e-prints
Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, A&A, 369, 574
Waldman, R., Sauer, D., Livne, E., [et al.]{} 2011, ApJ, 738, 21
Wheeler, J. C., Harkness, R. P., Clocchiatti, A., [et al.]{} 1994, ApJ, 436, L135
, D. S., [Park]{}, S., [Shimasaki]{}, K., & [Filippenko]{}, A. V. 2005, Central Bureau Electronic Telegrams, 235, 1
Woosley, S. E., & Heger, A. 2002, RMP, 74, 1015
, S. E., [Taam]{}, R. E., & [Weaver]{}, T. A. 1986, , 301, 601
, S. E., & [Weaver]{}, T. A. 1980, , 238, 1017
Yoon, S.-C., Podsiadlowski, P., & Rosswog, S. 2007, MNRAS, 380, 933
Yun, M. S., & Carilli, C. L. 2002, ApJ, 568, 88
, D., [Kennicutt]{}, Jr., R. C., & [Huchra]{}, J. P. 1994, , 420, 87
[lc cccc]{} 2005 Sep. 26 &53639.3 &18.25 (0.02) & 17.83 (0.02) & 17.46 (0.01) & 17.20 (0.02)\
2005 Sep. 27 &53640.3 &18.38 (0.03) & 18.03 (0.03) & 17.41 (0.02) & 17.13 (0.04)\
2005 Sep. 28 &53641.3 &18.65 (0.02) & 17.92 (0.01) & 17.60 (0.01) & 17.18 (0.02)\
2005 Sep. 29 &53642.2 & & & 17.69 (0.02) &\
2005 Sep. 30 &53643.2 &19.10 (0.05) & 18.24 (0.02) & 17.86 (0.01) & 17.47 (0.03)\
2005 Oct. 02 &53645.3 &19.71 (0.07) & 18.66 (0.02) & 18.18 (0.01) & 17.71 (0.02)\
2005 Oct. 03 &53646.5 &20.07 (0.07) & 18.93 (0.02) & &\
2005 Oct. 05 &53648.0 & & 19.48 (0.06) & 18.83 (0.03) & 18.13 (0.06)\
2005 Oct. 06 &53649.2 &20.67 (0.04) & 19.63 (0.03) & 19.03 (0.02) & 18.26 (0.02)\
2005 Oct. 07 &53650.4 &20.90 (0.04) & 19.86 (0.03) & 19.26 (0.02) & 18.51 (0.02)\
2005 Oct. 08 &53651.3 &21.05 (0.04) & 19.98 (0.04) & 19.48 (0.02) & 18.61 (0.02)\
2005 Oct. 09 &53652.5 && 20.35 (0.05) & 19.75 (0.04) & 18.74 (0.03)\
2005 Oct. 11 &53654.2 &21.74 (0.12) & 20.60 (0.08) & 20.08 (0.05) & 19.01 (0.05)\
2005 Oct. 12 &53655.2 && 20.74 (0.10) & &\
2005 Oct. 13 &52656.2 && 20.88 (0.08) & 20.47 (0.08) & 19.47 (0.06)\
2005 Oct. 14 &53657.2 && 21.22 (0.13) & &
[lc ccc]{} 2005 Sep. 18 & 53631.5 & KAIT & clear & $\lesssim$ 19\
2005 Sep. 24 & 53637.5 & KAIT & clear & 17.82 (0.08)\
2005 Sep. 25 & 53638.4 & KAIT & clear & 17.64 (0.09)\
2005 Sep. 29 & 53642.5 & KAIT & clear & 17.82 (0.07)\
2005 Oct. 11 &53654.9 & P200 & $g'$ & 21.02 (0.01)\
2005 Oct. 11 &53654.9 & P200 & $r'$ & 20.28 (0.01)\
2005 Oct. 11 &53654.9 & P200 & $i'$ & 19.96 (0.01)\
2005 Dec. 05 &53709.5 & P200 & $g'$ & $<$22.7\
2005 Dec. 05 &53709.5 & P200 & $r'$ & $<$22.3\
2005 Dec. 05 &53709.5 & P200 & $i'$ & 21.56 (0.03)
[lc cccccc]{} 2005 Sep. 28 &53641.8 & $<$20.6 & $<$20.1 & $<$19.8 && & 18.10 (0.17)\
2005 Sep. 29 &53642.0 & $<$20.5 & $<$20.3 & $<$20.2 && &\
2005 Sep. 29 &53642.9 & $<$20.6 & & & 19.28 (0.15) & 18.97 (0.10) & 18.40 (0.11)\
2005 Sep. 30 &53643.5 & $<$20.9 & $<$20.5 & $<$20.2 & 19.10 (0.19) & 19.24 (0.18) & 18.48 (0.19)\
2005 Oct. 06 &53649.3 & $<$21.1 & & $<$20.6 & & & 19.70 (0.22)\
2005 Oct. 10 &53653.5 & $<$21.0 & $<$20.5& $<$20.3 & $<$19.8 & $<$ 19.8 & $<$19.6\
2005 Oct. 16 &53659.9 & $<$21.5 & $<$19.7 & & & & $<$19.9
[lccccc]{} 2005 Sep. 26 & 53639 & -1 & SN2005ek & Shane 3-m & Kast\
2005 Sep. 27 & 53640 & 0 & SN2005ek & Tillinghast 60-in& FAST\
2005 Sep. 28 & 53641 & +1 & SN2005ek & Tillinghast 60-in& FAST\
2005 Sep. 30 & 53643 & +3 & SN2005ek & Tillinghast 60-in& FAST\
2005 Oct. 01 & 53644 & +4 & SN2005ek & Tillinghast 60-in& FAST\
2005 Oct. 03 & 53646 & +6 & SN2005ek & Tillinghast 60-in& FAST\
2005 Oct. 06 & 53649 & +9 & SN2005ek & Keck-II & DEIMOS\
2005 Oct. 08 & 53651& +11 & SN2005ek & HET & LRS\
2011 Feb. 22 & 55614 & & UGC 2526 & MMT & Blue Channel\
2011 Feb. 23 & 55615 & & UGC 2526 & MMT & Blue Channel
[lccccc]{} $B$ & 18.25 (0.02) & -16.72 (0.15) & 3.51 (0.13) & 0.24 (0.01) & 4.7 (0.1)\
$V$ & 17.83 (0.02) & -16.96 (0.15) & 2.79 (0.07) & 0.22 (0.01) & 6.9 (0.1)\
$R$ & 17.41 (0.02) & -17.26 (0.15) & 2.88 (0.05) & 0.21 (0.01) & 6.3 (0.1)\
$I$ & 17.13 (0.04) & -17.38 (0.15) & 2.13 (0.06) & 0.15 (0.01) & 8.5 (0.2)\
$J$ & 17.10 (0.09) & -17.26 (0.17) & 1.17 (0.50) & 0.11 (0.01) & 9.6 (0.5)\
$H$ & 16.85 (0.10) & -17.45 (0.18) & 1.62 (0.77) & 0.09 (0.01) & 12.6 (0.7)\
$K$ & 16.49 (0.10) & -17.78 (0.18) & 2.51 (1.12) & 0.09 (0.02) & 10.2 (0.9)
[llc]{} $L_{\rm bol,peak}$ & $10^{42}$ ergs s$^{-1}$ & $1.2 \pm 0.2$\
$E_{\rm rad}$ & $10^{47}$ ergs & $8.2 \pm 0.3$\
$M_{\rm ej}$ & M$_\odot$ & 0.3–0.7\
$E_K$ & 10$^{51}$ erg & 0.25–0.52\
$M_{Ni}$ & M$_\odot$ & 0.02–0.03\
$v_{\rm phot,max}$ & km s$^{-1}$ & $8500 \pm 500$
[^1]: We adopt the NED distance after correction for Virgo, Great Attractor, and Shapley Supercluster Infall and assuming $\rm H_0 = 73$ km s$^{-1}$ Mpc$^{-1}$ [@mhf+00].
[^2]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation.
[^3]: VLA Intensive Survey of Naked Supernovae; VISiONS [@ams07].
[^4]: The large Na I D absorption seen in Figure \[fig:SpecEvol\] is along the line of site to the galaxy core.
[^5]: Each data point in each realization is a random variable chosen from a normal distribution with a mean and variance determined by its counterpart in our initial data set.
[^6]: “Transitional” refers to the transition from optically thick to thin, which, in this case, we characterize by an increased prominence of the emission component of the Ca II near-IR triplet)
[^7]: This is an updated version of SYNOW; https://c3.lbl.gov/es/ .
[^8]: 12 $+\log{\rm (O/H)}_{\rm solar} = 8.69$ on the PP04N2 scale
[^9]: Its distance of 66 Mpc and classification as a SN Ic placed it just slightly outside the relevant sample volume (60 Mpc for core-collapse SN).
|
---
abstract: 'It is challenging to construct metrology schemes which harness quantum features such as entanglement and coherence to surpass the standard quantum limit. We propose an ansatz for devising adaptive-feedback quantum metrology (AFQM) strategy which reduces greatly the searching space. Combined with the Markovian feedback assumption, the computational complexity for designing AFQM would decrease from $N^7$ to $N^4$, for $N$ probing systems. The feedback scheme devising via machine learning such as particle-swarm optimization and derivative evolution requires much less time and produces equally well imprecision scaling. We have thus devised an AFQM for $207$-partite system. The imprecision scaling would persist steadily for $N>207$ when the parameter settings for $207$-partite system is employed without further training.'
author:
- Yi Peng
- Heng Fan
bibliography:
- 'Bibliography.bib'
title: 'Feedback Ansatz for Adaptive-Feedback Quantum Metrology Training with Machine Learning'
---
[GB]{}
*Introduction*— Given $N$ entangled probing systems, quantum metrology promises parameter estimation with imprecision below the lowest limit $1/\sqrt{N}$ allowed by classical theory. This is known as the [*standard quantum limit (SQL)*]{}. The lowest imprecision permitted by quantum mechanics is $1/N$, i.e. the so-called [*Heisenberg limit (HL)*]{} [@braunstein1994statistical; @giovannetti2004quantum-enhanced; @giovannetti2006quantum; @giovannetti2011advances; @Degen2017; @Pezze2018]. Such kind of quantum superiority over the classical schemes attracts much attention in both academic and industry communities. Because it has wide range of applications including spectroscopy [@Bollinger1996], accurate clock construction [@Kessler2014; @Derevianko2011; @Zhang2016], gravitational wave detection [@LIGO2013; @Schnabel2010], fundamental biology research and medicine development [@Taylor2016], and others [@giovannetti2004quantum-enhanced; @Demkowicz-Dobrzanski2015; @Degen2017; @Pezze2018].
To harness the advantage offered by quantum mechanics for practical metrology, there exist at least three prominent challenges. a) Both SQL and HL are asymptotic and require great amount of data to approach. It is a serious limitation in many circumstances. For instance, the gravitation detection window is very narrow [@LIGO2013; @Schnabel2010] while many biological samples are too fragile to endure much photon bombardment [@Taylor2013; @Taylor2016]. Thus, we need to finish the interference in a restricted time period and with limited number of prob systems (such as photons). b) Environment noise which is inevitable in practical platforms can completely demolish such quantum advantage [@Escher2011general; @Demkowicz-Dobrzanski2012; @Haase2016]. c) Many metrology schemes proposed previously require input states or final measurements which are difficult to realize. For example, the Greenberger-Horne-Zeilinger (GHZ) state has ability to asymptotically achieve HL [@Leibfried2004; @giovannetti2004quantum-enhanced; @giovannetti2006quantum; @giovannetti2011advances; @Monz2011; @Liu2015]. Synthesising GHZ state is well recognized as highly complicated and inefficient in many platforms [@Monz2011; @Wang2018; @Zhang2018; @Wei2006; @Barends2014; @Song2019; @Omran2019; @Wei2019]. In typical phase estimation tasks, canonical positive-operator-valued measure (POVM) based on the so-called *phase state* $\ket{\phi}\bra{\phi}$ and the sine input state (\[sine\_state\_def\]) have been frequently utilized to demonstrate asymptotic HL. To our knowledge, there is no clear way to realize either of them for $N\ge3$ [@Sanders1995; @Wiseman1995; @DAriano1998; @VanDam2007; @Hassani2017].
The adaptive-feedback quantum metrology (AFQM) is believed to be a promising candidate capable of giving good parameter estimation with limited number of measurements and thus resolve issue a). As an example, the so-called Berry-Wiseman-Breslin scheme (BWB) can provide single-shot estimation achieving imprecision below SQL. Besides, BWB employs local projective measurements which partially resolves issue c). The limitation of BWB is its employment of the sine input state which is hard to generate [@berry2000optimal; @berry2001optimal]. BWB is a well-educated heuristic strategy. Devising AFQM is highly challenging. Considering the AFQM employing local projective measurement described in Fig. \[feedback\_qmetr\_circ\], the total measurement outcome combinations as well as the feedbacks would amount to $2^N$ if $N$ qubits are employed. It indicates plenty flexibility of this type of AFQM scheme as well as a great challenge of optimizing it.
![(Color online) Quantum circuit of AFQM employing local projective measurement for $N=4$. $\ket{\psi_\mathrm{in}}$ is the total input state. The interference process $\hat{U}_\phi$ is controlled by $\phi$. $\phi_0$ is the initial random guess generated by a random number generator (RNG). $\phi_1$ is feedback information gathered from the first measurement, $\phi_2$ from the first and second measurements while $\phi_3$ from the first three measurements. $\phi_4$ is the final single-shot estimation of $\phi$ determined by all the measurements. Case of arbitrary $N$ is similar.[]{data-label="feedback_qmetr_circ"}](feedbackQMetr.pdf){width="45.00000%"}
It was firstly shown by Hentschel and Sanders that we can reduce the dimension of the feedback parameter space to $N$ assuming a Markovian feedback sequence. By employing machine learning such as particle-swarm optimization (PSO) and differential evolution (DE), one can devise promising AFQM autonomously. We call such a scheme devising procedure as the Hentschel-Sanders approach (HS) [@Hentschel2010; @Hentschel2011]. Without noise, the achievable imprecision breaches SQL and shows superiority over BWB (cf. Table \[sum\_previous\_results\]).
------------------ ------------------ ------------------ ------------------ ------------------ --------------- -------------------------- --------------------------- --------------------------
Ref. [@Hentschel2010] [@Hentschel2011] [@Hentschel2010] [@Hentschel2011] [@Lovett2013] [@Palittapongarnpim2016] [@Palittapongarnpim2017b] [@Palittapongarnpim2018]
$\alpha$ $0.704$ $0.708$ $0.736$ $0.747$ $0.74$ $0.71$ $0.7198$ $0.729$
$N_\mathrm{max}$ $\le14$ $\le50$ $\le98$ $\le100$ $\le100$ $\le100$
------------------ ------------------ ------------------ ------------------ ------------------ --------------- -------------------------- --------------------------- --------------------------
: Summary of previous results. $\alpha$ is the inverse-scaling power of the imprecision $\delta\phi$ with respect to $N$. $N_\mathrm{max}$ is maximum prob number for which AFQM can be obtained via HS machine learning approach.
\[sum\_previous\_results\]
Given permutation symmetric input state, schemes thus devised have remarkable resilience against environment noise. They can also provide single-shot estimation. Though only the sine and product input states have been considered up to now, HS can be applied to other types of input states. Hence HS approach can solve a), b) and c) simultaneously. Generating such an AFQM would consume time $\mathcal{O}(N^7)$ and memory space $\mathcal{O}(N)$ for computation. AFQM for up to $N=100$ has thus been devised [@Hentschel2010; @Hentschel2011; @Hentschel2011b; @Lovett2013; @Palittpongarnpim2016; @Palittapongarnpim2017; @Palittapongarnpim2017b; @Palittapongarnpim2019; @Hayes2014].
Here we introduce an ansatz for devising AFQM. Drawn from HL and SQL, the ansatz suggests that the adjustment of the feedback in every step should be a polynomial of the inverse powers of the step. It can reduce the feedback parameter space dimension from $N$ to a chosen constant, if we adopt HS. The memory space for parameter storing would also be a constant. With the parameter space dimension reduced to a constant, we can ensure persistent imprecision for at least $N=207$ without increasing the training time of our policy for bigger $N$. Further, we can generate an $N$-partite scheme without knowing schemes for fewer qubits which is required in the previous HS implementations [@Hentschel2010; @Hentschel2011; @Hentschel2011b; @Lovett2013; @Palittpongarnpim2016; @Palittapongarnpim2017; @Palittapongarnpim2017b; @Palittapongarnpim2019; @Hayes2014]. The computation time thus scales as $\mathcal{O}(N^4)$. We test the ansatz for devising AFQM via PSO as well as DE. Both the previously studied sine state (\[sine\_state\_def\]) and the spin-squeezed state (SSS) are considered. It is widely believed that SSS has admirable resilience against environment noise [@sorensen2001many; @Dunningham2002; @ma2011quantum; @Duan2011; @Zhang2013; @Pezze2013] and its synthesis has been realized in many labs [@Hald1999; @Fernholz2008; @Takano2009; @Gross2010; @Leroux2010; @Hamley2012; @Sewell2012; @Muessel2014; @Hosten2016; @Zou2018]. The performance of AFQM thus devised is as good as the performance of the AFQM devised via previous HS. One of the most intriguing part is that when applying the parameters obtained for $207$-partite system to bigger systems $N>207$, the imprecision scaling persists for an admirable range.
*Feedback ansatz for AFQM—* Suppose there are $N$ spin-$\frac{1}{2}$ probes. The interference process characterized by $\phi$ is $$\hat{U}_\phi = e^{-i\phi\hat{J}_y},\quad\textrm{with}\quad
\hat{J}_y = \sum_{n=1}^N\hat{s}_y^{(n)}.$$ $\hat{J}_{x,y,z}$ denotes total angular momentum along $x$, $y$ and $z$ direction respectively while $\hat{s}_{x,y,z}^{(n)}$ are spin operators of the $k$th probe. After the probes have passed through the parameter channel $\hat{U}_\phi$, we apply feedback adjustment $\hat{U}_{\phi_{n-1}}^\dag$ to compensate $\hat{U}_\phi$ as closely as possible. The initial compensation $\phi_0$ is a random guess between $-\pi$ and $\pi$. Note that we assume $\phi\in[\pi,\pi)$. Then we locally measure $\hat{s}_z^{(1)}$, the result of which would be used to estimate the next compensation $\phi_1$. The estimation-compensation-measurement procedure carries on until we obtain the final estimation $\phi_N$. The $n$th compensation $\phi_n$ can be regarded as an update of $\phi_{n-1}$ with an adjustment depending on the $n$ previous measurement outcomes $s_1,\ldots,s_n$ of $\hat{s}_z^{(1)},\ldots,\hat{s}_z^{(n)}$ $$\phi_n = \phi_{n-1} - \Delta_n(s_1,\ldots,s_n).$$ We want $\Delta_n(s_1,\ldots,s_n)$ to bring $\phi_n$ closer to $\phi$ in each step and $|\phi_n-\phi|$ decreases with respect to $n$. One can regard $\phi_1$,…, and $\phi_{N}$ as a serial of estimations of $\phi$. We would expect $|\phi_n-\phi|$ to be of the order of $1/n^\alpha$ with $\alpha$ being some positive constant between $1/2$ and $1$. $\alpha=1/2$ correspond to SQL while $\alpha=1$ to HL. We cannot allow $\Delta_n(s_n,\ldots,s_n)$ being too big compared with $|\phi_{n-1}-\phi|$. $\Delta_n(s_n,\ldots,s_n)$ cannot be too small either. Consider the case when $\Delta_n(s_n,\ldots,s_n)=s_n/2^{n-1}$. If the measurement result $s_n$ makes $\Delta_n(s_n,\ldots,s_n)$ move $\phi_n$ away from $\phi$, the best of all the later adjustments $\Delta_{n'}(s_n,\ldots,s_n)$ with $n'>n$ can achieve is to neutralize the detrimental effect of $\Delta_n(s_n,\ldots,s_n)$. In such circumstances, the final estimation $\phi_N$ would be worse than $\phi_n$. It seems setting $\Delta_n(s_n,\ldots,s_n)$ around the order of $|\phi_n-\phi|$ would be reasonable. Another fact should be noted is that $|\phi_{n}-\phi|$ and $|\phi_{n+1}-\phi|$ are about the same order. Thus $\Delta_n(s_n,\ldots,s_n)$ should be smaller than $1/|\phi_n-\phi|$. Now comes the feedback ansatz $$\Delta_n(s_n,\ldots,s_n) \propto 1/(n+1)^{\wp_n(s_1,\ldots,s_n)}.
\label{feedbackAnsatz}$$ where $\wp_n(s_1,\ldots,s_n)$ can be out of the range $[1/2,1]$ bounded by by SQL and HL. We used $1/(n+1)^{\wp_n(s_1,\ldots,s_n)}$ instead of $1/n^{\wp_n(s_1,\ldots,s_n)}$ to ensure that the variation of $\wp_1(s_1,\ldots,s_n)$ matters. So far the ansatz (\[feedbackAnsatz\]) can only reduce the volumn of the AFQM parameter space. Combined with other assumptions, it can reduce the parameter space drastically as shown in the following.
*Hentschel-Sanders AFQM parameter space reduced by feedback ansatz.—* Here we provide an example of application of the feedback ansatz (\[feedbackAnsatz\]) in the HS AFQM devising approach. HS indicates that the adjustment of $\phi_n$ from the immediate former compensation $\phi_{n-1}$ depends only on the measurement result $s_n$ of $\hat{s}_z^{(n)}$ $$\phi_n = \phi_{n-1}-2s_n\Delta_n.$$ $\Delta_1$,…, and $\Delta_N$ thus constitutes the AFQM parameter search space [@Hentschel2010; @Hentschel2011; @Lovett2013; @Palittpongarnpim2016; @Palittapongarnpim2017; @Palittapongarnpim2017b; @Palittapongarnpim2019; @Hayes2014]. By invoking ansatz (\[feedbackAnsatz\]), we would have $$\phi_n = \phi_{n-1}-2s_n/(n+1)^{\wp_n}.$$ The parameter space becomes that of $\wp_1$, …, and $\wp_N$. If $|\phi_n-\phi|$ is about the scale of $1/n^\alpha$ and $\alpha$ is nearly the same for almost every $\phi_n$, then we may expect $\Delta_n$ to be of the order $1/(n+1)^\wp$ with $\wp$ being almost the same for every $\Delta_n$. $\wp_1$, …, and $\wp_N$ are located in an small zone around $\wp$. Generally, we expect the adjustment to be a polynomial of the inverse power of $n$ $$\Delta_n = \sum_{\ell=0}^{N_\mathrm{s}-1}\frac{c_\ell\pi}{(n+1)^{\wp+\ell}}.
\label{inverse_scaling_adjustment}$$ The coefficients $c_0$, …, and $c_{N_\mathrm{s}-1}$ gives us enough flexibility in AFQM generation. $N_\mathrm{s}$ is our choice of number of terms in the expansion. (\[inverse\_scaling\_adjustment\]) can be seen as a derivative of our feedback ansatz (\[feedbackAnsatz\]). Including $\wp$, there are $N_\mathrm{s}+1$ control parameters, the combination of which we call an [*inverse-scaling policy*]{} $\mathscr{P}$. This reduces the search space dimension to $N_\mathrm{s}$ independent of $N$. It enables a reduction of computation complexity. The memory space required to store $\mathscr{P}$ is also constant.
*Devising AFQM via machine learning: cost function and complexity.*—Following the HS approach [@Hentschel2010; @Hentschel2011; @Lovett2013; @Palittpongarnpim2016; @Palittapongarnpim2017; @Palittapongarnpim2017b; @Palittapongarnpim2019], we implement machine learning algorithm such as PSO and DE to generate AFQM under the guidance of our ansatz derivative (\[inverse\_scaling\_adjustment\]). Employing machine learning to optimize the AFQM with the feedback ansatz is very much like a treasure hunting under the guidance of SQL and HL.
*Cost function.* We employ Holevo variance to quantify the imprecision $\delta\phi$ of the final estimation $\phi_N$ as before $$V_\phi = (\delta\phi)^2 = \frac{1}{S^2} - 1,
\,\textrm{with}\,
S = \left|\int_{-\pi}^\pi{\mathrm{d}}{\phi}P(\phi)e^{i(\phi-\phi_N)}\right|.$$ Note that $V_\phi$ is an good approximation of the traditional variance in statistics when $\phi_N$ is very close to $\phi$ [@berry2001adaptive]. One can simulate $K=10N^2$ trials of experiment and obtain thus many estimations $\phi_N^{(k)}$. $S$ is the so-called [*sharpness*]{} and can be estimated via Monte-Carlo method as $$S = \left|\frac{1}{K}\sum_{k=1}^Ke^{i(\phi-\phi_N^{(k)})}\right|.$$
*Complexity.* Both PSO and DE employ a group of searching agents and record the best strategy found by the agents throughout their evolving [@Kennedy1995; @Storn1996]. With greater number of searching agents, one can find better outcome at the cost of adding computation complexity. By our ansatz derivative (\[inverse\_scaling\_adjustment\]), we need $\Xi=20(N_\mathrm{s}+1)$ searching agents instead of $20N$ [@Hentschel2010; @Hentschel2011; @Lovett2013; @Palittpongarnpim2016; @Palittapongarnpim2017; @Palittapongarnpim2017b; @Palittapongarnpim2019] when the input state is the so-called sine state $$\ket{\psi_\mathrm{\sin}}
= \frac{1}{\sqrt{j+1}}\sum_{\mu=-j}^{j}\sin\left\lbrack{\frac{(\mu+j+1)\pi}{2(j+1)}}\right\rbrack\ket{j\mu}_y.
\label{sine_state_def}$$ $j=N/2$ and $\ket{j\mu}_{x,y,z}$ is the eigenstate of $\hat{J}_{x,y,z}$ respectively, belonging to eigenvalue $\mu$. We have also considered feeding the spin-squeezed state [@kitagawa1993] $$\ket{\psi_\mathrm{sss}} = e^{i\hat{J}_x\delta_\mathrm{adj}}e^{-i\hat{J}_z^2T_\mathrm{s}}\ket{jj}_x
\,\,\textrm{with}\,\,
\delta_\mathrm{adj}=\frac{1}{2}\arctan\frac{B}{A},$$ where $A = 1-\left(\cos2T_\mathrm{s}\right)^{N-2}$ and $B = 4\sin{T_\mathrm{s}}\left(\cos{T_\mathrm{s}}\right)^{N-2}$. Adding $T_\mathrm{s}$, the parameter space dimension would be $N_\mathrm{s}+2$ and thus we dispatch $\Xi=20(N_\mathrm{s}+2)$ agents to search if SSS has been fed to the interferometer. Given $N_\mathrm{s}=4$, the search space boundaries has been chosen according to Table \[search\_space\_boundary\_table\].
--------- ---------- ------------------
$\wp$ $c_\ell$ $T_\mathrm{s}$
$[0,5]$ $[-5,5]$ $[0,2/\sqrt{N}]$
--------- ---------- ------------------
: Boundaries of inverse-scaling policy parameters. Since we suspect $\wp$ to be very close to the region between $1/2$ (SQL) and $1$ (HL), we choose the search zone that covers the region between SQL and HL and $10$ times bigger. The boundaries for $c_0$,…, and $c_{N_\mathrm{s}-1}$ are empirical which provides good results but not guaranteed to be optimal. We choose the upper bound $2/\sqrt{N}$ for spin squeezing time $T_\mathrm{s} $ since $1/\sqrt{N}$ is the minimum time needed to ensure maximal quantum Fisher information for $\ket{\psi_\mathrm{sss}}$ [@pezze2009entanglement]. Recall that quantum Fisher information quantifies the metrology prowess of $\ket{\psi_\mathrm{sss}}$ [@braunstein1994statistical].
\[search\_space\_boundary\_table\]
We iterate both PSO and DE for $N_\mathrm{I}=300$ times as has been done in Ref. . The $N$-partite inverse-scaling policy can be generated directly, without knowing any $(N-k)$-partite policy for $1\le{k}\le{N-1}$. To generate a $N$-partite inverse-scaling policy we hence need time of $\mathcal{O}(K\Xi{N}^2)=\mathcal{O}(N^4)$. Recall that the number $\Xi$ of searching agents is constant independent of $N$ while each simulation of the adaptive feedback metrology progress consumes time of $N^2$ [@Hentschel2011b].
*Results and analysis.*— As has been mentioned previous, we consider both the sine state and SSS as input state. We consider the sine state for comparison with previous results. SSS is considered due to its well recognized noise-resisting ability and proven synthesis procedure in labs [@Hald1999; @Fernholz2008; @Takano2009; @Gross2010; @Leroux2010; @Hamley2012; @Sewell2012; @Muessel2014; @Hosten2016; @Zou2018]. We generate AFQM for both kinds of input states via PSO as well as DE. There are four groups of data which we analyze and present in the following: PSO-SSS, PSO-Sine, DE-SSS and DE-Sine.
{width="\textwidth"}
We summarized eight main conclusions drawn from our numerical data. i) AFQM generated with our feedback ansatz is equally good as the previous AFQM generated via HS. Given sine inpute state, this is clear from Fig. \[hpv\_invSFeedback\_sss\](a,c) and Table \[search\_space\_boundary\_table\]. ii) We can obtain inverse-scaling policy for bigger $N$ in shorter time. For example, we generated the $207$-partite inver scaling policy for the sine state via PSO at the cost of approximately $200$ hours running of $120$ CPUs at 2.6 GHz. iii) Since the parameter space having a much small dimension $N_\mathrm{s}$, PSO and DE produces almost equally good AFQM. There is no breakdown of PSO up to $N=207$. iv) By optimizing the squeezing time as well, feeding SSS state to the interferometer can outperform AFQM with sine input state. v) The inverse-scaling policy trained for $N=207$ can also sever as a good policy for AFQM with bigger $N$. As shown in Fig. \[hpv\_invSFeedback\_sss\], the power-law scaling of imprecision $\delta\phi$ does not breakdown immediately for $N>207$ if the inverse-scaling policy of $N=207$ is applied without further training. In the case of sine state, the imprecision scaling of the $207$-partite policy for $N>207$ would gradually broke but would stay below SQL for a considerable range (cf. Fig. \[hpv\_invSFeedback\_sss\] (a,c)). Given SSS, the performance scaling would endure for much bigger $N$ when $207$-partite policy is employed without further training (cf. Fig. \[hpv\_invSFeedback\_sss\] (b,d)). vi) As a matter of fact, we can see a general trending of the leading inverse-scaling exponentiate $\wp$ (cf. Fig. \[wp\_afqm\](a)).
![(Color online) General Trending of inverse-scaling policy. (a)Leading inverse-scaling exponentiate $\wp$ of the adjustment $\Delta_n$ of feedback and (b)optimal squeezing time for AFQM.[]{data-label="wp_afqm"}](wp_squeezing.pdf){width="50.00000%"}
From the fairly success of the $207$-partite inverse-policy applying to bigger prob ensemble as well as the general trending of $\wp$, one can see the validity and merit of our ansatz (\[feedbackAnsatz\]) and its derivative (\[inverse\_scaling\_adjustment\]). vii) In fact we also see a general trending the optimal spin-squeezing time $T_\mathrm{s}$ (cf. Fig. \[wp\_afqm\](b)). For $N$ big enough ($N{\gtrsim}100$), our numerical result suggests that the optimal spin-squeezing time should be approximately $0.6/N^{2/3}$. In fact, we have been optimizing $c_\mathrm{s} = T_\mathrm{s}N^{2/3}$ in our simulation. In applying the $207$-partite inverse-policy for SSS with $N>207$, it is $c_\mathrm{s}$ that has been inherited instead of $T_\mathrm{s}$. This hints that the optimal squeezing time for employing SSS in AFQM should scale as $1/N^{2/3}$. viii) As long as the inverse-policy has been trained employing either PSO or DE, the scaling of the imprecision $\delta\phi$ would not break up to at least $N=207$. This upper limit for $N$ would be much bigger, since we can see the scaling persistence in a moderate range when the $207$-partite inverse-policy has been applied without training for $N>207$.
*Conclusion and discussion.*— We have proposed the feedback ansatz (\[feedbackAnsatz\]) for devising AFQM. When combined with the previous HS scheme, we demonstrated the prowess of our ansatz via numerical simulation. Though only the noise-less situation has been tested here, we can expect similar application in devising AFQM infested by environment noise. It may also be useful in devising multi-parameter estimation schemes. It is interesting to see that HL and SQL can be used as guidelines for AFQM designing instead of being mere metrology performance borderlines. Our method may provide more insight on this direction of research.
We thank B.C. Sanders for stimulating discussions. This work was supported National Key R & D Program of China (Grant Nos. 2016YFA0302104 and 2016YFA0300600), National Natural Science Foundation of China (Grant Nos. 11774406 and 11934018), Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000), and Beijing Academy of Quantum Information Science (Grant No. Y18G07).
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abstract: 'We investigate the evolution of galaxy masses and star formation rates in the Evolution and Assembly of Galaxies and their Environment () simulations. These comprise a suite of hydrodynamical simulations in a $\Lambda$CDM cosmogony with subgrid models for radiative cooling, star formation, stellar mass loss, and feedback from stars and accreting black holes. The subgrid feedback was calibrated to reproduce the observed present-day galaxy stellar mass function and galaxy sizes. Here we demonstrate that the simulations reproduce the observed growth of the stellar mass density to within 20 per cent. The simulation also tracks the observed evolution of the galaxy stellar mass function out to redshift $z=7$, with differences comparable to the plausible uncertainties in the interpretation of the data. Just as with observed galaxies, the specific star formation rates of simulated galaxies are bimodal, with distinct star forming and passive sequences. The specific star formation rates of star forming galaxies are typically 0.2 to 0.5 dex lower than observed, but the evolution of the rates track the observations closely. The unprecedented level of agreement between simulation and data across cosmic time makes a powerful resource to understand the physical processes that govern galaxy formation.'
author:
- |
M. Furlong $^{1}$[^1], R. G. Bower$^1$, T. Theuns$^{1, 2}$, J. Schaye$^3$, R. A. Crain$^3$, M. Schaller$^1$, C. Dalla Vecchia$^{4, 5}$, C. S. Frenk$^1$, I. G. McCarthy$^6$, J. Helly$^1$, A. Jenkins$^1$, and Y. M. Rosas-Guevara$^{7, 8}$\
$^{1}$Institute for Computational Cosmology, Durham University, South Road, Durham, DH1 3LE\
$^2$Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium\
$^{3}$Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, the Netherlands\
$^4$Instituto de Astrofsíca de Canarias, C/ Vía Láctea s/n,38205 La Laguna, Tenerife, Spain\
$^5$Departamento de Astrofsíca, Universidad de La Laguna, Av. del Astrofsícasico Franciso Sánchez s/n, 38206 La Laguna, Tenerife, Spain\
$^{6}$Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF\
$^{7}$ Université de Lyon, Lyon, F-69003, France\
$^{8}$ CNRS, UMR 5574, Centre de Recherche Astrophysique de Lyon, Ecole Normale Supérieure de Lyon, Lyon, F-69007, France\
bibliography:
- '../../mybib.bib'
date: Accepted 15 April 2015
title: Evolution of galaxy stellar masses and star formation rates in the simulations
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\[1\] \[1\][[~~\#1~~]{}]{} \[1\] \[1\]
\[firstpage\]
galaxies: abundances, evolution, formation, high-redshift, mass function, star formation
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are very grateful for the endless technical support provided by Dr. Lydia Heck during the preparation of these simulations and during post processing. MF thanks Violeta Gonzalez-Perez and Peter Mitchell for providing observational data sets.
RAC is a Royal Society University Research Fellow. This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. We also gratefully acknowledge PRACE for awarding us access to the resource Curie based in France at Trés Grand Centre de Calcul. This work was sponsored by the Dutch National Computing Facilities Foundation (NCF) for the use of supercomputer facilities, with financial support from the Netherlands Organization for Scientific Research (NWO). The research was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreements 278594-GasAroundGalaxies, GA 267291 Cosmiway, and 321334 dustygal, the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy OWNce (\[AP P7/08 CHARM\]), the National Science Foundation under Grant No. NSF PHY11-25915, the UK Science and Technology Facilities Council (grant numbers ST/F001166/1 and ST/I000976/1), Rolling and Consolidating Grants to the ICC, Marie Curie Reintegration Grant PERG06-GA-2009-256573, Marie Curie Initial Training Network Cosmocomp (PITN-GA-2009-238356).
\[lastpage\]
[^1]: E-mail: michelle.furlong@durham.ac.uk
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author:
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\
[CVLab, EPFL, Switzerland]{}\
[{firstname.lastname}@epfl.ch]{}\
bibliography:
- 'string.bib'
- 'graphics.bib'
- 'vision.bib'
- 'learning.bib'
- 'biomed.bib'
title: Shape Reconstruction by Learning Differentiable Surface Representations
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abstract: 'We consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.'
author:
- Kallol Paul and Gopal Das
date:
title: Cosine of angle and center of mass of an operator
---
**2000 Mathematics subject classification:** 47B44, 47A63.\
**Keywords and Phrases:** Antieigenvalues, accretive operator, orthogonality of bounded linear operators.
Introduction:
=============
Suppose T is a bounded linear operator on a complex Hilbert space H with inner product ( , ) and norm $ \| ~\|$. A bounded linear operator T is called strongly accretive if $ Re \langle Tx,x \rangle \geq m_T > 0 $ for all unit vectors x in H. For a strongly accretive, bounded operator T on a Hilbert space\
$$\sup_{\|x\| = 1}~~ \inf _{\epsilon > 0} \| (\epsilon T - I ) x \|^2 = \inf _{\epsilon > 0} ~~\sup_{\|x\| = 1}\| (\epsilon T - I ) x \|^2 ~.$$ This is the Min-max equality in operator trigonometry and was obtained by Gustafson[@5; @9] in 1968, Asplund and Pt$\acute{a}$k[@1] in 1971.\
The angle of an operator was introduced in 1968 by Gustafson[@6] while studying the problems in the perturbation theory of semi-group generators in [@7; @8]. The cosine of the angle of $T$ was defined by Gustafson as follows:$$\begin{aligned}
cos\phi(T) & = & \inf_{\left\Vert Tf\right\Vert \neq0}\frac{Re(Tf,f)}{\left\Vert Tf\right\Vert \,\left\Vert f\right\Vert }.\label{eq:1}\end{aligned}$$ The properties of $\cos\phi\left(T\right)$ are dependent on the real part of numerical range $W(T)$ of $T$.\
The quantity $\cos\phi\left(T\right)$ has another interpretation as the first antieigenvalue of $T$,$$\begin{aligned}
\mu_{1}\left(T\right) & = & \inf_{\left\Vert Tf\right\Vert \neq 0}\frac{Re(Tf,f)}{\left\Vert Tf\right\Vert \,\left\Vert f\right\Vert }.\label{eq:2}\end{aligned}$$ This concept was also introduced by Gustafson[@9] and studied by Gustafson[@10; @11; @12], Seddighin[@19], Gustafson and Seddighin[@15] and Gustafson and Rao[@13; @14]. In [@2] we studied the structure of of antieigenvectors of a strictly accretive operator and in [@16] we studied total antieigenvalues of a bounded normal operator. In [@17] we introduced the notion of symmetric antieigenvalues. The notion of the cosine of angle of an operator has a connection with the min-max equality according to Gustafson’s theory\[[@5] - [@12]\] as $$sin T = \sqrt{ 1 - cos^2T} = \min_{\epsilon > 0 } \| \epsilon T - I \| ~.$$ While studying the norm of the inner derivation Stampfli[@20] proved that for any bounded linear operator T there exists a complex scalar $ z_{0} $ such that $ \| T- z_{0}I\| \leq \|T-zI\| $ for all complex scalar z. He defined $z_0 $ as the center of mass ( or center) of T . In the Banach space B(H,H) for any two operators T and A , T is orthogonal to A in the sense of James\[4\] iff $\|T+\lambda A \| \geq \|T\| $ for all scalars $ \lambda $. Thus if $z_0$ is center of mass of an operator T, then $ T - z_{0}I $ is orthogonal to I. We studied in [@18] the the notion of orthogonality of two bounded linear operators T and A in B(H,H) and proved that T is orthogonal to A iff $ \exists $ $ \{x_{n}\}, \|x_{n}\| = 1 $ such that $ (A^{*}Tx_{n},x_{n}) \rightarrow 0 $ and $ \|Tx_{n}\| \rightarrow \|T\| $.We introduce the notion of real center of mass and total center of mass of an operator relative to another operator and explore their relation with cosine and total cosine of an operator. We also give an easy proof of the Min-max equality.\
\
Center of mass of an operator :
===============================
Before describing the definition of center of mass of an operator we first prove the following two results:\
**Lemma 2.1** \[L: lm1\]\
For any $ T,~A~\in~B(H)$ if $ |\epsilon | > 2 \frac{\|T\|}{\|A\|} ,~ \|A\| \ne 0,$ then $ \| T \| < \|T -\epsilon A \| $.\
***Proof.*** $$T,~A~\in~B(H), |\epsilon | > 2 \frac{\|T\|}{\|A\|} \Rightarrow \parallel T~-~\epsilon A \parallel \geq \mid \epsilon \mid~\parallel A \parallel~-~\parallel T \parallel
> 2 \frac{\|T\|}{\|A\|} \parallel A \parallel - \parallel T \parallel = \parallel T \parallel.$$ Hence the proof.\
**Lemma 2.2** \[L:lm2\]\
Let T and A be two bounded linear operators on a complex Hilbert space H. Then there exists a real scalar $ \epsilon_{0} $ such that $$\|T-\epsilon_{0}A\| \leq \|T-\epsilon A \| ~~~~~\forall \epsilon\in R.$$ ***Proof.*** We consider the function $ f:~G ~\rightarrow ~R $ defined by $$f(\epsilon) = \|T -\epsilon A \| ~\forall \epsilon \in G$$ where R is the set of real numbers and $ G \subseteq R $ is a closed interval centered at origin with radius $ 2 \frac{\|T\|}{\|A\|} $.\
Then it is clear that f is continuous, bounded and G is compact. So f attains its infimum. Thus there exists a $ \epsilon_{0} \in G$ such that $f(\epsilon) \geq f( \epsilon_{0} )~~\forall
\epsilon \in G \subseteq R $ which implies $ \|T-\epsilon_{0}A\| \leq \|T-\epsilon A \|~~
\forall \epsilon \in G .$\
If $ \epsilon \notin G $ then $ |\epsilon | > 2 \frac{\|T\|}{\|A\|} $ and so $ \|T-\epsilon A \|~>~\parallel T \parallel$ by the above lemma. Also $ f(0) \geq f( \epsilon_{0} ) $ i.e., $ \parallel T \parallel~\geq~\|T-\epsilon_{0}A\|.
$ Hence $\|T-\epsilon_{0}A\| \leq \|T-\epsilon A \| $ if $ \epsilon \notin G. $\
Thus $ \|T-\epsilon_{0}A\| \leq \|T-\epsilon A \|$ for all $\epsilon~ \in R $. Hence the proof.\
Thus for any two bounded linear operators T and A there exists a real scalar $ \epsilon_0 $ such that $$\|T-\epsilon_{0}A\| \leq \|T-\epsilon A \|~~ \forall ~\epsilon~ \in R .$$ We define $ \epsilon_0 $ as the **real center of mass** of operator T relative to the operator A.\
Likewise one can show that for any two bounded linear operators T and A there exists a complex scalar $ \lambda_0 $ such that $$\|T-\lambda_{0}A\| \leq \|T-\lambda A \|~~ \forall ~\lambda~ \in C .$$ We define $ \lambda_0 $ as the **total center of mass** of operator T relative to the operator A. The real center of mass and total center of mass of operator T relative to the operator A is not always uniquely defined. For A = I, $ \lambda_0$ is the center of mass of T introduced by Stampfli in [@20].
Orthogonality of two operators T and A in B(H,H) :
==================================================
In the Banach space B(H,H) for any two operators T and A , T is orthogonal to A in the sense of James[@4] iff $\|T+\lambda A \| \geq \|T\| $ for all scalars $ \lambda $ in C. We say that T is real orthogonal to A iff $\|T+\epsilon A \| \geq \|T\| $ for all scalars $ \epsilon $ in R. Thus if $ \epsilon_0$ and $ \lambda_0 $ are real center of mass and total center of mass of T relative to A respectively then $ T - \epsilon_0 A $ is real orthogonal to A and $ T - \lambda_0 A $ is orthogonal to A respectively. We next prove two theorems which is going to characterize the real center of mass.\
**Theorem 3.1** \[T:th1\]\
The set $ W_{0}(A) =\{ \epsilon \in R ~:~ \exists ~\{x_{n}\}~\subset
H,~ \|x_{n}\|=1,~ \| Tx_{n} \| \rightarrow \| T \| $ and $ Re( Tx_{n},Ax_{n} ) \rightarrow \epsilon $ } is non-empty, closed and convex.\
***Proof.*** Clearly $ W_{0}(A) $ is a non-empty subset.\
We now show that $ W_{0}(A) $ is closed. Let $\epsilon_{n} \in W_{0}(A) $ and $
\epsilon_{n}~\rightarrow~\epsilon $. As $\epsilon_{n}\in W_{0}(A)$ so there exists $ \{x_{k}^{n}\}_{k=1}^{\infty}
, \parallel x_{k}^{n}\parallel ~=~1$ for all k and n such that for each $n~=~1,2,3,......$ $$Re(A^{*}Tx_{k}^{n},x_{k}^{n}) ~\longrightarrow~\epsilon_{n}
~as~
k~\longrightarrow~\infty$$ and $$\|Tx_{k}^{n}\|~\longrightarrow~\|T\|~~as ~ k~\longrightarrow~\infty.$$ Now for each n there exists $ k_{n}$ such that\
$\mid Re(A^{*}Tx_{k}^{n},x_{k}^{n})
~-~\epsilon_{n}\mid~<~\frac{1}{n}$ and $\mid \|Tx_{k}^{n}\|~-~\|T\| \mid~<~\frac{1}{n}$ for all $
k~\geq~ k_{n}.$ So we get\
$\mid Re(A^{*}Tx_{k_{n}}^{n},x_{k_{n}}^{n})
~-~\epsilon \mid~\leq~\mid Re(A^{*}Tx_{k_{n}}^{n},x_{k_{n}}^{n})
~-~\epsilon_{n}\mid~+~\mid \epsilon_{n} - \epsilon \mid
~\longrightarrow~0 $ as $n~\longrightarrow~\infty$.\
So $ Re(A^{*}Tx_{k_{n}}^{n},x_{k_{n}}^{n})~\longrightarrow~\epsilon $ as $n~\longrightarrow~\infty$.\
Similarly $ \|Tx_{k_{n}}^{n}\|~\longrightarrow~\|T\| $ as $n~\longrightarrow~\infty.$ Hence $\epsilon \in W_{0}(A). $ Thus $ W_{0}(A)$ is closed.\
Following an idea of Das et al[@3] we next show that $W_{0}(A)$ is convex. Let $ \mu $ and $ \eta \in W_{0}(A)$ , $ \mu \neq \eta $ and t be any real scalar $ 0<t<1 $. Then there exist $ \{x_{n}\}$ and $\{y_{n}\} $ in H with $\|x_{n}\| = 1, \|y_{n}\| =
1~ $ for all n , such that $$Re(A^{*}Tx_{n},x_{n}) \rightarrow \mu ~ and ~ \|Tx_{n}\| \rightarrow
\|T\|,$$ $$Re(A^{*}Ty_{n},y_{n}) \rightarrow \eta ~ and ~ \|Ty_{n}\| \rightarrow \|T\|.$$ For any real scalar $\epsilon$ and for any subsequence $\{x_{n_{k}}\},~\{y_{n_{k}}\}$ of sequences $
\{x_{n}\},~\{y_{n}\}$ respectively. Suppose $$\lim_{k \longrightarrow \infty} \parallel x_{n_{k}}~\pm~\epsilon y_{n_{k}}\parallel ~=~ 0$$ $\Rightarrow~ \lim_{k \longrightarrow \infty} \{
\parallel x_{n_{k}} \parallel^{2}~+~\epsilon^{2}\parallel y_{n_{k}}
\parallel^{2}~\pm~2\epsilon Re(x_{n_{k}},y_{n_{k}})\} ~=~ 0.$\
$\Rightarrow~\lim_{k \longrightarrow \infty} Re(x_{n_{k}},y_{n_{k}}) ~=~ \mp~\frac{1+\epsilon^{2}}{2\epsilon} $\
$\Rightarrow~\epsilon^{2}~ =~ 1$, otherwise $ \mid {\frac{1+ \epsilon^2}{2 \epsilon}} \mid > 1 $ which contradicts the fact that $ \mid Re(x_{n_{k}},y_{n_{k}}) \mid \leq 1 $.\
Let $\delta_{n_{k}}~=~x_{n_{k}}~\pm~\epsilon y_{n_{k}}.$ Then $\delta_{n_{k}}~\longrightarrow~0$ as $k~\longrightarrow~\infty.$\
Now $$\begin{array}{l}
\mu~=~\lim_{k \longrightarrow
\infty}Re(A^{*}Tx_{n_{k}},x_{n_{k}}) \\
~~~=~\lim_{k \longrightarrow
\infty}Re(A^{*}T(\delta_{n_{k}}~\mp~\epsilon
y_{n_{k}}),\delta_{n_{k}}~\mp~\epsilon y_{n_{k}}) \\
~~~=~\epsilon^{2}\lim_{k \longrightarrow
\infty}Re(A^{*}Ty_{n_{k}},y_{n_{k}})\\
~~~=~\epsilon^{2} \eta \\
~~~ =~\eta. \\
\end{array}$$ Thus we can assume that for any real scalar $\epsilon$ and for any subsequence $\{x_{n_{k}}\},~\{y_{n_{k}}\}$ of sequences $
\{x_{n}\},~\{y_{n}\}$ respectively $$\lim_{k
\longrightarrow \infty}
\parallel x_{n_{k}}~\pm~\epsilon y_{n_{k}}\parallel~\neq~0.$$ Now, $\{Re(A^{*}T(x_{n}+\epsilon y_{n}),x_{n}+\epsilon y_{n})\}$ is a bounded sequence and so it has a convergent subsequence, say, $\{Re(A^{*}T(x_{n_{k}}+\epsilon y_{n_{k}}),x_{n_{k}}+\epsilon
y_{n_{k}})\}$. Also $ \{\parallel x_{n_{k}}~+~\epsilon
y_{n_{k}}\parallel \}, $ being a bounded sequence, has a convergent subsequence, say $ \{\parallel x_{n_{k}}^{'}~+~\epsilon
y_{n_{k}}^{'}\parallel \} $. We show that there exists a real scalar $\epsilon$ for which\
$$\lim_{k \rightarrow \infty} \frac{Re(A^{*}T(x_{n_{k}}^{'} + \epsilon y_{n_{k}}^{'}),
x_{n_{k}}^{'} + \epsilon y_{n_{k}}^{'})}{ \|x_{n_{k}}^{'} + \epsilon y_{n_{k}}^{'}\|^{2}}
= t\mu + (1-t)\eta$$ $$\begin{array}{l}
\Leftrightarrow \lim_{k \rightarrow \infty} [
Re(A^{*}Tx_{n_{k}}^{'},x_{n_{k}}^{'})~+~\ \epsilon^{2}
Re(A^{*}Ty_{n_{k}}^{'},y_{n_{k}}^{'})~+~ \epsilon \{
Re(A^{*}Tx_{n_{k}}^{'},y_{n_{k}}^{'}) + Re(A^{*}Ty_{n_{k}}^{'},x_{n_{k}}^{'}) \}\\\\
~-~\{t\mu + (1-t)\eta\}\|x_{n_{k}}^{'} + \epsilon
y_{n_{k}}^{'}\|^{2}]~=~0\\\\
\Leftrightarrow \mu~+~\epsilon^{2} \eta~+~\epsilon \lim_{k \rightarrow
\infty} \{Re(A^{*}Tx_{n_{k}}^{'},y_{n_{k}}^{'}) + Re(A^{*}Ty_{n_{k}}^{'},x_{n_{k}}^{'})\}\\\\
~=~\{t\mu + (1-t)\eta\} \{1 + \epsilon^{2}+~2\lim_{k \rightarrow
\infty}Re \epsilon (x_{n_{k}}^{'},y_{n_{k}}^{'}) \}
\end{array}$$ This on simplification yields $$\epsilon^{2}~+~C\epsilon~~-~\frac{1-t}{t}~=~0,~~~~~~~~~~(3)$$ where C is a real constant independent of $\epsilon $.\
Equation(3) has two nonzero roots of different signs, say $r_{1}$ and $-r_{2}$ ( $ r_1,~r_2 > 0 )$.\
Now $ \{\parallel T(x_{n_{k}}^{'}~\pm~r
y_{n_{k}}^{'})\parallel \} $ is a bounded sequence and so it has a convergent subsequence, say, $ \{\parallel T(
\tilde{x}_{n_{k}}~\pm~r \tilde{y}_{n_{k}})\parallel \}
$ where r is any positive real number.\
So, $$\begin{array}{l}
\|T\|^2 \geq \lim_{k \rightarrow \infty} \frac{ \parallel T
(\tilde{x}_{n_{k}}~\pm~r \tilde{y}_{n_{k}})\parallel
^{2}}{ \parallel \tilde{x}_{n_{k}}~\pm~r
\tilde{y}_{n_{k}}\parallel^{2}}\\
~~~~~=~\parallel T \parallel ^{2}~\pm~2r~\lim_{k \rightarrow
\infty}~\frac {Re(T^{*}T\tilde{x}_{n_{k}}~-~\parallel T
\parallel
^{2}\tilde{x}_{n_{k}},\tilde{y}_{n_{k}})}{\parallel
\tilde{x}_{n_{k}}~\pm~r\tilde{y}_{n_{k}}\parallel^{2}}.
\end{array}$$ The sign of the second term on the R.H.S. is independent of r and so for all positive real r,\
either $$\parallel T (\frac{\tilde{x}_{n_{k}}~+~r
\tilde{y}_{n_{k}}}{\parallel \tilde{x}_{n_{k}}~+~r
\tilde{y}_{n_{k}}\parallel} )\parallel
~\longrightarrow~\parallel T \parallel ~as
~k~\longrightarrow~\infty$$ or $$\parallel T (\frac{\tilde{x}_{n_{k}}~-~r
\tilde{y}_{n_{k}}}{\parallel \tilde{x}_{n_{k}}~-~r
\tilde{y}_{n_{k}}\parallel} ) \parallel
~\longrightarrow~\parallel T \parallel ~as
~k~\longrightarrow~\infty.$$ Let $$z_{n_{k}}~=~\frac{\tilde{x}_{n_{k}}~+~r
\tilde{y}_{n_{k}}}{\parallel \tilde{x}_{n_{k}}~+~r
\tilde{y}_{n_{k}}\parallel}$$ and $$w_{n_{k}}~=~ \frac{\tilde{x}_{n_{k}}~-~r
\tilde{y}_{n_{k}}}{\parallel \tilde{x}_{n_{k}}~-~r
\tilde{y}_{n_{k}}\parallel}.$$ From above we see that either $ \|Tz_{n_{k}}\|~\longrightarrow~\|T\| $ and $
Re(A^{*}Tz_{n_{k}},z_{n_{k}})~\longrightarrow~t\mu +
(1-t)\eta~as~k~\longrightarrow~\infty,$ or $\|Tw_{n_{k}}\|~\longrightarrow~\|T\| $ and $Re(A^{*}Tw_{n_{k}},w_{n_{k}})~\longrightarrow~t\mu +
(1-t)\eta~as~k~\longrightarrow~\infty.$\
Hence $ t\mu + (1-t)\eta~\in~W_{0}(A)$. Thus $ W_{0}(A)$ is convex.\
This completes the proof of the theorem.\
We now prove the following Theorem :\
**Theorem 3.2** \[T:th2\]\
$ \|T \| \leq \|T - \epsilon A \| $ $ \forall \epsilon \in $ R iff $ \exists $ $ \{x_{n}\}, \|x_{n}\| = 1 $ such that $
Re(A^{*}Tx_{n},x_{n}) \rightarrow 0 $ and $ \|Tx_{n}\| \rightarrow
\|T\| $.\
***Proof.*** Suppose $ \exists~ \{x_{n}\}, \|x_{n}\| = 1 $ such that $ (A^{*}Tx_{n},x_{n})
\rightarrow 0 $ and $ \|Tx_{n}\| \rightarrow \|T\| $.\
Then $ {\|(T - \epsilon A)x_{n}\|}^{2} ={\|Tx_{n}\|}^{2} +{\epsilon}^{2}
{\|Ax_{n}\|}^{2} $ - 2Re $ \epsilon (Tx_{n},Ax_{n}) $, $ \forall \epsilon
\in $ R. So $ {\|T-\epsilon A \|}^{2} \geq \limsup_{n \rightarrow \infty}
{\|(T-\epsilon A)x_{n}\|}^{2} \geq {\|T\|}^{2} ~~ \forall \epsilon \in $ R. Hence $ \|T-\epsilon A \| \geq \|T\|~~ \forall \epsilon \in $ R.\
Conversely let $ \|T\| \leq \|T- \epsilon A \| $ $ \forall \epsilon \in $ R. We need to show $ 0 \in W_{0}(A) $.\
Without loss of generality we can assume that $ \|A\| = 1 $.\
Suppose $ 0 \not\in W_{0}(A) $ . Then as $ W_{0}(A) $ is closed and convex by rotating T suitably we can assume that $ W_{0}(A) > \eta > 0 $.\
Let M = {$x\in H ~:~ \|x\|$ = 1 and Re $ (Tx,Ax) \leq \eta $/2} and $\beta $ = $ sup_{x \in M} \|Tx\| $. We first claim that $ \beta < \|T\| $. Suppose $ \beta = \|T\|$. Then there exists $ x_n \in M $ such that $ \| Tx_n \| \rightarrow \|T\| $. As $ x_n \in M $ so $ Re (Tx_n,Ax_n)
\leq \eta/2 $ and $ \|x_n \|=1$. Now $ \{ Re(Tx_n,Ax_n)\} $ is a bounded sequence and so it has a convergent subsequence, without loss of generality we can assume that $ \{ Re(Tx_n,Ax_n)\} $ is convergent and converges to some point $ \mu $ (say). Then $ \mu \in W_0(A)$. Now $ Re(Tx_n,Ax_n) \leq \eta/2 $ and so $ \mu \leq \eta / 2$. This contradicts the fact that $ W_{0}(A) > \eta $.\
Let $ \epsilon_0 = \min \{ \eta, \frac{\|T\| - \beta}{2 \|A\|} \} $.\
Let $ x \in M $ . Then $ \|(T-\epsilon_{0}A)x\| \leq \|Tx\| + \mid \epsilon_{0} \mid \|Ax\| \leq \beta + \{( \|T\| - \beta )/ (2 \|A\|) \} \|A\| = \|T\|/2 + \beta /2 $. So $ \sup_{x \in M ~and~ \|x\|=1 } \| (T-\epsilon_0 A)x \| \leq (\|T\| + \beta)/2 < \|T\| $.\
Again let $ x \notin M $. Then let $ Tx = (a+ib)Ax + y $, where $ (Ax,y) $ = 0. Now $ 2a \geq 2a ||Ax\|^2 = 2 Re (Tx,Ax) > \eta \geq \epsilon_0 $ and so $ 2a - \epsilon_0 > 0 $. So $$\begin{aligned}
{\|(T-\epsilon_{0}A)x\|}^{2}&=&\{{(a-\epsilon_{0})}^{2} + b^{2}\} {\|Ax\|}^{2}+{\|y\|}^{2}\hspace{2cm} \\
&=&\mbox{}{\|Tx\|}^{2} + ({\epsilon_{0}}^{2} - 2a \epsilon_{0}) {\|Ax\|}^{2}\\
& \leq & {\|Tx\|}^{2} + ({\epsilon_{0}}^{2} - 2a \epsilon_{0}) \\
& \leq & {\|T\|}^{2} + ({\epsilon_{0}}^{2} - 2a \epsilon_{0})\end{aligned}$$ Hence $ \sup_{x \notin M ~and~ \|x\|=1 } \| (T-\epsilon_0 A)x \|^2 \leq \|T\|^2 + ({\epsilon_{0}}^{2} - 2a \epsilon_{0}) $. Thus in all cases $ \| T - \epsilon_0 A \| < \|T\| $. Hence $ \exists \hspace{.15cm} \{x_{n}\}, \|x_{n}\|$ = 1 $ \forall n,~ (Tx_{n},Ax_{n}) \rightarrow 0 $ and $ \|Tx_{n}\| \rightarrow \|T\| $.\
So far we proved that for any two operators T and A in B(H) with $ \| A\| \leq 1 $, $ \|T \| \leq \|T - \epsilon A \| $ $ \forall \epsilon \in $ R implies that $ \exists $ $ \{x_{n}\}, \|x_{n}\| = 1 $ such that $
Re(A^{*}Tx_{n},x_{n}) \rightarrow 0 $ and $ \|Tx_{n}\| \rightarrow \|T\| $.\
This completes the proof.\
From the last theorem it follows that if $ \epsilon_0 $ is the real center of mass of T relative to A then $ \exists $ $ \{x_{n}\}, \|x_{n}\| = 1 $ such that $ Re((T - \epsilon_0 A)x_{n},A x_{n}) \rightarrow 0 $ and $ \|(T - \epsilon_0 A)x_{n}\| \rightarrow \|T - \epsilon_0 A\| $.\
We next show the uniqueness of $ \epsilon_0 $ under the assumption that the approximate point spectrum of A, $\sigma_{app}(A)$ does not contain 0. Suppose $$\| T \| = \| T - \epsilon_0 A\| \leq \|T- \epsilon A \| ~ \forall ~\epsilon ~\in R ~~and~~ \epsilon_{0} \neq 0.$$ Then $ \exists \hspace{.15cm}\{x_{n}\},\|x_{n}\|$ = 1 such that $ ( (T - \epsilon_{0}A)x_{n}, Ax_{n} ) \rightarrow 0 $ and $ \|(T - \epsilon_{0}A)x_{n} \| \rightarrow \|T - \epsilon_{0}A \| $ .\
So $$\begin{aligned}
{\|T - \epsilon_{0} A\|}^2 &=& \lim \{ {\|(T - \epsilon_{0} A)x_{n}\|}^{2} \} \\
&=& \lim \{ {\|Tx_{n}\|}^{2} +\epsilon_{0}^{2}{\|Ax_{n}\|}^{2}
- 2Re(\epsilon_{0} (Tx_{n},Ax_{n})) \}\\
&=& \lim \{ {\|Tx_{n}\|}^{2} -\epsilon_{0}^{2}{\|Ax_{n}\|}^{2} \} \\
&=& \lim \{ {\|Tx_{n}\|}^{2}\} -{\mid\epsilon_{0}\mid}^{2} \lim \{{\|Ax_{n}\|}^{2} \} \\
&<& \lim \{ {\|Tx_{n}\|}^{2}\}~~,~~~~~ since~0~~\notin~~ \sigma_{app}(A)\\
&\leq& { \|T\|}^2\end{aligned}$$ This contradicts the fact that $ \| T \| = \| T - \epsilon_0 A\| $. Hence $ \epsilon_0 = 0 $.\
We give an example to show that $ \epsilon_{0} $ may not be unique if $ 0~~\in~~ \sigma_{app}(A) $.\
Let T and A be two bounded linear operators defined on $ R^{2} $ as T (x,y) = (x,0) and A(x,y) = (0,y), $ \forall ~(x,y) \in R^{2} $. Then $ \| T \| = \| T - A \| = \| T - (-1) A \| \leq \| T - \epsilon A \| ~~ \forall ~\epsilon \in R $.\
We are now in a position to prove\
**Theorem 3.3** \[T:th3\]\
**(Min-max equality)** For a strongly accretive, bounded operator T on a Hilbert space\
$$\sup_{\|x\| = 1}~~ \inf _{\epsilon > 0} \| (\epsilon T - I ) x \|^2 = \inf _{\epsilon > 0} ~~\sup_{\|x\| = 1}\| (\epsilon T - I ) x \|^2 ~.$$ ***Proof.*** For a fixed but arbitrary $ y \in H,~ \|y\| = 1 $ we have\
$$\begin{aligned}
\| (\epsilon T - I ) y \|^2 & \leq &~~\sup_{\|x\| = 1}\| (\epsilon T - I ) x \|^2 \\
\Rightarrow \inf _{\epsilon > 0} \| (\epsilon T - I ) y \|^2 & \leq & \inf _{\epsilon > 0} ~~\sup_{\|x\| = 1}\| (\epsilon T - I ) x \|^2\end{aligned}$$ This holds for all y in H with $ \|y\| = 1 $. So $$\sup_{\|x\| = 1}~~ \inf _{\epsilon > 0} \| (\epsilon T - I ) x \|^2 \leq \inf _{\epsilon > 0} ~~\sup_{\|x\| = 1}\| (\epsilon T - I ) x \|^2 ~.$$ For the converse part we use the notion of center of mass of an operator. As T is strongly accretive so there exists a real scalar $ \epsilon_0 > 0 $ such that $$\| \epsilon_0 T - I \| = \inf _{\epsilon > 0} \| (\epsilon T - I ) \|.$$ Further there exists a sequence $ \{x_{n}\}, \|x_{n}\| = 1 $ such that $ Re((\epsilon_0 T - I)x_{n}, Tx_{n}) \rightarrow 0 $ and $ \|(\epsilon_0 T - I)x_{n}\| \rightarrow \|\epsilon_0 T - I\| $.\
Now $$\begin{aligned}
\| \epsilon_0 T - I \|^2 & = & \lim_{n\rightarrow \infty} \|(\epsilon_0 T - I)x_n \|^2 \\
& = & \lim_{n\rightarrow \infty} \{ 1 - 2 \epsilon_0 Re \langle T x_n, x_n \rangle + {\epsilon_0}^2 \|Tx_n\|^2 \} \\
& = & \lim_{n\rightarrow \infty} \{ 1 - \frac{{ Re \langle T x_n, x_n \rangle }^2}{ \|Tx_n\|^2 } \}\\
& \leq & \sup_{\|x\| = 1} \{ 1 - \frac{{ Re \langle T x, x \rangle }^2}{ \|Tx\|^2 } \}~~~~~(*)\end{aligned}$$ For a fixed y in H with $ \|y \| = 1 $ we see that $$\| (\epsilon T - I ) y \|^2 = 1 - 2 \epsilon Re \langle T y, y \rangle + {\epsilon}^2 \|Ty\|^2$$ achieves its minimum at $ \epsilon(y) = \frac{{ Re \langle T y, y \rangle }}{ \|Ty\|^2 } $ and the minimum value is $$\inf_{\epsilon > 0 } \| (\epsilon T - I ) y \|^2 = 1 - \frac{{ Re \langle T y, y \rangle }^2}{ \|Ty\|^2 } .$$ Using this in (\*) we get $$\| \epsilon_0 T - I \|^2 \leq \sup_{\|x\| = 1} \inf_{\epsilon > 0 } \| (\epsilon T - I ) x\|^2$$ and so $$\inf _{\epsilon > 0} ~~\sup_{\|x\| = 1}\| (\epsilon T - I ) x \|^2 \leq \sup_{\|x\| = 1}~~ \inf _{\epsilon > 0} \| (\epsilon T - I ) x \|^2 .$$ Thus we have the Min-max equality $$\sup_{\|x\| = 1}~~ \inf _{\epsilon > 0} \| (\epsilon T - I ) x \|^2 = \inf _{\epsilon > 0} ~~\sup_{\|x\| = 1}\| (\epsilon T - I ) x \|^2 ~.$$ Next, if $\epsilon_{0}$ is the real center of mass of a bounded linear operator $T$ , then there exists a sequence $\{ x_{n} \}$ in H, $\|x_{n}\|=1$, $Re((I-\epsilon_{0}T)x_{n},Tx_{n}) \rightarrow 0$ and $\|(I-\epsilon_{0}T)x_{n}\| \rightarrow \|I-\epsilon_{0}T\|$.\
For a bounded linear operator T the antieigenvalue is defined as\
$$\cos T = \inf_{\| Tx \| \neq 0 } \frac{ Re \langle Tx,x \rangle } { \|Tx\| \|x\|}~.$$ We next prove the theorem\
**Theorem 3.4** \[T:th4\]\
Suppose $ \epsilon_{0} $ is the real center of mass of a strictly accretive, bounded linear operator T. Then $\cos T = \lim_{n \rightarrow \infty} \frac{Re(Tx_{n},x_{n})}{\|Tx_{n}\|}$ where $\{ x_{n} \}$ is a sequence of unit vectors in H, $Re((I-\epsilon_{0}T)x_{n},Tx_{n}) \rightarrow 0$ and $\|(I-\epsilon_{0}T)x_{n}\| \rightarrow \|I-\epsilon_{0}T\|$.\
***Proof.*** As $ \epsilon_0 $ is the real center of mass of T so there exists a sequence $ \{x_{n}\}, \|x_{n}\| = 1 $ such that $ Re((I - \epsilon_0 T )x_{n}, Tx_{n}) \rightarrow 0 $ and $ \|(I - \epsilon_0 T )x_{n}\| \rightarrow \|I - \epsilon_0 T\| $ and $$\|I - \epsilon_0 T\| = \inf_{\epsilon > 0} \| I - \epsilon T \| = \inf _{\epsilon > 0} ~~\sup_{\|x\| = 1}\| (I - \epsilon T ) x \| ~.$$ As in Theorem 3.3 we have $$\begin{aligned}
\| \epsilon_0 T - I \|^2 & = & \lim_{n\rightarrow \infty} \{ 1 - \frac{{ Re \langle T x_n, x_n \rangle }^2}{ \|Tx_n\|^2 } \}\\
& \leq & \sup_{\|x\| = 1} \{ 1 - \frac{{ Re \langle T x, x \rangle }^2}{ \|Tx\|^2 } \} \\
& \leq & \sup_{\|x\| = 1} \inf_{\epsilon > 0 } \| (I - \epsilon T ) x \|^2 \\
& = & \| \epsilon_0 T - I \|^2\end{aligned}$$ This shows that $$\inf_{\|x\| = 1 } \frac{{ Re \langle T x, x \rangle }}{ \|Tx\| } = \lim_{n\rightarrow \infty} \frac{{ Re \langle T x_n, x_n \rangle }}{ \|Tx_n\| } = \cos T.$$ We next give an example to calculate antieigenvalue for a finite dimensional operator using the above method.\
**Example 3.5**\
$T=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1+i \\
\end{array}
\right)
$ be an operator on a two dimensional complex Hilbert space H. Let, $z=(z_{1}, z_{2})^{t} \in H$, where $\left|z_{1}\right|^{2} + \left| z_{2}\right|^{2} = 1$. Then, $Re(Tz,z)=1$ and $\|Tz\|= \sqrt{1+\left|z_{2}\right|^{2}}$. Now, $\sup_{\|z\| = 1}\inf_{\epsilon >0}\|(\epsilon T - I)z\| =\sqrt{0.5}$ and this supremum is attained by the vector $z_{0}=(z_{1},0)^{t}$, where $\left|z_{1}\right|=1$. Then, $\epsilon_{0}=0.5$ and $\|\epsilon_{0}T-I\|=\sqrt{0.5}$. Now, $\cos T = \frac{Re(Tz_{0},z_{0})}{\|Tz_{0}\|}=\frac{1}{\sqrt{2}}$.\
In [@18] we proved that if $ \lambda_0 $ is the total center of mass of T then there exists a sequence $ \{x_{n}\}, \|x_{n}\| = 1 $ such that $ ((I - \lambda_0 T )x_{n}, Tx_{n}) \rightarrow 0 $ and $ \|(I - \lambda_0 T )x_{n}\| \rightarrow \|I - \lambda_0 T\| $.\
For a bounded linear operator T the total antieigenvalue is defined as\
$$\mid \cos \mid T = \inf_{\| Tx \| \neq 0 } \frac{ \mid \langle Tx,x \rangle \mid } { \|Tx\| \|x\|}~.$$ In [@16] we also studied the total antieigenvalue of a bounded linear operator.\
We now prove the theorem\
**Theorem 3.6**\
For a bounded linear operator T on a Hilbert space\
$$\sup_{\|x\| = 1}~~ \inf _{\lambda \in C} \| (\lambda T - I ) x \|^2 = \inf _{\lambda \in C} ~~\sup_{\|x\| = 1}\| (\lambda T - I ) x \|^2 ~.$$ ***Proof.*** For a fixed but arbitrary $ y \in H,~ \|y\| = 1 $ we have\
$$\begin{aligned}
\| (\lambda T - I ) y \|^2 & \leq &~~\sup_{\|x\| = 1}\| (\lambda T - I ) x \|^2 \\
\Rightarrow \inf _{\lambda \in C} \| (\lambda T - I ) y \|^2 & \leq & \inf _{\lambda \in C} ~~\sup_{\|x\| = 1}\| (\lambda T - I ) x \|^2\end{aligned}$$ This holds for all y in H with $ \|y\| = 1 $. So $$\sup_{\|x\| = 1}~~ \inf _{\lambda \in C} \| (\lambda T - I )x \|^2 \leq \inf _{\lambda \in C} ~~\sup_{\|x\| = 1}\| (\lambda T - I ) x \|^2 ~.$$ For the converse part we use the notion of center of mass of an operator. As T is a bounded linear operator so there exists a scalar $ \lambda_0 $ such that $$\| \lambda_0 T - I \| = \inf _{\lambda \in C} \| (\lambda T - I ) \|.$$ Further there exists a sequence $ \{x_{n}\}, \|x_{n}\| = 1 $ such that $ ((\lambda_0 T - I)x_{n}, Tx_{n}) \rightarrow 0 $ and $ \|(\lambda_0 T - I)x_{n}\| \rightarrow \|\lambda_0 T - I\| $.\
Now $$\begin{aligned}
\| \lambda_0 T - I \|^2 & = & \lim_{n\rightarrow \infty} \|(\lambda_0 T - I)x_n \|^2 \\
& = & \lim_{n\rightarrow \infty} \{ 1 - 2 Re \lambda_0 \langle T x_n, x_n \rangle + {\lambda_0}^2 \|Tx_n\|^2 \} \\
& = & \lim_{n\rightarrow \infty} \{ 1 - \frac{{ \mid \langle T x_n, x_n \rangle \mid}^2}{ \|Tx_n\|^2 } \}\\
& \leq & \sup_{\|x\| = 1} \{ 1 - \frac{{ \mid \langle T x, x \rangle \mid }^2}{ \|Tx\|^2 } \}~~~~~(*)\end{aligned}$$ For a fixed y in H with $ \|y \| = 1 $ we see that $$\| (\lambda T - I ) y \|^2 = 1 - 2 Re \lambda \langle T y, y \rangle + {\lambda}^2 \|Ty\|^2$$ $$\Rightarrow \| (\lambda T - I ) y \|^2 = \|Ty\|^2 \{ \mid \lambda - \frac{ \langle y,Ty \rangle}{\|Ty\|^2} \mid ^2 \} + \{ 1 - \frac{ \mid \langle Ty,y \rangle \mid ^2}{ \|Ty\|^2 }\} .$$ achieves its minimum at $ \lambda(y) = \frac{ \langle y,Ty \rangle}{\|Ty\|^2} $ and the minimum value is $$\inf_{\lambda \in C } \| (\lambda T - I ) y \|^2 = 1 - \frac{{ \mid \langle T y, y \rangle \mid}^2}{ \|Ty\|^2 } .$$ Using this in (\*) we get $$\| \lambda_0 T - I \|^2 \leq \sup_{\|x\| = 1} \inf_{\lambda \in C} \| (\lambda T - I ) x \|^2$$ and so $$\inf _{\lambda \in C} ~~\sup_{\|x\| = 1}\| (\lambda T - I ) x \|^2 \leq \sup_{\|x\| = 1}~~ \inf _{\lambda \in C} \| (\lambda T - I ) x \|^2 .$$ Thus we have the equality $$\sup_{\|x\| = 1}~~ \inf _{\lambda \in C} \| (\lambda T - I ) x \|^2 = \inf _{\lambda \in C} ~~\sup_{\|x\| = 1}\| (\lambda T - I ) x \|^2 ~.$$ **Theorem 3.7**\
Suppose $ \lambda_0 $ is the total center of mass of a bounded linear operator T. Then, $\left|\cos\right| T = \lim_{n \rightarrow \infty} \frac{\mid (Tx_{n},x_{n})\mid}{\|Tx_{n}\|}$ where $\{ x_{n} \}$ is a sequence of unit vectors in H, $((I- \lambda_{0}T)x_{n},Tx_{n}) \rightarrow 0$ and $\|(I-\lambda_{0}T)x_{n}\| \rightarrow \|I-\lambda_{0}T\|$\
***Proof.*** In [@18] we proved that if $ \lambda_0 $ is the total center of mass of T then there exists a sequence $ \{x_{n}\}, \|x_{n}\| = 1 $ such that $ ((I - \lambda_0 T )x_{n}, Tx_{n}) \rightarrow 0 $ and $ \|(I - \lambda_0 T )x_{n}\| \rightarrow \|I - \lambda_0 T\| $ and $$\|I - \lambda_0 T\| = \inf_{\lambda \in C} \| I - \lambda T \| = \inf _{\lambda \in C} ~~\sup_{\|x\| = 1}\| (I - \lambda T ) x \| ~.$$ As in Theorem 3.6 we have $$\begin{aligned}
\| \lambda_0 T - I \|^2 & = & \lim_{n\rightarrow \infty} \{ 1 - \frac{{ \mid \langle T x_n, x_n \rangle \mid}^2}{ \|Tx_n\|^2 } \}\\
& \leq & \sup_{\|x\| = 1} \{ 1 - \frac{{ \mid \langle T x, x \rangle \mid}^2}{ \|Tx\|^2 } \} \\
& \leq & \sup_{\|x\| = 1} \inf_{\lambda \in C } \| (I - \lambda T ) x \|^2 \\
& = & \| \lambda_0 T - I \|^2\end{aligned}$$ This shows that $$\inf_{\|x\| = 1 } \frac{{ \mid \langle T x, x \rangle \mid }}{ \|Tx\| } = \lim_{n\rightarrow \infty} \frac{{ \mid \langle T x_n, x_n \rangle \mid}}{ \|Tx_n\| } = \left|\cos \right| T$$\
We next give an example to calculate total antieigenvalue for a finite dimensional operator using the above method.\
**Example 3.8**\
$T=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1+i \\
\end{array}
\right)
$ be an operator on a two dimensional complex Hilbert space H. Let, $z=(z_{1}, z_{2})^{t} \in H$, where $\left|z_{1}\right|^{2} + \left| z_{2}\right|^{2} = 1$. Then, $\left|(Tz,z)\right|=\sqrt{1+\left|z_{2}\right|^{4}}$ and $\|Tz\|=\sqrt{ 1+\left|z_{2}\right|^{2}}$. Now, $\sup_{\|z\| = 1}\inf_{\lambda \in C}\|(\lambda T - I)z\| =\sqrt{2}-1$ and this supremum is attained by the vector $z_{0}=(z_{1},z_{2})^{t}$, where $\left|z_{2}\right|^{2}=\sqrt{2}-1$. Then, $\lambda_{0}=\frac{1-i(\sqrt{2}-1)}{\sqrt{2}}$ and $\|\lambda_{0}T-I\|=\sqrt{2}-1$. Now, $\left|\cos\right| T = \frac{Re(Tz_{0},z_{0})}{\|Tz_{0}\|}=\sqrt{2\sqrt{2}-2}$.\
**Acknowledgement**: We would like to thank Professor T. K. Mukherjee and Professor K. C. Das for their suggestion while preparing the paper. Second author would like to thank CSIR, India for supporting his research.
[1]{} Asplund E. and Pt$\acute{a}$k V., *A Minimax Inequality for Operators and a Related Numerical Range*, Acta. Math, **126** (1971), 53-62. Das K.C., DasGupta M. and Paul K., *Structure of the antieigenvectors of a strictly accretive operator* International J. Math. and Math. Sci., **21** No. 4, (1998),761-766. Das K.C., Majumder S. and Sims B., *Restricted numerical range and weak convergence on the boundary of numerical range*, Jour. Math. Phy. Sci., **21** No.1, (1987) , 35-42. James R.C., *Orthogonality and linear functionals in normed linear spaces*, Trans. Amer. Math. Soc. **61** (1947), 265-292. Gustafson K., *A Min-Max Theorem*, Notices Amer. Math. Soc. **15** (1968d), 799. Gustafson K., *Angle of an Operator and Positive Operator Products*, Bull. Amer. Math. Soc., **74** (1968a), 488-492. Gustafson K., *Positive(noncommutating) Operator Products and Semigroups*, Math. Zeit. **105** (1968b), 160-172. Gustafson K., *A note on left multiplication of semigroup generators*, Pacific J. Math. **24** (1968c), 463-465. Gustafson K., *Antieigenvalue Inequalities in Operator Theory* , Inequalities III, Proceedings Los Angeles Symposium, 1969 ed. O. Shisha, Academic Press (1972), 115-119. Gustafson K., *An extended operator trigonometry*, Linear Algebra Appl. **319** (2000), 117-135. Gustafson K., *Operator Trigonometry*, Linear and Multilinear Algebra, **37** (1994), 139-159. Gustafson K., *Matrix Trigonometry*, Linear Algebra Appl., **217** (1995), 117-140. Gustafson, K., *Interaction antieigenvalues*, J. Math. Anal. Appl., **299** (2004), 174-185.
Gustafson K. and Rao D., *Numerical Range and Accretivity of Operator Products*, J. Math. Anal. Appl., **60** (1977), 693-702. Gustafson K. and Rao D., *Numerical Range: The Field Values of Linear Operators and Matrices*, Springer, New York, 1997. Gustafson K. and Seddighin M., *Antieigenvalue Bounds*, J. Math. Anal. Appl., **143** (1989), 327-340.
Hossein Sk.M., Das K.C., Debnath L. and Paul K., *Bounds for total antieigenvalue of a normal operator* International J. Math. and Math. Sci., **70** (2004), 3877-3884. Hossein Sk.M., Paul K., Debnath L. and Das K.C., *Symmetric Anti-eigenvalue and Symmetric Anti-eigenvector* J. Math. Analysis and Applications., **345** (2008), 771-776. Paul K., Hossein Sk.M. and Das K.C., *Orthogonality on B(H,H) and Minimal-norm Operator*, Journal of Analysis and Applications, **6** (2008), 169-178. Seddighin M., *Antieigenvalues and total antieigenvalues of normal operators*, J. Math. Anal. Appl., **274** (2002) no.1, 239-254. Stampfli J.G., *The Norm of A Derivation*, Pacific Journal of Mathematics, **33(3)** (1970), 737-747.
[Kallol Paul]{}\
[Department of Mathematics]{}\
[Jadavpur University, Kolkata 700032, West Bengal, INDIA]{}\
[email:]{} *kalloldada@yahoo.co.in*\
\
[Gopal Das]{}\
[Department of Mathematics, Jadavpur University, Kolkata 700032, INDIA ]{}\
[email:]{} *gopaldasju@gmail.com*\
|
---
abstract: 'Highlighting is a powerful tool to pick out important content and emphasize. Creating summary highlights at the sub-sentence level is particularly desirable, because sub-sentences are more concise than whole sentences. They are also better suited than individual words and phrases that can potentially lead to disfluent, fragmented summaries. In this paper we seek to generate summary highlights by annotating summary-worthy sub-sentences and teaching classifiers to do the same. We frame the task as jointly selecting important sentences and identifying a single most informative textual unit from each sentence. This formulation dramatically reduces the task complexity involved in sentence compression. Our study provides new benchmarks and baselines for generating highlights at the sub-sentence level.'
author:
- |
Kristjan Arumae$^\spadesuit$, Parminder Bhatia$^\diamondsuit$, Fei Liu$^\spadesuit$\
$^\spadesuit$Computer Science Department, University of Central Florida\
$^\diamondsuit$Amazon, USA\
[{arumae,parmib}@amazon.com feiliu@cs.ucf.edu]{}\
bibliography:
- 'kristjan.bib'
- 'fei\_highlights.bib'
nocite: '[@Mann:1988]'
title: 'Towards Annotating and Creating Sub-Sentence Summary Highlights'
---
Introduction {#sec:intro}
============
Highlighting at an appropriate level of granularity is important to emphasize salient content in an unobtrusive manner. A small collection of keywords may be insufficient to deliver the main points of an article, while highlighting whole sentences often provide superfluous information. In domains such as newswire, scholarly publications, legal and policy documents [@Kim:2010; @Sadeh:2013; @Hasan:2014], people are tempted to write long and complicated sentences. It is particularly desirable to pick out only *important sentence parts* as opposed to whole sentences.
Generating highlights at the sub-sentence level has not been thoroughly investigated in the past. A related thread of research is extractive and compressive summarization [@Daume:2002; @Zajic:2007; @Martins:2009; @Filippova:2010; @Kirkpatrick:2011; @Thadani:2013; @Wang:2013; @Li:2013:EMNLP; @Li:2014:EMNLP; @Durrett:2016]. The methods select representative sentences from source documents, then delete nonessential words and constituents to form compressed summaries. Nonetheless, making multiple interdependent decisions on word deletion can render summaries ungrammatical and fragmented. In this paper, we investigate an alternative formulation that can dramatically reduce the task complexity involved in sentence compression.
We frame the task as jointly selecting representative sentences from a document and identifying a *single* most informative textual unit from each sentence to create sub-sentence highlights. This formulation is inspired by rhetorical structure theory (RST; Mann and Thompson, 1988) where sub-sentence highlights resemble the *nuclei* which are text spans essential to express the writer’s purpose. The formulation also mimics human behavior on picking out important content. If multiple parts of a sentence are important, a human uses a single stroke to highlight them all, up to the whole sentence. If only a part of the sentence is relevant, she only picks out that particular sentence part.
Generating sub-sentence highlights is advantageous over abstraction [@See:2017; @Chen:2018:ACL; @Gehrmann:2018; @Lebanoff:2018; @Celikyilmaz:2018] in several aspects. The highlights can be overlaid on the source document, allowing them to be interpreted in context. The number of highlights is controllable by limiting sentence selection. In contrast, adjusting summary length in an end-to-end, abstractive system can be difficult. Further, highlights are guaranteed to be true-to-the-original, while system abstracts can sometimes “hallucinate” facts and distort the original meaning. Our contributions in this work include the following:
[ll]{} **(i)**:& , -lrb- cnn -rrb- prosecutor leading an investigation into the of flight\
&wednesday that was not aware any from board the plane .\
\
**(ii)**:& prosecutor leading an investigation into the\
&\
\
**(iii)**:&[ ]{} prosecutor leading an investigation into the [ ]{}\
& [ ]{}\
- we introduce a new task formulation of creating sub-sentence summary highlights, then describe an annotation scheme to obtain binary sentence labels for extraction, as well as start and end indices to mark the most important textual unit of a positively labeled sentence;
- we examine the feasibility of using neural extractive summarization with a multi-termed objective to identify summary sentences and their most informative sub-sentence units. Our study provides new benchmarks and baselines for highlighting at the sub-sentence level.
Annotating Sub-Sentence Highlights {#sec:annotation}
==================================
We propose to derive gold-standard sub-sentence highlights from human-written abstracts that often accompany the documents [@Hermann:2015]. However, the challenge still exists, because abstracts are very loosely aligned with source documents and they contain unseen words and phrases. We define *a summary-worthy sub-sentence unit* as the longest consecutive subsequence that contains content of the abstract. We obtain gold-standard labels for sub-sentence units by first establishing word alignments between the document and abstract, then smoothing word labels to generate sub-sentence labels.
**Word Alignment**The attention matrix of neural sequence-to-sequence models provides a powerful and flexible mechanism for word alignment. Let $S$=$\{w_i\}_{i=1}^\textsf{M}$ be a sequence of words denoting the document, and $T$=$\{w_t\}_{t=1}^\textsf{N}$ denoting the abstract. The attention weight $\alpha_{t,i}$ indicates the amount of attention received by the $i$-th document word in order to generate the $t$-th abstract word. All attention values ($\boldsymbol\alpha$) can be automatically learned from parallel training data. After the model is trained, we identify *a single document word* that receives the most attention for generating each abstract word, as denoted in Eq. (\[eq:alignment\]) and illustrated by Figure \[figure:annot\] (i). This step produces a set of source words containing the content of the abstract but possibly with distinct word forms.[^1] $$\begin{aligned}
w_i^{(t)} = \operatorname*{arg\,max\,}_{i\in\textsf{M}} \alpha_{t,i} \quad \forall t
{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:alignment}\end{aligned}$$
**Smoothing**Our goal is to identify sub-sentence units containing content of the abstract by smoothing word labels obtained in the previous step. We extract a single most informative textual unit from a sentence. As a first attempt, we obtain start and end indices of sub-sentence units using heuristics, which are described as follows:
- connecting two selected words if there is a small gap ($<$5 words) between them. For example, in Figure \[figure:annot\] (ii), the gap between “*crash*” and “*germanwings*” is bridged by labelling all gap words as selected;
- the longest consecutive subsequence after filling gaps is chosen as the most important unit of the sentence. In Figure \[figure:annot\] (iii), we select the longest segment containing 22 words. When a tie occurs, we choose the segment appearing first;
- creating gold-standard labels for sentences and sub-sentence units. If a segment is the most informative, i.e., longest subsequence of a sentence and $>$5 words, we record its start and end indices. If a segment is selected, its containing sentence is labelled as 1, otherwise 0.
----------- -------------- ----------- --------- ---------- -------- --------- ----------
\#TotalSents %PosSents \#Sents \#Tokens %CompR \#Sents \#Tokens
**Train** 5,312,010 24.42 4.51 51.46 0.47 3.68 56.47
**Valid** 211,022 30.85 4.87 57.11 0.47 4.00 62.73
**Test** 182,663 29.63 4.72 54.47 0.46 3.79 59.56
----------- -------------- ----------- --------- ---------- -------- --------- ----------
Dataset and Statistics {#sec:data}
----------------------
We conduct experiments on the CNN/DM dataset released by See et al. containing news articles and human abstracts. We choose the pointer-generator networks described in the same work to obtain attention matrices used for word alignment. The model was trained on the training split of CNN/DM, then applied to all train/valid/test splits to generate gold-standard sub-sentence highlights. At test time, we compare system highlights with gold-standard highlights and human abstracts, respectively, to validate system performance.
In Table \[table:stats\], we present data statistics of the gold-standard sub-sentence highlights. We observe that gold-standard highlights and human abstracts are of comparable length in terms of tokens. On average, 28% of document sentences are labelled as positive. Among these, 47% of the words belong to gold-standard sub-sentence highlights. In our processed dataset we retain important document level information such as original sentence placement and document ID. We consider each document sentence as a data instance, and introduce a neural model to predict (i) a binary sentence level label, and (ii) start and end indices of a consecutive subsequence for a positive sentence. We are particularly interested in predicting start and end indices to encourage sub-sentence segments to remain self-contained. Finally, we leverage the document ID to re-combine model output to still generate summaries at the document level.
------------------------------- ------- ------- ----------- ------- ------- ----------- ------- ------- -----------
Model P R F$_1$ P R F$_1$ P R F$_1$
Oracle (sent.) 36.63 69.52 46.58 20.24 37.76 25.55 25.59 47.84 32.34
Oracle (segm.) 59.71 50.95 53.82 34.42 29.60 **31.16** 43.23 36.89 38.95
Pointer Gen.[ [@See:2017]]{} – – 39.53 – – 17.28 – – 36.38
QASumm+NER[ [@Arumae:2019]]{} – – 25.89 – – 11.65 – – 22.06
Sent 30.91 48.61 34.84 13.31 21.40 15.09 20.14 31.44 22.55
Sent + posit. 31.31 56.53 37.72 14.45 26.70 **17.53** 20.51 37.05 24.63
Segm 32.58 44.97 34.73 13.79 19.36 14.75 21.36 29.03 22.51
Segm + posit. 33.11 52.74 37.99 14.96 24.30 **17.26** 21.69 34.41 24.75
Sent 38.93 58.49 42.81 28.88 44.49 31.96 32.92 50.14 36.32
Sent + posit. 39.97 68.59 [47.02]{} 31.38 55.31 **37.19** 34.58 60.30 [40.86]{}
Segm 41.31 54.27 42.83 30.29 40.38 31.43 34.81 46.01 36.07
Segm + posit. 42.43 64.09 [47.43]{} 32.75 50.40 **36.76** 36.43 55.58 [40.80]{}
------------------------------- ------- ------- ----------- ------- ------- ----------- ------- ------- -----------
Models {#sec:approach}
======
We provide initial modeling for our data with a single state-of-the-art architecture. The purpose is to build meaningful representations that allow for joint prediction of summary-worthy sentences and their sub-sentence units. Our model receives an input sequence as an individualized sentence denoted as $S$=$\{w^s_i\}_{i=1}^\textsf{M}$, where $s$ denotes the sentence index in the original document. The model learns to predict the sentence label and start/end index of a sub-sentence unit based on contextualized representations.
For each token $w^s_i$ we leverage a combined representation $E_{\mbox{\scriptsize tok}}$, $E_{\mbox{\scriptsize s-pos}}$, and $E_{\mbox{\scriptsize d-pos}}$, i.e., a token embedding, sentence level positional embedding, and a document level positional embedding. Here *s-pos* denotes the token position in a sentence, *d-pos* denotes the sentence position in a document, and $E(w^s_i) \in \mathbb{R}^d$. We justify the last embedding by noting that the sentence position within that document plays an important role since generally there is a higher probability of positive labels towards the beginning. The final input representation is an element-wise addition of all embeddings (Eq. ). This input is encoded using a bi-directional transformer [@NIPS2017_7181; @devlin2018bert], denoted as $\mathbf{h}$. $$\label{equation:enc}
E(w^s_i) \coloneqq E_{\mbox{\scriptsize tok}}(w^s_i) + E_{\mbox{\scriptsize s-pos}}(w^s_i) + E_{\mbox{\scriptsize d-pos}}(w^s_i)$$
Objectives
----------
We use the transformer output to generate three labels: sentence, start and end positions of the sub-sentence unit. First we obtain the sequence representation via the token.[^2] We apply a linear transformation to this vector and a softmax layer to obtain a binary label for the entire sentence.
For the indexing objective we transform the encoder output, $\mathbf{h}$, to account for start and end index classification. $\mathbf{a} = \textrm{MLP}_{\mbox{\scriptsize start/end}}(\mathbf{h}) \in \mathbb{R}^{\textsf{M} \times 2}$. Again we make use of a single linear transformation, here it is applied across the encoder temporally giving each time-step two channels. The two channels are individually passed through a softmax layer to produce two distributions, for the start and end index. Finally we use a combined loss term which is trained end-to-end using a cross entropy objective: $$\mathcal{L} = \lambda (\mathcal{L}_{\mbox{\scriptsize start}} + \mathcal{L}_{\mbox{\scriptsize end}}) + \mathcal{L}_{\mbox{\scriptsize sent}}.$$ For negatively labeled sentences $\mathcal{L}_{\mbox{\scriptsize start}}$ and $\mathcal{L}_{\mbox{\scriptsize end}}$ are not utilized during training. $\lambda$ is a coefficient balancing between two task objectives.
Experimental Setup
------------------
The encoder hidden state dimension is set at $768$, with $12$ layers and $12$ attention heads (BERT~BASE~ uncased). We utilize dropout [@srivastava2014dropout] with $p=0.1$, and $\lambda$ is empirically set to $0.1$. We use Adam [@kingma2014adam] as our optimizer with a learning rate of $3e^{-5}$, and implement early stopping against the validation split. Devlin et al. suggest that fine-tuning takes only a few epochs with large datasets. Training was conducted on a GeForce GTX 1080 Ti GPU, and each model took at most three days to converge with a maximum epoch time of 12 hours.
At inference time we only extract start and end indices when the sentence label is positive. Additionally if the system produced an end index occurring before the start index we ignore it and select the argmax of the distribution for end indexes which are located after the start index.
Results {#sec:results}
=======
In Table \[table:rouge\] we report results on the CNN/DM test set evaluated by ROUGE [@lin2004rouge]. We examine to what extent our summary sentences and sub-sentence highlights, annotated using the strategy presented in §\[sec:annotation\], have matched the content of human abstracts. These are the *oracle* results for sentences and segments, respectively. Despite that abstracts can contain unseen words, we observe that 70% of the abstract words are covered by gold-standard sentences, and 51% of abstract words are included in sub-sentence units, suggesting the effectiveness of our annotation method on capturing summary-worthy content.
We proceed by evaluating our method against state-of-the-art extractive and abstractive summarization systems. Arumae and Liu present an approach to extract summary segments using question-answering as supervision signal, assuming a high quality summary can serve as document surrogate to answer questions. See et al. present pointer-generator networks, an abstractive summarization model and a reliable baseline for being both state-of-the-art, and also a vital tool for guiding our data creation. We show that the performance of oracle summaries is superior to these baselines in terms of R-2, with sub-sentence highlights achieving the highest R-2 F-score of 31%, suggesting extracting sub-sentence highlights is a promising direction moving forward.
Modeling
--------
Our models are shown in the bottom two sections of Table \[table:rouge\]. We obtain system-predicted whole sentences (*Sent*) and sub-sentence segments (*Segm*); then evaluate them against both human abstracts (<span style="font-variant:small-caps;">Abstract</span>) and gold-standard highlights (<span style="font-variant:small-caps;">Sub-sent</span>). We test the efficacy of document positional embeddings (Eq. (\[equation:enc\])), denoted as *+posit*.
Using R-2 as a defining metric, our model outperforms or performs competitively with both the abstractive and extractive baselines. We find that the use of document level positional embeddings is beneficial and that for both summary types, models with these embeddings have a competitive edge against those without. Notably sub-sentence level ROUGE scores consistently outmatch sentence level values. These results are nontrivial, as segment level modeling is highly challenging, often resulting in increased precision but drastically reduced recall [@cheng2016neural].
Our model (*+posit*) positively labeled $22.27\%$ of sentences, with an average summary length of $3.54$ sentences. The segment model crops selected sentences, exhibiting a compression ratio of $0.77$. Comparing to gold-standard ratio of $0.47$, there is a $67.4\%$ increase, pointing to future work on highlighting sub-sentence segments.
Conclusion {#sec:conclusion}
==========
We introduced a new task and dataset to study sub-sentence highlight extraction. We have shown the dataset provides a new upper bound for evaluation metrics, and that the use of sub-sentence segments provides more concise summaries over full sentences. Furthermore, we evaluated our data using a state-of-the-art neural architecture to show the modeling capabilities using this data.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the anonymous reviewers for their valuable suggestions. This research was supported in part by the National Science Foundation grant IIS-1909603.
[^1]: Aligning multiple document words with a single abstract word is possible by retrieving document words whose attention weights exceed a threshold. But the method can be data- and model-dependent, increasing the variability of alignment.
[^2]: is fine-tuned as a class label for the entire sequence, and always positioned at $\mathbf{h}_1$
|
---
abstract: |
In this paper we show that Quiescent Cosmology [@Barrow; @Goode; @Scott] is consistent with Penrose’s Weyl Curvature Hypothesis and the notion of gravitational entropy [@Penrose]. Gravitational entropy, from a conceptual point of view, acts in an opposite fashion to the more familiar notion of entropy. A closed system of gravitating particles will coalesce whereas a collection of gas particles will tend to diffuse; regarding increasing entropy, these two scenarios are identical. What has been shown previously [@Goode; @Scott] is that gravitational entropy at the initial singularity predicted by Quiescent Cosmology - the Isotropic Past Singularity (IPS) - tends to zero. The results from this paper show that not only is this the case but that gravitational entropy increases as this singularity evolves.\
In the first section of this paper we present relevant background information and motivation. In the second section of this paper we present the main results of this paper. Our third section contains a discussion of how this result will inspire future research before we make concluding remarks in our final section.
address: |
Centre for Gravitational Physics,\
College of Physical Sciences,\
The Australian National University,\
Canberra ACT 0200\
Australia
author:
- Philip Threlfall and Susan M Scott
bibliography:
- 'Bibliography.bib'
title: The Monotonicity of the Gravitational Entropy Scalar within Quiescent Cosmology
---
Background and Motivation
=========================
Quiescent Cosmology
-------------------
Barrow introduced the world to Quiescent Cosmology in 1978 [@Barrow] as an attempt to explain the current large scale isotropy and homogeneity of the Universe. Quiescent Cosmology effectively states that the Universe began in a highly ordered state and has evolved away from its highly regular and smooth beginning because of gravitational attraction [^1]. This means that the reason we continue to observe large scale regularity is because we exist in an early stage of cosmological evolution. In order for Quiescent Cosmology to be compatible with a Big Bang type singularity, it is necessary that that singularity is one that is isotropic. This type of initial isotropic singularity was given a rigorous mathematical definition in 1985 by Goode and Wainwright [@Goode] using a conformal relationship between two spacetimes. The definition given in this paper is due to Scott [@Scott2] who removed the inherent technical redundancies of the original definition.\
Goode and Wainwright based their analysis on the beginning of the universe, but recently there has been increasing interest in possible future evolutions of the Universe. Höhn and Scott [@Scott] introduced different isotropic and anisotropic definitions that describe possible future end states of the Universe. Following Goode and Wainwright they also exploited conformal relationships between spacetimes.
Conformal Structures
--------------------
In this paper we primarily deal with isotropic structures and thus we require the conformal definitions that relate to isotropic initial and final states of the universe. In order for this paper to be fully appreciated, however, results pertaining to isotropic structures will be put in context with anisotropic structures; these definitions will also be presented in this introduction.\
The isotropic definitions comprise of the Isotropic Past Singularity, the Isotropic Future Singularity and the Future Isotropic Universe. The anisotropic definitions of Quiescent Cosmology are the Anisotropic Future Endless Universe and the Anisotropic Future Singularity. Any ancillary definitions that are needed will also be included.
A metric $\mathbf{g}$ is said to be conformally related to a metric $\mathbf{\tilde{g}}$ on a manifold $\mathcal{M}$ if there exists a conformal factor $\Omega$ such that $$\begin{aligned}
\mathbf{g} &=& \Omega^{2}\mathbf{\tilde{g}}\textrm{, where $\Omega$
is a strictly positive function on $\mathcal{M}$.}\end{aligned}$$
For a space-time $(\mathcal{M},g)$, a cosmic time function is a function $T$ on the manifold $\mathcal{M}$ which increases along every future-directed causal curve.
### Isotropic Definitions
It should be noted that we will henceforth denote relevant quantities for past cosmological frameworks with a tilde $(\sim)$ and for future cosmological frameworks we will use a bar $(-)$.
A space-time $(\mathcal{M},\mathbf{g})$ admits an Isotropic Past Singularity if there exists a space-time $(\tilde{\mathcal{M}},
\mathbf{\tilde{g}})$, a smooth cosmic time function $T$ defined on $\tilde{\mathcal{M}}$ and a conformal factor $\Omega(T)$ which satisfy
1. $\mathcal{M}$ is the open submanifold $T > 0$,
2. $\mathbf{g} = \Omega^{2}(T)\mathbf{\tilde{g}}$ on $\mathcal{M}$, with $\mathbf{\tilde{g}}$ regular (at least $C^{3}$ and non-degenerate) on an open neighbourhood of $T =
0$,
3. $\Omega(0) = 0$ and $\exists\; b > 0$ such that $\Omega \in C^{0}[0,b] \cap C^{3}(0,b]$ and $\Omega(0,b] >
0$,
4. $\lambda \equiv \lim\limits_{T\rightarrow 0^{+}}L(T)$ exists, $\lambda \neq 1$, where $L \equiv
\frac{\Omega''}{\Omega}\left(\frac{\Omega}{\Omega'}\right)^{2}$ and a prime denotes differentiation with respect to $T$.
\[IPSDefinition\]
It was demonstrated by Goode and Wainwright [@Goode] that, in order to ensure initial asymptotic isotropy, it is also necessary to introduce a constraint on the cosmological fluid flow.
With any unit timelike congruence $\mathbf{u}$ in $\mathcal{M}$ we can associate a unit timelike congruence $\mathbf{\tilde{u}}$ in $\tilde{\mathcal{M}}$ such that $$\begin{aligned}
\mathbf{\tilde{u}} &=& \Omega\mathbf{u}\qquad \textrm{in }
\mathcal{M}\textrm{.}\end{aligned}$$
- If we can choose $\mathbf{\tilde{u}}$ to be regular (at least $C^{3}$) on an open neighbourhood of $T = 0$ in $\tilde{\mathcal{M}}$, we say that $\mathbf{u}$ is regular at the IPS.
- If, in addition, $\mathbf{\tilde{u}}$ is orthogonal to $T = 0$, we say that $\mathbf{u}$ is orthogonal to the IPS.
\[IPSFluidDefinition\]
In figure \[IPS\] we present a pictorial interpretation of the IPS.
Below is given the analogous definition of an Isotropic Future Singularity introduced by Höhn and Scott [@Scott], followed by the constraint on the fluid flow required to ensure final asymptotic isotropy. The IFS is not compatible with the fundamental ideals of Quiescent Cosmology (that the end state of the Universe is anisotropic) but it remains a structure worth analysing.
A space-time $(\mathcal{M},\mathbf{g})$ admits an Isotropic Future Singularity if there exists a space-time $(\bar{\mathcal{M}},
\mathbf{\bar{g}})$, a smooth cosmic time function $\bar{T}$ defined on $\bar{\mathcal{M}}$, and a conformal factor $\bar{\Omega}(\bar{T})$ which satisfy
1. $\mathcal{M}$ is the open submanifold $\bar{T} < 0$,
2. $\mathbf{g} =
\bar{\Omega}^{2}(\bar{T})\mathbf{\bar{g}}$ on $\mathcal{M}$, with $\mathbf{\bar{g}}$ regular (at least $C^{2}$ and non-degenerate) on an open neighbourhood of $\bar{T} = 0$,
3. $\bar{\Omega}(0) = 0$ and $\exists\; c > 0$ such that $\bar{\Omega} \in C^{0}[-c, 0] \cap C^{2}[-c,0)$ and $\bar{\Omega}$ is positive on $[-c,0)$,
4. $\bar{\lambda} \equiv \lim\limits_{\bar{T}\rightarrow
0^{-}}\bar{L}(\bar{T})$ exists, $\bar{\lambda} \neq 1$, where $\bar{L} \equiv
\frac{\bar{\Omega}''}{\bar{\Omega}}\left(\frac{\bar{\Omega}}{\bar{\Omega}'}\right)^{2}$ and a prime denotes differentiation with respect to $\bar{T}$.
\[IFSDefinition\]
With any unit timelike congruence $\mathbf{u}$ in $\mathcal{M}$ we can associate a unit timelike congruence $\mathbf{\bar{u}}$ in $\bar{\mathcal{M}}$ such that $$\begin{aligned}
\mathbf{\bar{u}} &=& \bar{\Omega}\mathbf{u}\qquad \textrm{in }
\mathcal{M}\textrm{.}\end{aligned}$$
- If we can choose $\mathbf{\bar{u}}$ to be regular (at least $C^{2}$) on an open neighbourhood of $\bar{T} = 0$ in $\bar{\mathcal{M}}$, we say that $\mathbf{u}$ is regular at the IFS.
- If, in addition, $\mathbf{\bar{u}}$ is orthogonal to $\bar{T} = 0$, we say that $\mathbf{u}$ is orthogonal to the IFS.
\[IFSFluidDefinition\]
In figure \[IFS\] we present a pictorial interpretation of the IFS.
\[IFS\]
Finally we give below the definition of a Future Isotropic Universe introduced by Höhn and Scott [@Scott]. This definition covers the further possibility for a conformal structure with an isotropic future behaviour, which does not necessarily lead to a future singularity; for example, some open FRW universes satisfy this definition [@Threlfall].
A space-time $(\mathcal{M},\mathbf{g})$ is said to be a Future Isotropic Universe if there exists a space-time $(\bar{\mathcal{M}}, \mathbf{\bar{g}})$, a smooth cosmic time function $\bar{T}$ defined on $\bar{\mathcal{M}}$, and a conformal factor $\bar{\Omega}(\bar{T})$ which satisfy
1. $\lim\limits_{\bar{T}\rightarrow
0^{-}}\bar{\Omega}(\bar{T}) = +\infty$ and $\exists\; c > 0$ such that $\bar{\Omega} \in C^{2}[-c,0)$ and $\bar{\Omega}$ is strictly monotonically increasing and positive on $[-c,0)$,
2. $\bar{\lambda}$ as defined above exists, $\bar{\lambda} \neq 1,2$, and $\bar{L}$ is continuous on $[-c,0)$ and
3. otherwise the conditions of definitions \[IFSDefinition\] and \[IFSFluidDefinition\] are satisfied.
\[FIUDefinition\]
### Anisotropic Definitions
All anisotropic definitions[^2] refer to the future and hence their unphysical quantities are denoted by a bar ($-$). It is conceivable, however, that these definitions could be recast for past cosmological states (similar to the IPS/IFS scenarios).
Consider $p\in\mathcal{M}$. Let $\gamma_{p}(s)$ be a future inextendible causal curve such that $\gamma_{p}(s):[0,a)\to\mathcal{M}$, where $a\in\mathbb{R}^{+}\cup\{\infty\}$, such that $p =
\gamma_{p}(a)\equiv\mathop{\lim}\limits_{s\to a}\gamma_{p}(s)$ with limiting tangent vector $\gamma'_{p}\neq0$ at $p$.The metric $\bar{g}$ is said to be causally degenerate at $p$ if there exists a curve $\gamma_{p}$ which satisfies $\bar{g}(\gamma'_{p},X) =
0\;\forall\; X\in T_{p}\bar{M}$. (Note that this assumes the metric is continuous on an open neighbourhood of $p$).
A spacetime, $\left(\mathcal{M}, g\right)$ is said to be an Anisotropic Future Endless Universe if there exists
1. a larger manifold $\bar{\mathcal{M}}\supset\mathcal{M}$,
2. a smooth function $\bar{T}$ defined on $\bar{\mathcal{M}}$ (with $\bar{\nabla}\bar{T}\neq0$ everywhere on $\bar{\mathcal{M}})$ such that $\mathcal{M}$ is the open submanifold $\bar{T} < 0$,
3. a $C^{0}$ tensor field $\mathbf{\bar{g}}$ of type $(0,2)$ defined on $\mathcal{M} \cup \mathcal{N}$, where $\mathcal{N}$ is an open neighbourhood of $\bar{T} = 0$ in $\bar{\mathcal{M}}$, and
4. a conformal factor $\bar{\Omega}({\bar{T}})$ defined on $\mathcal{M}$, which satisfies
1. $\bar{T}$ is a cosmic time function on $\mathcal{M}\:
\cup\: \mathcal{N}$,
2. $\mathbf{g} =
\bar{\Omega}^{2}\left(\bar{T}\right)\mathbf{\bar{g}}$ on $\mathcal{M}$ and $\mathbf{\bar{g}}$ is degenerate on $\bar{T} = 0$,
3. $\mathop{\lim}\limits_{\bar{T} \to
0^{-}}\bar{\Omega}\left(\bar{T}\right) = +\infty$ and $\exists\: c > 0$ such that $\bar{\Omega}\in
C^{2}[-c,0)$ and $\bar{\Omega}$ is strictly monotonically increasing and positive on $[-c,0)$,
4. $\bar{L}$ as defined above is continuous on $[-c,0)$, $\bar{\lambda}$ exists, $\bar{\lambda} \neq 1$, and
5. $\mathop {\lim}\limits_{\bar{T} \to
0^{-}}\bar{\Omega}^{6}|\bar{g}| = \infty $, where $\bar{g}$ is the determinant of $\mathbf{\bar{g}}$.
Thus, the next figure we present is that which represents the AFEU, seen in figure \[AFEUDiagram\].
\[AFEUDiagram\]
A spacetime, $\left(\mathcal{M}, g\right)$ is said to be an Anisotropic Future Singularity if there exists
1. a larger manifold $\bar{\mathcal{M}}\supset\mathcal{M}$,
2. a smooth function $\bar{T}$ defined on $\bar{\mathcal{M}}$ (with $\bar{\nabla}\bar{T}\neq0$ everywhere on $\bar{\mathcal{M}})$ such that $\mathcal{M}$ is the open submanifold $\bar{T} < 0$,
3. a $C^{0}$ tensor field $\mathbf{\bar{g}}$ of type $(0,2)$ defined on $\mathcal{M}\: \cup\: \mathcal{N}$, where $\mathcal{N}$ is an open neighbourhood of $\bar{T} = 0$ in $\bar{\mathcal{M}}$, and
4. a conformal factor $\bar{\Omega}({\bar{T}})$ defined on $\mathcal{M}$, which satisfies
1. $\bar{T}$ is a cosmic time function on $\mathcal{M}
\cup \mathcal{N}$,
2. $\mathbf{g} =
\bar{\Omega}^{2}\left(\bar{T}\right)\mathbf{\bar{g}}$ on $\mathcal{M}$ and $\mathbf{\bar{g}}$ is degenerate on $\bar{T} = 0$,
3. $\mathop{\lim}\limits_{\bar{T} \to
0^{-}}\bar{\Omega}\left(\bar{T}\right) = +\infty$ and $\exists\: c > 0$ such that $\bar{\Omega}\in C^{2}[-c,0)$ and $\bar{\Omega}$ is strictly monotonically increasing and positive on $[-c,0)$,
4. $\bar{L}$ as defined above is continuous on $[-c,0)$, $\bar{\lambda}$ exists, $\bar{\lambda} \neq 1$, and
5. $\mathop{\lim}\limits_{\bar{T} \to
0^{-}}\bar{\Omega}^{8}|\bar{g}| = 0$, where $\bar{g}$ is the determinant of $\mathbf{\bar{g}}$.
Figure \[AFSDiagram\] demonstrates the degenerate nature of the AFS.
\[AFSDiagram\]
Gravitational Entropy
=====================
The Weyl Curvature Hypothesis (WCH) of Penrose [@Penrose] states that the Weyl curvature at an initial (Big Bang) singularity must be bounded and has been increasing ever since. Gravitational entropy [@GoodeB; @BarrowB; @Lim; @Scott] is closely linked to the WCH as will be explained here. A common mental image that comes to mind when thinking of entropy involves a gas expanding within a chamber - thus maximising the entropy of the system. If this imagery were applied to a collection of gravitating particles the particles would attract one another and end up in a system that seems to have less entropy than when it started. Penrose [@Penrose] addressed this problem by postulating the gravitational entropy of a system reaches a maximum when gravitational collapse results in a black hole. This means that a collection of particles that coalesce are actually increasing the entropy available and thus gravity can be consistent with thermodynamics. In General Relativity a quantity that is hypothesised to be a measure of gravitational entropy [@Penrose] is the ratio between the Weyl and Ricci curvature invariants $$\begin{aligned}
\label{GravEntropy}
K &=& \frac{C^{abcd}C_{abcd}}{R^{ef}R_{ef}}\textrm{.}\end{aligned}$$ Where it is understood that Weyl curvature describes the curvature purely due to the gravitational field and the Ricci scalar will describe the curvature due to matter. In order for entropy to have been globally increasing (as is expected) from the Big Bang, the entropy at the Big Bang must have been low, i.e. $K = 0$ at the Big Bang. The original interpretation for this was that the Weyl scalar must have been identically zero at the Big Bang but this constraint was too strict [@Tod] as it would have ruled out all cosmological models apart from the Friedmann-Robertson-Walker (FRW) solutions. The compromise is that the Weyl scalar must be asymptotically dominated by the Ricci scalar at the Big Bang. The IPS has been shown to be consistent with this hypothesis when Goode and Wainwright [@Goode] proved that, as an IPS is approached, $\mathop{\lim}\limits_{\bar{T}\to 0} K = 0$.
The K Theorem
-------------
When Höhn and Scott [@Scott] expanded the Quiescent Cosmology framework to consider possible future cosmological states, they were able to show that Goode and Wainwright’s result is able to be expanded to all isotropic structures; this result was published as The K Theorem.
\[KTheorem\] Let $\mathbf{(\mathcal{M},g)}$ and $\mathbf{(\bar{\mathcal{M}},\bar{g})}$ be two spacetimes which are related via the conformal structure $\mathbf{g =
\bar{\Omega}^{2}(\bar{T})\bar{g}}$, where $\bar{T}$ is a smooth cosmic time function defined on $\mathbf{(\bar{\mathcal{M}},\bar{g})}$ and $\bar{g}$ is non-degenerate and at least $C^{2}$ on an open neighbourhood of $\bar{T}=0$. Let one of the following conditions be true
1. $\bar{T}\to 0^{-}$, $\mathop {\lim}\limits_{\bar{T} \to
0^{-}}\bar{\Omega}(0)=\infty$ and $\bar{\Omega}$ is positive, $C^{2}$ and strictly increasing on some interval $(0,c]$
2. $\bar{T}\to 0^{-}$, $\mathop {\lim}\limits_{\bar{T} \to
0^{-}}\bar{\Omega}(0)=0$ and $\bar{\Omega}$ is positive, $C^{2}$ and strictly decreasing on some interval $[-c,0)$
3. $\bar{T}\to 0^{+}$, $\mathop {\lim}\limits_{\bar{T} \to
0^{+}}\bar{\Omega}(0)=\infty$ and $\bar{\Omega}$ is positive, $C^{2}$ and strictly decreasing on some interval $(0,c]$
4. $\bar{T}\to 0^{+}$, $\mathop {\lim}\limits_{\bar{T} \to
0^{+}}\bar{\Omega}(0)=0$ and $\bar{\Omega}$ is positive, $C^{2}$ and strictly increasing on some interval $(0,c]$
and $\bar{\lambda}\neq 1$ then $\mathop {\lim}\limits_{\bar{T} \to 0^{\pm}}
\frac{C^{abcd}C_{abcd}}{R^{ef}R_{ef}} = 0$.
Theorem \[KTheorem\] is important as it shows that, with a few assumptions placed upon the conformal factor, all conformally regular spacetimes give asymptotic isotropic behaviour. This demonstrates that the Quiescent Cosmology is, at least asymptotically, consistent with Penrose’s ideas about Weyl curvature at an initial singularity. The other half of Penrose’s hypothesis (that gravitational entropy increases after a Big Bang) is yet to be answered. It is important to know this answer because if $K$ is always asymptotically zero according to Quiescent Cosmology then it means that Quiescent Cosmology is not compatible with Penrose’s ideas in full generality. The most telling indicator of how $K$ may or may not increase is shown by considering the derivative with respect to cosmic time. If this value is positive as $\bar{T}$ increases from zero, it means that $K$ will be monotonically increasing away from the IPS; this is in line with the prediction of Penrose [@Penrose].
The Derivative of the Gravitational Entropy Scalar
==================================================
Recall that $K$ is given by $$\begin{aligned}
K &=& \frac{C^{abcd}C_{abcd}}{R^{ef}R_{ef}}\\
&=&\frac{\bar{\Omega}^{-4}\bar{C}^{abcd}\bar{C}_{abcd}}{R^{ef}R_{ef}}\textrm{.}\end{aligned}$$ We can now take the partial derivative of this scalar with respect to cosmic time to obtain the following $$\begin{aligned}
K' &=& \frac{(\bar{\Omega}^{-4}\bar{C}^{abcd}\bar{C}_{abcd})_{,m}R^{ef}R_{ef}
- \bar{\Omega}^{-4}(R^{ef}R_{ef})_{,m}\bar{C}^{abcd}\bar{C}_{abcd}
}{\left(R^{ef}R_{ef} \right)^{2}}\end{aligned}$$ It is clear that we will need to know the derivative of the physical Ricci scalar and the unphysical Weyl scalar; we present those now.
The Unphysical Weyl Scalar’s Derivative
---------------------------------------
The derivative of the physical Weyl scalar is $$\begin{aligned}
\left(C^{abcd}C_{abcd}\right)' &=& -4\bar{\Omega}^{-5}\bar{C}^{abcd}\bar{C}_{abcd}\bar{T}_{,m}
+ \bar{\Omega}^{-4}\left(\bar{C}^{abcd}\bar{C}_{abcd}\right)'\textrm{.}\end{aligned}$$ Where the unphysical Weyl tensor’s derivative is $$\begin{aligned}
(\bar{C}_{abcd})'&=&(\bar{R}_{abcd,m} -\frac{1}{4}\bar{g}^{ij}_{\phantom{ij},m}\left((\bar{g}_{ac}\bar{R}_{idjb} -
\bar{g}_{ad}\bar{R}_{icjb})- (\bar{g}_{bc}\bar{R}_{idja} -
\bar{g}_{bd}\bar{R}_{icja})\right)\nonumber\\
&-&\frac{1}{4}\bar{g}^{ij}((\bar{g}_{ac,m}\bar{R}_{idjb} + \bar{g}_{ac}\bar{R}_{idjb,m} -
\bar{g}_{ad,m}\bar{R}_{icjb}-\bar{g}_{ad}\bar{R}_{icjb,m})\nonumber\\
&-& (\bar{g}_{bc,m}\bar{R}_{idja} +\bar{g}_{bc}\bar{R}_{idja,m} -
\bar{g}_{bd}\bar{R}_{icja}- \bar{g}_{bd,m}\bar{R}_{icja,m}))\nonumber\\
&+& \frac{1}{6}(\bar{g}^{ij}_{\phantom{ij},m}\bar{g}^{kl}\bar{R}_{ikjl} + \bar{g}^{ij}\bar{g}^{kl}_{\phantom{kl},m}\bar{R}_{ikjl}
+ \bar{g}^{ij}\bar{g}^{kl}\bar{R}_{ikjl,m} )\nonumber\\
&\cdot&(\bar{g}_{ac,m}\bar{g}_{db} + \bar{g}_{ac}\bar{g}_{db,m}
-\bar{g}_{ad,m}\bar{g}_{cb} - \bar{g}_{ad}\bar{g}_{cb,m}))\end{aligned}$$ As such the unphysical Weyl scalar’s derivative is $$\begin{aligned}
(\bar{C}^{abcd}\bar{C}_{abcd})_{,m}&=&
(\bar{g}^{an}_{\phantom{an},m}\bar{g}^{bo}\bar{g}^{cp}\bar{g}^{dq} + \bar{g}^{an}\bar{g}^{bo}_{\phantom{bo},m}\bar{g}^{cp}\bar{g}^{dq}
+ \bar{g}^{an}\bar{g}^{bo}\bar{g}^{cp}_{\phantom{cp},m}\bar{g}^{dq}\nonumber\\
& +& \bar{g}^{an}\bar{g}^{bo}\bar{g}^{cp}\bar{g}^{dq}_{\phantom{dq},m})
(\bar{C}_{abcd,m}\bar{C}_{nopq} + \bar{C}_{abcd}\bar{C}_{nopq,m})\end{aligned}$$ This is the easier derivative to calculate because of the simple relationship between the unphysical Weyl scalar and its physical counterpart.
The Physical Ricci Scalar’s Derivative
--------------------------------------
A calculation of the physical Ricci scalar’s derivative is more involved than the Weyl scalar’s but the process is similar enough. With this in mind, first recall the physical Ricci scalar $$\begin{aligned}
& &R^{ef}R_{ef}=\bar{\Omega}^{-4}\Big(12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L} +
1\right)\nonumber\\
&-&
2\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{3}\left(4\left(2-\bar{L}\right)\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,f}\bar{T}_{:ij}
-2\left(4\bar{L} -
1\right)\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,i}\bar{T}_{:fj}\right)\nonumber\\
&+& \left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{2}\Big(
4\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{:ef}\bar{T}_{:ij} + 8\bar{g}^{ei}\bar{g}^{ei}\left(\bar{T}_{:ei}
\right)^{2} +
4\bar{g}^{ei}\bar{g}^{fj}\left(2-\bar{L}\right)\bar{R}_{ef}\bar{T}_{,i}\bar{T}_{,j}\nonumber\\
&-&2\bar{g}^{ei}\left(1+ \bar{L}\right)\bar{R}\bar{T}_{,e}\bar{T}_{,i}
\Big)\nonumber\\
&-&2\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)\left(2\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{T}_{:ij}
+ \bar{g}^{ei}\bar{R}\bar{T}_{:ei}\right) +
\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{R}_{ij}\Big)\textrm{.}\end{aligned}$$ This means that the Ricci scalar’s derivative is given by $$\begin{aligned}
& & \left(R^{ef}R_{ef}\right)_{,m} = \left(\bar{\Omega}^{-4}\right)_{,m}\bar{\Omega}^{4}R^{ef}R_{ef}\nonumber\\
&+& \bar{\Omega}^{-4}\Big(12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L} +
1\right)\nonumber\\
&-&
2\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{3}\left(4\left(2-\bar{L}\right)\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,f}\bar{T}_{:ij}
- 2\left(4\bar{L} -
1\right)\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,i}\bar{T}_{:fj}\right)\nonumber\\
&+& \left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{2}\Big(
4\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{:ef}\bar{T}_{:ij} + 8\bar{g}^{ei}\bar{g}^{ei}\left(\bar{T}_{:ei}
\right)^{2} +
4\bar{g}^{ei}\bar{g}^{fj}\left(2-\bar{L}\right)\bar{R}_{ef}\bar{T}_{,i}\bar{T}_{,j}\nonumber\\
&-&2\bar{g}^{ei}\left(1+ \bar{L}\right)\bar{R}\bar{T}_{,e}\bar{T}_{,i}
\Big)\nonumber\\
&-&2\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)\left(2\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{T}_{:ij}
+ \bar{g}^{ei}\bar{R}\bar{T}_{:ei}\right) +
\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{R}_{ij}\Big)_{,m}\textrm{.}\end{aligned}$$ This equation is simpler if we analyse it one term at a time. To aid the reader following along with this derivation, the following two equations may prove helpful $$\begin{aligned}
\left(\frac{\bar{\Omega}'}{\bar{\Omega}} \right)^{n}_{\phantom{n},m} &=&
\left(\frac{\bar{\Omega}'}{\bar{\Omega}} \right)^{n+1}\left(\bar{L} - 1\right)\bar{T}_{,m}\\
\bar{L}_{,m} &=& \bar{L}\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\left(1-2\bar{L} \right)
+\frac{\bar{\Omega}'''}{\bar{\Omega}''}\right)\bar{T}_{,m}\end{aligned}$$ For simplicity’s sake, the derivative of the physical Ricci scalar is given one line at a time. Beginning with the top line, $$\begin{aligned}
\left(\bar{\Omega}^{-4}\right)_{,m}\bar{\Omega}^{4}R^{ef}R_{ef} &=& -4\frac{\bar{\Omega}'}{\bar{\Omega}^{5}}\bar{T}_{,m}\bar{\Omega}^{4}R^{ef}R_{ef}\end{aligned}$$ Now the more interesting lines, beginning with the second $$\begin{aligned}
& &\bar{\Omega}^{-4}\left(12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L} +
1\right)\right)_{,m}\nonumber\\
&=& 48\bar{\Omega}^{-4}\Big(\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{5}\left(\bar{L}-1\right)\left(\bar{L}^{2}-\bar{L} +
1\right)
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\bar{T}_{,m}\nonumber\\
&+& 12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}\Big(
2\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)\left(\bar{g}^{ei}_{\phantom{ei},m}\bar{T}_{,e}\bar{T}_{,i}
+ \bar{g}^{ei}\left(\bar{T}_{,em}\bar{T}_{,i} + \bar{T}_{,e}\bar{T}_{,im} \right)\right)\nonumber\\
&\cdot&\left(\bar{L}^{2}-\bar{L} +1\right) + \left(2\bar{L}-1\right)\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\bar{L}_{,m}\Big)\Big)\textrm{,}\end{aligned}$$ now the third line $$\begin{aligned}
& &\bar{\Omega}^{-4}\Big(2\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{3}\Big(\left(8-4\bar{L}\right)\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,f}\bar{T}_{:ij}\nonumber\\
&-&\left(8\bar{L} +
2\right)\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,i}\bar{T}_{:fj}\Big) \Big)_{,m}\nonumber\\
&=& 6\bar{\Omega}^{-4}\Big(\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}\left(\bar{L}-1\right)\Big(\left(8-4\bar{L}\right)\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,f}\bar{T}_{:ij}\nonumber\\
&-&\left(8\bar{L} +
2\right)\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,i}\bar{T}_{:fj}\Big)\bar{T}_{,m}\nonumber\\
&-& 2\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{3}\Big(4\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,f}\bar{T}_{:ij}\bar{L}_{,m}\nonumber\\
&+& \left(8-4\bar{L}\right)\Big(
\left(\bar{g}^{ei}_{\phantom{ei},m}\bar{g}^{fj} + \bar{g}^{ei}\bar{g}^{fj}_{\phantom{fj},m}\right)\bar{T}_{,e}\bar{T}_{,f}\bar{T}_{:ij}\nonumber\\
&+&
\bar{g}^{ei}\bar{g}^{fj}\Big(\bar{T}_{,em}\bar{T}_{,f}\bar{T}_{:ij} + \bar{T}_{,e}\bar{T}_{,fm}\bar{T}_{:ij}\nonumber\\
&+& \bar{T}_{,e}\bar{T}_{,f}\bar{T}_{:ij,m}\Big)\Big)- 8\left(\bar{L}_{,m}\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{,e}\bar{T}_{,i}\bar{T}_{:fj}\right)\nonumber\\
&-& \left(8\bar{L} +
2\right)\Big(\left(\bar{g}^{ei}_{\phantom{ei},m}\bar{g}^{fj} + \bar{g}^{ei}\bar{g}^{fj}_{\phantom{fj},m}\right)\bar{T}_{,e}\bar{T}_{,i}\bar{T}_{:fj}\nonumber\\
&-& \bar{g}^{ei}\bar{g}^{fj}\left(\bar{T}_{,em}\bar{T}_{,i}\bar{T}_{:fj}
+ \bar{T}_{,e}\bar{T}_{,im}\bar{T}_{:fj} + \bar{T}_{,e}\bar{T}_{,i}\bar{T}_{:fj,m}\right)\Big)\Big)\Big)\textrm{,}\end{aligned}$$ the fourth line now $$\begin{aligned}
& & \bar{\Omega}^{-4}\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{2}\Big(
4\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{:ef}\bar{T}_{:ij} + 8\left(\bar{g}^{ei}\bar{T}_{:ei}
\right)^{2} +
4\left(2-\bar{L}\right)\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{T}_{,i}\bar{T}_{,j}\nonumber\\
&-&2\left(1+ \bar{L}\right)\bar{g}^{ei}\bar{R}\bar{T}_{,e}\bar{T}_{,i}
\Big)_{,m}\nonumber\\
&=& \bar{\Omega}^{-4}\Big(\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{3}\left(\bar{L}-1 \right)\Big(
4\bar{g}^{ei}\bar{g}^{fj}\bar{T}_{:ef}\bar{T}_{:ij} + 8\left(\bar{g}^{ei}\bar{T}_{:ei}
\right)^{2}\nonumber\\
&+&4\left(2-\bar{L}\right)\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{T}_{,i}\bar{T}_{,j}
-2\left(1+ \bar{L}\right)\bar{g}^{ei}\bar{R}\bar{T}_{,e}\bar{T}_{,i}
\Big)\bar{T}_{,m}\nonumber\\
&+& 4\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{2}\Big(
\left(\bar{g}^{ei}_{\phantom{ei},m}\bar{g}^{fj} +\bar{g}^{ei}\bar{g}^{fj}_{\phantom{fj},m} \right)\bar{T}_{:ef}\bar{T}_{:ij}
+
\bar{g}^{ei}\bar{g}^{fj}\left(\bar{T}_{:ef,m}\bar{T}_{:ij} + \bar{T}_{:ef}\bar{T}_{:ij,m}\right)\nonumber\\
&+& 4\bar{g}^{ei}\bar{T}_{:ei}
\left(\bar{g}^{ei}_{\phantom{ei},m}\bar{T}_{:ei} + \bar{g}^{ei}\bar{T}_{:ei,m}\right)-
\left(\bar{L}_{,m}\right)\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{T}_{,i}\bar{T}_{,j}\nonumber\\
&+&
\left(2-\bar{L}\right)\Big(\left(\bar{g}^{ei}_{\phantom{ei},m}\bar{g}^{fj} + \bar{g}^{ei}\bar{g}^{fj}_{\phantom{fj},m}\right)\bar{R}_{ef}\bar{T}_{,i}\bar{T}_{,j}\nonumber\\
&+&\bar{g}^{ei}\bar{g}^{fj}\Big(\bar{R}_{ef,m}\bar{T}_{,i}\bar{T}_{,j}
+
\bar{R}_{ef}\left(\bar{T}_{,im}\bar{T}_{,j} + \bar{T}_{i}\bar{T}_{,jm}\right)\Big)\Big)\nonumber\\
&-&\frac{1}{2}\Big(\bar{L}_{,m}\bar{g}^{ei}\bar{R}\bar{T}_{,e}\bar{T}_{,i}
+\left(1+ \bar{L}\right)\Big(\bar{g}^{ei}_{\phantom{ei},m}\bar{R}\bar{T}_{,e}\bar{T}_{,i}\nonumber\\
&-&\bar{g}^{ei}\Big(\left(\bar{R}_{,m}\bar{T}_{,e}\bar{T}_{,i}\right)
+\bar{R}\left(\bar{T}_{,em}\bar{T}_{,i} + \bar{T}_{,e}\bar{T}_{,im}\right)\Big)\Big)\Big)\Big)\Big)\textrm{,}\end{aligned}$$ to the last line $$\begin{aligned}
& & \bar{\Omega}^{-4}\left(2\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)\left(2\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{T}_{:ij}
+ \bar{g}^{ei}\bar{R}\bar{T}_{:ei}\right) +
\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{R}_{ij} \right)_{,m}\nonumber\\
&=& \bar{\Omega}^{-4}\Big(2\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{2}\left(\bar{L}-1 \right)\left(2\bar{g}^{ei}\bar{g}^{fj}\bar{R}_{ef}\bar{T}_{:ij}
+ \bar{g}^{ei}\bar{R}\bar{T}_{:ei}\right)\bar{T}_{,m}\nonumber\\
&+& 2\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)
\Big(2\left(\bar{g}^{ei}_{\phantom{ei},m}\bar{g}^{fj} + \bar{g}^{ei}\bar{g}^{fj}_{\phantom{fj},m}\right)\bar{R}_{ef}\bar{T}_{:ij}\nonumber\\
&+& 2\bar{g}^{ei}\bar{g}^{fj}\left(\bar{R}_{ef,m}\bar{T}_{:ij} + \bar{R}_{ef}\bar{T}_{:ij,m}\right)
+ \bar{g}^{ei}_{\phantom{ei},m}\bar{R}\bar{T}_{:ei} + \bar{g}^{ei}\left(\bar{R}_{,m}\bar{T}_{:ei} + \bar{R}\bar{T}_{:ei,m}\right) \Big)\nonumber\\
&+& \left(\bar{g}^{ei}_{\phantom{ei},m}\bar{g}^{fj}
+ \bar{g}^{ei}\bar{g}^{fj}_{\phantom{fj},m}\right)\bar{R}_{ef}\bar{R}_{ij} + \bar{g}^{ei}\bar{g}^{fj}\left(\bar{R}_{ef,m}\bar{R}_{ij} + \bar{R}_{ef}\bar{R}_{ij,m}\right)\Big)\textrm{.}\end{aligned}$$
Asymptotic Monotonicity of $K$
==============================
Thanks to the work in the last section, we are now in a position to determine the monotonic behaviour of $K$. It is explicit in the below theorem but to be clear - we will be dealing with a regular unphysical metric and hence all unphysical metric components, and their derivatives, will be well behaved at the hypersurface $\bar{T}=0$. Furthermore, this means that theorem \[KPrime\] does not apply to the AFS and AFEU. It is also important to remember that for a regular unphysical metric (an IPS/IFS or PIU/FIU) $\bar{\Omega}$ is $C^{3}$ and as such $\bar{L}_{,m}$ will be well behaved.
\[KPrime\] Let $\mathbf{(\mathcal{M},g)}$ and $\mathbf{(\bar{\mathcal{M}},\bar{g})}$ be two spacetimes which are related via the conformal structure $\mathbf{g =
\bar{\Omega}^{2}(\bar{T})\bar{g}}$, where $\bar{T}$ is a smooth cosmic time function defined on $\mathbf{(\bar{\mathcal{M}},\bar{g})}$ and $\bar{g}$ is non-degenerate and at least $C^{2}$ on an open neighbourhood of $\bar{T}=0$. If $$\begin{aligned}
\bar{C}^{abcd}\bar{C}_{abcd}\not\equiv 0\end{aligned}$$ and one of the following conditions are satisfied
1. $\bar{T}\to 0^{+}$, $\mathop {\lim}\limits_{\bar{T}
\to 0^{+}}\bar{\Omega}=0$ and $\bar{\Omega}$ is positive, $C^{3}$ and strictly decreasing on some interval $[-c,0)$,
2. $\bar{T}\to 0^{-}$, $\mathop {\lim}\limits_{\bar{T}
\to 0^{-}}\bar{\Omega}=0$ and $\bar{\Omega}$ is positive, $C^{3}$ and strictly decreasing on some interval $[-c,0)$,
3. $\bar{T}\to 0^{-}$, $\mathop {\lim}\limits_{\bar{T}
\to 0^{-}}\bar{\Omega}=+\infty$ and $\bar{\Omega}$ is positive, $C^{3}$ and strictly increasing on some interval $(0,c]$,
then $$\begin{aligned}
\mathop{\lim}\limits_{\bar{T} \to
0^{\pm}}K'>0\textrm{.}\end{aligned}$$ If, however
1. $\bar{T}\to 0^{+}$, $\mathop {\lim}\limits_{\bar{T}
\to 0^{+}}\bar{\Omega}=+\infty$ and $\bar{\Omega}$ is positive, $C^{3}$ and strictly increasing on some interval $(0,c]$
then $$\begin{aligned}
\mathop{\lim}\limits_{\bar{T} \to
0^{-}}K'<0\textrm{.}\end{aligned}$$
*Proof*\
For subcases i) and ii) the dominant term in the Ricci scalar’s derivative is $$\begin{aligned}
\left(R^{ef}R_{ef}\right)_{,m}
&\approx& 48\bar{M}^{5}\bar{\Omega}\left(\bar{L}-1\right)\left(\bar{L}^{2}-\bar{L} +
1\right)
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\bar{T}_{,m}\end{aligned}$$ because this contains the highest power of $\bar{M} :=
\frac{\bar{\Omega}'}{\bar{\Omega}^{2}}$ (which is divergent for these subcases [@Scott]); all other terms are either regular or bounded.\
For the subcases iii) and iv) the dominant term of the Ricci scalar’s derivative is $$\begin{aligned}
\left(R^{ef}R_{ef}\right)_{,m}
&\approx& -48\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{5}\left(\bar{L}-1\right)\left(\bar{L}^{2}-\bar{L} +
1\right)
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\bar{T}_{,m}\end{aligned}$$ because it contains the highest power of $\bar{\Omega}'/\bar{\Omega}$[^3] and all other terms will be regular or bounded.\
While for the Ricci scalar, the dominant term is always going to be $$\begin{aligned}
R^{ef}R_{ef}&\sim&12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)\textrm{.}\end{aligned}$$
Initially we consider cases i) and ii). The entropy scalar’s derivative, in this case is give by, $$\begin{aligned}
K_{,m} &=&
\frac{-4\bar{\Omega}^{-5}(\bar{C}^{abcd}\bar{C}_{abcd})_{,m}R^{ef}R_{ef}\bar{T}_{,m}
- \bar{\Omega}^{-4}(R^{ef}R_{ef})_{,m}\bar{C}^{abcd}\bar{C}_{abcd}
}{\left(R^{ef}R_{ef} \right)^{2}}\\
&\sim& \frac{-4\bar{\Omega}^{-5}(\bar{C}^{abcd}\bar{C}_{abcd})_{,m}(12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right))\bar{T}_{,m}}{\left(12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)\right)^{2}}\nonumber\\
&-& \frac{\bar{\Omega}^{-4}\left(48\bar{M}^{5}\bar{\Omega}\left(\bar{L}-1\right)\left(\bar{L}^{2}-\bar{L} +
1\right)
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\bar{T}_{,m}\right)\bar{C}^{abcd}\bar{C}_{abcd}
}{\left(12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)\right)^{2}}\\
&=& \frac{-(\bar{C}^{abcd}\bar{C}_{abcd})_{,m}\bar{T}_{,m}}
{3\bar{\Omega}^{5}\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)}\nonumber\\
&-& \frac{\left(\bar{L}-1\right)\bar{C}^{abcd}\bar{C}_{abcd}\bar{T}_{,m}}
{3\bar{\Omega}^{7}\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{3}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)}\\
&\sim& \frac{\left(1-\bar{L}\right)\bar{C}^{abcd}\bar{C}_{abcd}\bar{T}_{,m}}
{3\bar{\Omega}^{7}\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{3}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)}\\
&=& \frac{1}{\bar{\Omega}}\frac{1}{\frac{\bar{\Omega}'}{\bar{\Omega}}}
\frac{\bar{C}^{abcd}\bar{C}_{abcd}\left(1-\bar{L}\right)\bar{T}_{,m}}
{3\bar{\Omega}^{6}\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{2}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)}\textrm{.}\end{aligned}$$ The reason for writing it in this form becomes clear when we note that $\mathop{\lim}\limits_{\bar{T} \to
0^{\pm}}\bar{\Omega}(\bar{T})\to 0$, $\bar{\lambda} < 1$. Therefore the sign of $K_{,m}$ solely depends on the signs of $\bar{\Omega}$ and $\bar{\Omega}'/\bar{\Omega}$ because all other terms are positive and nonzero.\
The reader will recall that for case i), $\mathop
{\lim}\limits_{\bar{T}
\to 0^{+}}1/\bar{\Omega}\to
\infty^{+}$ and $\mathop {\lim}\limits_{\bar{T}
\to 0^{+}}\bar{\Omega}'/\bar{\Omega}\to +\infty$. Hence the above is always positive and so is the entropy scalar’s derivative.\
For case ii), $\mathop {\lim}\limits_{\bar{T}
\to 0^{-}}1/\bar{\Omega}\to \infty^{-}$ and $\mathop {\lim}\limits_{\bar{T}
\to 0^{-}}\bar{\Omega}'/\bar{\Omega}\to -\infty$ and hence the above is positive and so is $K_{,m}$.\
We turn to cases iii) and iv) now. The entropy scalar’s derivative is $$\begin{aligned}
& &K_{,m} =
\frac{-4\bar{\Omega}^{-5}(\bar{C}^{abcd}\bar{C}_{abcd})_{,m}R^{ef}R_{ef}\bar{T}_{,m}
- \bar{\Omega}^{-4}(R^{ef}R_{ef})_{,m}\bar{C}^{abcd}\bar{C}_{abcd}
}{\left(R^{ef}R_{ef} \right)^{2}}\\
&\sim& \frac{-4\bar{\Omega}^{-5}(\bar{C}^{abcd}\bar{C}_{abcd})_{,m}(12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right))\bar{T}_{,m}}{\left(12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)\right)^{2}}\nonumber\\
&+& \frac{\bar{\Omega}^{-4}\left(48\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{5}\left(\bar{L}-1\right)\left(\bar{L}^{2}-\bar{L} +
1\right)
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\bar{T}_{,m}\right)\bar{C}^{abcd}\bar{C}_{abcd}
}{\left(12\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)\right)^{2}}\\
&=& \frac{-\bar{\Omega}^{-5}(\bar{C}^{abcd}\bar{C}_{abcd})_{,m}\bar{T}_{,m}}{3\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)}\nonumber\\
&+& \frac{\bar{\Omega}^{-4}\frac{\bar{\Omega}'}{\bar{\Omega}}\bar{C}^{abcd}\bar{C}_{abcd}\bar{T}_{,m}
}{3\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{4}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}
+ 1\right)}\\
&\sim&\frac{\bar{C}^{abcd}\bar{C}_{abcd}\bar{T}_{,m}
}{3\bar{\Omega}^{4}\left(\frac{\bar{\Omega}'}{\bar{\Omega}}\right)^{3}
\left(\bar{g}^{ei}\bar{T}_{,e}\bar{T}_{,i}\right)^{2}\left(\bar{L}^{2}-\bar{L}+ 1\right)}\end{aligned}$$ As we saw before, this mathematical form is helpful because $\mathop{\lim}\limits_{\bar{T} \to
0^{\pm}}\bar{\Omega}^{4}(\bar{T})\to +\infty$, $\bar{\lambda} > 1$. So the sign of $K_{,m}$ solely depends on the sign of $\bar{\Omega}'/\bar{\Omega}$ because all other terms are positive and nonzero.\
For case iii) $\mathop {\lim}\limits_{\bar{T}
\to 0^{+}}\bar{\Omega}'/\bar{\Omega}\to -\infty$ and hence the above is negative.\
For case iv) $\mathop {\lim}\limits_{\bar{T}
\to 0^{-}}\bar{\Omega}'/\bar{\Omega}\to +\infty$ and hence the above is positive. $\Box$\
This now completes the proof. In order to guide the reader in visualising this behaviour, we present three representations of the monotonicity of $K$. The first represents cases i) and iii), the second is case ii) and the last is case iv).
{width="45.00000%"} \[IPSPIUKPrime\]
{width="45.00000%"} \[IFSKPrime\]
{width="45.00000%"} \[FIUKPrime\]
This is fundamentally important because it means for initial isotropic structures, their measure of gravitational entropy will increase away from zero and for final isotropic states, their gravitational entropy will decrease towards zero. What we want to ascertain now is how $K$ behaves for anisotropic future states. At this stage, all example cosmologies that admit an AFEU or AFS have $K > 0$ but we have not been able to prove this in all generality. As the direction of this study will be somewhat different to this paper, we defer this discussion to an upcoming paper apart from the following remarks.\
All observational evidence indicates that, at least locally, entropy is ever increasing and if Quiescent Cosmology is to be consistent with observational evidence (as well as Penrose’s WCH) then the AFEU and AFS should have a measure of gravitational entropy that is nonzero. This will serve to demonstrate that a universe that begins with an isotropic structure and ends in an anisotropic state will have a net increase of gravitational entropy. This will not serve to demonstrate monotonicity in the intermediate region as gravitational entropy may be oscillatory in nature during this region but it will show a net increase.
Conclusions and Further Outlook
===============================
The work in this paper is pivotal to prove not only the physical plausibility of an IPS but also to demonstrate that the IPS is truly compatible with the WCH. We have been able to show that the gravitational entropy scalar will, in a local neighbourhood of the IPS at $\bar{T} = 0$, monotonically increase for non conformally flat spacetimes. This is in direct agreement with Penrose’s conjecture regarding the dominance of the Weyl scalar.\
Furthermore, we have also been able to prove that, if the Universe did not start with a Big Bang but rather was a uniform distribution of matter, corresponding to a PIU then this too has zero gravitational entropy that monotonically increases as cosmic time increases. Although classical General Relativity seems to predict that the Universe started with a Big Bang, it is reassuring nevertheless, that Quiescent Cosmology and the WCH is compatible with this structure.\
If the Universe were to end in an isotropic singularity then the gravitational entropy will be locally monotonically increasing towards zero. This seems to indicate that $K$ would obtain a maximum (negative) value before increasing to zero. This is somewhat similar to the case when the Universe ends as an FIU because in this case the entropy scalar decreases monotonically as the FIU is approached. Both of these scenarios indicate that at some stage prior to the isotropic end, the Universe had a maximum, nonzero value of gravitational entropy and that it will tend to zero in the future. This is not compatible with the second law of thermodynamics but it means that if the end of the Universe is going to be isotropic then it means gravitational entropy will have to decrease from some finite maximal value.\
As mentioned at the end of our main results section, the obvious extension for this type of work is to consider our anisotropic futures and see if their gravitational entropy scalar’s are monotonically increasing as they are approached. As will be seen in future papers, the problems caused by the degenerate nature of the AFEU and AFS will force us to address the question of gravitational entropy in a different manner.
References {#references .unnumbered}
==========
[^1]: This is in contrast to the ideals of Chaotic Cosmology, made famous by Misner [@Misner]
[^2]: The definitions given here are slightly modified from the original ones [@Scott]. The modifications are due to the removal of the limiting causal future at $\bar{T} = 0$. This has been replaced with our open neighbourhood on $\bar{T} = 0$.
[^3]: the behaviour of this function has been well described previously [@Scott]
|
---
bibliography:
- 'Krivorotko\_reference.bib'
title: Differential evolution algorithm of solving an inverse problem for the spatial Solow mathematical model
---
Introduction
============
As the Solow growth model [@Solow_1956] was build using production function (naturally Cobb-Douglas production function), law of motion for the stock of capital and saving/investment function, model can be easily extended to include a households problem (the Ramsey-Cass-Koopmans model). Usually the interest of the Solow model is that it perpetual growth, that can be obtained using balanced growth path and technological progress over time. Output per worker can grow only as long as capital per worker grows and the key to constant growth is the existence of non-diminishing marginal product of capital. Another way of perpetual economical growth is letting technological progress change in model, it means allowing technological parameter to grow exogenously over time.
As it has been written in [@Enkhbat_2010], the Solow growth model was considered with assumptions as concave homogeneous production function (instead of Cobb-Douglas production function), exponentially growing labor and constant saving function. In addition, they considered the per capita consumption maximization problem subject to economic equilibria. Authors considered two cases when production function is logistic and the labor grow exponentially and when both of them are logistic, to reduce them to one variable parametric maximization problem. After making sure that production function is nonconvex and satisfies the Lipschitz conditions, authors solved nonconvex optimization problem by global optimization techniques, considered on a sufficiently large interval. The solution can be found by the method of piecewise linear function [@Horst_1995].
In the article [@Smirnov_2018] by Smirnov and Wang, the work of Ryuzo Sato [@Sato_1980], devoted to the development of economic growth models within the framework of the Lie group theory, was extended to a new growth model based on the assumption of logistic growth by using the Solow economic growth model as a starting point. Authors claimed that the Cobb-Douglas function can no longer adequately describe the growth of the economy over a long-run, it was aimed to develop a new mathematical paradigm that can be used to study the current state of economy and to replace neoclassical growth model in the sense of Sato representing exponential growth with a logistic growth. Also they used the new “S-shaped” production function, the consequence of logistic growth in factors, to solve maximization problem of profit under condition of perfect competition, using the same arguments of subject to relevant changes by assuming that the revenue of the firm from sales is determined.
The logistic growth in other words can be described by spatial Solow model and that was used in [@Engbers_2014], where they did identification of production function using (noisy) data that is an ill-posed inverse problem, using non-parametric approach and applied Tikhonov regularization to stabilize the computations. As there is no clear choice which production function will fit the situation best, it was proposed to identify production function from data about the capital distribution of some spatial economy, further they obtained the following optimization problem that has to be minimized. To solve the minimization problem authors applied the gradient descent algorithm. As the objective function was Freshet differentiable, they used the directional derivatives of the Langrangian and to find the minimum of the functional the simple steepest descent method and a backtracking line search method [@Nocedal_1999] were used. So they reconstructed the production function to a spatial Solow model with the different noise levels and different technology terms, when it is constant and space-dependent.
We use the spatial Solow mathematical model as in [@Engbers_2014] and investigate the inverse problem for its using stochastic approach for global optimization.
The paper is organized as follows. Section 2 presents the derivation of the original Solow mathematical model described by ordinary differential equation and the statement of inverse problem for the Solow model described by partial differential equation. The formulation of an inverse problem as the optimization problem and numerical algorithm for solving inverse problem is presented in Section 3. The results of numerical calculations for spatial Solow model are presented and discussed in Section 4. Conclusions are given in Section 5, followed by a list of references.
Statement of the problem
========================
In this Section the derivation of neoclassical Solow mathematical model for ordinary differential equation is demonstrated at subsection \[ODE\_Solow\]. Based on that derivation the statement of the spatial Solow model is considered in subsection \[spatial\_Solow\] and the inverse problem statement for spatial Solow model is formulated at subsection \[inverse\_Solow\].
Solow mathematical model for ODE {#ODE_Solow}
--------------------------------
Neoclassical economical Solow model describes evolution of gross output – $Y(t)$, using next (due to such) indicators as: used labor resources – $L(t)$, saving capital – $K(t)$ and technological progress – $A(t)$. And since the output parameter of the model should be a stable indicator of a productive economy, then the gross domestic product (GDP) is taken, which is a macroeconomic index reflecting the market value of all final goods and services produced during the year in the state [@McConnellBrueFlynn]. A mathematical notation connecting these variables is $Y(t)=A(t)Q(K(t),L(t))$, where $Q$ represents production function. It is assumed that the production function is homogeneous, which means $Q(\alpha K(t),\alpha L(t))=\alpha Q(K(t),L(t))$. Also it can be noted that the production function satisfies the following condition $$\begin{aligned}
Q( 0, L(t))=0= Q(K(t),0).\end{aligned}$$ It is worth saying that the description of the development of any economy only due to the absolute value of any gross output is useless, it is hard to say whether the economy is doing well or not. Simon Kuznets, one of the architects of the US national accounting system, the man who first introduced the concept of GDP in 1934, warned against identifying GDP growth with increasing economic or social welfare.What we are interested in is the rate of economic growth [@Kuznets_1934; @Kuznets_1941]. Therefore, we consider the rate of change in capital, which looks like $$\begin{aligned}
\label{eq:change_capital}
\dfrac{dK}{dt}=Y(t)-C(t)-\delta K(t).\end{aligned}$$ It means the change in fixed capital stock negatively depends on the volume of consumption $C(t)$ and on the amount of depreciation that is supposed to occur with the rate $\delta$. Moreover, we assume that the difference in production and consumption persists for each period of time, namely $$\begin{aligned}
\label{eq:prod_cons}
Y(t)-C(t)=sY(t).\end{aligned}$$ Then using that and inserting (\[eq:prod\_cons\]) into (\[eq:change\_capital\]), we have the following $$\begin{aligned}
\label{eq:gross_output_sub}
\dfrac{dK}{dt}=sA(t)Q(K(t),L(t))-\delta K(t).\end{aligned}$$ Next, we introduce a new variable, namely $k(t)=\dfrac{K(t)}{L(t)}$, the capital per capita. Then it turns out, using the homogeneous of function $Q$ we can write the following $$\begin{aligned}
q(k(t))=\frac{1}{L(t)}Q(K(t),L(t))=Q\left(\frac{K(t)}{L(t)},1\right) \end{aligned}$$ and calculate $$\begin{aligned}
\frac{dk(t)}{dt}=\frac{d}{dt}\left(\frac{K(t)}{L(t)}\right)=\frac{\frac{dK(t)}{dt}}{L(t)}-n\frac{K(t)}{L(t)},\end{aligned}$$ where $n=\frac{\frac{dL(t)}{dt}}{L(t)}$ denotes a constant growth rate of labor costs (labor intensity). With these designations and abbreviations, we can rewrite (\[eq:gross\_output\_sub\]) as follows $$\begin{aligned}
\label{eq:gross_output_sub_rewrite}
\dfrac{dk(t)}{dt}=sA(t)q(k(t))-(\delta+n)k,\end{aligned}$$ which is the basic equation for spatial structured Solow model [@Solow_1956; @Krugman_1991; @Mossay_2003].
It is worthy to clarify that we are interested in the change in capital for a work unit (that is, an employee) - the capital-labor ratio, or more precisely, the situation where the capital per work unit reaches its steady state. To do this, consider a stationary solution of equation (\[eq:gross\_output\_sub\_rewrite\]) $$\begin{aligned}
0=sA(t)q(k(t))-(\delta+n)k.\end{aligned}$$ If we assume that $A(t)=1$, then it means that there is no technological progress at all. Then there are only three variables describing the capital-labor ratio: saving rate - $s$, depreciation rate - $\delta$ and the rate of population growth or unit of labor used - $n$. Consequently, capital intensity will increase (grow) if $$\begin{aligned}
sq(k(t))>(\delta+n)k\end{aligned}$$ and decrease (fall) otherwise. Thus, if the capital ratio is a constant number, then economy tends to its steady state $k_E$, i.e. there are enough savings to cover the costs associated with population growth and the amount of capital lost due to depreciation. Moreover, the economic growth rate in steady state equals to rate of population growth (i.e. $n$). Further, we assume that parameters such as population growth rate and depreciation coefficient are always constant, then the only variable affecting the model is the savings rate - $s$. It is also assumed that when saving changes from $s_0$ to $s_1$ at ($s_1>s_0$), the function shows a sharp rise, and then the steady state increases from $k_0$ to $k_1$. It is good for economy for a short period of time, because economic growth occurs faster, but in the long run the economy will tend to a new steady state and then the economic growth rate will again be equal to $n$. So $n$ is not only constant, but also equals zero, since the population does not change at all. The rate of savings over a large time interval, in turn, does not have any effect either on the rate of economic growth. The only option to obtain economic growth is a technological progress [@Engbers_2009]. Thus, if the parameter as $n$ does not have any effect on model - $n=0$, noting that we set $s=1$ for simplification, then the equation (\[eq:gross\_output\_sub\_rewrite\]) should be rewritten as $$\begin{aligned}
\label{eq:basic_rewr}
\dfrac{dk(t)}{dt}=A(t)q(k(t))-\delta k.\end{aligned}$$
The spatial Solow model {#spatial_Solow}
-----------------------
Consider the scaled initial-boundary value problem for the mathematical model described dynamic of the capital stock held by the representative household located at $x$ at date $t$ [@Camacho_2008; @Engbers_2014]. Then the mathematical model (\[eq:basic\_rewr\]) with adding initial and boundary conditions is rewritten as follows: $$\begin{aligned}
\label{eq:spartial_Solow_m}
\left\{\begin{array}{ll}
\dfrac{\partial k}{\partial t} - d {\mathop{}\!\mathbin\bigtriangleup}k(x,t) = g(k, x, t), \quad & x\in \varOmega , t\in [0,T],\\
k(x,0) = k_0(x) > 0, & x\in \varOmega,\\
\nabla k\cdot n = 0, & \text{on}\,\,\partial \varOmega\times [0,T].
\end{array}\right.\end{aligned}$$ Here $d=\frac{1}{\delta L^2}$ is a scaled coefficient, $\delta$ is the depreciation rate, $g(k,x,t) = \frac{A(x,t)}{\delta}q(k) - k$, $A(x,t)$ denotes the technological level at $x$ and time $t$. The standard neoclassical production function is assumed to be non-negative, increasing and concave, and verifies the Inada conditions, that is, $$\begin{aligned}
\lim_{k\to 0}q^\prime (k) = +\infty, \quad \lim_{k\to \infty}q^\prime (k) = 0,\quad q(0)=0.\end{aligned}$$ We will depart from the assumptions with respect to concavity in particular around zero as well as the first Inada condition and allow for general convex-concave production functions, an example being [@Engbers_2014] $$\begin{aligned}
\label{eq:prouction_func}
q(k)=\dfrac{\alpha_1 k^p}{1+\alpha_2 k^p},\quad \alpha_1,\alpha_2 \geq 0, p > 1.\end{aligned}$$ Such examples of $q$ are of particular interest, because they are related to the potential existence of poverty traps. Define the set of admissible production functions $$\begin{aligned}
\begin{array}{cc}
Q_{\mbox{adm}} = \left\{ q\in H^1(0,K)\ |\ q(0)=0, 0\leq q^\prime(k)\leq q^\prime_{\max}\,\, \text{for}\,\, k\in(0,K),\right.\\
\qquad\qquad\qquad\quad \left. q^\prime(k) = 0\,\, \text{else} \right\},
\end{array}\end{aligned}$$ where $q^\prime_{\max}$ being a fixed constant, which can be understood as the maximal growth that an economy is capable of.
The technological level $A(x,t)$ is determined via a diffusion equation of the form $$\begin{aligned}
\label{eq:tech_level_A}
\left\{\begin{array}{ll}
\dfrac{\partial A}{\partial t} - {\mathop{}\!\mathbin\bigtriangleup}A = Ag_A, \quad & x\in \varOmega , t\in [0,T],\\
A(x,0) = A_0(x), & x\in \varOmega,\\
\frac{\partial A}{\partial x} = 0, & \text{on}\,\,\partial \varOmega\times [0,T],
\end{array}\right.\end{aligned}$$ with $g_A$ being either constant, a function depending only on space or a function depending on space as well as on time.
The Neumann boundary condition in problem (\[eq:spartial\_Solow\_m\]) represents no capital flow through the boundary and thereby a closed economy.
In paper [@Engbers_2014] authors proved a well-posedness of direct problem (\[eq:spartial\_Solow\_m\]) at space $L^2([0,T], H^1(\varOmega))\cap H^1([0,T], H^{-1}(\varOmega))$ if $k_0\in L^\infty(\varOmega)$, $q\in Q_{\mbox{adm}}$ and $A\in C(\varOmega\times [0,T])$. A more detailed analysis of this model can be found in [@Capasso_2010].
Inverse problem statement for spatial Solow model {#inverse_Solow}
-------------------------------------------------
The choice of the production function is crucial for an economic model, as its shape will greatly influence the capital distribution. In general, data about the economic situation, such as the gross domestic product (GDP), of different regions and different countries are readily available. Suppose, that we have additional information about GDP of some spatial economy at fixed space and time points: $$\begin{aligned}
\label{eq:ip_data}
k(x_m, t_j) = f_{mj} + \varepsilon_{mj},\; x_m\in \varOmega, t_j\in [0,T], \, m=1,\ldots,M, j=1,\ldots, N.\end{aligned}$$ Here $\varepsilon_{mj}$ are Gaussian noise in measurements.
The *inverse problem* (\[eq:spartial\_Solow\_m\]), (\[eq:ip\_data\]) consists in identification of production function (\[eq:prouction\_func\]) (or identification of parameters $\alpha_1,\alpha_2,p$) of initial-boundary value problem (\[eq:spartial\_Solow\_m\]) using additional measurements (\[eq:ip\_data\]). It means that we have the nonlinear parameter-to-solution map $A:\ q\in Q_{\text{adm}} \mapsto f^\varepsilon\in E^{MN}$ mapping the production function $q$ to the respective capital distribution $f^\varepsilon = \{f_{mj} + \varepsilon_{mj}\}_{\substack{m=1,\ldots,M,\\ j=1,\ldots, N }}$, i.e. $A(q) = f^\varepsilon$. Here $E$ is an Euclidean space of measurements.
The inverse problem (\[eq:spartial\_Solow\_m\]), (\[eq:ip\_data\]) is ill-posed [@KashtanovaVNKabanikhin2011], i.e. the solution $q(k)$ is non-unique and can be unstable [@Engbers_2014]. That is we apply the regularization technique described in Section \[sec:optimization\].
Optimization problem and numerical algorithm {#sec:optimization}
============================================
Reduce our inverse problem (\[eq:spartial\_Solow\_m\]), (\[eq:ip\_data\]) to an optimization problem that consists in minimization of the misfit function $$\begin{aligned}
\label{misfit_func}
J(q) = \Vert A(q) - f^\varepsilon\Vert^2_{L^2_\chi(\varOmega\times [0,T])} := \int\limits_0^T\int\limits_\varOmega \chi(x,t) (A(q) - f^\varepsilon)^2\, dxdt.\end{aligned}$$ Here $\chi (x,t)$ is a characteristic function of incomplete measurements (\[eq:ip\_data\]). In our case the misfit function (\[misfit\_func\]) has the form: $$J(q) = \dfrac{1}{N M}\sum\limits_{j=1}^N\sum\limits_{m=1}^M (k(x_m,t_j;q) - f_{mj}^\varepsilon)^2.$$
Optimization problem can be solved by various methods such as gradient approaches, stochastic methods, etc [@KOI_KSI_arXiv2019]. The misfit function (\[misfit\_func\]) has a lot of local minimums due to ill-posedness of inverse problem (\[eq:spartial\_Solow\_m\]), (\[eq:ip\_data\]). In paper [@Engbers_2014] authors applied the Tikhonov regularization approach based on gradient method with Tikhonov regularization term. The main weaknesses of this approach are the difficulty of choosing the regularization parameter and the dependence of the convergence of the gradient method on the choice of the initial approximation (local convergence). We choose the stochastic algorithm of global optimization based on solving more simple evolutionary problems from biology named differential evolution algorithm [@Storn_Price_1995].
Differential evolution algorithm {#DE_alg}
--------------------------------
Differential evolution algorithm (DE), a class of evolutionary algorithms, was introduced by Storn and Price at 1995 [@Storn_Price_1995; @Storn_Price_1997; @Price_2005; @Storn_2008] for solving a polynomial fitting problem. The algorithm is generally called as a very simple but very powerful population-based meta-heuristic algorithm [@Qing_2009]. The algorithm is generally characterized by the features of simplicity, effectiveness and robustness. Also, it is easy-to-use, and it requires few controlling parameters, and it has fast convergence characteristic [@Storn_Price_1997]. Due to these advantages, it presents a wide range of implementation examples in different areas such as acoustics, biology, material science, mechanic, medical imaging, optic, mathematics, physics, seismology, economics etc. More details and examples about the implementation of DE to solve various problems are given in [@Qing_2009]. Even though previous comprehensive studies over real-world problems have shown that DE performs better in terms of convergence rate and robustness [@Sambarta2009_DE_convergence; @Hahn_1963] than the other evolutionary algorithms such as genetic algorithm, particle swarm optimization [@PSO_1995], simulated annealing [@Kirkpatrick_1983], etc.
An algorithm of differential evolution is follows:
1. *Initialization*. Create an initial population of target vectors of parameters $q_{i,G} = \left( q_{i,G}^1, q_{i,G}^2, q_{i,G}^3 \right)$, $i=1,\ldots,Np$, where $Np$ is the population size, $G$ denotes current generation. Here $q_{i,G}^1 = \alpha_{1_{i,G}}$, $q_{i,G}^2 = \alpha_{2_{i,G}}$, $q_{i,G}^3 = p_{i,G}$. The algorithm is initialized by a randomly created population within a predefined search space considering the upper (index $u$) and lower (index $l$) bounds of each parameter $q_{i,G}^j \in [q_{l}^j, q_{u}^j]$, $j=1,2,3$.
2. *Choose stopping criteria*. Set the stopping parameter $\varepsilon_{stop}$ for misfit function and maximum number of iterations $G_{\max}$. If $J(q_{i,G}) < \varepsilon_{stop}$ for some $i=1,\ldots, Np$ or $G=G_{\max}$ then stop iterations and choose $i$ with minimum value of misfit function $J(q_{i,G})$. Otherwise go to step 3.
3. *Mutation*. At each iteration, the algorithm generates a new generation of vectors, randomly combining vectors form the previous generation. For each new generation ($G+1$) of a vector from a given target vector $q_{i}$ from the old generation ($G$) algorithm randomly selects three vectors $q_{r_1,G}$, $q_{r_2,G}$ and $q_{r_3,G}$ such that $i, r_1, r_2, r_3$ are distinct and creates a donor vector $$v_{i,G+1} = q_{r_1,G} + F(q_{r_2,G} - q_{r_3,G}), \quad F\in [0,2] \; \text{is a differential weight.}$$
4. *Crossover (recombination)*. Create the trial vector $u_{i,G}$ from the elements of the target vector $q_{i,G}$ and donor vector $v_{i,G+1}$ with probability $Cr\in [0,1]$ using formula: $$\begin{aligned}
u_{i,G+1}^j = \left\{\begin{array}{ll}
v_{i,G+1}^j, \quad & \text{if}\;\; \mbox{rand}_{i,j}\leq Cr\;\; \text{or}\;\; j=j_{\mbox{rand}},\\
q_{i,G}^j, & \text{otherwise}
\end{array}\right., \; j=1,2,3.\end{aligned}$$ Here $\mbox{rand}_{i,j}$ represents a uniformly distributed random variable in the range of $[0,1)$, $j_{\mbox{rand}}$ is a randomly chosen integer in the range $[1, 3]$ to provide that the trial vector does not duplicate the target vector.
5. *Selection*. The vector obtained after crossover is the test vector. If it is better than the base vector, then in the new generation the base vector is replaced by trial one, otherwise the base vector is stored in the new generation. Choose the next generation as follows: $$\begin{aligned}
q_{i,G+1} = \left\{\begin{array}{ll}
u_{i,G+1}, \quad & J(u_{i,G+1})\leq J(q_{i,G}),\\
q_{i,G}, & \text{otherwise}.
\end{array}\right.,\quad i=1,\ldots, Np\end{aligned}$$ and go to step 2 till $G+1<G_{max}$.
Numerical experiments
=====================
We will show some numerical results of the inverse problem (\[eq:spartial\_Solow\_m\]), (\[eq:ip\_data\]) using DE algorithm described is Section \[DE\_alg\]. We start by giving some details about the simulated dataset used for the calculations and then show the identification results for a constant technological level $A$ (Section \[constantA\]) and for a space-dependent technology term $A(x)$ (Section \[spacedepA\]).
Simulated dataset {#simulated_dataset}
-----------------
Consider the modelling scaled domain $[0,L]\times [0,T]$ with $L=50$ and $T=150$ (here $L$ can numerated regions with different GDP and $T$ described time in years). After nondimensionalization we get new computational domain $[0,1]\times [0,\delta T]$ where mathematical problem (\[eq:spartial\_Solow\_m\]) is formulated. We put $\delta = 0.05$. We set an equidistant grid with $N_x=26$ nodal points in space and $N_t=251$ nodal points in time, which leads to a spatial-step size $h_x = 1/N_x = 0.04$ and a time-step size $h_t =\delta T/N_t = 0.03$. The classic second-order difference approximation has been used to discretize the diffusion. The time derivative is approximated by backward difference of the first-order.
We put an initial condition $k_0(x)$ as a piece-wise function on the interval $[0,1]$: $$\begin{aligned}
k_0(x) = \left\{\begin{array}{ll}
0, & x\in [0,0.3),\\
25(x-0.3), & x\in[0.3, 0.7],\\
10, &x\in(0.7,1].
\end{array}\right.\end{aligned}$$ We obtain the synthetic data $f_{mj}$ from (\[eq:ip\_data\]) for different $M$ and $N$ by solving the direct problem (\[eq:spartial\_Solow\_m\]) with the production function $$q_{ex}(k) = \dfrac{0.0005 k^4}{1+0.0005 k^4}$$ presented at figure \[ris:func\_q\] (left) and two types of technological terms $A(x,t)$ (see below). Measurements are uniformly distributed in space on $[0,1]$ and time on $[\delta T/2, \delta T]$ (see example for $M=5$, $N=6$ at figure \[ris:func\_q\] right).
![The exact production function $q_{ex}(k)$ (left) and map of direct problem solution $k(x,t;q_{ex})$ with points $m=1,\ldots,M$, $j=1,\ldots,N$ of measurements (\[eq:ip\_data\]) for $M=5$, $N=6$ (right).[]{data-label="ris:func_q"}](Inidata_multiplot){width="1\linewidth"}
Then we add the Gaussian noise to inverse problem data (\[eq:ip\_data\]) as follows $$f_{mj}^\varepsilon = f_{mj} + \varepsilon f_{mj} \xi_{mj},\quad m=1,\ldots,M, j=1,\ldots,N.$$ Here $\xi_{mj} \sim N_{0,1}$ is a normally distributed modeled random variables with zero mean and unit dispersion, $\varepsilon$ is an error level.
For DE algorithm we put population size $Np=100$ and we choose parameters $F=0.7$ and $Cr=0.9$ as the best combination for convergence features for the algorithm [@BALKAYA2013160]. We set maximum number of iterations $G_{\max} = 5000$ and $\varepsilon_{stop} = 10^{-4}$. For getting the optimized solution of the inverse problem we launch the DE algorithm 1000 times for all decribed numerical calculations using the cluster NKS-30T in the Siberian Supercomputer Center in the Institute of Computational Mathematics and Mathematical Geophysics of the SB RAS and then take the arithmetic average.
Numerical results for constant technological term $A(x,t)=1$ {#constantA}
------------------------------------------------------------
We solve optimization problem (\[misfit\_func\]) with constant technological term $A(x,t) = 1$ using DE described in Section \[DE\_alg\]. For $\varepsilon =0.1$ in data (\[eq:ip\_data\]) we get the inverse problem solution $q_\varepsilon(k)$ for four variants of $M$ and $N$. Figure \[ris:func\_q\_difMN\] (left) demonstrates the difference $\delta(k) = q_{ex}(k)-q_\varepsilon(k)$ of exact and calculated solutions of inverse problem with four variants of measurements. Table \[tab:tab1\] shows that the $M=5$, $N=6$ is sufficient for reconstruction of production function with necessary accuracy in relative error $\rho_\varepsilon = \Vert k(\cdot,\cdot;q_{ex}) - k(\cdot,\cdot;q_\varepsilon)) \Vert_{L^2}/\Vert k(\cdot,\cdot;q_{ex}) \Vert_{L^2}$. The smaller number of measurement points the greater the difference $\delta(k)$ (see figure \[ris:func\_q\_difMN\] left).
![The difference $\delta (k)$ of exact and approximate solutions for different points of measurements $M$ and $N$ for fixed error level in data (\[eq:ip\_data\]) $\varepsilon=0.1$ (left). The difference $\delta (k)$ of exact and approximate solutions for different noise levels $\varepsilon=0, 0.05, 0.1$ in measurements for $M=5$, $N=6$ (right).[]{data-label="ris:func_q_difMN"}](IPsol_q_multiplot){width="1\linewidth"}
Values of $M$ and $N$ $\max|\delta(k)|$ $\rho_\varepsilon$ $J(q_\varepsilon)$
----------------------- ------------------- -------------------- --------------------
$M=3$, $N=2$ 0.171 0.024 0.102
$M=4$, $N=4$ 0.053 0.015 0.479
$M=5$, $N=6$ 0.02 0.004 0.293
$M=13$, $N=10$ 0.01 0.006 0.199
: Relative errors and value of the misfit function $J(q_\varepsilon)$ for different number of measurements (\[eq:ip\_data\]) with error level $\varepsilon = 0.1$ and constant technological level $A(x,t)=1$.[]{data-label="tab:tab1"}
For $\varepsilon =0$, $0.05$ and $0.1$ in data (\[eq:ip\_data\]), $M=5$, $N=6$, we get the inverse problem solution $q_\varepsilon(k)$ (the differences $\delta(k)$ are plotted on figure \[ris:func\_q\_difMN\] right).Table \[tab:tab2\] shows the reconstructed parameters $\alpha_1$, $\alpha_2$ and $p$ in function (\[eq:prouction\_func\]) for different error level in measured data. If we have noise free data of inverse problem then the difference $\delta(k)$ is close to zero. It means that reconstruction of parameters $\alpha_1, \alpha_2, p$ is close to the tested ones (the maximum of the absolute difference $\delta(k)$ is equal to 0.005 as given in table \[tab:tab2\]). Note, that maximum absolute error of inverse problem solutions for $M=5$, $N=6$ is less than 2%, i.e. $\max|\delta(k)| \leq 0.02$ for maximum error level in inverse problem data (\[eq:ip\_data\]) $\varepsilon = 0.1$.
------------------- -------- ------------------- ---------------------- ---------------------
$\varepsilon = 0$ $\varepsilon = 0.05$ $\varepsilon = 0.1$
$\alpha_1$ 0.0005 0.00057 0.00048 0.00036
$\alpha_2$ 0.0005 0.00057 0.00048 0.00036
$p$ 4 3.9226 4.0202 4.1806
$\max|\delta(k)|$ 0.005 0.019 0.02
------------------- -------- ------------------- ---------------------- ---------------------
: Reconstructed parameters in function $q_\varepsilon(k)$ for different error levels $\varepsilon = 0, 0.05, 0.1$ in measurements (\[eq:ip\_data\]) for $M=5$, $N=6$ and constant technological term $A(x,t) = 1$.[]{data-label="tab:tab2"}
The solution $k(x,t;q_\varepsilon)$ of spatial Solow mathematical model for reconstructed $q_\varepsilon(k)$ and measured data (\[eq:ip\_data\]) for $\varepsilon = 0.1$ is demonstrated on figure \[ris:direct\_pr\_solution\].
![The solution $k(x,t_j;q_\varepsilon)$ of the direct problem (\[eq:spartial\_Solow\_m\]) for reconstructed $q_\varepsilon(k)$ with error level in measurements $\varepsilon = 0.1$ and points of measurement are $M=5$, $N=6$ for constant technological term. Here $t_1=75$, $t_3=100.2$, $t_6=138$.[]{data-label="ris:direct_pr_solution"}](Dir_pr_multiplot){width="1\linewidth"}
Figure \[ris:direct\_pr\_solution\] shows the compliance of model solution $k(x,t;q_\varepsilon)$ (red line for fixed time point) with measured synthetic noisy data with noise level $\varepsilon = 0.1$ (black triangles).
### Sensitivity analysis of spatial Solow mathematical model.
Investigate the influence of parameters $\alpha_1$, $\alpha_2$ and $p$ to the mathematical model (\[eq:spartial\_Solow\_m\]) namely to the right-hand side $$g(k, x,t) = \gamma \frac{\alpha_1 k^p}{1+\alpha_2 k^p} - k,\quad \gamma = \frac{A(x,t)}{\delta}.$$ For this function $g$, consider its gradient by parameters: $$\frac{\partial g}{\partial\alpha_1} = \frac{\gamma k^p}{1+\alpha_2 k^p}, \; \frac{\partial g}{\partial\alpha_2} = -\frac{\gamma\alpha_1 k^{2p}}{(1+\alpha_2 k^p)^2}, \; \frac{\partial g}{\partial p} = -\frac{\gamma\alpha_1\mbox{ln}(k) k^{p}}{(1+\alpha_2 k^p)^2}.$$
For different values of function $k(x,t)$ we construct the gradient field of function $g$. Figure \[ris:grad\_sensitivity\] shows that the maximum rate of gradient variability for small values of capital stock $k(x,t)$ corresponds to parameters $\alpha_1$ and $\alpha_2$. For bigger values of $k(x,t)$ (figure \[ris:grad\_sensitivity\] right) and for small values of parameters $\alpha_1$ and $\alpha_2$ gradient grows to the direction of parameter $p$, but when the values of parameters $\alpha_1$ and $\alpha_2$ became bigger the gradient growth turns to parameters $\alpha_1$ and $\alpha_2$ again.
Numerical results for space dependent technological level $A(x)$ {#spacedepA}
----------------------------------------------------------------
We consider a space-dependent technological level $A(x)$ demonstrated at figure \[ris:inidata\_A(x)\] (left). Then the solution of the direct problem (\[eq:spartial\_Solow\_m\]) for the exact function $q_{ex}$ demonstrates on figure \[ris:inidata\_A(x)\] (right).
Using the same simulated dataset (see Section \[simulated\_dataset\]) the inverse problem (\[eq:spartial\_Solow\_m\]), (\[eq:ip\_data\]) is solved for number of measurements $M=5$, $N=6$ and differents error level $\varepsilon = 0, 0.05, 0.1$. The results are collected to table \[tab:tab3\] and demonstrated at figure \[ris:func\_q\_A(x)\]. Note, that the results of inverse problem solution are the same as for constant technological term $A$ (see Section \[constantA\]), i.e. accuracy in relative error $\rho_\varepsilon$ is less than $10^{-2}$, maximum of absolute difference of exact and approximated solutions of inverse problem $\max|\delta(k)|$ is the same order of $10^{-2}$. The difference of exact and approximated solutions of inverse problem $\delta (k)$ for $\varepsilon = 0, 0.05, 0.1$ in inverse problem data (\[eq:ip\_data\]) is plotted at figure \[ris:func\_q\_A(x)\] (right). We can see that such error in reconstruction of parameters $\alpha_1$, $\alpha_2$ and $p$ (see table \[tab:tab3\]) is not critical to the behavior of the function $q(k)$ (see figure \[ris:func\_q\_A(x)\] from the left that demonstrated the exact and reconstructed solutions of inverse problem for error level in data (\[eq:ip\_data\]) $\varepsilon=0.1$ and $M=5$, $N=6$).
-------------------- -------- ------------------- ---------------------- ---------------------
$\varepsilon = 0$ $\varepsilon = 0.05$ $\varepsilon = 0.1$
$\alpha_1$ 0.0005 0.00055 0.00065 0.00018
$\alpha_2$ 0.0005 0.00055 0.00064 0.00018
$p$ 4 3.9409 3.843 4.5525
$\max|\delta(k)|$ 0.005 0.02 0.05
$\rho_\varepsilon$ 0.001 0.005 0.009
$J(q_\varepsilon)$ $9\cdot 10^{-5}$ $3.8\cdot 10^{-2}$ 0.118
-------------------- -------- ------------------- ---------------------- ---------------------
: Reconstructed parameters in function $q_\varepsilon(k)$ for different error levels $\varepsilon = 0, 0.05, 0.1$ in measurements (\[eq:ip\_data\]) for $M=5$, $N=6$ and space-dependent technological term $A(x)$.[]{data-label="tab:tab3"}
![The exact $q_{ex}(k)$ and approximate $q_\varepsilon(k)$ solutions of inverse problem for spatial Solow model for error level in data (\[eq:ip\_data\]) $\varepsilon=0.1$ (left) and difference $\delta (k)$ of exact and approximate solutions of inverse problem for different noise levels $\varepsilon=0, 0.05, 0.1$ in measurements (right) for $M=5$, $N=6$ for space-dependent technological level $A(x)$.[]{data-label="ris:func_q_A(x)"}](IPsol_q_multiplot_A_x){width="1\linewidth"}
The solution $k(x,t;q_\varepsilon)$ of spatial Solow mathematical model for reconstructed $q_\varepsilon(k)$ and measured data (\[eq:ip\_data\]) for $\varepsilon = 0.1$ is demonstrated on figure \[ris:direct\_pr\_solution\_A(x)\] in case of space-dependent technological level $A(x)$. Note that capital stocks $k(x,t_j;q_\varepsilon)$, $j=1,2,3$ (purple lines) are close to the measurement points $f^\varepsilon$ (black triangles) as expected.
![The solution $k(x,t_j;q_\varepsilon)$ of the direct problem (\[eq:spartial\_Solow\_m\]) for reconstructed $q_\varepsilon(k)$ with error level in measurements $\varepsilon = 0.1$ and points of measurement are $M=5$, $N=6$ for space-dependent technological term. Here $t_1=75$, $t_3=100.2$, $t_6=138$.[]{data-label="ris:direct_pr_solution_A(x)"}](Dir_pr_multiplot_A_x){width="1\linewidth"}
Conclusion and outlook
======================
Today, economists use Solow’s sources-of-growth accounting to estimate the separate effects on economic growth of technological change, capital, and labor. One important use of the Solow growth model is to estimate the share of observed growth that has resulted from growth in Total Factor Productivity (TFP), rather than from the application of increased inputs - labor, capital, and human capital (increased productive skills resulting from education and training.) Using the Solow model to approximate the output that would result in the absence of any change in TFP, you can then subtract this value from the output actually produced, and attribute the difference to TFP growth. The growth Solow model is the starting point of all analyses in modern economic growth theories, thus understanding of the model is essential to understanding the theories of the Solow growth.
The differential evolution algorithm is applied to the optimization problem of the production function $q(k)$ reconstruction for the spatial Solow model using additional measurements of GDP type for fixed space and time. Despite the fact that the considered inverse problem is ill-posed, numerical calculations show a good result with the accuracy of recovery of the production function is more than 95% (in case of error level in measured data $10\%$). We compare the results with calculations from paper [@Engbers_2014] where the authors applied Tikhonov regularization and gradient method for solving the regularized optimization problem. In the case of full measured data (that means $f^\varepsilon(x,t) = k(x,t) + \varepsilon (x,t)$) and error level $10\%$ the accuracy of reconstruction of production function was 80% for both cases of technological levels. The reason consists in sensitivity of local regularization methods to an initial approximation while the reconstruction results for DE approach do not depend on initial population.
|
---
abstract: 'Random graphs with prescribed degree sequences have been widely used as a model of complex networks. Comparing an observed network to an ensemble of such graphs allows one to detect deviations from randomness in network properties. Here we briefly review two existing methods for the generation of random graphs with arbitrary degree sequences, which we call the “switching” and “matching” methods, and present a new method based on the “go with the winners” Monte Carlo method. The matching method may suffer from nonuniform sampling, while the switching method has no general theoretical bound on its mixing time. The “go with the winners” method has neither of these drawbacks, but is slow. It can however be used to evaluate the reliability of the other two methods and, by doing this, we demonstrate that the deviations of the switching and matching algorithms under realistic conditions are small compared to the “go with the winners” algorithm. Because of its combination of speed and accuracy we recommend the use of the switching method for most calculations.'
author:
- 'R. Milo'
- 'N. Kashtan'
- 'S. Itzkovitz'
- 'M. E. J. Newman'
- 'U. Alon'
title: On the uniform generation of random graphs with prescribed degree sequences
---
Introduction
============
In the rapidly growing literature on the modeling of complex networks one of the most important classes of network models is the random graph [@Bollobas2001]. One well-studied such model is the model consisting of the ensemble of all graphs that have a given degree sequence [@Bender; @Molloy; @1995; @Molloy; @1998; @Newman; @2001; @Chung_diameter], and this model has proved useful in understanding a variety of network properties. Realistic applications often require that we restrict ourselves to graphs with no multiple edges between any vertex pair and no self-edges. Unfortunately, both the analytic and numerical study of such networks is known to present challenges [@Bender; @Snijders; @Rao; @Roberts; @Kannan; @chen; @Itzkovitz; @MSZ02; @Park]. In this short paper we consider computer algorithms for generating graphs uniformly from this ensemble. We are concerned primarily with directed graphs, since the examples we will consider are directed, but the concepts discussed generalize in a straightforward fashion to the undirected case also.
There are two algorithms in common use for the generation of random graphs with single edges. We will refer to them as the *switching algorithm* [@Rao; @Roberts; @Newman; @2002; @Maslov; @2002; @Stone; @Shen; @or; @2002; @Milo; @2002] and the *matching algorithm* [@Molloy; @1995; @Newman; @2001; @Milo; @2002]. We argue that, under certain circumstances, both of these algorithms can generate a nonuniform sample of possible graphs. We then present a new algorithm based on the Monte Carlo procedure known as *go with the winners* [@Aldous; @Grassberger], which generates uniformly sampled graphs. We compare the three methods in the context of a particular network problem—estimation of the density of commonly occurring subgraphs or *motifs*—and show that, in this context, the difference between them is small. This result is of some practical importance, since the “go with the winners” algorithm, although statistically correct, is slow, while the other two algorithms are substantially faster.
----------------------- --------- --------- --------- ---------- --------- --------- --------- --------- -------- --------- --------- --------
Network
mean s.d. $Z$ mean s.d. $Z$ mean s.d. $Z$ mean s.d. $Z$
“go with the winners” 7.57(5) 3.05(3) 10.6(1) 11.06(6) 3.60(4) 14.1(2) 88(1) 10.7(7) 3.4(3) 2.20(5) 1.48(3) 284(6)
switching 7.63(9) 3.05(6) 10.5(2) 11.0(1) 3.71(7) 13.7(3) 88.3(3) 10.1(2) 3.6(1) 2.24(5) 1.47(3) 286(6)
matching 7.67(9) 2.98(6) 10.8(2) 11.1(1) 3.67(7) 13.8(3) 94.5(3) 10.0(2) 3.0(1) 2.21(5) 1.45(3) 290(6)
----------------------- --------- --------- --------- ---------- --------- --------- --------- --------- -------- --------- --------- --------
Algorithms
==========
In this section we describe the three algorithms under consideration.
Switching algorithm
-------------------
First, we describe the switching algorithm, which uses a Markov chain to generate a random graph with a given degree sequence [@Rao; @Roberts; @Newman; @2002; @Maslov; @2002; @Stone; @Shen; @or; @2002; @Milo; @2002]. For simplicity, we discuss directed networks with no mutual edges (vertex pairs with edges running in both directions between them). The case with mutual edges is a simple generalization [@Roberts].
The method starts from a given network and involves carrying out a series of Monte Carlo switching steps whereby a pair of edges $(A\to B,C\to D)$ is selected at random and the ends are exchanged to give $(A\to D,C\to B)$. However, the exchange is only performed if it generates no multiple edges or self-edges; otherwise it is not performed. The entire process is repeated some number $QE$ times, where $E$ is the number of edges in the graph and $Q$ is chosen large enough that the Markov chain shows good mixing. (Exchanges that are not performed because they would generate multiple or self-edges are still counted to insure detailed balance [@footnote].)
This algorithm works well but, as with many Markov chain methods, suffers because in general we have no measure of how long we need to wait for it to mix properly. Theoretical bounds on the mixing time exist only for specific near-regular degree sequences [@Kannan]. We empirically find, however, that for many networks, values of around $Q=100$ appear to be more than adequate (see Fig. \[fig2\]).
Matching algorithm
------------------
An alternative approach is the matching algorithm [@Molloy; @1995; @Newman; @2001; @Milo; @2002], in which each vertex is assigned a set of “stubs” or “spokes”—the sawn-off ends of incoming and outgoing edges—according to the desired degree sequence. (One can also assign mutual-edge stubs for networks that include such edges.) Then in-stubs and out-stubs are picked randomly in pairs and joined up to create the network edges. If a multiple or self-edge is created, the entire network is discarded and the process starts over from scratch.
This process will correctly generate random directed graphs with the desired properties. Unfortunately, however, many real-world networks have a heavy-tailed degree distribution that includes a small minority of vertices with high degree. All other things being equal, the expected number of edges between two such vertices will often exceed one, making it unlikely that the procedure above will run to completion, except in the rarest of cases. To obviate this problem a modification of the method can be used in which, following selection of a stub pair that creates a multiple edge, the network is not discarded, and an alternative stub pair is selected at random. In general this method generates a biased sample of possible networks [@King] but, as we will show, not significantly so for our purposes (see Table \[Table1\]).
Go with the Winners algorithm
-----------------------------
The “go with the winners” algorithm is a non-Markov-chain Monte Carlo method for sampling uniformly from a given distribution [@Aldous; @Grassberger]. When applied to the problem of graph generation, the method is as follows. We consider a colony of $M$ graphs. As with the matching algorithm, we start with the appropriate number of in-stubs and out-stubs for each vertex and repeatedly choose at random one in-stub and one out-stub from the graph and link them together to create an edge. If a multiple edge or self-edge is generated, the network containing it is removed from the colony and discarded. To compensate for the resulting slow decline in the size of the colony, its size is periodically doubled by cloning each of the surviving graphs; this cloning step is carried out at a predetermined rate chosen to keep the size of the colony roughly constant on average. The process is repeated until all stubs have been linked, then one network is chosen at random from the colony and assigned a weight: $$\begin{aligned}
W_i=2^{-c}\frac{m}{M},
\label{eq2}\end{aligned}$$ where $c$ is the number of cloning steps made and $m$ is the number of surviving networks. The mean of any quantity $X$ (for example, the number of occurrences of a given subgraph) over a set of such networks is then given by $$\begin{aligned}
\frac{\sum_{i}{{W_i}{X_i}}}{{\sum_{i}{W_i}}},
\label{eq3}\end{aligned}$$ where $X_i$ is the value of $X$ in network $i$.
Comparison of algorithms
========================
In Fig. \[hub1\] we show a comparison of the performance of our three algorithms when applied to a simple toy network. The network consists of an out-hub with ten outgoing edges, an in-hub with ten incoming edges, and ten nodes with one incoming edge and one outgoing edge each. Given this degree sequence, there are just two distinct network topologies with no multiple edges, as shown in Fig. \[hub1\]a and \[hub1\]b. There is only a single way to form the network in \[hub1\]a, but there are 90 different ways to form \[hub1\]b.
We generated $100\,000$ random networks using each of the 3 methods described here and the results are summarized in Fig. \[hub1\]c. As the figure shows, the matching algorithm introduces a bias, undersampling the configuration of Fig. \[hub1\]a. This is a result of the dynamics of the algorithm, which favors the creation of edges between hubs. The switching and “go with the winners” algorithms on the other hand sample the configurations uniformly, generating each graph an equal number of times within the measurement error on our calculations. The “go with the winners” algorithm truly samples the ensemble uniformly but is far less efficient than the two other methods. The results given here indicate that the switching algorithm produces essentially identical results while being a good deal faster. The matching algorithm is faster still but samples in a measurably biased way.
Now consider the study of network motifs. We are interested in knowing when particular subgraphs or motifs appear significantly more or less often in a real-world network than would be expected on the basis of chance, and we can answer this question by comparing motif counts to random graphs. Some results for the case of the “feed-forward loop” motif [@Shen; @or; @2002; @Milo; @2002] are given in Table \[Table1\]. In this case the densities of motifs in the real-world networks are many standard deviations away from random, which suggests that any of the present algorithms is adequate for generating suitable random graphs to act as a null model, although the “go with the winners” and switching algorithms, while slower, are clearly more satisfactory theoretically. The matching algorithm was measurably nonuniform for our toy example above, but seems to give better results on the real-world problem.
Overall, our results appear to argue in favor of using the switching method, with the “go with the winners” method finding limited use as a check on the accuracy of sampling. Accuracy checks are also supplied by analytical estimates for subgraph numbers [@Itzkovitz]. Numerical results in [@Milo; @2002; @Milo; @2004; @Itzkovitz] were done using the switching algorithm.
Conclusions
===========
In this paper we have compared three algorithms for generating random graphs with prescribed degree sequences and no multiple edges or self-edges. Two of the three have been used previously, but suffer from nonuniformity in their sampling properties, while the third, a method based on the “go with the winners” Monte Carlo procedure, is new and samples uniformly but is quite slow. Of the two older algorithms, we show that one, which we call the “matching” algorithm, has measurable deviations from uniformity when compared to the “go with the winners” method, although for graphs typical of practical studies these deviations are small enough to make no significant difference to most previously published results. The other older algorithm, which we call the “switching” algorithm and which is based on a Markov chain Monte Carlo method, samples correctly in the limit of long times and in practice is found to give good results when compared with the “go with the winners” method. Overall, therefore, we conclude that the switching algorithm is probably the algorithm of choice, with the “go with the winners” algorithm finding a supporting role as a check on uniformity, although its slowness makes it impractical for large-scale use.\
We thank Oliver D. King for discussions and for pointing out and demonstrating that the matching algorithm of the supplementary online material of [@Milo; @2002] does not uniformly generate simple graphs.
[99]{} B. Bollobas, *Random Graphs*, 2nd edition, Academic Press, New York (2001). E. Bender and E. Canfield, The asymptotic number of labelled graphs with given degree sequences, *J. Combin. Theory Ser. A* **24**, 296–307 (1978). M. Molloy and B. Reed, The size of the giant component of a random graph with a given degree sequence, *Combinatorics, Probability and Computing* **7**, 295–305 (1998). M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, *Random Structures and Algorithms* **6**, 161–179 (1995). M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Random graphs with arbitrary degree distribution and their applications, *Phys. Rev. E* **64**, 026118 (2001). F. Chung and L. Lu, The average distances in random graphs with given expected degrees, *Proc. Natl. Acad. Sci. U.S.A.* **99**, 15879–15882 (2002). T. A. B. Snijders, Enumeration and simulation methods for 0–1 matrices with given marginals, *Psychometrika* **56**, 397–417 (1991). A. R. Rao, R. Jana, and S. Bandyopadhya, A Markov chain Monte Carlo method for generating random $(0,1)$-matrices with given marginals, *Indian J. of Statistics* **58**, 225–242 (1996). J. M. Roberts, Jr., Simple methods for simulating sociomatrices with given marginal totals, *Social Networks* **22**, 273–283 (2000). R. Kannan, P. Tetali, and S. Vempala, Simple Markov-chain algorithms for generating bipartite graphs and tournaments, *Proceedings of the ACM Symposium on Discrete Algorithms* (1997). Y. Chen, P. Diaconis, S. P. Holmes, and J. S. Liu, Sequential Monte Carlo Methods for Statistical Analysis of Tables, *Discussion Paper 03-22*, Institute of Statistics and Decision Sciences, Duke University (2003). S. Itzkovitz, R. Milo, N. Kashtan, G. Ziv, and U. Alon, Subgraphs in random networks, *Phys. Rev. E* **68**, 026127 (2003). S. Maslov, K. Sneppen, and A. Zaliznyak, Pattern detection in complex networks: Correlation profile of the Internet, preprint cond-mat/0205379 (2002). J. Park and M.E.J. Newman, The origin of degree correlations in the Internet and other networks, *Phys. Rev. E* **68**, 026112 (2003). M. E. J. Newman, Assortative mixing in networks, *Phys. Rev. Lett.* **89**, 208701 (2002). S. Maslov and K. Sneppen, Specificity and stability in topology of protein networks, *Science* **296**, 910–913 (2002). A. Roberts, L. Stone, Island-sharing by archipelago species. *Oecologia* **83**, 560-567 (1990). S. Shen-Orr, R. Milo, S. Mangan, and U. Alon, Network motifs in the transcriptional regulation network of Escherichia coli, *Nature Genetics* **31**, 64–68 (2002). R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan., D. Chklovskii, and U. Alon, Network motifs: Simple building blocks of complex networks, *Science* **298**, 824–827 (2002). R. Milo, S. Itzkovitz, N. Kashtan, R. Levitt, S. Shen-Orr, I. Ayzenshtat, M. Sheffer and U. Alon, Superfamilies of designed and evolved networks, *Science* **303**, 1538–42 (2004). O. D. King, Private communication. D. Aldous and U.V. Vazirani, “Go With the Winners” algorithms, *Proceedings of the IEEE Symposium on Foundations of Computer Science*, pp. 492–501 (1994). P. Grassberger and W. Nadler, “Go-with-the-winners” simulations, preprint cond-mat/0010265 (2000). F. Brglez, D. Bryan, and K. Kozminski, Combinatorial Profiles of Sequential Benchmark Circuits, *Proceedings of IEEE International Symposium on Circuits and Systems*, 1929 (1989). There are singular networks where the Markov process is not ergodic. This lack of ergodicity can be removed by making small modifications as in [@Rao], and by choosing the number of switching steps per edge to be randomly distributed around Q.
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---
abstract: 'A covariant calculus for the construction of effective string theories is developed. Effective string theory, describing quantum string-like excitations in arbitrary dimension, has in the past been constructed using the principles of conformal field theory, but not in a systematic way. Using the freedom of choice of field definition, a particular field definition is made in a systematic way to allow an explicit construction of effective string theories with manifest exact conformal symmetry. The impossibility of a manifestly invariant description of the Polchinski-Strominger Lagrangian is demonstrated and its meaning is explained.'
author:
- 'N.D. Hari Dass'
- Peter Matlock
title: Covariant Calculus for Effective String Theories
---
Introduction {#intro}
============
Although fundamental string theory is of course confined to certain critical dimensions, string-like phenomena do indeed appear as defects, solitons or effective descriptions in a variety of physical situations. Since these situations generically are of non-critical dimension, an effective theory of strings must exist in order to describe them.
Polchinski and Strominger (PS) proposed the construction of such a theory in [@PS]. As in other constructions of effective theories, the formulation is required to exhibit the correct symmetries, while dropping such requirements as renormalisability and polynomial lagrangian, which are usually taken as minimal for a ‘fundamental’ theory expected to be valid at all energy scales. In particular, PS treated an expansion around a long-string vacuum, where the characteristic string length $R$ is taken as a large parameter. The effective action is thus expanded in inverse powers of $R$. The notable difference with fundamental string theory is that the effective PS theory contains a variable central charge, which can be adjusted for consistency in any dimension. Although Polchinski and Strominger showed that the price one has to pay for this quantum consistency in any dimension is the allowance of nonpolynomial terms in the action, in such a perturbative expansion around the long-string vacuum, such terms are perfectly acceptable.
PS were able to calculate the excitation spectrum including in the effective action the first correction after the leading Polyakov-type (equivalently Nambu-Goto) term. Surprisingly, the spectrum does not deviate from that of Nambu-Goto theory at this order. It has been shown in [@Drum] and [@HDPM2] using an action valid to order $R^{-3}$, that at even the next relevant order after this, the spectrum does not differ from that of Nambu-Goto theory.
In the original formulation [@PS] of PS, the choice was made to omit terms in the effective action proportional to the leading-order equations of motion (EOM), which may be removed to appropriate order by a field redefinition. In fact, as we have shown in [@HDPM2], dropping or including a particular set of such ‘irrelevant terms’[^1] amounts to a particular choice of field definition; the PS field definition corresponds to the omission of all EOM terms. It was pointed out in [@HDPM2] that different such choices of field definition will correspond to actions invariant under different transformation laws; different field definitions are related by some field redefinition transformation, and this of course relates potentially different transformation laws, each representing the conformal symmetry of the theory.
The effective action proposed by Polchinski and Strominger is $$\begin{aligned}
\label{PSaction}
S_{PS} &=& \frac{1}{4\pi} \int {\textup{d}}\tau^+ {\textup{d}}\tau^- \bigg\{
\frac{1}{a^2} {\partial}_+ X^\mu {\partial}_- X_\mu \nn\\
&&\mbox{}+\beta \frac{{\partial}_+^2 X\cdot{\partial}_- X {\partial}_+ X\cdot{\partial}_-^2 X}{({\partial}_+X\cdot{\partial}_-X)^2}
+\Pcm{O}(R^{-3})
\bigg\}
.\end{aligned}$$ The quantity $R$ signifies the length of the string and in what follows, is taken to be large. Consideration is restricted to fluctuations around the classical background. The leading-order equation of motion $
{\partial}_+{\partial}_- X^\mu = 0
$ has the solution $
X^\mu_{\textup{cl}} = e^\mu_+R\tau^+ + e^\mu_- R \tau^-,
$ where $e_-^2=e_+^2=0$ and $e_+\cdot e_- = -1/2$.
The action of is invariant, to the appropriate order, under the transformation $$\label{modtrans}
\delta^{\textup{PS}}_- X^\mu = \epsilon^-(\tau^-){\partial}_- X^\mu - \frac{\beta a^2}{2}{\partial}_-^2 \epsilon^-(\tau^-)
\frac{{\partial}_+ X^\mu}{{\partial}_+X\cdot{\partial}_-X}$$ (and another: $\delta_+X$ with $+$ and $-$ interchanged).
PS proposed an algorithm for extending their analysis to higher orders which can be stated as follows. Firstly, write down all possible $(1,1)$ terms which according to PS simply amounts to keeping terms whose net number of $\pm$-derivatives (terms in the denominator count negatively) is $(1,1)$. Secondly, discard all terms proportional to the leading-order constraints ${\partial}_\pm X\cdot{\partial}_\pm X$ and their derivatives. Finally, use integration by parts to relate equivalent terms.
At this point one will have terms with and without ‘mixed derivatives’, terms sporting mixed derivatives being what we have called *irrelevant* in [@HDPM2]. The PS prescription then is to discard all irrelevant terms and *find transformation laws* that leave the relevant terms in the action invariant.
Clearly, generalisation of the PS formalism requires not only finding the right action to the desired order, but also determining the appropriate transformation laws. This is reminiscent of the early days of supergravity theories, and this procedure becomes tedious and unwieldy with increasing order. Not only does the procedure become tedious, more importantly it does not lend itself to a systematic method of construction and analysis at higher orders. It is the purpose of the present paper to propose a simplified formalism, in both a technical and conceptual sense. We propose to achieve this through a formulation wherein the transformation laws are independent of the particular action chosen. We start by demonstrating how this can be done for the PS action itself.
Recall that the PS proposal for the leading correction was based on a comparison with the Liouville action for subcritical strings $$S_{Liou} = \frac{26-D}{48\pi}\int~d\tau^+~d\tau^-~~ \partial_+\phi\partial_-\phi$$ They argued that in effective string theories the conformal factor $e^\phi$ should be replaced by the component $\partial_+X\cdot\partial_-X$ (in the conformal gauge) of the induced metric on the worldsheet. They had also proposed replacing $(26-D)/12$ by a parameter $\beta$ which was to be determined by requiring the vanishing of the total central charge in all dimensions, though they eventually found $\beta$ to be just the same as in the Liouville theory [^2]. A direct application of this idea would have suggested the total action $$\begin{aligned}
\label{action2}
S_{2} &=& \frac{1}{4\pi} \int {\textup{d}}\tau^+ {\textup{d}}\tau^- \bigg\{
\frac{1}{a^2} {\partial}_+ X^\mu {\partial}_- X_\mu \nn\\
&&\mbox{}+\beta
\frac {{\partial}_+({\partial}_+X\cdot{\partial}_-X){\partial}_-({\partial}_+X\cdot{\partial}_-X)}{({\partial}_+X\cdot{\partial}_-X)^2}
\bigg\}
.\end{aligned}$$ It is easily shown that $S_{(2)}$ is invariant under the transformations $$\label{urtrans}
\delta^{0}_\pm X^\mu = \epsilon^\pm(\tau^\pm){\partial}_\pm X^\mu
.$$ For the purposes of the present discussion, we consider the ‘$-$’ alternative, without loss of generality. More explicitly, if we write the second part of $S_2$, $S_2^{(2)}$, as $$\label{defL2}
S^{(2)}_2 = \int {\textup{d}}\tau^+{\textup{d}}\tau^- {L}_2$$ it is easy to show that $$\label{psanom}
\delta^{0}_- L_2 = {\partial}_-(\epsilon^- L_2) + {\partial}_-^2\epsilon^- {\partial}_+ L$$ The first term is what one would have expected if $L_2$ had transformed as a scalar density, and the second term is a departure from this. We shall explain this important point later; for the moment it suffices to note that the additional term can be rewritten as $${\partial}_-^2\epsilon^- {\partial}_+ L = {\partial}_+({\partial}_-^2\epsilon^- L)$$ ensuring the invariance of $S_2$ if we neglect integrals of total derivative terms.
Polchinski and strominger [@PS] build their effective action while discarding all total derivatives. This has generally been done in the literature; treatment of total derivative terms in the action is a subtle and important issue that in principle needs careful scrutiny. In this paper, we shall nevertheless proceed with the premise that such total derivative terms can be ignored.
The algebra of the PS transformations of is $$[\delta^{PS}_-(\epsilon_1^-) , \delta^{PS}_-(\epsilon_2^-)] = \delta^{PS}_-(\epsilon_{12}^-) +\Pcm{O}(R^{-4})
,$$ where $\epsilon_{12}^- = \epsilon_1^-{\partial}_-\epsilon_2^- - \epsilon_2^-{\partial}_-\epsilon_1^-$. On the other hand, the algebra of the transformations of is $$[\delta^{0}_-(\epsilon_1^-) , \delta^{0}_-(\epsilon_2^-)] = \delta^{0}_-(\epsilon_{12}^-)
.$$ Thus both generate the same group of symmetry transformations, namely the conformal group. While the PS transformations realise this only approximately, to $\Pcm{O}(R^{-4})$ which however is sufficient in context as the PS action is defined to $\Pcm{O}(R^{-3})$, the transformations leaving $S_{2}$ invariant do so *exactly*.
It should be noted that field redefinitions do not change the algebra of transformations, though the transformation laws are themselves changed. This is indeed what is happening here and to understand this note $$\label{2ps}
S_{PS}-S_{2} = \frac{\beta}{4\pi}\int L^{-2}\partial_{+-}X\cdot\partial_-X
\partial_+L$$ where $L=\partial_+X\cdot\partial_-X$. Thus the additional terms are proportional to the EOM of the leading part of the action, and can be removed through field redefinitions to appropriate order. A detailed discussion of how the field redefinition corresponding to indeed connects and , as well as an alternate description of effective string theories based on the action $S_{2}$ can be found in [@HDPM2].
Simplified Formalism
====================
The above discussion points to a much simpler formulation of effective string theories whereby the transformation law is always of the form . Furthermore, $S_{2}$ provides an example of an effective string theory which is in principle valid to all orders in $1/R$, and would provide an important test case for understanding higher-order corrections to the spectrum of Nambu-Goto theory. Before beginning construction, we briefly discuss here the merits of such a formulation.
Covariant Formalism
-------------------
Since the transformation is fixed and does not have to be fine-tuned to a given action we now have the possibility of a systematic covariant calculus for the construction of invariant actions. One is always assured of obtaining invariant actions this way whereas a generalisation of the PS algorithm based on the naïve $(1,1)$ counting there is no guarantee that to a given action one can always find a suitable transformation law. In a given case one has either to use trial and error or to identify a field redefinition, and even then the results are valid only up to some prescibed order. On the other hand each construction based on a covariant calculus will yield actions valid to all orders in $1/R$. In the present paper, we approach the construction of a covariant formulation in two ways, explicitly given in sections and .
Measures and Quantum Equivalences
---------------------------------
An important issue tied up with field redefinitions in Quantum Field Theory is that of the quantum equivalence of theories related by them. This has been addressed to some extent in [@HDPM2]. In the path integral formulation, to which the canonical formulation should be equivalent after all care has been exercised, this concerns the invariant (under the symmetry transformations) measure to be adopted as well as its transformation under field redefinitions. Both these issues are naturally taken care of in a covariant formulation based on . As the ‘naïve’ measure $\prod_\sigma {\textup{d}}X(\sigma)$ is invariant modulo irrelevant regularisation-dependent factors, specifying the action specifies everything. This is a great simplification. Covariance fixes the irrelevant terms also thereby fixing a field definition also.
The last point also means that in the covariant formulation one cannot simply drop the irrelevant terms as that would amount to changing the field definition which generically would result in a change of measure, as well as the transformation law which would necessitate changing the covariant calculus itself. If, however, it can be shown that the resulting field redefinition to a certain order does not spoil quantum equivalence (i.e. the measure is left unchanged to the relevant order), irrelevant terms can indeed be dropped. However, the resulting changes in the transformation law have to be taken into account.
Two paths to covariance
=======================
Symmetry Content
----------------
It is the symmetry content of the theory, more precisely the symmetry variations (transformation laws) that determine the covariant calculus. Clearly, this is dictated by the physics of the system and is not a matter of formalism. We are seeking a covariant formalism for the symmetry variations of . Before doing so it is worthwhile understanding why these should embody the symmetry content of effective string theories. Justifiably one could have taken the view that this depends on the details of systems with effective string behaviour. For example, it may have been so that only the ‘global’ version of as against the more restrictive ‘local’ version correctly captures the relevant symmetry. It just so happens that for both the leading order effective action as well as for the PS terms, the global invariance also implies the local invariance. Clearly at high orders this will no longer be true. Then it will become a matter of ‘phenomenology’ to find out which will be a better description. Nevertheless, we shall develop a covariant formulation for the local invariance. Should phenomenology prefer the global invariance as the true symmetry the rationale for such a covariant formulation would be considerably weakened. It should be pointed out that even then such a covariant formulation will be useful as a framework for any systematic phenomenological analysis.
In what follows we shall actually seek something more general. We shall seek the most general coordinate invariant version of the transformation laws of and develop the corresponding covariant calculus.
Two paths {#twopaths}
---------
We have mentioned that we will construct our covariant formalism in two alternate ways. The two distinct approaches are similar to what has been followed in the case of fundamental string theories.
The first, the Nambu-Goto method, is to start with the action $$\label{nambu}
S_{\textup{NG}} = \int \sqrt{\det({\partial}_\alpha X\cdot{\partial}_\beta X)}
,$$ invariant under . This approach is characterised by the absence of an intrinsic metric on the worldsheet. The composite operator, ${\partial}_\alpha
X\cdot{\partial}_\beta X$, also the induced metric on the worldsheet due to the flat geometry of the target space, transforming exactly as the metric, acts as a substitute metric in realising general coordinate invariance.
The second, the Polyakov method, introduces an auxiliary metric field $h_{\alpha\beta}$. The action equivalent to is the Polyakov action $$\label{polyact}
S=\int {\textup{d}}^2 \sigma \sqrt{h} h^{\alpha\beta} {\partial}_\alpha X \cdot {\partial}_\beta X
,$$ invariant under and . The metric field $h_{\alpha\beta}$ is independent. The Polyakov action is also general-coordinate invariant, although the real symmetry content is reflected in the invariance of the action under the Weyl transformations, $h_{\alpha\beta}\rightarrow \lambda(\sigma)
h_{\alpha\beta}$.
In effective string theories one will necessarily have to consider higher derivative terms in the action and these may not in general be Weyl invariant. This will require some additional technical structures which are developed in . In fundamental string theories one did not need these.
Although conceptually distinct, both these approaches lead to identical content for the final effective string theory they are designed to produce. This will be shown in detail in section .
Reparametrisation Invariance vs. Symmetry {#concepts}
-----------------------------------------
It can be seen from the above that reparametrisation invariance plays radically different rôles in the two approaches. It is worthwhile understanding this important difference. Generically reparametrisation invariance is considered in situations with an intrinsic metric for the space-time manifold. In such situations, any action can be made reparametrisation invariant, and consequently the latter is devoid of physical content. It is only the statement that specific choice of coordinates is immaterial, and that it is desirable to write the theory in a *form* which reflects this. It is not a symmetry of the physical system.[^3] This is best illustrated by the following elementary example. Consider a theory with a scalar field $\phi$ on a flat background. The action could look something like $$\label{scalarex}
S=\int{\textup{d}}^d x \big( {\partial}_\mu\phi {\partial}_\mu\phi - m^2\phi^2 + \cdots + \phi^4 + \cdots \big)
.$$
Since it doesn’t matter what coordinates we choose, it is desirable to represent the theory in a way that under general coordinate transformations, the action is invariant, in the sense that it *has the same form in any coordinate system*: $$\label{scalarexgen}
S=\int{\textup{d}}^d x \sqrt{g} \big( g^{\mu\nu}\nabla_\mu\phi \nabla_\nu\phi - m^2\phi^2 + \cdots + \phi^4 + \cdots \big)
.$$
It should be noted that this does not *change* the theory at all. The crucial point is that this applies to *any* theory. Any theory can thus be written in such diffeomorphism-invariant form, so that a particular choice of coordinates can be postponed or avoided.
Turning our attention to symmetries, the situation is conceptually different. Symmetries, unlike diffeomorphisms in the above context, restrict the physical content of the theory and not any theory can be made invariant under the symmetry transformations. A trivial example in the context of the above mentioned scalar field theory is the symmetry under $\phi\rightarrow -\phi$. This restricts the form of the action to have only even powers of $\phi$ and not all actions possess this feature irrespective of the choice of coordinates.
While any *theory* must be diffeomorphism invariant, and therefore can be written down in a covariant way which reflects this, only certain theories have a particular symmetry. There is no way to take an arbitrary theory and somehow *make* it symmetric.
Conformal Symmetry from Reparametrisation Invariance {#demon}
----------------------------------------------------
As will be seen later, in the first approach conformal symmetry emerges as residual invariance of the conformal gauge choice . Since in this approach this symmetry arises from the underlying reparametrisation invariance, which has been argued above to be generically void of physical content, it is important to understand the precise connection between this emergent conformal symmetry and reparametrisation invariance.
Does the group of reparametrisations contain the group of conformal transformations? Strictly, it does indeed; A mapping (assumed invertible, differentiable, etc..) from $x$ coordinates to $x'$ is a general coordinate transformation, and of course the mappings which correspond to conformal transformations are of the same kind. It is crucial to realise that reparametrisation invariance does not always result in conformal symmetry upon choosing the conformal gauge.
This is best exemplified again by the scalar field example of in two dimensions where coordinates can generically (at least locally) be chosen so that the intrinsic metric is of the form $$\label{confgmet}
g_{\alpha\beta} = \left[ \begin{array}{cc} 0 & \varphi \\ \varphi & 0 \end{array}\right]$$ in coordinates $\sigma^\pm$. This does not use up all available freedom, and residual coordinate transformations which preserve this form are easily seen to comprise the conformal group. The action of is indeed invariant under the action of these transformations; yet the physical content of the theory is exactly that of . What is more, *any* scalar field theory can be made to have this invariance, and it therefore does not represent any physical symmetry.
In the second approach what does represent a symmetry is the invariance under Weyl-scaling. In the scalar field example also one sees that not all actions possess this invariance in keeping with what a symmetry is.
In our first approach, in which we do not treat the metric as an independent field, we do not make any assumption of Weyl symmetry. Nevertheless there is a symmetry in this case and that is traceable entirely to reparametrisation invariance. The rôle of an intrinsic metric is instead played by suitable composite fields constructed out of the physical fields. The only degrees of freedom in the theory are taken to be the $X$ scalar fields. Since now not every action can be reparametrisation invariant, reparametrisation invariance in this case becomes a physical symmetry.
Going to the equivalent of by a coordinate choice, where $g_{\alpha\beta}$ is now a composite field transforming like a metric, and choosing $$\varphi \equiv {\partial}_+ X {\partial}_- X$$ one realises the conformal gauge of the first formalism with conformal invariance as the residual symmetry.
Thus in both approaches conformal invariance emerges as the residual invariance of the conformal gauge; but in the first case it emerges as a true physical symmetry, while in the second approach it is like a generic reparametrisation invariance but not a symmetry. It is the underlying Weyl-scaling invariance that finally results in the conformal invariance being elevated to a symmetry in the precise sense that not all actions are invariant. It should also be emphasised that in other gauges, like for example the transverse gauge $X^0=\tau,
X^1=\sigma$ there will be nothing like conformal symmetry in either of the two approaches. In that sense, this is true of fundamental string theory also, there is nothing intrinsic to conformal symmetry per se; what is important is the symmetry content of the gauge-unfixed theory.
The Denominator Principle {#denomprin}
-------------------------
What is being developed in this work is for effective string theories as opposed to fundamental string theories. The allowed actions for effective string theories can sometimes become singular for certain string configurations but for long strings fluctuating about a classical background such action terms should be sensible. However even this requirement should preclude terms in effective string actions whose denominators can become singular for some flucuation of the effective string. This becomes an important guiding principle for effective string theories. In particular, it needs to be evoked while restricting substitute metrics in section , the Weyl connection in section as well as restricting the Weyl-weight compensators in section .
Covariant Calculus I: non-intrinsic metric {#covMet}
==========================================
In this section we make one of our proposals for a covariant formulation. To attain final covariance under conformal transformations, we shall use initially covariance under worldsheet general coordinate transformations only. *A priori*, a metric field is needed for any covariant formulation. In the spirit of PS we shall not introduce any intrinsic metric on the worldsheet. It suffices to have an object that transforms the same way as a metric under general coordinate transformations. One natural choice for such a *metric substitute* is the [*induced metric*]{} on the worldsheet
$$\label{induced}
g_{\alpha\beta} = {\partial}_\alpha X\cdot{\partial}_\beta X
.$$
Strictly speaking, any quantity built out of the basic variables $X^\mu$ with the correct $2-d$ tensor structure is also a *bona fide* candidate. In fact, any such object would lead to a formulation in which covariance is manifest, and effective actions could be constructed. The choice of is in a sense the *simplest* one can make and it is also the choice that PS made.
Finally, the quantity we choose here will later appear, in gauge-fixed form, in various denominators. As we require the effective theory to be valid on any fluctuation, is the simplest choice, just as $L$ was for denominators in the initial PS formulation and subsequent treatments [@Drum; @HDPM2]. These choices are also consistent with the Denominator Principle enunciated above. All other choices, upon resorting to perturbation in $R^{-1}$, are essentially equivalent to this.
Once the metric substitute is chosen, the rest of the construction is along standard lines of Riemannian Geometry. Various covariant derivatives $D_{\alpha\beta\gamma..}X$ can be written, and invariants made out of the $g$ and these objects. In addition, tensors containing only the derivatives of $g$ can only enter through the Riemann curvature tensor $R_{\alpha\beta\gamma\delta}$ and its covariant derivatives. Since in two dimensions $$\label{rtensor}
R_{\alpha\beta\gamma\delta} = (g_{\alpha \gamma}g_{\beta\delta} - g_{\beta\gamma}g_{\alpha\delta})\frac{R}{2}
,$$ where $R$ is the Ricci scalar, one need consider $R$ and its covariant derivatives only. This vastly simplifies the construction of actions.
Some Manifestly Covariant Actions - I {#covacts1}
-------------------------------------
In this section we provide a few examples of manifestly covariant action terms, more specifically, terms that transform as scalar densities. A systematic procedure for construction of such terms to any desired order in $1/R$ will be given later, in section . One could begin with $$I_{\textup{cov}} = \sqrt{g}D_{\alpha_1\beta_1..}X^{\mu_1}D_{\alpha_2\beta_2..}X^{\mu_2}\cdot A^{\alpha_1\beta_1\cdots\alpha_2\beta_2\cdots}B_{\mu_1\mu_2\cdots}$$ where $A^{\alpha_1\beta_1\cdots\alpha_2\beta_2\cdots}$ is composed of suitable factors of Levi-Civita and metric tensors on the two-dimensional worldsheet and $B_{\mu_1\mu_2\cdot}$ made up of $\eta_{\mu\nu}$ and Levi-Civita tensors in target space. In the conformal gauge this construction can be done even more simply by stringing together a number of covariant derivatives so that there are equal net numbers of $(+,-)$ indices, and finally use sufficient inverse powers of $L$ to make the expression $(1,1)$.
Now we illustrate these methods by covariantising some terms proposed by Drummond. The PS term itself is at leading order $R^{-2}$, and Drummond [@Drum] found four possibilities for the next relevant order-$R^{-6}$ part of the action. These are $$\begin{aligned}
\label{dterms1}
M_1 & = & \frac{1}{L^3} {\partial}_+^2X\cdot{\partial}_+^2X~{\partial}_-^2X\cdot{\partial}_-^2 X ,\\
\label{dterms2}
M_2 & = & \frac{1}{L^3} {\partial}_+^2X\cdot{\partial}_-^2X~{\partial}_+^2X\cdot{\partial}_-^2 X ,\\
\label{dterms3}
M_3 &=& \frac{1}{L^4} {\partial}_-^2 X \cdot {\partial}_+^2 X {\partial}_- X \cdot {\partial}_+^2 X {\partial}_-^2 X \cdot {\partial}_+ X, \\
\label{dterms4}
M_4 &=& \frac{1}{L^5} ({\partial}_- X \cdot {\partial}_+^2 X)^2 ({\partial}_-^2 X \cdot {\partial}_+ X)^2
.\end{aligned}$$
Considering the first two terms, we can expect these to be contained in the covariant forms $$\label{cov1ex1}
{\cal M}_1 = \sqrt{g}D_{\alpha_1\beta_1}X\cdot D_{\alpha_2\beta_2}X~D^{\alpha_1\beta_1}X\cdot D^{\alpha_2\beta_2}X$$ $$\label{cov1ex2}
{\cal M}_2 = \sqrt{g}D_{\alpha_1\beta_1}X\cdot D^{\alpha_1\beta_1}X~D_{\alpha_2\beta_2}X\cdot D^{\alpha_2\beta_2}X$$
Conformal Gauge and Conformal Transformations in Calculus-I {#conf1}
-----------------------------------------------------------
The PS formulation specifically hinged on the use of the conformal transformations $$\label{conftr}
\tau^\pm~\rightarrow \tau^\pm + \epsilon^\pm; \qquad \quad {\partial}_\pm~\epsilon^\mp = 0$$ In the context of general coordinate invariance, these transformations arise as the residual transformations maintaining the conformal gauge $$\label{cgauge1}
g_{++} = g_{--} =0$$ In this gauge $g_{+-}=g_{-+}=L$ transforms as a true $(1,1)$-tensor under the conformal transformations. Importantly, $g^{+-}=g^{-+}=L^{-1}$ transforms as a $(-1,-1)$ tensor. It is straightforward to work out the non-vanishing components of the Christoffel connection as well as the Riemann curvature tensor: $$\label{Gamma1}
{\Gamma^{(1)}}^+_{++} = {\partial}_+ \ln L;~~~~{\Gamma^{(1)}}^-_{--} = {\partial}_- \ln L$$ $$\begin{aligned}
R^+_{+-+} &=& -R^+_{++-} = {\partial}_+{\partial}_- \ln L\nn\\
R^-_{-+-} &=& -R^-_{--+} = {\partial}_+{\partial}_- \ln L\end{aligned}$$ All the remaining components are zero. The resulting scalar curvature is $$\label{ricciscalar}
R = -2 \frac{{\partial}_+{\partial}_- \ln L}{L};~~~~~~\sqrt{g} R = -2{\partial}_+{\partial}_- \ln L$$ We next give explicit expressions for some covariant derivatives: $$\begin{aligned}
\label{explicit}
D_\pm~X^\mu &=& {\partial}_\pm~X^\mu\nn\\
D_{++}X^\mu &=& {\partial}_{++}X^\mu - {\partial}_+\ln L{\partial}_+ X^\mu\nn\\
D_{--}X^\mu& =& {\partial}_{--}X^\mu - {\partial}_-\ln L{\partial}_-X^\mu\nn\\
D_{+-} X^\mu &=& D_{-+}X^\mu = {\partial}_{+-}X^\mu\nn\\
D_{++-} X^\mu &=& D_{+-+} X^\mu = {\partial}_{++-} X^\mu -{\partial}_+\ln L~{\partial}_{+-} X^\mu\nn\\
D_{-++} X^\mu &=& {\partial}_{-++} X^\mu - {\partial}_-({\partial}_+\ln L~{\partial}_{+} X^\mu) \end{aligned}$$ The last two of these equations show that i) just the number of $\pm$ indices does not fully characterise a tensor; their order is important. ii) not all tensors with mixed indices are proportional to leading order EOM. The latter will alter the rules for constructing general actions in comparison to what was discussed in [@HDPM2; @Drum2]. However the last but one equation displays mixed-indices tensors that are indeed proportional to leading order EOM. This is a consequence of the following two important relations:
If $T_{\mu_1\dots\mu_n}$ is a tensor with $m_\pm$ indices of type $\pm$, $$\begin{aligned}
D_{+}T_{\mu_1\dots\mu_n} &=& {\partial}_+~T_{\mu_1\dots\mu_n} - m_+{\partial}_+\ln L T_{\mu_1\dots\mu_n}\nn\\
D_{-}T_{\mu_1\dots\mu_n} &=& {\partial}_-~T_{\mu_1\dots\mu_n} - m_-{\partial}_-\ln L T_{\mu_1\dots\mu_n}\nn\\\end{aligned}$$ Hence covariant derivatives of tensors which are a combination of leading order EOM and its derivatives are also combinations of leading order EOM and its derivatives.
Another important property is that $D_{\pm\pm}X\cdot D_\pm X$ are linear combinations of leading order constraints ${\partial}_\pm X\cdot{\partial}_\pm X$ and their derivatives. That is, $$\label{pmpmconst}
D_{\pm\pm}X\cdot D_\pm X = \frac{1}{2}{\partial}_\pm({\partial}_\pm X\cdot{\partial}_\pm X) - {\partial}_\pm\ln L ({\partial}_\pm X\cdot
{\partial}_\pm X)
.$$ This too follows trivially from the second and third eqns of .
Covariant Calculus II: Intrinsic Metric and Weyl Symmetry {#covWeyl}
=========================================================
In this section, we develop the covariant calculus based on the Polyakov approach which is both general coordinate invariant and Weyl-invariant.
We constuct covariant derivatives with respect not only to the diffeomorphisms, but also to the Weyl-scaling symmetry, and use these objects to construct covariant terms. Although this approach is quite different from the non-intrinsic metric approach of section , we will show in the end that the two approaches give identical results.
conformal symmetry {#review}
------------------
Beginning with the Polyakov action $$\label{polyact2}
S=\int {\textup{d}}^2 \sigma \sqrt{h} h^{\alpha\beta} {\partial}_\alpha X \cdot {\partial}_\beta X$$ since $X^\mu$ is a worldsheet scalar, this construction ensures two-dimensional worldsheet reparametrisation invariance. The infinitesimal such transformation generated by $\sigma \rightarrow \sigma' = \sigma - \epsilon (\sigma)$ is given by $$\label{covtrans1}
\delta_\epsilon X^\alpha = \epsilon^\gamma {\partial}_\gamma X^\alpha
.$$ $$\label{covtrans2}
\delta_\epsilon h^{\alpha \beta} =
\epsilon^\gamma{\partial}_\gamma h^{\alpha\beta}
-{\partial}_\gamma\epsilon^\alpha h^{\gamma\beta}
-{\partial}_\gamma\epsilon^\beta h^{\alpha\gamma}
.$$ The important symmetry of is of course the local Weyl Scaling, which only affects the metric, $$\label{finWeyl}
h_{\alpha\beta} \rightarrow h'_{\alpha\beta} = \omega(\sigma) h_{\alpha\beta}$$ whose infinitesimal version with $\omega(\sigma)=1+\lambda(\sigma)$ reads $$\label{inflWeyl}
\delta_\lambda h_{\alpha\beta} = \lambda h_{\alpha\beta}
.$$
A combination of the reparametrisation and Weyl symmetries is used to bring the worldsheet metric to the form $h_{\alpha\beta}=\eta_{\alpha\beta}$, called conformal gauge.
This choice of $h_{\alpha\beta}$ does not fix the coordinates and the freedom of Weyl scaling completely; A combined Weyl scaling and coordinate transformation such that $$\label{wctcomb}
\lambda h_{\alpha\beta} = {\partial}_\beta\epsilon_\alpha + {\partial}_\alpha\epsilon_\beta$$ preserves $h_{\alpha\beta}=\eta_{\alpha\beta}$. This residual symmetry is worldsheet conformal symmetry. Defining coordinates $\tau^{\pm} = \tau \pm \sigma$, the remaining infinitesimal symmetries are parametrised by arbitrary functions $$\epsilon^+(\tau^+) , \qquad \textup{and} \qquad \epsilon^-(\tau^-)
.$$ This is of course just the symmetry of .
Generalised Covariant Derivatives {#covf}
----------------------------------
What we need are quantities that transform covariantly under both general coordinate transformations as well as local Weyl scalings. Hence we need tensors with definite Weyl-scaling dimensions. A tensor $\phi$ of Weyl-scaling dimension $j$ transforms under Weyl-scalings as $$\label{finWeyl2}
\phi \rightarrow \phi '= \omega(\sigma)^j \phi$$ The Weyl-weight of $h_{\alpha\beta}$ is $1$ according to (this is a matter of convention without any loss of generality). To see the issues involved, consider a worldsheet vector $V_\beta$ with Weyl-weight $j_V$; its covariant derivative with respect to reparametrisations is $$\label{covD1}
\nabla_\alpha V_\beta = {\partial}_\alpha V_\beta - \Gamma_{\alpha\beta}^\gamma V_\gamma$$ with the connection $\Gamma_{\alpha\beta}^\gamma$ given by the standard Christoffel symbol $$\label{connection}
\Gamma_{\alpha\beta}^\gamma = -\frac12 h^{\gamma\delta} \big(
{\partial}_\delta h_{\alpha\beta}-{\partial}_\alpha h_{\beta\delta}-{\partial}_\beta h_{\alpha\delta}
\big)
.$$
Clearly under Weyl-scalings of $V_\alpha$ the covariant derivative of does not scale in any simple way. In this particular example, there are two sources for this; the occurrence of the derivative of $V$ on the one hand, and the occurrence of the derivatives of $h_{\alpha\beta}$ on the other. Ordinary derivatives of a tensor $\phi$ with definite Weyl-weight $j$ do not simply scale when $\phi$ is locally scaled.
This motivates the definition of a new *Weyl-covariant derivative*; for a tensor field $\phi$ of Weyl-scaling dimension $j$, we set $$\label{weylD}
\Delta_\alpha \phi \equiv {\partial}_\alpha \phi - j \chi_\alpha \phi$$ Restricting to the case when $\phi$ is a scalar, one sees that $\chi_\alpha$ must transform as a worldsheet vector under reparametrisations. The Weyl-covariant derivative of a field with Weyl-scaling dimension $j$ should again be a field with the same Weyl-scaling dimension $j$: $$\label{wcdscalar}
(\Delta_\alpha \phi) '= \omega^j \Delta_\alpha \phi$$ under the transformation . This requires the following inhomogeneous transformation of $\chi_\alpha$ under Weyl-scaling $$\label{chitrans}
\chi'_\alpha = \chi_\alpha + {\partial}_\alpha \ln\omega$$
This immediately leads to the following generalisation of the Christoffel symbol that is appropriate for the present context: $$\begin{aligned}
\label{Gconnection}
G_{\alpha\beta}^\gamma &=& \frac12 h^{\gamma\delta}
\big(
\Delta_\alpha h_{\beta\delta}
+\Delta_\beta h_{\alpha\delta}
-\Delta_\delta h_{\alpha\beta}
\big)\nn\\
&\equiv& \Gamma_{\alpha\beta}^\gamma + W_{\alpha\beta}^\gamma\end{aligned}$$ where $$\label{Wtensor}
W_{\alpha\beta}^\gamma
= \frac12(h_{\alpha\beta}\chi^\gamma - \delta_\alpha^\gamma \chi_\beta - \delta_\beta^\gamma \chi_\alpha )
.$$
From it is easy to see that $G_{\alpha\beta}^\gamma$ is invariant under Weyl-scalings (it has Weyl-weight $0$) while neither $\Gamma$ nor $W$ has well-defined Weyl-weight. Since $W_{\alpha\beta}^\gamma$ transforms as a proper tensor under reparametrisations, it follows that $G_{\alpha\beta}^\gamma$ also transforms as a proper connection. Putting these observations together, we define the Weyl-reparametrisation covariant derivative $\Pcm{D}_\alpha$ of a rank-$n$ worldsheet tensor $T_{\beta_1\dots\beta_n}$ of Weyl-scaling dimension $j$ by $$\begin{aligned}
\label{totcovder}
\Pcm{D}_\alpha T_{\beta_1\dots\beta_n} &\equiv&
\Delta_\alpha T_{\beta_1\dots\beta_n} \nn\\
&-& G_{\alpha\beta_1}^\gamma T_{\gamma\beta_2\dots\beta_n} \nn\\
&-& \cdots \nn\\
&-& G_{\alpha\beta_n}^\gamma T_{\beta_1\dots\beta_{n-1}\gamma} \end{aligned}$$ It is useful to rewrite this in the suggestive form $$\begin{aligned}
\label{totcovder2}
\Pcm{D}_\alpha T_{\beta_1\dots\beta_n} &\equiv&
D_\alpha T_{\beta_1\dots\beta_n} -j\chi_\alpha T_{\beta_1\dots\beta_n}\nn\\
&-& W_{\alpha\beta_1}^\gamma T_{\gamma\beta_2\dots\beta_n} \nn\\
&-& \cdots \nn\\
&-& W_{\alpha\beta_n}^\gamma T_{\beta_1\dots\beta_{n-1}\gamma} \end{aligned}$$ In every term has the same Weyl-weight as the tensor $T$ and consequently so does $\Pcm{D} T$, but none of these terms transforms as a tensor under reparametrisations. On the other hand in every term transforms as a tensor under reparametrisations while none of them has a definite Weyl-weight. Together equations and imply that $\Pcm{D} T$ is covariant under both Weyl-scalings and reparametrisations.
The various covariant derivatives obey a Leibniz rule, just as ${\partial}_\alpha$ does: $$\begin{aligned}
\label{leibnitz}
\nabla_\alpha ( T_1 T_2 ) &=& \nabla_\alpha T_1 T_2 + T_1 \nabla_\alpha T_2 ,\\
\Delta_\alpha ( T_1 T_2 ) &=& \Delta_\alpha T_1 T_2 + T_1 \Delta_\alpha T_2 ,\\
\Pcm{D}_\alpha ( T_1 T_2 ) &=& \Pcm{D}_\alpha T_1 T_2 + T_1 \Pcm{D}_\alpha T_2
.\end{aligned}$$ where $T_1$ and $T_2$ are tensors, each with definite Weyl dimension, but not necessarily of the same rank.
$\Pcm{D}$ sports the important property $$\label{dhpzero}
\Pcm{D}_\alpha h_{\gamma\delta} = 0
.$$
Weyl Connection {#weylconnect}
---------------
All the features discussed above hold for any choice of $\chi_\alpha$ as long it responds to Weyl-scalings according to . In fact, according to that equation, a connection of the form $$\label{connectform}
\chi_\alpha = \frac1{W_\Phi} {\partial}_\alpha \log \Phi$$ where $\Phi$ is any worldsheet *scalar* of Weyl-scaling dimension $W_\Phi$, would be acceptable. It follows that $$\Pcm{D}_\alpha \Phi = 0
.$$ We shall choose $\Phi$ to be constructed from $h$ and derivatives of $X$.
We are still free to choose a form for $\Phi$. We are not constrained to use only one form for $\Phi$; anything will do so long as it is a scalar with non-zero Weyl-dimension, and also that it is conformity with the Denominator Principle of . This constrains $\Phi$ to be of the form $$\Phi = \Pcm{L} + \textup{higher order in $1/R$}.$$ where $$\Pcm{L} \equiv h^{\alpha\beta} \Pcm{D}_\alpha X \Pcm{D}_\beta X \equiv X_{;\alpha} X_{;\alpha}$$
By arguments identical to the ones that led to as the simplest choice for the metric substitute in the first approach, we conclude that the simplest choice for $\Phi$ is $$\label{Phichoice}
\Phi = \Pcm{L} \qquad W_\Phi = -1$$ In section we shall see that there is indeed an intimate connection between these two choices.
Manifestly Covariant Action Terms - II {#covacts2}
--------------------------------------
After constructing all the Weyl-reparametrisation covariant derivatives $\Pcm{D}_{\alpha\beta\dots}~X^\mu$ with Weyl-weight $0$, the Weyl-reparametrisation covariant generalised Riemann tensor $$\label{griemann}
\Pcm{R}_{\beta\gamma\delta}^\alpha\equiv \Delta_\gamma G_{\beta\delta}^\alpha
-\Delta_\delta G_{\beta\gamma}^\alpha +G_{\gamma\eta}^\alpha G_{\delta\beta}^\eta
-G_{\delta\eta}^\alpha G_{\gamma\beta}^\eta$$ and its Weyl-reparametrisation covariant derivatives, all of Weyl-weight $0$, one can construct action integrands which are scalar densities under reparametrisation and invariant under Weyl-scalings. We shall do this as a two-step process to highlight important differences from the corresponding construction in section ; first we shall construct scalar densities under reparametrisation and use Weyl-scaling covariance to eventually obtain our quantities of interest. The first step is very similar to what was done in section . Let us illustrate this by working out the analog of of section ; $$\label{cov2ex0}
\bar{\Pcm{N}}_1 = \sqrt{h}\{ \Pcm{D}_{\alpha_1\beta_1}X\cdot\Pcm{D}_{\alpha_2\beta_2}X
h^{\alpha_1\alpha_2}h^{\beta_1\beta_2}\}^2$$ The Weyl-weight of $\bar{\Pcm{N}}_1$ is $-3$ and that brings us to the second step; in order to get a term of Weyl-weight $0$ one has to multiply by something with Weyl-weight $3$. Clearly there are many ways of doing so. We call these [*Weyl-weight Compensators*]{}. We now show that if the ‘total divergence’ property of covariant derivatives is to be extended to the Weyl-reparametrisation covariant derivatives, these compensators have to be appropriate powers of $\Phi$ of .
Consider a contravariant vector $V^\alpha$ of Weyl-weight $J$. Its Weyl-reparametrisation covariant derivative is given by $$\Pcm{D}_\alpha V^{\beta} = \nabla_\alpha V^\beta - J \chi_\alpha V^\beta + W_{\alpha\gamma}^\beta V^\gamma
.$$ Hence $$\Pcm{D}_\alpha V^{\alpha} = \nabla_\alpha V^\alpha - J \chi_\alpha V^\alpha + W_{\alpha\gamma}^\alpha V^\gamma
.$$ On recalling $\nabla_\alpha V^\alpha = \frac{1}{\sqrt{h}}{\partial}_\alpha (\sqrt{h}V^\alpha)$ and $W_{\alpha\gamma}^\gamma = -\chi_\gamma$, one gets $$\label{totder2}
\Pcm{D}_\alpha V^\alpha = \frac{\Phi^{\frac{J+1}{W_\Phi}}}{\sqrt{h}}{\partial}_\alpha
(\sqrt{h}\Phi^{-{\frac{J+1}{W_\Phi}}}V^\alpha)$$ Thus in order to convert the scalar density $\sqrt{h}\Pcm{D}_\alpha
V^\alpha$ of Weyl-weight $J+1$ into a scalar density with Weyl-weight $0$ so that the total divergence property is maintained, it has to be multiplied only by $\Phi^{-(J+1)/W_\Phi}$ and not by just any expression with Weyl-weight $-(J+1)$. In other words, the Weyl-weight Compensators have to be appropriate powers of $\Phi$. With the specific choice of these compensators are powers of $\Pcm{L}$. Thus the final desired expression for our example is $$\label{cov2ex}
{\Pcm{N}}_1 = \sqrt{h}{\Pcm{L}}^{-3}\{ \Pcm{D}_{\alpha_1\beta_1}X\cdot\Pcm{D}_{\alpha_2\beta_2}X
h^{\alpha_1\alpha_2}h^{\beta_1\beta_2}\}^2$$ We will show later that and eqn are the same.
Conformal Gauge and Conformal Transformations in Calculus-II {#conf2}
-------------------------------------------------------------
As explained in detail above, we begin with both Weyl and coordinate invariance and intend to fix both to end up with something written in “$+/-$” notation. The resultant actions will be invariant under the conformal transformation .
A choice of coordinates $\sigma^\pm$ and Weyl scaling is made to set $$\label{cgauge2}
h_{+-} = h_{-+} = 2 , \qquad h^{+-} = h^{-+} = 1/2
.$$
We write gauge-fixed quantities using a ‘check’ and covariant quantities in script letters. For example, $$\Pcm{L} \equiv h^{\alpha\beta} \Pcm{D}_\alpha X \Pcm{D}_\beta X \equiv X_{;\alpha} X_{;\alpha}
\quad\rightarrow\quad \check{\Pcm{L}} \sim L \equiv {\partial}_+ X {\partial}_- X
.$$ here we write the covariant $\Pcm{D}$ derivative with a “$;$” to save space, and also assume that repeated indices are summed using the metric.
In this gauge ${\Gamma^{(2)}}^\gamma_{\alpha\beta}=0$ and the W-tensor is given by $$\check{W}_{\alpha\beta}^\gamma = -\frac{1}{2L}
\big( h_{\alpha\beta} {\partial}^\gamma - \delta_\alpha^\gamma{\partial}_\beta - \delta_\beta^\gamma{\partial}_\alpha \big) L
.$$
$+$ and $-$ skeletal forms {#pmsf}
--------------------------
In this section we explore some of the consequences of this gauge fixing. Suppose $T_{\alpha\beta\dots}$ is a gauge-fixed tensor of Weyl-dimension $j$. $$\Pcm{D}_+ T_{\dots}^{(j)} = {\partial}_+ T_{\dots} - j \chi_+ T_{\dots} - t_+ W_{++}^+ T_{\dots}$$ where we have used that $W_{++}^- = 0$ and $W_{+-}^\pm=0$, and $t_+$ is the number of $+$ indices on $T$. Evaluating the gauge-fixed $W$-connection, the only components which do not vanish are $$\label{Wfix}
W_{++}^+=- \chi_+ = {\partial}_+\ln L \qquad W_{--}^-=- \chi_- = {\partial}_-\ln L$$ and thus $$\Pcm{D}_+ T_{\dots}^{(j)} = {\partial}_+ T_{\dots} - j \chi_+ T_{\dots} + t_+ \chi_+ T_{\dots}
.$$ Similarly for $+ \leftrightarrow -$, $$\Pcm{D}_- T_{\dots}^{(j)} = {\partial}_- T_{\dots} - j \chi_- T_{\dots} + t_- \chi_- T_{\dots}
.$$ Evidently, $$\begin{aligned}
\label{dcomm}
[\Pcm{D}_+,\Pcm{D}_-] T_{\dots}^{(j)} &=& (t_- -t_+) ({\partial}_+\chi_-) T_{\dots}^{(j)} \\
&=& (t_- - t_+) ({\partial}_+{\partial}_-\log \Phi) T_{\dots}^{(j)}
,\end{aligned}$$ in fact consistent with our earlier calculations, despite the difference in formalism.
We now show that the Weyl-reparametrisation covariant derivatives $\Pcm{D}_{\alpha\beta\dots}~X^\mu$ are identical to the covariant derivatives $D_{\alpha\beta\dots}~X^\mu$ of section . We show this recursively by first proving that covariant derivatives of zero Weyl-weight tensors are the same in both methods. Consider such zero weight tensors $T_{\beta_1\dots\beta_n}$. Then $$\begin{aligned}
\label{identity}
\Pcm{D}_\alpha~T_{\beta_1\dots\beta_n} &=&
{\partial}_\alpha T_{\beta_1\dots\beta_n}-G^\gamma_{\alpha\beta_1}T_{\gamma\beta_2\dots}-\dots\nn\\
&=&
{\partial}_\alpha T_{\beta_1\dots\beta_n}-{\check W}^\gamma_{\alpha\beta_1}T_{\gamma\beta_2\dots}-\dots\nn\\
&=&
{\partial}_\alpha T_{\beta_1\dots\beta_n}-{\Gamma^{(1)}}^\gamma_{\alpha\beta_1}T_{\gamma\beta_2\dots}-\dots\nn\\
&=& D_\alpha~T_{\beta_1\dots\beta_n}
.\end{aligned}$$ We have used the important fact that the components of gauge fixed W-tensor given by are identical to those of the Christoffel connection of section given in and that the Christoffel symbols $\Gamma^{(2)}$ of covariant calculus-II in its conformal gauge are all $0$. In other words, the $G^\gamma_{\alpha\beta}$ is the same in the two conformal gauges. This is easily understood as the metric choices of and are related by the Weyl-scaling factor $L$, and the tensor $G^\gamma_{\alpha\beta}$ is itself of Weyl-weight $0$.
An immediate and important corollary is that all the components of generalised Riemann tensor $\Pcm{R}^\alpha
_{\beta\gamma\delta}$ of the second approach are identical to the standard Riemann tensor $R^\alpha_{\beta\gamma
\delta}$ of section . In the light of all the covariant derivatives of the Riemann tensors are also the same in the two approaches. This means that the tensor ingredients of the two approaches in their conformal gauges are the same. However, what are different are the metric tensors needed to construct scalar densities, and the compensators in the second approach. We shall however show in section that even these conspire to match perfectly. It is shown in that section that this is not just an accident of the choices made in and but is a more general feature. It is also shown in that section that the said equivalence continues to hold even when action terms are constructed using tensors of nonzero Weyl-weight.
Equivalence of the two conformal gauge formalisms {#equiv}
=================================================
We shall now show that the two conformal gauge formalisms are equivalent and that this equivalence is more general than the explicit choices made in and . We illustrate this equivalence by again considering the covariant actions of and . We have already shown that all the covariant derivatives of $X^\mu$ are all the same. Evaluating in the conformal gauge of one gets $$\label{2exconf}
{\check{\Pcm{N}}_1 =
\frac{1}{2} L^{-3}(D_{++}X\cdot D_{--}X + D_{+-}X\cdot D_{+-}X})^2$$ On the other hand evaluating in the conformal gauge one gets $$\label{1ex2conf}
{\check{\Pcm{M}}_2 = 4 L^{-3}
(D_{++}X\cdot D_{--}X + D_{+-}X\cdot D_{+-}X})^2$$ Thus the two terms are equal modulo an irrelevant constant.
Instead of the special choices and consider the pair $$\label{Phistar}
g_{\alpha\beta} = g^*_{\alpha\beta} \qquad \Phi^* = h^{\alpha\beta}g^*_{\alpha\beta}
\quad W_{\Phi^*}=-1$$ If $g^*$ satisfies the Denominator Principle of so will $\Phi^*$, and vice versa. Let us denote the corresponding Weyl connection by $\chi^*_\alpha$. Now consider the pair of conformal gauge metrics related by a Weyl-scaling $$\label{starmap}
h'_{+-} = \Phi^* h_{+-}$$ If $h_{+-}$ is the metric of , the metric $h'_{+-}$ according to is the metric substitute $g^*_{+-}$. The choices , , and are specific realisations of this general scheme. On using one finds $$\label{chiprime}
{\chi^*}'_\alpha = 0$$ Furthermore $$\label{Gprime}
G'^\gamma_{\alpha\beta} = G^\gamma_{\alpha\beta}\quad {\check W}'^\gamma_{\alpha\beta}=0\quad {\Gamma ^{(2)}} '^\gamma_{\alpha\beta}= {\Gamma^{(1)}}^\gamma_{\alpha\beta}$$ and $$\label{Gconf2}
{\Gamma^{(2)}}^\gamma_{\alpha\beta} =0 \qquad {\check W}^\gamma_{\alpha\beta} = {\Gamma^{(1)}}^\gamma_{\alpha\beta}$$ Now following the same strategy as in proving one shows that the Weyl-reparametrisation covariant derivatives of zero weight tensors are identical to the covariant derivatives of . It is also easy to see that the way the metric factors and compensators matched in the example discussed earlier in this section continues to work even for the general case and also for any action term considered. This establishes the complete equivalence of the two formalisms as long as all tensors considered are of zero Weyl weight.
This equivalence continues to hold even when we construct actions with tensors of non-zero Weyl weights. Firstly, the Weyl weight compensators pick up an additional factor ${\Phi^*}^J$ where $J$ is the sum of the Weyl weights of all the tensor factors. In place of one has, when the tensors are of non-zero weight, $$\label{identitygen}
\Pcm{D}_{\alpha_1\dots\alpha_n}~T_{\beta_1\dots\beta_n}={\Phi^*}^{-j}D_{\alpha_1\dots\alpha_n}~T_{\beta_1\dots\beta_n}$$ This results in an exact compensation of the $j$-dependent factors and one ends up with the equality of the action terms (modulo irrelevant constant factors) just as in the earlier case of the construction with zero weight tensors.
Systematic Construction of Effective Actions in Conformal Gauge {#highercovterms}
===============================================================
We have shown how to construct manifestly covariant action terms in both the approaches in sections and . At a classical level this is all that is required. At a quantum level, one has to work with gauge-fixed actions. As long as the symmetries are not violated through quantum corrections, any gauge is as good as any other. We shall restrict ourselves to the conformal gauge and discuss the procedure for a systematic construction of effective actions. Nevertheless, often it is instructive to work in different gauges both because of technical simplicity as well as for demonstration of gauge invariance.
As we have already demonstrated the complete equivalence of the conformal gauges of the two approaches in we shall use the form of the results of section ; one could equally well have used section .
The systematic construction of effective action terms that are manifestly covariant under conformal transformations proceeds more or less along the lines of what has already been presented in [@HDPM2] with some improvements suggested in [@Drum2], but with some very important differences which we address here. Before that, we draw attention to the fact that these earlier methods were based on using skeletal forms which were ordinary derivatives of $X^\mu$. Because of this the transformation laws that left these actions invariant had to be discovered each time, and by trial and error. Our constructions in this paper now allow the skeletal forms to be built out of covariant derivatives and because of this, invariance of the action terms is guaranteed.
As before the method of construction involves stringing together covariant derivatives of $X^\mu$ duly contracted with target space invariant tensors $\eta_{\mu\nu}, \epsilon_{\mu_1\dots\mu_D}$ and then rendered into $(1,1)$ worldsheet tensors by dividing with appropriate powers of $L$. As before, terms proportional to the constraints and their derivatives are dropped. Covariantly this amounts to dropping terms proportional to $D_{\pm\pm}X\cdot D_\pm X$ and their covariant derivatives.
The main difference from what was presented in [@HDPM2; @Drum2] comes in the treatment of terms proportional to EOM and its derivatives. There they were simply dropped. As shown in detail in [@HDPM2] dropping such terms amounts to a field redefinition which can affect the transformation laws as well as the measure (in the path integral approach). The covariant calculi presented here are based on the fixed form of transformation laws . Therefore in the systematic construction of terms such EOM terms can not be dropped.
Hence mixed covariant derivative terms (in the sense of having both $+$ and $-$ indices) have to be considered in the general construction in contrast to [@HDPM2; @Drum2]. Even apart from the EOM issue, shows that not all mixed covariant derivatives, unlike mixed ordinary derivatives, are proportional to EOM.
It was shown in [@HDPM2] that as long as one is interested in terms up to order $R^{-3}$, such field redefinitions can be safely carried out without worrying about the invariant measure or the Jacobians for transformation. The transformation laws, however, have to be modified. A practical way out of the latter is to first work out the full equations of motion and the full stress tensor for covariant actions constructed by our covariant calculus and then express these in terms of the new fields. Even when working with action term of higher than $R^{-3}$ order, it may prove desirable from a calculational point of view to drop such EOM terms and carry out the concommitant changes. The details depend on the particular case at hand.
In the next two subsections we show how this systematic method may be applied at the level of the PS action terms as well as the Drummond terms at order $R^{-6}$.
As we shall see, the integrand of the PS term does not appear at all in the covariant formulation. In fact, it has to be treated and understood in a different way. The PS term is of course essential to ‘adjust’ the central charge of the theory; without the PS term the effecive string construction is not consistent outside the usual critical dimension. We discuss this peculiar situation and the impossibility of covariantising the integrand of the PS term in section . However, as noted there and as already known from earlier works, the [*a*ction]{} represented by the PS term is indeed conformally invariant.
Attempts at covariantising the PS Terms
---------------------------------------
In this subsection we make an attempt at covariantising the integrand of the PS term. As discussed at length in [@HDPM2] there are two (in particular) equivalent forms for the PS term that differ by total derivative and EOM terms. These are $$\label{psterm1}
I_{PS}^{(1)}=\frac{1}{L}{\partial}_+^2~X\cdot{\partial}_-^2~X$$ and $$\label{psterm2}
I_{PS}^{(2)}=\frac1{L^2}{\partial}_+^2~X\cdot{\partial}_-X~{\partial}_-^2~X\cdot{\partial}_+~X
.$$ This second expression is the form given in [@PS], appearing in , and we generally refer to it as “the PS term”.
Let us consider the first of these. The obvious conformal-gauge candidate for this is $$\label{confpscand}
I_{PSConf}^{(1)}= L^{-1}~D_{++}~X\cdot D_{--}~X
.$$
On using can be expanded as $$\begin{aligned}
\label{prel}
&& {\partial}_+^2X\cdot{\partial}_-^2X - L^{-1}{\partial}_+L{\partial}_-L \nn\\
&+& \frac{{\partial}_+L{\partial}_-X\cdot {\partial}_{+-}X+{\partial}_-L{\partial}_+X\cdot{\partial}_{+-}X}{L}
.\end{aligned}$$ On recalling the following identity from [@HDPM2] $$\begin{aligned}
& &\frac{\partial_+^2 X\cdot\partial_-^2 X}{L} =
\frac {\partial_+^2 X\cdot\partial_- X
\partial_-^2 X\cdot\partial_+ X}{L^2}\nonumber\\
&+&\frac {\partial_{+-}X\cdot\partial_{+-}X}{L}
- \frac{\partial_{+-}X\cdot\partial_+ X
\partial_{+-}X\cdot\partial_-X}{L^2}\nonumber\\
&+&\partial_-\big(\frac{\partial_+^2X\cdot\partial_-X}{L}\big)
-\partial_+\big(\frac{\partial_{+-}X\cdot\partial_-X}{L}\big)\end{aligned}$$ we see that $$\label{prelim}
L^{-1}D_{++}X\cdot D_{--}X = \textup{Total Derivative} + \textup{EOM}$$
Thus though appears to be a candidate for covariant form of it ends up being a linear combination of total derivative terms and EOM. Through a more tedious calculation it can be shown that the second term meets the same fate. In fact, using the systematic procedure for constructing actions discussed above, it is easily seen that it is not possible to write any covariant term reproducing the PS terms. A clue to this ‘anomalous’ behaviour is already present in . We shall prove this impossibility in a different and more fundamental way in section .
Covariantising the Drummond Terms {#covIdrum}
---------------------------------
Before proceeding, we make a few statements on terms proportional to EOM. At this order one has to explicitly verify whether EOM terms can be dropped or not and they can not be dropped generically. However, in [@Drum] EOM terms were dropped in arriving at . Thus a comparison can only be made if we examine the terms modulo EOM, but otherwise we emphasise that the general construction proposed in this paper is the more legitimate.
Let us start with and . It is easy to work out these expressions in the conformal gauge : $$\begin{aligned}
{\cal M}_1 &=& 2 \frac{D_{++}X\cdot D_{++}X~D_{--}X\cdot D_{--}X}{L^3} \nn\\
&+& 2\frac{(D_{++}X\cdot D_{--}X)^2}{L^3}\end{aligned}$$ $${\cal M}_2 = \frac{4}{L^3}(D_{++}X\cdot D_{--}X)^2$$ We consider the particular combination $${\cal M}_1-\frac{{\cal M}_2}{2} = \frac{2}{L^3}(D_{++}X\cdot D_{++}X)(D_{--}X\cdot D_{--}X)
,$$ and it is easy to show that, modulo terms that are leading-order constraints and their derivatives, this is just $M_1$. To understand ${\cal M}_2$ let us display slightly differently as $$\label{prel3}
L^{-1}D_{++}X\cdot D_{--}X = {\partial}_-(L^{-1}{\partial}_+^2X\cdot{\partial}_-X) + \textup{EOM}$$ Then it follows that $$\begin{aligned}
{\cal M}_2 &=& L^{-1}[L^{-1}{\partial}_+^2X\cdot{\partial}_-^2X-L^{-2}{\partial}_-L~{\partial}_+^2X\cdot{\partial}_-X]^2\nn\\
&=& M_2-2M_3 +M_4\end{aligned}$$ This way we are able to obtain two independent linear combinations of . It can be shown, through straightforward but tedious algebra, that the covariant calculus can not produce any other combinations. The obvious approach to covariantising the by replacing ordinary derivatives by covariant derivatives only produces, apart from these combinations, EOM and derivatives, constraints and their derivatives, and total derivatives. This is completely analogous to the situation with PS terms discussed in the previous secion.
The present formalism, while representing quite a general way of formulating covariance, is thus extremely restrictive. The only possible gauge-fixed action to $R^{-6}$ order is, up to irrelevant terms, $$\label{covallowed}
\int\frac{{\textup{d}}^2 \sigma}{4\pi} \bigg( \frac{L}{a^2} + \beta \frac{{\partial}_+ L{\partial}_- L}{L^2} + \beta_1 ( M_2 - 2M_3 +M_4 ) +\beta_2 M_1 \bigg)
.$$ Drummond’s terms appear, but only in these particular combinations. This is one of the main results of the present paper; which we now emphasise. The effective string action has only three parameters $\beta,\beta_1,\beta_2$ (in addition to the string tension $a^2$ which merely sets a physical scale) at order $R^{-6}$. Of these the second term, which corresponds to the PS term, does not transform covariantly. Nevertheless, the PS action is invariant under the conformal transformations. Of the three parameters, the leading order analysis as given by PS already fixes $\beta$ leaving only two free parameters $\beta_1,\beta_2$. It would be interesting to see if higher order analysis would further fix some of the remaining parameters.
Impossibility of Covariantising the PS Integrand {#impossible}
================================================
We shall prove that for WZNW effective actions for a conformal anomaly in two dimensions defined by $$\label{WZNW}
\delta_\lambda S_{WZNW} = \int {\textup{d}}^2\xi \lambda(\xi)\sqrt{g}R(\xi)$$ the integrand of $S_{WZNW}$ can never be manifestly covariant under coordinate transformations.
Here, $R(\xi)$ is the Ricci scalar. The action proposed by Polyakov [@polyagrav] in the context of two dimensional quantum gravity, $$S_{\textup{Polya}} = \int R \frac{1}{\nabla^2} R$$ is such a WNZW action.
Written out explicitly, $$\label{pol}
S_{\textup{Polya}} = \int {\textup{d}}^2 x \sqrt{g(x)} R(x) (\frac{1}{\nabla^2} R)(x)$$ where $R(x)$ is the scalar curvature in two dimensions. The value of the integrand in the conformal gauge of is $$\begin{aligned}
I_{\textup{Polya}} &=& ({\partial}_+{\partial}_-~\ln L)\cdot\frac{1}{{\partial}_+{\partial}_-}\cdot({\partial}_+{\partial}_- \ln L)\nn\\
&=& ({\partial}_+{\partial}_-~\ln L)\ln L\end{aligned}$$ This is the same as the integrand $L_2$ of upto total derivative terms. The variation of this under conformal transformations is $$\label{polvar}
\delta I_{\textup{Polya}}
= {\partial}_-(\epsilon^-~I_{\textup{Polya}})+{\partial}_-\epsilon^-~{\partial}_+{\partial}_-\ln L$$ Since ${\partial}_+\epsilon ^-=0$, it follows that the Polyakov action is indeed invariant under conformal transformations.
The Weyl scalings are defined by $$\delta_{\lambda(\xi)}~g_{\alpha\beta} = \lambda(\xi)~g_{\alpha\beta}$$ and the infintesimal coordinate transformations are given by $$\delta_\epsilon~g_{\alpha\beta} = \epsilon^\gamma{\partial}_\gamma g_{\alpha\beta}+{\partial}_\alpha\epsilon^\gamma
g_{\gamma\beta}+{\partial}_\beta\epsilon^\gamma g_{\alpha\gamma}$$ Now we look for a combination of Weyl scaling and coordinate transformation that leaves the form of the metric unchanged; $$\label{totdef}
\bar\lambda g_{\alpha\beta} =
\epsilon^\gamma{\partial}_\gamma g_{\alpha\beta}+{\partial}_\alpha\epsilon^\gamma
g_{\gamma\beta}+{\partial}_\beta\epsilon^\gamma g_{\alpha\gamma}
.$$ The strategy is to consider $$\delta_{tot} = \delta_{\bar\lambda}-\delta_\epsilon
,$$ and by construction $$\label{total}
\delta_{tot} \tilde{\cal L} = 0
.$$ Although we are talking about the same transformations we discussed in section , it is worth emphasising that here does not say anything about the invariance or lack of invariance of the action under any of the said transformations. On the other hand, we have, from $$\label{weylvar}
\delta_{\textup{Weyl}}(\bar\lambda) \tilde{\cal L} = \bar\lambda \sqrt{g} R(\xi) + {\partial}_+ X^+(g,\bar\lambda) + {\partial}_-X^-(g,\bar\lambda)$$ It should be noted that the dependence of $X^\pm$ is explicitly on $\bar\lambda$ and its derivatives.
For the remainder we work explicitly in the conformal gauge of and without loss of generality, restrict our attention to only $\xi^-$ diffeomorphisms ($\epsilon^+=0$). In this case reads $$\label{totcgauge}
{\partial}_+\epsilon^-=0;~~~~\bar\lambda = {\partial}_-\epsilon^-+\epsilon^-~{\partial}_-\ln L$$ Using and we rewrite as $$\begin{aligned}
\label{explicit2}
\delta_{\textup{Weyl}}(\bar\lambda)~\tilde{\cal L}
&=&{\partial}_-\{-2\epsilon^-{\partial}_{+-}\ln L+X^-\} \\
&+&{\partial}_+\{2\epsilon^-{\partial}_{--}\ln L -\epsilon^-({\partial}_-\ln L)^2+X^+\} \nn\end{aligned}$$ Now, this must equal $\delta(\epsilon^-)\tilde{\cal L}$. If $\tilde{\cal L}$ were transforming as a scalar density, the ${\partial}_+$ terms in the last line of must vanish identically. This can happen only if $$\label{xplus}
{\partial}_+X^+ = {\partial}_+\{\epsilon^-(-2{\partial}_{--}\ln L+({\partial}_-\ln L)^2)\}$$ As we have already noted $X^+$ must have an explicit dependence on $\bar\lambda$ and its derivatives. Part of can indeed be cast into this form (which part can be so cast is not unique); $$\begin{aligned}
{\partial}_+X^+ &=& {\partial}_+\{-2{\partial}_-(\epsilon^-{\partial}_-\ln L+{\partial}_-\epsilon^-) \\
&+&(\epsilon^-{\partial}_-\ln L+{\partial}_-\epsilon^-){\partial}_-\ln L+{\partial}_-\epsilon^-{\partial}_-\ln L\} \nn
.\end{aligned}$$ The last term ${\partial}_-\epsilon^-\cdot{\partial}_{+-}\ln L$ makes it evident that $\tilde{\cal L}$ fails to be a scalar density. It fails by precisely the same term as obtained through explicit calculations with $I_{\textup{Polya}}$, which we have seen in . Nevertheless, $S_{Polya}$ is invariant as shown above.
Conclusions
===========
We have given in this paper a vastly simplified approach to the theory of effective strings in comparison with the PS formalism. The essential simplification is that the transformation law is always the same as the standard transformation law for free bosonic string action. The transformation law does not have to be tuned to the action. In the conformal gauge, these represent conformal transformations exactly, and not approximately as in the case of the PS transformation law( only to order $R^{-3}$ to be precise).
Consequently every action constructed by our covariant calculus, and in particular $S_{2}$ of , is in principle valid to *all orders* in $R^{-1}$. Whether phenomenologically any action or combination of actions is correct or not is a different issue.
A further consequence of this covariantisation is the restriction of the effective action to order $R^{-6}$ to include only two free parameters $\beta_1$ and $\beta_2$. Our result for the complete effective action to this order, from our first approach, is the truncation to order $R^{-6}$ of $$\begin{aligned}
\label{covallowed2}
S &=&\frac{\beta}{4\pi} S_{\textup{Polya}}+\int\frac{{\textup{d}}^2\sigma}{4\pi}\big[\frac{\sqrt{g}}{a^2}+
\beta_1 X^{;\alpha\beta}\cdot X^{;\delta\gamma} X_{;\alpha\beta}\cdot X_{;\delta\gamma} \nn\\
&+& \beta_2 X^{;\alpha\beta}\cdot X_{;\alpha\beta}X^{;\gamma\delta} \cdot X_{;\gamma\delta}
\big] ,\nn\\\end{aligned}$$ The conformal gauge expression for is given by and that entire action is by construction *exactly* conformally invariant under the transformation law . Both the approaches yield the same actions in the conformal gauge. As already emphasised before, one can use the entire , without truncation, if one so wishes. Further terms presumably begin to appear at $\Pcm{O}(R^{-8})$.
Finally, it is worth emphasising that of course without a covariant formulation, one can only identify the relevant terms which may be included in the action, up to a given order, and adorn these terms with coefficients which are then the parameters of the theory. These must then be determined phenomenologically (using lattice QCD, for example). In contrast, in our covariant construction, in general the number of new free parameters introduced at each order in $1/R$ is fewer than the number which would obtain given such independent insertion of all relevant terms. This highly desirable reduction in parameters is also a conclusion which could be subject to verification in simulations or experiments.
acknowledgement {#acknowledgement .unnumbered}
===============
NDH acknowledges the award of the DAE Raja Ramanna Fellowship enabling his stay at IISc.
[99]{} Joseph Polchinski and Andrew Strominger, *Effective String Theory*, PRL 67, 1681 (1991). J. M. Drummond, *Universal Subleading Spectrum of Effective String Theory*, `hep-th/0411017`. N. D. Hari Dass and Peter Matlock, *Field Definitions, Spectrum and Universality in Effective String Theories*, `hep-th/0612291`. E. Kretschmann, Ann. Phys. Leipzig. [**53**]{}, 575(1917). J. M. Drummond, `hep-th/0608109v1`. A.M. Polyakov, *Quantum Gravity in Two Dimensions*, Mod.Phys.letts. [**A2**]{} 893(1987).
[^1]: We retain the terminology used in [@PS; @HDPM2], whereby terms proportional to the leading-order EOM are called *irrelevant*, and terms irreducible to this form are deemed *relevant*.
[^2]: NDH thanks Hikaru Kawai regarding why this has to be so.
[^3]: Almost immediately after Einstein had formulated his General Theory of Relativity, Kretschmann [@kretsch] had pointed this out but it is not widely appreciated even at present.
|
---
abstract: 'Let $C$ be a complex smooth projective algebraic curve endowed with an action of a finite group $G$ such that the quotient curve has genus at least $3$. We prove that if the $G$-curve $C$ is very general for these properties, then the natural map from the group algebra ${{\mathbb Q}}G$ to the algebra of ${{\mathbb Q}}$-endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation acts ${{\mathbb Q}}$-irreducibly in a $G$-isogeny space of $H^1(C; {{\mathbb Q}})$ and with image often a ${{\mathbb Q}}$-almost simple group.'
address:
- 'Departamento de Matemática, UFMG, Belo Horizonte - MG (Brasil)'
- 'Yau Mathematical Sciences Center, Tsinghua University Beijing (China) and Mathematisch Instituut, Universiteit Utrecht (Nederland)'
author:
- Marco Boggi
- Eduard Looijenga
title: Curves with prescribed symmetry and associated representations of mapping class groups
---
[^1]
Introduction {#introduction .unnumbered}
============
A classical theorem of Hurwitz (1886) asserts that for a *very general* complex projective smooth curve $C$ of genus $\ge 2$, the endomorphism ring of its Jacobian $J(C)$ is as small as possible, namely ${{\mathbb Z}}$. Lefschetz [@lefschetz] proved in 1928 that this is still true if we restrict ourselves to hyperelliptic curves and we refer to Ciliberto’s survey [@Ciliberto] for what other results of that type were known before 1989. As of then the theorem of Lefschetz has been refined and generalized in several directions. For example, Ciliberto, van der Geer and Teixidor i Bigas [@CGT] studied (and to some extent characterized) the locus of curves for which the endomorphism ring of the Jacobian is strictly larger than ${{\mathbb Z}}$, while Zarhin addressed this problem over other base fields and also considered in [@zahrin2004] and [@zahrin2012] curves with a specified automorphism of a particular type other than a hyperelliptic involution: he proved that for a general such curve the endomorphism ring of its Jacobian is indeed as small as it could possibly be.
In the present paper we obtain a natural generalization of these results, at least over the base field ${{\mathbb C}}$, by proving such a minimality property for curves endowed with an action of a given (but arbitrary) finite group. More precisely, we prove that for a very general complex smooth projective curve $C$ with automorphism group $G$ whose quotient curve has genus at least $3$, the natural map from ${{\mathbb Q}}G$ to the rational endomorphism algebra of the Jacobian of $C$ is an isomorphism (Corollary \[endo2\]). We also obtain a result of this kind in the hyperelliptic case (Theorem \[endo\] and Theorem \[theorem:GN\]).
Although we believe that this has an interest of its own, our motivation has been of a more topological nature, since it has implications for certain virtual linear representations of the (pointed) mapping class groups, as we shall now explain. Let $S$ be a closed oriented connected surface of genus $\ge 2$ endowed with an action of a finite group $G$ by orientation preserving diffeomorphisms. Then $G$ embeds in the mapping class group ${\operatorname{Mod}}(S)$ of $S$. The centralizer ${\operatorname{Mod}}(S)^G$ of $G$ in ${\operatorname{Mod}}(S)$ maps to a finite index subgroup of the mapping class group ${\operatorname{Mod}}(S_G)$ of the orbifold surface $S_G$ (these are mapping classes of the underlying surface which take orbifold points to orbifold points of the same type), with kernel the center $Z(G)$ of $G$. Thus ${\operatorname{Mod}}(S)^G$ and ${\operatorname{Mod}}(S_G)$ have in common a subgroup of finite index so that the natural representation of ${\operatorname{Mod}}(S)^G$ on $H^1(S,{{\mathbb Q}})$ can be regarded as a virtual linear representation of the mapping class group ${\operatorname{Mod}}(S_G)$. This representation takes its values in the centralizer of $G$ in the symplectic group ${\operatorname{Sp}}(H^1(S,{{\mathbb Q}}))$ and respects the isotypical decomposition of $H^1(S,{{\mathbb Q}})$ into irreducible ${{\mathbb Q}}G$-modules. Then the above minimality property implies that when $S_G$ has genus $\ge 3$, a ${{\mathbb Q}}G$-isotypical summand of $H^1(S,{{\mathbb Q}})$ is essentially also isotypical for the representation of ${\operatorname{Mod}}(S)^G$ (to be more precise, isotypical for the *connected monodromy group*, that is, the identity component of the Zariski closure of the image of this representation). In particular, the subspace of $H^1(S,{{\mathbb Q}})$ spanned by the finite ${\operatorname{Mod}}(S)^G$-orbits is a sum of ${{\mathbb Q}}G$-isotypical summands. This subspace comes with a natural polarizable Hodge structure which is independent of any complex structure on $S$, but let us add Putman and Wieland conjecture that this subspace is trivial.
Our main applications are through the use of Theorem \[topology1\]. For example, Corollary \[cor:primmon\] states that in case a ${{\mathbb Q}}G$-isotypical summand is not fixed by all the $G$-invariant multi-Dehn twists, and the genus of the quotient curve is at least $3$, then the connected monodromy group acts irreducibly on that summand. Theorem \[topology1\] gives rise to a number of questions, which we collected in \[quest:opology1\]. The paper also ends in this spirit: with a conjecture (\[conj:liegeneration\]) and a question (\[quest:arithmetic2\]).
We thank Ben Moonen for drawing our attention to a paper by Masa-Hiko Saito and we thank Tyakal Venkataramana for helpful consultations.
*Conventions*. If a group $G$ acts properly discretely on a smooth object $S$ in a category where this makes sense, then $S_G$ denotes the associated orbifold object.
When $G$ is a finite group and $k$ is a field of characteristic zero, then we denote by $X(k G)$ the set of irreducible characters of $k G$. For every $\chi\in X(k G)$, we let $V_\chi $ be a finitely generated (irreducible) $k G$-module with that character. Is $H$ is a $k G$-module of finite rank, then we write $H[\chi]$ for the $V_\chi$ isogeny space ${\operatorname{Hom}}_{kG}(V_\chi,H)$. The *$\chi$-isotypical summand* of $H$ is the image of the evaluation map $V_\chi\otimes_k H[\chi]\to H$ and denoted by $H_\chi$. We indeed have $H=\oplus_{\chi\in X(k G)} H_\chi$.
We write ${{\mathbb K}}$ for the Hamilton quaternions, considered as a division ring containing ${{\mathbb R}}$.
Endomorphisms of Jacobians of $G$-curves {#endomorphisms}
========================================
A central role in this paper is played by a moduli stack whose definition we will give in a moment. We do this in some detail, as it involves some subtleties. More on this can be found in the paper by Collas and Maugeais [@cm] which also addresses the question over which number field such a stack is defined.
Let $S$ be a fixed closed oriented surface of genus $g\ge 2$, $G$ a finite group and $\phi$ a monomorphism of $G$ in the mapping class group ${\operatorname{Mod}}(S)$ of $S$. By a theorem of Nielsen-Kerckhoff, we can lift $\phi$ to a monomorphism $\Phi$ of $G$ to a group of orientation preserving diffeomorphisms of $S$. This implies that $S$ admits a complex structure also invariant under $G$ so that we obtain a smooth projective complex curve $C$. We consider the $G$-curves which so arise. To be precise, we consider pairs $(C,G{\hookrightarrow}{\operatorname{Aut}}(C))$, where $C$ is a smooth projective complex algebraic curve $C$ of genus $\geq 2$ and $G{\hookrightarrow}{\operatorname{Aut}}(C)$ a monomorphism of groups such that there exists an orientation preserving $G$-equivariant homeomorphism of $C$ onto $S$. Observe that the last property only depends on the ${\operatorname{Mod}}(S)$-conjugacy class of $\phi$. So if we precompose $\phi$ with an automorphism $\sigma$ of $G$, then we get an isomorphic object if and only if there exists an orientation preserving diffeomorphism $h$ of $S$ such that $h \phi (g)h^{-1}=\phi\sigma(g) $ for all $g\in G$. We denote a pair as above simply by $(C,G)$ (which is admittedly somewhat of an abuse of notation, as it hides the datum of the ${\operatorname{In}}({\operatorname{Mod}}(S))$-orbit of $\phi$) and we will often refer to it as a *$G$-curve* of topological type $\phi$.
The $G$-curves of topological type $\phi$ are parameterized by a D-M stack ${{\mathscr M}}^\phi$, which is defined as follows. Recall that the space of isotopy classes of complex structures on $S$ compatible with the given orientation is parameterized by a connected complex manifold, the Teichmüller space ${\operatorname{Teich}}(S)$, which supports a holomorphic family of curves ${{\mathscr C}}_{{\operatorname{Teich}}(S)}/{\operatorname{Teich}}(S)$. The mapping class group ${\operatorname{Mod}}(S)$ acts on this family. We let $G$ act on ${\operatorname{Teich}}(S)$ via $\phi$ (or what amounts to the same, via the natural map $G\to {\operatorname{Mod}}(S)$) and then its fixed point locus ${\operatorname{Teich}}(S)^G$ ([^2]) parameterizes the curves of topological type $\phi$. Since the $G$-invariant complex structures on $S$ are in bijective correspondence with the complex structures on $S_G$, we may identify ${\operatorname{Teich}}(S)^G$ with the Teichmüller space ${\operatorname{Teich}}(S_G)$ of the orbifold $S_G$.
The centralizer of $G$ in ${\operatorname{Mod}}(S)$, ${\operatorname{Mod}}(S)^G$, meets ${\operatorname{Mod}}(S)$ in the center $Z(G)$ of $G$ and $G\cdot{\operatorname{Mod}}(S)^G$ is a subgroup of the normalizer of $G$ in ${\operatorname{Mod}}(S)$. The center $Z(G)$ acts trivially on ${\operatorname{Teich}}(S)^G$ and the quotient ${\operatorname{Mod}}(S)^G/Z(G)$ can be identified with a finite index subgroup of the mapping class group ${\operatorname{Mod}}(S_G)$ (this is the group of mapping classes of the underlying surface which respect the type of the orbifold points). Note that an element of ${\operatorname{Mod}}(S_G)$ lies in this image if and only if it lifts to a mapping class of $S$ which commutes with the $G$-action.
The group $G\cdot{\operatorname{Mod}}(S)^G\subset {\operatorname{Mod}}(S)$ acts on the restriction ${{\mathscr C}}_{{\operatorname{Teich}}(S)^G}/{\operatorname{Teich}}(S)^G$ of ${{\mathscr C}}(S)/{\operatorname{Teich}}(S)$ to ${\operatorname{Teich}}(S)^G$ and since $G$ acts trivially on ${\operatorname{Teich}}(S)^G$, this turns the restriction ${{\mathscr C}}_{{\operatorname{Teich}}(S)^G}/{\operatorname{Teich}}(S)^G$ into a $G$-curve over ${\operatorname{Teich}}(S)^G$. It is universal as a marked $G$-curve, the marking being given by an equivariant isotopy class of diffeomorphisms with $(S,G)$ (so $\phi$ is here still being suppressed in the notation). If we divide out this universal marked $G$-curve by the action of ${\operatorname{Mod}}(S)^G$, that is, if we pass to the stack or orbifold quotient, then we obtain a smooth complex D-M stack, denoted ${{\mathscr M}}^\phi$, which parameterizes the $G$-curves of topological type $\phi$. It is characterized by the property that if $Y$ is a complex variety (or scheme for that matter), then a smooth $G$-curve over $Y$ of topological type $\phi$ gives rise to a (stack) morphism $Y\to {{\mathscr M}}^\phi$, and that thus is obtained a bijection between the isomorphism classes of smooth $G$-curves over $Y$ of topological type $\phi$ and the morphisms from $Y$ to ${{\mathscr M}}^\phi$. Beware that if we ignore the stack structure by passing to the underlying variety, the fiber over every closed point is a quotient of a $G$-curve by a subgroup of $G$ which contains $Z(G)$.
It is clear that ${{\mathscr M}}^\phi$ only depends on the ${\operatorname{In}}({\operatorname{Mod}}(S))$-orbit of $\phi$. Since ${\operatorname{Teich}}(S)^G$ is a simply-connected manifold, ${{\mathscr M}}^\phi$ is irreducible and the fundamental group of the underlying analytic orbifold is ${\operatorname{Mod}}(S)^G$. Precomposition with an automorphism $\rho$ of $G$ may lead to a different D-M complex stack ${{\mathscr M}}^{\phi\rho}$ to which we shall refer as the *$\rho$-twist* of ${{\mathscr M}}^\phi$. If however, $\rho$ is inner and defined by conjugation with some $g\in G$, say, then $g: C\to C$ makes the $G$-curve isomorphic to its $\rho$-twist. So the union ${{\mathscr M}}^{[G]}$ of ${\operatorname{Aut}}(G)$-twists (denoted by Collas-Maugeais in [@cm] by ${{\mathscr M}}_{g}[G]$, where $g$ is the genus of $S$) comes with an action of ${\operatorname{Aut}}(G)$, which permutes its irreducible components through ${\operatorname{Out}}(G)$. If we pass to the stack quotient with respect to this ${\operatorname{Aut}}(G)$-action, we obtain a D-M stack ${{\mathscr M}}^{(G)}$ which parameterizes curves $C$ endowed with a subgroup of ${\operatorname{Aut}}(C)$ isomorphic to $G$ (but with no such isomorphism specified).
The forgetful morphism ${{\mathscr M}}^{\phi}\to {{\mathscr M}}(S)$ is finite and the universal $G$-curve ${{\mathscr C}}_{{{\mathscr M}}^\phi}/{{\mathscr M}}^\phi$ fits in a forgetful cartesian diagram which maps to the universal curve ${{\mathscr C}}_{{{\mathscr M}}(S)}/{{\mathscr M}}(S)$. Note that we have a factorization ${{\mathscr M}}^{\phi}\to{{\mathscr M}}^{(G)}\to {{\mathscr M}}(S)$. In fact, ${{\mathscr M}}^{(G)}$ can be regarded as the normalization of the image of ${{\mathscr M}}^{\phi}\to {{\mathscr M}}(S)$ (this expresses the fact that a curve $C$ whose automorphism group contains a copy of $G$, will in general have only one such copy) and ${{\mathscr M}}^{\phi}\to{{\mathscr M}}^{(G)}$ will be étale.
The following observation is important for what follows. For every $G$-curve $(C,G)$ of type $\phi$ (so defining a point of ${{\mathscr M}}^\phi$), $G$ acts on $H^0(C,\Omega_C)$ but the character of this action will be independent of the point we choose, simply because ${{\mathscr M}}^\phi$ is irreducible. In other words, this only depends on $\phi$, so that this character may be regarded as a topological invariant. We are therefore justified in denoting it by $\chi^\phi$. It is clear that for every $\rho\in {\operatorname{Aut}}(G)$, $\chi^{\phi\sigma}=\rho^*\chi^{\phi}$.
Let us say that a geometric point $\{x\}\to {{\mathscr M}}^\phi$ is a *very general point* of ${{\mathscr M}}^\phi$ if it is not contained in a given countable union of closed proper substacks of ${{\mathscr M}}^\phi$.
\[endo\]Let $G$ be a finite group and $\phi : G\to {\operatorname{Mod}}(S)$ a monomorphism such that the centralizer of the image of $\phi$ does not contain a hyperelliptic involution. Assume that the associated moduli stack ${{\mathscr M}}^\phi$ has positive dimension and that for a $G$-curve $C$ representing a point of ${{\mathscr M}}^\phi$ the action of $G$ on $H^0(C, \Omega_C)$ satisfies the following two properties:
1. every irreducible symplectic representation occurring in $H^0(C, \Omega_C)$ occurs with multiplicity $\ge 3$ and
2. every non-self-dual representation occurring in $H^0(C, \Omega_C)$ occurs along with its dual and both do so with multiplicity $\ge 2$.
Then the natural homomorphism ${{\mathbb Q}}G\to{\operatorname{End}}_{{\mathbb Q}}(J(C))$ is surjective, provided the $G$-curve $C$ represents a very general member of ${{\mathscr M}}^\phi$.
\[rem:verygeneral=dominant\] We can avoid the notion of a very general point by using the following more concrete (and equivalent) formulation of the conclusion of this theorem: if ${{\mathscr C}}/Y$ is a $G$-curve over a complex variety $Y$ of type $\phi$ such that the associated morphism $Y\to{{\mathscr M}}^\phi$ is dominant, then ${{\mathbb Q}}G\to {\operatorname{End}}_{{\mathbb Q}}(J({{\mathscr C}}/Y))$ is onto. This is essentially what we shall prove and it is also the formulation through which we will use it.
\[notinjective\]If the genus of the quotient curve is zero, then the natural homomorphism ${{\mathbb Q}}G\to{\operatorname{End}}_{{\mathbb Q}}(J(C))$ is not injective, as the trivial $G$-module is not represented in $H^0(C,\Omega_C)$. But as we will see, the Chevalley-Weil formula implies that this map is injective when the genus of $C_G$ is at least $2$. This formula will also show that the conditions we imposed on the $G$-action on $H^0(C, \Omega_C)$ are topological, in the sense that they only depend on how $G$ acts on $\phi$ (as we should expect).
The assumption that ${{\mathscr M}}^\phi$ has positive dimension translates into asking that the moduli space of the quotient orbifold curve (so endowed with the finite subset representing irregular $G$-orbits) is not rigid. This simply amounts to demanding that the regular $G$-orbit space of $C$ has negative Euler characteristic, but is not a $3$-punctured sphere. It is also equivalent to $H^0(C, \Omega_C^{\otimes 2})^G$ being nonzero.
The hypothesis that $G$ does not contain a hyperelliptic involution in its center is needed in order to ensure that some point of ${{\mathscr M}}^\phi$ is defined by a nonhyperelliptic curve (this will actually imply that every point of ${{\mathscr M}}^\phi$ has that property).
Observe that if a nontrivial automorphism of $C$ acts trivially on its canonical image in $\check{{\mathbb P}}(H^0(C,\Omega_C))$, then $C$ is hyperelliptic and this automorphism is the hyperelliptic involution. The latter acts as $-1$ on $H^0(C,\Omega_C)$, and so $G$ always acts faithfully on $H^1(C,{{\mathbb C}})$.
The hypothesis on the $G$-module structure of $H^0(C,\Omega_C)$ comes from the use of the following lemma in the proof.
\[lemma:quadratic\] Let $F$ be a finite dimensional ${{\mathbb C}}G$-module with the property that
1. every irreducible symplectic representation occurring in $F$ occurs with multiplicity $\ge 3$ and
2. every non-self-dual representation occurring in $F$ occurs along with its dual and both do so with multiplicity $\ge 2$.
Let $L$ be a compact Lie subgroup of ${\operatorname{GL}}(F)$ which contains the image of $G$ and has the property that $({\operatorname{Sym}}^2F)^L= ({\operatorname{Sym}}^2F)^G$. Then $L$ is contained in the image of ${{\mathbb R}}G$ in ${\operatorname{End}}(F)$.
The proof uses the decomposition of ${{\mathbb R}}G$ as an ${{\mathbb R}}$-algebra in terms of the decomposition of ${{\mathbb C}}G$ as a ${{\mathbb C}}$-algebra and so we need to recall how this comes about. Remember that $X({{\mathbb C}}G)$ stands for the set of characters of irreducible ${{\mathbb C}}G$-modules, that for every $\chi \in X({{\mathbb C}}G)$ we have chosen a representative ${{\mathbb C}}G$-module $V_\chi$, and that the obvious map $$\textstyle {{\mathbb C}}G\to \prod_{\chi\in X({{\mathbb C}}G)} {\operatorname{End}}(V_\chi)$$ is an isomorphism of ${{\mathbb C}}$-algebras. ‘Taking the dual’ defines an involution $\chi\mapsto \chi^*$ in $X({{\mathbb C}}G)$. Then for all $\chi, \mu \in X({{\mathbb C}}G)$, the space of $G$-invariant bilinear forms $V_\chi\times V_{\mu}\to {{\mathbb C}}$ is nonzero unless $\mu=\chi^*$, in which case it is generated by a nondegenerate pairing $\alpha_\chi: V_\chi\times V_{\chi^*}\to {{\mathbb C}}$. The fixed point set of this involution decomposes into the orthogonal characters $X({{\mathbb C}}G)_+$ (the $\chi$ for which $\alpha_\chi$ is symmetric) and the symplectic characters $X({{\mathbb C}}G)_-$ (the $\chi$ for which $\alpha_\chi$ is anti-symmetric). Let $X({{\mathbb C}}G)_o\subset X({{\mathbb C}}G)$ be a system of representatives for the free orbits, so that $X({{\mathbb C}}G)$ is the disjoint union of $X({{\mathbb C}}G)_+$, $X({{\mathbb C}}G)_-$, $X({{\mathbb C}}G)_o$ and $X({{\mathbb C}}G)_o^*$. Choose a $G$-invariant inner product $h_\chi$ in $V_\chi$. When $\chi \in X({{\mathbb C}}G)_\varepsilon$ with $\varepsilon=\pm$, let $c_\chi$ be the anti-linear automorphism of $V_\chi$ characterized by the property that $\alpha_\chi(x,y)=h_\chi(c_\chi(x), y)$. Note that $c_\chi$ is $G$-equivariant. Hence so is $c_\chi^2$. As $c_\chi^2$ is also ${{\mathbb C}}$-linear, Schur’s lemma (applied to the irreducible ${{\mathbb C}}G$-module $V_\chi$) implies that $c_\chi^2$ must be a scalar, $\lambda$, say. The identity $$\lambda h_\chi(x, x)=h_\chi(c_\chi^2(x), x)=\alpha_\chi(c_\chi(x),x)=\varepsilon \alpha_\chi(x, c_\chi(x))=\varepsilon h_\chi(c_\chi(x), c_\chi(x))$$ then shows that $\lambda$ is real and has the sign $\varepsilon$. Upon replacing $\alpha_\chi$ by $|\lambda|^{-1/2}\alpha_\chi$, we arrange that $c_\chi^2=\varepsilon 1$. In the orthogonal case ($c_\chi^2=1$), the fixed point space of $c_\chi$ is a real form $V_\chi({{\mathbb R}})$ of $V_\chi$ and so ${{\mathbb R}}G$ maps to ${\operatorname{End}}_{{\mathbb R}}(V_\chi({{\mathbb R}}))$. In the symplectic case ($c_\chi^2=-1$), $c_\chi$ endows $V_\chi$ with the structure of a right ${{\mathbb K}}$-module (recall that ${{\mathbb K}}$ denotes the Hamilton quaternions: here we let $c_\chi$ act as $j$), which extends the ${{\mathbb C}}$-vector space structure, so that ${{\mathbb R}}G$ maps to ${\operatorname{End}}_{{\mathbb K}}(V_\chi)$. It is known that the disjoint union of $\{V_\chi({{\mathbb R}})\}_{\chi\in X({{\mathbb C}}G)_+}$, $\{V_\chi\}_{\chi\in X({{\mathbb C}}G)_-}$ and $\{V_\chi\}_{\chi\in X({{\mathbb C}}G)_o}$ represent the distinct elements of $X({{\mathbb R}}G)$ (complexification as a ${{\mathbb R}}G$-module reproduces resp. $V_\chi$, $V_\chi\oplus V_\chi$ and $V_\chi\oplus V_{\chi*}$) and that the obvious ${{\mathbb R}}$-algebra homomorphism $$\label{eqn:rdec}
\textstyle {{\mathbb R}}G\to \big(\prod_{\chi\in X({{\mathbb C}}G)_+} {\operatorname{End}}_{{\mathbb R}}(V_\chi({{\mathbb R}})) \big) \times \big(\prod_{\chi\in X({{\mathbb C}}G)_-} {\operatorname{End}}_{{\mathbb K}}(V_\chi)\big)
\times \big(\prod_{\chi\in X({{\mathbb C}}G)_o} {\operatorname{End}}(V_\chi)\big).$$ is an isomorphism (see for instance [@Serre], §13.2).
For $\chi\in X({{\mathbb C}}G)$ we put $F[\chi]:={\operatorname{Hom}}_{{{\mathbb C}}G}(V_\chi, F)$, so that the obvious evaluation map $\oplus_{\chi\in X({{\mathbb C}}G)} V_\chi\otimes F[\chi]\to F$ is an isomorphism of $G$-modules. Via this isomorphism, the image of ${{\mathbb C}}G$ in ${\operatorname{End}}(F)$ is identified with the product of the ${\operatorname{End}}(V_\chi)$ taken over all $\chi\in X({{\mathbb C}}G)$ with $F[\chi]\not=0$. The generators $\alpha_\chi$ determine an isomorphism $$({\operatorname{Sym}}^2 F)^G
\cong \big(\underset{\chi\in X({{\mathbb C}}G)_+}\oplus {\operatorname{Sym}}^2 F[\chi]\big) \oplus \big(\underset{\chi\in X({{\mathbb C}}G)_-}\oplus \wedge^2 F[\chi]\big)
\oplus \big(\underset{\chi\in X({{\mathbb C}}G)_o} \oplus F[\chi]\otimes F[\chi^*]\big).$$ We are given that this space is also equal to $({\operatorname{Sym}}^2 F)^L$. We claim that for every $\chi\in X({{\mathbb C}}G)$ and every line $\ell \subset F[\chi]$, $L$ preserves $V_\chi\otimes\ell$. To see this, note that with any $q\in {\operatorname{Sym}}^2 F$ is associated a subspace $F_q\subset F$, namely the smallest subspace of $F$ such that $q\in {\operatorname{Sym}}^2F_q$ (so that $q$ is nondegenerate, when regarded as a quadratic form on the dual of $F_q$). It is clear that any linear transformation of $F$ which fixes $q$, leaves $F_q$ invariant. In our case this means that $L$ leaves invariant every subspace $F_q$ with $q\in ({\operatorname{Sym}}^2 F)^G$. Among the nonzero subspaces $F_q$ we thus obtain are those of the form $V_\chi\otimes \ell$ with $\chi\in X({{\mathbb C}}G)_+$ and $\ell$ any line in $F[\chi]$, $V_\chi\otimes P$ with $\chi \in X({{\mathbb C}}G)_-$ and $P$ any plane in $F[\chi]$, and $(V_\chi\otimes \ell)\oplus (V_{\chi^*}\otimes \ell')$ with $\chi\in X({{\mathbb C}}G)_o$ and $\ell\subset F[\chi]$ and $\ell'\subset F[\chi^*]$ arbitrary lines. Then our assumptions (i) and (ii) imply that for every $\chi\in X({{\mathbb C}}G)$ and every line $\ell$ in $F[\chi]$, $V_\chi\otimes\ell$ is an intersection of such subspaces (and hence preserved by $L$). It follows that $L$ acts trivially on each ${{\mathbb P}}(F[\chi])$, or rather, that $L$ is contained in $\prod_{\chi\in X({{\mathbb C}}G)} {\operatorname{GL}}(V_\chi)\otimes \mathbf{1}_{F[\chi]}$, so that $L$ is at least contained in the image of ${{\mathbb C}}G$ in ${\operatorname{End}}(F)$. If $F[\chi]=0$, then the factor ${\operatorname{GL}}(V_\chi)\otimes \mathbf{1}_{F[\chi]}$ is of course trivial, but otherwise $L$ will act on $V_\chi$ in such a way that it preserves $\alpha_\chi$.
Assume that $F[\chi]\not=0$. Since $L$ is compact, it leaves invariant an inner product in $V_\chi$. Since this inner product is also left invariant by $G$, we may assume that this is the one we used to define the $c_\chi$ above in case $\chi\in X({{\mathbb R}}G)_\pm$. So then $L$ preserves $c_\chi$ when defined. Hence in all cases $L$ lands in the corresponding factor of the decomposition given by Equation (\[eqn:rdec\]) so that it is contained in the image of ${{\mathbb R}}G$ in ${\operatorname{End}}(F)$.
\[rem:\] This lemma can fail when a symplectic representation occurs with multiplicity $2$. To see this, let $V$ be an irreducible ${{\mathbb R}}G$-module which is of quaternionic type. This means that the ${{\mathbb R}}$-algebra $k:={\operatorname{End}}_{{{\mathbb R}}G}(V)$ is isomorphic to ${{\mathbb K}}$ and the image of ${{\mathbb R}}G$ in ${\operatorname{End}}_{{\mathbb R}}(V)$ is equal to ${\operatorname{End}}_k(V)$. Let $s: V\times V\to {{\mathbb R}}$ be a $G$-invariant positive definite quadratic form. Then $L:={\operatorname{O}}(V, s)$ is clearly not contained in the image of ${{\mathbb R}}G$. The ${{\mathbb C}}G$-module $F:={{\mathbb C}}\otimes_{{\mathbb R}}V$ is a direct sum of two copies of a symplectic irreducible ${{\mathbb C}}G$-module, but both $G$ and $L$ have the same subspace of invariants in ${\operatorname{Sym}}^2(V_{{\mathbb C}})$, namely the line spanned by the tensor determined by $s$.
Before we begin the proof of Theorem \[endo\], we recall that for a polarized abelian variety $A$, the polarization determines in the algebra ${\operatorname{End}}_{{\mathbb Q}}(A)$ an anti-involution (the Rosati involution) $b\mapsto b^\dagger$ which is positive in the sense that for every nonzero $b\in {\operatorname{End}}_{{\mathbb Q}}(A)$, ${\operatorname{tr}}(bb^\dagger)> 0$. This remains so after tensoring with ${{\mathbb R}}$. We recall a few facts about such algebras.
Given a finite dimensional ${{\mathbb R}}$-algebra ${{\mathscr B}}$ with a positive anti-involution $\dagger$, then by the classification of such algebras, ${{\mathscr B}}$ naturally decomposes into a product of ordinary matrix algebras with coefficients ${{\mathbb R}}$, ${{\mathbb C}}$, or ${{\mathbb K}}$, and $\dagger$ is then in each factor equivalent to taking the conjugate transpose. So the set of $f\in {{\mathscr B}}$ with $ff^\dagger=1$ is a subgroup $L ({{\mathscr B}})$ of the group of units in ${{\mathscr B}}$. It is a finite extension of a product of orthogonal groups, unitary groups and unitary quaternion groups, so in particular, compact. Moreover, it generates ${{\mathscr B}}$ as an ${{\mathbb R}}$-algebra. Given a finite dimensional ${{\mathscr B}}$-module $F$, then we say that an element of ${\operatorname{Sym}}^2F$ is ${{\mathscr B}}$-invariant if for all $b\in {{\mathscr B}}$, it is killed by $b\otimes 1-1\otimes b^\dagger$. Since $L({{\mathscr B}})$ generates ${{\mathscr B}}$ as an algebra, this is equivalent to this element being $L({{\mathscr B}})$-invariant in the usual sense. There is similar notion of covariance, but in view the reductive nature of ${{\mathscr B}}$ it is not worthwhile to make the distinction.
This situation shows up when we consider a polarized abelian variety $A$, endowed with a homomorphism ${{\mathscr B}}\to {\operatorname{End}}_{{\mathbb R}}(A)$ of algebras with involution. If we put $F:=H^0(A,\Omega_A)$, then ${\operatorname{End}}_{{\mathbb R}}(A):={{\mathbb R}}\otimes {\operatorname{End}}(A)$ (and hence ${{\mathscr B}}$) acts on $F$. As is well-known, the space of first order deformations of $A$ as a polarized variety can be identified with the complex dual of ${\operatorname{Sym}}^2F$. The first order deformations for which ${{\mathscr B}}$ still maps to the deformed object can be identified with the dual of the ${{\mathscr B}}$-coinvariants of ${\operatorname{Sym}}^2F$, i.e., the dual of $({\operatorname{Sym}}^2F)_{L({{\mathscr B}})}\cong ({\operatorname{Sym}}^2F)^{L({{\mathscr B}})}$. The more precise statement is that we have locally a universal deformation in the sense of Schlessinger with smooth base and whose cotangent space at the closed point can be identified with $({\operatorname{Sym}}^2F)^{L({{\mathscr B}})}$.\
We now turn to the proof of Theorem \[endo\]. We first reduce it to Lemma \[lemma:endo\] below. We write ${{\mathscr B}}$ for ${\operatorname{End}}_{{\mathbb R}}(J(C))$ and $F$ for $H^0(C, \Omega_C)$. The property that ${{\mathbb Q}}G\to {\operatorname{End}}_{{\mathbb Q}}(J(C))$ be surjective is equivalent to ${{\mathbb R}}G\to {{\mathscr B}}$ being surjective and since $L({{\mathscr B}})$ generates ${{\mathscr B}}$, the latter is equivalent to $L({{\mathscr B}})$ being contained in the image of ${{\mathbb R}}G$. This is what we will show.
Since $L({{\mathscr B}})$ is compact, $({\operatorname{Sym}}^2 F)^{L({{\mathscr B}})}$ has a ${L({{\mathscr B}})}$-invariant supplement $({\operatorname{Sym}}^2F)^{{L({{\mathscr B}})}\not=1}$ in ${\operatorname{Sym}}^2 F$. This is also $G$-invariant and so $$({\operatorname{Sym}}^2F)^{G,{L({{\mathscr B}})}\not=1}:=({\operatorname{Sym}}^2F)^G\cap ({\operatorname{Sym}}^2F)^{{L({{\mathscr B}})}\not=1}$$ is a $G$-invariant supplement of $({\operatorname{Sym}}^2 F)^{L({{\mathscr B}})}$ in $({\operatorname{Sym}}^2 F)^G$. In view of our hypotheses and Lemma \[lemma:quadratic\], Theorem \[endo\] will follow from:
\[lemma:endo\] We have $({\operatorname{Sym}}^2 F)^{G,{L({{\mathscr B}})}\not= 1}=0$.
The proof of Lemma \[lemma:endo\] involves a Hilbert scheme. We write ${{\mathbb P}}$ for the projective space $\check{{\mathbb P}}(F)$ of hyperplanes of $F$. The very general $G$-curve yields a canonical embedding $C\subset{{\mathbb P}}$. Let us denote by ${{\mathscr H}}$ the open subscheme of the irreducible component of ${\operatorname{Hilb}}({{\mathbb P}})$ which parameterizes canonically embedded smooth projective curves. The $G$-equivariant canonical embeddings of $G$-curves in ${{\mathbb P}}$ of our given topological type define an irreducible closed subscheme ${{\mathscr H}}^G$ of ${{\mathscr H}}$. The action of $G$ on $F$ induces one on ${{\mathscr H}}$ and then ${{\mathscr H}}^G$ is an irreducible component of the fixed point locus for this action.
\[quadric\]Let $X$ be a $G$-invariant quadric hypersurface of ${{\mathbb P}}$ containing the $G$-invariant canonical curve $C$. Then the $G$-invariant Hilbert scheme ${{\mathscr H}}_X^G$ (of $G$-curves contained in $X$) is a proper subscheme of the $G$-invariant Hilbert scheme ${{\mathscr H}}^G$.
In what follows we use the following notational convention. Given an immersion $Y\hookrightarrow Z$ of schemes, we write ${{\mathcal C}}_{Y/Z}$ for its conormal sheaf on $Y$. This is an ${{\mathscr O}}_Y$-module. The normal sheaf ${{\mathcal N}}_{Y/Z}$ of this immersion is by definition the ${{\mathscr O}}_Y$-dual of ${{\mathcal C}}_{Y/Z}$.
In the proof of Lemma 2.2 in [@bl], we observed that to the nested embeddings $C\subset X\subset{{\mathbb P}}$ is associated a short exact sequence $$0\to(\Omega_C^{\otimes 2})^\vee\to{\mathcal C}_{C/{{\mathbb P}}}\to{\mathcal C}_{C/X}\to 0,$$ which, by Lemma 2.2 in [@bl], determines an exact sequence $$0\to H^0(C,{{\mathcal N}}_{C/X})\to H^0(C,{{\mathcal N}}_{C/{{\mathbb P}}})\to H^0(C, \Omega^{\otimes 2}_C)\to {\operatorname{Ext}}_C^1({{\mathcal C}}_{C/X}, {{\mathscr O}}_C)\to 0.$$ Passing to $G$-invariants is an exact functor, and so we get the exact sequence $$\label{exact}
0\to H^0(C,{{\mathcal N}}_{C/X})^G\to H^0(C,{{\mathcal N}}_{C/{{\mathbb P}}})^G\to H^0(C, \Omega^{\otimes 2}_C)^G\to {\operatorname{Ext}}_C^1({{\mathcal C}}_{C/X}, {{\mathscr O}}_C)^G\to 0,$$ where $H^0(C,{{\mathcal N}}_{C/X})^G$ and $H^0(C,{{\mathcal N}}_{C/{{\mathbb P}}})^G$ identify with the tangent space at $[C\subset X]$ to the scheme ${{\mathscr H}}_X^G$ resp. the tangent space at $[C\subset{{\mathbb P}}]$ to the scheme ${{\mathscr H}}^G$, while $H^0(C, \Omega^{\otimes 2}_C)^G$ identifies with the cotangent space to ${{\mathscr M}}^\phi$ at $s$. By assumption $H^0(C, \Omega^{\otimes 2}_C)^G\neq\{ 0\}$. It is well-known (see [@bl], Lemma 2.1) that ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/{{\mathbb P}}},{{\mathscr O}}_C)^G= H^1(C,{{\mathcal N}}_{C/{{\mathbb P}}})^G=0$, so that the $G$-invariant Hilbert scheme ${{\mathscr H}}^G$ is smooth at $[C\subset{{\mathbb P}}]$. In case ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G=0$, then the exact sequence (\[exact\]) shows that the inclusion ${{\mathscr H}}_X^G\subseteq{{\mathscr H}}^G$ is strict.
When ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G\not=0$, we need the interpretation of this space as an obstruction space, as provided by Lemma \[obstruction3\] below. It then tells us that the $G$-equivariant deformation theory of $C$ in the quadric $X$ is obstructed. So the $G$-invariant Hilbert scheme ${{\mathscr H}}_X^G$ is singular at $[C\subset X]$ and then the inclusion ${{\mathscr H}}_X^G\subseteq{{\mathscr H}}^G$ is also strict.
\[obstruction3\]A minimal obstruction space for the $G$-equivariant deformation theory of $C$ in the quadric $X$ is ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G$.
We proved in Theorem 1.1 in [@bl] that ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)$ is a minimal obstruction space for the deformation theory of $C$ in the quadric $X$ and so we only need to check that this retains this interpretation in a $G$-equivariant setting. This is a matter of carefully going through the definitions.
Let $\omega$ be the map which assigns to every small extension $0\to I\to A'\to A\to 0$ of Artin ${{\mathbb C}}$-algebras and every deformation $\xi\co{{\mathscr C}}\subset X_A$ over $A$ of the embedding $C\subset X$ (where $X_A=X\times {\operatorname{spec}}(A)$) an element $\omega(\xi,A')\in{\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)\otimes I$ which is the obstruction to lifting $\xi$ to $A'$ (cf. Definition 5.1 in [@TV]).
The group $G$ acts on deformations of the embedding $C\subset X$ by changing the identification of the central fiber. Let us assume that $0\to I\to A'\to A\to 0$ is a tiny extension, i.e. that $I\cong{{\mathbb C}}$. Then, from the canonicity of minimal obstruction spaces (cf. Exercise 5.8 in [@TV]), it follows that $\omega(\alpha\cdot\xi,A')=\alpha\cdot\omega(\xi,A')$, for all $\alpha\in G$. Therefore, obstructions arising from $G$-equivariant lifting problems are contained in ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G$.
We have to show that all elements of ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G$ are obtained as obstructions to $G$-equivariant lifting problems. For this, it is enough to observe the following. Every element of ${\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)$ is of the form $\omega(\xi,A')$, for some tiny extension $0\to I\to A'\to A\to 0$ and some deformation $\xi\co{{\mathscr C}}\subset X_A$ over $A$ of the embedding $C\subset X$. By the construction appearing in the proof of Proposition 5.6 in [@TV], the sum of the elements appearing in the $G$-orbit of $\omega(\xi,A')$: $$\textstyle \sum_{\alpha\in G}\alpha\cdot\omega(\xi,A')=\sum_{\alpha\in G}\omega(\alpha\cdot\xi,A')\in{\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G$$ is realized as the obstruction to lifting the $G$-equivariant deformation $\times_{\alpha\in G}\alpha\cdot\xi$ over $\otimes^{|G|}A$ of the embedding $C\subset X$ in a $G$-equivariant fashion.
For this we use an interpretation of $({\operatorname{Sym}}^2 F)^{G,{L({{\mathscr B}})}\not= 1}$ as a conormal space of moduli spaces. We first note that the restriction map $$\Phi: ({\operatorname{Sym}}^2F)^G=({\operatorname{Sym}}^2 H^0(C, \Omega_C))^G \to H^0(C, \Omega^{\otimes 2}_C)^G$$ can be understood as follows: the dual of $H^0(C, \Omega^{\otimes 2}_C)^G$ is the space of first order deformations of $C$ as a $G$-curve and the dual of $({\operatorname{Sym}}^2F)^G$ is the space of first order deformations of deformations of the Jacobian $J(C)$ as a ppav with $G$-action, and $\Phi$ is then the obvious map. (By Max Noether’s theorem, the natural map ${\operatorname{Sym}}^2F={\operatorname{Sym}}^2H^0(C, \Omega_C)\to H^0(C, \Omega^{\otimes 2}_C)$ is onto and hence the same is true for $\Phi$.) As mentioned above, the dual of $({\operatorname{Sym}}^2F)^{L({{\mathscr B}})}$ has the interpretation of the space of first order deformations of $J(C)$ as a ppav endowed with an involution preserving homomorphism of ${{\mathscr B}}$ to the endomorphism algebra tensored with ${{\mathbb R}}$. In other words, $({\operatorname{Sym}}^2 F)^{G,{L({{\mathscr B}})}\not= 1}$ vanishes on the first order deformations of $J(C)$ as $G$-ppav which retain ${{\mathscr B}}$ in their endomorphism algebra tensored with ${{\mathbb R}}$. By construction, the first order deformations of $J(C)$ as a $G$-Jacobian are of this type. It follows that the kernel of $\Phi$ contains $({\operatorname{Sym}}^2 F)^{G,{L({{\mathscr B}})}\not= 1}$. As $C$ is very general, it is also very general as a member of ${{\mathscr H}}^G$ and so this will then be true for all members parameterized by ${{\mathscr H}}^G$ (we here use that the associated morphism ${{\mathscr H}}^G\to {{\mathscr M}}(G)$ is dominant, see Remark \[rem:verygeneral=dominant\]). It follows that any nonzero element of $({\operatorname{Sym}}^2 F)^{G,{L({{\mathscr B}})}\not= 1}$ gives us a $G$-invariant quadric $X$ for which ${{\mathscr H}}_X^G={{\mathscr H}}^G$. As this would contradict Lemma \[quadric\], we must have $({\operatorname{Sym}}^2 F)^{G,{L({{\mathscr B}})}\not= 1}=0$.
The hypotheses of Theorem \[endo\] are satisfied in most of the situations of interest. For instance, if the genus of the quotient orbifold curve $C_G$ is positive, then so is the dimension of the corresponding moduli stack ${{\mathscr M}}^\phi$ and the center of $G$ cannot contain a hyperelliptic involution. In order to establish when the condition on the structure of the $G$-module of abelian differentials $H^0(C,\Omega_C)$ is also satisfied, we use the classical Chevalley-Weil formula, which we now recall.
Let $C\to C_G$ be a Galois covering defined by a smooth projective $G$-curve and let $\chi\in X({{\mathbb C}}G)$ be *nontrivial* (that is, not the character of the trivial irreducible ${{\mathbb C}}G$-module ${{\mathbb C}}$, which is constant equal to $1$). Denote by $B\subset C_G$ the inertia locus (a finite set). For every $Q\in B$, we choose a representative $\tilde Q\in C$ and let ${\gamma}_{\tilde Q}$ be the positive generator of the (cyclic) stabilizer $G_{\tilde Q}\subset G$. Since the conjugacy class of ${\gamma}_{\tilde Q}$ only depends on $Q$, the same is true for the characteristic polynomial of ${\gamma}_{\tilde Q}$ in $V_\chi$. For $a\in {{\mathbb Q}}\cap (0,1]$, let $N_{\chi, Q}(a)$ be the multiplicity of $\exp(2\pi\sqrt{-1}a)$ as a root of this polynomial and put $N_\chi(a):=\sum_{Q\in B} N_{\chi, Q}(a)$. Then the Chevalley-Weil formula [@cw] asserts that $$\label{eqn:CW1}
\textstyle {\operatorname{dim}}_{{\mathbb C}}H^0(X,\Omega_X)[\chi] =(g(C_G)-1) \chi (1)+ \sum_{a\in {{\mathbb Q}}\cap (0,1]} N_{\chi}(a)(1-a).$$ If $\chi$ is the trivial character, the left hand side is of course equal to $g(C_G)$. This shows that when $g(C_G)\ge 2$, all irreducible characters of $G$ are afforded by the ${{\mathbb C}}G$-module $H^0(C,\Omega_C)$, so that the natural homomorphism ${{\mathbb Q}}G\to{\operatorname{End}}_{{\mathbb Q}}(J(C))$ is injective.
\[endo2\] For a very general $G$-curve $C$ for which the genus of the orbit curve $C_G$ is at least $3$, the natural homomorphism ${{\mathbb Q}}G\to{\operatorname{End}}_{{\mathbb Q}}(J(C))$ is an isomorphism.
The Chevalley-Weil formula shows that every $\chi\in X({{\mathbb C}}G)$ appears in $H^0(C,\Omega_C)$ with multiplicity $\ge 2$ and even with multiplicity $\ge 4$, unless $\chi$ is the character of the trivial representation. Now apply Theorem \[endo\].
We shall see (Remark \[g=2\]) that this corollary can fail if $g(C_G)=2$.
Since the dual of $H^0(C,\Omega_C)$ can be identified with its Hodge supplement in $H^1(C; {{\mathbb C}})$, the character of $H^1(C; {{\mathbb C}})$ must be equal to $\chi^\phi +\overline{\chi^\phi}$. The character of $H^1(X,{{\mathbb C}})$ can also be computed by means of a $G$-invariant cellular decomposition of the underlying oriented surface $S$: choose one such that the induced decomposition $S_G$ has as zero skeleton the union of $B$ and a base point of $S_G{\smallsetminus}B$, has $2g(S_G)+|B|$ 1-cells and has one $2$-cell. The associated cellular complex is then $$\label{display:complex}
0\to {{\mathbb Z}}G \to ({{\mathbb Z}}G)^{2g(C_G)}\oplus ({{\mathbb Z}}G)^B\to {{\mathbb Z}}(G)\oplus\underset{Q\in B}\oplus{{\mathbb Z}}(G/G_{\tilde Q}) \to 0.$$ The complex cohomology of $S$ as a ${{\mathbb C}}G$-module is that of the complex obtained by applying ${\operatorname{Hom}}(\; , {{\mathbb C}})$ and from this we see that for a nontrivial irreducible ${{\mathbb C}}G$-module $V_\chi$, $$\label{formula2}
\textstyle {\operatorname{dim}}_{{\mathbb C}}H^1(X,{{\mathbb C}})[\chi] =2( g(S_G)-1)\chi (1)+\sum_{Q\in B} {\operatorname{dim}}(V_\chi/V_\chi^{G_{\tilde Q}}).$$ So if $\chi^\phi$ is self-dual, then the character $H^0(C,\Omega_C)$ will then be half that of $H^1(C,{{\mathbb C}})$. Conversely, when $G$ acts freely on $S$, then a comparison with the Chevalley-Weil formula (\[eqn:CW1\]) shows that $\chi^\phi$ is half the character of $H^1(C,{{\mathbb C}})$, so that $\chi^\phi$ must be self-dual.
\[rem:\] It is not hard to see that complex conjugation sends ${{\mathscr M}}^\phi$ to a ${{\mathscr M}}^{\bar\phi}$, where $\bar\phi$ is represented by retaining our $\phi : G\hookrightarrow {\operatorname{Mod}}(S)$ and changing the orientation of $S$. We then have $\chi^{\bar\phi}=\overline{\chi^\phi}$. In particular, if ${{\mathscr M}}^\phi$ is real, then so is $\chi^\phi$. Whether this is a useful result is another matter. Let us just note that it fits in a more general Galois setting: if ${{\mathscr M}}^{(G)}$ is defined over the number field $K$ (Collas and Maugeais have shown in [@cm] and [@cm2014] that we can take $K={{\mathbb Q}}$ for $G$ cyclic), then the individual ${{\mathscr M}}^\phi$ are defined over a (common) finite extension $L/K$ so that the Galois group of $\overline K/K$ permutes them through ${\operatorname{Gal}}(L/K)$. The induced action of ${\operatorname{Gal}}(L/K)$ on the characters $\chi^\phi$ will then be the obvious one (via a cyclotomic character, since all the characters of $G$ take their values in the extension of ${{\mathbb Q}}$ generated by the $|G|$-th roots of unity).
Even though we cannot expect that the hypotheses of Theorem \[endo\] are always satisfied by $G$-curves $C$ whose orbit curve has genus zero, there is a particular case for which this happens:
\[theorem:GN\] Let $C$ be a smooth projective $G$-curve and $N\subset G$ a subgroup of index 2 which acts freely on $C$ and is such that $C_N$ is of genus $\geq 2$ with $G/N$ acting on $C_N$ as a hyperelliptic involution (so that $C_G$ is of genus zero). Then
1. $G$ is a semi-direct product of $N$ and a group of order $2$ and the $G$-invariant subspace $H^0(C_N,\Omega_{C_N})$ of $H^0(C,\Omega_C)$ is acted on via $G/N$ with the nontrivial element acting as scalar multiplication by $-1$.
2. Every irreducible ${{\mathbb C}}G$-submodule $V$ of $H^0(C,\Omega_C)$ not contained in $H^0(C_N,\Omega_{C_N})$ decomposes into two non-isomorphic irreducible ${{\mathbb C}}N$-modules, which are characterized by the property that if we induce them up to $G$, we get a ${{\mathbb C}}G$-module equivalent to $V$. The two are exchanged by the outer action of $G/N$ and $V$ appears in $H^0(C,\Omega_C)$ with multiplicity $\tfrac{1}{2}(g(C_{N})-1) {\operatorname{dim}}V$.
3. If $N$ is nontrivial, then $C$ is nonhyperelliptic.
Moreover, if $C_N$ is of genus $\ge 4$ and $(C, G)$ is very general as a $G$-curve, then the natural homomorphism ${{\mathbb Q}}G\to{\operatorname{End}}_{{\mathbb Q}}(J(C))$ is surjective.
In case $N$ is trivial (so that $C$ is hyperelliptic), the three assertions are trivial or empty, and the last clause follows from Theorem 1.2 in [@CGT]. We will therefore assume in the proof below that $N$ is nontrivial.
We first show that the involution $\iota$ of $C_{N}$ induced by the nontrivial element of $G/N$ lifts to an involution $\tilde\iota$ in $G$. To see this, let $x\in C$ be such that its image $x_{N}$ in $C_{N}$ is fixed under $\iota$. Since $N$ acts freely on $C$, the $G$-stabilizer $G_x$ of $x$ meets $N$ trivially. So $G_x$ is of order two and its nontrivial element $\tilde\iota$ is a lift of $\iota$. In particular, $G$ is a semi-direct product $\langle \tilde\iota\rangle\ltimes N$. Part (i) now follows, because the hyperelliptic involution $\iota$ acts on $H^0(C_N,\Omega_{C_N})$ as multiplication by $-1$.
Since $N$ is of index 2 in $G$, we have for every irreducible $G$-submodule $V$ of $H^0(C,\Omega_C)$ two possibilities, according to whether the outer action of $\iota\in G/N$ fixes the character of $V$ or not (cf. Proposition 5.1 in [@FH]):
(irr)
: $V$ is also irreducible as a ${{\mathbb C}}N$-module (with ${\tilde}{\iota}$ acting on it trivially or by multiplication by $-1$) or
(red)
: $V$ splits into two non-isomorphic irreducible ${{\mathbb C}}N$-modules $V'\oplus V''$, where $V''$ is obtained from $V'$ by precomposition of $N$ with conjugation by $\tilde\iota$, and ${\operatorname{Ind}}^G_N V'$ and ${\operatorname{Ind}}^G_N V''$ are equivalent to $V$.
We now invoke the Chevalley-Weil formula (\[eqn:CW1\]). The group $N$ acts freely on $C$, while the involution $\iota$ in $C_N$ is hyperelliptic, and so $G$ has $2g(C_{N})+2$ irregular orbits in $C$, each of which contains a fixed point of $\tilde\iota$. So the sum in the right hand side of (\[eqn:CW1\]) is in fact a single term $N_V(a)(1-a)$ with $N_V(a)=(2g(C_{N})+2){\operatorname{dim}}V$ and either $a=1$ or $a=\tfrac{1}{2}$, according to whether $\tilde\iota$ acts in $V$ as $1$ or as $-1$. We find that in the first case, the multiplicity of $V$ in $H^0(C,\Omega_C)$ is $-{\operatorname{dim}}V$ (which is of course absurd) and so $\tilde\iota$ must act in $V$ as $-1$ and $V$ appears with multiplicity $-{\operatorname{dim}}V+ (2g(C_{N})+2){\operatorname{dim}}V.\frac{1}{2}=g(C_{N}){\operatorname{dim}}V$. But $N$ acts freely on $C$ and hence the Chevalley-Weil formula (\[eqn:CW1\]) applied to the $N$-action implies that the only irreducible representation of $N$ which appears in $H^0(C, \Omega_C)$ with multiplicity $g(C_{N})$ is the trivial one, in other words, $V$ is then contained in $H^0(C_N, \Omega_{C_N})$.
The above dichotomy also shows that if the irreducible ${{\mathbb C}}G$-module $V$ is not contained in $H^0(C_N, \Omega_{C_N})$, then $V$ decomposes as a ${{\mathbb C}}N$-module as in (ii): $V=V'\oplus V''$. Since $V'$ appears in $H^0(C, \Omega_C)$ with multiplicity $(g(C_{N})-1){\operatorname{dim}}V'$, the ${{\mathbb C}}G$-module $V\cong {\operatorname{Ind}}^G_N V'$ appears in $H^0(C, \Omega_C)$ with the same multiplicity, i.e., $\frac{1}{2}(g(C_{N})-1){\operatorname{dim}}V$. This proves (ii). Since $\tilde\iota$ has the effect of exchanging $V'$ and $V''$, this also shows that $\tilde\iota$ is not hyperelliptic.
Suppose now $g(C_N)\ge 4$ and $(C, G)$ is very general. The one-dimensional nontrivial representation of $G/N$ appears in $H^0(C, \Omega_C)$ and its isotypical subspace is $H^0(C_N, \Omega_{C_N})$. So it has multiplicity $\ge 4$ in $H^0(C, \Omega_C)$. According to (ii), any other nontrivial irreducible representation $V$ which appears in $H^0(C, \Omega_C)$ is of the form $V'\oplus V''$ and hence appears with multiplicity $(g(C_{N})-1){\operatorname{dim}}V'\ge 3$. It then remains to apply Theorem \[endo\].
\[g=2\] Theorem \[theorem:GN\] implies that the image ${{\mathbb Q}}N\to{\operatorname{End}}_{{\mathbb Q}}(J(C))$ cannot contain the lift $\tilde\iota$ of the hyperelliptic involution (for $\tilde\iota$ permutes non-isomorphic irreducible ${{\mathbb Q}}N$-modules), so that in particular, ${{\mathbb Q}}N\to{\operatorname{End}}_{{\mathbb Q}}(J(C))$ is not surjective. Since it is easy to find nontrivial unramified covers of a closed genus 2 surface such that a hyperelliptic involution lifts, we see that Corollary \[endo2\] can fail when the genus of the orbit curve is $2$.
Symplectic $G$-modules and associated unitary groups
====================================================
We here collect (or derive) some facts regarding symplectic $G$-modules and the algebraic groups that they give rise to, insofar they are needed in this paper.
Irreducible $G$-modules {#irreducible-g-modules .unnumbered}
-----------------------
We review the basics of the theory, as can be found for instance in [@isaacs] and use this opportunity to establish notation. Let $k$ be a number totally real field or be equal to ${{\mathbb R}}$ (but for us only the cases $k={{\mathbb Q}}$ and $k={{\mathbb R}}$ will be relevant). The set $X(k G)$ of irreducible characters of $k G$ is a set of $k$-valued class functions on $G$, which is invariant under the Galois group of $k/{{\mathbb Q}}$. For every $\chi\in X({{\mathbb Q}}G)$, there is always a positive integer $m$ and an orbit in the Galois group of ${{\mathbb R}}/{{\mathbb Q}}$ in $X({{\mathbb R}}G)$ such that $\chi$ is $m$ times the sum over this Galois orbit (see e.g., Theorem (9.21) in [@isaacs]). We shall express this shortly in terms of the representations themselves.
For a given $\chi\in X(k G)$, Schur’s lemma implies that ${\operatorname{End}}_{k G}(V_\chi)$ is a division algebra. We denote the opposite algebra by $D_\chi$ and regard $V_\chi$ as a left $k G$-module and as a right $D_\chi$-module. We also fix a positive definite $G$-invariant inner product $s_\chi :V_\chi\times V_\chi\to k$. For any bilinear form $\phi: V_\chi\times V_\chi\to k$, there exists a unique $\sigma \in {\operatorname{End}}_k(V_\chi)$ such that $\phi(v, v')= s_\chi(\sigma(v), v')$. The form $\phi$ is $G$-invariant if and only if $\sigma$ is $G$-equivariant, so that the latter will be given by right multiplication with some $\lambda\in D_\chi$. For the same reason there exists a $\lambda^\dagger\in D_\chi$ such that $\phi (v, v')= s_\chi(v, v'\lambda^\dagger)$. Note that thus is defined an anti-involution $\lambda \mapsto \lambda^\dagger$ on $D_\chi$ characterized by the property that $s_\chi(v\lambda , v')=s_\chi(v, v'\lambda^\dagger)$. So if $\lambda\in D_\chi$ is symmetric in the sense that $\lambda=\lambda^\dagger$, then $(v, v')\mapsto s_\chi(v\lambda , v')=s_\chi(v, v'\lambda)$ is also a $G$-invariant symmetric bilinear form with values in $k$. In particular, $s_\chi$ need not be unique up to scalar. The identity $s(v\lambda, v'\lambda)=s(v\lambda\lambda^\dagger, v')$, shows that $\dagger$ is positive in the following sense. Denote by $L_\chi$ the center of $D_\chi$ and by $K_\chi\subset L_\chi$ the subfield fixed under the anti-involution $\dagger$. Then $K_\chi$ is finite extension of $k$ that is totally real and for every nonzero $\lambda\in D_\chi$, the trace ${\operatorname{tr}}_{D_\chi/K_\chi}(\lambda\lambda^\dagger)\in K_\chi$ has positive image under every field embedding $K_\chi\hookrightarrow {{\mathbb R}}$.
Recall that for a finitely generated $k G$-module $H$ and $\chi\in X(kG)$, we write $H[\chi]$ for ${\operatorname{Hom}}_{k G}(V_\chi, H)$. The right $D_\chi$-module structure on $V_\chi$ determines a left $D_\chi$-module structure on $H[\chi]$ by the rule $(du)(v):=u(vd)$ and then the natural map $$\oplus_{\chi\in X(k G)} V_\chi\otimes_{D_\chi} H[\chi]\to H, \quad v\otimes_{D_\chi} u\in V_\chi\otimes_{D_\chi} H[\chi]\mapsto u(v)$$ is an isomorphism of $k G$-modules. This is the $kG$-isotypical decomposition of $H$ with the image $H_\chi$ of $V_\chi\otimes_{D_\chi} H[\chi]$ being its $\chi$-isotypical summand. Any $k G$-linear automorphism of $H$ preserves its $G$-isotypical decomposition and acts in the summand $H_\chi\cong V_\chi\otimes_{D_\chi} H[\chi]$ through a $D_\chi$-linear transformation of $H[\chi]$. Thus, the $G$-centralizer ${\operatorname{End}}_k(H)^G\subset {\operatorname{End}}_k(H)$ is identified with $\prod_{\chi\in X(k G)} {\operatorname{End}}_{D_\chi}(H[\chi])$. The natural map $$\textstyle kG\to \prod_{\chi\in X(k G)} {\operatorname{End}}^{D_\chi}(V_\chi),$$ where ${\operatorname{End}}^{D_\chi}(V_\chi)$ stands for the algebra of endomorphisms of $V_\chi$ as a right $D^\chi$-module, is an isomorphism of $k$-algebras (this can be regarded as an instance of one of Wedderburn’s structure theorems).
We discuss the passage from $k={{\mathbb Q}}$ to $k={{\mathbb R}}$. For this, let us for now fix $\chi\in X( kG)$ and temporarily omit $\chi$ in the notation. So we write $D=D_\chi$, $L=L_\chi$, $V=V_\chi$ etc.
When $k={{\mathbb R}}$, we have the three classical cases: $D={{\mathbb R}}$ (so $\dagger$ is then the identity) or $D$ is isomorphic to ${{\mathbb K}}$ or ${{\mathbb C}}$, with $\dagger$ in both cases corresponding to the usual conjugation. (These three cases correspond to $\chi\in X({{\mathbb R}}G)$ being resp. an element of $X({{\mathbb C}}G)$, twice an element of $X({{\mathbb C}}G)$, the sum of a complex conjugate distinct pair in $X({{\mathbb C}}G)$.)
When $k={{\mathbb Q}}$ there are in some sense also three cases. Given a field embedding $\sigma :K\hookrightarrow {{\mathbb R}}$, let us use write $\sigma^*$ for the associated base change: if $F$ is a $K$-vector space, then $\sigma^*F={{\mathbb R}}\otimes_\sigma F$. For the irreducible ${{\mathbb Q}}G$-module $V=V_\chi$, the ${{\mathbb R}}G$-module $\sigma^*V$ is in general no longer irreducible, but it will be isotypical for an irreducible ${{\mathbb R}}G$-module $V^\sigma$ with character $\chi^\sigma\in X({{\mathbb R}}G)$, say with multiplicity $m^\sigma$. The characters $\{\chi^\sigma\}_\sigma$ are pairwise distinct and make up an orbit of the Galois group of ${{\mathbb R}}/{{\mathbb Q}}$ and $m^\sigma$ is independent of $\sigma$: $m^\sigma=m$. Denote by $D^\sigma$ the opposite of ${\operatorname{End}}_{{{\mathbb R}}G} (V^\sigma)$. It is a real division algebra, which will be isomorphic to ${{\mathbb R}}$, ${{\mathbb C}}$ or ${{\mathbb K}}$ and whose isomorphism type is independent of $\sigma$. We put $I^\sigma={\operatorname{Hom}}_{{{\mathbb R}}G} (V^\sigma, \sigma^*V)$ (so we could also write this as $ \sigma^*V[\chi^\sigma]$). Then $I^\sigma$ is a left $D^\sigma$-module (of rank $m$) and a right $\sigma^*D$-module and we have the isotypical decomposition as ${{\mathbb R}}G$-modules $$V_{{\mathbb R}}(:={{\mathbb R}}\otimes_{{\mathbb Q}}V)=\oplus_\sigma \, \sigma^*V \cong \oplus_\sigma \, V^\sigma\otimes_{D^\sigma} I^\sigma.$$ This also identifies $\sigma^*D$ with the opposite of ${\operatorname{End}}_{{{\mathbb R}}G}(\sigma^*V)={\operatorname{End}}_{D^\sigma}(I^\sigma)$ and gives rise to the isomorphism $$\begin{gathered}
\sigma^*(H[\chi])=\sigma^*({\operatorname{Hom}}_{{{\mathbb Q}}G}(V, H))\cong{\operatorname{Hom}}_{{{\mathbb R}}G}(\sigma^*V, H_{{\mathbb R}})\cong {\operatorname{Hom}}_{{{\mathbb R}}G}(V^\sigma\otimes_{D^\sigma} I^\sigma, H_{{\mathbb R}})\cong\\
\cong {\operatorname{Hom}}_{D^\sigma} (I^\sigma, {\operatorname{Hom}}_{{{\mathbb R}}G}(V^\sigma, H_{{\mathbb R}}))= {\operatorname{Hom}}_{D^\sigma} (I^\sigma, H_{{\mathbb R}}[\chi^\sigma])
\end{gathered}$$ as left $\sigma^*D$-modules. The anti-involution $\dagger$ on $D$ yields one on $\sigma^*D$ and via the identification of the latter with ${\operatorname{End}}_{D^\sigma}(I^\sigma)^{\mathrm{opp}}$ becomes ‘taking the adjoint’ with respect to a hermitian form $h^\sigma$ on $I^\sigma$ obtained as follows. There is up to scalar exactly one $G$-invariant inner product on the ${{\mathbb R}}G$-module $V^\sigma$. Let us fix one and denote it $s^\sigma$. Then the $G$-invariant symmetric bilinear form $s$ on $V$ determines a positive definite hermitian form $h^\sigma: I^\sigma\times I^\sigma\to D^\sigma$ (relative to the standard conjugation in $D^\sigma$) characterized by the property that for all $v, v'\in V^\sigma$ and $w, w'\in I^\sigma$, $s^\sigma( w(v), w'(v'))=s^\sigma(vh^\sigma (w, w'),v')$.
Symplectic $G$-modules {#symplectic-g-modules .unnumbered}
----------------------
Suppose that our $kG$-module $H$ comes with a nondegenerate $G$-invariant symplectic form $(a,b)\in H\times H\mapsto {\langle}a, b{\rangle}\in k$ (for example $H=H^1(S; k)$, and the symplectic form being the intersection product). We then first observe that we have a decomposition $$\textstyle {\operatorname{Sp}}(H)^G=\prod_{\chi\in X(k G)} {\operatorname{Sp}}(H_\chi)^G.$$ Let us now fix some $\chi\in X(k G)$. Given $u,u'\in H[\chi]$, then $(v, v')\mapsto {\langle}u(v), u'(v'){\rangle}$ is a $G$-invariant bilinear form on $V_\chi$. Since $s_\chi$ is nondegenerate, there is an $h\in {\operatorname{End}}(V_\chi)$ such that ${\langle}u(v), u'(v'){\rangle}=s_\chi (h(v), v')$ for all $v, v'\in V_\chi$. The $G$-invariance of the form implies that $h$ is right multiplication with some $h_\chi (u, u')\in D_\chi$: $${\langle}u(v), u'(v'){\rangle}=s_\chi(vh_\chi (u, u'),v').$$ It is then straightforward to verify that $h_\chi$ is $D_\chi$-linear in the first variable and satisfies $h_\chi(u, u')=-h_\chi(u',u)^\dagger$ (so that $h_\chi(u, du')=h_\chi(u',u)d^\dagger$). In other words, it is a skew-hermitian form on the $D_\chi$-module $H[\chi]$. We shall therefore denote this form by ${\langle}\; ,\; {\rangle}_\chi$. It is clearly nondegenerate. We thus find an identification $${\operatorname{Sp}}(H_\chi)^G\cong {\operatorname{U}}_{D_\chi}(H[\chi]),$$ where ${\operatorname{Sp}}(H_\chi)^G$ stands for the $G$-centralizer in ${\operatorname{Sp}}(H_\chi)$ and ${\operatorname{U}}_{D_\chi} (H[\chi])$ denotes the group of automorphisms of $H[\chi]$ as a skew-hermitian $D_\chi$-module (but we write ${\operatorname{U}}(H[\chi])$ when $D_\chi$ is clear from the context). The group on the left is in a natural manner the group of $k$-points of a reductive $k$-algebraic group ${\operatorname{{\mathscr Sp}}}(H_\chi)^G$ and the group on the right can be regarded as the group of $K_\chi$-points of an algebraic group ${{\mathscr U}}(H[\chi])$ defined over $K_\chi$. In other words, ${\operatorname{{\mathscr Sp}}}(H_\chi)^G$ is obtained from ${{\mathscr U}}(H[\chi])$ by restriction of scalars (from $K_\chi$ to $k$).
This enables us to understand what the group of real points of ${\operatorname{{\mathscr Sp}}}(H_\chi)^G$ is like when $k={{\mathbb Q}}$. Given a field embedding $\sigma :K_\chi\to {{\mathbb R}}$, then, as we noted, the ${{\mathbb R}}$-algebra $\sigma^*D_\chi$ is isomorphic to a matrix algebra over ${{\mathbb R}}$, ${{\mathbb C}}$ or ${{\mathbb K}}$. Clearly, $\sigma^*H[\chi]$ is a left $\sigma^*D_\chi$-module and ${\langle}\; , \: {\rangle}$ determines a skew-hermitian form $\sigma^*{\langle}\; , \: {\rangle}$ on it. The above identification then yields an isomorphism $$\textstyle {\operatorname{{\mathscr Sp}}}(H_\chi)^G({{\mathbb R}})\to \prod_\sigma {\operatorname{U}}_{\sigma^*D_\chi} (\sigma^*H[\chi]),$$ where the product is over all field embeddings $\sigma: K_\chi\hookrightarrow {{\mathbb R}}$.
The factor ${\operatorname{U}}_{\sigma^*D_\chi} (\sigma^*H[\chi])$ can be identified with a unitary group over the classical division algebra $D^\sigma=D_{\chi^\sigma}$ as follows. Let us recall that we write $V^\sigma$ for $V_{\chi^\sigma}$ and $I^\sigma$ for ${\operatorname{Hom}}_{{{\mathbb R}}G} (V^\sigma, \sigma^*V)$ and that we have isomorphisms $$\sigma^*D_\chi \cong {\operatorname{End}}_{D^\sigma}(I^\sigma)^{\mathrm{opp}}\text{ (of algebras)}, \quad \sigma^*H[\chi]\cong {\operatorname{Hom}}_{D^\sigma} (I^\sigma, H_{{\mathbb R}}[\chi^\sigma]) \text{ (of $\sigma^*D_\chi$-modules)}.$$ If we apply the preceding to the case $k={{\mathbb R}}$, then we see that $H_{{{\mathbb R}}, \chi^\sigma}$ is a nondegenerate subspace of $H_{{\mathbb R}}$ for the symplectic form and that the $D^\sigma$-module $H_{{\mathbb R}}[\chi^\sigma]$ comes with a nondegenerate $D^\sigma$-valued skew-hermitian form. Let us denote the latter by ${\langle}\; ,\; {\rangle}^\sigma$. So if $\phi, \phi'\in \sigma^*H[\chi]\cong {\operatorname{Hom}}_{D^\sigma} (I^\sigma, H_{{\mathbb R}}[\chi^\sigma]) $, then a skew-hermitian form on $I^\sigma$ is given by $$(u,u')\mapsto {\langle}\phi (u) ,\phi' (u') {\rangle}^\sigma$$ We had already defined a positive definite hermitian form $h^\sigma: I^\sigma\times I^\sigma\to D^\sigma$. Hence there is a unique $g(\phi, \phi')\in {\operatorname{End}}_{D^\sigma}(I^\sigma)$ such that $${\langle}\phi (u) ,\phi' (u') {\rangle}^\sigma= h^\sigma(g(\phi, \phi')(u), u')$$ for all $u, u'\in I^\sigma$. It is straightforward to verify that $g$ defines a ${\operatorname{End}}_{D^\sigma}(I^\sigma)^{\mathrm{opp}}$-valued skew-hermitian form on ${\operatorname{Hom}}_{D^\sigma} (I^\sigma, H_{{\mathbb R}}[\chi^\sigma])$. Via the isomorphisms above this is up to a scalar the $\sigma^*D_\chi$-valued skew-hermitian form on $\sigma^*H[\chi]$. We thus find:
\[cor:\] We have a natural identification ${\operatorname{U}}_{\sigma^*D_\chi} (\sigma^*H[\chi])\cong {\operatorname{U}}_{D_{\chi^\sigma}}(H_{{\mathbb R}}[\chi^\sigma])$ and hence an identification ${\operatorname{{\mathscr Sp}}}(H_\chi)^G({{\mathbb R}})\cong \prod_\sigma {\operatorname{U}}_{D_{\chi^\sigma}}(H_{{\mathbb R}}[\chi^\sigma])$.
So if $r$ denotes the rank of $H_{{\mathbb R}}[\chi^\sigma]$ as a $D_{\chi^\sigma}$-module, then
real
: $H_{{\mathbb R}}[\chi^\sigma]$ is a real symplectic vector space so that ${\operatorname{U}}(\sigma^*H[\chi])$ is isomorphic to a symplectic group in $r$ real variables.
quaternion
: $H_{{\mathbb R}}[\chi^\sigma]$ is a skew-hermitian module over $D_{\chi^\sigma}\cong {{\mathbb K}}$ and ${\operatorname{U}}(\sigma^*H[\chi])$ is isomorphic to a unitary group in $r$ quaternionic variables.
complex
: $H_{{\mathbb R}}[\chi^\sigma]$ is a skew-hermitian module $D_{\chi^\sigma}=L_{\chi^\sigma}\cong {{\mathbb C}}$ and ${\operatorname{U}}(\sigma^*H[\chi])$ is isomorphic to a unitary group in $r$ complex variables.
Virtual linear representations of the mapping class group {#virtual}
=========================================================
Hodge group and connected monodromy group {#hodge-group-and-connected-monodromy-group .unnumbered}
-----------------------------------------
We recall that a weight $m$ Hodge structure on a ${{\mathbb Q}}$-vector space $W$ amounts to giving an action of the circle group ${\operatorname{U}}(1)$ on $W_{{\mathbb R}}$ so that $W_{{\mathbb C}}=\oplus_{p+q=m}W^{p,q}$ can be regarded as the eigenspace decomposition of an ${\operatorname{U}}(1)$-action, where $\lambda\in {\operatorname{U}}(1)$ acts as multiplication by $\lambda^{q-p}$. The *Hodge group* of this Hodge structure is defined to be the smallest ${{\mathbb Q}}$-subgroup ${\mathscr{H}\! g}(W)$ of ${\operatorname{GL}}(W)$ whose group of real points contains the image of this action. It is a connected reductive group. Note that $W$ is indecomposable as a Hodge structure, if and only if ${\mathscr{H}\! g}(W)$ acts ${{\mathbb Q}}$-irreducibly on $W$.
Now let ${{\mathbb W}}$ be a *variation of polarizable Hodge structure* ${{\mathbb W}}$ over a connected complex manifold $Y$ with Hodge numbers $(1,0)$ and $(0,1)$ (our definition assumes that there exists a sublattice ${{\mathbb W}}_{{\mathbb Z}}\subset{{\mathbb W}}$, but this is a property and not part of the data). Since the Hodge group gets smaller as the Hodge structure becomes more special and is locally constant on a dense subset, we have a locally constant family of subgroups $\{{\mathscr{H}\! g}({{\mathbb W}})_y\subseteq {\operatorname{GL}}({{\mathbb W}}_y)\}_{y\in Y}$ characterized by the property that ${\mathscr{H}\! g}({{\mathbb W}}_y)\subseteq {\mathscr{H}\! g}({{\mathbb W}})_y$ for all $y\in Y$ and with equality for some $y$. This is what is called the *generic Hodge group* (of ${{\mathbb W}}$ at $y$). Deligne’s semi-simplicity theorem ([@Deligne], Thm. 4.2.6, Cor. 4.2.8 and Cor. 4.2.9) tells us that ${{\mathbb W}}$ is semi-simple as a variation of Hodge structure and that its complexification ${{\mathbb W}}_{{\mathbb C}}$ is semisimple as a local system. The last property implies that the identity component of the Zariski closure of the monodromy group of ${{\mathbb W}}$ in ${\operatorname{GL}}({{\mathbb W}}_y)$—which we shall denote by ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ and refer to as the *connected monodromy group*—is semisimple as well. Note that ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ does not change under base change by an unramified finite cover of $Y$.
The endomorphism algebra of the local system underlying ${{\mathbb W}}$, ${\operatorname{End}}({{\mathbb W}})$, has a natural Hodge structure with Hodge numbers $(1,-1)$, $(0,0)$ and $(-1, 0)$. Note that the subalgebra ${\operatorname{End}}_{HS}({{\mathbb W}})$ of endomorphisms preserving the Hodge structure is ${\operatorname{End}}({{\mathbb W}})^{(0,0)}({{\mathbb Q}})=
{\operatorname{End}}({{\mathbb W}}_{{\mathbb C}})^{(0,0)}\cap {\operatorname{End}}({{\mathbb W}})$. If ${{\mathbb W}}$ is indecomposable as a VHS, this will be a division algebra.
Taking into account the results quoted above, then the essential part of the following lemma is second part, which is due to Masa-Hiko Saito [@mhsaito] (see also the discussion in section (1.5) of [@MZ]).
\[lemma:rigid\] Let ${{\mathbb W}}$ be a polarized variation of Hodge structure over a connected complex manifold $Y$ with Hodge numbers $(1,0)$ and $(0,1)$. Suppose ${{\mathbb W}}$ is indecomposable and has infinite monodromy (or equivalently, is not isotrivial). Then:
1. ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ is a product of ${{\mathbb Q}}$-simple factors of the derived Hodge group ${{\mathscr {DH}}\!g}({{\mathbb W}})_y$, and its representation on ${{\mathbb W}}_y$ is isotypical. The group ${\mathscr{H}\! g}({{\mathbb W}})_y$ acts on ${\operatorname{End}}({{\mathbb W}})$ via ${\mathscr{H}\! g}({{\mathbb W}})_y/{{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ and the latter’s action on ${\operatorname{End}}({{\mathbb W}})$ has as its kernel a central subgroup.
2. The variation of Hodge structure ${{\mathbb W}}$ is *rigid* in the sense that ${\operatorname{End}}({{\mathbb W}})$ is of type $(0,0)$ if and only if ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ acts ${{\mathbb Q}}$-irreducibly on ${{\mathbb W}}_y$; this is also equivalent to ${{\mathscr {DH}}\!g}({{\mathbb W}})_y={{\mathscr M}{on}}^\circ({{\mathbb W}})_y$.
3. When ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ has positive ${{\mathbb Q}}$-rank, then ${{\mathscr {DH}}\!g}({{\mathbb W}})_y={{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ and this group is ${{\mathbb Q}}$-almost simple. In particular, ${{\mathbb W}}$ is rigid.
We merely show how this follows from the literature. We may assume that ${{\mathbb W}}_y$ is indecomposable as Hodge structure and that ${\mathscr{H}\! g}({{\mathbb W}}_y)={\mathscr{H}\! g}({{\mathbb W}})_y$. It is a general result of André [@Andre] that ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ is normal in ${{\mathscr {DH}}\!g}({{\mathbb W}})_y$. Since ${{\mathscr {DH}}\!g}({{\mathbb W}})_y$ is a semisimple ${{\mathbb Q}}$-algebraic group, ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ will be a product of ${{\mathbb Q}}$-simple factors of it. The isotypical decomposition of ${{\mathbb W}}_y$ as a representation of ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ will respect the Hodge decomposition on ${{\mathbb W}}_y$ and hence must be trivial in the sense that ${{\mathbb W}}_y$ will be an isotypical ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$-module. We can state this as follows. The isogeny type of ${{\mathbb W}}$ is represented by an irreducible local system ${{\mathbb V}}$ of ${{\mathbb Q}}$-vector spaces over $Y$ such that ${{\mathbb W}}$ is as local system isomorphic a sum of copies of ${{\mathbb V}}$. More precisely, $D:={\operatorname{End}}({{\mathbb V}})$ is a division algebra and if we put $U:={\operatorname{Hom}}({{\mathbb V}}, {{\mathbb W}})$, then $U$ is in a natural manner a right $D$-module such that the evident map $U\otimes_D {{\mathbb V}}\cong {{\mathbb W}}$ is an isomorphism of local systems. This isomorphism also identifies ${\operatorname{End}}({{\mathbb W}})$ with the matrix algebra ${\operatorname{End}}^D(U)$ (the endomorphisms of $U$ as a right $D$-module). But note that the tensor decomposition ${{\mathbb W}}\cong U\otimes_D {{\mathbb V}}$ cannot respect the Hodge structure, unless this decomposition is trivial. In particular, $U$ need not have a Hodge structure. On the other hand, ${\operatorname{End}}^D(U)\cong {\operatorname{End}}({{\mathbb W}})$ has one. Since this Hodge structure is invariant under the monodromy, ${\mathscr{H}\! g}({{\mathbb W}})_y$ acts on it through ${\mathscr{H}\! g}({{\mathbb W}})_y/{{\mathscr M}{on}}^\circ({{\mathbb W}})_y$. It is clear that the kernel of this action is a central subgroup of ${\mathscr{H}\! g}({{\mathbb W}})_y/{{\mathscr M}{on}}^\circ({{\mathbb W}})_y$.
We now turn to the second assertion. Since Saito does not express his results in terms of the Hodge group, we must make the translation. We address the nontrivial direction: if ${{\mathbb W}}$ is rigid, then we must show that every ${{\mathbb Q}}$-almost simple factor ${{\mathscr G}}$ of ${\mathscr{H}\! g}({{\mathbb W}})_y$ is contained in ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$. Let ${{\mathscr G}}'$ be a complementary normal ${{\mathbb Q}}$-subgroup of ${\mathscr{H}\! g}({{\mathbb W}})_y$ so that ${{\mathscr G}}$ and ${{\mathscr G}}'$ commute with each other and have finite intersection. The Hodge structure on ${{\mathbb W}}_y$ is given by a morphism ${\operatorname{U}}(1)\to{\mathscr{H}\! g}({{\mathbb W}})_y$. Denote its composite with the projection ${\mathscr{H}\! g}({{\mathbb W}})_y\to {\mathscr{H}\! g}({{\mathbb W}})_y/{{\mathscr G}}'$ by $\rho_y$. This composite is independent of $y$ if and only if ${{\mathscr G}}$ is not an irreducible constituent of ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ (we could paraphrase this by saying that variations of Hodge structures manufactured out of ${{\mathbb W}}$ which do not involve ${{\mathscr G}}'$ will be constant). But as Saito makes clear, this just means that $\rho_y$ describes a (necessarily) nontrivial Hodge structure on ${\operatorname{End}}({{\mathbb W}})$. As this precludes rigidity, this shows that ${{\mathscr G}}\subseteq {{\mathscr M}{on}}^\circ({{\mathbb W}})_y$.
His argument also shows that when every factor of ${{\mathscr G}}({{\mathbb R}})$ is noncompact, we have in fact equality: ${{\mathscr G}}={\mathscr{H}\! g}({{\mathbb W}})_y$. (Apply Cor. 6.27 to the family of abelian varieties which deforms the Hodge structures in the ${{\mathscr G}}$-direction only: it implies that the group denoted there by $G'_{{\mathbb R}}$—in our situation $G'$ is essentially the centralizer of ${{\mathscr G}}$ in the the symplectic group of ${{\mathbb W}}_y$—has to be compact, so that the defining homomorphism ${\operatorname{U}}(1)\to {\operatorname{Sp}}({{\mathbb W}}_y)$ actually lands in ${{\mathscr G}}({{\mathbb R}})$.) If ${{\mathscr M}{on}}^\circ({{\mathbb W}})_y$ has positive ${{\mathbb Q}}$-rank, then it has a ${{\mathbb Q}}$-simple factor ${{\mathscr G}}$ of positive ${{\mathbb Q}}$-rank. Since every factor of such a ${{\mathscr G}}({{\mathbb R}})$ is noncompact, (iii) also follows.
Application to $G$-curves {#application-to-g-curves .unnumbered}
-------------------------
We are going to apply the preceding to $H^1(S; {{\mathbb Q}})$, where $S$ is as before: a closed connected oriented surface with a faithful $G$-action. So then for every $\chi\in X({{\mathbb Q}}G)$, we have a skew-hermitian $D_\chi$-module $H^1(S; {{\mathbb Q}})[\chi]$.
We equip $V_\chi$ with the trivial Hodge structure of bidegree $(0,0)$; it is then polarized by $s_\chi$ (but this can be one of many polarizations). When we endow $S$ with a $G$-invariant complex structure, resulting in a $G$-curve $C$, then $H^1(C; {{\mathbb Q}})[\chi]$ comes with a polarizable $D_\chi$-invariant Hodge structure with Hodge numbers $(1,0)$ and $(0,1)$ and $D_\chi$ acts by endomorphisms of type $(0,0)$. If the natural algebra homomorphism ${{\mathbb Q}}G\to {\operatorname{End}}_{{\mathbb Q}}(J(C))\cong{\operatorname{End}}_{HS} (H^1(C; {{\mathbb Q}}))$ is onto, then the $G$-isotypical decomposition cannot be refined any further in a $G$-equivariant manner such that it also respects the Hodge structure: otherwise we would get a projector in ${\operatorname{End}}_{{\mathbb Q}}(J(C))$ that is not in the image of ${{\mathbb Q}}G$. In other words, the endomorphism algebra of the Hodge structure on $H^1(C; {{\mathbb Q}})[\chi]$ is then $D_\chi$.
This becomes more interesting in a relative setting. We aim to formulate the consequences in topological language.
Let $(S, G)$ be a closed connected oriented $G$-surface. Let us denote by ${{\mathscr M}{on}}^\circ(S)^G$ the identity component of the Zariski closure of the image of the action of ${\operatorname{Mod}}(S)^G$ on $H^1(S;{{\mathbb Q}})$. It respects the isotypical decomposition $H^1(S; {{\mathbb Q}})\cong \oplus_\chi V_\chi\otimes_{D_\chi} H^1(S; {{\mathbb Q}})[\chi]$ and ${{\mathscr M}{on}}^\circ(S)^G$ acts on the $\chi$-summand via $H^1(S; {{\mathbb Q}})[\chi]$. We denote the image of this action by ${{\mathscr M}{on}}^\circ(S)[\chi]$. A $G$-invariant conformal structure on $S$ yields a $G$-curve $C$. The Hodge group of $H^1(C; {{\mathbb Q}})$ is a reductive ${{\mathbb Q}}$-subgroup of ${\operatorname{Sp}}(H^1(S; {{\mathbb Q}}))^G$. Since these conformal structures are (up to $G$-equivariant isotopy) parameterized by the (connected) complex manifold ${\operatorname{Teich}}(S)^G={\operatorname{Teich}}(S_G)$, the preceding discussion shows that for a very general choice of a $G$-invariant conformal structure, this Hodge group is in fact independent of that choice. We therefore denote it by ${\mathscr{H}\! g}(G,S)$. This group respects the isotypical decomposition and like ${{\mathscr M}{on}}^\circ(S)^G$, it acts on the $\chi$-summand via $H^1(S; {{\mathbb Q}})[\chi]$. We denote the image of this action by ${\mathscr{H}\! g}(S)[\chi]$.
Our main application is Theorem \[topology1\] below. We have already shown that its hypotheses are fulfilled when $S_G$ has genus at least $3$ or when we are in the situation of Theorem \[theorem:GN\] (where $G$ contains a subgroup $N$ of index $2$ which acts freely on $S$ such that $S_N$ is of genus $\ge 4$ and $G/N$ acts on $S_N$ as a hyperelliptic involution).
\[topology1\] Let $(S, G)$ be a closed connected oriented $G$-surface. Assume that for a very general $G$-invariant conformal structure on $S$, the resulting $G$-curve $C$ has the property that ${{\mathbb Q}}G$ maps onto ${\operatorname{End}}_{HS}(H^1(C; {{\mathbb Q}}))$. Then:
1. The centralizer of ${\mathscr{H}\! g}(S)[\chi]$ in ${\operatorname{End}}(H^1(S; {{\mathbb Q}})[\chi])$ is $D_\chi$.
2. ${\operatorname{Mod}}(S)[\chi]$ is a normal subgroup of the derived group ${{\mathscr {DH}}\!g}(S)[\chi]$ of ${\mathscr{H}\! g}(S)[\chi]$ and the quotient ${\mathscr{H}\! g}(S)[\chi]/{{\mathscr M}{on}}^\circ(S)[\chi]$ acts on ${\operatorname{End}}_{{{\mathscr M}{on}}^\circ(S)[\chi]}(H^1(S; {{\mathbb Q}})[\chi])$ with finite central kernel.
3. The group ${\mathscr{H}\! g}(S)[\chi]$ (resp. ${{\mathscr M}{on}}^\circ(S)[\chi]$) acts ${{\mathbb Q}}$-irreducibly (resp. ${{\mathbb Q}}$-isotypically) on the $D_\chi$-module $H^1(S;{{\mathbb Q}})[\chi]$.
4. If ${{\mathscr M}{on}}^\circ(S)[\chi]$ has positive ${{\mathbb Q}}$-rank, then ${{\mathscr M}{on}}^\circ(S)[\chi]={{\mathscr {DH}}\!g}(S)[\chi]$ and this group is ${{\mathbb Q}}$-almost simple.
5. If ${{\mathscr M}{on}}^\circ(S)[\chi]$ is trivial, then ${\operatorname{Mod}}(S)^G$ acts through a finite group on $H^1(S; {{\mathbb Q}})_\chi$ and all $G$-invariant conformal structures on $S$ define the same indecomposable Hodge structure on $H^1(S;{{\mathbb Q}})[\chi]$.
\[rem:\] It is not known whether the case (v) of finite monodromy occurs at all. Putman and Wieland conjecture (1.2 of [@P-W]) that it does not (even when the genus of $S_G$ is $\ge 2$), and prove that this non-occurrence is essentially equivalent to a conjecture of Ivanov, which states that the first Betti number of a finite index subgroup of a mapping class group of genus $\ge 3$ is trivial.
In order to prove the theorem for a cofinite subgroup ${\Gamma}\subset {\operatorname{Mod}}(S)^G$, there is no loss of generality in passing to a smaller one and so we may assume that ${\Gamma}\subset {\operatorname{Mod}}(S)^G$ is a normal subgroup which acts trivially on $H^1(S, {{\mathbb Z}}/3)$. By a classical observation of Serre, ${\Gamma}$ is then torsion free, so if we divide out ${{\mathscr C}}_{{\operatorname{Teich}}^G}/{\operatorname{Teich}}^G$ by the freely acting ${\Gamma}$ we obtain a $G$-curve $f: {{\mathscr C}}\to Y$, where $Y$ has the structure of a smooth quasi-projective variety étale over ${{\mathscr M}}^\phi$. For $y\in Y$, we have an isomorphism $\pi_1(Y,y)\cong {\Gamma}$, unique up to inner automorphism and via such an isomorphism, the $\pi_1(Y,y)\times G$-action on $H^1(C_y)$ is equivalent to the ${\Gamma}\times G$-action on $H^1(S)$. We are now in a situation in which we can apply Lemma \[lemma:rigid\] to ${{\mathbb W}}=R^1f_*{{\mathbb Q}}[\chi]$ and all the assertions then follow.
\[quest:opology1\] We do not know whether in the case of infinite monodromy (under the hypotheses of Theorem \[topology1\]) which of the inclusions $${{\mathscr M}{on}}^\circ(S)[\chi]\subseteq {{\mathscr {DH}}\!g}(C)[\chi]\subseteq {\mathscr {D\!U}}(H^1(S, {{\mathbb Q}})[\chi])$$ can be strict. (According to Theorem \[topology1\]-iv, the first inclusion is an isomorphism when ${{\mathscr M}{on}}^\circ(S)[\chi]$ has positive ${{\mathbb Q}}$-rank, but we do not know whether that is always the case; see also Corollary \[cor:primmon\] below.) Nor do we know whether the image of the ${\operatorname{Mod}}(S)^G$-action on $H^1(S;{{\mathbb Q}})[\chi])$ is an arithmetic subgroup. When $G$ is abelian and acts freely on $S$ and is such that $g(S_G)\ge 2$, then it was verified in [@L] that the answer is yes; the essential case is then $G$ cyclic, which means that $S\to S_G$ is trivial over a connected subsurface of genus $g(S_G)-1$. Grünewald, Larsen, Lubotzky and Malestein [@GLLM] generalized this to the case when $S\to S_G$ is trivial over a connected subsurface of genus one that is the complement of $g(S_G)-1$ pairwise disjoint loops in $S_G$ (with $G$ still acting acts freely and $g(S_G)\ge 2$). This implies that $G$ has $\le g(S_G)-1$ generators, but can be very much non-abelian. Venkataramana [@venky] proved that the answer is also yes when $g(S_G)=0$, $G$ abelian and such that the number of irregular orbits does not exceed half the order of $G$. In none of these cases a nontrivial $H^1(S, {{\mathbb Q}})[\chi]$ occurs with finite monodromy group.
Another question is whether these representations are independent, in the sense that the natural map ${{\mathscr M}{on}}^\circ(S)^G\to \prod_\chi {{\mathscr M}{on}}^\circ(S)[\chi]$ is an isomorphism (again assuming that $g(S_G)\ge 3$). The results in [@L] imply that this is so when $B=\emptyset$ and $G$ is abelian.
Virtual representations {#virtual-representations .unnumbered}
-----------------------
Let us now explain the title of this section. For this we need to elaborate a bit more on the structure of the D-M stack ${{\mathscr M}}^\phi$. Choose a base point $y\in S_G$ away from the set of inertia points. The data of $S_G$ as the $G$-orbifold quotient of $S$, determines and is determined by an epimorphism $p:\pi_1(S_G, y)\to G$ up to an inner automorphism of $G$. Here $\pi_1(S_G, y)$ is to be understood as the fundamental group of the orbifold $S_G$. The kernel $K$ of $p$ is of course intrinsic to this situation, but it does not quite determine the topological type $\phi$ up to a $G$-isomorphism, because it is also necessary to specify an ${\operatorname{In}}(G)$-orbit of isomorphisms $\pi_1(S_G, y)/K\cong G$. However, the relevant stabilizer subgroup in ${\operatorname{Mod}}(S_G)$, that is, the group of mapping classes of ${\operatorname{Mod}}(S_G)$ which, when viewed as a group of outer automorphisms of $\pi_1(S_G, y)$, preserve $K$ and change $p$ by an inner automorphism of $G$, only depends on $K$, and can therefore be denoted by ${\operatorname{Mod}}(S_G)[K]$. It consists of the mapping classes which lift to $S$ and commute with the $G$-action. It is of finite index in ${\operatorname{Mod}}(S_G)$. We thus obtain an étale covering ${{\mathscr M}}(S_G)[K]\to {{\mathscr M}}(S_G)$ and a short exact sequence: $$\label{shortexact}
1\to Z(G)\to {\operatorname{Mod}}(S)^G\to {\operatorname{Mod}}(S_G)[K]\to 1.$$ (This exact sequence expresses the fact that the smooth D–M stack ${{\mathscr M}}^\phi$ is a $Z(G)$-gerbe over ${{\mathscr M}}(S)[K]$.) If $\overline g$ is the genus of $S_G$ and $\overline n$ the cardinality of the inertia set, then ${\operatorname{Mod}}(S_G)$ contains ${\operatorname{Mod}}_{\overline g, \overline n}$ as a subgroup of finite index. Since $Z(G)$ is finite and the group ${\operatorname{Mod}}(S_G)[K]$ is of finite index in ${\operatorname{Mod}}(S_G)$, the group ${\operatorname{Mod}}(S)^G$ has a subgroup of finite index which is also of finite index in ${\operatorname{Mod}}_{\overline g, \overline n}$. So in that sense, the monodromy representation of ${\operatorname{Mod}}(S)^G$ appearing in Theorem \[topology1\] can be regarded as a virtual representation of ${\operatorname{Mod}}_{\overline g, \overline n}$.
The Dehn groups {#the-dehn-groups .unnumbered}
---------------
Given an oriented surface $\Sigma$, we shall denote by ${\operatorname{\mathscr{X}}}(\Sigma)$ the vertices of the curve complex of $\Sigma$, that is, the set of isotopy classes of embedded circles in $\Sigma$ which do not bound a disk or a once punctured disk. We recall that every $\alpha\in {\operatorname{\mathscr{X}}}(\Sigma)$ defines a Dehn twist $\tau_\alpha\in {\operatorname{Mod}}(\Sigma)$ and carries, after an orientation, a homology class $[\alpha]\in H_1(\Sigma)$ (which is therefore only given up to sign). The action of $\tau_\alpha$ on $H_1(\Sigma)$ is given by the transvection $T_{[\alpha]}: c\in H_1(\Sigma)\mapsto c+ {\langle}c ,[\alpha]{\rangle}[\alpha]$, where ${\langle}\; , \; {\rangle}$ is the intersection form on $H_1(\Sigma)$. Here the choice of orientation of $\alpha$ clearly doesn’t matter.
Let us write $S_G^\circ$ for $S_G{\smallsetminus}B$, where $B\subset S_G$ denotes the set of inertia points. We observe that every $\bar\alpha\in {\operatorname{\mathscr{X}}}(S_G^\circ)$ determines a $G$-orbit $A=A_{\bar\alpha}\subset {\operatorname{\mathscr{X}}}(S)$, namely the set of connected components of its preimage of a representative of $\bar\alpha$. We denote by ${\operatorname{\mathscr{X}}}(S, G)\subset {\mathscr {X}}(S)$ the union of the $G$-orbits $A$ so obtained. It is clear that a $G$-orbit $A$ as above spans an isotropic subspace for the intersection pairing. The Dehn twists $\{\tau_{\alpha}\}_{\alpha\in A}$ commute pairwise and their product $\tau_{A}$ is an element of ${\operatorname{Mod}}(S)^G$ which canonically lifts $\tau_{\bar \alpha}^{m_{\bar\alpha}}\in {\operatorname{Mod}}(S_G)$, where $m_{\bar\alpha}$ denotes the common degree of the members of $A$ over $\bar\alpha$. Then $T_{A}:=\tau_{A*}\in {\operatorname{Sp}}(H_1(S))^G$ is given by $$\textstyle T_{A}(x)=x+\sum_{\alpha\in A} {\langle}x,[\alpha]{\rangle}[\alpha]$$ and hence generates the one-parameter subgroup $$\textstyle {\mathscr {T}}_{A}:{\operatorname{\mathbb{G}}}_a\to {\operatorname{{\mathscr Sp}}}(H)^G, \quad {\mathscr {T}}_{A}(t) (x)= x+ \sum_{\alpha\in A} {\langle}x,[\alpha]{\rangle}t[\alpha].$$ It is clearly contained in the connected monodromy group ${{\mathscr M}{on}}^\circ(S)^G$.
\[cor:primmon\] Assume that we are in the situation of Theorem \[topology1\] and that $\chi\in X({{\mathbb Q}}G)$ is such that ${\operatorname{\mathscr{X}}}(S,G)$ has a nonzero image in $H_1(S;{{\mathbb Q}})_\chi$. Then this image spans $H_1(S;{{\mathbb Q}})_\chi$ over ${{\mathbb Q}}$. Moreover, ${\operatorname{Mod}}(S)^G$ acts irreducibly on $H_1(S;{{\mathbb Q}})[\chi]$ and the identity component of the Zariski closure of this representation, ${{\mathscr M}{on}}^\circ(S)[\chi]$, is ${{\mathbb Q}}$-almost simple.
Let $A\subset {\mathscr {X}}(S,G)$ be a $G$-orbit whose image in $H_1(S;{{\mathbb Q}})_\chi$ is nontrivial. Then the one-parameter subgroup ${\mathscr {T}}_{A}$ of ${{\mathscr M}{on}}^\circ(S)^G$ has a nontrivial image in ${{\mathscr M}{on}}^\circ(S)[\chi]$ and hence ${{\mathscr M}{on}}^\circ(S)[\chi]$ has positive ${{\mathbb Q}}$-rank. It then follows from Theorem \[topology1\]-iv that ${{\mathscr M}{on}}^\circ(S)[\chi]$ is ${{\mathbb Q}}$-almost simple and acts irreducibly on $H_1(S;{{\mathbb Q}})[\chi]$. It then also follows that $G\times {\operatorname{Mod}}(S)^G$ acts irreducibly on $H_1(S;{{\mathbb Q}})_\chi$. Since the ${{\mathbb Q}}G$-submodule of $H_1(S;{{\mathbb Q}})_\chi$ spanned by the image of ${\operatorname{\mathscr{X}}}(S,G)$ is $G\times {\operatorname{Mod}}(S)^G$-invariant, it must be all of $H_1(S;{{\mathbb Q}})_\chi$.
\[rem:\] Suppose that $S_G{\smallsetminus}B$ contains an embedded circle $\gamma$ over which the covering is trivial and is such that its preimage $\tilde\gamma$ in $S$ has the property that $S{\smallsetminus}\tilde\gamma$ is still connected. This implies that that ${{\mathbb Z}}G\cong H_1(\tilde\gamma)\to H_1(S)$ is an injection of ${{\mathbb Z}}G$-modules. Hence, if in addition $g(S_G)\ge 3$, then the hypotheses of Corollary \[cor:primmon\] are satisfied for all $\chi$, so that for *every* $\chi\in X({{\mathbb Q}}G)$, ${\operatorname{Mod}}(S)^G$ acts ${{\mathbb Q}}$-irreducibly on $H_1(S;{{\mathbb Q}})[\chi]$ and ${{\mathscr M}{on}}^\circ(S)[\chi]$ is ${{\mathbb Q}}$-almost simple.
We close the paper with asking a question and proposing a conjecture.
\[conj:liegeneration\] In the situation of Corollary \[cor:primmon\], the group generated isotropic transvections $\{T_{A}[\chi]\}_{A\in G{\backslash}{\operatorname{\mathscr{X}}}(S,G)}$ maps onto an arithmetic subgroup of ${{\mathscr M}{on}}^\circ (S)[\chi]$.
\[quest:arithmetic2\] Malestein and Putman [@mp] have recently shown that it can happen that ${\mathscr {X}}(S, G)$ does not span $H_1(S;{{\mathbb Q}})$. It then follows from Corollary \[cor:primmon\] that for some $\chi\in X({{\mathbb Q}}G)$, the image of ${\mathscr {X}}(S, G)$ in $H_1(S;{{\mathbb Q}})[\chi]$ is trivial. So then any Dehn twist in $S_G{\smallsetminus}B$ admits a power which lifts to a multi-Dehn twist in ${\operatorname{Mod}}(S)^G$ that acts trivially on $H_1(S;{{\mathbb Q}})_\chi$. Since a Dehn twist in $S_G{\smallsetminus}B$ represents a loop in the moduli space of curves of orbifold type $S_G$, there exists then an étale cover $Y\to {{\mathscr M}}^\phi$ of our moduli space ${{\mathscr M}}^\phi$ with the property that for the resulting pull-back $f:{{\mathscr C}}_Y\to Y$ (a family of $G$-curves), the local variation of Hodge structure $R^1f_*{{\mathbb Q}}[\chi]$ extends across the Deligne-Mumford compactification of $Y$. Since the kind of representations of ${\operatorname{Mod}}(S_G)$ which come from the theory of conformal blocks (the so-called quantum representations) also have this property, one of us [@Looij:OW] has raised the question whether in such a case the (virtual) representation of ${\operatorname{Mod}}(S_G)$ on $H_1(S; {{\mathbb Q}})[\chi]$ is of this type. (See also [@ks].) It is likely that the Zariski closure of such a representation is anisotropic (i.e., has ${{\mathbb Q}}$-rank zero).
[PP]{} Y. André: *Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part*. Compositio Mathematica [**82**]{} (1992), 1–24.
M. Boggi, E. Looijenga: *Deforming a canonical curve inside a quadric*. Int. Math. Res. Not. IMRN,\
<https://doi.org/10.1093/imrn/rny027> (2018).
C. Chevalley, A. Weil, E. Hecke: *Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers.* Abh. Math. Sem. Univ. Hamburg [**10**]{} (1934), 358–361.
C. Ciliberto: *Endomorfismi di Jacobiane*. Rend. Sem. Mat. Fis. Milano [**59**]{} (1989), 213–242.
C. Ciliberto, G. van der Geer, M. Teixidor i Bigas: *On the number of parameters of curves whose Jacobians possess nontrivial endomorphisms*. J. Alg. Geom. [**1**]{} (1992), 215–229.
B. Collas, S. Maugeais: *Composantes irréductibles de lieux spéciaux d’espaces de modules de courbes, action galoisienne en genre quelconque.* Ann. Inst. Fourier (Grenoble) **65** (2015), 245–276, <https://arxiv.org/pdf/1403.2517.pdf>
B. Collas, S. Maugeais: *On Galois action on stack inertia of moduli space of curves.*\
<https://arxiv.org/pdf/1412.4644v1.pdf> (2014).
P. Deligne: *Th[é]{}orie de Hodge, II*. Publ. Math. IHÉS [**40**]{} (1971), 5–57.
W. Fulton, J. Harris. *Representation Theory: A First Course*. Graduate Texts in Mathematics [**129**]{}, Springer-Verlag (1991).
F. Grünewald, M. Larsen, A. Lubotzky, J. Malestein: *Arithmetic quotients of the mapping class group.* Geom. Funct. Anal. **25** (2015) 1493–1542, <https://arxiv.org/pdf/1307.2593v2>
I.M. Isaacs: *Character theory of finite groups.* Academic Press (1976).
T. Koberda, R. Santharoubane: *Quotients of surface groups and homology of finite covers via quantum representations.* Invent. Math. **206** (2016), 269–292, <https://arxiv.org/pdf/1510.00677>
S. Lefschetz: *A Theorem on Correspondences on Algebraic Curves.* Am. J. of Math., [**50**]{} (1928), 159–166.
E. Looijenga: *Prym Representations of Mapping Class Groups*. Geometriae Dedicata [**64**]{} (1997), 69–83.
E. Looijenga: *Some algebraic geometry related to the mapping class group,* abstract of a talk at the Oberwolfach workshop *New perspectives on the the Interplay between Discrete Groups in Low-Dimensional Topology and Arithmetic Lattices* (June 2015), available at <http://www.staff.science.uu.nl/~looij101/eduardlooijengaOWjuni2015.pdf>
A. Putman, J. Malestein: *Simple closed curves, finite covers of surfaces, and power subgroups of ${\operatorname{Out}}(F_n)$,* <https://arxiv.org/pdf/1708.06486v3.pdf>
B.J.J. Moonen, Yu.G. Zarhin: *Hodge classes on abelian varieties of low dimension*. Math. Annalen [**315**]{} (1999), 711–733.
A. Putman, B. Wieland: *Abelian quotients of the mapping class group and higher Prym representations*, J. Lond. Math. Soc. (2) [**88**]{} (2013), 79–96, <https://arxiv.org/pdf/1106.2747.pdf>
Masa-Hiko Saito: *Classification of nonrigid families of abelian varieties,* Tohoku Math. J. [**45**]{} (1993), 159–189.
J-P. Serre: *Linear Representations of Finite Groups*. Graduate Texts in Mathematics [**42**]{}, Springer-Verlag (1977).
T.A. Springer: *Linear algebraic groups,* 2nd ed., Progress in Mathematics, **9** Birkhäuser Boston, Inc., Boston, MA (1998).
M. Talpo, A. Vistoli: *Deformation theory from the point of view of fibered categories.* The Handbook of Moduli (G. Farkas, I. Morrison, editors), Vol. III, pp. 281–397, Adv. Lect. in Math. [**25**]{}, Int. Press (2013), <https://arxiv.org/pdf/1006.0497.pdf>
T. N. Venkataramana: *Monodromy of cyclic coverings of the projective line.* Invent. Math. 197 (2014), 1–45. <https://arxiv.org/abs/1204.4778>
Y. Zarhin: *Hodge classes on certain hyperelliptic prymians,* in *Arithmetic, geometry, cryptography and coding theory.* 171–183, Contemp. Math. [**574**]{}, Amer. Math. Soc. Providence, RI (2012), <https://arxiv.org/pdf/1012.3731.pdf>
Y. Zarhin: *The endomorphism rings of Jacobians of cyclic covers of the projective line.* Math. Proc. Cambridge Philos. Soc. [**136**]{} (2004), 257–267, <https://arxiv.org/pdf/math/0103203.pdf>
[^1]: E.L. was supported by the Chinese National Science Foundation.
[^2]: When a notion only depends on the image of $\phi$ in ${\operatorname{Mod}}(S)$, we usually suppress $\phi$ in the notation and write $G$ instead.
|
---
abstract: 'Cosmologists have embraced a particular ad hoc formula for the primordial power spectrum from inflation for universes with $\Omega_0 < 1$. However, the so-called “Open Inflation” models, which are attracting renewed interest in the context of the “string theory landscape” give a different result, and offer a more fully developed picture of the cosmology and fundamental physics basis for inflation with $\Omega_0 < 1$. The Open Inflation power spectrum depends not only on $\Omega_0$, but on the parameters of the effective fields that drive the universe [*before*]{} the Big Bang (in “another part of the landscape”). This paper considers the search for features in CMB temperature anisotropy data that might reflect a primordial spectrum of the Open Inflation form. We ask whether this search could teach us about high energy physics that described the universe before the onset of the Big Bang, and perhaps even account for the low CMB quadrupole. Unfortunately our conclusion is that the specific features we consider are unobservable even with future experiments although we note a possible loophole connected with our use of the thin wall approximation.'
author:
- Michael Barnard
- Andreas Albrecht
title: 'On Open Inflation, the string theory landscape and the low CMB quadrupole'
---
*[**Note Added**]{}: Since this paper was completed we became aware of a large body of existing literature which treats the problem of perturbations in Open Universe models with a much greater degree of sophistication than we do here (including working away from the thin wall limit). See [@Garriga:1996pg; @Sasaki:1996qn; @Garcia-Bellido:1996gd; @Sasaki:1997ex; @Garcia-Bellido:1997hy; @Garriga:1997wz; @Garcia-Bellido:1997te; @Garriga:1998he; @Garcia-Bellido:1998wd; @Linde:1999wv] and references therein.*
An up-to-date treatment of the important questions raised in this paper (about the possible universality of open inflation in the string theory landscaped and resulting observational signatures) requires the application of these more sophisticated methods and results, a process we are now undertaking. We apologies to the authors of this impressive earlier work for our ignorance about it in the first version of this paper posted on the archive. We also thank Jaume Garriga and Thomas Hertog for bringing this work to our attention.
Introduction {#introduction .unnumbered}
============
One of the great achievements of modern cosmology is the ability to calculate detailed predictions for the cosmological perturbations from specific models of the early universe. This, along with impressive new data such as the WMAP survey[@Bennett:2003bz] has allowed significant constraints to be placed on early universe physics as well as on a number of cosmological parameters.
One of the key cosmological parameters is $\Omega_0$, the ratio of the current cosmic density (including the dark energy) to the critical density. A well-known problem is that for cosmological models with $\Omega_0 < 1$ the perturbation calculation is more problematic, particularly on large scales. This is because for typical models of cosmic inflation to make precise predictions for perturbations on all observed scales they must also predict $\Omega_0 = 1$ to about one part in $10^5$. In the context of these models, to calculate the large scale perturbations in the $\Omega_0 <
1$ case one must answer the question “what physics other than inflation determined the perturbations on the largest observable scales?”. This issue has been recognized since the first papers on inflation with $\Omega_0 < 1$[@Lyth:1990dh; @Ratra:1994dm].
For the most part, the cosmology community has “resolved” this problem by simply assuming a particular formula for perturbations in cosmologies with $\Omega_0 < 1$. This formula appears in all the main software packages (such as CMBfast) which determine the perturbation spectra for $\Omega_0 < 1$ models. It is only because of this particular choice that it even seems possible to determine $\Omega_0$ to high precision. One is left open to the possibility that a deeper understanding of early universe physics could shift our preference to different pictures of $\Omega_0 < 1$ cosmology which could yield different formulas for the perturbation spectrum. For $\Omega_0 < 1$ models with different spectra, the same data might well lead to a different preferred value of $\Omega_0$ as well as other parameters.
In fact, we may be in the midst of such a shift right now. Recent work [@Kachru:2003aw] suggests that string theory (our best hope for a realistic quantum gravity theory) predicts a landscape of different “vacua” which are highly stable, but which have some non-zero probability of tunneling into one another. This picture suggests a cosmology strikingly similar to the so-called “Open Inflation” models of Bucher et al. [@BT1; @BT2])
The Open Inflation models were first invented to address the ambiguities of the perturbation spectra for $ \Omega_0 < 1$ cosmologies discussed above. Bucher et al. consider a cosmological model with an initial phase of inflation that defines the cosmological state on a range of length scales that spans many orders of magnitude and drives the global state of the universe toward $\Omega_0 = 1$. Bucher et al. modeled this phase of inflation with a field trapped in false vacuum, in the manner of “old inflation”[@Guth:1980zm].
This initial period of inflation ends with a tunneling process that produces a bubble universe which is open from the point of view of observers within it. The field that tunnels can experience a shorter period of slow-roll inflation[@new] after the tunneling event which can bring the bubble universe close to $\Omega =
1$ and define the perturbation spectrum on smaller scales. Because of the early period of old inflation the pre-tunneling cosmic state is uniquely determined, and this allows the perturbations in the bubble universe to be well determined on [*all*]{} observable scales with no ambiguities.
When first introduced the open inflation models seemed a bit artificial (although it really was a matter of taste whether one considered them more so than “typical” slow-roll inflation models). Today, the landscape picture that is emerging from string theory suggests that the cosmology for a universe in any one of the many metastable vacua universally starts with a tunneling event preceded by a long period of old inflation in the (false) vacuum of the previous landscape location. Although there still are a number of unresolved questions, this picture certainly suggests that the Open Inflation model of Bucher et al. may well be [*the*]{} universal cosmology seen by an observer in the string theory landscape.[^1]
Our main motivation is the string theory landscape, but we also note that the puzzling low quadrupole and octopole ($C_2$ and $C_3$) in the WMAP first year data suggest that interesting information might be lurking in the cosmic perturbations on large scales[@WMAP1; @WMAP2]. Since the open inflation perturbation spectrum depends not only on $\Omega_0$, but on the curvature of the inflaton potential during the period of old inflation (before tunneling) in principle we could read information about the physics of the universe before the big bang from large scale cosmological data.
With these motivations, we have undertaken a calculation of the CMB temperature anisotropies in Open Inflation models. Unfortunately, our results show that the differences between the open inflation results and the generic formula used in most cosmology papers is immeasurably small for realistic cosmological parameters. Thus we have nothing new to add to the interpretation of cosmological data. In particular, at least as far as the Open Universe models go, the standard determination of the value of $\Omega_0$ and other cosmological parameters is unaltered, and there is no opportunity to measure new parameters from other parts of the string theory landscape. Of course this also means that we cannot rule out open inflation models with realistic values of $\Omega_0$. The one caveat is that our work assumes that the thin wall approximation gives a valid treatment of the tunneling event. It is possible that corrections to this approximation could lead to a more interesting result.
It is also possible that a deeper understanding of the string theory landscape could lead to other kinds of predictive power in connection with open inflation. For example, a “most likely” form for the inflaton driving the post-tunneling period of new inflation could emerge, which in turn could lead to specific signature in the CMB power. This is not the effect we consider in this paper, which is devoted to effects generic to [*all*]{} open inflation models.
The primordial power spectrum
=============================
The primordial power spectrum for Open Inflation presented in [@BT2] is $$\begin{aligned}
P_{\chi}(\beta)=&&
\frac{9}{4\pi^2}
\left( \frac{H^3}{V,_\phi} \right)^2
\frac{1}{\beta (\beta^2+1)} \nonumber\\
&&\times
\left[\frac{
e^{\pi\beta}+e^{-\pi\beta}
+\frac{|{\cal C}_2|}{{\cal C}_1}
\left(
\frac{\beta+i}{\beta-i}e^{i\bar{\varphi}}
+\frac{\beta-i}{\beta+i}e^{-i\bar{\varphi}}
\right)
}{e^{\pi\beta}-e^{-\pi\beta}}
\right]
\nonumber \\\end{aligned}$$ where k is the co-moving wavenumber which is related to $\beta$ and the curvature $K$ by $k^2=\beta^2-K$. With the usual normalization, $k^2=\beta^2+1$. The field variable that gives the density fluctuations is $\chi$, and ${\cal C}_1$ and ${\cal C}_2$ are parameterized by $${\cal C}_1=2\pi\cosh^2[\bar{\xi}(\beta)]$$ $${\cal C}_2=2\pi\cosh[\bar{\xi}(\beta)]
\sinh[\bar{\xi}(\beta)]e^{i\bar{\varphi}}$$ The definitions of ${\cal C}_1$ and ${\cal C}_2$ are
$${\cal C}_1=2\pi\left[
1+\frac{\sin^2(\pi\nu')}{\sinh^2(\pi\beta)}
\right]$$
$$\begin{aligned}
{\cal C}_2=&&
2\pi\frac
{\sin(\pi\nu')\Gamma(i\beta-\nu')\Gamma(1-i\beta)}
{\sinh^2(\pi\beta)\Gamma(-i\beta-\nu')\Gamma(1+i\beta)}
\nonumber \\
&&\times
\left(
\cosh(\pi\beta)\sin(\pi\nu')-i\sinh(\pi\beta)\cos(\pi\nu')
\right)\end{aligned}$$
with $\nu'=\sqrt{\frac94-m^2}-\frac12$. Here, $m^2$ is the false vacuum effective mass squared (the second derivative of the potential during the false vacuum inflation) in plank mass units. It is then more direct to express the power spectrum as
$$\begin{aligned}
P_{\chi}(\beta)=&&
\frac{9}{4\pi^2}
\left( \frac{H^3}{V,_\phi} \right)^2
\frac{1}{\beta (\beta^2+1)}
\nonumber\\
&&\times
\left[
\coth(\pi\beta)+
\frac{(\beta^2-1)Re({\cal C}_2)-2\beta Im({\cal C}_2)}
{{\cal C}_1 (\beta^2+1) \sinh(\pi\beta)}
\right]
\nonumber \\\end{aligned}$$
In open inflation, rather than having $k^2\chi$ relating to the density fluctuations, we have $(\beta^2-4K)\chi$ so the primordial power spectrum with be
$$\begin{aligned}
P(\beta)=&&
\frac{(\beta^2+4)^2}{\beta (\beta^2+1)}
\nonumber \\
&&\times
\left[
\coth(\pi\beta)+
\frac{(\beta^2-1)Re({\cal C}_2)-2\beta Im({\cal C}_2)}
{{\cal C}_1 (\beta^2+1) \sinh(\pi\beta)}
\right]
\nonumber \\\end{aligned}$$
compared to the standard [@ZS1]
$$P(\beta)=
\frac{(\beta^2+4)^2}{\beta (\beta^2+1)}$$
Thus, all that is necessary to compare Open Inflation predictions with the standard results is to insert the bracketed term into the initial power spectrum in the cmbopen subroutine of CMBfast[@CMBFast]. The bracketed term quickly approaches unity for $\beta > 1$ (wavelengths smaller than the curvature scale), so it effects the very largest scales with out changing anything on small scales. For concreteness we take a tilt of unity ($n_s = 1$).
Evaluation
==========
To evaluate the CMB anisotropies from Open Inflation, the program CMBfast [@CMBFast] was used to calculate CMB temperature power spectra, and an expression for the bracketed term was inserted into the subroutine cmbopen. As illustrated in Fig. \[fig1\], we explored the dependence of the bracketed term on the false vacuum mass and found that it controlled an oscillation in the bracketed term with respect to $\beta$. We found that $m^2=4.5 M_p^2$ yielded the strongest suppression of power at small wave number (of interest because of the WMAP anomalies).
An important feature to note is that the open power spectrum tends toward zero at very small wave numbers without the bracketed term, leaving only a limited window of wavenumbers for which the bracketed term has any effect. The bracketed term does diverge for most choices of $m^2$ as wavenumber goes to zero, but not fast enough to overcome the rest of the power spectrum. This limits the effect of increasing the curvature on the power spectrum. Also note that, for $m^2=4.5 M_p^2$, the bracketed term does not appear to diverge, but rather tends toward zero with no oscillation.
A wide range of curvatures were tested, comparing power spectra obtained from identical parameters with and without the correction. A best fit for parameters with a prior on $\Omega_{tot}$ not being readily available[^2], the choice of the parameters for these trials is a bit arbitrary. However, the effect of Open Inflation should be independent of all but the curvature scale, and we are comparing spectra that differ only by the inclusion or exclusion of the extra term that distinguishes Open Inflation. Given the small size of the difference the bracketed term generated, which are summarized in Table \[table1\], these concerns are largely unimportant.
$\Omega_{tot}$ % decrease in $C_2$
---------------- ---------------------
.99 $\sim$.01%
.98 .02%
.95 .08%
.90 .20%
.85 .25%
.80 .27%
: This table details the percent decrease in $C_2$, the $l=2$ value of power, caused by including the open inflation corrections with false vacuum mass $m^2=4.5$ in plank units, as this value has the most effect on the primordial power spectrum. The first two entries were done using CMBfast with the best fit parameters given by the WMAP team, with the dark energy density reduced to achieve the stated total density. The rest were done simply using the CMBfast default settings with dark energy reduced. All had no re-ionization. We chose $C_2$ to show here because the effect on the other multipoles was even less significant. The effect on the $C_l$’s is much less dramatic than on the quantities shown in the plots because the $C_l$’s depend on the power at many values of $\beta$, not just at $\beta
\approx 1$ where the effect on the power is most pronounced. The effect of the correction should depend on curvature and mass alone, so the values given will at least approximate those for any model with that curvature. []{data-label="table1"}
Conclusions
===========
The effects of Open Inflation on the CMB power spectrum are very small compared to the cosmic variance for the effected observables (Table 1 shows that the most effected observable, $C_2$, experiences less than a 1% change, while it has a cosmic variance $O(50\%)$). We conclude that the generic form for perturbations from Open Inflation are not distinguishable in the CMB temperature anisotropy power spectrum from perturbations given by the standard formula used throughout the literature. Thus this data cannot be used to identify evidence for or against Open Inflation or measure parameters in other vacuua in the proposed string theory landscape that might be reflected in the Open Inflation primordial spectrum. Also, the general differences between the Open Inflation power spectrum and the standard version are so small that simply choosing between the two will not significantly impact constraints on cosmological parameters from CMB data. However, if our theoretical understanding evolves to the point where specific inflaton potentials are strongly preferred, a greater distinguishability between the two types of inflation might possibly emerge. We note that the power spectrum derived in [@BT2] uses a thin wall approximation that may not be valid in many theories, and the effect of relaxing this assumption is unknown.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge helpful discussions with M. Bucher, N. Kaloper, M. Kaplinghat, L. Knox and especially with L. Susskind who stimulated our interest in this topic. This work was supported in part by DOE grant DE-FG03-91ER40674.
[10]{}
J. Garriga, Phys. Rev. D [**54**]{}, 4764 (1996) \[arXiv:gr-qc/9602025\]. M. Sasaki and T. Tanaka, Phys. Rev. D [**54**]{}, 4705 (1996) \[arXiv:astro-ph/9605104\]. J. Garcia-Bellido and A. R. Liddle, model,” Phys. Rev. D [**55**]{}, 4603 (1997) \[arXiv:astro-ph/9610183\]. models,” Phys. Rev. D [**56**]{}, 616 (1997) \[arXiv:astro-ph/9702174\]. J. Garcia-Bellido, CMB,” Phys. Rev. D [**56**]{}, 3225 (1997) \[arXiv:astro-ph/9702211\]. J. Garriga, X. Montes, M. Sasaki and T. Tanaka, bubble open Nucl. Phys. B [**513**]{}, 343 (1998) \[arXiv:astro-ph/9706229\]. J. Garcia-Bellido, J. Garriga and X. Montes, Phys. Rev. D [**57**]{}, 4669 (1998) \[arXiv:hep-ph/9711214\]. J. Garriga, X. Montes, M. Sasaki and T. Tanaka, universe,” Nucl. Phys. B [**551**]{}, 317 (1999) \[arXiv:astro-ph/9811257\].
J. Garcia-Bellido, J. Garriga and X. Montes, Phys. Rev. D [**60**]{}, 083501 (1999) \[arXiv:hep-ph/9812533\]. A. D. Linde, M. Sasaki and T. Tanaka, Phys. Rev. D [**59**]{}, 123522 (1999) \[arXiv:astro-ph/9901135\]. C. L. Bennett [*et al.*]{}, Astrophys. J. Suppl. [**148**]{}, 1 (2003) \[arXiv:astro-ph/0302207\].
D. H. Lyth and E. D. Stewart, Phys. Lett. B [**252**]{}, 336 (1990). B. Ratra and P. J. E. Peebles, Astrophys. J. [**432**]{}, L5 (1994). S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, Phys. Rev. D [**68**]{}, 046005 (2003) \[arXiv:hep-th/0301240\]. M. Bucher, A Goldhaber, and N. Turok, [*Open universe from inflation*]{}, Phys. Rev. D [**52**]{} 3314 (1995) M. Bucher, and N. Turok, [*Open inflation with an arbitrary false vacuum mass*]{}, Phys. Rev. D [**52**]{} 5538 (1995)
A. H. Guth, Phys. Rev. D [**23**]{}, 347 (1981). A. Linde, Phys. Lett. B [**108**]{}, 389 (1982); A. Albrecht and P. Steinhardt, Phys. Rev. Lett. [**48**]{}, 1220 (1982).
S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L. McAllister and S. P. Trivedi, JCAP [**0310**]{}, 013 (2003) \[arXiv:hep-th/0308055\]. G. Dvali and S. Kachru, arXiv:hep-th/0309095.
J. P. Hsu, R. Kallosh and S. Prokushkin, JCAP [**0312**]{}, 009 (2003) \[arXiv:hep-th/0311077\].
A. Buchel and R. Roiban, Phys. Lett. B [**590**]{}, 284 (2004) \[arXiv:hep-th/0311154\]. H. Firouzjahi and S. H. H. Tye, Phys. Lett. B [**584**]{}, 147 (2004) \[arXiv:hep-th/0312020\]. L. Pilo, A. Riotto and A. Zaffaroni, arXiv:hep-th/0401004. A. Linde, arXiv:hep-th/0402051. M. R. Garousi, M. Sami and S. Tsujikawa, massive scalar field on the arXiv:hep-th/0402075. C. P. Burgess, J. M. Cline, H. Stoica and F. Quevedo, arXiv:hep-th/0403119. M. Berg, M. Haack and B. Kors, in string theory,” arXiv:hep-th/0404087. A. Buchel and A. Ghodsi, arXiv:hep-th/0404151. K. Dasgupta, J. P. Hsu, R. Kallosh, A. Linde and M. Zagermann, arXiv:hep-th/0405247.
G. Hinshaw et. all, astro-ph/0302217. D. N. Spergel et. all, astro-ph/0302209
M. Zaldarriaga, U. Seljak, E. Bertshinger, astro-ph/9704265
Data available at http://lambda.gsfc.nasa.gov M. Zaldarriaga, U. Seljak, CMBFast4.5.1
[^1]: Others have considered inflation in the string theory landscape (see for example [@Kachru:2003sx; @Dvali:2003vv; @Hsu:2003cy; @Buchel:2003qj; @Firouzjahi:2003zy; @Pilo:2004mg; @Linde:2004kg; @Garousi:2004uf; @Burgess:2004kv; @Berg:2004ek; @Buchel:2004qg; @Dasgupta:2004dw]). Linde gives a nice review in sections 11 and 12 of [@Linde:2004kg]. For the most part these authors have sought a sufficiently large amount of new inflation at the onset of the big bang to make the issues raised here unimportant (that is, to make $\Omega_0$ very close to unity). We have no argument against that approach, although in general it seems no easier to arrange potentials for long periods new inflation in the landscape picture than in other frameworks. The motivation for this paper is that models with slightly smaller values of $\Omega_0$ might be considerably more interesting. In fact, any argument that the generic instance of inflation would tend to be short (as some have made in the landscape context) would favor a large effect of the sort we discuss here.
[^2]: Given the overall disappointing nature of our results, it was not worth the effort to undertake a full-scale optimization in a large parameter space
|
---
address: 'Oskar Sultanov, Institute of Mathematics, Ufa Scientific Center, Russian Academy of Sciences, 112, Chernyshevsky str., Ufa, Russia, 450008'
author:
- Oskar Sultanov
title: Capture into parametric autoresonance in the presence of noise
---
Introduction {#introduction .unnumbered}
============
Autoresonance is the phenomenon of continuous phase locking of nonlinear oscillator with slowly varying parametric pumping that leads to significant growth of the energy of the oscillator. This phenomenon was first suggested in the problem of acceleration of relativistic particles [@V44; @M45]. Later, it was observed that autoresonance occurs widely in nature and plays the important role in many problems of nonlinear physics [@FGF00; @FF01; @LF09]. A wide range of applications requires the study of the effect of perturbations on different mathematical models of autoresonance. The influence of additive noise on the capture into the autoresonance (non-parametric) was analysed in [@BFSSh09], where the effect of perturbations was considered only at the initial time. The problem of capture into the parametric autoresonance for a quantum anharmonic oscillator with initial disturbances was discussed in [@BF14]. The effect of persistent perturbations with random jumps on the stability of autoresonance models was investigated in [@OS14]. In this paper we consider the deterministic model of parametric autoresonance [@KM01] and we study the effect of persistent perturbations of white noise type on the captured solutions.
The paper is organized as follows. In section \[sec0\], we give the mathematical formulation of the problem. In section \[sec1\] we discuss the stability of autoresonant solutions with respect to perturbations of initial data. Section \[sec2\] deals with a more general problem of stochastic stability of a class of locally stable dynamical systems. In section \[sec3\] these results are applied in the study of the capture into parametric autoresonance in the presence of stochastic perturbations.
Problem statement {#sec0}
=================
We consider the system of primary parametric autoresonance equations $$\begin{gathered}
\label{eq0}
\frac{dr}{d\tau}= r \sin \psi-\gamma r, \ \ \frac{d\psi}{d\tau}= r-\lambda\tau + \cos \psi, \ \ \tau>0,\end{gathered}$$ where $0<\gamma<1$ and $\lambda>0$ are parameters. This system appears after the averaging of equations, describing the behaviour of nonlinear oscillators in the presence of small slowly changing parametric pumping. The real-valued functions $r(\tau)$ and $\psi(\tau)$ represent the amplitude and phase shift of harmonic oscillations. Solutions with an infinitely growing amplitude $r(\tau)\approx\lambda \tau$ and bounded phase shift $\psi(\tau)=\mathcal O(1)$ as $\tau\to\infty$ correspond to the capture into the parametric autoresonance. The existence and asymptotic behaviour of captured solutions of system were discussed in [@KM01; @AM05; @KG07; @LK08]. In this paper we study the effect of white noise on the stability of such solutions.
The asymptotic solution of system with growing amplitude at infinity can be constructed in the following form $$\begin{gathered}
\label{as}
r(\tau)={\lambda\tau}+\sum_{k=0}^{\infty} r_k \tau^{-k}, \quad \psi(\tau)=\psi_0+\sum_{k=1}^\infty \psi_k \tau^{-k},\end{gathered}$$ where $r_k$ and $\psi_k$ are constant coefficients. Substituting these series in system and grouping the expressions of same power of $\tau$ give the following recurrence relations for determining the coefficients $r_k$, $\psi_k$: $\sin\psi_0=\gamma$, $\psi_1=(\cos\psi_0)^{-1}$, $r_0=-\cos\psi_0$, $r_1=-\tan\psi_0$, etc. Note that there are no free parameters in the asymptotic series. The existence of exact particular solutions of system with constructed asymptotics follows from [@AK89]. The stability of these isolated solutions determines the presence of capture into autoresonance. We show that in the case of stability, solutions with asymptotics attract many other autoresonant solutions with more complicated asymptotic expansions (see Fig. \[Pic1\]). Note that system has also non-resonant solutions with the slipping phase and the bounded amplitude. The existence of such solutions excludes the global stability of autoresonant solutions for all initial data. We also note that the structure of the capture region (the set of initial points such that the corresponding solutions possess the unboundedly growing amplitude) for system remains unknown, and we do not discuss this problem here.
![The evolution of the amplitude $r(\tau)$ and the phase shift $\psi(\tau)$ for the solutions of with $\lambda=1$, $\gamma=0.1$ and different initial data.[]{data-label="Pic1"}](p1.eps "fig:"){width="45.00000%"} ![The evolution of the amplitude $r(\tau)$ and the phase shift $\psi(\tau)$ for the solutions of with $\lambda=1$, $\gamma=0.1$ and different initial data.[]{data-label="Pic1"}](p2.eps "fig:"){width="45.00000%"}
Consider the perturbed system in the form $$\begin{gathered}
\label{eq1}
\frac{dr}{d\tau}= \big[1 + \mu \xi_1(\tau)\big]r \sin \psi-\gamma r, \ \
\frac{d\psi}{d\tau}= r-\lambda\tau + \big[1 + \mu \xi_1(\tau)]\cos \psi+ \mu \xi_2(\tau), \quad \tau>0,\end{gathered}$$ where the stochastic processes $\xi_1(\tau)$ and $\xi_2(\tau)$ defined on a probability space $(\Omega, \mathcal F, \mathbb P)$ play the role of perturbations. It is assumed that $\mathbb E[\xi_i(\tau)]=0$, $\mathbb E[\xi_i(\tau)\xi_j(0)]=\delta_{ij}\sigma^2_i(\tau)\delta(\tau)$ for all $i,j\in\{1,2\}$ and $\tau\geq 0$, where $\delta_{ij}$ is the Kronecker delta, $\delta(\tau)$ is the Dirac delta function, and the deterministic functions $\sigma_i(\tau)$ together with the small parameter $0<\mu<1$ are used to control the intensity of the perturbations. Let $\xi_i(\tau)=\sigma_i(\tau) \dot w_i(\tau)$, where $w_1(\tau)$ and $w_2(\tau)$ are independent Wiener processes. Then we can consider the perturbed system in the form of Itô stochastic differential equations. Our goal is to find constraints on the functions $\sigma_1(\tau)$, $\sigma_2(\tau)$, such that the capture into parametric autoresonance is preserved in the perturbed system with probability tending to one. Since the persistent perturbation of white noise type leads to the loss of stability of solutions for all $\tau>0$ (see [@FV; @RKh; @PRK] and Fig. \[Fig2\]), we consider a weaker problem. Specifically, our goal is to find the largest possible time interval on which the stability of autoresonant solutions is preserved.
![Sample paths of the amplitude $r(\tau)$ for solutions of system with $\lambda=1$, $\gamma=0.1$, $r(0)=1.09$, $\psi(0)=2.15$, $\sigma_1(\tau)\equiv 0$, $\sigma_2(\tau)\equiv 1$ and $\mu\in \{0.1, 0.35, 0.55\}$.[]{data-label="Fig2"}](w4.eps){width="50.00000%"}
Note that system is of universal character in the description of parametric autoresonance in nonlinear systems. It describes long-term evolution of different nonlinear oscillations under small parametric driving. As but one example let us consider the following equation $$\begin{gathered}
\label{eq02}
\frac{d^2u}{dt^2}+\big(1+\varepsilon A(t) \cos 2\Phi(t) \big)\sin u +\vartheta \frac{du}{dt}= 0,\end{gathered}$$ where $A(t)=1+\mu \eta_1(\varepsilon t)$, $\Phi'(t)=1-\alpha t+ \mu \varepsilon \eta_2(\varepsilon t)$, $0<\varepsilon,\alpha,\vartheta,\mu\ll1$. The functions $\eta_1(s)$, $\eta_2(s)$ play the role of perturbations. Solutions of equation with $\mu=0$ whose amplitude increase with time from small values $|u(0)|+|u'(0)|\ll 1$ to quantities of order one are associated with the capture into parametric autoresonance. For the asymptotic description of such solutions at the initial stage of the capture we use the method of two scales. We introduce a slow time $\tau= \varepsilon t/2$ and a fast variable $\phi(t)=t-\alpha t^2/2$. Then the asymptotic substitution $$u(t)= \sqrt{4 \varepsilon r(\tau)}\cos\Big(\frac{\psi(\tau)}{2}+\Phi(t)\Big)+\mathcal O(\varepsilon^{3/2})$$ in equation and the averaging procedure over the fast variable $\phi(t)$ lead to system for the slowly varying functions $r(\tau)$ and $\psi(\tau)$ with $\lambda=8\alpha\varepsilon^{-2}$, $\gamma=2\vartheta \varepsilon^{-1}$, $\xi_1(\tau)=\eta_1(2\tau)$, $\xi_2(\tau)=4 \eta_2(2\tau)$. In the case $\mu=0$, we get system .
Perturbations of initial data for autoresonant solutions {#sec1}
========================================================
Note that the unperturbed system has two different asymptotic solutions in the form distinguished by the choice of a root to the equation $\sin \psi_0=\gamma$. It can easily be checked that the solution with $\psi_0=\arcsin \gamma$ is linearly unstable. However, linear stability analysis fails for the captured solution $r_\ast(\tau)$, $\psi_\ast(\tau)$ with asymptotics , $\psi_0=\pi-\arcsin \gamma$. To study the stability of this solution, we need to take into account high-order terms of the equations. In our analysis we use only the first terms of the asymptotic expansion for the solution, $$\begin{gathered}
r_\ast(\tau)=\lambda \tau+\nu +\mathcal O(\tau^{-1}), \quad \psi_\ast(\tau)=\pi-\arcsin\gamma-(\nu\tau)^{-1}+\mathcal O(\tau^{-2}), \quad \nu:=\sqrt{1-\gamma^2}.
\label{as2}\end{gathered}$$ We have
Suppose that the coefficients of system satisfy the inequalities $\lambda>0$, $0<\gamma<1$. Then there exists $\tau_0>0$ and for all $\varepsilon>0$ there exists $\delta_0>0$ such that for all $(\varrho_0, \varphi_0)$: $(\varrho_0-r_\ast(\tau_0))^2+(\varphi_0-\psi_\ast(\tau_0))^2\leq \delta_0^2$ the solution $r(\tau)$, $\psi(\tau)$ to system with initial data $r(\tau_0)=\varrho_0$, $\psi(\tau_0)=\varphi_0$ satisfies the inequalities $$\begin{gathered}
\label{est}
\sup_{\tau>\tau_0}\Big\{|r(\tau)-r_\ast(\tau)|\tau^{-1/2}\Big\}\leq\varepsilon,\quad \sup_{\tau>\tau_0}\Big\{|\psi(\tau)-\psi_\ast(\tau)|\Big\}\leq\varepsilon.\end{gathered}$$
In system we make the change of variables $r=r_\ast(\tau)+R(\tau)$, $\psi=\psi_\ast(\tau)+\Psi(\tau)$, and for new functions $R(\tau)$, $\Psi(\tau)$ we study the stability of the trivial solution $R(\tau)\equiv 0$, $\Psi(\tau)\equiv 0$ to the following system close to Hamiltonian system $$\begin{gathered}
\label{eq4}
\frac{dR}{d\tau}=-\partial_\Psi H(R,\Psi,\tau)-\gamma R,\quad \frac{d\Psi}{d\tau}=\partial_R H(R,\Psi,\tau),\end{gathered}$$ where $$\begin{aligned}
H(R,\Psi,\tau)& = &\frac{R^2}{2}+\big(R+r_\ast(\tau)\big) \Big[\cos\big(\Psi+\psi_\ast(\tau)\big)-\cos\psi_\ast(\tau)\Big]+\Psi r_\ast(\tau) \sin\psi_\ast(\tau).\end{aligned}$$ By taking into account the asymptotics of the captured solution $r_\ast(\tau)$, $\psi_\ast(\tau)$ one can readily write out the asymptotics of the function $H(R,\Psi,\tau)$ as $\tau\to\infty$ and $d=\sqrt{R^2+\Psi^2}\to 0$: $$H(R,\Psi,\tau)= \frac{\nu\tau\Psi^2}{2}\Big[1+\mathcal O(\Psi)\Big]+\frac{R^2}{2}+\mathcal O(d^3)+\mathcal O(d^2)\mathcal O(\tau^{-1}).$$ The asymptotic estimates are uniform with respect to $(R,\Psi,\tau)$ in the domain $ D(d_1,\tau_1)=\{(R,\Psi,\tau)\in\mathbb R^3: d\leq d_1, \tau\geq \tau_1\}$ with positive constants $d_1>0$ and $\tau_1>1$. It is clear that the function $H(R,\Psi,\tau)$ is positive definite function in the neighbourhood of the equilibrium $(0,0)$. A Lyapunov function candidate for system is constructed of the form $$V(R,\Psi,\tau)=(\nu\tau)^{-1} \Big[ H(R,\Psi,\tau)+ \frac{\gamma R\Psi}{2} \Big].$$ From the properties of the function $H(R,\Psi,\tau)$ it follows that there exist $0<d_0\leq d_1$ and $\tau_0\geq \tau_1$ such that $$\begin{gathered}
\begin{split}
\label{eq5}
\frac{1}{4}\Big[(\nu\tau)^{-1}R^2+\Psi^2\Big]\leq V(R,\Psi,\tau)\leq \frac{3}{4}\Big[(\nu\tau)^{-1}R^2+\Psi^2\Big],\\
\frac{dV}{d\tau}\Big|_{\eqref{eq4}} = \partial_\tau V+\partial_R V [-\partial_\Psi H -\gamma R] +\partial_\psi V \partial_R H \leq \\
\leq -\frac{\gamma}{4} \Big[(\nu \tau)^{-1}R^2+\Psi^2\Big][1+\mathcal O(d)+\mathcal O(\tau^{-1})]\leq -\frac{\gamma}{6} V\leq 0
\end{split}\end{gathered}$$ for all $(R,\Psi,\tau)\in D(d_0,\tau_0)$. Integrating the last expression with respect to $\tau$, we obtain the following estimates $$\begin{gathered}
\frac{1}{4}\Big[(\nu\tau)^{-1} R^2(\tau)+\Psi^2(\tau)\Big]\leq V\big(R(\tau),\Psi(\tau),\tau)\leq V\big(R(\tau_0),\Psi(\tau_0),\tau_0)\leq \frac{3}{4\nu}\Big[R^2(\tau_0)+ \Psi^2(\tau_0)\Big]\end{gathered}$$ as $\tau\geq \tau_0$, where $R(\tau)$, $\Psi(\tau)$ is the solution to system with initial data $R^2(\tau_0)+\Psi^2(\tau_0)\leq \delta^2_0$. Therefore, for all $\varepsilon>0$ $(\varepsilon<d_0)$ there exists $\delta_0=\varepsilon\sqrt{\nu/3}>0$ such that $|R(\tau)|\tau^{-1/2}\leq \varepsilon$ and $|\Psi(\tau)|\leq \varepsilon$ for all $\tau\geq \tau_0$. By means of change of variables we derive the estimates .
Suppose that the coefficients of system satisfy the inequalities $\lambda>0$, $0<\gamma<1$. Then the solution $r_\ast(\tau)$, $\psi_\ast(\tau)$ with asymptotics is the attractor for two-parametric family of captured solutions.
From the inequality for the total derivative of the Lyapunov function $V(R,\Psi,\tau)$ along the trajectories of system it follows that $$0\leq V(R(\tau),\Psi(\tau),\tau)\leq V(R(\tau_0),\Psi(\tau_0),\tau_0)\exp\big(-\gamma(\tau-\tau_0)/6\big)\leq \frac{3d_0^2}{4} \exp\big(-\gamma(\tau-\tau_0)/6\big)$$ as $\tau\geq \tau_0$, where $R^2(\tau_0)+\Psi^2(\tau_0)\leq d^2_0$. The change-of-variables formula implies the following asymptotic estimates for solutions to system with initial data from the $\delta_0$-neighbourhood of the isolated autoresonant solution $r(\tau)=r_\ast(\tau)+\mathcal O(\tau^{1/2}\exp(-\gamma \tau/12))$, $\psi(\tau)=\psi_\ast(\tau)+\mathcal O(\exp(-\gamma \tau/12))$ as $\tau\geq \tau_0$.
Stochastic perturbations of locally stable systems {#sec2}
==================================================
In the study of stochastic perturbations of system and other similar equations with locally stable solutions it is convenient to consider the system of differential equations $$\frac{d{\bf z}}{d t}={\bf f}({\bf z}, t), \quad {\bf z}=({\bf x},{\bf y})=(x_1,\dots,x_l,y_1,\dots,y_m)\in\mathbb R^n, \quad t\geq t_0>1,
\label{2eq1}$$ where $l+m=n$, $1\leq m\leq n-1$ and ${\bf f}(0, t)\equiv 0$. Suppose that the vector-valued function ${\bf f}({\bf z},t)=(f_1({\bf z},t),\dots, f_n({\bf z},t))$ is continuous and for all $T>0$ satisfies a Lipschitz condition: $|{\bf f}({\bf z}_1, t)-{\bf f}({\bf z}_2, t)|\leq M_1 |{\bf z}_1-{\bf z}_2|$ for all ${\bf z}_1, {\bf z}_2\in\mathbb R^n$, $t_0\leq t\leq t_0+T$ with positive constant $M_1$. Assume that there exists a local Lyapunov function $U({\bf z}, t)$ for system satisfying the inequalities: $$\begin{array}{c}
\displaystyle |{\bf x}|^2 + a t^{-b} |{\bf y}|^2\leq U({\bf z}, t)\leq A \Big[|{\bf x}|^2+a t^{-b} |{\bf y}|^2\Big], \quad |\partial_{ {\bf z}} U|^2\leq B U, \quad |\partial_{ z_i}\partial_{z_j} U|\leq C,\\
\displaystyle
\frac{dU}{d t}\Big|_{\eqref{2eq1}}\stackrel{def}{=}\frac{\partial U}{\partial t}+\sum\limits_{k=1}^{n}\frac{\partial U}{\partial z_k} f_k\leq - q U
\end{array}
\label{2eq2}$$ in the domain $\{ ({\bf z},t)\in\mathbb R^{n+1}: |{\bf z}|\leq \rho_0, t\geq t_0\}$ with parameters $A, B, C, q, \rho_0, b>0$, $a\geq 0$. The existence of such Lyapunov function guaranties that the trivial solution ${\bf z}(t)\equiv 0$ is locally stable with respect to variables ${\bf x}=(x_1,\dots,x_l)$ (if $m=0$, the trivial solution is stable with respect to all variables). Note that if $a>0$, then the solution ${\bf z}(t)\equiv 0$ is stable with respect to variables ${\bf y}=(y_1,\dots,y_m)$ in some weighted norm. Let us remark that the Lyapunov function, constructed in the previous section for the system of primary parametric autoresonance, possesses the similar estimates (cp. with ). Note also that such Lyapunov functions are constructed in the stability analysis of nonlinear non-autonomous systems of differential equations (see, for example, [@Vor; @HK02; @LK14]).
Together with system we consider the perturbed system in the form of Itô stochastic differential equations $$\begin{gathered}
d{\bf z}(t)={\bf f}({\bf z}(t), t)\, d t+\mu\, G({\bf z}(t), t)\,d{\bf w}({ t}), \quad {\bf z}(t_0)={\bf z}_0\in\mathbb R^n,
\label{2eq0}
\end{gathered}$$ where ${\bf w}( t)=(w_1( t),\dots,w_n( t))$ is $n$-dimensional Wiener process defined on a probability space $(\Omega,\mathcal F,\mathbb P)$, $G({\bf z}, t)=\{g_{ij}({\bf z}, t)\}_{n\times n}$ is a continuous matrix which is independent of $\omega\in\Omega$ and for all $T>0$ satisfies the following conditions $\|G({\bf z}, t)\|\leq M_2(1+|{\bf z}|)$, $\|G({\bf z}_1, t)-G({\bf z}_2, t)\|\leq M_3 |{\bf z}_1 -{\bf z}_2|$ for all ${\bf z}, {\bf z}_1, {\bf z}_2\in\mathbb R^n$, $t_0\leq t\leq t_0+T$ with positive constants $M_2, M_3>0$. We assume that ${\bf z}_0$ does not depend on $\omega\in\Omega$. These constraints on the coefficients of system guarantee the existence and uniqueness of solution ${\bf z}(t)$ for all $t\geq t_0$ and for all ${\bf z}_0\in\mathbb R^n$ (see, for instance, [@BO98 §5.2], [@RH12 §3.3]). We assume that the perturbed system does not preserve the trivial solution, $G(0,t)\not\equiv 0$. Define the class of perturbations $\mathcal A_h$ as a set of matrices $G({\bf z}, t)$ such that $|\sigma_{ij}({\bf z},t)|\leq h$ for all $|{\bf z}|\leq \rho_0$ and $t\geq t_0$, where $\sigma=G\cdot G^\ast/2=\{\sigma_{ij}\}_{n\times n}$.
We study the stability of the solution ${\bf z}(t)\equiv 0$ of system with respect to stochastic perturbations on a finite time interval. One variant of this approach is to find the largest possible time interval $[t_0; t_0+T_\mu]$ on which solutions to the perturbed system are close to the equilibrium of the deterministic system (see, for instance, [@FV Chap. 9] and [@MH88 Chap. 7]). We have
\[Th2\] Suppose that for system there exists a Lyapunov function $U({\bf z},t)$, possessing estimates . Then, for all $N\in\mathbb N$, $h>0$, $0<\varkappa<1$, $\varepsilon_1, \varepsilon_2 > 0$ there exist $\delta, \Delta>0$ such that $\forall\, \mu<\Delta$, $G\in\mathcal A_h$, ${\bf z}_0 = ({\bf x}_0, {\bf y}_0):$ $|{\bf z}_0|<\delta$ the solution ${\bf z}(t)$ of the unperturbed system with initial data ${\bf z}(t_0)={\bf z}_0$ satisfies the inequalities $$\begin{gathered}
\label{Pest}
\mathbb P\Big(\sup_{t_0 \leq t\leq t_0+ T_\mu}|{\bf x}( t)|\geq \varepsilon_1\Big)\leq \varepsilon_2, \quad \mathbb P\Big(\sup_{t_0 \leq t\leq t_0+ T_\mu} a t^{-b/2}|{\bf y}( t)|\geq \varepsilon_1\Big)\leq \varepsilon_2\end{gathered}$$ with $T_\mu=\mu^{-2N(1-\varkappa)}$.
Let us fix the parameters $h>0$, $0<\varkappa<1$, $\varepsilon_2>0$ and $0<\varepsilon_1<r_0$. Let ${\bf z}( t)$ be a solution of system with $G\in\mathcal A_h$ and initial data ${\bf z}(t_0)={\bf z}_0=({\bf x}_0, {\bf y}_0)$, $|{\bf z}_0|<\delta$, and let $t_{\mathcal D}$ be the first exit time of the solution ${\bf z}( t)$ from the domain $$\mathcal D \stackrel{def}{=}\{({\bf z}, t)\in\mathbb R^{n+1}: |{\bf z}|< \varepsilon_1, \ \ t_0 < t < t_0+ T\}.$$ We define the function $s_t=\min\{ t_{\mathcal D}, t\}$, then ${\bf z}(s_t)$ is the process stopped at first exit time from the domain $\mathcal D$. Positive parameters $\delta$, $T$ will be specified later. Let us first consider the case $N=1$. The Lyapunov function for the stochastic system is constructed in following the form $$U_1({\bf z}, t; T)=U({\bf z}, t)+ \mu^2 h n^2 C \cdot (T+t_0-t).$$ In the study of stability of solutions to stochastic differential equations the following operator plays the role of the total derivative along the trajectories [@RH12 §3.6]: $\mathcal L:=\partial_ t+\sum_{i=1}^n f_i({\bf z}, t) \partial_{z_i} + \mu^2 \sum_{i,j=1}^n \sigma_{ij}({\bf z}, t)\partial_{z_i}\partial_{z_j}.$ It easy to see that $U_1({\bf z},t;T)\geq U({\bf z},t)\geq 0$ and $$\begin{gathered}
\mathcal L U_1 = \frac{dU}{d t}\Big|_{\eqref{2eq1}} + \mu^2\sum_{i,j=1}^n \sigma_{ij} \, \partial_{z_i}\partial_{z_j} U - \mu^2 h n^2 C \leq -q U \leq 0
\end{gathered}$$ for all $({\bf z}, t)\in \mathcal D$. These estimates guarantee that $U_1({\bf z}(s_ t),s_ t)$ is a nonnegative supermartingale [@RH12 §5.2]. Using the properties of the function $U({\bf z},t)$ and Doob’s inequality for supermartingales, we get the following estimates $$\label{2eq6}
\begin{array}{lll}
\displaystyle \mathbb P\Big(\sup_{ t_0\leq t\leq t_0+ T} |{\bf x}( t)|\geq \varepsilon_1\Big)
& = & \displaystyle \mathbb P\Big(\sup_{ t_0\leq t\leq t_0+T} |{\bf x}( t)|^2\geq \varepsilon_1^2\Big) \leq \\
&\leq & \displaystyle \mathbb P\Big(\sup_{t_0\leq t\leq t_0+T} U({\bf z}(t), t)\geq \varepsilon_1^2\Big) \leq \\
& \leq & \displaystyle \mathbb P\Big(\sup_{ t_0\leq t\leq t_0+T} U_1({\bf z}(t), t; T)\geq \varepsilon_1^2\Big) = \\
& = & \displaystyle \mathbb P\Big(\sup_{ t\geq t_0} U_1({\bf z}(s_ t),s_t;T)\geq \varepsilon_1^2\Big) \leq \\
& \leq & \displaystyle \frac{ U_1({\bf z}_0,t_0;T)}{\varepsilon_1^2}\leq \frac{A\big[|{\bf x}_0|^2+a t_0^{-b}|{\bf y}_0|^2\big]+\mu^2 n^2 h C T}{\varepsilon_1^2}.
\end{array}$$ Define $T=\mu^{-2(1-\varkappa)}$ and the parameters $\delta=({\varepsilon_1^2 \varepsilon_2}/{2 A}(1+a))^{1/2}$ and $\Delta=({ \varepsilon_1^2 \varepsilon_2 }/{2\, n^2 h \, C})^{1/2\varkappa}$. Then $A\big[|{\bf x}_0|^2+a t_0^{-b}|{\bf y}_0|^2\big] +\mu^{2\varkappa} n^2 h \, C \leq \varepsilon_1^2 \varepsilon_2$ for all $|{\bf z}_0|<\delta$, $\mu<\Delta$. If $a=0$, this estimate holds for all ${\bf y}_0\in\mathbb R^n$. Taking into account , we obtain the estimate $$\begin{gathered}
\label{2eq7}
\mathbb P(\sup_{t_0\leq t\leq t_0+T}|{\bf x}( t)|\geq \varepsilon_1)\leq \varepsilon_2.
\end{gathered}$$
The stability for $0 \leq t\leq \mu^{-2N(1-\varkappa)}$ is proved by using the Lyapunov function $U_N({\bf z},t;T)$ in the following form [@OS17] $$\begin{aligned}
U_N({\bf z},t;T) & = & \big( U({\bf z},t)\big)^N+\mu^2 a_{N-1} U_{N-1}({\bf z},t;T), \\
U_k({\bf z},t;T) & = & \big( U({\bf z},t)\big)^k+\mu^2 a_{k-1} U_{k-1}({\bf z},t;T), \quad k=2,\dots,N-1,\\
U_1({\bf z}, t; T)& = & U({\bf z}, t)+ \mu^2 n^2 h C \cdot (T+t_0-t),
\end{aligned}$$ where $a_k=(k+1) n^2 h (B+C)q^{-1}$. It is easy to check that the following inequalities hold $$\begin{aligned}
\mathcal L U_1 & \leq & -q U, \\
\mathcal L U_2 & = & 2 U \,\mathcal L U + 2 \mu^2 \sum_{i,j=1}^n \sigma_{ij}\, \partial_{z_i}U\partial_{z_j}U + \mu^2 a_1 \mathcal L U_1 \leq \\
& \leq & - 2 q U^2 + \mu^2 \big(2 n^2 h (B+C) - a_1 q \big) U=-2q U^2, \\
\mathcal L U_3 & \leq & 3 U^2 \,\mathcal L U +6\mu^2 U \sum_{i,j=1}^n \sigma_{ij}\, \partial_{z_i}U\partial_{z_j}U + \mu^2 a_2 \mathcal L U_2 \leq \\
& \leq & -3q U^3+2 \mu^2 \Big(3 n^2 h\, (B+C) - a_2 q \Big) U^2 = - 3q U^3, \\
\mathcal L U_{N} & \leq & N U^{N-1} \,\mathcal L U +N (N-1)\mu^2 U^{N-2} \sum_{i,j=1}^n \sigma_{ij}\, \partial_{z_i}U\partial_{z_j}U + \mu^2 a_{N-1} \mathcal L U_{N-1} \leq \\
& \leq & -N q U^{N}+(N-1) \mu^2 \Big(N n^2 h\, (B+C) - a_{N-1} q \Big) U^{N-1}=- N q U^{N}\leq 0
\end{aligned}$$ for all $({\bf z},t)\in\mathcal D$ and any natural number $N\geq 1$. Since $U_N({\bf z},t;T)\geq \big(U({\bf z},t)\big)^N\geq 0$ for all $({\bf z}, t)\in \mathcal D$, we see that the function $U_N({\bf z}(s_t),s_t;T)$ is a nonnegative supermartingale and the following estimates hold $$\label{2eq8}
\begin{array}{lll}
\displaystyle \mathbb P\Big(\sup_{ t_0\leq t\leq t_0+ T} |{\bf x}( t)|\geq \varepsilon_1\Big)
& = & \displaystyle \mathbb P\Big(\sup_{ t_0\leq t\leq t_0+T} |{\bf x}( t)|^{2N}\geq \varepsilon_1^{2N}\Big) \leq \\
&\leq & \displaystyle \mathbb P\Big(\sup_{t_0\leq t\leq t_0+T} \big(U({\bf z}(t), t)\big)^N\geq \varepsilon_1^{2N}\Big) \leq \\
& \leq & \displaystyle \mathbb P\Big(\sup_{ t_0\leq t\leq t_0+T} U_N({\bf z}(t), t; T)\geq \varepsilon_1^{2N}\Big) = \\
& = &\displaystyle \mathbb P\Big(\sup_{ t\geq t_0} U_N({\bf z}(s_ t),s_t;T)\geq \varepsilon_1^{2N}\Big) \leq \\
& \leq & \displaystyle \frac{ U_N({\bf z}_0,t_0;T)}{\varepsilon_1^{2N}}.
\end{array}$$ Now we define $T=\mu^{-2N(1-\varkappa})$; then $|U_N({\bf z}_0,t_0;T)|\leq M_N\big[|{\bf x}_0|^{2N}+t_0^{-b N}|{\bf y}_0|^{2N}+\mu^{2N\varkappa}\big]$ as $|{\bf z}_0|\to 0$ and $\mu\to 0$ with a positive constant $M_N$. Therefore, for all $\varepsilon_1,\varepsilon_2>0$ there exist $\delta>0, \Delta>0$ such that $U_N({\bf z}_0,t_0;T)\leq \varepsilon_1^{2N} \varepsilon_2$ for all $|{\bf z}_0|< \delta$ and $\mu<\Delta$ (if $a=0$, we can choose $|{\bf x}_0|< \delta$ and ${\bf y}_0\in\mathbb R^n$). Taking into account , we obtain . Thus, for any natural $N$ and for all $h>0$ the trivial solution to system is stable with respect to variables ${\bf x}=(x_1,\dots,x_l)$ under stochastic perturbations on the interval $t_0\leq t\leq t_0+\mu^{-2N(1-\varkappa)}$ uniformly for $G\in\mathcal A_h$. Note that if $a>0$, then from similar arguments it follows that $$\mathbb P\Big( \sup_{t_0\leq t\leq t_0+T} a t^{-b/2}|{\bf y}(t)|\geq \varepsilon_1\Big)\leq \varepsilon_2.$$
Note that if $(1+t)^{-\beta} G\in\mathcal A_h$ with $\beta>0$ and $G\not\in\mathcal A_h$, then it can be proved that the stochastic stability of the trivial solution ${\bf z}(t)\equiv 0$ holds on the time interval $t_0\leq t\leq t_0+\mu^{(-2+\varkappa)/(1+\beta)}$. In this case, the Lyapunov function has the form $U_\beta({\bf z},t)=U({\bf z},t)+\mu^2 M_\beta (T+t_0-t)^{1+\beta}$ with a positive constant $M_\beta>0$.
Stochastic perturbations of stable autoresonant solution {#sec3}
========================================================
In this section we study the stability of the autoresonant solution $r_\ast(\tau)$, $\psi_\ast(\tau)$ with asymptotics under stochastic perturbations. In system we make the change of variables $r=r_\ast(\tau)+R(\tau)$, $\psi=\psi_\ast(\tau)+\Psi(\tau)$; then for the functions $R(\tau)$, $\Psi(\tau)$ we have the following system of stochastic differential equations $$\begin{gathered}
\begin{split}
\label{3eq1}
& dR(\tau)=\big[-\partial_\Psi H(R,\Psi,\tau)-\gamma R\big]d\tau+\mu g_{11}(R,\Psi,\tau) d w_1(\tau), \\
& d\Psi(\tau)= \partial_R H(R,\Psi,\tau) d\tau+\mu g_{21}(\Psi,\tau)dw_1(\tau) + \mu g_{22}(\tau)dw_2(\tau),
\end{split}\end{gathered}$$ where $g_{11}=\sigma_1(\tau)[r_\ast(\tau)+R]\sin(\psi_\ast+\Psi)$, $g_{21}=\sigma_1(\tau)\cos(\psi_\ast+\Psi)$, $g_{22}=\sigma_2(\tau)$. Thus the problem is reduced to the stability analysis of the equilibrium $(0,0)$ of system with respect to stochastic perturbations of the form with matrix $G=\{g_{i,j}(R,\Psi,\tau)\}$. We have
Suppose that the coefficients of system satisfy the inequalities $\lambda>0$, $0<\gamma<1$. Then for all $N\in\mathbb N$, $h, \varepsilon_1, \varepsilon_2 > 0$ there exist $\delta, \Delta>0$ such that $\forall\, \mu<\Delta$, $(\varrho_0, \varphi_0)$[:]{} $(\varrho_0-r_\ast(\tau_0))^2+(\varphi_0-\psi_\ast(\tau_0))^2\leq \delta_0^2$, $(\sigma_1,\sigma_2)$[:]{} $\sup_{\tau>\tau_0}\big\{ |\sigma_1(\tau)|\tau+|\sigma_2(\tau)|\big\}\leq h$ the solution $r_\mu(\tau)$, $\psi_\mu(\tau)$ to system with initial data $r_\mu(\tau_0)=\varrho_0$, $\psi_\mu(\tau_0)=\varphi_0$ satisfies the inequalities $$\begin{gathered}
\label{prest}
\begin{split}
& \mathbb P\Big(\sup_{\tau_0 \leq \tau\leq \tau_0+ \mu^{-N}}|\psi_\mu(\tau)-\psi_\ast(\tau)|\geq \varepsilon_1\Big)\leq \varepsilon_2, \\ & \mathbb P\Big(\sup_{\tau_0 \leq \tau\leq \tau_0+ \mu^{-N}} \tau^{-1/2}|r_\mu(\tau)-r_\ast(\tau)|\geq \varepsilon_1\Big)\leq \varepsilon_2.
\end{split}\end{gathered}$$
Note that system has the Lyapunov function $V(R,\Psi,\tau)$ satisfying with $a=\nu^{-1}$, $b=1$ and $q=\gamma/3$. The restrictions of the coefficients $\sigma_i(\tau)$ imply that $G=\{g_{ij}\}\in\mathcal A_h$. If we combine this with Theorem \[Th2\], we get the stochastic stability of the trivial solution to system and the estimates with ${\bf x}=\Psi$ and ${\bf y}=R$. By means of change of variables we derive the inequalities .
Thus, the stability of the isolated autoresonant solution $r_\ast(\tau)$, $\psi_\ast(\tau)$ is preserved in the perturbed system on asymptotically long time interval $\tau_0\leq \tau\leq \tau_0+\mu^{-N}$, $N\geq 1$. Therefore, the perturbation of white noise type with moderate intensity cannot destroy the stability of the capture into parametric autoresonance.
[99]{}
V. I. Veksler, A new method of acceleration of relativistic particles, J. Phys. USSR, 9, 153–158 (1945)
E. M. McMillan, The synchrotron - a proposed high energy particle accelerator, Phys. Rev., 68, 143–144 (1945)
J. Fajans, E. Gilson, and L. Friedland, Second harmonic autoresonant control of the $l=1$ diocotron mode in pure-electron plasmas, Phys. Rev. E, 62, 4131 (2000)
J. Fajans and L. Friedland, Autoresonant (nonstationary) excitation of pendulums, Plutinos, plasmas, and other nonlinear oscillators, Am. J. Phys., 69, 1096–1102 (2001)
L. Friedland, Scholarpedia 4, 5473 (2009).
I. Barth, L. Friedland, E. Sarid, and A. G. Shagalov, Autoresonant transition in the presence of noise and self-fields, Phys. Rev. Lett., 103, 155001 (2009)
I. Barth and L. Friedland, Quantum phenomena in a chirped parametric anharmonic oscillator, Phys. Rev. Lett., 113, 040403 (2014)
O. A. Sultanov, Stability of autoresonance models subject to random perturbations for systems of nonlinear oscillation equations, Comput. Math. and Math. Phys., 54, 59–73 (2014)
E. Khain and B. Meerson, Parametric autoresonance, Phys. Rev. E, 64, 036619 (2001)
M. Assaf and B. Meerson, Parametric autoresonance in Faraday waves, Phys. Rev. E, 72, 016310 (2005)
O.M. Kiselev and S.G. Glebov, The capture into parametric autoresonance, Nonlinear Dynam., 48, 217–230 (2007)
L. A. Kalyakin, Asymptotic analysis of autoresonance models, Russian Math. Surveys, 63, 791–857 (2008)
A. N. Kuznetsov, Existence of solutions entering at a singular point of an autonomous system having a formal solution, Funct. Anal. Appl., 23, 308–317 (1989)
M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems. New York, Heidelberg, Berlin : Springer-Verlag, 1998.
R. Khasminskii, Stochastic Stability of Differential Equations. Berlin, Heidelberg: Springer-Verlag, 2012.
A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences. Cambridge: Cambridge University Press, 2001.
V. I. Vorotnikov, Partial Stability and Control. Boston, Basel, Berlin : Birkhäuser, 1998.
H. K. Khalil, Nonlinear systems. Prentice Hall, Upper Saddle River, NJ (2002)
L. A. Kalyakin, Synchronization in a nonisochronous nonautonomous system, Theoret. and Math. Phys., 181:2, 1339–1348 (2014)
B. [Ø]{}ksendal, Stochastic Differential Equations. An Introduction with Applications. New York, Heidelberg, Berlin : Springer-Verlag, 1998.
R. Khasminskii, Stochastic Stability of Differential Equations. Berlin, Heidelberg : Springer-Verlag, 2012.
M. M. Hapaev, Averaging in Stability Theory: A Study of Resonance Multi-Frequency Systems. Dordrecht, Boston : Kluwer Academic Publishers, 1993.
O. Sultanov, White noise perturbation of locally stable dynamical systems, Stochastics and Dynamics, 17:1, 1750002 (2017)
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---
abstract: 'In this paper we consider finite-dimensional constrained Hamiltonian systems of polynomial type. In order to compute the complete set of constraints and separate them into the first and second classes we apply the modern algorithmic methods of commutative algebra based on the use of bases. As it is shown, this makes the classical Dirac method fully algorithmic. The underlying algorithm implemented in Maple is presented and some illustrative examples are given.'
author:
- 'Vladimir P. Gerdt'
- 'Soso A. Gogilidze'
title: 'Constrained Hamiltonian Systems and Bases[^1] '
---
Introduction
============
The generalized Hamiltonian formalism invented by Dirac [@Dirac] for constrained systems has become a classical tool for investigation of gauge theories in physics [@GT; @HT; @PS], and a platform for numerical analysis of constrained mechanical systems [@Seiler1]. Finite-dimensional constrained Hamiltonian systems are part of differential algebraic equations whose numerical analysis is of great research interest over last decade [@Campbell] because of importance for many applied areas, for instance, multi-body mechanics and molecular dynamics.
In physics, the constrained systems are mainly of interest for purposes of quantization of gauge theories which play a fundamental role in modern quantum field theory and elementary particle physics. Dirac devised his methods to study constrained Hamiltonian systems just for those quantization purposes. Having this in mind, he classified the constraints in the first and second classes. A first class constrained physical system possesses gauge invariance and its quantization requires gauge fixing whereas a second class constrained system does not need this. The effect of the second class constraints may be reduced to a modification of a naive measure in the path integral. The presence of gauge degrees of freedom (first class constraints) indicates that the general solution of the system depends on arbitrary functions. Hence, the system is underdetermined. To eliminate unphysical gauge degrees of freedom one usually imposes gauge fixing conditions whereas for elimination of other unphysical degrees of freedom occurring because of the second class constraints, one can use the Dirac brackets [@GT; @HT; @Sundermeyer]. In some special cases one can explicitly eliminate the unphysical degrees of freedom [@Soso1].
Unlike physics, where constrained systems are singular, as they contain internal constraints, mechanical systems are usually regular with externally imposed constraints [@Arnold]. Such a system is equivalent to a singular one whose Lagrangian is that of the regular system enlarged with a linear combination of the externally imposed constraints whose coefficients (multipliers) are to be treated as extra dynamical variables. The latter system may reveal extra constraints for the former system providing the consistency of its dynamics.
Therefore, to investigate a constraint Hamiltonian system one has to detect all the constraints involved, and separate them, for physical models, into first and second classes. In his theory [@Dirac] Dirac gave the receipt for computation of constraints which is widely known as [*Dirac algorithm*]{}, and it has been implemented in computer algebra software [@Alain]. However, the Dirac approach, as a method for computation of constraints, is not yet an algorithm. Even computation of the primary constraints, given a singular Lagrangian, is not generally algorithmic. Moreover, in generation of the secondary, tertiary, etc., constraints by the Dirac method one must verify if a certain function of the phase space variables vanishes on the constraint manifold. Generally, the latter problem is algorithmically unsolvable. Similarly, there are no general algorithmic schemes for separation of constraints into the first and second classes. In physical literature one can find quite a number of particular methods developed for the constraint separation (see, for example, [@Lusanna; @Pons]). But all of them have non-algorithmic defects. Thereby, being successfully applied to one constrained system, those methods may be failed for another system even of a similar type.
In practice, many constrained physical and mechanical problems are described by polynomial Lagrangians that lead to polynomial Hamiltonians. In this case, as we show in the present paper, one can apply bases which nowadays have become the most universal algorithmic tool in commutative algebra [@BW] and algebraic geometry [@CLO1; @CLO2]. The combination of the Dirac method with the bases technique makes the former fully algorithmic and, thereby, allows to compute the complete set of constraints. Moreover, the constraint separation is also done algorithmically. We show this and present the underlying algorithm which we call [*algorithm Dirac-Gröbner*]{}. This algorithm has been implemented in Maple V Release 5, and we illustrate it by examples both from physics and mechanics.
Dirac Method
============
In this section we shortly describe the computational aspects of the Dirac approach to constrained finite-dimensional Hamiltonian systems [@Dirac; @HT].
Let us start with a Lagrangian $L(q,\dot{q})\equiv L(q_i,\dot{q}_j)$ $(1\leq i,j\leq n)$ as a function of the generalized coordinates $q_i$ and velocities $\dot{q}_j$[^2]. If the Hessian $\partial^2 L/{\partial \dot{q}_i}{\partial \dot{q}_j}$ has the full rank $r=n$, then the system is [*regular*]{} and it has no internally hidden constraints. Otherwise, if $r<n$, the Euler-Lagrange equations $$\dot{p}_i=\frac{\partial L}{\partial q_i}\quad (1\leq i\leq n) \label{Lag_eq}$$ with $$p_i=\frac{\partial L}{\partial \dot{q}_i} \label{def_p}$$
are [*singular*]{} or [*degenerate*]{}, as not all differential equations (\[Lag\_eq\]) are of the second order. There are just $n-r$ such independent lower order equations. By the Legendre transformation [^3] $$H_c(p,q)=p_iq_i - L \label{def_H_c},$$ we obtain the [*canonical Hamiltonian*]{} with momenta $p_i$ defined in (\[def\_p\]). In the degenerate case there are [*primary constraints*]{} denoted by $\phi_\alpha$, which form the [*primary constraint manifold*]{} denoted by $\Sigma_0$ $$\Sigma_0\ :\quad \phi_\alpha (p,q)=0\quad (1\leq \alpha\leq n-r),
\label{pr_constr}$$ Thus, the dynamics of the system is determined only on the constraint manifold (\[pr\_constr\]). To take this fact into account, Dirac defined the [*total Hamiltonian*]{} $$H_t=H_c+u_\alpha \phi_\alpha \label{def_h_t}$$ with [*multipliers*]{} $u_\alpha$ as arbitrary (non-specified) functions of the coordinates and momenta. The corresponding Hamiltonian equations determine the system dynamics together with the primary constraints $$\dot{q}_i=\{H_t,q_i\},\ \ \dot{p}_i=\{H_t,p_i\},\ \
\phi_\alpha(p,q)=0\ \ (1\leq i\leq n,\ 1\leq \alpha\leq n-r), \label{H_eq}$$ where the [*Poisson brackets*]{} are defined for any two functions $f,g$ of the dynamical variables $p$ and $q$ as follows $$\{f,g\}=\frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i}-
\frac{\partial g}{\partial p_i} \frac{\partial p}{\partial q_i}. \label{p_b}$$ In order to be consistent with the system dynamics, the primary constraints must satisfy the conditions $$\dot{\phi}_\alpha=\{H_t,\phi_\alpha\} \stackrel{\Sigma_0}{=} 0\quad
(1\leq \alpha\leq n-r), \label{cons_cond}$$
where $\stackrel{\Sigma_0}{=}$ stands for the equality, called [*a week equality*]{}, on the primary constraint manifold (\[pr\_constr\]). The Poisson bracket in (\[cons\_cond\]) must be a linear combination of the constraint functions [@HT]. Given a constraint function $\phi_\alpha$, the consistency condition (\[cons\_cond\]), unless it is satisfied identically, may lead either to a contradiction or to a new constraint. The former case signals that the given Hamiltonian system is inconsistent. In the latter case, if the new constraint does not involve any of multipliers $u_\alpha$, it must be added to the constraint set, and, hence, the constraint manifold must involve this new constraint. Otherwise, the consistency condition is considered as defining the multipliers, and the constraint set is not enlarged with it.
The iteration of this consistency check ends up with the [*complete set of constraints*]{} such that for every constraint in the set condition (\[cons\_cond\]) is satisfied. This is the Dirac method of the constraint computation. As shown in [@Seiler2], the method is nothing else than completion of the initial Hamiltonian system to involution, and the constraints generated are just [*the integrability conditions*]{}. For general systems of PDEs, the completion process is done [@Pommaret] by sequential prolongations and projections. For Hamiltonian systems, the time derivative of a constraint is its prolongation whereas projection of the prolonged constraint is realized in (\[cons\_cond\]) by computing the Poisson bracket on the constraint manifold.
Let now $\Sigma$ be the constraint manifold for the complete set of constraints $$\Sigma\ :\quad \phi_\alpha (p,q)=0\quad (1\leq \alpha\leq k).
\label{constr_man}$$
If a constraint function $\phi_\alpha$ satisfies the condition $$\{\phi_\alpha(p,q),\phi_\beta(p,q)\} \stackrel{\Sigma}{=} 0\quad
(1\leq \beta\leq k), \label{fc_cons}$$
it is of [*the first class*]{}. Otherwise, the constraint function is of [*the second class*]{}. The number of the second class constrains is equal to rank of the following $(k\times k)$ [*Poisson bracket matrix*]{}, whose elements must be evaluated on the constraint manifold
$$M_{\alpha \beta} \stackrel{\Sigma}{=} \{\phi_\alpha,\phi_\beta \}.
\label{PBM}$$
Note that matrix $M$ has even rank because of its skew-symmetry.
If a Lagrangian system $L_0(q,\dot{q})$ is regular with externally imposed [*holonomic*]{} constraints $\psi_\alpha(q)=0$, the system is equivalent [@Seiler1] to the singular one with Lagrangian $L=L_0+\lambda_\alpha \phi_\alpha$ and extra generalized coordinates $\lambda_\alpha$. Furthermore, the Dirac method can be applied for finding the other constraints inherent in the initial regular system and, hence, not involving the extra dynamical variables.
Therefore, the problem of constraint computation and separation is reduced to manipulation with functions of the coordinates and momenta on the constraint manifold. Generally, there is no algorithmic way for such a manipulation. However, for polynomial functions all the related computations can be done algorithmically by means of bases, as we show in the next section.
Algorithm Description
=====================
Here we describe an algorithm which, given a polynomial Lagrangian whose coefficients are rational numbers, computes the complete set of constraints and separates them into the first and second classes. The algorithm combines the above described Dirac method with the bases technique. By this reason we call it algorithm Dirac-Gröbner. All the below used concepts, definitions and constructive methods related to bases are explained, for instance, in textbooks [@BW; @CLO1; @CLO2].
At first we present the algorithm under assumption that a polynomial ideal generated by constraints is radical. This is true for most of real practical problems. Next, we indicate how to modify the algorithm to treat the most general (non-radical) case. 0.3cm
**Algorithm Dirac-**
0.2cm [**Input:**]{} $L(q,\dot{q})$, a polynomial Lagrangian $(L\in Q[q,\dot{q}])$ 0.1cm [**Output:**]{} $\Phi_1$ and $\Phi_2$, sets of the first and second class constraints, respectively.
1. Computation of the canonical Hamiltonian and primary constraints: 0.1cm
1. Construct the polynomial set $F=\cup_{i=1}^n
\{p_i-\partial L/\partial \dot{q}_i\}$ in variables $p,q,\dot{q}$.
2. Compute the basis $G$ of the ideal in ring $Q[p,q,\dot{q}]$ generated by $F$ with respect to an ordering[^4] which eliminates $\dot{q}$. Then compute the canonical Hamiltonian as the normal form of (\[def\_H\_c\]) modulo $G$.
3. Find the set $\Phi$ of primary constraint polynomials as $G\cap Q[p,q]$. If $\Phi=\emptyset$, then stop since the system is regular. Otherwise, go to the next step.
0.2cm
2. Computation of the complete set of constraints:
0.1cm
1. Take $G=\Phi$ for the basis $G$ of the ideal generated by $\Phi$ in $Q[p,q]$ with respect to the ordering induced by that chosen at Step 1(b). Fix this ordering in the sequel.
2. Construct the total Hamiltonian in form (\[def\_h\_t\]) with multipliers $u_\alpha$ treated as symbolic constants (parameters).
3. For every element $\phi_\alpha$ in $\Phi$ compute the normal form $h$ of the Poisson bracket $\{H_t,\phi_\alpha\}$ modulo $G$. If $h\neq 0$ and no multipliers $u_\beta$ occur in it, then enlarge set $\Phi$ with $h$, and compute the basis $G$ for the enlarged set.
4. If $G=\{1\}$, stop because the system is inconsistent. Otherwise, repeat the previous step until the consistency condition (\[cons\_cond\]) is satisfied for every element in $\Phi$ irrespective of multipliers $u_\alpha$. This gives the complete set of constraints $\Phi=\{\phi_1,\ldots,\phi_k\}$.
0.2cm
3. Separation of constraints into first and second classes:
0.1cm
1. Construct matrix $M$ in (\[PBM\]) by computing the normal forms of its elements modulo $G$, and determine rank $r$ of $M$. If $r=k$, stop with $\Phi_1=\emptyset$, $\Phi_2=\Phi$. If $r=0$, stop with $\Phi_1=\Phi$ and $\Phi_2=\emptyset$. Otherwise, go to the next step.
2. Find a basis $A=\{a_1,\ldots,a_{k-r}\}$ of the null space (kernel) of the linear transformation defined by $M$. For every vector $a$ in $A$ construct a first class constraint as $a_\alpha \phi_\alpha$. Collect them in set $\Phi_1$.
3. Construct $(k-r)\times k$ matrix $(a_j)_\alpha$ from components of vectors in $A$ and find a basis $B=\{b_1,\ldots,b_{r}\}$ of the null space of the corresponding linear transformation. For every vector $b$ in $B$ construct a second class constraint as $b_\alpha \phi_\alpha$. Collect them in set $\Phi_2$.
The correctness of Steps 1, 2 and 3(a) of the algorithm is provided by the properties of bases [@BW; @CLO1; @CLO2] and by the following facts: (i) the definition (\[def\_H\_c\]) of the canonical Hamiltonian implies its independence of $\dot{q}$ on the primary constraint manifold (\[pr\_constr\]); (ii) whenever a multiplier $u_\alpha$ in (\[def\_h\_t\]) is differentiated when the Poisson bracket in (\[cons\_cond\]) is evaluated, the corresponding term vanishes on the constraint manifold. The correctness of Steps 3(b) and 3(c) follows from definition (\[fc\_cons\]) of the first class constraints and the correctness of Step 3(a). The termination of algorithm Dirac-Gröbner follows from the finiteness of the basis $G$ which is constructed at Step 2(c).
Now consider the most general case when the constraints obtained from (\[cons\_cond\]) lead to a non-radical ideal. It should be noted that the ideal generated by the primary constraint polynomials (Step 1) is always radical. This is provided by linearity of (\[def\_p\]) in momenta. However, already the first secondary constraint added may destroy this property of the ideal. Therefore, the algorithm needs one more step, namely, Step 2(e), where the basis $G$ of the radical ideal for the polynomial set $\Phi$ is computed. Next, every constraint polynomial in $\Phi$ is replaced by its normal form modulo $G$. All the elements with zero normal forms are eliminated from the set. The extra step is also algorithmic. There are algorithms for construction of a basis, and, hence, a basis, of the radical of a given ideal, which are built-in in some computer algebra systems (see [@BW; @CLO1; @CLO2] for more details and references). One can also check the radical membership of $h$ at Step 2(c) before its adding to $\Phi$. This check is easily done [@BW; @CLO1], but in any case Step 2, for the correctness of Step 3, must end up with the radical sets $\Phi$ and $G$.
We implemented algorithm Dirac-Gröbner, as it presented above for the radical case, in Maple V Release 5. The implementation is relied on the built-in system facilities for computation and manipulation with bases and for linear algebra. Using our Maple code for different examples from physics and mechanics, we experimentally observed that in those infrequent cases when the constraint ideals are non-radical this can easily be detected from the structure of the output set.
Examples
========
In this section we illustrate, by examples from physics and mechanics, the application of algorithm Dirac-Gröbner.
$SU(2)$ Yang-Mills mechanics in $0+1$ dimensional space-time [@Soso1]. This is a constrained physical model with gauge symmetry. The model Lagrangian is given by $L=\frac{1}{2}(D_t)_i(D_t)_i$, $(D_tx)_i=\dot{x}_i+g\epsilon_{ijk}y_jx_k$ $(1\leq i,j,k\leq 3)$. Here $x_i$ and $y_i$ are the generalized coordinates and tensor $\epsilon_{ijk}$ is anti-symmetric in its indices with $\epsilon_{123}=1$. Respectively, the primary constraints and the canonical Hamiltonian are $p_i^y=0$ and $H_c=\frac{1}{2}-\epsilon_{ijk}x_jp_ky_i$ with the momenta given by $p_i^y=\partial L/\partial \dot{y}_i$ and $p_i=\partial L/\partial \dot{x}_i$. The other constraints in the complete set computed by the algorithm are $\phi_i=\epsilon_{ijk}x_jp_k=0$, and all the six constraints found are of the first class.
Point particle of mass $m$ moving on the surface of a sphere (rigid rotator). The movement is described by the regular Lagrangian $L_0=\frac{1}{2}m^2(\dot{q_1}^2+\dot{q_2}^2+\dot{q_3}^2)/2
\equiv \frac{1}{2}m^2\dot{q}^2$ with the externally imposed holonomic constraint $\phi(q)=q^2-1=0$. This system is equivalent to the singular Lagrangian system $ L=L_0 + \lambda \phi$, where $\lambda$ is an extra coordinate. There is the only primary constraint $p_\lambda=0$ $(p_\lambda=\partial L/\partial \lambda)$, and the canonical Hamiltonian is $H_c=\frac{1}{2}m^2p^2-\lambda \phi(q)$ $(p_i=\partial L/\partial q_i)$. The complete set of constraint polynomials for the singular system contains four second class polynomials $\{p_\lambda,\phi(q),p_iq_i,2m\lambda + p^2\}$. Coming back to the initial regular system, the first and the last polynomials in the set must be omitted since they determine the extra dynamical variables.
Singular physical system with both first and second class constraints[^5]. The system Lagrangian is $L=q_1(\dot{q}_2-q_3) - \dot{q}_1q_2$. There are three primary constraint polynomials $\{p_1+q_2,p_2-q_1,p_3\}$. The canonical Hamiltonian is $H_c=q_1q_2$. One more constraint polynomial $q_1$ is found by the Dirac-Gröbner algorithm. The sets $\Phi_1$ and $\Phi_2$ of the first and second classes are $\{p_2+q_1,p_3\}$ and $\{p_1+q_2,q_1\}$, respectively. Note that this system has no physical degrees of freedom (c.f. [@Seiler2]).
Inconsistent singular system [@PS]: $L=\frac{1}{2}\dot{q}_1^2+q_2$. There is the single primary constraint $p_2=0$. The canonical Hamiltonian is $H_c=p_1^2/2-q_2$. At Step 2(c) of algorithm Dirac-Gröbner the inconsistency $\dot{p}_2=1$ occurs. The algorithm detects this inconsistency and stops.
The above examples are rather small and can be treated by hand. With our Maple code we have already tried successfully much more nontrivial examples. For instance, we computed and separated the constraints for the $SU(2)$ Yang-Mills mechanics in $3+1$ dimensional space-time [@Soso1]. Surprisingly, this computation took only a few seconds on an Pentium 100 personal computer though the model Lagrangian and the canonical Hamiltonian are rather cumbersome polynomials of the 4th degree in 21 variables.
[99]{}
Dirac, P.A.M.: Generalized Hamiltonian Dynamics. [*Canad. J. Math.*]{} [**2**]{} (1950), 129-148; [*Lectures on Quantum Mechanics*]{}, Belfer Graduate School of Science, Monographs Series, Yeshiva University, New York, 1964.
Gitman, D.M., Tyutin, I.V.: [*Quantization of Fields with Constraints*]{}, Springer-Verlag, Bonn, 1990.
Henneaux, M., Teitelboim, C.: [*Quantization of Gauge Systems*]{}, Princeton University Press, Princeton, New Jersey, 1992.
Prokhorov, L.V., Shabanov, S.V.: [*Hamiltonian Mechanics of Gauge Systems*]{}, St. Petersburg University, 1997 (in Russian).
Seiler, W.M.: Numerical Integration of Constrained Hamiltonian Systems Using Dirac Brackets. [*Math. Comp.*]{} [**68**]{} (1999) 661-681.
Brenan, K.E., Campbell, S.L., Petzold, L.R.: [ *Numerical Solution of Intial-Value Problems in Differential-Algebraic Equations*]{}, Classics in Applied Mathematics [**14**]{}, SIAM, Philadelphia, 1996.
Sundermeyer, K.: [*Constrained Dynamics*]{}, Lecture Notes in Physics [**169**]{}, Springer-Verlag, New York, Berlin, 1982.
Gogilidze, S.A., Khvedelidze, A.M., Mladenov, D.M., Pavel, H.-P.: Hamiltonian Reduction of $SU(2)$ Dirac-Yang-Mills Mechanics, [*Phys. Rev.*]{} [**D57**]{} (1998) 7488-7500.
Arnold, V.I.: [*Mathematical Methods of Classical Mechanics*]{}, Graduate Texts in Mathematics [**60**]{}, Springer-Verlag, New York, 1978.
Tombal, Ph., Moussiaux, A.: MACSYMA Computation of the Dirac-Bergman Algorithms for Hamiltonian Systems with Constraints. [*J. Symb. Comp.*]{} [**1**]{} (1985) 419-421.
Chaichian, M., Martinez, D.L., Lusanna, L.: Dirac’s Constrained Systems: The Classification of Second Class Constraints. [*Ann. Phys. $($N.Y.$)$*]{} [**232**]{} (1994) 40-60.
Battle, C., Comis, J., Pons, J.M., Roman-Roy, N.: Equivalence Between the Lagrangian and Hamiltonian Formalism for Constrained Systems. [*J. Math. Phys.*]{} [**27**]{} (1986) 2953-2962.
Becker, T., Weispfenning, V., Kredel, H.: [*Gröbner Bases. A Computational Approach to Commutative Algebra*]{}, Graduate Texts in Mathematics [**141**]{}, Springer-Verlag, New York, 1993.
Cox, D., Little, J., O’Shea, D.: [*Ideals, Varieties and Algorithms*]{}, 2nd Edition, Springer-Verlag, New York, 1996.
Cox, D., Little, J., O’Shea, D.: [*Using Algebraic Geometry*]{}, Graduate Texts in Mathematics [**185**]{}, Springer-Verlag, New York, 1998.
Seiler, W.M., Tucker, R.W.: Involution and Constrained Dynamics. [*J. Phys. A.*]{} [**28**]{} (1995) 4431-4451.
Pommaret, J.F.: [*Partial Differential Equations and Group Theory. New Perspectives for Applications*]{}, Kluwer, Dordrecht, 1994.
[^1]: This work was supported in part by Russian Foundation for Basic Research, grant No. 98-01-00101.
[^2]: We consider only autonomous systems, and there is no loss of generality since time $t$ may be treated as an additional variable.
[^3]: In this paper summation over repeated indices is we always assumed.
[^4]: An elimination ordering which induced the degree-reverse-lexicographical one for monomials in $p$ and $q$ is heuristically best for efficiency reasons.
[^5]: A.Burnel. Private communication.
|
---
abstract: 'Let $V$ be a symplectic vector space over a finite or local field. We compute the character of the Weil representation of the metaplectic group $\Mp(V)$. The final formulas are overtly free of choices (e.g. they do not involve the usual choice of a Lagrangian subspace of $V$). Along the way, in results similar to those of K. Maktouf, we relate the character to the Weil index of a certain quadratic form, which may be understood as a Maslov index. This relation also expresses the character as the pullback of a certain simple function from $\Mp(V\oplus V)$.'
author:
- Teruji Thomas
date: 1 January 2007
title: The Character of the Weil Representation
---
Introduction
============
Let $F$ be a field of characteristic not $2$: it may be the real numbers $\RRR$, the complex numbers $\CCC$, a non-archimedean local field, or a finite field. Fix a non-trivial additive character $\psi\colon F\to\CCC^\times$. Let $V$ be a vector space over $F$ with symplectic form $\<-\,,-\>$.
The symplectic group $\Sp(V)$ has a well known projective representation $\rho$, depending on $\psi$, called the ‘Weil representation’ after A. Weil’s seminal paper [@We]. When $F$ is finite or complex, $\rho$ can be realized as a true representation of $\Sp(V)$, and in the other cases as a true representation of a two-fold cover $\Mp(V)$ of $\Sp(V)$. In this article we compute the characters $\Tr\rho$ of these representations.
The standard constructions of $\rho$ involve a choice: for example, that of a Lagrangian subspace $\l\subset V$. Our goal, more precisely, is to present formulas for $\Tr\rho$ completely free of such choices.
Formulation of the Main Results.
--------------------------------
### {#first}
First suppose $F$ is finite of cardinality $|F|$, so $\Tr\rho$ is a function on $\Sp(V)$.
Given $g\in\Sp(V)$, the endomorphism $(g-1)\in\operatorname{End}(V)$ plays a key role in the formula for $\Tr\rho(g)$ (cf. [@GH] §2.1 and [@Howe] p. 294). Let us denote by $\sigma_g$ the induced isomorphism $$\sigma_g\colon V/\ker(g-1)\overset\sim\too (g-1)V.$$ It is easy to check that $v\otimes w\mapsto\<\sigma_g v,w\>$ defines a nondegenerate bilinear form on $V/\ker(g-1)$. Let $\det\sigma_g\in F^\times/(F^\times)^2$ be its discriminant (see §\[orient\]); if $(g-1)$ is invertible then $\det\sigma_g$ is just the usual determinant $\det(g-1)$ mod $(F^\times)^2$.
The second ingredient we need is the ‘Weil index,’ a character $\gamma$ of the Witt group $W(F)$ of quadratic forms over $F$ (see §\[WittWeil\] and [@We], [@Pe]). If $a\in F^\times$ then denote by $\gamma(a)$ the value of $\gamma$ on the one-dimensional quadratic form $x\mapsto ax^2$. In this finite field case, $$\gamma(a)=|F|^{-1/2}\sum_{x\in F}\psi(\tfrac12ax^2).$$ It depends only on $a\bmod(F^\times)^2.$
\[THM1A\] If $F$ is a finite field, then $$\Tr\rho(g)=|F|^{\tfrac12\dim\ker(g-1)}\gamma(1)^{\dim V-\dim\ker(g-1)-1}\gamma(\det\sigma_g).$$
Restricted to the case where $\ker(g-1)=0$, an equivalent formula has been independently obtained by S. Gurevich and R. Hadani [@GH], as a corollary to their algebraic-geometric approach to the Weil representation over a finite field.
\[remHowe\] R. Howe [@Howe] understood many aspects of Theorem 1A without, apparently, finding a closed formula. For example, one can determine the absolute value of $\Tr\rho(g)$ from the fact that $\rho\otimes\rho^*$ is the natural action of $\Sp(V)$ on $L^2(V)$.
\[remchiit\] The values of $\gamma$ depend on $\psi$. For $a\in F^\times$, $\gamma(a)$ is calculated for standard choices of $\psi$ in the appendix of [@Pe]. It is well known that, in this finite field case, $\chi\colon a\mapsto \gamma(-1)\gamma(a)$ is the unique non-trivial character of $F^\times/(F^\times)^2\cong\ZZZ/2\ZZZ$ (see e.g. [@Bu], Exercise 4.1.14), and consequently $\gamma(a)\gamma(b)=\gamma(1)\gamma(ab)$. With this in mind one can re-write Theorem 1A in various ways, for example as$$\Tr\rho(g)=|F|^{\tfrac12\dim\ker(g-1)}\gamma(1)^{\dim V-\dim\ker(g-1)}\chi(\det\sigma_g).$$ In §\[example\] we consider in detail what happens when $\dim V=2$.
### {#section}
Now suppose $F=\CCC$ is the field of complex numbers. Then $\Tr\rho$ is defined as a generalized function on $\Sp(V)$ (see §\[char\]). It is known by the work of Harish Chandra to be (represented by) a locally integrable function, smooth (i.e. $C^\infty$) on the open set of regular semisimple elements. Let $$\label{Sp''}\Sp(V)'':=\{g\in\Sp(V)\,|\,\det(g-1)\neq0\}.$$ In [@Ma], p. 293, it is shown that $\Sp(V)''$ contains the regular semisimple elements, but it obviously contains much more.
\[THM1B\] If $F=\CCC$ then $\Tr\rho$ is smooth on $\Sp(V)''$, and indeed $$\Tr\rho(g)=\dens{(g-1)}{-1}.$$
This statement appears as Theorem 1 in part II of [@Torasso]. Perhaps it is worth remarking that $\gamma\equiv 1$ when $F=\CCC$.
### {#section-1}
Finally, suppose $F$ is the field of real numbers or else a non-archimedean local field. Now $\rho$ is a representation not of $\Sp(V)$ but of a double cover $\Mp(V)$. A standard construction of $\Mp(V)$ is recalled in §\[Meta\]. Let $\pi\colon \Mp(V)\to\Sp(V)$ be the projection and $$\label{DMp''}\Mp(V)'':=\pi\inv(\Sp(V)'')=\{\tilde g\in\Mp(V)\,|\,\det(\pi(\tilde g)-1)\neq0\}.$$
The character $\Tr\rho$ is a generalized function on $\Mp(V)$, again known to be represented by a locally integrable function, smooth on the set of regular semisimple elements. Here $\tilde g\in\Mp(V)$ is said to be regular semisimple if $\pi(\tilde g)$ itself is; ‘smooth’ means ‘$C^\infty$’ in the real and ‘locally constant’ in the non-archimedean cases.
If $\tilde 1\in\Mp(V)$ is the non-identity element over $1\in\Sp(V)$, then $\rho(\tilde 1)=-1$. Thus, given $\tilde g\in\Mp(V)$, $\Tr\rho(\tilde g)$ is determined up to sign by $g:=\pi(\tilde g)\in \Sp(V)$.
\[THM1C\] If $F$ is real or non-archimedean, then $\Tr\rho$ is smooth on $\Mp(V)''$, and given up to sign by $$\Tr\rho(\tilde g)=\pm\frac{\gamma(1)^{\dim V-1}\gamma(\det(g-1))}{\dens{\(g-1\)}{1/2}}.$$
Various aspects of $\Tr\rho$ were previously understood, including some formulas (see e.g. [@Ad], [@Howe], [@Ma], [@Torasso], as well as Remark \[remHowe\] and §\[mak1\]). What seems to be new in Theorem 1C is that it is overtly independent of choices, as explained at the beginning of this article. The right-hand side is also easy to compute using the values of $\gamma$ from [@Pe].
The Character (Without Sign Ambiguity) via the Maslov Index.
------------------------------------------------------------
In Theorem 2, stated in this section and proved beginning in §\[overview\], we express $\Tr\rho$ in terms of the Maslov index $\tau$ (see §\[RMI\] and [@LV],[@Th]). Alternatively, as we explain in §\[zowee\], Theorem 2 can be understood to describe $\Tr\rho$ as the pullback of a simple function from a larger metaplectic group. Either way, this computes $\Tr\rho$ with no sign ambiguity, in contrast to Theorem 1C; but to get a number, one must choose a Lagrangian subspace of $V$.
We deduce Theorem 1 from Theorem 2 in §\[implication\].
### {#evaluate}
To give a uniform approach, define $\Mp(V)$, when $F$ is complex or finite, as the trivial extension of $\Sp(V)$ by $\ZZZ/2\ZZZ$. In general, we we will use the following description of $\Mp(V)$ taken from [@LV],[@Ma],[@Pe].
Let $\Grass(V)$ be the set of Lagrangian subspaces in $V$. An element of $\Mp(V)$ is a pair $(g,t)$ with $g\in\Sp(V)$ and $t$ a function $t\colon \Grass(V)\to \CCC^\times$ satisfying certain conditions (we will recall more details in §\[Meta\]).
\[splitting\] This description makes sense over any of our fields. When $F=\CCC$ the splitting of $\Mp(V)\to\Sp(V)$ is given by $g\mapsto(g,1)$. However, when $F$ is finite, the splitting is more complicated (see Proposition \[split\]).
### {#ee}
Fix $\l\in\Grass(V)$. Let $\overline V$ be $V$ equipped with minus the given symplectic form. Let $\Gamma_g$ be the graph of $g\colon\overline V\to V$, so $\Gamma_1$ is the diagonal in $\overline V\oplus V$. Then $\Gamma_g$, $\Gamma_1$, and $\l\oplus \l$ are all Lagrangian subspaces of $\overline V\oplus V$. For $(g,t)\in\Mp(V)$ set $$\label{eqTheta2}
\Theta_\l(g,t):=t(\l)\cdot\gamma(\tau(\Gamma_g,\Gamma_1,\l\oplus \l))$$ where $\tau$ is the Maslov index. In §\[constancy\] we prove
\[indep\] $\Theta_\l$ is independent of $\l$ and locally constant on $\Mp(V)''$.
$\Mp(V)''$ was defined by formula .
Neither factor in the definition of $\Theta_\l$ is itself continuous on $\Mp(V)''$, and both depend on the choice of $\l$. A more canonical description of $\Theta_\l$ is given in §\[zowee\].
\[thm2A\] Suppose $F$ is infinite. Then $$\Tr\rho(g,t)=\left\|\det(g-1)\right\|^{-1/2}\cdot \Theta_\l(g,t)$$ where $\|\cdot\|$ denotes the usual norm when $F$ is real or archimedean, and the square of the usual norm when $F=\CCC$.
If $F=\CCC$ then $\gamma\equiv1$ so $\Theta_\l(g,1)=1$. Thus Theorem 1B already follows from Theorem 2A and Remark \[splitting\].
\[thm2b\] Suppose $F$ is finite. Then $$\Tr\rho(g,t)=|F|^{\tfrac12\dim\ker(g-1)}\cdot \Theta_\l(g,t).$$
### An Explicit Quadratic Form. {#defq1}
In §\[theform\] we describe a canonical quadratic space $(S'\gl,q'\gl)$ representing the Maslov index $\tau(\Gamma_g,\Gamma_1,\l\oplus\l)$ that appears in . For $g\in\Sp(V)''$ the answer is particularly simple (and this is the only case needed for Theorem 2A):
For any fixed $\l\in\Grass(V)$ and any $g\in\Sp(V)''$ define $$\label{eqq1}q'\gl(a,b)=\<a,(g-1)\inv b\>$$ for all $a,b\in \l$. Then $q'\gl$ is a symmetric bilinear form on $\l$, and $S'\gl$ is defined to be the quotient of $\l$ on which $q'\gl$ is nondegenerate. (The symmetry of $q'\gl$ is explained in Proposition \[aresym\].)
### Relation to the Work of Maktouf. {#mak1}
K. Maktouf [@Ma] proved a theorem in the $p$-adic case very similar to Theorem 2A. As explained in §\[maktouf\], our formulas are identical when $g$ is semisimple and $\l$ is appropriately chosen—in general, Maktouf’s version of $\Theta_\l$ (denoted $\Phi$ in [@Ma]), and even $\l$ itself, is constructed from the semisimple part of $g$. Since the semisimple elements are dense in $\Mp(V)$, one can deduce the $p$-adic Theorem 2A from the main theorem in [@Ma] and Proposition \[indep\]. However, our proof is different and in some sense more direct.
The Character as the Pullback of a Function from $\Mp(\overline V\oplus V)$. {#zowee}
----------------------------------------------------------------------------
We use the notation of §\[evaluate\]–§\[ee\]. Let $f$ be the embedding $f\colon\Sp(V)\to\Sp(\overline V\oplus V)$ defined by $f(g)=(1,g)$. There is a unique[^1] homomorphic embedding $\tilde f\colon\Mp(V)\to\Mp(\overline V\oplus V)$ covering $f$, described explicitly in §\[constancy\]. Consider the function $\mathrm{ev}_{\Gamma_1}$ on $\Mp(\overline V\oplus V)$ defined by $(g',t')\mapsto t'(\Gamma_1)$.
\[zow\] $\Theta_\l(g,t)=\mathrm{ev}_{\Gamma_1}\circ \tilde f$.
Along with Theorem 2, this gives another description of $\Tr\rho$. The proof of Proposition \[zow\] is given in §\[constancy\], where we use it to deduce Proposition \[indep\].
I would like to thank J. Adams for inspiring this work, and him and R. Kottwitz for some useful comments. I am also grateful to the referee for a careful reading.
Example: $\SL_2$ over a Finite Field {#example}
====================================
Suppose $F$ is a finite field (but the infinite case is similar); let $V=F^2$ with basis $\{e_1,e_2\}$ such that $\<e_1,e_2\>=1$. Then $\Sp(V)=\SL_2(F)$. Suppose $g=\(\begin{smallmatrix} a&b\\c&d\end{smallmatrix}\),$ with $ad-bc=1$. Then $\det(g-1)=2-a-d$. With similar calculations, and using Theorem 1A and Remark \[remchiit\], one finds:
1. If $a+d\neq2$ then $\Tr\rho(g)=\gamma(1)^2\chi(2-a-d)=\chi(a+d-2)$.
2. If $a+d=2$, $b\neq 0$, then $\det\sigma_g=b\bmod(F^\times)^2$, $\Tr\rho(g)=|F|^{1/2}\gamma(1)\chi(b)$.
3. If $a+d=2$, $c\neq 0$, then $\det\sigma_g=-c\bmod(F^\times)^2$, $\Tr\rho(g)=|F|^{1/2}\gamma(1)\chi(-c)$.
4. If $a+d=2,$ $b=c=0$, then $g=1$, $\det\sigma_g=1\bmod(F^\times)^2$, $\Tr\rho(g)=|F|$.
Let us test these formulas on some standard elements of $\Sp(V)$. The results may be verified using explicit formulas for $\rho$, like those in [@Bu], Exercises 4.1.13–4.1.14.
1. $\Tr\rho\(\begin{smallmatrix} a&b\\0&1/a\end{smallmatrix}\)=\chi(a)$ if $a\neq 1$ (note that $a+a\inv-2\equiv a\bmod(F^\times)^2$).
2. $\Tr\rho\(\begin{smallmatrix} 1&b\\0&1\end{smallmatrix}\)=|F|^{1/2}\gamma(1)\chi(b)$ if $b\neq 0$.
3. $\Tr\rho\(\begin{smallmatrix} 0&1\\-1&0\end{smallmatrix}\)=\chi(-2)$. In case $|F|=p$ prime, Quadratic Reciprocity implies that $\chi(-2)=1$ if $p\equiv1,3\bmod 8$, and $\chi(-2)=-1$ if $p\equiv5,7\bmod 8$.
Recollections I: Weil Index
===========================
We call a [*quadratic space*]{} a vector space equipped with a nondegenerate symmetric bilinear form. Let $W(F)$ be the Witt group (or ring, but we are only interested in the additive structure) of quadratic spaces over $F$ (see [@Lam]).
\[WittWeil\]
Definition and Properties. {#sg}
--------------------------
The next proposition/definition is due to Weil [@We] (see also [@LV] p. 111). If $(A,q)$ is a quadratic space, we write $f_{q}$ for the function $x\mapsto\psi(\tfrac12q(x,x))$ on $A$; $dq$ is the self-dual measure on $A$, so that the Fourier transform $(f_{q}\,dq)^\wedge$ is a generalized function on $A^*$; and $q^*$ is the dual quadratic form on $A^*$. That is, $q^*(x,y):=\<x,\Phi^{-1}(y)\>$ where $\Phi\colon A\to A^*$ is the isomorphism $a\mapsto q(a,-)$.
\[pd\]There exists a number $\gamma(q)$ such that $$\label{eqg}
(f_{q}\,dq)^\wedge=\gamma(q)\cdot f_{-q^*}$$ as generalized functions on $A^*$. Moreover, $(A,q)\mapsto\gamma(q)$ is a (unitary) character $W(F)\to\CCC^\times$. If $a\in F^\times$ then denote by $\gamma(a)$ the value of $\gamma$ on the one-dimensional quadratic form $x\mapsto ax^2$. Then:
1. For $a\in F^\times$, $\gamma(a)$ depends only on $a\bmod(F^\times)^2$.
2. $\gamma(a)\gamma(b)=\pm\gamma(1) \gamma(ab)$ for all $a,b\in F^\times$, with a plus if $F$ is finite or complex.
3. For a quadratic space $(A,q)$, $\gamma(q)=\pm\gamma(1)^{\dim A-1}\gamma(\det q)$, with a plus when $F$ is finite or complex; here $\det q$ is the discriminant of $q$.
To be precise, in (ii) the sign is the Hilbert symbol $(a,b)$: $(a,b)=1$ if $a$ is a norm from $F[\sqrt b]$, and $(a,b)=-1$ otherwise. In (iii) the sign is the Hasse invariant of $q$.
Integral Formulas.
------------------
First suppose $F$ is finite. We allow here the case when $q$ may be degenerate on $A$. Then to define $\gamma(q)$ one should consider $q$ as a nondegenerate form on $A/\ker q$.
\[pg1\] Let $q$ be a possibly degenerate quadratic form on $A$. Then $$\gamma(q)=|F|^{-\tfrac12(\dim A/\ker q)-\dim\ker q}\sum_{x\in A} \psi(\tfrac12q(x,x)).$$
If $\ker q=0$ then the right side is just $(f_q\,dq)^\wedge$, evaluated at $0$. In general, use that $f_q$ is constant along the cosets of $\ker q$ in $A$.
### {#Dirac}
Now we treat $F$ infinite. Let $h$ be a Schwartz function on $A$ such that $h(0)=1$ and such that the Fourier transform $h^\wedge$ is a positive measure. For $s\in F$ set $h_s=h(sx)$; then the Fourier transform $h_s^\wedge$ approaches a delta-measure as $s\to 0$.
\[pg\] Suppose $F$ is infinite and $q$ nondegenerate. For $h$ as above, $$\gamma(q)=\lim_{s\to 0}\int_{x\in A} h_s(x)\cdot \psi(\tfrac12q(x,x))\,dq.$$
Certainly $\displaystyle\int_{x\in A} h_s(x)\cdot\psi(\tfrac12q(x,x))\,dq=
\displaystyle\int_{x\in A^*} h_s^\wedge(x)\cdot(f_q\,dq)^\wedge(x).$ Now apply and take the limit $s\to0$.
Recollections II: Maslov Index {#RMI}
==============================
Let $U$ be a symplectic vector space (in our applications, $U=V$ or $U=\overline V\oplus V$).
Basic Properties. {#mi}
-----------------
For any $n>0$, the Maslov index is a function $$\tau\colon\Grass(U)^n\to W(F),$$ invariant under the diagonal action of $\Sp(U)$ on $\Grass(U)^n$. We recall the following properties (for which see [@Th], [@KS]):
1. Dihedral symmetry: $$\tau(\l_1,\ldots,\l_n)=-\tau(\l_n,\ldots,\l_1)=\tau(\l_n,\l_1,\ldots,\l_{n-1}).$$
2. Chain condition: For any $j$, $1<j<n$, $$\tau(\l_1,\l_2,\ldots, \l_j)+\tau(\l_1,\l_j,\ldots,\l_n)=\tau(\l_1,\l_2,\ldots,\l_n).$$
3. Additivity: If $U,U'$ are symplectic spaces, $\l_1,\ldots,\l_n\in\Grass(U),$ $\l'_1,\ldots,\l'_n\in\Grass(U')$, so that $\l_i\oplus \l_i'\in\Grass(U\oplus U')$, we have $$\tau(\l_1\oplus \l_1',\ldots, \l_n\oplus \l_n')=\tau(\l_1,\ldots,\l_n)+\tau(\l_1',\ldots,\l_n').$$
Rank and Discriminant.
----------------------
In [@Th] we constructed a canonical quadratic space representing $\tau(\l_1,\ldots,\l_n)$. We now will give formulas for the rank (stated in [@Th]) and the discriminant (essentially found in [@Pe]). For the latter we need the notion of an oriented Lagrangian.
### Orientations and Discriminants. {#orient}
An oriented vector space is a pair $(A,o)$ where $A$ is a vector space and $o$ is an element of $\det A$, consided up to multiplication by $(F^\times)^2$. Write $\ooo(A)$ for the set of all orientations on $A$.
1. For any subspace $B\subset A$, the wedge product gives a natural map $$\wedge\colon \ooo(B)\times_{F^\times}\ooo(A/B)\to\ooo(A).$$
2. For any perfect pairing $\beta\colon A\otimes A'\to F$, there is also a pairing $$\beta\colon\ooo(A)\times_{F^\times}\ooo(A')\to F^\times/(F^\times)^2.$$
Namely, if $\{e_1,\ldots,e_m\}$ is a basis for $A$ and $\{e_1^*,\ldots,e_m^*\}$ is the dual basis for $A'$, then $\beta(e_1\wedge\cdots\wedge e_m,e^*_1\wedge\cdots\wedge e_m^*)=1$.
In case $A=A'$, the number $\beta(o,o)\in F^\times/(F^\times)^2$ is independent of the orientation $o$ on $A$, and is called the [*discriminant*]{} of $\beta$. (If $A=0$, then the discriminant is defined to be $1$.)
### {#DA}
Suppose $(\l_1,o_1)$ and $(\l_2,o_2)$ are Lagrangians with orientations. Choose any orientation $o$ of $\l_1\cap\l_2$. For $i=1,2$ there is a unique orientation $\bar o_i$ of $\l_i/\l_1\cap\l_2$ such that $o\wedge\bar o_i=o_i$. The symplectic form induces a perfect pairing $(\l_1/\l_1\cap\l_2)\otimes (\l_2/\l_1\cap\l_2)\to F$. Set $$O\lll:=\<\bar o_1,\bar o_2\>.$$ It is independent of $o$ but obviously depends on $o_1,o_2$.
### {#section-2}
From [@Th] in conjunction with the calculations in [@Pe] §1.6.1, we obtain
\[rankdisc\] The Maslov index $\tau(\l_1,\ldots,\l_n)$ can be represented by a canonically defined quadratic space $(T,q)$ with dimension $$\dim T=\frac{n-2}{2}\dim V-\sum_{i\in\ZZZ/n\ZZZ}\dim (\l_i\cap \l_{i+1})+2\dim \bigcap_{i\in\ZZZ/n\ZZZ} \l_i$$ and discriminant $$\det {q}=
(-1)^{\tfrac12\dim V+\dim\bigcap_{i\in\ZZZ/n\ZZZ} \l_i}\prod_{i\in\ZZZ/n\ZZZ} O_{\l_i,\l_{i+1}}$$ for arbitrarily chosen orientations on $\l_1,\ldots,\l_n$.
Weil Index.
-----------
Fix an arbitrary orientation on each $\l_i$. Define $$\label{eqm}m(\l_i,\l_{i+1}):=
\gamma(1)^{\tfrac12\dim V-\dim\l_i\cap \l_{i+1}-1}\gamma(O_{\l_{i},\l_{i+1}}).$$ Applying Propositions \[rankdisc\] and \[pd\], one deduces as in [@Pe]
\[cor:m\] $\gamma(\tau(\l_1,\ldots,\l_n))=\pm \prod_{i\in\ZZZ/n\ZZZ}m(\l_i,\l_{i+1})$, with a plus when $F$ is finite or complex.
On the right side of Corollary \[cor:m\], the factors $m(\l_i,\l_{i+1})$ depend on the orientations on $\l_i,\l_{i+1}$, but the overall product does not.
\[remA\] Regarding the compatibility of Corollary \[cor:m\] with §\[mi\](i,ii,iii) and the symplectic invariance of $\tau$, one can easily check $$\label{minv}\begin{aligned}
&m(g\l_i,g\l_{i+1})=m(\l_i,\l_{i+1})=\pm m(\l_{i+1},\l_i)\inv\\
&m(\l_i\oplus\l'_i,\l_{i+1}\oplus\l'_{i+1})=\pm m(\l_i,\l_{i+1})\cdot m(\l'_i,\l'_{i+1})\end{aligned}$$ with plusses as usual if $F$ is finite or complex.
Recollections III: Metaplectic Group {#Meta}
====================================
In this section we recall the explicit model of the metaplectic group developed in [@Pe],[@LV]. Again let $U$ be a symplectic vector space; in practice $U=V$ or $U=\overline V\oplus V$.
The Maslov Cocycle.
-------------------
Fix $\l\in\Grass(U)$. The properties of the Maslov index recalled in §\[mi\](i,ii) imply that the function $$\label{cgh}
(g,h)\mapsto c_{g,h}(\l):=\gamma(\tau(\l,g\l,gh\l))$$ is a $2$-cocycle on $\Sp(U)$ with values in $\CCC^\times$. Corollary \[cor:m\] implies that one can change this cocycle by a coboundary to take values in $\{\pm 1\}\subset\CCC^\times$. The metaplectic group $\Mp(U)$ is defined to be the corresponding central extension of $\Sp(U)$ by $\{\pm1\}$. Let us now make this construction explicit.
Definition of the Metaplectic Group. {#constr}
------------------------------------
Observe that, in the notation of , $$\label{eqs}
m_g(\l):=m(g\l,\l)=\gamma(1)^{\tfrac12\dim U-\dim g\l\cap\l-1}\gamma(O\gll)$$ is independent of the choice of orientation on $\l$, if we give $g\l$ the same orientation transported by $g$.
The metaplectic group $\Mp(U)$ consists of pairs of the form $(g,\pm t_g)$ with $g\in\Sp(U)$ and $t_g\colon\Grass(U)\to\CCC$ any function satisfying the following conditions:
1. $t_g(\l)^2=m_g(\l)^2$ and
2. $t_g(\l')=\gamma(\tau(\l,g\l,g\l',\l'))\cdot t_g(\l)$ for any $\l,\l'\in\Grass(U)$.
Multiplication in $\Mp(U)$ is given by $$\label{multiply}(g,s)\cdot(h,t)=(gh,s t c_{g,h})$$ where $c_{g,h}$ is defined by .
This particular form of the construction appears in [@LP] and apparently goes back to M. Duflo.
Properties.
-----------
\[exactlytwo\] $\Mp(U)$ is a two-fold cover of $\Sp(U)$, i.e. there are exactly two functions $t_g\colon\Grass(U)\to\CCC$ satisfying conditions (i) and (ii). In case $F$ is finite or complex, we may take $t_g=\pm m_g$.
There are clearly zero or two such functions. To show they do exist, one has to verify the following two facts for all $\l,\l',\l''\in\Grass(U)$.
1. $m_g(\l')=\pm \gamma(\tau(\l,g\l,g\l',\l'))\cdot m_g(\l)$ with a plus if $F$ is finite or complex.
2. $\tau(\l,g\l,g\l',\l')+\tau(\l',g\l',g\l'',\l'')=\tau(\l,g\l,g\l'',\l'').$
Use Corollary \[cor:m\] and property to prove (a). As for (b), §\[mi\](i,ii) imply $$\tau(\l,g\l,g\l',\l')+\tau(\l',g\l',g\l'',\l'')=\tau(\l,g\l,g\l'',\l'')+\tau(g\l,g\l',g\l'')+\tau(\l,\l'',\l').$$ But the last two terms cancel by §\[mi\](i) and the symplectic invariance of $\tau$.
\[split\] If $F$ is finite or complex then $g\mapsto(g,m_g)$ is a group homomorphism splitting the projection $\Mp(U)\to\Sp(U)$. If $F$ is complex then $m_g\equiv 1$.
For the first statement we must check that $m_gm_hc_{gh}=m_{gh}$. This is immediate from , , and Corollary \[cor:m\]. The second statement is obvious since $\gamma\equiv 1$ when $F=\CCC$.
\[splitrem\] The splitting of Proposition \[split\] is known to be unique except when $|F|=3$ and $\dim U=2$.
\[topology\] The topology on $\Mp(U)$ is determined by the fact that $\pi\colon\Mp(U)\to\Sp(U)$ is a local homeomorphism, and by the following property. For each $\l\in\Grass(U)$ and $n\geq 0$, set $N_{\l,n}:=\{g\in\Sp(U)\,|\,\dim \l\cap g\l=n\}$. Then the function $\mathrm{ev}_\l\colon(g,t)\mapsto t(\l)$ is locally constant over each $N_{\l,n}$.
That the property holds is implicit in [@LV], §1.9.11. To determine the topology on the group, it suffices to specify a neighbourhood $N$ of a single point $(g,t)$ such that $\pi$ is injective on $N$. We can choose $g$ to lie in the open set $N_{\l,0}$, and take $N=\{(h,s)\,|\, h\in N_{\l,0},\, \mathrm{ev}_\l(h,s)=\mathrm{ev}_\l(g,t)\}$.
Proof of Propositions \[indep\] and \[zow\] {#constancy}
===========================================
We use the notation of §\[zowee\]. To define $\tilde f$ explicitly, write $\tilde f(g,t)=((1,g),f_g(t))$, where $f_g(t)\colon\Grass(\overline V\oplus V)\to\CCC$ is determined by §\[constr\](ii) and the condition $$f_g(t)(\l\oplus\l):=t(\l)$$ for any fixed $\l\in\Grass(V)$.
\[embed\] The map $\tilde f$ is a homomorphic embedding and is independent of $\l$.
According to §\[constr\](ii), $\tilde f$ is independent of $\l$ as long as $$\tau(\l\oplus\l,\l\oplus g\l,\l'\oplus g\l',\l'\oplus\l')=\tau(\l,g\l,g\l',\l'),$$which holds by §\[mi\](iii) and the fact that $\tau(\l,\l,\l',\l')=0$. Indeed, $\tau(\l,\l,\l',\l')$ is represented by a quadratic space of dimension zero, by Proposition \[rankdisc\].
According to , $\tilde f$ is a homomorphism if $c_{(1,g),(1,h)}(\l\oplus\l)=c_{g,h}(\l)$, that is, if $$\tau(\l\oplus\l,\l\oplus g\l,\l\oplus gh\l)=\tau(\l,g\l,gh\l).$$ This again follows from §\[mi\](iii) and Proposition \[rankdisc\].
Finally, $\tilde f$ is obviously injective. To show it is continuous, it is enough to notice that for every $\l\in\Grass(V)$ and every $n\geq 0$, $f$ maps $N_{\l,n}$ into $N_{\l\oplus\l,n+\dim \l}$; as a result, the subspace topology on $\Mp(V)\subset\Mp(\overline V\oplus V)$ satisfies Proposition \[topology\].
We have $$\mathrm{ev}_{\Gamma_1}\circ\tilde f:=f_g(t)(\Gamma_1)=f_g(t)(\l\oplus\l)\cdot\gamma(\tau(\l\oplus\l,\l\oplus g\l,\Gamma_g,\Gamma_1))$$ by §\[constr\](ii). Since $f_g(t)(\l\oplus\l)=t(\l)$, it remains only to check $$\label{expando}
\tau(\Gamma_g,\Gamma_1,\l\oplus\l)=\tau(\l\oplus\l,\l\oplus g\l,\Gamma_g,\Gamma_1).$$ By §\[mi\](i,ii) the difference between the two sides is $\tau(\Gamma_g,\l\oplus\l,\l\oplus g\l)$. This vanishes by Proposition \[rankdisc\]: it is represented by a quadratic space of dimension zero.
We note that $f_g(t)(\Gamma_1)$ is independent of $\l$. Moreover, by Proposition \[topology\], $(g',t')\mapsto t'(\Gamma_1)$ is locally constant for $g'$ in the open set $N_{\Gamma_1,0}.$ Therefore $(g,t)\mapsto f_g(t)(\Gamma_1)$ is locally constant for $(1,g)\in N_{\Gamma_1,0}$, or, equivalently, for $g\in\Sp(V)''$.
Deduction of Theorem 1 from Theorem 2 {#implication}
=====================================
To deduce Theorem 1 from Theorem 2, we have to check $$\label{tbc}\Theta_\l(g,t)\overset?=\pm\gamma(1)^{\dim V-\dim\ker(g-1)-1}\gamma(\det\sigma_g)$$ with a plus when $F$ is finite or complex and we use the splitting of Proposition \[split\].
According to Proposition \[zow\], $\Theta_\l(g,t)=f_g(t)(\Gamma_1)$; but, by §\[constr\](i), we have $f_g(t)(\Gamma_1)=\pm m_{(1,g)}(\Gamma_1)=\pm m(\Gamma_g,\Gamma_1)$, with plusses when $F$ is finite or complex. Here we must choose the orientations on $\Gamma_1$ and $\Gamma_g$ to be related by the isomorphism $(1,g)$. By definition of $m(\Gamma_g,\Gamma_1)$, it remains to prove
With orientations chosen as above, $O_{\Gamma_g,\Gamma_1}=\det\sigma_g\bmod(F^\times)^2$.
Consider the isomorphisms $\Gamma_g \cong V \cong \Gamma_1$ defined by $(x,gx) \mapsto x \mapsto (x,x)$. The hypothesis is just that the orientations of $\Gamma_g$ and $\Gamma_1$ correspond to the same orientation of $V$. These same isomorphisms induce $$\Gamma_g/\Gamma_g\cap\Gamma_1\cong V/\ker(g-1)\cong \Gamma_1/\Gamma_g\cap\Gamma_1.$$ The symplectic pairing $(\Gamma_g/\Gamma_g\cap\Gamma_1)\otimes(\Gamma_1/\Gamma_g\cap\Gamma_1)\to F$, used to define $O_{\Gamma_g,\Gamma_1}$ in §\[DA\], induces the pairing $x\otimes y\mapsto\<\sigma_g(y),x\>$ on $V/\ker(g-1)\otimes V/\ker(g-1).$ But $\det\sigma_g$ is by its definition in §\[first\] the discriminant of this bilinear form.
The proof above is closely related to the following proposition, which is an adaptation of Proposition \[rankdisc\].
\[rts\] There is a quadratic space $(T,q)$ representing $\tau(\Gamma_g,\Gamma_1,\l\oplus \l)$ with rank $$\dim T=\tfrac12\dim V-\dim\ker(g-1)-\dim g\l\cap \l+2\dim\l\cap\ker(g-1)$$ and discriminant $$\det {q}=(-1)^{\dim\l\cap(g-1)\l}O\gll\det\sigma_g.$$
Formula then follows, alternatively, from Proposition \[pd\](iii).
Overview of the Proof of Theorem 2 {#overview}
==================================
The idea used to prove Theorem 2 is to represent the operator $\rho(g,t)$ by an integral kernel and then to compute the trace as the integral along the diagonal (this works quite literally over a finite field).
Integral Operators.
-------------------
Recall that for each $\l\in\Grass(V)$ the representation $\rho$ is realized in a Hilbert space completing the space of Schwartz sections $\SSS(\HHH_\l)$ of a certain sheaf (complex line bundle) $\HHH_\l$ on $V/\l$. We recall how to define $\HHH_\l$ in §\[defH\]. Let $\HHH_\l^\vee$ be the ‘dual’ sheaf such that a section of $\HHH_\l^\vee\otimes\HHH_\l$ is a measure on $V/\l$.
We define in §\[secQ\] a generalized section $K\lll$ of $\HHH_{\l_1}^\vee\boxtimes\HHH_{\l_2}$ on $V/\l_1\times V/\l_2$. In [@Th] we proved that the convolution $$\FF_{21}\colon\SSS(\HHH_{\l_1})\to\SSS(\HHH_{\l_2})\qquad f\mapsto f*K\lll$$ is the operator defined in [@LV], meaning that, for any $\l_1,\ldots,\l_n\in\Grass(V)$, we have $$\label{compos}
\FF_{1n}\circ\FF_{n(n-1)}\circ\cdots\circ\FF_{21}=\gamma(-\tau(\l_1,\ldots,\l_n))$$ and we can realize $\rho$ in the following way.
Let $\alpha_g:V/\l\times V/\l\to V/g\l\times V/\l,$ $\alpha_g(x,y)=(gx,y). $ Then $$\label{rho1}
\rho(g,t)\colon\SSS(\HHH_{\l})\to\SSS(\HHH_{\l})\qquad\mbox{is}\qquad f\mapsto t(\l)\cdot f * \alpha_g^*K\gll.$$
\[pullback\] The expression $\alpha_g^*K\gll$ should be understood as follows: the sheaves $\HHH_{\l_1}^\vee\boxtimes\HHH_{\l_2}$, for varying $\l_1,\l_2$, define a sheaf on $$\{(\l_1,\l_2;w_1,w_2)\,|\,\l_i\in\Grass(V),\,w_i\in V/\l_i\}$$ with a natural $\Sp(V)\times\Sp(V)$-equivariant structure. So $\alpha_g^*K\gll$ is a generalized section of $\HHH_{\l}^\vee\boxtimes\HHH_{\l}$ on $V/\l\times V/\l$.
Restriction to the Diagonal.
----------------------------
Now let $\Delta\colon V/\l\to V/\l\times V/\l$ be the diagonal, so $\Delta^*(\HHH_{\l}^\vee\boxtimes\HHH_{\l})=\HHH_{\l}^\vee\otimes\HHH_{\l}$ is the sheaf of measures on $V/\l$. Naively, it would follow from that $$\label{bad}\mbox{``}\Tr\rho(g,t)=t(\l)\cdot\int_{V/\l}\Delta^*\alpha_g^*K\gll.\mbox{''}$$ This is quite correct when $F$ is finite; let us first consider that case. In doing so, we identify measures and functions on finite sets, using counting measure as a standard.
### {#finconc}
We define in §\[theform\] (and for any $F$) a quadratic space $(S\gl,q\gl)$ whose class in $W(F)$ is $\tau(\Gamma_g,\Gamma_1,\l\oplus\l)$. By definition, $S\gl$ is a quotient of a certain subspace $\hat S\gl\subset V/\l$ (see §\[hatS\]).
\[finrel\] Suppose $F$ is finite. Fix $g\in\Sp(V)$. Then $\Delta^*\alpha^*K\gll$ is supported on $\hat S\gl$, and, as functions there, $$\Delta^*\alpha^*K\gll(x)=\psi(\tfrac12q\gl(x,x))\cdot |F|^{-\tfrac12\dim \l/g\l\cap\l}.$$
Theorem 2B now follows almost immediately from and Proposition \[pg1\]—the details, and the proof of Proposition \[finrel\], are worked out in §\[finitefield\].
### {#section-3}
Now suppose $F$ is infinite. We henceforth restrict ourselves, as we are entitled, to $g$ in the dense open set $$\label{Spl}
\Sp(V)^\l:=\{g\in\Sp(V)''\,|\, g\l\cap \l=0\}$$ with $\Sp(V)''$ defined by . From the definition of $K\gll$ (more precisely, from formula ) one knows
\[smooth\] The generalized section $(g,x,y)\mapsto\alpha^*K\gll(x,y)$ is smooth when restricted to $\Sp(V)^\l\times V/\l\times V/\l.$
Thus the restriction $\Delta^*\alpha_g^*K\gll$ to the diagonal is a well defined and smooth measure on $V/\l$. Let $dq\gl$ be the self-dual measure on $S\gl$. In §\[restrict\] we prove the following analogue of Proposition \[finrel\].
\[rel\] Fix $g\in\Sp(V)^\l$. Then $S\gl=V/\l$, and, as measures on $V/\l$, $$\begin{aligned}
\Delta^*\alpha^*K\gll(x)&=\psi(\tfrac12q\gl(x,x))\cdot\left\|\det(g-1)\right\|^{-1/2}\, dq\gl.\end{aligned}$$
Following §\[finconc\], we should apply the correct infinite versions of and Proposition \[pg1\]. These are given by Lemma \[pt\] and Proposition \[pg\], and a little analysis in §\[proof\] completes the proof of Theorem 2A.
\#1\#2[\^\_[\#1]{}$#2$]{} \#1\#2[\_[\#1]{}$#2$]{}
Definition of the Sheaf $\HHH_\l$ and the Convolution Kernel $K\lll$ {#lb}
====================================================================
We must first fix some notation.
Conventions on Measures and Densities. {#hd}
--------------------------------------
For $\alpha\in\RRR$, the space of $\alpha$-densities on an $F$-vector space $X$ is defined to be the one-dimensional $\RRR$-vector space $$\OR\alpha{X}=\{\nu\colon\det X\to\RRR\,|\,\nu(\lambda x)=|\lambda|^\alpha \nu(x),\,\forall x\in\det X,\,\lambda\in F\}.$$
We identify $\OR{1}{X}$ with the space of real invariant measures on $X$: $\nu\in\OR{1}{X}$ corresponds to the invariant measure that assigns to $\{a_1v_1+\cdots+a_kv_k\,|\,a_i\in F, |a_i|\leq 1\}$ the volume $\nu(v_1\wedge\ldots\wedge v_k)$, for any basis $v_1,\ldots,v_k$ of $X$.
### {#OR:aut}
An isomorphism $f\colon X\to Y$ induces an isomorphism $\OR{\alpha}{X}\to\OR{\alpha}{Y}$, such that $g\in\GL(X)$ acts on $\nu\in\OR{\alpha}{X}$ by $g\cdot\nu=\left\|\det g\right\|^{-\alpha}\nu$, with $\left\|\cdot\right\|$ defined in Theorem 2A.
We can identify $\OR{\alpha}{X}\otimes\OR{\beta}{X}=\OR{\alpha+\beta}{X}$ and $\OR{-\alpha}{X}=\OR{\alpha}{X}^*=\OR{\alpha}{X^*}.$ If $Y\subset X$ then one can identify $\OR{\alpha}{X}=\OR{\alpha}{Y}\otimes\OR\alpha{X/Y}$.
### {#section-4}
Set $\OC\alpha{X}:=\OR\alpha{X}\otimes_\RRR\CCC$. Then §\[OR:aut\] works for $\Omega_{\alpha}$ as well as for $\Omega^\RRR_\alpha$.
### {#sqrt}
For $\nu\in\OR{1}{X}$ a positive measure, define $\nu^{1/2}\in\OR{1/2}{X}$ by $\nu^{1/2}(x):=\left|\nu(x)\right|^{1/2}.$ Then $\nu^{1/2}\otimes\nu^{1/2}=\nu$ using $\OR{1/2}{X}\otimes\OR{1/2}{X}=\OR{1}{X}$.
Definition of $\HHH_\l$. {#defH}
------------------------
Let $\psi$ be our fixed additive character. For open $U\subset V/\l$ let $\tilde U$ be its pre-image in $V$. Let $\HHH_\l$ be the sheaf on $V/\l$ such that a smooth section $f$ over $U$ is a smooth function $\tilde f\colon \tilde U\to\OC{1/2}{V/\l}$ satisfying the condition $$\tilde f(v+a)=\psi(\tfrac12\<v,a\>)\cdot \tilde f(v)\qquad\forall v\in V,a\in \l.$$
Definition of $K\lll$. {#gs}
----------------------
\[secQ\] A generalized section $K\lll$ of $\HHH_{\l_1}^\vee\boxtimes\HHH_{\l_2}$ is ‘the same’ as a generalized function $\tilde K\lll\colon V\times V\to
\OC{1/2}{V/\l_1}\otimes \OC{1/2}{V/\l_2}$ satisfying the $\l\times \l$-equivariance condition $$\tilde K\lll(v+a,w+b)=\psi(\tfrac12\<a,v\>)\cdot \tilde K\lll(v,w)\cdot\psi(\tfrac12\<w,b\>)$$ for any $a\in \l_1$ and $b\in \l_2$.
Our $K\lll$ will be smooth on its support $$\label{suppK}
\operatorname{support}K\lll=\{(x,y)\in V/\l_1\times V/\l_2\,|\,x-y\in \l_1+\l_2\}.$$ In other words, $\operatorname{support}\tilde K\lll$ will be the $\l_1\times \l_2$-invariant subspace $T\lll\subset V\times V$: $$\label{eqT}T\lll:=\{(x,y)\in V\times V\,|\,x-y\in \l_1+\l_2\}=\ker\left[V\times V\overset\partial\too V/(\l_1+\l_2)\right].$$ $\tilde K\lll$ is constructed from the following quadratic form $Q\lll$ on $T\lll$.
### {#defQ}
Given $(x,y)\in T\lll$ write $x-y=a_1+a_2$ with $a_i\in \l_i$. Then $$\label{Q}Q\lll((x,y)):=\<a_1,x\>+\<a_2,y\>.$$
\[lemQ\]
1. $Q\lll((x,y))$ is independent of the choice of $a_1,a_2$.
2. For $b_i\in \l_i$, $Q\lll((x+b_1,y+b_2))=Q\lll((x,y))+\<b_1,x\>+\<y,b_2\>.$
For (i), if $x-y=(a_1+\epsilon)+(a_2-\epsilon)$, with $\epsilon\in \l_1\cap \l_2$, then $$\<a_1+\epsilon,x\>+\<a_2-\epsilon,y\>=\<a_1,x\>+\<a_2,y\>+\<\epsilon,x-y\>$$ but $\<\epsilon,x-y\>=0$ since $x-y\in\l_1+\l_2$. The second statement is similarly easy.
### Definition of $\tilde K\lll$. {#defK}
Let $\delta\lll$ be the extension-by-zero of the constant function $1$ from $T\lll$ to $V\times V$. It is naturally a generalized function on $V\times V$ with values in $\OC{1}{(V\times V)/T\lll}=\OC{1}{V/(\l_1+\l_2)}$.
Let $\mu\lll$ be the element of $\OC{1}{(\l_1+\l_2)/\l_1\cap\l_2}$ corresponding to self-dual measure on the symplectic space $(\l_1+\l_2)/\l_1\cap \l_2$. Let $\mu\lll^{1/2}$ be its square root (§\[sqrt\]).
Set $$\tilde K\lll(v,w):=\psi(\tfrac12Q\lll((v,w)))\cdot \delta\lll(v,w)\cdot\mu\lll^{1/2}.$$ Since $\OC{1}{(\l_1+\l_2)/\l_1\cap\l_2}=\OC{1}{(\l_1+\l_2)/\l_2}\otimes\OC{1}{(\l_1+\l_2)/\l_1},$ one sees that $\tilde K\lll$ is a generalized function on $V\times V$ with values in $\OC{1/2}{V/\l_1}\otimes\OC{1/2}{V/\l_2}$. By Lemma \[lemQ\](ii) and the first sentence of §\[gs\], $\tilde K\lll$ determines a generalized section $K\lll$ of $\HHH_{\l_1}^\vee\boxtimes\HHH_{\l_2}$.
### Definition of $\eta\lll$. {#eta}
For technical purposes in §\[restrict\] we will also need: $$\eta\lll(v,w):=\delta\lll(v,w)\cdot\mu\lll^{1/2}.$$ Since $\eta\lll$ depends on $(v,w)$ only modulo $\l_1\times \l_2$, we may consider it as a generalized function on $V/\l_1\times V/\l_2$, supported on , with values in $\OC{1/2}{V/\l_1}\otimes\OC{1/2}{V/\l_2}$. The pull-back $\alpha_g^*\eta\gll$ should be understood much as in Remark \[pullback\]: it is a generalized function on $V/\l\times V/\l$ with values in $\OC{1}{V/\l}$.
Two Quadratic Spaces {#theform}
====================
Now we define the quadratic spaces $(S\gl,q\gl)$ and $(S'\gl,q\gl)$ mentioned in §\[defq1\] and §\[finconc\] (they are dual, hence isometrically isomorphic to one another). We first give an abstract definition, and then explicit formulas in §\[expform\]. In §\[MaslovInterp\], we show that both forms represent $\tau(\Gamma_g,\Gamma_1,\l\oplus\l)$ in $W(F)$. Finally, in §\[maktouf\] we detail the connection to Maktouf’s work [@Ma] mentioned in §\[mak1\].
Definition of $S\gl$ and $S'\gl$. {#Dq}
---------------------------------
Let $A,B,C,D$ denote the rows (they are complexes) in the commutative diagram $$\xymatrix@R=0.3in@C=.3in{
& g\l\cap\l \ar[r]^{1-g\inv}\ar[d]^{g\inv}\ar[d]
& {\l} \ar[r]\ar[d]
& V/(g-1)V\ar@{=}[d]
\\
& {\l} \ar[r]^{g-1}\ar[d]
& g\l+\l \ar[r]\ar[d]
& V/(g-1)V\ar@{=}[d]
\\
{\ker(g-1)}\ar[r]\ar@{=}[d]
& V \ar[r]^{g-1}\ar[d]
& V \ar[r]\ar[d]
& V/(g-1)V
\\
{\ker(g-1)}\ar[r]
& V/\l \ar[r]^{g-1}
& V/(g\l+\l)
}$$
Let $S\gl$ be the cohomology of $D$ at its center term, and $S'\gl$ be the cohomology of $A$ at its center term. Then $S\gl$ and $S'\gl$ are dual to one another under the symplectic pairing, because indeed $A$ and $D$ themselves are dual.
Definition of $q\gl$ and $q'\gl$. {#DPhi}
---------------------------------
Let $W$ denote the cohomology of the complex $B$ at its center term. The map $A\to B$ is a quasi-isomorphism of complexes, giving an isomorphism $S'\gl\to W$. $B\to C\to D$ is a short exact sequence of complexes, and $C$ is acyclic, so the boundary map gives an isomorphism $S\gl\to W$ on cohomology. Together we have an isomorphism $$\Phi\gl\colon S\gl\to W\leftarrow S'\gl.$$
Set $$q\gl(x,y):=\<\Phi\gl(x),y\>\qquad q'\gl(a,b):=\<a,\Phi\gl\inv(b)\>$$ for all $x,y\in S\gl$, $a,b\in S'\gl.$ In particular, $\Phi\gl\colon S\gl\to S'\gl$ is an isometric isomorphism.
Explicit Formulas. {#expform}
------------------
Let us now give explicit formulas for $q\gl$ and $q'\gl$ and show that they are symmetric forms.
### {#hatS}
First observe that $S\gl$ is a quotient of $$\label{eqS}\hat S\gl:=\{x\in V/\l\,|\,(g-1)x\in g\l+\l\}\subset V/\l$$ and $S'\gl$ is a quotient of $$\label{eqS'}\hat S'\gl:=\l\cap(g-1)V\subset \l.$$ We will really give formulas for $q\gl$ and $q'\gl$ pulled back to $\hat S\gl$ and $\hat S'\gl$.
\[remS\] For $g\in\Sp(V)''$, as in , $S\gl=\hat S\gl$ and $\hat S'\gl=\l$. Moreover, if $g\in\Sp(V)^\l$, as in , then $S\gl=\hat S\gl=V/\l$ and $S'\gl=\hat S'\gl=\l$.
### {#section-5}
Given $x,y\in \hat S\gl$, there exist $a,b\in\l$ with $(g-1)x\equiv(ga+b)\bmod (g-1)\l$. Then $$\label{expq}q\gl(x,y)=\<a+b,y\>.$$ (Indeed, $\Phi\gl(x)=a+b$ because $a+b\equiv ga+b\bmod (g-1)\l$.)
Given $a,b\in \hat S'\gl$, suppose $b=(g-1)y$. Then $$\label{expq'}q'\gl(a,b)=\<a,y\>.$$
\[aresym\] Both $q\gl$ and $q'\gl$ are symmetric forms.
The two forms are isometric, so it suffices to consider $q'\gl$. Given $a,b\in \hat S'\gl$, suppose $a=(g-1)x, b=(g-1)y$. Then $$q'\gl(a,b)=\<(g-1)x,y\>=\<x,(g\inv-1)y\>=\<x,-g\inv b\>=\<gx,-b\>.$$ Now $gx-x=a\in\l$, so $\<gx,-b\>=\<x,-b\>=\<b,x\>=q'\gl(b,a).$
Interpretation via the Maslov Index. {#MaslovInterp}
------------------------------------
\[MI\] The class of $(S\gl,q\gl)$ and therefore of $(S'\gl,q'\gl)$ in $W(F)$ equals $\tau(\Gamma_g,\Gamma_1,\l\oplus\l)$. In fact, these quadratic spaces satisfy Proposition \[rts\].
Recall from [@Th] that $\tau(\Gamma_g,\Gamma_1,\l\oplus \l)$ is represented by a degenerate quadratic form $q$ on $$\hat T:=\{(x,y,z)\in\Gamma_g\times\Gamma_1\times(\l\oplus \l)\,|\,x+y+z=0\in\overline V\oplus V\}.$$ Namely, $q((x,y,z))=\<x,z\>$, where we pair using the symplectic form on $\overline V\oplus V$.
We claim that the map $f\colon \hat T\to \hat S'\gl$, $$f\colon((x,gx),(y,y),(a,b))\mapsto a-b$$ descends to an isometric isomorphism on the nondegenerate quotients.
First we observe that the condition $s:=((x,gx),(y,y),(a,b))\in \hat T$ implies that $a-b=(g-1)x$, so $f$ does have the right target. It is also an isometry since $$q(s)=\<gx,b\>-\<x,a\>=\<x,b\>-\<x,a\>=\<a-b,x\>=q'\gl(f(s),f(s)).$$ The second equality holds because $gx\equiv x\bmod\l$, the fourth from .
Any isometry descends to an injective map of nondegenerate quotients, so it remains to observe that the nondegenerate quotient $S'\gl$ of $\hat S'\gl$, defined in §\[Dq\], has the same dimension as the nondegenerate quotient $T$ of $\hat T$ (cf. Proposition \[rts\]), namely, $$\label{dim}\begin{aligned}
\dim S'\gl=\dim S\gl=\tfrac12&\dim V-\dim\ker(g-1) \\
& -\dim g\l\cap\l+2\dim{\l\cap \ker(g-1)}.\end{aligned}$$
Relation with Maktouf’s Construction. {#maktouf}
-------------------------------------
Let $(s,t_s)$ be the semisimple part of $(g,t)\in\Mp(V)$. Assume for simplicity that $(s-1)$ is invertible. Maktouf [@Ma] asserts that there exists an $s$-stable symplectic decomposition $V=V_1\oplus V_2$ and a Lagrangian $\l_i\in V_i$ such that $s\l_1=\l_1$ and $s\l_2\cap \l_2=0$. He then defines a quadratic form $q_{\mathrm{Mak}}$ on $\l_2$ by $q(a,a)=\<a,(s\inv-1)\inv a\>$ and sets $$\Phi(g,t):=t_s(\l_1\oplus\l_2)\cdot\gamma(-q_{\mathrm{Mak}}).$$
Suppose that $g=s$ is semisimple and $g-1$ is invertible. Choose $\l:=\l_1\oplus\l_2\subset V$, and consider $q'\gl$ as a form on $\hat S'\gl=\l$. We will show that $q_{\mathrm{Mak}}=-q'\gl$, so $\Phi=\Theta_\l$.
\[kerq2\] We have $\l_1=\ker q'\gl$. Thus $q'\gl$ defines a nondegenerate bilinear form on $\l_2$.
Clearly $g\l\cap \l=\l_1$, and $\l_1$ is $(1-g\inv)$-stable. But $q'\gl$ was defined to be nondegenerate on $S'\gl=\l/((1-g\inv)(g\l\cap\l))$.
\[qq\] We have $q'\gl=-q_{\mathrm{Mak}}$ on $\l_2$.
Suppose $a\in \l_2$. Then $$\begin{aligned}
q_{\mathrm{Mak}}(a,a)&&=\<a,(g\inv-1)\inv a\>=\<(g-1)(g-1)\inv a,(g\inv-1)\inv a\>\\
&&=\<(g-1)\inv a,(g\inv-1)(g\inv-1)\inv a\>\\&&=\<(g-1)\inv a,a\>=-q'\gl(a,a).\end{aligned}$$
Proof of Proposition \[finrel\] and Theorem 2B {#finitefield}
==============================================
Let $F$ be a finite field; we again identify measures and functions on finite sets.
Proof of Proposition \[finrel\]. {#finrestrict}
--------------------------------
For any $g\in\Sp(V)$, consider the composition $$\label{dag}
\begin{CD}
V/\l@>\Delta>> V/\l\times V/\l @>{\alpha_g}>> V/g\l\times V/\l\end{CD}\qquad x\mapsto(x,x)\mapsto(gx, x).$$ Proposition \[finrel\] amounts to the following lemma.
\[finlrestrict\] Fix any $g\in\Sp(V)$.
1. $\Delta\inv\circ\alpha_g\inv(\operatorname{support}K\gll)=\hat S\gl\subset V/\l.$
2. $\Delta^*\alpha_g^*Q\gll(x)=q\gl(x,x)$ for any $x\in \hat S\gl$.
3. $\Delta^*\alpha^*\eta\gll(x)=|F|^{-\tfrac12\dim{\l/g\l\cap\l}}$ for any $x\in \hat S\gl$.
(Notation: see §\[defQ\] for $Q\gll$ and §\[eta\] for $\eta\gll$.)
The first statement follows from , . The second statement follows from and §\[defQ\]. For the third, recall that $\eta\gll$ is defined in §\[eta\] to be constant on its support (which is $\hat S\gl$ by part (i)), with value $$\mu\gll^{1/2}=\(|F|^{-\tfrac12\dim\((g\l+\l)/g\l\cap\l\)}\)^{1/2}= |F|^{-\tfrac12\dim \l/g\l\cap\l}.$$ Here we use that if $A$ has a nondegenerate bilinear form, then the self-dual measure on $A$ is $|F|^{-\dim A/2}$.
Proof of Theorem 2B.
--------------------
Using and Proposition \[finrel\] above, we have $$\Tr\rho(g,t)=t(\l)\cdot\sum_{x\in \hat S\gl} \psi(\tfrac12q\gl(x,x))\cdot |F|^{-\tfrac12\dim \l/g\l\cap\l}.$$ Applying Propositions \[pg1\] and \[MI\], we obtain $$\begin{aligned}
\label{almost}\Tr\rho(g,t)&=t(\l)\cdot |F|^{\tfrac12\(\dim S\gl-\dim \l/g\l\cap\l\)+\dim\ker q\gl}\cdot \gamma(q\gl)\\
&=|F|^{\tfrac12\(\dim S\gl-\dim \l/g\l\cap\l\)+\dim\ker q\gl}\cdot\Theta_\l(g,t)
\end{aligned}$$ where $\ker q\gl$ is the kernel of $q\gl$ as a form on $\hat S\gl$. Now, by construction of $S\gl$ in §\[Dq\], we have $$\dim \ker q\gl=\dim\ker(g-1)-\dim\l\cap\ker(g-1)$$ as well as formula for $\dim S\gl$. This with establishes Theorem 2B.
Proof of Proposition \[rel\] and Theorem 2A
===========================================
Proof of Proposition \[rel\]. {#diagonal}
-----------------------------
\[restrict\] Proposition \[rel\] amounts to Lemma \[finlrestrict\](i,ii), which still hold, Remark \[remS\], and the following infinite version of Lemma \[finlrestrict\](iii).
\[lrestrict\] If $g\in\Sp(V)^\l$ then $\Delta^*\alpha_g^*\eta\gll=\left\|\det (g-1)\right\|^{-1/2}dq\gl$ as measures on $V/\l$.
It is clear that $\nu_\l:=\Delta^*\alpha_g^*\eta\gll$ is an invariant measure on $\hat S\gl=S\gl=V/\l$; that is, $\nu_\l\in\OC{1}{V/\l}$ in the notation of §\[hd\]. Let $\Phi\gl\colon V/\l \to \l$ be the isomorphism defining $q\gl$, as in §\[DPhi\]. The claim is that $$\label{muit1}\<\nu_\l,{\Phi\gl}_*\nu_\l\>=\left\|\det (g-1)\right\|\inv$$ under the pairing of $\Omega_1(V/\l)$ with $\Omega_1(\l)$ induced by the symplectic form.
Let $\mu_V$ be the self-dual measure on $V$. Fix $\omega\in\det \l$ and consider $g\omega\in\det(V/\l)$. It is easy to see that $\nu_\l$, as a function $\det(V/\l)\to\RRR$ in the sense of §\[hd\], satisfies $$\nu_\l(g\omega)=\left(\mu_V(g\omega\wedge\omega)\right)^{1/2}.$$ On the other hand $\Phi\gl\inv(x)=(g-1)\inv x\bmod \l$ for any $x\in\l$; thus $${\Phi\gl}_*\nu_\l(\omega)=\nu_\l((g-1)\inv\omega)=
\nu_\l(g\omega)\cdot\frac{\mu_V((g-1)\inv\omega\wedge\omega)}{\mu_V(g\omega\wedge\omega)},$$ the fraction here being the absolute value of the ratio of $(g-1)\inv\omega$ and $g\omega$ as elements of $\det(V/\l)$. Since $(g-1)\inv\omega\wedge\omega=(g-1)\inv(\omega\wedge(g-1)\omega)=(g-1)\inv(\omega\wedge g\omega)$, we have all together $${\Phi\gl}_*\nu_\l(\omega)=\left\|\det (g-1)\right\|\inv\left(\mu_V(g\omega\wedge \omega)\right)^{1/2}.$$ Since $\<\nu_\l,{\Phi\gl}_*\nu_\l\>=\frac{\nu_\l(g\omega)\cdot{\Phi\gl}_*\nu_\l(\omega)}{\mu_V(g\omega\wedge\omega)}$, we obtain .
Proof of Theorem 2A. {#proof}
--------------------
### {#char}
First recall how $\Tr\rho$ is defined as a generalized function on $\Mp(V)$. If $M$ is a compactly supported smooth function on $\Mp(V)$ then $$\rho(M)\colon f\mapsto\int_{(g,t)\in\Mp(V)} M(g,t)\rho(g,t)f$$ is a trace-class operator on $\SSS(\HHH_{\l_1})$, and $$\<\Tr\rho,M\>:=\Tr\rho(M).$$
### {#Mpl}
Assume from now on that $M$ is supported over $\Sp(V)^\l$, see . Here is a valid version of .
\[pt\] $\displaystyle{{\Tr\rho(M)=\int_{x\in V/\l}\int_{(g,t)\in\Mp(V)} \!\!\!\!M(g,t)\cdot t(\l)\cdot \Delta^*\alpha_g^*K\gll(x).}}$
As in , the operator $\rho(M)$ is represented by the integral kernel $$\label{rhoM}(x,y)\mapsto\int_{(g,t)\in\Mp(V)} M(g,t)\cdot t(\l)\cdot \alpha_g^*K\gll(x,y)$$ on $V/\l\times V/\l$. We know $\rho(M)$ is trace class. For its trace to be the integral of along the diagonal, it suffices that be smooth. This is clear from Lemma \[smooth\].
### {#section-6}
Now choose a function $h$ on $A:=V/\l$ as in §\[Dirac\], and again set $h_s(x):=h(sx)$. We claim that $$\label{lim}\Tr\rho(M)=\lim_{s\to0}\Tr(h_s\cdot\rho_M).$$ Indeed, adapting Lemma \[pt\], we have $$\label{e}\Tr (h_s\cdot\rho(M))=\int_{x\in V/\l}\int_{(g,t)\in \Mp(V)} M(g,t)\cdot h_s(x)\cdot t(\l)\cdot \Delta^*\alpha_g^*K\gll(x).$$ The outer integral converges absolutely and uniformly in $s$, being dominated by $\Tr\rho(M)$. Therefore $$\lim_{s\to 0} \Tr (h_s\cdot\rho(M))=\int_{x\in V/\l}\int_{(g,t)\in\Mp(V)} M(g,t)\cdot t(\l)\cdot \Delta^*\alpha_g^*K\gll(x)=\Tr \rho(M)$$ as desired.
### {#section-7}
On the other hand, we may exchange the order of integration in . $$\label{f}\Tr (h_s\cdot\rho(M))= \int_{(g,t)\in\Mp(V)} M(g,t)\cdot\int_{x\in V/\l} h_s(x)\cdot t(\l)\cdot \Delta^*\alpha_g^*K\gll(x).$$ Now, from Propositions \[rel\] and \[pg\] and Remark \[remS\], we find, for $g\in\Sp(V)^\l$, $$\lim_{s\to0} \int_{x\in V/\l} h_s(x)\cdot\Delta^*\alpha_g^*K\gll(x) =\left\|\det(g-1)\right\|^{-1/2}\cdot \gamma(q\gl).$$ Moreover, due to Lemma \[smooth\], this limit converges uniformly for $g$ in a compact set. Uniformity allows us to calculate $\lim_{s\to 0}\Tr(h_s\cdot\rho(M))$ from by taking the limit $s\to 0$ inside the first integral to obtain $$\lim_{s\to0}\Tr(h_s\cdot\rho(M))=\int_{(g,t)\in\Mp(V)} M(g,t)\cdot t(\l)\cdot\left\|\det(g-1)\right\|^{-1/2}\cdot \gamma(q\gl).$$ This combined with and Proposition \[MI\] completes the proof of Theorem 2A.
[19]{} , ‘Character of the oscillator representation’, [*Israel J. Math.*]{} [98]{} (1997) 229–252. , [*Automorphic forms and representations*]{}, Cambridge Studies in Advanced Mathematics 55 (Cambridge University Press, 1998). , ‘The geometric Weil representation’, to appear. Preprint at www.arxiv.org/math.RT/0610818. , ‘On the character of Weil’s representation’, [*Trans. Amer. Math. Soc.*]{} [177]{} (1973) 287–298. , [*Sheaves on Manifolds*]{}, Grundlehren der mathematischen Wissenschaften 292 (Springer, Berlin, 1990). , [*The algebraic theory of quadratic forms*]{} (W.A. Benjamin, Reading, Mass., 1973). , [*The Weil representation, Maslov index and Theta series*]{}, Progress in Mathematics 6 (Birkha" user, Boston, 1980). , ‘Extension des representations de groupes unipotents $p$-adiques: calculs d’obstructions’, [*Non Commutative Harmonic Analysis and Lie Groups (Marseille-Luminy, 1980)*]{} (eds J. Carmona and M. Vergne), Lecture Notes in Mathematics 880 (Springer, Berlin, 1981), pp. 337–356. , ‘Le caractère de la représentation métaplectique et la formule du caractère pour certaines représentations d’un groupe de Lie presque algébrique sur un corps $p$-adique’, [*J. Functional Analysis*]{} [164]{} (1999) 249–339. , ‘Représentations de Schrödinger, indice de Maslov et groupe metaplectique’, [*Non Commutative Harmonic Analysis and Lie Groups (Marseille-Luminy, 1980)*]{} (eds J. Carmona and M. Vergne), Lecture Notes in Mathematics 880 (Springer, Berlin, 1981), pp. 370–407. , ‘The Maslov index as a quadratic space.’ [*Math. Res. Lett.*]{} [13]{} (2006) 985–999. Expanded electronic version at www.arxiv.org/math.SG/0505561/. , ‘Sur le caractère de la représentation de Shale-Weil de $\Mp(n,\RRR)$ et $\Sp(n,\CCC)$’, [*Math. Ann.*]{} [252]{} (1980) 53–86. , ‘Sur certains groupes d’opérateurs unitaires’, [*Acta Math.*]{} [111]{} (1964) 143–211.
[^1]: Unless $|F|=3$ and $\dim V=2$, when $\Mp(V)$ has automorphisms over $\Sp(V)$—see Remark \[splitrem\].
|
---
abstract: |
We propose genuine ($k$, $m$)-threshold controlling schemes for controlled teleportation via multi-particle entangled states, where the teleportation of a quantum state from a sender (Alice) to a receiver (Bob) is under the control of $m$ supervisors such that $k$ ($k\leq m$) or more of these supervisors can help Bob recover the transferred state. By construction, anyone of our quantum channels is a genuine multipartite entangled state of which any two parts are inseparable. Their properties are compared and contrasted with those of the well-known Greenberger-Horne-Zeilinger, W, and linear cluster states, and also several other genuine multipartite entangled states recently introduced in literature. We show that our schemes are secure against both Bob’s dishonesty and supervisors’ treacheries. For the latter case, the game theory is utilized to prove that supervisors’ cheats can be well prevented. In addition to their practical importance, our schemes are also useful in seeking and exploring genuine multipartite entangled states and opening another perspective for the applications of the game theory in quantum information science.
PACS number(s): 03.67.Dd, 03.67.Hk, 03.67.Mn
address: |
$^1$Department of Physics, and Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China\
$^2$School of Physics and Material Science, Anhui University, Hefei 230039, China
author:
- 'Xin-Wen Wang$^{1}$, Da-Chuang Li$^{2}$, and Guo-Jian Yang$^{1}$[^1]'
title: 'Genuine ($k$, $m$)-threshold controlled teleportation and its security'
---
introduction
============
In quantum information science, information is encoded in quantum states. Quantum information processing is in fact the manipulation or (and) transfer of quantum states. Quantum teleportation [@70PRL1895] is a typical quantum information processing task, which functions as transferring a quantum state from one site to another one via previously shared entanglement assisted by classical communications and local operations. Quantum teleportation can not only be directly used to realize quantum communication but also construct a primitive of a quantum computer [@primitive]. Quantum teleportation has been realized in many experiments (see e.g., [@390N575]). Since the end of last century, a new quantum teleportation idea, i.e., controlled teleportation (CT), has been attracting much interest [@58PRA4394; @70PRA022329; @72PRA022338; @75PRA052306; @40JPB1767; @0609026; @68PRA022321; @341PLA55]. CT functions as teleporting a quantum state from a sender’s (Alice) site to a receiver’s (Bob) site under the control of multiple supervisors (Charlie 1, Charlie 2,$\cdots$). In other words, Alice and Bob need the cooperation of Charlies in order to realize the teleportation (communication) successfully. A CT scheme has already been demonstrated in an optical experiment [@430N54].
CT is useful in the context of networked quantum communication, quantum computation, and cryptographic conference [@72PRA022338; @59PRA1829; @79PRA062315; @79PRA062313; @08011544; @0401076; @cryptography]. For instance, CT can be used as a secret sharing to hide a quantum state as a secret [@72PRA022338; @79PRA062313]. In addition, CT has many similarities with the secure multi-party quantum computation (MPQC) protocol [@08011544] which allows multiple players to compute an agreed quantum circuit where each player has access only to his own quantum input. A MPQC protocol has two phases, sharing phase and reconstruction phase. In the sharing phase, dealers provide many agents with their initial state; in the reconstruction phase, one agent is designated to reconstruct the final state of the protocol with the help of the other ones. CT may have other interesting applications, such as in opening a credit account on the agreement of multiple managers in a quantum network.
The previous CT schemes [@58PRA4394; @70PRA022329; @72PRA022338; @75PRA052306; @40JPB1767; @0609026; @68PRA022321; @341PLA55] are focused on the ($m$, $m$)-threshold controlling schemes where the achievement of teleportation is conditioned on the collaboration of all the supervisors. In other words, it is impossible to realize teleportation between Alice and Bob if anyone of Charlies does not cooperate for subjective or objective reasons. However, a more general CT scheme should consider the ($k$, $m$)-threshold case ($k\leq m$) where $k$ or more of the supervisors can help Bob successfully recover the transferred state, but less than $k$ of them cannot. Recently, different ($k$, $m$)-threshold controlling schemes were discussed in Refs. [@Wang; @79PRA062313]. The scheme in Ref. [@Wang] needs lowering the fidelity of teleportation and its successful probability for enduring the uncooperation of part of supervisors. In Ref. [@79PRA062313], authors pointed out that a ($k$, $m$)-threshold controlling scheme can be constructed by using secret sharing. That is, the teleportation is controlled by a classical key which is shared by the supervisors such that $k$ or more of them can recover the key. However, as mentioned in Ref. [@79PRA062313], a classical key can be easily copied, and Charlies cannot stop Bob from recovering Alice’s original state if Bob manages to obtain as least $k$ shares of the key without consent of Charlies. More importantly, the classical ($k$, $m$)-threshold controlling scheme can not prevent Charlies’ cheats as will be shown. They also proposed another “($k$, $m$)-threshold” CT scheme which is a combination of a ($m$, $m$)-threshold CT scheme and the ($k$, $m$)-threshold secret sharing scheme mentioned above. Evidently, it is not a genuine ($k$, $m$)-threshold controlling scheme, because Bob still needs the assistance of all the supervisors for recovering Alice’s original state. In principle, a ($k$, $m$)-threshold controlling scheme can be constructed by using the quantum polynomial codes [@83PRL648] as mentioned in Ref. [@79PRA062313]. However, it needs the supervisors and Bob to come together and perform nonlocal operations (multi-particle joint operations).
In this article, we propose genuine ($k$, $m$)-threshold controlling schemes for CT. In these schemes, the supervisors (Charlies) only need to perform single-particle measurements and announce their outcomes. If the recipient receives $k$ correct outcomes, he or she can reconstruct the original state that the sender wants to transfer by appropriate local operations. We first consider the CT of a single-particle state via a multipartite entangled sate. Then the CT of an $n$-particle state can be directly realized by using $n$ such multipartite entangled states. However, the directly generalized method requires considerably large auxiliary particle resources and local operations, as well as classical communications, especially when $n$ is very large. We propose a much more economical scheme for CT of an arbitrary $n$-particle state with a single multipartite entangled state. By construction, our quantum channels are genuine multipartite entangled states in which any two parts are inseparable. Their properties are compared and contrasted with those of the well-known Greenberger-Horne-Zeilinger, W, and linear cluster states, and also several other genuine multipartite entangled states recently introduced in literature. We show that our schemes are secure against both Bob’s dishonesty and supervisors’ treacheries. For the latter case, the game theory is utilized to prove that supervisors’ cheats can be well prevented. In addition to the potential applications in networked quantum communication and quantum computation, our schemes are also useful in seeking and exploring genuine multipartite entangled states and opening another perspective for the applications of the game theory in quantum information science.
The paper is organized as follows. In Sec. II, we describe the ($k$, $m$)-threshold CT protocols, and briefly analyze the features of the entanglement channels. In Sec. III, we discuss the security of our schemes against Bob’s dishonesty and supervisors’ treacheries. Concluding remarks appear in Sec. IV.
($k$, $m$)-threshold controlling scheme for controlled teleportation
====================================================================
A brief review of the teleportation scheme with a Bell state
------------------------------------------------------------
Quantum teleportation was first proposed by Bennett *et al.* [@70PRL1895]. In their original scheme, the state to be teleported is an arbitrary single-particle state given by $$\label{psi}
|\psi\rangle_T=\alpha|0\rangle_T+\beta|1\rangle_T$$ with $|\alpha|^2+|\beta|^2=1$, and the quantum channel shared by the sender Alice and the receiver Bob is an EPR singlet state. In fact, the quantum channel can be anyone of the four Bell basis states $$\begin{aligned}
\label{Bell}
|\mathcal{B}^1\rangle_{AB}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)_{AB},\nonumber\\
|\mathcal{B}^2\rangle_{AB}=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)_{AB},\nonumber\\
|\mathcal{B}^3\rangle_{AB}=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)_{AB},\nonumber\\
|\mathcal{B}^4\rangle_{AB}=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)_{AB}.\end{aligned}$$ Note that the four Bell states can be transformed into each other by local operations on one particle. For instance, $|\mathcal{B}^1\rangle_{AB}=\sigma^z_B|\mathcal{B}^2\rangle_{AB}
=\sigma^x_B|\mathcal{B}^3\rangle_{AB}=i\sigma^y_B|\mathcal{B}^4\rangle_{AB}$, where $\sigma^j$ ($j=x,y,z$) are the conventional Pauli matrices given by $$\begin{aligned}
\sigma^x=\left(
\begin{array}{cc}
0 & 1 \\ 1 & 0 \\
\end{array}
\right),~
\sigma^y=\left(
\begin{array}{cc}
0 & -i \\ i & 0 \\
\end{array}
\right),~
\sigma^z=\left(
\begin{array}{cc}
1 & 0 \\ 0 & -1 \\
\end{array}
\right).\end{aligned}$$ As an example, we assume that the quantum channel is $|\mathcal{B}^1\rangle_{AB}$. Then the state of the total system is $$\begin{aligned}
\label{Psitotal}
|\Psi\rangle_{total}&=&|\psi\rangle_T\otimes|\mathcal{B}^1\rangle_{AB} \nonumber\\
&=&\frac{1}{2}\left[|\mathcal{B}^1\rangle_{TA}(\alpha|0\rangle +\beta|1\rangle)_B \right.\nonumber\\
&&+ |\mathcal{B}^2\rangle_{TA}(\alpha|0\rangle -\beta|1\rangle)_B\nonumber\\
&&+|\mathcal{B}^3\rangle_{TA}(\alpha|1\rangle + \beta|0\rangle)_B\nonumber\\
&&\left.+ |\mathcal{B}^4\rangle_{TA}(\alpha|1\rangle - |0\rangle)_B\right]\nonumber\\
&=&\frac{1}{2}\left[|\mathcal{B}^1\rangle_{TA}|\psi\rangle_B
+\sigma_A^z|\mathcal{B}^1\rangle_{TA}\sigma_B^z|\psi\rangle_B\right.\nonumber\\
&&\left.+\sigma_A^x|\mathcal{B}^1\rangle_{TA}\sigma_B^x|\psi\rangle_B
+(-i\sigma_A^y)|\mathcal{B}^1\rangle_{TA}(-i\sigma_B^y)|\psi\rangle_B\right].\end{aligned}$$ Alice performs a Bell-basis measurement on particles $T$ and $A$ and broadcasts the outcomes, after which Bob applies the required Pauli rotation to transform the state of his particle $B$ into an accurate replica of the original state of Alice’s particle $T$. The one-to-one correspondence between Alice’s possible measurement outcomes and the required Pauli rotations can be easily obtained from Eq. (\[Psitotal\]). It can be easily proved that if the quantum channel is another Bell state $|\mathcal{B}^j\rangle_{AB}$ ($j=2$, 3, or 4), the state of the total system can also be expanded as $$\begin{aligned}
|\Psi\rangle_{total}&=&|\psi\rangle_T\otimes|\mathcal{B}^j\rangle_{AB} \nonumber\\
&=&\frac{1}{2}\left[|\mathcal{B}^j\rangle_{TA}|\psi\rangle_B
+\sigma_A^z|\mathcal{B}^j\rangle_{TA}\sigma_B^z|\psi\rangle_B\right.\nonumber\\
&& \left.+\sigma_A^x|\mathcal{B}^j\rangle_{TA}\sigma_B^x|\psi\rangle_B
+(-i\sigma_A^y)|\mathcal{B}^j\rangle_{TA}(-i\sigma_B^y)|\psi\rangle_B\right].\end{aligned}$$ Thus the one-to-one correspondence between Alice’s possible measurement outcomes and the required Pauli rotations can always be easily obtained.
($k$, $m$)-threshold controlled teleportation for an arbitrary single-particle state
------------------------------------------------------------------------------------
Before discussing the ($k$, $m$)-threshold schemes, we first give a general description on the basic idea of CT. Assume that there is a community which is composed of $m+2$ members, Alice, Bob, Charlie 1, Charlie 2, $\cdots$, and Charlie $m$. The members are distributed in a network and connected by a quantum channel, i.e., a multipartite entangled state, and one or more classical channels (can be considered as the conventional classical communication facilities). One of Alice and Bob is the sender of a quantum state (the carrier of quantum information), and the other one is the receiver. Charlies act as the supervisors who can decide whether or not to allow Alice and Bob to carry out the teleportation. In a word, the teleportation of a quantum state between Alice and Bob is supervised by Charlies and needs their approval. Without loss of generality, we assume Alice is the sender and Bob is the receiver. In order to realize the CT of the single-particle state $|\psi\rangle_T$, the quantum channel shared by them can be in the form of $$\begin{aligned}
\label{Phi}
|\Phi\rangle_{2+m}&=&x_1|\mathcal{B}^1\rangle_{AB}|\phi^1\rangle_{C_1C_2\cdots C_m}
+ x_2|\mathcal{B}^2\rangle_{AB}|\phi^2\rangle_{C_1C_2\cdots C_m}\nonumber\\
&& +x_3|\mathcal{B}^3\rangle_{AB}|\phi^3\rangle_{C_1C_2\cdots C_m}
+x_4|\mathcal{B}^4\rangle_{AB}|\phi^4\rangle_{C_1C_2\cdots C_m},\end{aligned}$$ where $\sum_{i=1}^4|x_i|^2=1$, $|\phi^i\rangle_{C_1C_2\cdots C_m}$ are normalized and their forms depend on the concrete schemes but should satisfy $\langle\phi^{i'}|\phi^i\rangle=\delta_{ii'}$ and can be distinguished by local measurements and classical communications. Here, particle $A$ belongs to Alice, particle $B$ to Bob, and particle $C_j$ to Charlie $j$ ($j=1,2,\cdots,m$). It has been shown in the above subsection that anyone of the four Bell states can be competent for realizing the teleportation of the state $|\psi\rangle_T$. However, Alice and Bob can carry out the teleportation only if they can ascertain which Bell state their subsystem is in. With the quantum channel $|\Phi\rangle_{2+m}$, the identification of the Bell states can be achieved by the following method: Charlies make measurements with appropriate bases on their own particles and inform Bob the outcomes; then Bob can distinguish the states $\{|\phi^i\rangle_{C_1C_2\cdots C_m}, i=1,2,3,4\}$ and thus can identify the Bell states. The one-to-one correspondence between $\{|B^i\rangle_{AB}\}$ and $\{|\phi^i\rangle_{C_1C_2\cdots
C_m}\}$ is clearly shown in Eq. (\[Phi\]). Without the cooperation of Charlies, the subsystem of Alice and Bob will be in the mixed state $\rho_{AB}=\mathrm{tr}_{C_1C_2\cdots
C_m}\left(|\Phi\rangle_{2+m}\langle\Phi|\right)
=|x_1|^2|\mathcal{B}^1\rangle_{AB}\langle\mathcal{B}^1|+|x_2|^2|\mathcal{B}^2\rangle_{AB}\langle\mathcal{B}^2|
+|x_3|^2|\mathcal{B}^3\rangle_{AB}\langle\mathcal{B}^3|+|x_4|^2|\mathcal{B}^4\rangle_{AB}\langle\mathcal{B}^4|$. The mixed state cannot be used to implement perfect teleportation [@60PRA1888].
In the conventional CT schemes which use the Greenberger-Horne-Zeilinger (GHZ)-type entangled states [@GHZ] as the quantum channel, two terms of $\{x_i,i=1,2,3,4\}$ are set to zero, and the other two are not and their corresponding $|\phi\rangle_{C_1C_2\cdots C_m}$ states are different Dicke states. For example, the quantum channel $|\Phi\rangle_{2+m}$ is a GHZ state $|GHZ\rangle_{2+m}=(1/\sqrt{2})(|0000\cdots 0\rangle+|1111\cdots
1\rangle)_{ABC_1C_2\cdots C_m}$, then $x_3=x_4=0$, $x_1=x_2=1/\sqrt{2}$, $|\phi^1\rangle_{C_1C_2\cdots
C_m}=(1/\sqrt{2^m})\left[\sum_{l=0}^{m^+}S_m^{2l}|-\rangle^{\otimes
2l}|+\rangle^{\otimes(m-2l)}\right]$, and $|\phi^2\rangle_{C_1C_2\cdots
C_m}=(1/\sqrt{2^m})\left[\sum_{l=0}^{m^-}S_m^{2l+1}|-\rangle^{\otimes
(2l+1)}|+\rangle^{\otimes(m-2l-1)}\right]$, where $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$, $S_m^{\tilde{l}}=m!/[\tilde{l}!(m-\tilde{l})!]$ ($\tilde{l}=2l,2l+1$) is the combinational coefficient, $|-\rangle^{\otimes \tilde{l}}|+\rangle^{\otimes(m-\tilde{l})}$ denotes that $\tilde{l}$ particles are in the state $|-\rangle$ and $m-\tilde{l}$ particles are in the state $|+\rangle$, and when $m$ is odd $m^-=m^+=(m-1)/2$, otherwise, $m^-=m/2-1$ and $m^+=m/2$. That is, $|\phi^1\rangle_{C_1C_2\cdots C_m}$ and $|\phi^2\rangle_{C_1C_2\cdots C_m}$ are the Dicke states with even $|-\rangle$ and odd $|-\rangle$, respectively. Thus Charlies can perform single-particle measurements on their own particles with the basis $\{|\pm\rangle\}$ and inform Bob the outcomes, and Bob can identify the Bell states with the outcomes, even or odd $|-\rangle$. Evidently, such a CT scheme is a ($m$, $m$)-threshold controlling scheme, i.e., Alice and Bob can implement the teleportation if and only if all Charlies agree and cooperate.
Now, let us move on to the ($k$, $m$)-threshold controlling scheme. For simplicity, we first consider the case $k=1$. That is, Alice and Bob can realize successfully teleportation if anyone of Charlies cooperate with them. We can set $x_3=x_4=0$, $x_1=x_2=1/\sqrt{2}$, $|\phi^1\rangle_{C_1C_2\cdots C_m}=|00\cdots 0\rangle_{C_1C_2\cdots
C_m}$, and $|\phi^2\rangle_{C_1C_2\cdots C_m}=|11\cdots
1\rangle_{C_1C_2\cdots C_m}$ in Eq. (\[Phi\]). Then the quantum channel is $$\begin{aligned}
\label{Phi1}
|\Phi^1\rangle_{2+m}&=&\frac{1}{\sqrt{2}}\left(|\mathcal{B}^1\rangle_{AB}|00\cdots 0\rangle_{C_1C_2\cdots C_m}
+|\mathcal{B}^2\rangle_{AB}|11\cdots 1\rangle_{C_1C_2\cdots C_m}\right)\nonumber\\
&=& \frac{1}{2}(|0000\cdots 0\rangle+|0011\cdots 1\rangle+|1100\cdots 0\rangle
-|1111\cdots 1\rangle)_{ABC_1C_2\cdots C_m}.\end{aligned}$$ It can be seen that if anyone of Charlies performs a measurement on his particle with the basis $\{|0\rangle,|1\rangle\}$ (i.e., in the $z$ direction) and informs Bob the outcome, Bob can know particles $A$ and $B$ are in the Bell state $|\mathcal{B}^1\rangle_{AB}$ for the outcome $|0\rangle$ or $|\mathcal{B}^2\rangle_{AB}$ for $|1\rangle$. In other words, anyone of Charlies suffices to help Alice and Bob achieve the teleportation of the state $|\psi\rangle_T$. However, if all of Charlies do not collaborate with them, they cannot achieve the teleportation. Note that any combination of $\{|00\cdots 0\rangle_{C_1C_2\cdots C_m},|11\cdots
1\rangle_{C_1C_2\cdots C_m}\}$ with two of the four Bell states can construct a quantum channel which can realize the (1, $m$)-threshold CT mentioned above. For instance, we can also construct a suitable quantum channel by setting $x_1=x_2=0$, $|\phi^3\rangle_{C_1C_2\cdots C_m}=|00\cdots 0\rangle_{C_1C_2\cdots
C_m}$, and $|\phi^4\rangle_{C_1C_2\cdots C_m}=|11\cdots
1\rangle_{C_1C_2\cdots C_m}$ in Eq. (\[Phi\]).
For the case $k>1$, the quantum channel can be constructed as $$\begin{aligned}
\label{Phik}
|\Phi^k\rangle_{2+m}&=&\frac{1}{\sqrt{2}}\left(|\mathcal{B}^1\rangle_{AB}|\phi^1\rangle_{C_1C_2\cdots
C_m} + |\mathcal{B}^2\rangle_{AB}|\phi^2\rangle_{C_1C_2\cdots C_m}\right)\nonumber\\
|\phi^1\rangle_{C_1C_2\cdots C_m}&=&|00\cdots 0\rangle_{C_1C_2\cdots C_m}\nonumber\\
|\phi^2\rangle_{C_1C_2\cdots
C_m}&=&\frac{1}{\sqrt{S_m^{k-1}}}|k-1,m-k+1\rangle_{C_1C_2\cdots C_m},\end{aligned}$$ where $S_m^{k-1}=m!/[(m-k+1)!(k-1)!]$ is the combinational coefficient, $|k-1,m-k+1\rangle_{C_1C_2\cdots C_m}$ denotes all the totally symmetric states including $k-1$ zeros and $m-k+1$ ones. For example, $m=3$ and $k=2$, then $|\phi^2\rangle_{C_1C_2
C_3}=(1/\sqrt{3})(|011\rangle+|101\rangle+110\rangle)_{C_1C_2
C_3}$. As a matter of fact, $|\phi^2\rangle_{C_1C_2\cdots
C_m}$ is then a symmetric Dicke state with $m-k+1$ excitations. By the way, the symmetric six-qubit Dicke state with three excitations has recently been realized in experiment [@103PRL020503]. Note that when $k=1$, the state of Eq. (\[Phik\]) reduces to that of Eq. (\[Phi1\]). We consider that $l$ ($l\leq m$) of Charlies perform single-particle measurements on their own particles with the basis $\{|0\rangle, |1\rangle\}$. There are two cases. (a) $l\geq
k$, if all of them get the outcome $|0\rangle$, the subsystem of particles $A$ and $B$ collapses into $|\mathcal{B}^1\rangle_{AB}$, otherwise, it collapses into $|\mathcal{B}^2\rangle_{AB}$. (b) $l<k$, if all of them get the outcome $|0\rangle$, the subsystem of particles $A$ and $B$ collapses into a mixed state of $|\mathcal{B}^1\rangle_{AB}$ and $|\mathcal{B}^2\rangle_{AB}$. Thus we can conclude that $k$ or more of Charlies can help Alice and Bob deterministically distinguish between the two Bell states $|\mathcal{B}^1\rangle_{AB}$ and $|\mathcal{B}^2\rangle_{AB}$, while less than $k$ of them cannot. In other words, Alice can deterministically teleport the state $\psi\rangle_T$ to Bob if and only if $k$ or more of Charlies collaborate with them. The procedure of such a CT protocol is as follows.
\(i) Alice performs a Bell-basis measurement on particles $T$ and $A$, and informs Bob the outcome, one of $\{|\mathcal{B}^1\rangle_{TA},|\mathcal{B}^2\rangle_{TA},|\mathcal{B}^3\rangle_{TA},|\mathcal{B}^4\rangle_{TA}\}$.
\(ii) Bob sends his petition to Charlies.
(iii)Charlies talk over whether or not to allow Bob to recover the original state of Alice’s particle $T$. If more than a certain number of Charlies (e.g., $2/3$ of them) vote for allowing, a collective decision should be made that permitting Bob to recover Alice’s original state. Then all Charlies should perform single-particle measurements on their own particles with the basis $\{|0\rangle,|1\rangle\}$ and broadcast their outcomes.
\(iv) According to Alice’s and Charlies’ measurement outcomes, Bob performs a corresponding Pauli rotation on particle $B$ and recovers Alice’s original state on it.
Note that we need all of Charlies instead of $k$ of them to broadcast their outcomes in step (iii) is based on the consideration that there may exist treacherous Charlies who will cheat Bob and send him the false measurement outcomes. The detailed proof for the security of our scheme against Charlies’ cheats will be given in Sec. III.
($k$, $m$)-threshold controlled teleportation for an arbitrary multi-particle state
-----------------------------------------------------------------------------------
As a direct generalization of the teleportation of a single-particle state, teleportation of an arbitrary $n$-particle state $$\label{psiTn}
|\psi\rangle_{T_1T_2\cdots T_n}=\sum\limits_{j_1,j_2,\cdots,j_n=0}^{1}y_{j_1j_2\cdots j_n}|j_jj_2\cdots
j_n\rangle_{T_1T_2\cdots T_n}$$ can be achieved with $n$ Bell states. In fact, the teleportation of a two-particle state with two Bell states has already been demonstrated in an optical experiment [@0609129].
Thus, one can use $n$ copies of the state of Eq. (\[Phi\]) to realize the CT of an arbitrary $n$-particle state. Also, we can directly use $n$ copies of the state $|\Phi^1\rangle_{2+m}$ or $|\Phi^k\rangle_{2+m}$ \[see Eqs. (\[Phi1\]) and (\[Phik\])\] to accomplish the ($k$, $m$)-threshold CT of $|\psi\rangle_{T_1T_2\cdots T_n}$. However, this method requires considerably large auxiliary particle resources and local operations, as well as classical communications, especially when the number of “teleported” qubits is very large. Particularly, each Charlie needs to hold $n$ controlling particles, perform $n$ single-particle measurements, and send Bob $n$ bits of classical information about the measurement outcomes.
We now propose a much more economical way to implement the ($k$, $m$)-threshold CT of an arbitrary $n$-particle state. The quantum channel is the multipartite entangled state $$\begin{aligned}
\label{Phik2nm}
|\Phi^k\rangle_{2n+m}&=&\frac{1}{\sqrt{2}}\left(\prod\limits_{i=1}^{n}|\mathcal{B}^1\rangle_{A_iB_i}
\otimes|00\cdots 0\rangle_{C_1C_2\cdots C_m}\right.\nonumber\\
&&\left.+\prod\limits_{i=1}^{n}|\mathcal{B}^2\rangle_{A_iB_i}\otimes\frac{1}{\sqrt{S_m^{k-1}}}|k-1,m-k+1\rangle_{C_1C_2\cdots C_m}\right),\nonumber\\\end{aligned}$$ where particles $A_i$ are held by Alice, $B_i$ held by Bob. In order to successfully implement the teleportation, Alice and Bob need Charlies to help them identify the two sequences of Bell states. Particularly, the procedure is as follows.
\(i) Alice performs a sequence of Bell-basis measurements on the pairs of particles $\{(T_i,A_i),i=1,2,\cdots,n\}$, and informs Bob the outcomes.
\(ii) and (iii) are the same as that of the CT protocol for a single-particle state.
\(iii) According to Alice’s and Charlies’ measurement outcomes, Bob applies the corresponding Pauli rotations on particles $\{B_i,i=1,2,\cdots,n\}$ and reconstructs the state of Eq. (\[psiTn\]).
As shown above, regardless of the number of qubits to be teleported, the proposed approach only requires that each supervisor holds one particle, performs one single-particle measurement on his or her particle, and send one bit of classical message to the receiver Bob. Therefore, compared with the directly generalized method mentioned above, this method is much more economical, because the required auxiliary particle resources, the number of measurements, and the quantity of classical communications are greatly reduced.
We notice that any two of the four Bell states can be distinguished by local (single-particle) measurements with appropriate measurement bases and classical communications. For instance, we can distinguish between the two sets $\{|\mathcal{B}^1\rangle,|\mathcal{B}^2\rangle\}$ and $\{|\mathcal{B}^3\rangle,|\mathcal{B}^4\rangle\}$ by using the measurement basis $\{|0\rangle,|1\rangle\}$, which can be evidently seen from Eq. (\[Bell\]). In order to show how to distinguish between the two sets $\{|\mathcal{B}^1\rangle,|\mathcal{B}^3\rangle\}$ and $\{|\mathcal{B}^2\rangle,|\mathcal{B}^4\rangle\}$ by local measurements and classical communication, we rewrite them as $$\begin{aligned}
\label{Bell1}
|\mathcal{B}^1\rangle=\frac{1}{\sqrt{2}}(|++\rangle+|--\rangle),\nonumber\\
|\mathcal{B}^2\rangle=\frac{1}{\sqrt{2}}(|-+\rangle+|+-\rangle),\nonumber\\
|\mathcal{B}^3\rangle=\frac{1}{\sqrt{2}}(|++\rangle-|--\rangle),\nonumber\\
|\mathcal{B}^4\rangle=\frac{1}{\sqrt{2}}(|-+\rangle-|+-\rangle).\end{aligned}$$ Obviously, if two participants perform, respectively, a single-particle measurement on different particles with the basis $\{|\pm\rangle\}$, they can discriminate between the two sets $\{|\mathcal{B}^1\rangle,|\mathcal{B}^3\rangle\}$ and $\{|\mathcal{B}^2\rangle,|\mathcal{B}^4\rangle\}$ by exchanging the outcomes. That is, if their outcomes are anticorrelated, the state of the whole system is initially in the set $\{|\mathcal{B}^2\rangle,|\mathcal{B}^4\rangle\}$, otherwise, it is in the set $\{|\mathcal{B}^1\rangle,|\mathcal{B}^3\rangle\}$. With this method, Alice and Bob can measure anyone of $n$ pairs of particles $\{(A_i,B_i),i=1,2,\cdots,n\}$ and identify the states of the other $n-1$ pairs of particles in the quantum channel of Eq. (\[Phik2nm\]). Then Alice and Bob can realize the teleportation of an $n$-particle state with a high fidelity when $n$ is large, out of the control of Charlies. Especially, when the $n$-particle state $|\psi\rangle_{T_1T_2\cdots T_n}$ \[see Eq. (\[psiTn\])\] is separable, such as $y_{j_1j_2\cdots
j_n}=y_{j_1}y_{j_2}\cdots y_{j_n}$, Alice and Bob can realize perfect teleportation of $n-1$ qubits information escaping from the control of Charlies.
However, this drawback can be avoided by the following methods. We can establish two sequences of states chosen from the four Bell states for the $n$ pairs of particles $\{(A_i,B_i)\}$, and make one-to-one correspondence between them and the two states $|00\cdots
0\rangle_{C_1C_2\cdots C_m}$ and $(1/\sqrt{S_m^{k-1}})|k-1,m-k+1\rangle_{C_1C_2\cdots C_m}$. That is, we can use the following entangled state, instead of $|\Phi^k\rangle_{2n+m}$, to act as the quantum channel: $$\begin{aligned}
\label{Phik2nm1}
|\Phi'^k\rangle_{2n+m}&=&\frac{1}{\sqrt{2}}\left(\prod\limits_{i=1}^{n}|\mathcal{B}^{r_i}\rangle_{A_iB_i}
\otimes|00\cdots 0\rangle_{C_1C_2\cdots C_m}\right.\nonumber\\
&&\left.+\prod\limits_{i=1}^{n}|\mathcal{B}^{s_i}\rangle_{A_iB_i}\otimes\frac{1}{\sqrt{S_m^{k-1}}}|k-1,m-k+1\rangle_{C_1C_2\cdots
C_m}\right),\end{aligned}$$ where $r_i$ ($s_i$) $=1,2,3,$ or $4$. Note that Alice and Bob can know the $n$ pairs of particles $\{(A_i,B_i)\}$ are in the sequence of states $\prod\limits_{i=1}^{n}|\mathcal{B}^{r_i}\rangle_{A_iB_i}$ or $\prod\limits_{i=1}^{n}|\mathcal{B}^{s_i}\rangle_{A_iB_i}$ if and only if they know the particles $\{C_j,j=1,2,\cdots,m\}$ are in the state $|00\cdots 0\rangle_{C_1C_2\cdots C_m}$ or $(1/\sqrt{S_m^{k-1}})|k-1,m-k+1\rangle_{C_1C_2\cdots C_m}$ by Charlies’ help. In other words, they cannot ascertain which sequence of states the subsystem of their $n$ pairs of particles is in without the cooperation of Charlies. The teleportation of an arbitrary $n$-particle state can also be implemented by using a genuine $2n$-particle entangled state as shown in Refs. [@96PRL060502; @74PRA032324; @364PLA7]. Thus the quantum channel of the ($k$, $m$)-threshold CT of an $n$-particle state can also be constructed as the following form for avoiding the aforementioned drawback: $$\begin{aligned}
\label{Phik2nm2}
|\Phi''^k\rangle_{2n+m}&=&\frac{1}{\sqrt{2}}\left(|MES\rangle_{A_1\cdots A_nB_1\cdots B_n}
\otimes\frac{1}{\sqrt{S_m^{k-1}}}|k-1,m-k+1\rangle_{C_1C_2\cdots C_m}\right.\nonumber\\
&& \left.+\sigma^{j_1}_{A_1}\cdots\sigma^{j_n}_{A_n}|MES\rangle_{A_1\cdots A_nB_1\cdots B_n}
\otimes|00\cdots 0\rangle_{C_1C_2\cdots C_m}\right),\end{aligned}$$ where $|MES\rangle_{A_1\cdots A_nB_1\cdots B_n}$ is a genuine $2n$-particle entangled state showed in Eq. (18) of Ref. [@96PRL060502] (for $n=2$) or Eq. (10) of Ref. [@74PRA032324], $j_i=0,x,y$, or $z$ ($i=1,2,\cdots,n$) with $\sigma^{0}$ being the two-dimensional identity operator. However, when $n\geq 4$, $|MES\rangle_{A_1\cdots A_nB_1\cdots B_n}$ was not explicitly constructed in Ref. [@74PRA032324]. As shown in Ref. [@364PLA7], $|MES\rangle_{A_1\cdots A_nB_1\cdots B_n}$ can be replaced by a $2n$-qubit cluster state $|Cluster\rangle_{A_1B_1\cdots
A_nB_n}=(|0\rangle_{A_1}+|1\rangle_{A_1}\sigma^z_{B_1})(|0\rangle_{B_1}+|1\rangle_{B_1}\sigma^z_{A_2})\cdots
(|0\rangle_{A_n}+|1\rangle_{A_n}\sigma^z_{B_n})(|0\rangle_{B_n}+|1\rangle_{B_n})$ [@86PRL910]. Then the quantum channel reads $$\begin{aligned}
\label{Phik2nm3}
|\Phi'''^k\rangle_{2n+m}&=&\frac{1}{\sqrt{2}}\left(|Cluster\rangle_{A_1B_1\cdots A_nB_n}
\otimes\frac{1}{\sqrt{S_m^{k-1}}}|k-1,m-k+1\rangle_{C_1C_2\cdots C_m}\right.\nonumber\\
&& \left.+\sigma^{j_1}_{A_1}\cdots\sigma^{j_n}_{A_n}|Cluster\rangle_{A_1B_1\cdots A_nB_n}
\otimes|00\cdots 0\rangle_{C_1C_2\cdots C_m}\right).\end{aligned}$$ Note that $\sigma^{j_1}_{A_1}\cdots\sigma^{j_n}_{A_n}$ in Eqs. (\[Phik2nm2\]) and (\[Phik2nm3\]) can not be set to $\sigma^{0}_{A_1}\cdots\sigma^{0}_{A_n}$, i.e., $j_i$ can not be simultaneously equal to zero.
The features of the entanglement channels
-----------------------------------------
It is known that the complexity of multipartite entanglement increases greatly with the increase of the number of parties involved. So far, the properties of multipartite entanglement are not very clear. The classification and quantification of genuine three-qubit [@threequbit] and four-qubit [@fourqubit] entangled states were intensively studied. The classification and quantification of genuine entangled states involving more than four qubits were also discussed [@multiqubit; @74PRA022314]. Although several typical multipartite entangled states, such as GHZ states [@GHZ], W states [@threequbit], and cluster states [@86PRL910], were presented, the inequivalent types of genuine multipartite entangled states for more than four particles are still very vague. It will need a long-term effort to well understand the entanglement involving many parties. To seek for genuine multipartite entangled states we can resort to particular quantum schemes since sharing a unique entanglement may allow ones to do some things that ones cannot otherwise do. Teleportation is a well example, with which some genuine multipartite entangled states were found [@75PRA052306; @96PRL060502; @74PRA032324]. Obviously, all the states $|\Phi^k\rangle_{2+m}$ \[see Eq. (\[Phik\])\], $|\Phi^k\rangle_{2n+m}$ \[see Eq. (\[Phik2nm\])\], $|\Phi'^k\rangle_{2n+m}$ \[see Eq. (\[Phik2nm1\])\], $|\Phi''^k\rangle_{2n+m}$ \[see Eq. (\[Phik2nm2\])\], and $|\Phi'''^k\rangle_{2n+m}$ \[see Eq. (\[Phik2nm3\])\], which act as the quantum channels in our ($k$, $m$)-threshold CT schemes, are genuine multipartite entangled states, because any bipartite cut in them is inseparable [@74PRA022314]. Here, we roughly show the relationships or differences between them and other genuine multipartite entangled states presented in literature.
We begin with the state $|\Phi^1\rangle_{2+m}$. When $m=1$, $|\Phi^1\rangle_{2+1}=(1/2)(|00\rangle_{AB}|+\rangle_C+|11\rangle_{AB}|-\rangle_{C})$ is a three-qubit GHZ state; when $m=2$, $|\Phi^1\rangle_{2+2}=(1/2)(|0000\rangle+|0011\rangle+|1100\rangle-|1111\rangle)_{ABC_1C_2}$ is just a four-qubit linear cluster state [@86PRL910]. As to $m>2$, $|\Phi^1\rangle_{2+m}=\mathrm{l.u.}|G\rangle_{2+m}$, where “l.u.” indicates that the equality holds up to local unitary transformations on one or more of the qubits and $$\begin{aligned}
\label{G}
|G\rangle_{2+m}&=&(|0\rangle_A+|1\rangle_A\sigma^z_B)(|0\rangle_B+|1\rangle_B\sigma^z_{C_1})\nonumber\\
&&\otimes(|0\rangle_{C_1}+|1\rangle_{C_1}\sigma^z_{C_2}\cdots \sigma^z_{C_m})\prod\limits_{i=2}^{m}(|0\rangle_{C_i}+|1\rangle_{C_i})
\end{aligned}$$ is a graph state [@69PRA062311] shown in Fig. 1. Obviously, when $m>2$, $|\Phi^1\rangle_{2+m}$ is inequivalent to the well-known GHZ, W, and linear cluster states, in terms of stochastic local operations and classical communications (SLOCC). By the way, many schemes for generating multi-qubit graph states were presented (see e.g., [@97PRL143601]), and the six-qubit graph states are already achievable in the optical experiment [@3NP91].
![The graph state of Eq. (\[G\]). (**a**) $m=3$. (**b**) $m=4$. (**c**) $m=6$.[]{data-label="figure1"}](figure.eps){width="9cm" height="3cm"}
In order to compare the state $|\Phi^k\rangle_{2+m}$ with the corresponding GHZ state $|GHZ\rangle_{2+m}$, W state $|W\rangle_{2+m}$, and linear cluster state $|Cluster\rangle_{2+m}$, we resort to the concept of *persistency of entanglement* [@86PRL910]. The *persistency of entanglement* $P_e(|\Psi\rangle)$ of an entangled state $|\Psi\rangle$ of $N$ particles is the minimum number of local measurements such that, for all measurement outcomes, the state is completely disentangled. For pure states, a completely disentangled state means a product state of all $N$ particles [@86PRL910]. Evidently, for all $N$-qubit states $0\leq P_e\leq N-1$. As shown in Ref. [@86PRL910], two states with different $P_e$ are SLOCC inequivalent, but the inverse case needs further investigation. We now discuss the three cases as follows. (a) $k<m$ and $k\neq m/2$. We can prove that $P_e(|\Phi^k\rangle_{2+m})=k+1$ is different from $P_e(|GHZ\rangle_{2+m})=1$, $P_e(|W\rangle_{2+m})=m+1$, and $P_e(|Cluster\rangle_{2+m})=[(m+2)/2]$ [@86PRL910]. Thus the state $|\Phi^k\rangle_{2+m}$ is SLOCC inequivalent to the corresponding GHZ, W, and linear cluster states. (b) $k= m/2$. $P_e(|\Phi^k\rangle_{2+m})=m/2+1=P_e(|Cluster\rangle_{2+m})$. The relation of $|\Phi^k\rangle_{2+m}$ and $|Cluster\rangle_{2+m}$ needs further investigation. (c) $k=m$. $P_e(|\Phi^m\rangle_{2+m})=m+1=P_e(|W\rangle_{2+m})$. Then we cannot distinguish between $|\Phi^m\rangle_{2+m}$ and $|W\rangle_{2+m}$ by this method. However, we notice that $|\Phi^m\rangle_{2+m}$ belongs to the GHZ-W-type entangled states recently proposed by Chen *et al.* [@74PRA062310], and thus does not belong to the W-type states. On the other hand, $\mathrm{tr}_{C_1\cdots
C_m}(|\Phi^m\rangle_{2+m}\langle\Phi^m|)=\frac{1}{2}|00\rangle_{AB}\langle00|
+\frac{1}{2}|11\rangle_{AB}\langle11|$ is a separable state and $\mathrm{tr}_{C_1\cdots C_m}(|W\rangle_{2+m}\langle
W|)=\frac{m}{m+2}|00\rangle_{AB}\langle 00|
+\frac{2}{m+2}|\mathcal{B}^3\rangle_{AB}\langle\mathcal{B}^3|$ is a partially mixed entangled state, which also justifies the conclusion that $|\Phi^m\rangle_{2+m}$ and $|W\rangle_{2+m}$ are SLOCC inequivalent. By the way, a scheme for generating a GHZ-W-type state has been proposed lately [@79PRA062315]. Similarly, we can prove that all the states $\{|\Phi^k\rangle_{2+(m+2n-2)}~[\mathrm{see}~
\mathrm{Eq.}~(\ref{Phik})],|\Phi^k\rangle_{2n+m}~(\mathrm{or}~
|\Phi'^k\rangle_{2n+m}), |GHZ\rangle_{2n+m}, |W\rangle_{2n+m},
|Cluster\rangle_{2n+m} \}$ are generally SLOCC inequivalent to each other.
Now, let us pay attention to the states $|\Phi''^k\rangle_{2n+m}$ and $|\Phi'''^k\rangle_{2n+m}$. In the state $|\Phi''^k\rangle_{2n+m}$, $|MES\rangle_{A_1\cdots A_nB_1\cdots
B_n}$ is explicitly constructed for $n=2$ [@96PRL060502] and $n=3$ [@74PRA032324], respectively. That is, $$\begin{aligned}
|MES\rangle_{A_1A_2B_1B_2}&=&\frac{1}{2\sqrt{2}}(|0000\rangle-|0011\rangle-|0101\rangle+|0110\rangle\nonumber\\
&& +|1001\rangle+|1010\rangle+|1100\rangle+|1111\rangle)_{A_1A_2B_1B_2},\nonumber\\
|MES\rangle_{A_1A_2A_3B_1B_2B_3}&=&\frac{1}{2\sqrt{2}}
(|000000\rangle +|010110\rangle +|110100\rangle +|100010\rangle\nonumber\\
&&+|011011\rangle+|001101\rangle+|101111\rangle+|111001\rangle)_{A_1A_2A_3B_1B_2B_3}.\end{aligned}$$ Both the states were proved to be SLOCC inequivalent to the corresponding GHZ and W states [@96PRL060502; @74PRA032324]. By the way, a scheme for generating $|MES\rangle_{A_1A_2B_1B_2}$ has been proposed recently [@78PRA024301]. We notice that $|\Phi''^1\rangle_{4+1}$ is SLOCC equivalent to the state of Eq. (17) of Ref. [@75PRA052306] which was constructed also for implementing (1,1)-threshold CT of a two-particle state. In addition, $|\Phi''^1\rangle_{4+2}$ can be transformed into $|MES\rangle_{A_1A_2A_3B_1B_2B_3}$ by local operations with $C_1$ and $C_2$ replaced by $A_3$ and $B_3$, respectively; $|\Phi'''^1\rangle_{2n+1}$ is a $(2n+1)$-qubit linear cluster state. It can be proved that $P_e(|\Phi''^k\rangle_{2n+m})=P_e(|\Phi'''^k\rangle_{2n+m})=n+k$. Thus when $k\neq m/2$ ($m>1$), $|\Phi''^k\rangle_{2n+m}$ and $|\Phi'''^k\rangle_{2n+m}$ are SLOCC inequivalent to the corresponding GHZ, W, and linear cluster states. As to the case $k=m/2$, $|\Phi''^k\rangle_{2n+m}$ and $|\Phi'''^k\rangle_{2n+m}$ are also SLOCC inequivalent to the corresponding GHZ and W states, but the relation of them and linear cluster states needs further investigation.
Security of the ($k$, $m$)-threshold controlled teleportation
=============================================================
Our ($k$, $m$)-threshold CT schemes are secure against both Bob’s dishonesty and Charlies’ treacheries.
Security against Bob’s dishonesty
---------------------------------
Bob may manage to recover Alice’s original state out of the control of Charlies. Thus, during the distribution of the quantum channel, he intercepts $k$ or more of the particles $\{C_i,i=1,2,\cdots, m\}$ and performs them single-particle measurements with the basis $\{|0\rangle,|1\rangle\}$, and resends them or sends other $k$ or more auxiliary particles to corresponding Charlies, respectively. By this way, Bob can ascertain the state of the subsystem of pairs of particles $\{(A_j,B_j),j=1,2,\cdots\}$ and thus successfully recovers Alice’s original state without the cooperation of Charlies. However, the correlation among particles $A_j$, $B_j$, and $C_i$ is disturbed or destroyed. We take $k=1$ as an example. If Bob performs a measurement on one of the particles $\{C_i\}$ and directly resends it to corresponding Charlie, the subsystem of Charlies will be in a product state $|00\cdots 0\rangle_{C_1C_2\cdots C_m}$ or $|11\cdots
1\rangle_{C_1C_2\cdots C_m}$. Then there is no any correlation among particles $\{C_i\}$. This case can be can be easily found by Charlies. If Bob sends other $m$ auxiliary particles in a GHZ state $(1/2)(|00\cdots 0\rangle+|11\cdots 1\rangle)_{C'_1C'_2\cdots C'_m}$ to Charlies, the correlation between the subsystem of Alice and Bob and that of Charlies is destroyed. Thus such an action of Bob can also be detected. In fact, the correlation of any genuine multipartite entangled state will be disturbed or destroyed by any measurement on a subspace of it, and cannot be perfectly simulated by another entangled state involving less parties. As a consequence, Bob’s dishonest action can always be detected in our schemes. The detailed proof is so complicated and prolix, and will be given elsewhere. Note that Charlies should randomly choose a sufficient subset of quantum channels to check whether particles are intercepted during the distribution before carrying out the task of CT. The security checking process is similar to that of quantum secret sharing schemes (see, e.g. [@41JPA255309]). As a matter of fact, most of quantum communication schemes need ones to use this method to check the security of quantum channels against eavesdropper’s interception. Also, all the previous CT schemes [@58PRA4394; @70PRA022329; @72PRA022338; @75PRA052306; @0609026; @68PRA022321; @79PRA062313] are secure against Bob’s dishonesty if checking the security of quantum channels before carrying out the corresponding tasks.
Security against Charlies’ treacheries
--------------------------------------
When some Charlies are not satisfied with a collective decision, they may betray the community by three possible ways as follows. (a) They privately help Bob to reconstruct Alice’s original state. (b) They reject cooperating with Bob and making measurements on their particles. (c) They cheat Bob and send him the false measurement outcomes. We assume that any classical communication is open and insecure, and treacherous Charlies will be punished if their treacherous actions are detected. Then cases (a) and (b) will not occur. In the following, we show how case (c) can be prevented.
We first consider that there is only one treacherous Charlie, e.g., Charlie $j$, who cheats Bob and sends him the false measurement outcome. That is, when Charlie $j$ gets the measurement outcome $|0\rangle$ he broadcasts $|1\rangle$, when getting $|1\rangle$ he broadcasts $|0\rangle$. There are two cases. Case one: $k<m$. If the real measurement outcome on the subsystem of particles $\{C_i,i=1,2,\cdots, m\}$ is $|00\cdots 0\rangle_{C_1C_2\cdots
C_m}$, then the broadcasted outcome is $|00\cdots 010\cdots
0\rangle_{C_1C_2\cdots C_{j-1}C_{j}C_{j+1}\cdots C_m}$ because Charlie $j$ announced the opposite outcome. However, such an outcome should not appear when there is no treacherous Charlie. Thus the cheat action of Charlie $j$ is exposed. If the real measurement outcome is one term of $(1/\sqrt{S_m^{k-1}})|k-1,m-k+1\rangle_{C_1C_2\cdots C_m}$ involving $k-1$ zeros and $m-k+1$ ones, then the broadcasted outcome involves $k-2$ or $k$ zeros. In this case, Bob can also find that there exists a betrayer, although he cannot directly know which Charlie cheated him. In a word, Bob can always detect whether or not there exist treacherous Charlies who cheat him. The probability of exactly finding the cheat action of Charlie $j$ is $1/2$. Case two: $k=m$. If the real measurement outcome is $|00\cdots 0\rangle_{C_1C_2\cdots
C_m}$ or $|00\cdots 010\cdots 0\rangle_{C_1C_2\cdots
C_{j-1}C_{j}C_{j+1}\cdots C_m}$, the broadcasted outcome is $|00\cdots 010\cdots 0\rangle_{C_1C_2\cdots
C_{j-1}C_{j}C_{j+1}\cdots C_m}$ or $|00\cdots 0\rangle_{C_1C_2\cdots
C_m}$. Then the cheat action of Charlie $j$ cannot be found and Bob will obtain a wrong state instead of Alice’s original state. If the real measurement outcome is $|10\cdots 0\rangle_{C_1C_2\cdots C_m}$, then the broadcasted outcome is $|10\cdots 010\cdots
0\rangle_{C_1C_2\cdots C_{j-1}C_{j}C_{j+1}\cdots C_m}$. However, such an outcome should not appear when there is no treacherous Charlie. Thus Bob can find that there exists a betrayer. In a nutshell, the probability of finding the existence of treacherous Charlie is $(S_m^{k-1}-1)/(2S_m^{k-1})$. Note that they may randomly broadcast an artificial outcome without measurement. This way has no essential differences with the one discussed above.
For the case where there are $l$ ($l<m$) treacherous Charlies who send the false outcomes to Bob, when $l\neq m-k+1$, the probability of finding the treacherous Charlies is one ($l$ is odd) or $1-S_l^{l/2}/S_m^k$ ($l$ is even); when $l=m-k+1$, the probability is $(S_m^k-1)/(2S_m^k)$ ($l$ is odd) or $(S_m^k-S_l^{l/2}-1)/(2S_m^k)$ ($l$ is even).
According to the above analysis, when there is only one Charlie who cheats Bob and sends him the false measurement outcome, his cheat action can be directly detected with probability $1/2$. Because when the cheat action of any one of Charlies is found, he will be chastised, the case where one or more Charlies cheat Bob will not occur in practice. We now prove it by the game theory [@game]. Assume that there are $l$ potential treacherous Charlies who are not satisfied with a collective decision that permitting Bob to reconstruct Alice’s original state. They will play a multi-player *Prisoners-Dilemma-like* game. The so-called Prisoners’ Dilemma game [@prisoner] is as follows. Two or more perpetrators are caught by the police and are interrogated in separate cells *without communication among them*. Unfortunately, the police lacks enough proof to implead them. The chief policeman now makes the following offer to each prisoner: if one of them confesses to the crime, but the others do not, then he or she will be commuted by $r$ years and the others will increase $r$ years; if all of them deny, then each of them will be commuted by $s$ years ($s<r$); if all of them confess, then everyone will be commuted by $t$ years ($t<s<r$). The objective of each player (prisoner) is to maximize his or her individual payoff. The catch of the dilemma is that confessing (i.e., they defect from each other) is the dominant strategy, that is, rational reasoning forces each player to defect, and thereby doing substantially worse than if they would all decide to cooperate (deny). In terms of the game theory, such a mutual defection is a Nash equilibrium [@Nash] because each of the players comes to the conclusion that he or she could not have done better by unilaterally changing his or her own strategy. In our scheme, if one of the potential treacherous Charlies sends Bob the false outcome, and the others do not, he will be detected and chastised and they will achieve their purpose of preventing Bob from recovering Alice’s original state; if two or more of them send false outcomes, they can accomplish their purpose escaping from penalty; if all of them do not send false outcome, each will not be punished but they cannot achieve their aim. Thus each of potential treacherous Charlies wish their partners but not himself to send the false outcomes, because then he can accomplish his purpose but not be chastised. The rational reasoning and selfish gene force each Charlie to send correct outcome. This decision is a Nash equilibrium because each of Charlies could not do better by unilaterally changing his action.
In a word, our schemes are secure against Charlies’ cheats. It is worth pointing out that all previous CT schemes [@58PRA4394; @70PRA022329; @72PRA022338; @75PRA052306; @40JPB1767; @0609026], including the scheme of Ref. [@79PRA062313], are insecure when there exist treacherous Charlies. That is, the cheat action of Charlies can not be detected. Then Bob may obtain a wrong state with very low fidelity instead of Alice’s original state when one or more Charlies send him the false measurement outcomes. For instance, we consider the CT of a single-particle state $|\psi\rangle_T$ \[see Eq. (\[psi\])\] with a standard GHZ state. When there are odd Charlies who send the false measurement outcomes to Bob, he will get a wrong state with only the fidelity $F=(|\alpha|^2-|\beta|^2)^2$.
Concluding remarks
==================
In summary, we have proposed several ($k$, $m$)-threshold controlling schemes for CT, where the teleportation of a quantum state Alice to Bob is under the control of $m$ Charlies such that $k$ ($k\leq m$) or more of them can help Bob successfully recover the transferred state. We have also shown that our schemes are secure against both Bob’s dishonesty and Charlies’ treacheries. However, previous ($m$, $m$)-threshold schemes cannot prevent Charlies’ cheats. The presented schemes have potential applications in networked quantum information processing. For example, they can be used to implement the ($k$, $m$)-threshold quantum-secret-sharing without nonlocal operation among receivers and additional limitation for $k$, following the idea of Ref. [@79PRA062313]. Our schemes are also useful to seek and explore genuine multipartite entangled states. We utilized the game theory to prove the security of our schemes against Charlies’ cheats. This implies that our schemes may open another perspective for the applications of the game theory.
Although we only discussed the case where the quantum channels are pure entangled states, suitable mixed entangled states may also be competent for the ($k$, $m$)-threshold CT. In fact, the general form of the pure-entangled-state channel of Eq. (\[Phi\]) can be replaced by the mixed-state channel $$\begin{aligned}
\label{rho}
\rho_{2+m}&=&|x_1|^2|\mathcal{B}^1\rangle_{AB}\langle\mathcal{B}^1|\otimes|\phi^1\rangle_{C_1C_2\cdots
C_m}\langle\phi^1|\nonumber\\
&& + |x_2|^2|\mathcal{B}^2\rangle_{AB}\langle\mathcal{B}^2|\otimes|\phi^2\rangle_{C_1C_2\cdots
C_m}\langle\phi^2|\nonumber\\
&& +|x_3|^2|\mathcal{B}^3\rangle_{AB}\langle\mathcal{B}^3|\otimes|\phi^3\rangle_{C_1C_2\cdots
C_m}\langle\phi^3|\nonumber\\
&& +|x_4|^2|\mathcal{B}^4\rangle_{AB}\langle\mathcal{B}^4|\otimes|\phi^4\rangle_{C_1C_2\cdots
C_m}\langle\phi^4|.\end{aligned}$$ Then corresponding mixed-state channels of the ($k$, $m$)-threshold CT can be constructed by the same methods as in Sec. II B and Sec. II C. With the forms of the states of Eqs. (\[Phi\]) and (\[rho\]), one can construct different quantum channels for implementing ($k$, $m$)-threshold CT. Note that all the quantum channels should at least satisfy the following conditions. (a) They are symmetric under permutation of qubits $\{C_1,C_2,\cdots, C_m\}$. (b) The four states $\{|\phi^1\rangle_{C_1C_2\cdots
C_m},|\phi^2\rangle_{C_1C_2\cdots C_m},|\phi^3\rangle_{C_1C_2\cdots
C_m},|\phi^4\rangle_{C_1C_2\cdots C_m}\}$ can not be fully distinguished unless $k$ of supervisors perform single-particle measurements on their own particles with appropriate bases and combine the measurement outcomes. In addition, different methods may be needed to discuss the security of concrete schemes.
As mentioned above, SaiToh *et al.* [@79PRA062313] also proposed a “($k$, $m$)-threshold” CT scheme which is a combination of a ($m$, $m$)-threshold CT scheme and a ($k$, $m$)-threshold secret sharing scheme. In their scheme, however, the receiver Bob still needs receiving all of the supervisors’ correct measurement outcomes, i.e., needs the cooperation of all Charlies, for recovering the teleported state. Thus their scheme is not a genuine ($k$, $m$)-threshold controlling scheme and can not prevent Charlies’ cheats. They also mentioned that a ($k$, $m$)-threshold controlling scheme can be constructed by sharing a classical key among Charlies such that $k$ or more of them can recover the key. The distribution of the key can be achieved by quantum cryptography. However, they did not construct a concrete scheme. In addition, as shown in Ref. [@79PRA062313], a classical key can be easily copied, and Charlies cannot stop Bob from recovering Alice’s original state if Bob manages to obtain as least $k$ shares of the key without consent of Charlies. More importantly, the classical ($k$, $m$)-threshold controlling scheme can not prevent Charlies’ cheats. In principle, a ($k$, $m$)-threshold controlling scheme can be constructed by using the quantum polynomial codes [@83PRL648] as mentioned in Ref. [@79PRA062313]. However, it needs Charlies and Bob to come together and perform nonlocal operations (multi-particle operations). In contrast, our schemes do not need performing nonlocal operations and are secure against Charlies’ cheats of sending false measurement outcomes.\
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by National Natural Science Foundation of China, Project No. 10674018 and No. 10874019, and the National Fundamental Research Program of China, Projects No. 2004CB719903.
Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres A and Wootters W K 1993 *Phys. Rev. Lett.* **70** 1895 Gottesman D and Chuang I L 1999 *Nature* **402** 390\
Knill E, Laflamme L and Miburn G J 2001 *Nature* **409** 46\
Kok P, Munro W J, Nemoto K, Ralph T C, Dowling J P and Milburn G J 2007 *Rev. Mod. Phys.* **79** 135 Bouwmeester D, Pan J W, Mattle K, Eibl M, Weinfurter H and Zeilinger A 1997 *Nature* **390** 575\
Marcikic I, de Riedmatten H, Tittel W, Zbinden H and Gisin N 2003 *Nature* **421** 509\
Chen Y A, Chen S, Yuan Z S, Zhao B, Chuu C S, Schmiedmayer J and Pan J W 2008 *Nat. Phys.* **4** 103\
Olmschenk S, Matsukevich D N, Maunz P, Hayes D, Duan L M and Monroe C 2009 *Science* **323** 486 Karlsson A and Bourennane M 1998 *Phys. Rev.* A **58** 4394 Yang C P, Chu S and Han S 2004 *Phys. Rev.* A **70** 022329\
Yang C P, Han S 2005 *Phys. Lett.* A **343** 267 Deng F G, Li C Y, Li Y S, Zhou H Y and Wang Y 2005 *Phys. Rev.* A **72** 022338 Man Z X, Xia Y J and An N B 2007 *J. Phys. B: At. Mol. Opt. Phys.* **40** 1767\
Li X H, Zhou P, Li C Y, Zhou H Y and Deng F G 2006 *J. Phys. B: At. Mol. Opt. Phys.* **39** 1975 Man Z X, Xia Y J and An N B 2007 *Phys. Rev.* A **75** 052306 Kenigsberg D and Mor T 2006 *Preprint* quant-ph/0609028 An N B 2003 *Phys. Rev.* A **68** 022321 Zhang Z J and Man Z X 2005 *Phys. Lett.* A **341** 55\
Zhang Z J 2006 *Phys. Lett.* A **352** 55 Zhao Z, Chen Y A, Zhang A N, Yang T, Briegel H J and Pan J W 2004 *Nature* **430** 54 Hillery M, Bužek V and Berthiaume A 1999 *Phsy. Rev.* A **59** 1829\
Deng F G, Li X H, Li C Y, Zhou P and Zhou H Y 2005 *Phys. Rev.* A **72** 044301\
Markham D and Sanders B C 2008 *Phys. Rev.* A **78** 042309 Wang X W and Yang G J 2009 *Phys. Rev.* A **79** 062315 SaiToh A, Rahimi R and Nakahara M 2009 *Phys. Rev.* A **79** 062313 Ben-Or M, Crépeau C, Gottesman D, Hassidim A and Smith A 2006 *Proc. 47th Annual IEEE Symposium on the Foundations of Computer Science* (FOCS ’06) p249-260 (IEEE Press) Aoun B and Tarifi M 2004 *Preprint* quant-ph/0401076 Biham E, Huttner B and Mor T 1996 *Phys. Rev.* A **54** 2651\
Townsend P D 1997 *Nature* **385** 47\
Bose S, Vedral V and Knight P L 1998 *Phys. Rev.* A **57** 822 Wang X W and Yang G J 2009 *Quantum Inf. Process.* **8** 319 Cleve R, Gottesman D and Lo H K 1999 *Phys. Rev. Lett.* **83** 648 Horodecki M, Horodecki P and Horodecki R 1999 *Phys. Rev.* A **60** 1888\
Lee J and Kim M S 2000 *Phys. Rev. Lett.* **84** 4236\
Bandyopadhyay S and Sanders B C 2006 *Phys. Rev.* A **74** 032310 Greenberger D M, Horne M A, Shimony A and Zeilinger A 1990 *Am. J. Phys.* **58** 1131 Prevedel R, Cronenberg G, Tame M S, Paternostro M, Walther P, Kim M S and Zeilinger A 2009 *Phys. Rev. Lett.* **103** 020503\
Wieczorek W, Krischek R, Kiesel N, Michelberger P, Tóth G and Weinfurter H 2009 *Phys. Rev. Lett.* **103** 020504 Zhang Q, Goebel A, Wagenknecht C, Chen Y A, Zhao B, Yang T, Mair A, Schmiedmayer J and Pan J W 2006 *Nat. Phys.* **2** 678 Yeo Y and Chua W K 2006 *Phys. Rev. Lett.* **96** 060502 Chen P X, Zhu S Y and Guo G C 2006 *Phys. Rev.* A **74** 032324 Wang X W, Shan Y G, Xia L X, Lu M W 2007 *Phys. Lett.* A **364** 7 Briegel H J and Raussendorf R 2001 *Phy. Rev. Lett.* **86** 910 Dür W, Vidal G and Cirac J I 2000 *Phys. Rev.* A **62** 062314\
Cornelio M F and de Toledo Piza A F R 2006 *Phys. Rev.* A **73** 032314\
Acín A, Andrianov A, Costa L, Jané E, Latorre J I and Tarrach R 2000 *Phys. Rev. Lett.* **85** 1560 Verstraete F, Dehaene J, De Moor B and Verschelde H 2002 *Phys. Rev.* A **65** 052112\
Lamata L, León J, Salgado D and Solano E 2007 *Phys. Rev.* A **75** 022318\
Li D, Li X, Huang H and Li X 2007 *Phys. Rev.* A **76** 052311 Osterloh A and Siewert J 2005 *Phys. Rev.* A **72** 012337\
Lamata L, León J, Salgado D and Solano E 2006 *Phys. Rev.* A **74** 052336 Rigolin G, de Oliveira T R and de Oliveira M C 2006 *Phys. Rev.* A **74** 022314 Hein M, Eisert J and Briegel H J 2004 *Phys. Rev.* A **69** 062311\
Schlingemann D and Werner R F 2001 *Phys. Rev.* A **65** 012308 Bodiya T P and Duan L M 2006 *Phys. Rev. Lett.* **97** 143601\
Browne D E and Rudolph T 2005 *Phys. Rev. Lett.* **95** 010501\
Nielsen M A 2004 *Phys. Rev. Lett.* **93** 040503 Lu C Y, Zhou X Q, Gühne O, Gao W B, Zhang J, Yuan Z S, Goebel A, Yang T and Pan J W 2007 *Nat. Phys.* **3** 91 Chen L and Chen Y X 2006 *Phys. Rev.* A **74** 062310 Wang X W and Yang G J 2008 *Phys. Rev.* A **78** 024301 Chi D P, Choi J W, Kim J S, Kim T and Lee S 2008 *J. Phys. A: Math. Theor.* **41** 255309 von Neumann J and Morgenstern O 1947 *The Theory of Games and Economic Behaviour* (Princeton University Press, Princeton) Dawkins R 1976 *The Selfish Gene* (Oxford University Press, Oxford) Myerson R B 1991 *Game Theory: An Analysis of Conflict* (MIT Press, Cambridge)\
Nash J 1950 *Proc. Nat. Acad. Sci.* **36** 48
[^1]: E-mail: yanggj@bnu.edu.cn
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abstract: 'Recently, a new kind of $f(z)$ theory is proposed to provide a different perspective for the development of reliable alternative models of gravity in which the $f(R)$ Lagrangian terms are reformulated as a polynomial parameterizations $f(z)$. In the previous study, the parameters in the $f(z)$ models have been constrained by using cosmological data. In this paper, these models will be tested by the observations in the solar system. After solving the Ricci scalar as a function of the redshift, one could obtain $f(R)$ that could be used to calculate the standard Parameterized-Post-Newtonian (PPN) parameters. We find that some models are consistent with or favored by the tests, while other ones are not.'
author:
- 'Ji-Yao Wang'
- 'Chao-Jun Feng'
- 'Xiang-Hua Zhai'
- 'Xin-Zhou Li'
title: 'Solar System Tests of a New Class of $f(z)$ Theory'
---
Introduction
============
Einstein’s general relativity (GR) has been successful in predicting many phenomenologies in the universe and the solar system. In the past 20 years, more and more astronomical observations have strongly confirmed that the universe is under accelerating expansion[@Riess:1998cb; @Perlmutter:1998np; @Spergel:2003cb; @Eisenstein:2005su; @Kowalski:2008ez; @Aghanim:2018eyx; @Hinshaw:2012aka]. However, ordinary matters can only drive a decelerating universe. To explain the accelerating, a kind of exotic component in the universe is needed called the dark energy.
Another way to drive the accelerating expansion of the universe is to modify Einstein’s gravity theory. $f(R)$ theory is a kind of such modified gravity theories. In the $f(R)$ theory, the Einstein-Hilbert action is replaced by a function of $f(R)$. When $f(R)=R$, it is just the Einstein’ gravity. Such a modified theory gives a geometrical explanation for the accelerating expansion of the universe [@Sotiriou:2008ve; @Clifton:2006kc; @Dunsby:2010wg]. Some famous $f(R)$ models have been deeply studied, such as $R+\alpha^2 R$ [@Teyssandier:1983zz], $R+\mu/R$[@Dick:2003dw], see also [@Sawicki:2007tf]. Recently, a new kind of $f(R)$ theory is proposed[@Lazkoz:2018aqk], in which the $f(R)$ Lagrangian terms are reformulated as a polynomial parameterizations $f(z)$. It provides a new and different perspective for the development of reliable alternative models of gravity. Cosmological data have been used to constrain the parameters in the $f(z)$ models.
There are many experiments that could be used to test gravity theories in a relatively high accurate level, including those in the Solar system[@Jin:2006if; @Berry:2011pb; @Lin:2016nvj], such as the gravitational redshift[@Lebach:1995zz], the perihelion advance of Mercury[@Gai:2012ws], the Shapiro time delay[@Shapiro:2004zz] and the Nordevert Effect[@Nordtvedt:1968qr]. As is known to all, general relativity is well consistent with the solar system tests. The parametric post Newtonian[@Nordtvedt:1972zz] limit measures the deviations of modified theories of gravity with respect from the general relativity, and it connects the observations with some parameters in the gravitational potential, i.e. the Parameterized-Post-Newtonian (PPN) parameters. Therefore, it has become a useful framework to test the theories of gravity in the solar system.
In this paper, the $f(z)$ theory will be tested in the solar system. After solving the Ricci scalar as a function of the redshift, one could obtain $f(R)$ that could be used to calculate the standard PPN parameters. We find that some models are with or favored by the tests, while some ones are not, which may need the help of some mechanisms like the chameleon mechanism[@Waterhouse:2006wv] to pass the solar system tests.
The structure of this paper is as follows. In Section \[sec:fzr\], we obtain the equation for the Ricci scalar as a function of the redshift. And then, we solve this equation for each model proposed in Ref.[@Lazkoz:2018aqk], in which every model has an explicit formalism of $f(z)$. In Section\[sec:test\], we perform the solar system test on these models. The influence of the variations of parameters in the models is also discussed. Finally, discussions and conclusions will be given in Section \[sec:conclusion\].
From $f(z)$ model to $f(R)$ {#sec:fzr}
===========================
The most general $f(R)$ modified gravity theory is described by the following action: $$\mathcal{S}=\int d^4x \sqrt{-g}\bigg[f(R)+\mathcal{L}_m\bigg]$$ where $g$ is the metric determinant and $\mathcal{L}_m$ is the Lagrangian of matter component. Here we use the units $8\pi G=1$. By varying the action with respect to the metric $g_{\mu\nu}$, one obtains the equations of motion as $$\label{eq:eom}
R_{\mu\nu}f_R-\frac{1}{2}g_{\mu\nu}f+(g_{\mu\nu}\nabla_\alpha\nabla^{\alpha}-\nabla_\mu\nabla_\nu)f_R=T_{\mu\nu}^m\,,$$ where $f_R\equiv df/dR$ and $T^m_{\mu\nu}$ is the stress energy tensor of the matter. The FRW metric that describes a homogeneous and isotropic flat universe is given by $$ds^2 = -dt^2 + a(t)^2\bigg[dr^2+r^2(d\theta^2 + \sin^2\theta d\phi^2)\bigg]\,,$$ where $a(t)$ is the scale factor. From Eq.(\[eq:eom\]) with the FRW background, one can obtain the modified Friedmann equations: $$\begin{aligned}
H^2&=&\frac{1}{3f_R}\bigg(\rho_m+\frac{Rf_R-f}{2}-3H\dot{R}f_{2R}\bigg)\,,\label{eq:eom1}\\
-3H^2-2\dot{H}&=&\frac{1}{f_R}\bigg[\dot{R}^2f_{3R}+(2H\dot{R}+\ddot{R})f_{2R}+\frac{1}{2}(f-Rf_R)\bigg]\,.\label{eq:eom2}\end{aligned}$$
In Ref.[@Lazkoz:2018aqk], the authors have expressed $f(R)$ as a function of the redshift $z$, with $1+z=1/a$. The derivatives of $f(R)$ with respect to $R$, and of $R$ with respect to time are provided in terms of derivatives with respect to the redshift $z$. Therefore, one can obtain the Hubble parameter $H(z)$ from a given $f(z)$ model, and use cosmological observational data such as the Type Ia Supernovae to constrain the parameters in these $f(z)$ models. In Ref.[@Lazkoz:2018aqk], the authors have suggested eight ansatzes for $f(z)$: $$\begin{aligned}
f(z)_{\textbf{Model} 1}&=&f_0 + f_3 (1+z)^3 \,,\label{eq:m1}\\
f(z)_{\textbf{Model} 2}&=&f_0 + f_1(1+z)+f_2 (1+z)^{2} +f_3(1+z)^3\,,\label{eq:m2}\\
f(z)_{\textbf{Model} 3}&=&f_0 + f_2(1+z)^2+f_3(1+z)^3\,,\label{eq:m3}\\
f(z)_{\textbf{Model} 4}&=&f_0 + f_1(1+z) + f_3(1+z)^3\,,\label{eq:m4}\\
f(z)_{\textbf{Model} 5}&=&f_{12} (1+z) ^ {1/2}+f_3(1+z)^3\,,\label{eq:m5}\\
f(z)_{\textbf{Model} 6}&=&f_{12} (1+z) ^ {1/4}+f_1(1+z)+f_2(1+z)^2+f_3(1+z)^3\,,\label{eq:m6}\\
f(z)_{\textbf{Model} 7}&=&f_{14} (1+z) ^ {1/4}+f_3(1+z)^3\,,\label{eq:m7}\\
f(z)_{\textbf{Model} 8}&=&f_{14} (1+z) ^ {1/4}+f_1(1+z)+f_2(1+z)^2+f_3(1+z)^3\,,\label{eq:m8}\end{aligned}$$ where $f_{i}, i\in\{0,1,2,3,12,14\}$ are constant coefficients determined by observations. In the following, we call the above as Model $1\sim8$.
In this paper, we would like to test these $f(z)$ models in the solar system observations. So the function of $f(R)$ should be solved for a given $f(z)$ model. Then, by using the Eqs.(\[eq:eom1\]) and (\[eq:eom2\]), we eliminate the Hubble parameter $H(z)$ and get the equation of $R$ as the following: $$\begin{aligned}
&&D_0 (R_{3z}R_z^2-2 R_{2z}^2R_z)
+ R_{3z}R_zR
-3 R_{2z}^2R+ D_3 R_{2z} R_z^2+D_5 R_{2z}R_zR + D_6R_z^3 +D_7R_z^2R = 0\,,\label{eq:eqr}\end{aligned}$$ where $$\begin{aligned}
D_0&=& \frac{(2\rho_m-f)}{f_z}\,,\\
D_3&=&
\frac{4 f_{2 z} \rho _m}{f_z^2}+\frac{2 \rho _m}{f_z(1+z)}-\frac{4 f}{(1+z) f_z}-\frac{2 f f_{2 z}}{f_z^2}\,,\\
D_5&=& \frac{4 f_{2 z}}{f_z}+\frac{1}{1+z}\,,\\
D_6&=&\frac{2 \rho _m}{(1+z)^2 f_z}-\frac{2 f_{3 z} \rho _m}{f_z^2}-\frac{2 f_{2 z} \rho _m}{(1+z)f_z^2}+\frac{4 f f_{2 z}}{(1+z) f_z^2}+\frac{f f_{3 z}}{f_z^2}-\frac{4 f}{(1+z)^2 f_z}\,,\\
D_7&=&-\frac{f_{2 z}^2}{f_z^2}-\frac{f_{2 z}}{f_z(1+z)}-\frac{f_{3 z}}{f_z}+\frac{2}{(1+z)^2}\,.\end{aligned}$$ The subscript $z$ denotes the derivatives with respect to the redshift $z$, i.e. $f_{z}=df/dz, f_{2z}=d^2f/dz^2, f_{3z}=d^3f/dz^3$ and $R_{z}=dR/dz, R_{2z}=d^2R/dz^2, R_{3z}=d^3R/dz^3$. Therefore, for a given $f(z)$ model, one obtains $R(z)$ from Eq.(\[eq:eqr\]), then equations $(f,R)=(f(z), R(z))$ form a parametric representation of the function $f(R)$. Here we have used the following relations: $$\begin{aligned}
R
&=&-3(H^2)_z(1+z)+12H^2\,,\\
R_z&=&9(H^2)_z-3(1+z)(H^2)_{2z}\,,\\\
R_{2z}&=&6(H^2)_{2z}-3(1+z)(H^2)_{3z}\,.\end{aligned}$$ and $$\begin{aligned}
f_R &=& R_z^{-1}f_z\,,\\ \label{eq:fr}
f_{2R}& =& \left(f_{2z}R_z-f_zR_{2z} \right) R _z^{-3}\,,\\\label{eq:f2r}
f_{3R}& =& \frac {f_{3z}} {R_z^3} - \frac {f_z R_{3z} + 3f_{2z} R_{2z} } {R_z^4} + \frac {3f_z R_{2z}^ 2} {R_z^5}\,.\end{aligned}$$ The Friedmann equation becomes: $$\begin{aligned}
H^2 = \frac{D_0R_z+R}{6} \bigg[ 1- (1+z) \left(\frac{f_{2z}}{f_z}-\frac{R_{2z}}{R _z} \right) \bigg]^{-1} \,.\end{aligned}$$
Observational Tests from the Solar System {#sec:test}
=========================================
In this section, we will performance some observational tests on Model 2-8 from Eqs.(\[eq:m1\])$-$(\[eq:m8\]). Model $1$ has an exact solution with $f_0=6(1-\Omega_m), f_3=3\Omega_m$: $$\begin{aligned}
R(z)=12(1-\Omega_m)+3\Omega_m(1+z)^3 \,,\end{aligned}$$ where $\Omega_i$ are the relative densities of the components and hereafter the subscript $m$ denotes the dust matter. So the function of $f(R)$ is $$f(R) = R -6(1-\Omega_m) \,,$$ which is just the $\Lambda$CDM Model. For Model $2-8$, one usually can not obtain the exact solution of $R(z)$ through Eq.(\[eq:eqr\]), then the numerical approach is needed to solve this equation. To numerically solve Eq.(\[eq:eqr\]), we take the same initial conditions as those in Ref.[@Lazkoz:2018aqk]. Once the solution of $R(z)$ is found, one can obtain $f(R)$ immediately. To clearly see the differences between each model, we plot $\log f_i/\log f_1, (i\in {2\cdots 8})$ as a function of $\log R$ in Fig.\[fig:1\].
![The comparison of each solution to Model 1 is represented as $\log f_i/\log f_1, (i\in {2\cdots 8})$ v.s. $\log R$.[]{data-label="fig:1"}](1.png){width="0.6\linewidth"}
From Fig.\[fig:1\], one can clearly see that the differences of each model are obvious at $z\rightarrow 0$, but different models reach a same point at high red shift except model 1, as the modified terms works.
The GR theory is very successful in predicting the behavior of the gravitational phenomena in the Solar System, so every kind of generation of GR proposed to explain the accelerating expansion of the universe, such as the $f(R)$ theories, should be tested in the Solar System.
Usually, one expands about the GR solutions up to some perturbation orders when taking into the account deviation from GR. In the following, we take the standard PPN[@Nordtvedt:1972zz] expansion of the Schwarzschild metric: $$\begin{aligned}
ds^2=-\left[1-2\frac{GM}{r} + 2(\beta-\gamma) \left(\frac{GM}{r}\right) ^2 \right] dt^2+\left[1+2 \gamma \frac{GM}{r} \right]dr^2+ r^2 d\Omega^2 \,,\end{aligned}$$ where $\alpha$, $\beta$ and $\gamma$ are dimensionless parameters known as the Edditon parameters, which describe the deviations from GR. It is evident that the standard GR solution corresponds to the case $\beta=\gamma=1$. The parameter $\gamma$ measures how the space is curved by unit mass and it is also connected with time delay or the effect of light deflection, while the parameter $\beta$ measures how much the non-linearity is in gravitational superposition, which can be measured though Nordtvedt effect and the perihelion shift.
The expression of PPN-parameters can be extended from the definitions in the scalar-tensor theories[@Capozziello:2006jj], since they could be rigorously compared: $$\begin{aligned}
\gamma - 1& = & -\frac{\xi^2 }{f_zR_z^5+2\xi^2 } \label{eq:gamma}\\
\beta - 1 & =& \frac{1}{4}\bigg[ \frac{f_zR_z\xi}{2f_zR_z^5+3\xi^2}\bigg]\gamma_z \\
\gamma_z&\equiv& \frac{d \gamma}{d z} = -\frac{2\xi\xi_z }{f_zR_z^5+2\xi^2 }+ \frac{R_z^4\xi^2(\xi+ 6f_zR_{2z})+4\xi^3\xi_z}{(f_zR_z^5+2\xi^2)^2}\,,\end{aligned}$$ where we have used Eqs.(\[eq:fr\]) and (\[eq:f2r\]). Here the function $\xi$ is defined by $$\xi(z) \equiv f_{2z} R_z - f _ {z} R _{2z}\,,$$ and then we have $\xi_z = f_{3z} R_z-f_zR_{3z}$. As usual, the uncertainties of these parameters are given by $$\begin{aligned}
\label{eq:uncer}
\sigma_\gamma = \sqrt{\sum_{i} \left(\frac{\delta \gamma}{\delta f_i}\right)^2\sigma^2_{f_i} } \,,\quad \sigma_\beta = \sqrt{\sum_{i} \left(\frac{\delta \beta }{\delta f_i}\right)^2\sigma^2_{f_i} } \,,\quad i \in \{0,1,2,3,12,14\} \,.\end{aligned}$$ The variations of $\gamma, \beta$ can be obtained by using the following equations: $$\begin{aligned}
\delta\gamma &=&\frac{R_z^4\xi}{(f_zR_z^5+2\xi^2)^2}\bigg[-2 f_zR_z^2\delta f_{2z} + R_z(2 f_z R _{2z}+\xi )\delta f_{z} - f_z(2 f_{2z}R_z- 5\xi)\delta R_z +2 f_z^2R_z \delta R_{2z}\bigg] \,,\label{eq:v1} \\
\delta \beta & =&
\nonumber \frac{\gamma_z}{4(2f_zR_z^5+3\xi^2)^2}\bigg[ f_zR_z(2f_zR_z^5-3\xi^2)(R_z\delta f_{2z} - f_z \delta R_{2z} ) + R_z^2(3\xi^2f_{2z}-2f_z^2R_z^4R _{2z})\delta f_z \\
&&f_z^2(2R_z^6f_{2z}-3\xi^2R_{2z}-8R_z^5\xi)\delta R_z \bigg]+ \frac{1}{4}\bigg( \frac{f_zR_z\xi}{2f_zR_z^5+3\xi^2}\bigg)\delta\gamma_z \,, \label{eq:v2} \\
\delta\gamma_z
%&=& -\frac{2\delta\xi\xi_z+2\xi\delta\xi_z }{f_zR_z^5+2\xi^2 }+ \frac{2\xi\xi_z (\delta f_zR_z^5+ 5f_zR_z^4\delta R_z+4\xi\delta \xi )}{(f_zR_z^5+2\xi^2)^2 }\\
%&&+ \frac{4R_z^3\delta R_z\xi^2(\xi+ 6f_zR_{2z})+2R_z^4\xi\delta \xi(\xi+ 6f_zR_{2z})+R_z^4\xi^2(\delta \xi+ 6\delta f_zR_{2z}+6f_z\delta R_{2z})+12\xi^2\delta \xi\xi_z+4\xi^3\delta\xi_z}{(f_zR_z^5+2\xi^2)^2} \\
%&&-2\frac{R_z^4\xi^2(\xi+ 6f_zR_{2z})+4\xi^3\xi_z}{(f_zR_z^5+2\xi^2)^3}(\delta f_zR_z^5+5f_zR_z^4\delta R_z+4\xi\delta \xi)\\
\nonumber
&=&\frac{1}{{(f_zR_z^5+2\xi^2)^2}}\bigg[\bigg( 2\xi_z(f _zR_z^5+2\xi^2)+8\xi^2\xi_z-8\xi\frac{R_z^4\xi^2(\xi+6f_zR_{2z})+4\xi^3\xi_z}{f_zR_z^5+2\xi^2}\\
\nonumber
&&+(R_z^4\xi^2+2R_z^4\xi(\xi+6f_zR_{2z})+12\xi^2\xi_z) \bigg)\delta \xi +\bigg( 4\xi^3+2\xi \bigg)\delta \xi_z\\
\nonumber
&&+\bigg( 2\xi\xi_z R_z^5+6R_z^4 R_{2z}\xi^2-2\frac{R_z^4\xi^2(\xi+6f_zR_{2z})+4\xi^3\xi_z}{(f_zR_z^5+2\xi^2)}(R_z^5+4\xi) \bigg)\delta f_z\\
&&+\bigg( 10\xi\xi_z f_zR_z^4-10\frac{R_z^4\xi^2(\xi+6f_zR_{2z}+4\xi^3\xi_z)}{(f_zR_z^5+2\xi^2)}f_zR_z^4+4R_z^3\xi^2(\xi+6f_zR_{2z}) \bigg)\delta R_{z}+6f_z\xi^2R_z^4 \delta R_{2z}\bigg]\,, \label{eq:v3}\end{aligned}$$ where the variations of $f,f_z,f_{2z}$ and $f_{3z}$ could be easily obtained by using Eqs.(\[eq:m2\]-\[eq:m8\]). For instance, $$\begin{aligned}
\delta f&=& \delta f_0 + \delta f_1(1+z) + \delta f_3(1+z)^3 \,,\\
\delta f_z&=& \delta f_1+ 3\delta f_3(1+z)^2\,,\\
\delta f_{2z}&=& 6\delta f_3(1+z)\,,\\
\delta f_{3z}&=& 6 \delta f_3\,,\end{aligned}$$ for Model 4. However, to get the variations of $R,R_z,R_{2z}$ and $R_{3z}$, one needs to solve the following equation for $\delta R$: $$\label{eq:pert}
A_0+A_1 \delta R_{3z} + A_2 \delta R_{2z} + A_3 \delta R_z +A_4 \delta R = 0\,,$$ with the coefficients $$\begin{aligned}
A_0&=&-\bigg[ (\hat{R}_{3z}\hat{R}_z^2-2 \hat{R}_{2z}^2\hat{R}_z)\delta D_0+ \hat{R}_{2z} \hat{R}_z^2 \delta D_3+ \hat{R}_{2z}\hat{R}_z\hat{R} \delta D_5 + \hat{R}_z^3 \delta D_6 + \hat{R}_z^2\hat{R} \delta D_7 \bigg]\,,\\
A_1 &=& \hat{D}_0 \hat{R}_z^2+\hat{R}_z \hat{R}\,,\\
A_2 &=& -4\hat{D}_0\hat{R}_z\hat{R}_{2z}+\hat{D}_3 \hat{R}_z^2+\hat{D}_5\hat{R}_z\hat{R}-6\hat{R}_{2z} \hat{R}\,,\\
A_3 &=& -2\hat{D}_0 R_{2z}^2+2\hat{D}_3 \hat{R}_{2z}\hat{R}_z+\hat{D}_5\hat{R}_{2z}\hat{R}+3\hat{D}_6\hat{R}_z^2+2\hat{D}_7\hat{R}_z\hat{R}+\hat{R}_{3z}\hat{R}\,,\\
A_4 &=& \hat{D}_5\hat{R}_{2z}\hat{R}_z+\hat{D}_7\hat{R}_z^2+ \hat{R}_{3z}\hat{R}_z-3\hat{R}_{2z}^2 \,,\\\end{aligned}$$ where $$\begin{aligned}
\delta D_0&=& \bigg(\hat{D}_0\delta f_z +\delta f\bigg)/\hat{f_z}\,,\\
\delta D_3&=& 2 \frac{\hat f_{2z}}{\hat f_z^2}\delta f+\frac{4\delta f}{\hat f_z(1+z)}+\frac{(4\rho_m-2\hat f)\hat f_{2z}}{\hat f_z^2}\left(\frac{\delta f_z}{\hat f_z}- \frac{\delta f_{2z}}{\hat f_{2z}} \right)+\hat{D_3}\frac{\delta f_z}{\hat f_z}\,,\\
\delta D_5&=& \frac{4\hat f_{2z}}{\hat f_z^2}\delta f_z- \frac{4\delta f_{2z}}{\hat f_z} \,,\\
\delta D_6&=&(2\rho_m-\hat f) \bigg(\frac{\delta f_{3z}}{\hat f_z^2} -\frac{\hat f_{3z}}{\hat f_z^3}\delta f_z \bigg)
+ \delta f\bigg(\frac{4}{\hat f_z(1+z)^2} -\frac{4\hat f_{2z}}{\hat f_z^2(1+z)} -\frac{\hat f_{3z}}{\hat f_z^2} \bigg)\\\nonumber
&&+(2\rho_m- 4\hat f)\bigg(\frac{\delta f_{2z}}{\hat f_z^2(1+z)}-\frac{\hat f_{2z}\delta f_z}{\hat f_z^3(1+z)} \bigg)+ \hat{D_6}\frac{\delta f_z}{\hat f_z}\,,\\
\delta D_7&=& \frac{\delta f_{3z}}{\hat f_z} -\frac{\hat f_{3z}}{\hat f_z^2}\delta f_z +\frac{\delta f_{2z}}{\hat f_z(1+z)} -\frac{\hat f_{2z}}{\hat f_z^2(1+z)}\delta f_z +\frac{2\hat f_{2z}\delta f_{2z}}{\hat f_z^2}-2\frac{\hat f_{2z}^2}{\hat f_z^3}\delta f_z \,.\end{aligned}$$ It is hardly to solve Eq.(\[eq:pert\]) exactly, however, for Model 1, one could get the asymptotic solutions. In the limit of $z\rightarrow 0$, the coefficients $A_0\sim A_4$ all become constants, so we have a constant solution $$\delta R|_{z\rightarrow 0} = -\frac{A_0}{A_4} =\frac{ 4+15\Omega _m }{ \Omega _m }\delta f_{3} +6(\delta f_0+\delta f_3)\,.$$ In the limit of $z\rightarrow \infty$, $A_1$ is the most important coefficient, then Eq.(\[33\]) becomes $$\begin{aligned}
\label{33}
\delta f_{3z} + \delta R_{3z}=0\,,\end{aligned}$$ then we get $$\delta R|_{z\rightarrow \infty} = -\delta f \,.$$ Therefore, the asymptotic behavior of $\delta R$ is regular in Model 1. In fact, this conclusion is also valid in Model 2-8.
Data Description
----------------
To test the $f(z)$ models Eqs.(\[eq:m2\])-(\[eq:m8\]) in the solar system, we use the data from the Very Long Baseline Array (VLBA) at 43, 23 and 15 GHz, in which the gravitational bending of radio waves is observed and then the Eddington parameter $\gamma-1$ is constrained by [@Fomalont:2009zg]: $$\label{eq:gamma}
|\gamma-1| \leq 2\times10^{-4} \,.$$ From the observations of the the perihelion advance of Mercury, $\beta-1$ is constrained by [@Will:2005va]: $$\label{eq:beta}
|\beta-1|\leq 0.0023 \,.$$ As is known, the Nordtvert effect[@Nordtvedt:1968qr], as an effect that relates to the difference between the inertial mass $M(I)$ and the gravitational mass $M(G)$, $$\frac{M(G)_i }{M(I)_i}=1-\eta_{\mathrm{N}} \frac{1}{M_i c^2} \int \frac{G\rho(\vec{r})\rho (\vec{r}' )\:d^3 rd^3 r'}{2\:|\vec{r}-\vec{r} '|}\,,$$ can be described by the combination of $\gamma$ and $\beta$[@Williams:2004qba]: $$\label{eq:nord}
\eta_{\mathrm{N}}= 4 \beta-\gamma-3\,,$$ which could be observed by the Lunar Laser Ranging Tests (LLT). This parameter $\eta_{\mathrm{N}}$ could be regarded as as another PPN parameter, which is constrained by [@Williams:2004qba]: $$\label{eq:eta}
-1.3 \times 10^{-4} \leq\eta_{\mathrm{N}}\leq 0.9\times 10^{-4} \,.$$ We summarized these data in Table.\[tab:data\].
**PPN Parameters** **Related Phenomenon** **Experiment** **Result**
--------------------- -------------------------------------- ------------------------ ------------------------------------------------
Time Delay Cassini mission $(2.1\pm2.3)\times10^{-5}$ [@Bertotti:2003rm]
Gravitational Bending of Radio Waves VLBA $\pm 2\times10^{-4}$[@Fomalont:2009zg]
$\beta-1$ Perihelion Advance of Mercury Solar System Ephemeris $\pm 0.0023$[@Will:2005va]
$\eta_{\mathrm{N}}$ Nordtvert Effect LLT $(-0.2\pm1.1)\times 10^{-4}$[@Hofmann:2018myc]
: The observational values of PPN parameters. \[tab:data\]
Test Results
------------
By taking the values of $f_i$ in Table 1 of Ref.[@Lazkoz:2018aqk], one can obtain the values of PPN parameters with their uncertainty through Eqs.(\[eq:gamma\])-(\[eq:v3\]). We summarized the results in Table \[tab:res\].
**** **$\gamma-1$** **$\beta-1$** $\eta_{\mathrm{N}}$
--------------- -------------------------------- ------------------------------------ -----------------------------------
**Model $2$** $-0.00555624\pm0.0600337$ $0.000146599\pm 0.00210522 $ $0.00555624\pm 0.0598258 $
**Model $3$** $ -0.0002922154\pm 0.0036319$ $-0.000002163\pm0.0000389346 $ $0.0002835\pm 0.00031698 $
**Model $4$** $ -0.00161326\pm 0.00180823 $ $0.000025044\pm 0.00000000173558 $ $ -0.001713434 \pm 0.00180053 $
**Model $5$** $ -0.0448395\pm 0.0106936 $ $-0.00431988\pm 0.00180008 $ $ 0.02755998 \pm 0.0079627 $
**Model $6$** $ -0.0484702\pm 0.0137682 $ $-0.00485503\pm 0.00242667 $ $ 0.02905008 \pm 0.00976441 $
**Model $7$** $ -0.0253158 \pm 0.00887438 $ $-0.000210559\pm 0.00103595$ $ 0.024473564 \pm 0.00784753 $
**Model $8$** $ -0.0221699 \pm 0.0165683$ $-0.00143153 \pm 0.00292251$ $ 0.01644378 \pm 0.011741 $
: The values of PPN parameters and their $1\sigma$ errors. \[tab:res\]
.
In Table \[tab:res\], the central values of $\gamma-1, \beta-1 , \eta_{N}$ are obtained by using the best fitting values of $f_i$ in Ref.[@Lazkoz:2018aqk]. If the center value of one parameter, such as the $|\gamma-1|$, falls within the range given by Eq. (\[eq:gamma\]) (\[eq:beta\]) or (\[eq:eta\]), the corresponding model is regarded as being consistent with observations in respect of that parameter. From Table \[tab:res2\], one can see that Model 2-8 could be hardly favored by observations in respect of $\gamma-1$ and $\eta_N$. When the $1\sigma$ uncertainty of these parameters are taken into account, Model 4 is most favored by observations, while Model 5-8 are not consistent with the solar system observations. We summarized the results in Table.\[tab:res2\].
-------------- -------------- ---------------------- ---------------- ---------------------- --------------- ----------------------
**Best Fit** **Within $1\sigma$** **Best Fit** **Within $1\sigma$** **Best Fit** **Within $1\sigma$**
**Model$2$** $\triangle $ $\triangle $ $\checkmark $ $- $ $\triangle $ $\checkmark $
**Model$3$** $\triangle$ $\triangle $ $\checkmark $ $- $ $\checkmark $ $- $
**Model$4$** $\triangle$ $ \checkmark $ $\checkmark $ $- $ $\triangle $ $\checkmark $
**Model$5$** $\triangle$ $\triangle $ $\triangle $ $\triangle $ $\triangle $ $\triangle $
**Model$6$** $\triangle $ $\triangle $ $\triangle $ $\triangle $ $\triangle $ $\triangle $
**Model$7$** $\triangle$ $\triangle $ $\checkmark $ $- $ $\triangle $ $\triangle $
**Model$8$** $\triangle $ $\triangle $ $\triangle $ $\triangle $ $\triangle $ $\triangle $
-------------- -------------- ---------------------- ---------------- ---------------------- --------------- ----------------------
: The test results of $f(z)$ parametric models. The $\checkmark$ denotes that the model is in consistent with the solar system observations, while the $\triangle$ denotes that the model is not. The “$-$” sign means that the $1\sigma$ error is not necessarily considered while the center value is favored. \[tab:res2\]
From Eqs.(\[eq:m2\])-(\[eq:m8\]), one can see that all models have the parameter $f_3$. Therefore, we also plot the changes of the PPN parameters with respect to $\delta f_3$ in Fig.\[fig:12\] for some models. From Fig.\[fig:12\], one can see that the values of the PPN parameters changed little while $|\delta f_3|$ is larger than its $1\sigma $ error. Therefore, the uncertainty of $\delta f_3$ can hardly change our results. We also checked other parameters in Model 2-8, and got the same conclusion.
Conclusion and Discussion {#sec:conclusion}
=========================
In this paper, we have performed the solar system tests to the $f(z)$ models, which are proposed to explain the accelerating expansion of the universe in Ref.[@Lazkoz:2018aqk]. After solving the equation for the Ricci scalar (\[eq:eqr\]) numerically with $f(z)$ given by Model 2-8 in Eqs.(\[eq:m2\])-(\[eq:m8\]), we calculate the PPN parameters and compare them to recent data. We find that Model 4 in Eq.(\[eq:m4\]) is much more favored by the solar system observations, while Model 5-8 are hardly to be favored. According to the fitting values of the parameters in Model 4 in Ref.[@Lazkoz:2018aqk], one could see that $f_1=2.0\times10^{-4}$ is much smaller than $f_3=0.94$, so Model 4 has a slightly difference to Model 1, i.e., the $\Lambda$CDM model. In the future, the second term in Eq.(\[eq:m4\]) becomes much more important than the third one, i,e. $$1+z < \sqrt{\frac{f_1}{f_3}}-1 \approx 0.015\,.$$ However, the constant term $f_0=4.43$ is already much more larger than the second one at the same time. So the $f_1$ term may not be important.
As the authors of Ref.[@Lazkoz:2018aqk] stressed that $f(z)$ theories may not be the definitive answer to explain why the universe is under accelerating expansion, but it provides a different and interesting perspective on how to relate the modified gravity with observations. We also believe that even Model 5-8 can hardly be favored by the solar system test, there are some mechanisms like the chameleon mechanism that could help the theory to pass the solar system tests. And the future data of BepiColombo Mission[@Serra:2018irk] will improve the precision of PPN parameter and may help us to test the theories of gravitation.
This work is supported by National Science Foundation of China grant Nos. 11105091 and 11047138, “Chen Guang" project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation Grant No. 12CG51, and Shanghai Natural Science Foundation, China grant No. 10ZR1422000.
[999]{}
A. G. Riess [*et al.*]{} \[Supernova Search Team\], Astron. J. [**116**]{} (1998) 1009 doi:10.1086/300499 \[astro-ph/9805201\]. S. Perlmutter [*et al.*]{} \[Supernova Cosmology Project Collaboration\], Astrophys. J. [**517**]{} (1999) 565 doi:10.1086/307221 \[astro-ph/9812133\]. D. N. Spergel [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl. [**148**]{} (2003) 175 doi:10.1086/377226 \[astro-ph/0302209\].
D. J. Eisenstein [*et al.*]{} \[SDSS Collaboration\], Astrophys. J. [**633**]{} (2005) 560 doi:10.1086/466512 \[astro-ph/0501171\]. M. Kowalski [*et al.*]{} \[Supernova Cosmology Project Collaboration\], Astrophys. J. [**686**]{} (2008) 749 doi:10.1086/589937 \[arXiv:0804.4142 \[astro-ph\]\].
N. Aghanim [*et al.*]{} \[Planck Collaboration\], arXiv:1807.06209 \[astro-ph.CO\]. G. Hinshaw [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl. [**208**]{} (2013) 19 doi:10.1088/0067-0049/208/2/19 \[arXiv:1212.5226 \[astro-ph.CO\]\].
T. P. Sotiriou, J. Phys. Conf. Ser. [**189**]{} (2009) 012039 doi:10.1088/1742-6596/189/1/012039 \[arXiv:0810.5594 \[gr-qc\]\]. T. Clifton and J. D. Barrow, Class. Quant. Grav. [**23**]{} (2006) 2951 doi:10.1088/0264-9381/23/9/011 \[gr-qc/0601118\].
P. K. S. Dunsby, E. Elizalde, R. Goswami, S. Odintsov and D. S. Gomez, Phys. Rev. D [**82**]{}, 023519 (2010) doi:10.1103/PhysRevD.82.023519 \[arXiv:1005.2205 \[gr-qc\]\].
P. Teyssandier and P. Tourrenc, J. Math. Phys. [**24**]{} (1983) 2793. doi:10.1063/1.525659 R. Dick, Gen. Rel. Grav. [**36**]{} (2004) 217 doi:10.1023/B:GERG.0000006968.53367.59 \[gr-qc/0307052\].
I. Sawicki and W. Hu, Phys. Rev. D [**75**]{} (2007) 127502 doi:10.1103/PhysRevD.75.127502 \[astro-ph/0702278\].
R. Lazkoz, M. Ortiz-Banos and V. Salzano, Eur. Phys. J. C [**78**]{} (2018) no.3, 213 doi:10.1140/epjc/s10052-018-5711-6 \[arXiv:1803.05638 \[astro-ph.CO\]\].
X. H. Jin, D. J. Liu and X. Z. Li, astro-ph/0610854.
C. P. L. Berry and J. R. Gair, Phys. Rev. D [**83**]{} (2011) 104022 Erratum: \[Phys. Rev. D [**85**]{} (2012) 089906\] doi:10.1103/PhysRevD.85.089906, 10.1103/PhysRevD.83.104022 \[arXiv:1104.0819 \[gr-qc\]\].
R. H. Lin, X. H. Zhai and X. Z. Li, Eur. Phys. J. C [**77**]{}, no. 8, 504 (2017) doi:10.1140/epjc/s10052-017-5074-4 \[arXiv:1610.04956 \[gr-qc\]\]. D. E. Lebach, B. E. Corey, I. I. Shapiro, M. I. Ratner, J. C. Webber, A. E. E. Rogers, J. L. Davis and T. A. Herring, Phys. Rev. Lett. [**75**]{} (1995) 1439. doi:10.1103/PhysRevLett.75.1439
M. Gai, A. Vecchiato, S. Ligori, A. Sozzetti and M. G. Lattanzi, Exper. Astron. [**34**]{} (2012) 165 doi:10.1007/s10686-012-9304-3 \[arXiv:1203.5654 \[astro-ph.IM\]\].
S. S. Shapiro, J. L. Davis, D. E. Lebach and J. S. Gregory, Phys. Rev. Lett. [**92**]{} (2004) 121101. doi:10.1103/PhysRevLett.92.121101
K. Nordtvedt, Phys. Rev. [**169**]{} (1968) 1014. doi:10.1103/PhysRev.169.1014
K. J. Nordtvedt and C. M. Will, Astrophys. J. [**177**]{} (1972) 775. doi:10.1086/151755
T. P. Waterhouse, astro-ph/0611816.
S. Capozziello, A. Stabile and A. Troisi, Mod. Phys. Lett. A [**21**]{} (2006) 2291 doi:10.1142/S0217732306021633 \[gr-qc/0603071\].
B. Bertotti, L. Iess and P. Tortora, Nature [**425**]{} (2003) 374. doi:10.1038/nature01997
E. Fomalont, S. Kopeikin, G. Lanyi and J. Benson, Astrophys. J. [**699**]{} (2009) 1395 doi:10.1088/0004-637X/699/2/1395 \[arXiv:0904.3992 \[astro-ph.CO\]\]. C. M. Will, Living Rev. Rel. [**9**]{} (2006) 3 doi:10.12942/lrr-2006-3 \[gr-qc/0510072\].
F. Hofmann and J. M¨¹ller, Class. Quant. Grav. [**35**]{} (2018) no.3, 035015. doi:10.1088/1361-6382/aa8f7a J. G. Williams, S. G. Turyshev and D. H. Boggs, Phys. Rev. Lett. [**93**]{} (2004) 261101 doi:10.1103/PhysRevLett.93.261101 \[gr-qc/0411113\].
D. Bettoni, J. M. Ezquiaga, K. Hinterbichler and M. Zumalac¨¢rregui, Phys. Rev. D [**95**]{} (2017) no.8, 084029 doi:10.1103/PhysRevD.95.084029 \[arXiv:1608.01982 \[gr-qc\]\].
A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev. D [**79**]{} (2009) 064036 doi:10.1103/PhysRevD.79.064036 \[arXiv:0811.2197 \[hep-th\]\].
J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Phys. Rev. Lett. [**114**]{} (2015) no.21, 211101 doi:10.1103/PhysRevLett.114.211101 \[arXiv:1404.6495 \[hep-th\]\]. H. Rizwana Kausar, L. Philippoz and P. Jetzer, Phys. Rev. D [**93**]{} (2016) no.12, 124071 doi:10.1103/PhysRevD.93.124071 \[arXiv:1606.07000 \[gr-qc\]\]. D. Serra, V. Di Pierri, G. Schettino and G. Tommei, Phys. Rev. D [**98**]{} (2018) no.6, 064059. doi:10.1103/PhysRevD.98.064059
|
---
abstract: 'Online reinforcement learning agents are currently able to process an increasing amount of data by converting it into a higher order value functions. This expansion of the information collected from the environment increases the agent’s state space enabling it to scale up to a more complex problems but also increases the risk of forgetting by learning on redundant or conflicting data. To improve the approximation of a large amount of data, a random mini-batch of the past experiences that are stored in the replay memory buffer is often replayed at each learning step. The proposed work takes inspiration from a biological mechanism which act as a protective layer of human brain higher cognitive functions: active memory consolidation mitigates the effect of forgetting of previous memories by dynamically processing the new ones. The similar dynamics are implemented by a proposed augmented memory replay *AMR* capable of optimizing the replay of the experiences from the agent’s memory structure by altering or augmenting their relevance. Experimental results show that an evolved *AMR* augmentation function capable of increasing the significance of the specific memories is able to further increase the stability and convergence speed of the learning algorithms dealing with the complexity of continuous action domains.'
author:
- 'Mirza Ramicic$^1$[^1]'
- Andrea Bonarini$^2$
- |
$^1$Artificial Intelligence Center\
Faculty of Electrical Engineering\
Czech Technical University in Prague\
Prague, Czech Republic\
$^2$AI and Robotics Lab\
Dipartimento di Elettronica, Informazione e Bioingegneria\
Politecnico di Milano\
Milan, Italy\
ramicmir@fel.cvut.cz, andrea.bonarini@polimi.it
bibliography:
- 'main.bib'
title: Augmented Replay Memory in Reinforcement Learning With Continuous Control
---
Introduction
==============
Studies concerning human and animal learning have identified that the process of encoding new memories into long term storage is not so straightforward as previously thought. Recent studies have found that it involves a process of active memory consolidation, or AMC [@diekelmann2010memory; @rasch2013sleep; @feld2015sleep], that facilitates a better memory integration into the higher level cortical structures and also prevents forgetting previously encoded information. This process occurs while sleeping, a time when the brain is not encoding or perceiving new stimuli and relies on the memories stored in a short-term hippocampal structure when awake. Before their integration in the long term cortical structures, experiences are reactivated or replayed in the hippocampal memory as a part of the active consolidation. The word “active” in AMC implies that, in the process of consolidation, memories are altered in a way that their further integration into the existing knowledge wouldn’t induce forgetting of the previous ones.
The active structural modification of the consolidated memories is selective and, for the memories that are deemed to be the more important ones it will facilitate strengthening to reach a certain retrieval threshold. However, if the memory trace is deemed not strong enough for some memories it will result in their loss[@dumay2016sleep; @schreiner2018gain].
Biological architectures found in human brain and the computational reinforcement learning processes both use a functionally similar mechanism of replay memory. Along the introduction of artificial neural networks, or ANN, as function approximators in temporal-difference, or TD, learning [@lin1993reinforcement], the techniques that aim at their efficient training most commonly use a replay buffer of previous experiences out of which a mini-batch is sampled for re-learning at each time step. This technique has been recently revived in Deep Q-learning[@mnih2013playing; @mnih2015human]. Since in TD approaches the ANN is constantly updated to better represent the state-action value pairs $Q(s,a)$, which govern the agent’s policy $\pi$, the mechanisms involved in its training such as mini-batch replay became increasingly influential to the learning process itself.
Another advantage of the replay memory structure is that, when implemented, it acts as a form of agent’s cognition: depending on the way it is populated, it can alter how the agent perceives the information. In this way, a learning agent is not only concerned about the information it receives from its immediate environment, but also about the way in which this information is interpreted by this cognitive mechanism.
In the proposed approach an effective, but simple, mechanism of replay memory is extended with the ability to actively and dynamically process the information during the replay and thus bringing it closer to the functional characteristics of actual biological mechanisms. The dynamic processing mechanism of *Augmented Memory Replay* or *AMR* presented here is inspired by human *active memory consolidation* and it is capable of altering the importance of specific memories by altering their reward values, thus mimicking the AMC’s process of deeming the memory above the retrieval threshold. In the experiments reported in this paper, the *augmentation* dynamics are evolved over generations of learning agents performing reinforcement learning tasks in various environments. Their fitness function is defined in a straightforward way as their cumulative performance over a specific environment. Experimental results indicate that *AMR* type of memory buffer shows an improvement in learning performance over the standard static replay method in all of the tested environments.
Related Work
============
An extension of DDPG algorithm was proposed by Hausknecht and Stone [@hausknecht2015deep] allowing it to deal with a low level parameterized-continouos action space. However the evaluation of the approach was limited to a single simulated environment of RoboCup 2D Half-Field-Offense [@hausknecht2016half]. Hoothoft et. al [@houthooft2018evolved] proposed a meta-learning approach capable of evolving a specialized loss function for a specific task distribution that would provide higher rewards during its minimization by stochastic gradient descent. The algorithm is capable to produce a significant improvement of the agent’s convergence to the optimal policy but as its the case with the *AMR* approach the evolved improvements are task specific.
In contrast with the distributed methods like Apex which was proposed by Horgan et. al [@horgan2018distributed] which rely on a hundreds of actors learning in their own instance of the environment *AMR* algorithm works with a single instance just like vanilla DDPG [@lillicrap2015continuous] does. This fact has a significant impact on the computational time a specific algorithm induces to the problem.
Wang et. al [@wang2016sample] introduced an approach that is combining the importance or prioritized sampling techniques together with stochastic dueling networks in order to improve the convergence of some continuous action tasks such as Cheetah, Walker and Humanoid.
Another improvement of a vanilla DDPG is presented by Dai et. al [@dai2017boosting] as Dual-Critic architecture where the critic is not updated using the standard temporal-difference algorithms but it’s optimized according to the gradient of the actor.
An approach by Pacella et. al [@pacella2017basic] evolved basic emotions such as fear, used as a kind of motivational drive that governed the agent’s behavior by directly influencing action selection. Similar to the *AMR* approach a population of virtual agents were tested at each generation. In this process, each of the agents evolved a specific neural network that was capable of selecting its actions based on the input; this consisted of temporal information, visual perception and good and bad sensation neurons. Over time, the selection of best performing agents gave rise to populations that adopted specific behavioral drives such as being cautious or fearful as a part of a survival strategy. Contrary to the *AMR* which evolves a cognitive mechanism which only complements the main learning process, in [@pacella2017basic] the genetic algorithm represents the learning process itself.
Another evolutionary approach that is used to complement the main reinforcement learning algorithm was presented in [@singh2010intrinsically]. Similarly to *AMR*, it uses a genetic algorithm to evolve an *optimal reward function* which builds upon the basic reward function in a way that maximizes the agent’s fitness over a distribution of environments. Experimental results show the emergence of an intrinsic reward function that supports the actions that are not in line with the primary goal of the agent. [@schembri2007evolution] also presented an a reinforcement learning approach which relied on a evolved reinforcer in order to support learning atomic meta-skills. The reinforcement was evolved in a *childhood* phase, which equipped the agents with the meta-actions or skills for the use in the *adulthood* phase.
Persiani et al. [@persiani2018working] proposes a cognitive improvement through the use of replay memory structure like *AMR*. The algorithm makes it possible to learn which chunks of agent’s experiences are most appropriate for replay based on their ability to maximize the future expected reward.
A cognitive filter structure was proposed by Ramicic and Bonarini [@ramicic2019selective] able to improve the convergence of temporal-difference learning implementing discrete control rather than a continuous one. It was able to evolve the ANN capable to select whether a specific experience will be sampled into replay memory or not. Unlike *AMR* this approach did not modify the properties of the experiences.
Theoretical Background
======================
Temporal-difference learning
----------------------------
The goal of a reinforcement learning agent is to constantly update the function which maps its state to their actions i.e. its policy $\pi$ as close as possible to the *optimal policy* $\pi^*$. The *optimal policy* is a policy that selects the actions which maximize the future expected reward of an agent in the long run [@sutton1998reinforcement] and it is represented by a function, possibly approximated by an *Artificial Neural Network* or ANN. The process of updating the policy is performed iteratively after each of the consecutive discrete time-steps in which the agent interacts with its environment by executing its action $a_t$ and gets the immediate reward scalar $r_t$ defined by the *reinforcement function*. This iterative step is defined as a transition over *Markov Decision Process*, and represented it by a tuple $ \left ( s_t,a_t,r_t,s_{t+1} \right ) $. After each transition the agent corrects its existing policy $\pi$ according to the optimal action-value function shown in in order to maximize its expected reward within the existing policy. In the approaches that deal with discrete action spaces, such as [@watkins1992q], the agent can follow the optimal policy $\pi^*$ by taking an optimal action $a^*(s)$ which maximizes the optimal action-value function $Q^*(s,a)$ defined by .
$$Q^*(s,a) = \max_{\pi}\mathbb{E}[R_t \vert s_t = s,a_t = a,\pi]\label{eq:optimal}$$
$$\mu(s) = a^*(s) = \max_{a}Q^*(s,a)
\label{eq:action}$$
$$Q^\pi(s,a) = \mathbb{E} \left [ r + \gamma
\max_{a'}Q^\pi(s',a') \vert s,a \right ]
\label{eq:bellman}$$
The correction update to the policy $\pi$ starts by determining how wrong the current policy is with respect to the expectation, or value for the current state-action pair $Q(s,a)$. In case of a discrete action space this expectation of return is defined by the Bellman-optimality equation and it is basically the sum of the immediate reward $r$ and the discounted prediction of a maximum Q-value, given the state $s`$ over all of the possible actions $a`$.
Going continuous
----------------
Maximizing over actions in is not a problem when facing discrete action spaces, because the Q-values for each of the possible actions can be estimated and compared. However, when coping with continuous action values this approach is not realistic: we cannot just explore brute force the values of the whole action space in order to find the maximum. The more recent approach of [@lillicrap2015continuous] eliminates the maximization problem by approximating the optimal action $a^*(s)$ and thus creating a deterministic policy $\mu(s)$ in addition to the optimal state-value function $Q^*(s,a)$. Taking the new approximated policy into consideration the Bellman-optimality equation takes the form of and avoids the inner expectation.
$$Q^\mu(s,a) = \mathbb{E} \left [ r + \gamma
Q^\mu(s',\mu(s')) \vert s,a \right ]
\label{eq:deterministic}$$
In common among all the before mentioned approaches is the concept of *temporal difference*, or TD error, which is basically a difference between the current approximate prediction and the expectation of the Q value. The learning process performs an iterative reduction of a TD error using Bellman-optimality equation as a target, which guarantees the convergence of the agent’s policy to the optimal one given an infinite amount of steps [@sutton1998reinforcement].
Function approximation
----------------------
In order to deal with the increasing dimensionality and continuous nature of state and action spaces imposed by the real-life applications the aforementioned algorithms depend heavily on approximate methods usually implemented using ANN. A primary function approximation makes it possible to predict a $ Q $ value for each of the possible actions available to the agent by providing an agent’s current state as input of the ANN. After each time step, the expected Q value is computed using , and then compared to the estimate that the function approximator provides as its output $ Q(s,a;\Theta) \approx Q^*(s,a) $ by forwarding the state s0 as its input. The difference between the previous estimate of the approximator and the expectation is the TD error. This discrepancy is actually a loss function $ L_i(\Theta_i) $ that can bi minimized by performing a *stochastic gradient descent* on the parameters $\Theta$ in order to update the current approximation of $ Q^*(s,a) $ according to :
$$\label{eq:gradient} \nabla_{\Theta_i}L_i(\Theta_i) =
\left ( y_i - Q(s,a;\Theta_i) \right )\nabla_{\Theta_i}Q(s,a;\Theta_i),$$
where $ y_i = r + \gamma
Q^\mu(s',\mu(s'));\Theta_{i-1}) $ is in fact the Bellman equation defining the target value which depends on an yet another ANN that approximates the policy function $\mu(s)$ in policy-gradient approaches such as [@lillicrap2015continuous]. The update to the policy function approximator $\mu_{\Theta}(s)$ is more straightforward as it is possible to perform a gradient ascent on the respective network parameters $\Theta$ in order to maximize the $Q^\mu(s,a)$ as shown in .
$$\max_{\theta} \underset{s \sim {\mathcal D}}{{\mathrm E}}\left[ Q^\mu(s, \mu_{\theta}(s)) \right]
\label{eq:policy}$$
Model Architecture and Learning Algorithm
=========================================
In this section we propose a new model that combines the learning approaches of genetic algorithm with reinforcement learning order to improve the convergence of the latter. For clarity, the proposed model is separated in two main functional parts: evaluation and evolution.
The evaluation part is defined as a temporal-difference reinforcement problem where a reward function is dynamically modified by the proposed *AMR* block. *AMR* is a function approximator implemented by an ANN, which receives in input characteristics of experience that is perceived by the learning agent, and outputs a single scalar value , used to modify the reinforcement value of the transition.
The architecture of the AFB neural network approximator $ (f) $ consists of three layers: three input nodes fully connected to a hidden layer of four nodes, in turn connected to two softmax nodes to produce the final classification. This ANN is able to approximate the four parameters of the experience, respectively given in input, as TD error, reinforcement $ r_i $, entropy of the starting state $ s_t $, and entropy of the next transitioning state $ s_{t+1} $, to a regression output layer that provides a scalar *augmentation* value or $ A_t(s_t,s_{t+1},r_t) $. The *augmentation* process alters the reward value of each transition by an *augmentation rate* or $\beta$ as shown in .
$$r_t := r_t + \beta A_t
\label{eq:update}$$
{width="100.00000%"}
![Main function approximator ANN implemented in the (d) block of : it receives an N-dimensional state as its input and approximates it to $ Q $ values of each of $ A $ possible actions available to an agent at its output, therefore providing an approximation for $ Q(s,a) $ pairs. []{data-label="fig:neural"}](neural.pdf){width="50.00000%"}
While altering the reward scalar $r_t$ the *AMR* block is able to precisely and dynamically change the amount of influence each transition exerts on the learning process ant thus mimic the aforementioned biological processes [@diekelmann2010memory; @rasch2013sleep; @feld2015sleep].
The second component of the proposed architecture evolves the *AMR* block using a *genetic algorithm*, or $ GA $, in order to maximize its fitness function which is represented by the total learning score received by an agent during its evaluation phase.
Initialize critic network $Q(s,a\vert\Theta^Q)$, actor network $\mu(s\vert\Theta^\mu)$ and augmentation network $A(s,r\vert\Theta^\beta)$ with random weights $\Theta^Q$, $\Theta^\mu$ and $\Theta^\beta$ Initialize target network $Q'$ and $\mu'$ with weights $\Theta^{Q'} \leftarrow \Theta^Q$ and $\Theta^{\mu'} \leftarrow \Theta^\mu$ Initialize replay buffer $R$ Initialize a random process $N$ for action exploration Observe initial state $s_1$ Select action $a_t=\mu(s_t\vert\Theta^\mu) + N_t$ according to the current policy and exploration noise Execute action $a_t$ and observe reward $r_t$ and new state $s_{t+1}$ Augment the reward $r_t\leftarrow r_t + A_t(s_t,s_{t+1},r_t)$ according to the augmentation network parameters $\Theta^\beta$ Store transition $(s_t,a_t,r_t,s_{t+1})$ in R Sample a random minibatch of $S$ transitions $(s_t,a_t,r_t,s_{t+1})$ from R Set $y_i = r_i + \gamma Q'(s_{i+1},\mu'(s_{i+1}\vert\Theta^{\mu'})\vert\Theta^{Q'})$ Update critic by minimizing the loss $L=\frac{1}{S}\sum_{i}(y_i - Q(s_i,a_i\vert\Theta^{Q'}))^2$ Update the actor policy using the sampled policy gradient $\nabla_{\Theta^{\mu}}J\approx\frac{1}{S}\sum_{i}\nabla_{a}Q(s,a\vert\Theta^Q)\vert_{s=s_i,a=\mu(s_i)}\nabla_{\Theta^{\mu}}\mu(s\vert\Theta^\mu)\vert_{s_i}$ Update the networks $\Theta^{Q'}\leftarrow\tau\Theta^Q+(1-\tau)\Theta^{Q'}$ $\Theta^{\mu'}\leftarrow\tau\Theta^\mu+(1-\tau)\Theta^{\mu'}$
Experimental Setup
==================
Environment
-----------
The evaluation phase applied the proposed variations of the $DDPG$ learning algorithm to a variety of continuous control tasks running on an efficient and realistic physics simulator as a part of OpenAI Gym framework [@1606.01540] and shown in . The considered environments range from a relatively simple 2D robot (*Reacher-v2*), with a humble 11-dimensional state space, to a complex four-legged 3D robot such as *Ant-v2*[@schulman2015high], which boasts a total of 111-dimensional states coupled with 8 possible continuous actions. Various different tasks of intermediate complexity like making a 2D animal robot run (*HalfCheetah-v2*), and making a 2D snake-like robot move on a flat surface (*Swimmer-v2* [@coulom2002reinforcement]) have also be faced.
\[fig:env\]
{width=".9\linewidth"}
{width=".9\linewidth"}
{width=".9\linewidth"}
{width=".9\linewidth"}
Function Approximation
----------------------
An approximation of $ Q(s,a;\Theta) \approx Q^*(s,a) $ has been implemented using an ANN with one hidden fully connected layer of 50 neurons, able to take an agent’s state as an input and produce as output the Q values of all the actions available to the agent. The learning rate of an *critic* $Q$ approximator $\alpha$ is set to $0.002$.
The *actor* function approximator of $ a(s;\Theta) \approx a^*(s) $ is implemented using one hidden dense layer of 30 neurons which outputs a deterministic action policy based on the agent’s current state. The *actor* ANN has been trained using slightly higher learning rate of $0.001$ compared to the critic one.
The architecture of the *AMR* function approximator consists of three layers: four input nodes connected to a fully connected hidden layer of four nodes, in turn connected to a single regression node able to produce an *augmentation scalar* as output. This ANN is able to approximate four parameters of the current agent’s experience, respectively given in input as an absolute value of TD error, reinforcement, entropy of the starting state $ s_t $, entropy of the transitioning state $ s_{t+1} $, to scalar value $A_t$ that indicates how important the specific experience it to the learning algorithm.
Meta Learning Parameters
------------------------
During the evaluations phase at each learning step a batch of 32 experiences were replayed from the fixed capacity memory buffer of 10000. Learning steps per episode were limited to a maximum of 2000. Reward discount factor $\gamma$ was set to a high $0.9$ and soft replacement parameter $\tau$ was $0.01$. In order to achieve action space exploration an artificially generated noise is added to the deterministic action policy which is approximated by the *actor* ANN. The noise is gradually decreased or adjusted linearly from an initial scalar value $3.0$ to $0.0$ towards the end of the learning.
Experimental Results
====================
The proposed algorithm evolved the *AMR’s* neural network weights $\Theta^{AMR}$ trough a total of 75 generations. At each generation, the learning performance of 10 agents were evaluated based on their their total cumulative score during 200 learning episodes. Only the best 5 scoring agents of each generation had an opportunity to propagate their genotypes to the next generation in order to form a new population. As shown in this process involved common $GA$ techniques such as crossover and random mutation. The crossover of the genotypes, which are actually the *AMR* weights, were prioritized based on the agents cumulative score and the mutation was additionally applied at a rate of $0.25$ by adding a random scalar between $0.1$ and $-0.1$ to the weights. The obtained experimental results which are presented along the Figures \[fig:ant\], \[fig:reacher\], \[fig:cheetah\], \[fig:swimmer\] indicate that the most complex setup of *Ant-v2* improved its learning performance the most when using the proposed *AMR* approach when compared to the baseline approach that have not used *memory augmentation*. Regardless of the environment, it i evident that the proposed evolutionary approach with memory augmentation underperforms at the very first generations but quickly surpasses the baseline in less than 10 generations and further improves the total score of the agent in the following generations. As we can see from \[fig:ant\] the *AMR* evolutionary approach improves the *Ant’s* quad-legged robot learning about how to walk by a total of $18.9\%$ towards the end of the 75th generation. *AMR* algorithm have also showed a significant improvement in *Reacher*: the simplest of the environments. In this task the proposed evolutionary approach made the 2D robot hand with one actuating joint learn to fetch a randomly instantiated target faster and produced a $35.4\%$ increase in agent’s total cumulative score. Although not as significant as in *Ant* and *Reacher* setups, the *AMR* approach is also able to improve the performance in the *Cheetah* and *Swimmer* environments as evident from the and , respectively. We can also notice the difference of the score variance between the setups which can be attributed to distinctive robot/environment characteristics; while *Ant* and *Reacher* show relatively low variance in their scores, other problems like *Cheetah* and *Swimmer* have a very high variance.
![Average score or total reinforcement in Ant environment received over 75 generations of learning agents.[]{data-label="fig:ant"}](ant.pdf){width="50.00000%"}
![Average score or total reinforcement in Reacher environment received over 75 generations of learning agents.[]{data-label="fig:reacher"}](reacher.pdf){width="50.00000%"}
![Average score or total reinforcement in Cheetah environment received over 75 generations of learning agents.[]{data-label="fig:cheetah"}](cheetah.pdf){width="50.00000%"}
![Average score or total reinforcement in Swimmer environment received over 75 generations of learning agents.[]{data-label="fig:swimmer"}](swimmer.pdf){width="50.00000%"}
Discussion
==========
The presented approach represents yet another inspiration from biological systems, which implements a biologically inspired mechanisms that enables artificial learning agents to better adapt to a specific environment by selectively increasing the relevance of the information perceived. An agent implementing an *AMR* neural network is able to evolve its memory augmentation criteria to best fit the environment that is facing, in few generations. The evolved *AMR’s* augmentation criteria modifies the relevance of the information that an agents collects from its immediate environment into its replay memory allowing it to use the same data in a more efficient way during the learning process; this yields a direct improvement in the performance.
Thus, augmenting memory allows for the emergence of an *artificial cognition* as a intermediary dynamic filtering mechanism in learning agents, which opens a possibility for a variety of applications in the future.
References
==========
[^1]: Contact Author
|
---
abstract: 'We present the POL-2 850 $\mu$m linear polarization map of the Barnard 1 clump in the Perseus molecular cloud complex from the B-fields In STar-forming Region Observations (BISTRO) survey at the James Clerk Maxwell Telescope. We find a trend of decreasing polarization fraction as a function of total intensity, which we link to depolarization effects towards higher density regions of the cloud. We then use the polarization data at 850 $\mu$m to infer the plane-of-sky orientation of the large-scale magnetic field in Barnard 1. This magnetic field runs North-South across most of the cloud, with the exception of B1-c where it turns more East-West. From the dispersion of polarization angles, we calculate a turbulence correlation length of $5.0 \pm 2.5$ arcsec ($1500$ au), and a turbulent-to-total magnetic energy ratio of $0.5 \pm 0.3$ inside the cloud. We combine this turbulent-to-total magnetic energy ratio with observations of NH$_3$ molecular lines from the Green Bank Ammonia Survey (GAS) to estimate the strength of the plane-of-sky component of the magnetic field through the Davis-Chandrasekhar-Fermi method. With a plane-of-sky amplitude of $120 \pm 60$ $\mu$G and a criticality criterion $\lambda_c = 3.0 \pm 1.5$, we find that Barnard 1 is a supercritical molecular cloud with a magnetic field nearly dominated by its turbulent component.'
author:
- Simon Coudé
- Pierre Bastien
- Martin Houde
- Sarah Sadavoy
- Rachel Friesen
- James Di Francesco
- Doug Johnstone
- Steve Mairs
- Tetsuo Hasegawa
- Woojin Kwon
- 'Shih-Ping Lai'
- Keping Qiu
- 'Derek Ward-Thompson'
- David Berry
- 'Michael Chun-Yuan Chen'
- Jason Fiege
- Erica Franzmann
- Jennifer Hatchell
- Kevin Lacaille
- 'Brenda C. Matthews'
- 'Gerald H. Moriarty-Schieven'
- Andy Pon
- Philippe André
- Doris Arzoumanian
- Yusuke Aso
- 'Do-Young Byun'
- Eswaraiah Chakali
- 'Huei-Ru Chen'
- Wen Ping Chen
- 'Tao-Chung Ching'
- Jungyeon Cho
- Minho Choi
- Antonio Chrysostomou
- Eun Jung Chung
- Yasuo Doi
- 'Emily Drabek-Maunder'
- 'C. Darren Dowell'
- 'Stewart P. S. Eyres'
- Sam Falle
- Per Friberg
- Gary Fuller
- 'Ray S. Furuya'
- Tim Gledhill
- 'Sarah F. Graves'
- 'Jane S. Greaves'
- 'Matt J. Griffin'
- Qilao Gu
- 'Saeko S. Hayashi'
- Thiem Hoang
- Wayne Holland
- Tsuyoshi Inoue
- 'Shu-ichiro Inutsuka'
- Kazunari Iwasaki
- 'Il-Gyo Jeong'
- Yoshihiro Kanamori
- Akimasa Kataoka
- 'Ji-hyun Kang'
- Miju Kang
- 'Sung-ju Kang'
- 'Koji S. Kawabata'
- Francisca Kemper
- Gwanjeong Kim
- Jongsoo Kim
- 'Kee-Tae Kim'
- Kyoung Hee Kim
- 'Mi-Ryang Kim'
- Shinyoung Kim
- 'Jason M. Kirk'
- 'Masato I.N. Kobayashi'
- 'Patrick M. Koch'
- Jungmi Kwon
- 'Jeong-Eun Lee'
- Chang Won Lee
- 'Sang-Sung Lee'
- Dalei Li
- Di Li
- 'Hua-bai Li'
- 'Hong-Li Liu'
- Junhao Liu
- 'Sheng-Yuan Liu'
- Tie Liu
- Sven van Loo
- 'A-Ran Lyo'
- Masafumi Matsumura
- Tetsuya Nagata
- Fumitaka Nakamura
- Hiroyuki Nakanishi
- Nagayoshi Ohashi
- Takashi Onaka
- Harriet Parsons
- Kate Pattle
- Nicolas Peretto
- 'Tae-Soo Pyo'
- Lei Qian
- Ramprasad Rao
- 'Mark G. Rawlings'
- Brendan Retter
- John Richer
- Andrew Rigby
- 'Jean-François Robitaille'
- Hiro Saito
- Giorgio Savini
- 'Anna M. M. Scaife'
- Masumichi Seta
- Hiroko Shinnaga
- Archana Soam
- Motohide Tamura
- 'Ya-Wen Tang'
- Kohji Tomisaka
- Yusuke Tsukamoto
- Hongchi Wang
- 'Jia-Wei Wang'
- 'Anthony P. Whitworth'
- 'Hsi-Wei Yen'
- Hyunju Yoo
- Jinghua Yuan
- Tetsuya Zenko
- 'Chuan-Peng Zhang'
- Guoyin Zhang
- Jianjun Zhou
- Lei Zhu
bibliography:
- 'b1\_polarization.bib'
title: 'The JCMT BISTRO Survey: The Magnetic Field of the Barnard 1 Star-Forming Region'
---
Introduction {#sec:intro}
============
Magnetic fields, which are ubiquitous within the Galaxy [e.g., @Ordog2017; @Planck2015XIX], influence greatly the stability of molecular clouds and their dense filamentary structures in which star formation occurs [e.g., @Andre2014; @Andre2015]. Specifically, magneto-hydrodynamic simulations have shown that a combination of magnetism and turbulence is needed to slow the gravitational collapse of molecular clouds, and thus decrease the galactic star formation rate [e.g., @Padoan2014]. Measuring the amplitude of magnetic fields in dense interstellar environments is therefore crucial to our understanding of the physical processes leading to the formation of stars and their planets.
Interstellar magnetic fields are difficult to observe directly. Early studies hypothesized that polarization of background starlight through the interstellar medium was due to the alignment of irregularly-shaped dust grains with magnetic field lines [@Hiltner1949]. Subsequent observations of thermal dust emission in the far-infrared [@Cudlip1982] showed polarization orientations nearly orthogonal to measurements in the near-infrared, supporting the picture of elongated dust grains. Although magnetic fields are considered the most likely cause of dust alignment in interstellar environments, the grain alignment mechanisms themselves still remain a theoretical challenge [e.g., @Andersson2015 and references therein].
The Radiative Alignment Torque (RAT) theory of grain alignment is currently one of the most promising models to explain the polarization of starlight towards clouds and cores [@Lazarian2007_review]. In summary, this model predicts that asymmetric, non-spherical dust grains rotate due to radiative torques from their local radiation field and then align themselves with their long axis perpendicular to the ambient magnetic field [@Dolginov1976; @Draine1997; @Weingartner2003; @Lazarian2007a]. The degree of this alignment, however, depends on the quantity of paramagnetic material in the dust [@Hoang2016]. Submillimeter polarization observations of optically thin thermal dust emission will therefore lie perpendicular to the plane-of-sky component of the field.
The B-fields In STar-forming Region Observations (BISTRO) survey aims to study the role of magnetism for the formation of stars in the dense filamentary structures of giant molecular clouds [@Ward-Thompson2017]. This goal will be achieved by mapping the 850 $\mu$m linear polarization towards at least 16 fields (for a total of 224 hours) in nearby star-forming regions with the newly commissioned polarimeter POL-2 at the James Clerk Maxwell Telescope (JCMT). With the unprecedented single dish sensitivity of the Sub-millimetre Common-User Bolometer Array 2 (SCUBA-2) camera on which POL-2 is installed, the BISTRO survey will significantly expand on previously obtained polarization measurements at submillimeter and millimeter wavelengths [e.g., @Matthews2009; @Dotson2010; @Vaillancourt2012; @Hull2014; @Koch2014; @Zhang2014].
Several of the star-forming regions observed by BISTRO are part of the Gould Belt, a ring of active star-forming regions approximately $350$ pc-across that is centered roughly $200$ pc from the Sun [@Gould1879]. Here, we present the BISTRO observations of the Barnard 1 clump (hereafter Perseus B1, or B1) in the Perseus molecular cloud ($d \sim 295$ pc; @Ortiz2018). B1 is known to host several prestellar and protostellar cores at different evolutionary stages [e.g., @Hirano1997; @Hirano1999; @Matthews2006; @Pezzuto2012; @Carney2016]. This cloud was also a target of both the JCMT and *Herschel* Gould Belt surveys (from 70 $\mu$m to 850 $\mu$m), thus providing a characterization of its dust properties [@Sadavoy2013; @Chen2016].
This paper presents the BISTRO first-look analysis of the Perseus B1 star-forming region. In Section \[sec:observations\], we first describe the technical details of the polarization observations, and outline the spectroscopic data used in this work. In Section \[sec:results\], we show the POL-2 850 $\mu$m linear polarization map of B1 and its inferred plane-of-sky magnetic field morphology. We also characterize the relationship between the polarization fraction and the total intensity, and we compare the POL-2 data with previous SCUPOL observations. In Section \[sec:analysis\], we explain our methodology for measuring the magnetic field strength from the polarization data, and then present the results of this analysis. In Section \[sec:discussion\], we discuss the significance of these results for the role of the magnetic field on star formation within Perseus B1. Finally, we summarize our findings in Section \[sec:conclusion\].
Observations {#sec:observations}
============
Polarimetric Data {#sub:scuba2}
-----------------
The JCMT is a submillimeter observatory equipped with a 15 m dish that is located at an altitude of 4,092 m on top of Maunakea in Hawaii, USA. Its continuum instrument is SCUBA-2, a cryogenic $10,000$ pixel camera capable of simultaneous observing in the 450 $\mu$m and the 850 $\mu$m atmospheric windows [@Holland2013]. The SCUBA-2 beams can be approximated by a two-dimensional Gaussian with a full-width at half-maximum (FWHM) of $9.6$ arcsec at 450 $\mu$m and $14.6$ arcsec at 850 $\mu$m [@Dempsey2013].
The POL-2 polarimeter consists of a rotating half-wave plate and a fixed polarizer placed in the optical path of the SCUBA-2 camera (@Bastien2011; @Friberg2016; P. Bastien et al. in prep.). POL-2 is the follow-up instrument to the SCUBA polarimeter (SCUPOL), which had a similar basic design [@Greaves2003]. While SCUBA-2 always simultaneously observes at both 450 $\mu$m and 850 $\mu$m, only the 850 $\mu$m capabilities of POL-2 were commissioned at the time of writing. In brief, POL-2 observes by scanning the sky at a speed of 8 arcsec s$^{-1}$ in a daisy-like pattern over a field that is roughly 11 arcmin in diameter. Since the half-wave plate is rotated at a rate of 2 Hz, this scanning rate ensures a full rotation of the half-wave plate for every measurement of a $4$ arcsec box position in the map. For this paper, the Flux Calibration Factor (FCF) of POL-2 at 850 $\mu$m is assumed to be 725 Jy pW$^{-1}$ beam$^{-1}$ for each of the Stokes $I$, $Q$, and $U$ parameters(the Stokes parameters are defined in Section \[sub:polarization\]). This value was determined by multiplying the typical SCUBA-2 FCF of 537 Jy pW$^{-1}$ beam$^{-1}$ [@Dempsey2013] with a transmission correction factor of 1.35 measured in the laboratory and confirmed empirically by the POL-2 commissioning team using observations of the planet Uranus [@Friberg2016].
Perseus B1 was observed with POL-2 between 2016 September and 2017 March as part of the BISTRO large program at the JCMT (project ID: M16AL004). These observations total $14$ hours (or 20 individual sets of $\sim 40$-minutes observations) of integration in Grade 2 weather (i.e., for a 225 GHz atmospheric opacity, $\tau_{225}$, between $0.05$ and $0.08$). A 20-minute SCUBA-2 scan of B1 without POL-2 in the beam was also obtained on 2016 September 8 to serve as a reference for pointing corrections during data reduction.
The data were reduced using the <span style="font-variant:small-caps;">starlink</span> [@Currie2014] procedure *pol2map* [@POL2_Cookbook], which is adapted from the SCUBA-2 data reduction procedure *makemap* [@Chapin2013]. In particular, this routine is used to reduce POL-2 time-series observations into Stokes $I$, $Q$, and $U$ maps. We follow the convention set by the International Astronomical Union (IAU) for the definition of Stokes parameters. The default pixel size of the maps produced by *pol2map* is 4 arcsec. For the analysis presented in this paper, we have instead chosen a pixel size of 12 arcsec at the start of the data reduction process to improve the resulting signal-to-noise ratio (SNR) of the final Stokes $I$, $Q$, and $U$ maps.
The data reduction process is divided into three steps to optimize the SNR in the resulting maps: (1) the procedure *pol2map* is run a first time without applying any masks to obtain an initial Stokes $I$ intensity map directly from the POL-2 time-series observations; (2) this initial Stokes $I$ map is then used as the reference for the automatic masking process of *pol2map*, which is run a second time on the time-series observations to produce the final Stokes $I$ map; and (3) the masks obtained in Step 2 are also applied during a third run of *pol2map* to reduce the Stokes $Q$ and $U$ maps, which are automatically corrected for the instrumental polarization. The uncertainties in each pixel of the Stokes $I$, $Q$, and $U$ maps are taken directly from the variance maps provided by the *pol2map* procedure. The role of masking in the reduction of SCUBA-2 data, and incidentally POL-2 data, is discussed at length by @Mairs2015.
The correction for instrumental polarization is a crucial step in the analysis of any polarization measurement. If the instrumental polarization is not properly taken into account, then it may lead to erroneous results. For this reason, the latest model (January 2018) for the instrumental polarization of the JCMT at 850 $\mu$m was extensively tested by the POL-2 commissioning team with observations of Uranus and Mars (@Friberg2016 [@Friberg2018]; P. Bastien et al., in prep.). They found that the instrumental polarization can be accurately described using a two-components model combining the optics of the telescope and its protective wind blind. While the level of instrumental polarization is dependent on elevation, it is typically $\sim$ 1.5 per cent of the measured total intensity [@Friberg2018].
We also use 850 $\mu$m polarization data of Perseus B1 from the SCUPOL Legacy Catalog. @Matthews2009 built this legacy catalog by systematically re-reducing SCUPOL 850 $\mu$m observations towards 104 regions, including previously published observations of B1 [@Matthews2002], to provide reference Stokes cubes of comparable quality for all the astronomical sources with at least a 2 sigma detection of polarization. For this paper, the SCUPOL Stokes $I$, $Q$, and $U$ cubes for B1 were downloaded from the legacy catalog’s online archive hosted by the CADC. To match the POL-2 results, we resampled the SCUPOL polarization vectors onto a 12 arcsec pixel grid.
Spectroscopic Data {#sub:spectro}
------------------
The JCMT is also equipped with the HARP/ACSIS high-resolution heterodyne spectrometer capable of observing molecular lines between 325 GHz and 375 GHz (or 922 $\mu$m to 799 $\mu$m). The Heterodyne Array Receiver Program (HARP) is a $4 \times 4$ detector array that can be used in combination with the Auto-Correlation Spectral Imaging System (ACSIS) to rapidly produce large-scale velocity maps of astronomical sources [@Buckle2009]. In this paper, we use the previously published $\sim$14 arcsec resolution integrated intensity map of the $^{12}$CO J=3-2 molecular line towards Perseus B1 (project ID: S12AC01) [@Sadavoy2013]. This intensity map was integrated over a bandwidth of 1.0 GHz centered on the rest frequency of the $^{12}$CO J=3-2 line at 345.796 GHz. The noise added by integrating over such a large bandwidth has no effect on the results presented in this work since the $^{12}$CO J=3-2 data is used only to indicate the presence of outflows in Figure \[fig:fig1\_b1\_polarization\].
It is important to note that SCUBA-2, POL-2, and HARP are not sensitive to exactly the same spatial scales. This difference is due to a combination of the different scanning strategies for each instrument and their associated data reduction procedures [e.g., @Chapin2013]. Hence, this difference must be kept in mind when combining results from different instruments, such as correcting for molecular contamination using HARP or comparing source intensities between POL-2 and SCUBA-2. While this difference is not an issue for the results presented in this paper, it may need to be taken into account in future studies using BISTRO data (see Section \[sub:contamination\] for more details).
Finally, this project makes use of spectroscopic data from the Green Bank Ammonia Survey (GAS) [@Friesen2017]. GAS uses the K-Band Focal Plane Array (KFPA) and the VErsatile GBT Astronomical Spectrometer (VEGAS) at the Green Bank Telescope (GBT) to map ammonia lines, among others, in nearby star-forming regions. In this work, we specifically use measurements of the NH$_3$ (1,1) and (2,2) lines towards Perseus B1 (GAS Consortium, in prep.). These observations of NH$_3$ molecular lines at $\sim 23.7$ GHz have a spatial resolution of 32 arcsec and a velocity resolution of $\sim 0.07$ km s$^{-1}$.
Results {#sec:results}
=======
Polarization Properties {#sub:polarization}
-----------------------
The polarization vectors are defined by the polarization fraction $P$ and the polarization angle $\Phi$ measured eastward from celestial North. These properties are determined directly from the Stokes $I$, $Q$, and $U$ parameters, which is the commonly accepted parametrization for partially polarized light. The Stokes $I$ parameter is the total intensity of the incoming light, and the Stokes $Q$ and $U$ parameters are respectively defined as $ Q = I \, P \, \text{cos}\left( 2 \Phi \right)$ and $ U = I \, P \, \text{sin}\left( 2 \Phi \right)$.
When $Q$ and $U$ are near zero, these values will be dominated by the noise in our measurements. This noise contribution always leads to a positive bias in the calculation of the polarization fraction $P$ due to the quadratic nature of the polarized intensity $I_P = [Q^2+U^2]^{1/2}$ [e.g., @Wardle1974; @Montier2015; @Vidal2016]. The amplitude of this positive bias can be approximated from the uncertainty $\sigma_{I_P}$ given in Equation \[eq:polarised\_uncertainty\], which is used in Equation \[eq:fraction\] to de-bias the polarization fraction $P$ [e.g., @Naghizadeh_Clarke1993].
The de-biased polarization fraction $P$ (in per cent) can therefore be written as: $$P = \frac{100}{I} \; \sqrt[]{Q^2 + U^2 - \sigma_{I_P}^2 } = \frac{100}{I} \, I_P \, ,
\label{eq:fraction}$$ where we re-define $I_P$ as the de-biased polarized intensity with uncertainty $\sigma_{I_P}$. This uncertainty $\sigma_{I_P}$ is given by: $$\sigma_{I_P} = \, \left[ \frac{\left(Q \, \sigma_{Q} \right)^2 + \left(U \, \sigma_{U} \right)^2}{Q^2+U^2} \right]^{1/2} \, ,
\label{eq:polarised_uncertainty}$$ where $\sigma_Q$ and $\sigma_U$ are the uncertainties on the Stokes $Q$ and $U$ parameters respectively. The uncertainty $\sigma_P$ of the polarization fraction $P$ is given by: $$\sigma_{P} = \, P \, \left[ \left( \frac{\sigma_{I_P}}{I_P} \right)^2 + \left( \frac{\sigma_{I}}{I} \right)^2 \right]^{1/2} \, ,
\label{eq:fraction_uncertainty}$$ where $\sigma_I$ is the uncertainty on the Stokes $I$ total intensity.
Finally, the expression for the polarization angle $\Phi$ is: $$\Phi = \frac{1}{2} \, \arctan \left( \frac{U}{Q} \right) \, ,
\label{eq:angle}$$ where $\Phi$ is defined between 0 and $\pi$ (0$^\circ$ and 180$^\circ$) for convenience, and its related uncertainty $\sigma_\Phi$ is given by: $$\sigma_\Phi = \frac{1}{2} \, \frac{\sqrt{\left(U \, \sigma_{Q} \right)^2 + \left(Q \, \sigma_{U} \right)^2}}{Q^2 + U^2} \, .
\label{eq:error_angle}$$
BISTRO First-Look at Perseus B1 {#sub:pol2_perseusb1}
-------------------------------
{width="49.50000%"} {width="49.50000%"}
Figure \[fig:fig1\_b1\_polarization\] (left) shows the BISTRO 850 $\mu$m linear polarization map of Perseus B1 for a pixel size of 12 arcsec. The catalog of polarization vectors is calculated for every pixel of the POL-2 Stokes $I$, $Q$ and $U$ maps, but only vectors passing a set of pre-determined selection criteria are shown. These selection criteria are: a SNR of $I/\sigma_I\!>\!3$ for Stokes $I$ and its uncertainty $\sigma_I$, a SNR of $P/\sigma_P\!>\!3$ for the polarization fraction $P$ and its uncertainty $\sigma_P$, and an uncertainty $\sigma_P\!<\!5$ per cent for the polarization fraction. The criterion of $\sigma_P\!<\!5$ per cent was chosen arbitrarily as a precaution against potentially spurious vectors with anomalously high polarization fractions. These criteria provide a catalog of 224 polarization vectors for Perseus B1.
The mean values of the Stokes uncertainties $\sigma_I$, $\sigma_Q$, and $\sigma_U$ for the polarization vectors shown in Figure \[fig:fig1\_b1\_polarization\] are 1.6 mJy beam$^{-1}$, 1.3 mJy beam$^{-1}$, and 1.3 mJy beam$^{-1}$ respectively. At best, we achieve a sensitivity of $0.1$ per cent in polarization fraction and an uncertainty of $2.1^\circ$ in polarization angle, with mean values for $\sigma_P$ of $1.9$ per cent and for $\sigma_\Phi$ of $5.7^\circ$ for the entire catalog of vectors.
Assuming that interstellar dust grains are aligned with their long axis perpendicular to the magnetic field, the plane-of-sky field morphology in Perseus B1 is obtained by rotating the vectors in the polarization map by $90^{\circ}$. Figure \[fig:fig1\_b1\_polarization\] (right) shows the inferred plane-of-sky magnetic field map for B1. To help highlight the magnetic field structure, the rotated vectors are normalized to the same length. A contour plot of the HARP $^{12}$CO J=3-2 integrated intensity map from the JCMT Gould Belt Survey [@Sadavoy2013] is also included in the right panel of Figure \[fig:fig1\_b1\_polarization\].
Selected submillimeter sources are identified in both panels of Figure \[fig:fig1\_b1\_polarization\] to serve as references for the discussion in Section \[sec:discussion\] [@Bally2008]. These sources are embedded young stellar objects which have been associated with molecular outflows [@Hatchell2009; @Evans2009; @Hirano2014; @Carney2016]. Specifically, the lobes of the precessing molecular outflow originating from the protostellar core B1-c [@Matthews2006] are particularly well defined by the $^{12}$CO J=3-2 contour plot shown in the right panel of Figure \[fig:fig1\_b1\_polarization\].
![Depolarization of POL-2 observations towards Perseus B1. Each point represents one of the polarization vectors shown in the left panel of Figure \[fig:fig1\_b1\_polarization\]. The vertical and horizontal lines show the uncertainties for the plotted parameters in each panel. *Top*: De-biased polarization fraction $P$ as a function of the Stokes $I$ total intensity. *Bottom*: De-biased polarized intensity $I_P$ as a function of the Stokes $I$ total intensity. The solid line in the top panel is the power-law fit (with index $\alpha \sim -0.85$) between the polarization fraction $P$ and the Stokes $I$ total intensity ($P \propto I^\alpha$, see Section \[sub:pol2\_perseusb1\]). The solid line in the bottom panel is the same power-law fit as above, but multiplied by the Stokes $I$ total intensity ($I_P \propto I^{\alpha+1}$).[]{data-label="fig:fig2_depolarization"}](f2top-eps-converted-to.pdf "fig:"){width="\columnwidth"} ![Depolarization of POL-2 observations towards Perseus B1. Each point represents one of the polarization vectors shown in the left panel of Figure \[fig:fig1\_b1\_polarization\]. The vertical and horizontal lines show the uncertainties for the plotted parameters in each panel. *Top*: De-biased polarization fraction $P$ as a function of the Stokes $I$ total intensity. *Bottom*: De-biased polarized intensity $I_P$ as a function of the Stokes $I$ total intensity. The solid line in the top panel is the power-law fit (with index $\alpha \sim -0.85$) between the polarization fraction $P$ and the Stokes $I$ total intensity ($P \propto I^\alpha$, see Section \[sub:pol2\_perseusb1\]). The solid line in the bottom panel is the same power-law fit as above, but multiplied by the Stokes $I$ total intensity ($I_P \propto I^{\alpha+1}$).[]{data-label="fig:fig2_depolarization"}](f2bottom-eps-converted-to.pdf "fig:"){width="\columnwidth"}
The top panel of Figure \[fig:fig2\_depolarization\] compares the fraction of polarization $P$ with the Stokes $I$ total intensity for each of the POL-2 vectors shown on the left panel of Figure \[fig:fig1\_b1\_polarization\]. There is a clear trend of decreasing fraction $P$ as a function of increasing Stokes $I$. If the total intensity is correlated with the column density [@Hildebrand1983], this behavior can be understood as the result of a depolarization effect towards higher density regions of the cloud. The origin of this depolarization effect is discussed in Section \[sec:discussion\]. This trend does not mean, however, that the polarized intensity $I_P$ itself is decreasing. Indeed, the bottom panel of Figure \[fig:fig2\_depolarization\] shows that $I_P$ may be in fact increasing slowly with Stokes $I$.
We fitted a power-law ($P \propto I^\alpha$) to the data in Figure \[fig:fig2\_depolarization\] (top) using an error-weighted least-square minimization technique. We find a power index $\alpha = -0.85 \pm 0.01$, with a reduced chi-squared $\chi^2_r = 3.4$. This power-law is shown in both panels of Figure \[fig:fig2\_depolarization\] as a solid line. The spread of data points relative to their uncertainties is responsible for the large $\chi^2_r$ value obtained, which indicates that fitting a single power-law may not be sufficient to account for the entire data set. The detailed effects of measurement uncertainties on the power-law fit between $P$ and $I$ are currently under investigation (K. Pattle et al., in prep.).
The power index $\alpha \sim -0.85$ we find for B1 is nearly identical to the value measured in $\rho$ Ophiuchus B by @Soam2018 and relatively close to the index $\alpha \sim -0.8$ measured by @Kwon2018 in $\rho$ Ophiuchus A, both obtained from BISTRO data. Similarly, @Matthews2002 previously found a power index $\alpha \sim -0.8$ in B1 using SCUPOL 850 $\mu$m measurements. The differences between POL-2 and SCUPOL polarization maps of B1 are quantified in Section \[sub:scupol\].
![Relationship between the de-biased polarization fraction $P$ and the visual extinction $A_V$ in Perseus B1. Each point represents one of the polarization vectors from the left panel of Figure \[fig:fig1\_b1\_polarization\] that also have *Herschel*-derived opacity measurements. The visual extinction $A_V$ is derived from the $300$ $\mu$m $\tau_{300}$ opacity map from @Chen2016 assuming a reddening factor $R_V=3.1$. The figure covers a range of extinction $A_V$ from $30$ mag to $400$ mag. The vertical lines show the uncertainties for the polarization fraction $P$. The $8$ polarization vectors found towards B1-c are identified with squares. The solid line is the power-law fit (with index $\beta \sim -0.5$) between the polarization fraction $P$ and the visual extinction $A_V$ ($P \propto A_V^\beta$, see Section \[sub:pol2\_perseusb1\]).[]{data-label="fig:fig3_depolarization"}](f3-eps-converted-to.pdf){width="\columnwidth"}
However, in the context of grain alignment theory, it is more meaningful to take the optical depth into account when studying depolarization effects in molecular clouds. While an accurate modeling of the alignment efficiency of dust grains in Perseus B1 will require a detailed analysis beyond the scope of this work, we can nonetheless begin to characterize the relationship between the polarized dust thermal emission and the visual extinction $A_V$ in the cloud by fitting a power-law of the form $P \propto A_V^\beta$ [e.g., @Alves2014]. Specifically, we know that the polarization fraction $P$ of dust thermal emission obtained from submillimeter observations is proportional to the polarization efficiency $P_{ext}/A_V$ derived from measurements of the polarization fraction $P_{ext}$ due to extinction at visible wavelengths [@Andersson2015].
Figure \[fig:fig3\_depolarization\] shows the relation between the polarization fraction $P$ and the derived visual extinction $A_V$ for the polarization vectors shown the left panel of Figure \[fig:fig1\_b1\_polarization\] that also have an associated opacity measurement in the $300$ $\mu$m $\tau_{300}$ opacity map from @Chen2016. We estimate the visual extinction $A_V$ using Equation A5 from @Jones2015 and a version of the $\tau_{300}$ opacity map from @Chen2016 that has been re-gridded from a pixel scale of $14$ arcsec to $12$ arcsec to match our observations. We also assume a reddening $R_V$ of $3.1$ which may be more representative of the diffuse interstellar medium [@Weingartner2001], but should nonetheless serve as a reasonable lower limit for our estimation of the visual extinction $A_V$ across the cloud.
We fitted a power-law $P \propto A_V^\beta$ to the data shown in Figure \[fig:fig3\_depolarization\] using an error-weighted least-square minimization technique. We find a power index $\beta = -0.51 \pm 0.03$, with a reduced chi-squared $\chi^2_r = 26.3$. This power-law is shown in Figure \[fig:fig3\_depolarization\] as a solid line. The large reduced chi-squared $\chi^2_r$ value we find clearly indicates a poor fit to the data considering the spread of values and their uncertainties for the polarization fraction $P$ in Figure \[fig:fig3\_depolarization\]. This could be explained in part by our use of a single reddening value to derive the visual extinction $A_V$. Indeed, the reddening $R_V$ depends on the size distribution and composition of the dust grains, and so we do not expect this value to be constant across the cloud.
Nevertheless, the power index $\beta \sim -0.5$ we find in B1 is shallower than the power indices obtained from submillimeter observations in the Pipe-109 starless core ($\beta \sim -0.9$, @Alves2014 [@Alves2014corr]) and in the LDN 183 starless core ($\beta \sim -1.0$, @Andersson2015). In fact, a power index $\beta \sim -0.5$ is closer to the power index $\beta \sim -0.6$ measured towards lower extinction regions ($A_V < 20$) of LDN 183 using visible and near-infrared observations [@Andersson2015]. Although Figure \[fig:fig2\_depolarization\] clearly shows a depolarization effect with increasing total intensity $I$, the power index $\beta \sim -0.5$ we find using the data in Figure \[fig:fig3\_depolarization\] suggests that dust grains in Perseus B1 are aligned more efficiently than in starless cores with comparable measures of visual extinction $A_V$. Since B1 is a site of on-going star formation, this may provide evidence that radiation from embedded young stellar objects can compensate for the expected loss of grain alignment with increasing visual extinction.
Comparison with SCUPOL Legacy Data {#sub:scupol}
----------------------------------
![Comparison of dust polarization at 850 $\mu$m between POL-2 (red) and SCUPOL (blue) towards Perseus B1. The gray scale indicates the Stokes $I$ total intensity measured with POL-2. The length of each vector is determined by its associated polarization fraction $P$ (per cent). The SCUPOL polarization vectors from @Matthews2009 have been re-binned to match the exact position and pixel scale (from 10 arcsec to 12 arcsec) of the POL-2 observations.[]{data-label="fig:fig4_scupol"}](f4-eps-converted-to.pdf){width="\columnwidth"}
As mentioned in Section \[sub:scuba2\], Perseus B1 was previously observed at 850 $\mu$m with the SCUPOL polarimeter [@Matthews2002]. Here we specifically compare the BISTRO results presented in Section \[sub:pol2\_perseusb1\] to the polarization data of B1 found in the SCUPOL Legacy Catalog [@Matthews2009].
Figure \[fig:fig4\_scupol\] compares the BISTRO observations to their equivalent data set in the SCUPOL Legacy Catalog, with the POL-2 polarization vectors (same as Figure \[fig:fig1\_b1\_polarization\]) in red and the SCUPOL vectors in blue. To have a significant number of SCUPOL vectors for this comparison, we relaxed their selection criteria compared to POL-2. For the SCUPOL data, we use $I/\sigma_I\!>\!2$, $P/\sigma_P\!>\!2$, and $\sigma_P\!<\!10$ per cent. These relaxed criteria provide a total catalog of 69 vectors, compared to only 17 when applying the same selection criteria as for the POL-2 data.
At best, the relaxed catalog of SCUPOL vectors achieves a sensitivity of $0.5$ per cent in polarization fraction and an uncertainty of $5.5^\circ$ in polarization angle, with mean values for $\sigma_P$ of $2.7$ per cent and for $\sigma_\Phi$ of $10.3^\circ$.
![Histograms of polarization angles for Perseus B1 from POL-2 and SCUPOL. The number of vectors in each bin is normalized by the maximum value of the histogram ($N_{\text{bin}} / N_{\text{max}}$) for a given sample of polarization angles. *Top*: Histogram including all the POL-2 (224) and SCUPOL (69) polarization vectors shown in Figure \[fig:fig1\_b1\_polarization\] and Figure \[fig:fig4\_scupol\] respectively. *Bottom*: Histogram including only the 52 positions for which there exists both a POL-2 and a SCUPOL polarization vector in Figure \[fig:fig4\_scupol\]. In both panels, the range of polarization angles associated with the protostellar source B1-c is shown in gray.[]{data-label="fig:fig5_scupol_histo"}](f5top-eps-converted-to.pdf "fig:"){width="\columnwidth"} ![Histograms of polarization angles for Perseus B1 from POL-2 and SCUPOL. The number of vectors in each bin is normalized by the maximum value of the histogram ($N_{\text{bin}} / N_{\text{max}}$) for a given sample of polarization angles. *Top*: Histogram including all the POL-2 (224) and SCUPOL (69) polarization vectors shown in Figure \[fig:fig1\_b1\_polarization\] and Figure \[fig:fig4\_scupol\] respectively. *Bottom*: Histogram including only the 52 positions for which there exists both a POL-2 and a SCUPOL polarization vector in Figure \[fig:fig4\_scupol\]. In both panels, the range of polarization angles associated with the protostellar source B1-c is shown in gray.[]{data-label="fig:fig5_scupol_histo"}](f5bottom-eps-converted-to.pdf "fig:"){width="\columnwidth"}
Figure \[fig:fig5\_scupol\_histo\] shows the distribution of angles for both the POL-2 and SCUPOL polarization maps. The top panel shows the histogram including all the POL-2 and SCUPOL polarization vectors shown in Figure \[fig:fig4\_scupol\], normalized by the maximum value in each distribution. Both distributions peak between 65$^\circ$ and 85$^\circ$. The bottom panel shows the normalized distributions only for those vector positions that are common (i.e., spatially overlapping within the same pixel) to both SCUPOL and POL-2. There are 52 such positions in the maps.
We used a Kolmogorov-Smirnov test to compare the distributions shown at the bottom of Figure \[fig:fig5\_scupol\_histo\]. Specifically, a two-sample Kolmogorov-Smirnov test provides the probability that two independent data samples are drawn from the same intrinsic distribution by measuring the maximum distance between the cumulative probability distribution of each sample. For example, if both the SCUPOL and POL-2 values for the selected co-spatial vectors were exact measurements of the $850$ $\mu$m polarization towards Perseus B1, then we would expect the two catalogs of polarization angles, and therefore their respective cumulative probability distributions, to be identical and the Kolmogorov-Smirnov test to return a $100$ per cent probability that they are drawn from the same intrinsic distribution of polarization angles. In reality, the POL-2 and SCUPOL distributions shown in the bottom panel of Figure \[fig:fig5\_scupol\_histo\] are not identical even though they probe the same positions in B1, and so the Kolmogorov-Smirnov test becomes a way of quantifying the difference between them since it makes no assumption about the nature of the aforementioned intrinsic distribution.
In this case, we find a low likelihood ($0.6$ per cent) that both POL-2 and SCUPOL distributions in the bottom panel of Figure \[fig:fig5\_scupol\_histo\] are drawn from the same intrinsic distribution of polarization angles (with a maximum deviation $D = 0.39$ between the cumulative probability distributions). In other words, based only on the $52$ available co-spatial vectors in each sample, a two-sample Kolmogorov-Smirnov test shows that the distributions of POL-2 and SCUPOL polarization angles are significantly different from each other. If we set the selection criteria for POL-2 vectors to be identical to those applied for SCUPOL vectors, we find instead 64 positions with vectors common to both catalogs. This relaxed data set does not, however, improve the results of the Kolmogorov-Smirnov test.
![*Top*: Comparison of polarization angles for the 52 pairs of spatially overlapping POL-2 and SCUPOL vectors plotted in Figure \[fig:fig4\_scupol\]. The plain line follows the 1:1 correspondence, and the dotted and dashed lines respectively trace differences of a 45 degrees and 90 degrees in polarization angle. *Bottom*: Difference of polarization angle between each pair of POL-2 and SCUPOL vector ($\Delta\Phi = \left| \Phi_{\text{SCUPOL}} - \Phi_{\text{POL-2}} \right|$) as a function of the signal-to-noise ratio (SNR) of the polarization fraction measured with SCUPOL ($P_{\text{SCUPOL}}/\sigma_{P_{\text{SCUPOL}}}$). The vertical dashed line indicates a SNR of 3. In both panels, the color scale indicates the Stokes $I$ intensity of the POL-2 vector associated with each point.[]{data-label="fig:fig6_scupol_comparison"}](f6top-eps-converted-to.pdf "fig:"){width="\columnwidth"} ![*Top*: Comparison of polarization angles for the 52 pairs of spatially overlapping POL-2 and SCUPOL vectors plotted in Figure \[fig:fig4\_scupol\]. The plain line follows the 1:1 correspondence, and the dotted and dashed lines respectively trace differences of a 45 degrees and 90 degrees in polarization angle. *Bottom*: Difference of polarization angle between each pair of POL-2 and SCUPOL vector ($\Delta\Phi = \left| \Phi_{\text{SCUPOL}} - \Phi_{\text{POL-2}} \right|$) as a function of the signal-to-noise ratio (SNR) of the polarization fraction measured with SCUPOL ($P_{\text{SCUPOL}}/\sigma_{P_{\text{SCUPOL}}}$). The vertical dashed line indicates a SNR of 3. In both panels, the color scale indicates the Stokes $I$ intensity of the POL-2 vector associated with each point.[]{data-label="fig:fig6_scupol_comparison"}](f6bottom-eps-converted-to.pdf "fig:"){width="\columnwidth"}
Figure \[fig:fig6\_scupol\_comparison\] expands the comparison shown in Figure \[fig:fig5\_scupol\_histo\] (bottom) between the POL-2 and SCUPOL polarization angles for pairs of spatially overlapping vectors. The top panel of Figure \[fig:fig6\_scupol\_comparison\] shows that most outliers from the 1:1 correspondence line are found towards lower intensity regions ($I < 200$ mJy beam$^{-1}$), as measured from POL-2 Stokes $I$. Furthermore, in Figure \[fig:fig6\_scupol\_comparison\] (bottom), the vector pairs displaying the largest angular difference ($\left| \Phi_{\text{SCUPOL}} - \Phi_{\text{POL-2}} \right|$) are found near or below a SNR of 3 for the polarization fraction ($P_{\text{SCUPOL}}/\sigma_{P_{\text{SCUPOL}}} \lesssim 3$) measured with SCUPOL. Although the pairs of vectors at high SNR ($P_{\text{SCUPOL}}/\sigma_{P_{\text{SCUPOL}}} > 4$) also exhibit a non-negligible angular difference, this effect is not nearly as pronounced as for the low SNR vectors ($P_{\text{SCUPOL}}/\sigma_{P_{\text{SCUPOL}}} \lesssim 3$). This disparity between POL-2 and SCUPOL could therefore be explained by the relatively high noise levels in the SCUPOL Legacy data.
Analysis {#sec:analysis}
========
Angular Dispersion Analysis and Davis-Chandrasekhar-Fermi Method {#sub:dcf}
----------------------------------------------------------------
The magnetic field strength in molecular clouds can be estimated through the Davis-Chandrasekhar-Fermi (DCF) method [@Davis1951; @CF1953]. This technique relies on the assumption that turbulent motions in the gas will locally inject randomness in the observed morphology of a large-scale magnetic field. Since polarization vectors are expected to trace the plane-of-sky component of the magnetic field, we can infer the strength of this component by measuring the dispersion of polarization angles relative to the large-scale field orientation. This technique, however, also requires the velocity dispersion and the density of the gas in the cloud to be known beforehand.
According to @Crutcher2004, the DCF equation for the plane-of-sky magnetic field strength $B_{\text{pos}}$ can be written as: $$B_{\text{pos}} = A \; \sqrt[]{4 \pi \rho} \; \frac{\delta V}{\delta \Phi} \, ,
\label{eq:dcf}$$ where $\rho$ is the density, $\delta V$ is the velocity dispersion of the gas in the cloud, $\delta \Phi$ is the dispersion of polarization angles (in radians), and $A$ is a correction factor usually assumed to be $\sim 0.5$. The correction factor $A$ is included to account for the three-dimensional nature of the interplay between turbulence and magnetism [e.g., @Ostriker2001]. There is, however, a caveat to Equation \[eq:dcf\], namely that it cannot intrinsically account for changes in the large-scale field morphology. As a consequence, the technique from @Crutcher2004 was modified by @Pattle2017 to take large-scale variations in field morphology into account when calculating the magnetic field strength in Orion A.
Specifically, @Pattle2017 calculate the dispersion $\delta \Phi$ of polarization angles in Equation \[eq:dcf\] with an unsharp-masking technique. First, the large-scale component of the field is found by smoothing the map of polarization angles using $3 \times 3$-pixels boxes. This smoothed map is then subtracted from the original to obtain a map of the residual polarization angles. Finally, the dispersion $\delta \Phi$ is obtained from the mean value of the residual angles fitting a specific set of conditions. This approach therefore cancels the contribution of a changing field morphology to the dispersion of polarization angles at scales larger than the smoothed mean-field map.
In our work, we instead apply the improved DCF method developed by @Hildebrand2009 and @Houde2009, which was also adapted for polarimetric data obtained by interferometers such as the SMA and CARMA [@Houde2011; @Houde2016]. This technique avoids the problem of spatial changes in field morphology by using an angular dispersion function (sometimes called structure function) rather than the dispersion of polarization angles around a mean value. Furthermore, the angular dispersion technique from @Houde2009 was independently tested using both R-band [e.g., @Franco2010] and submillimeter [e.g., @Ching2017] polarimetric observations to characterize the magnetic and turbulent properties of star-forming regions.
This angular dispersion function is calculated by taking the angular difference between every pair of polarization vectors in a given map as a function of the distance between them. This technique effectively traces the ratio between turbulent and magnetic energies, which can then be fitted without any prior assumptions on the turbulence in the cloud or the morphology of the large-scale field [@Hildebrand2009]. As before, this analysis can be used to estimate the strength of the plane-of-sky magnetic field component if the density and velocity dispersion of the cloud are known. Additionally, it can be used to measure the effect of integrating turbulent cells along the line-of-sight within a telescope beam, effectively constraining the theoretical factor $A$ included in Equation \[eq:dcf\] [@Houde2009].
We first need to define the relevant quantities for the dispersion analysis presented in this paper. The difference in polarization angle between two vectors as a function of distance $\ell$ is defined as: $\Delta\Phi(\ell) \equiv \Phi(\textbf{x}) - \Phi(\textbf{x} + \bm{\ell})$, where $\Phi(\textbf{x})$ is the angle $\Phi$ of the polarization vector found at a position $\textbf{x}$ in the map and $\bm{\ell}$ is the angular displacement between two vectors. With this quantity, we can define the angular dispersion function as formulated by @Houde2009: $$1- \left\langle \cos[\Delta\Phi(\ell)] \right\rangle \, ,
\label{eq:dispersion}$$ where $\left\langle ... \right\rangle$ is the average over every pair of vectors separated by a distance $\ell$. Since Equation \[eq:dispersion\] is essentially a measure of the mean difference in polarization angles as a function of distance, it is accurate to describe it as an angular dispersion function.
The magnetic field $\textbf{B}(\textbf{x})$ in the cloud at a position $\textbf{x}$ can be written as a combination of a large-scale (or ordered) component $\textbf{B}_{o}(\textbf{x})$ and a turbulent component $\textbf{B}_{t}(\textbf{x})$, i.e., $\textbf{B}(\textbf{x}) = \textbf{B}_{o}(\textbf{x})+\textbf{B}_{t}(\textbf{x})$. Furthermore, we define the ratio between the average energy of the turbulent component to that of the large-scale component as $\left\langle B_t^2 \right\rangle / \left\langle B_o^2 \right\rangle$ and the ratio between the average energy of the turbulent component to that of the total magnetic field as $\left\langle B_t^2 \right\rangle / \left\langle B^2 \right\rangle$. Both quantities can be obtained from fitting the angular dispersion function.
To relate the magnetic fields and turbulence, we also need to define the turbulent properties of the cloud. Specifically, we require the number $N$ of independent magnetic turbulent cells observed for a column of dust along the line-of-sight and within a telescope beam from: $$N = \Delta' \, \frac{\left( \delta^2 + 2 W^2 \right)}{\sqrt[]{2 \pi}\, \delta^3} \, ,
\label{eq:turbulence}$$ where $\delta$ is the turbulent correlation length scale of the magnetic field, $W$ is the radius of the circular telescope beam (specifically, $\text{FWHM} = 2 \; \sqrt[]{2 \, \text{ln}2} \, W$), and $\Delta'$ is the effective thickness of the cloud [see Equation 52 in @Houde2009]. The turbulent correlation length scale $\delta$ can be understood as the typical size of a magnetized turbulent cell in the cloud. In this specific case, the turbulence is supposedly isotropic and the turbulent correlation length scale $\delta$ is assumed to be smaller than the thickness $\Delta'$ of the cloud.
If the physical depth of the cloud is not known beforehand, the effective thickness $\Delta'$ can be estimated from the autocorrelation function of the integrated polarized intensity across the cloud [see Equation 51 in @Houde2009]. This autocorrelation function is defined as: $$\left\langle I_P^2(\ell) \right\rangle \equiv \left\langle I_P(\textbf{x}) \, I_P(\textbf{x} + \bm{\ell}) \right\rangle\, ,
\label{eq:pi_autocorrelation}$$ from which we use the width at half-maximum to evaluate $\Delta'$. This approach, however, assumes that the spatial distribution of polarized dust emission on the plane-of-sky is an adequate probe of the cloud’s properties along the line-of-sight, which we believe to be reasonable in the case of dense molecular clouds.
The detailed derivations given by @Hildebrand2009 and @Houde2009 show that the relationship between the angular dispersion function and the magnetic and turbulent properties of a molecular cloud can be expressed by the following equation: $$1- \left\langle \cos[\Delta\Phi(\ell)] \right\rangle \simeq \frac{1}{N} \, \frac{\left\langle B_t^2 \right\rangle}{\left\langle B_o^2 \right\rangle} - b^2(\ell) + a \, \ell^2 \, ,
\label{eq:adf}$$ where $a$ is the first Taylor coefficient of the ordered autocorrelation function, and $b^2(\ell)$ is the autocorrelated turbulent component of the dispersion function [see Equations 53 and 55 in @Houde2009]. Specifically, the Taylor coefficient $a$ is related to the large-scale structure of the magnetic field. Additionally, we can write this autocorrelated turbulent component as: $$b^2(\ell) = \frac{1}{N} \, \frac{\left\langle B_t^2 \right\rangle}{\left\langle B_o^2 \right\rangle} \, e^{-\ell^2 / 2(\delta^2+2W^2)} \, .
\label{eq:autocorrelation}$$ Since the beam radius $W$ and the effective cloud thickness $\Delta'$ can be considered as known quantities, we only need to fit three parameters to the angular dispersion function: the ratio of turbulent energy to large-scale magnetic energy $\left\langle B_t^2 \right\rangle / \left\langle B_o^2 \right\rangle$, the turbulent correlation length scale $\delta$ of the magnetic field, and the first Taylor coefficient $a$ of the ordered autocorrelation function.
Finally, @Houde2009 rewrote the DCF equation (see Equation \[eq:dcf\]) for the plane-of-sky strength of the magnetic field to calculate it directly from the ratio of turbulent energy to total magnetic energy $\left\langle B_t^2 \right\rangle / \left\langle B^2 \right\rangle$ in the cloud. This new formulation of the DCF equation can be written as: $$B_{\text{pos}} \simeq \sqrt[]{4 \pi \rho} \: \delta V \, \left[ \frac{\left\langle B_t^2 \right\rangle}{\left\langle B^2 \right\rangle} \right]^{-1/2} \, ,
\label{eq:dcf_houde}$$ where as previously $\rho$ is the density and $\delta V$ is the one-dimensional velocity dispersion for the gas (see Equation 57 in @Houde2009 and Equation 26 in @Houde2016). The gas density $\rho$ takes the form $\rho~=~\mu \, m_{H} \, n(\text{H}_2)$, where $\mu = 2.8$ is the mean molecular weight of the gas [@Kauffmann2008], $m_{H}$ is the mass of an hydrogen atom, and $n(\text{H}_2)$ is the number density of hydrogen molecules in the cloud.
Once the strength of the plane-of-sky component of the magnetic field has been calculated with Equation \[eq:dcf\_houde\], it becomes possible to evaluate the magnetic critical ratio $\lambda_c$ of the studied molecular cloud [@Crutcher2004]. The critical ratio $\lambda_c$ can be estimated from the plane-of-sky amplitude of the magnetic field with the following equation: $$\lambda_c \simeq 7.6 \times 10^{-21} \, \frac{N(\text{H}_2)}{B_{\text{pos}}} \, ,
\label{eq:crit_ratio}$$ where $N(\text{H}_2)$ is the typical column density of molecular hydrogen in the cloud. If $\lambda_c < 1$, then the molecular cloud is magnetically subcritical and the magnetic field is sufficiently strong to stop its gravitational collapse. If $\lambda_c > 1$, the cloud is instead magnetically supercritical and the magnetic field alone cannot support the cloud against its self-gravity.
Cloud Characteristics and Magnetic Field Strength in Perseus B1 {#sub:dispersion_results}
---------------------------------------------------------------
![Dispersion of polarization angles for POL-2 observations of Perseus B1. *Top*: The angular dispersion function $[1-\text{cos}(\Delta \Phi)]$ as a function of the distance $\ell$. The fit of Equation \[eq:adf\] to the data is shown with (blue solid line) and without (black dashed line) including the autocorrelation function $b^2(\ell)$ defined in Equation \[eq:autocorrelation\]. *Bottom*: Signal-integrated turbulence autocorrelation function $b^2(\ell)$ as a function of distance $\ell$. The black dashed line shows the contribution of the telescope beam alone.[]{data-label="fig:fig7_dcf"}](f7-eps-converted-to.pdf){width="\columnwidth"}
Following Section \[sub:dcf\], we determine the angular dispersion function from the POL-2 data of Perseus B1. We include in this analysis all the POL-2 polarization vectors found in a 240 arcsec-wide square centered on the position (03$^{\text{h}}$ 33$^{\text{m}}$ 20$^{\text{s}}$.45, $+$31$^{\circ}$ 07$'$ 50$''$.16), as illustrated in the right panel of Figure \[fig:fig1\_b1\_polarization\]. This region covers most of the embedded young stellar objects in the densest parts of Perseus B1. The resulting angular dispersion function is shown in the top panel of Figure \[fig:fig7\_dcf\] as a function the distance $\ell$ in bins of 12 arcsec. The observed steady increase of this function with $\ell$ at small spatial scales (0.01 to 1.0 pc) is also a behavior seen in other studies using this technique [e.g., @Houde2009; @Houde2016; @Franco2010; @Ching2017; @Chuss2019].
The angular dispersion function was fitted with Equation \[eq:adf\] to obtain $\delta$ and $\left\langle B_t^2 \right\rangle / \left\langle B_o^2 \right\rangle$ using an effective cloud depth $\Delta'$ of $84$ arcsec, and a beam radius $W$ of $6.2$ arcsec (or a FWHM of $14.6$ arcsec) at 850 $\mu$m. The reduced chi-squared value for this fit is $\chi_r^2=1.5$. The results of the fit to the angular dispersion, including $\left\langle B_t^2 \right\rangle / \left\langle B^2 \right\rangle$, are given in Table \[tab:houde\_results\]. Additionally, the resulting turbulent autocorrelation function $b^2(\ell)$ is shown on the bottom panel of Figure \[fig:fig7\_dcf\].
At a distance of 295 pc [@Ortiz2018], the effective cloud depth $\Delta'$ of $84$ arcsec in B1 represents a physical depth of $\sim~0.1$ pc. While this effective cloud depth $\Delta' \sim 0.1$ pc was derived independently from the autocorrelation function of the polarized intensity $I_P$ (see Section \[sub:dcf\]), it is nonetheless comparable to the typical width of dense filaments in star-forming regions [e.g., @Arzoumanian2011; @Andre2014; @Koch2015; @Andre2016]. For reference, the square region shown in the right panel of Figure \[fig:fig1\_b1\_polarization\] has a width $\sim 0.4$ pc ($\sim 270$ arcsec).
The exact distance to the Perseus molecular cloud, and to B1 in particular, is still subject to some ambiguity. Indeed, different methods provide a wide range of values from 235 pc [22 GHz water maser parallaxes; @Hirota2008; @Hirota2011] to 315 pc [photometric reddening; @Schlafly2014]. Furthermore, @Schlafly2014 found a gradient of distances from the western (260 pc) to the eastern (315 pc) parts of the Perseus molecular cloud complex. However, recent parallaxes measurements with the *Gaia* space telescope instead suggest a smaller range of distances between NGC 1333 (295 pc) and IC 348 (320 pc) [@Ortiz2018]. According to these *Gaia* results, the distance to B1 is similar to that of NGC 1333 at 295 pc. This distance to B1 assumes that the young stellar objects used for these parallaxes measurements provide a good estimate of the clump’s true position along the line-of-sight.
Perseus B1 was mapped in emission from several NH$_3$ inversion transitions at $\sim$24 GHz by GAS (the first data release of the survey was presented by @Friesen2017). NH$_3$ is a commonly-used selective tracer of moderately dense gas ($n \gtrsim$ a few $10^3$ cm$^{-3}$; @Shirley2015). The NH$_3$ (1,1) emission closely follows the intensity detected with POL-2 across the cloud (GAS Consortium, in prep.). The velocity dispersion of the gas along each line-of-sight was obtained through simultaneous modeling of hyperfine structure of the detected NH$_3$ (1,1) and (2,2) inversion line emission. Assuming that the (1,1) and (2,2) lines share the same line-of-sight velocity, velocity dispersion, and excitation temperature, the analysis produces maps of the aforementioned parameters along with the gas kinetic temperature, and the total column density of NH$_3$. Further details of the modeling are given in @Friesen2017.
For the region delimited by the square in the right panel of Figure \[fig:fig1\_b1\_polarization\], we find an average velocity dispersion $\delta V = 0.29$ km s$^{-1}$, with a standard deviation $\sigma_{\delta V} = 0.11$ km s$^{-1}$. The uncertainties for individual line width measurements are typically $< 0.05$ km s$^{-1}$. We therefore use the velocity dispersion $\delta V = (2.9 \pm 1.1) \times 10^4$ cm s$^{-1}$ to calculate the plane-of-sky amplitude of the magnetic field with Equation \[eq:dcf\_houde\].
The number density $n(\text{H}_2)$ of the gas in Perseus B1 is also calculated from the same GAS NH$_3$ data (@Friesen2017; GAS Consortium, in prep.). Specifically, we follow the relation described by @Ho1983 between density, excitation temperature, and gas kinetic temperature to estimate the number density $n(\text{H}_2)$ in B1, assuming the NH$_3$ emission in B1 can be approximated by a two-level system. First, for the denser regions associated with polarized emission, we find a mean gas temperature of $11.6$ K with a standard deviation of $1.2$ K, and a mean excitation temperature of $6.5$ K with a standard deviation of $0.4$ K. Using these temperatures, we calculate a mean density $n(\text{H}_2) = (1.5 \pm 0.3) \times 10^5$ cm$^{-3}$. If the typical depth of the dense material in B1 is indeed $\sim 0.1$ pc, we then find a column density $N(\text{H}_2) = (4.7 \pm 0.9) \times 10^{22}$ cm$^{-2}$ in agreement with the values obtained from fitting far-infrared and submillimeter measurements of dust thermal emission [@Sadavoy2013; @Chen2016]. Finally, assuming a molecular weight $\mu = 2.8$ [@Kauffmann2008], we derive an average gas density $\rho = (7.0 \pm 1.4) \times 10^{-19}$ g cm$^{-3}$.
The ratio $\left\langle B_t^2 \right\rangle / \left\langle B^2 \right\rangle$ of turbulent-to-total magnetic energy given in Table \[tab:houde\_results\] can be used to calculate the plane-of-sky strength of the magnetic field in Perseus B1 using Equation \[eq:dcf\_houde\]. Combined with the values given previously for the density $\rho$ and velocity dispersion $\delta V$, we calculate the plane-of-sky strength of the magnetic field in Perseus B1 to be $120 \pm 60$ $\mu$G.
We compare the plane-of-sky strength of the magnetic field derived from the angular dispersion analysis [@Houde2009] with the one obtained from the classical DCF method [@Crutcher2004]. First, we fit a Gaussian curve to the histogram of POL-2 polarization angles shown in the top panel of Figure \[fig:fig5\_scupol\_histo\] and find a dispersion $\delta \Phi_{obs} = 0.213$ radians ($12.2^{\circ}$). We then evaluate the dispersion $\delta \Phi_{err}$ due to instrumental errors using the mean uncertainty in polarization angle of $0.099$ radians ($5.7^{\circ}$) given in Section \[sub:pol2\_perseusb1\]. This allows us to calculate the intrinsic angular dispersion $\delta \Phi = \sqrt{\delta \Phi_{obs}^2 - \delta \Phi_{err}^2} = 0.188$ radians ($10.8^{\circ}$). We then use Equation \[eq:dcf\], assuming a correction factor $A=0.5$ [e.g., @Pattle2017; @Soam2018; @Kwon2018], to derive a plane-of-sky magnetic field amplitude $B_{\text{pos}} \sim 230$ $\mu$G. This larger value for $B_{\text{pos}}$ suggests that a more appropriate correction factor for B1 would be $A \sim 0.25$. However, this derived field strength of $230$ $\mu$G could even be a lower limit (in the context of the classical DCF method) since the polarization vectors around B1-c are also included in the Gaussian fit, and so the appropriate correction factor to use would in fact be $A \lesssim 0.25$.
With the magnetic field amplitude $B_{\text{pos}}=120 \pm 60$ $\mu$G we have obtained from the angular dispersion analysis, it becomes possible to estimate the criticality criterion $\lambda_c$ of Perseus B1 with Equation \[eq:crit\_ratio\]. Using the hydrogen column density $N(\text{H}_2) = (4.7 \pm 0.9) \times 10^{22}$ cm$^{-2}$ derived previously, we find $\lambda_c = 3.0 \pm 1.5$. Since $\lambda_c > 1$, Perseus B1 is a magnetically supercritical molecular cloud, i.e., magnetic pressure alone cannot support the cloud against gravity.
Perseus B1 is among a few molecular clouds with a detection of OH Zeeman splitting, and thus a measurement of its magnetic field’s line-of-sight component. With observations of the OH lines at 1665 MHz and 1667 MHz using the Arecibo telescope and a beam width of $2.9$ arcmin, @Goodman1989 found a line-of-sight amplitude of $27 \pm 4$ $\mu$G for the magnetic field towards IRAS 03301+3057 (B1-a). While this value might have been overestimated relative to the line-of-sight amplitude of the magnetic field at large scales [@Crutcher1993; @Matthews2002], it nonetheless supports the idea that the orientation of the magnetic field in B1 might be mostly parallel to the plane of the sky (i.e., an inclination $\theta < 15^{\circ}$ relative to the plane of the sky).
[ccl]{} $\delta$ & $5.0 \pm 2.5$ arcsec & Turbulent correlation length scale\
$N$ & $27.3 \pm 0.3$ & Number of beam-integrated turbulent cells along the line-of-sight\
$\left\langle B_t^2 \right\rangle / \left\langle B_o^2 \right\rangle$ & $0.9 \pm 1.1$ & Turbulent-to-ordered magnetic energy ratio\
$\left\langle B_t^2 \right\rangle / \left\langle B^2 \right\rangle$ & $0.5 \pm 0.3$ & Turbulent-to-total magnetic energy ratio\
$a$ & $(2.4 \pm 0.2) \times 10^{-6}$ arcsec$^{-2}$ & First Taylor coefficient of the ordered auto-correlation function\
$\delta V$ & $(2.9 \pm 1.1) \times 10^4$ cm s$^{-1}$ & Velocity dispersion of the gas along the line-of-sight\
$n(\text{H}_2)$ & $(1.5 \pm 0.3) \times 10^{5}$ cm$^{-3}$ & Mean number density of the gas\
$N(\text{H}_2)$ & $(4.7 \pm 0.9) \times 10^{22}$ cm$^{-2}$ & Estimated column density for a cloud depth of $\sim 0.1$ pc\
$\rho$ & $(7.0 \pm 1.4) \times 10^{-19}$ g cm$^{-3}$ & Estimated density of the gas for a molecular weight $\mu=2.8$\
$B_{\text{pos}}$ & $120 \pm 60$ $\mu$G & Plane-of-sky amplitude of the magnetic field\
$\lambda_c$ & $3.0 \pm 1.5$ & Criticality ratio\
Discussion {#sec:discussion}
==========
Morphology of the Magnetic Field {#sub:morphology}
--------------------------------
The magnetic field in Perseus B1, as shown in the right panel of Figure \[fig:fig1\_b1\_polarization\], is seen to run roughly North-South (or $\sim 165^{\circ}$ East of North) across the whole region, including SMM3. The orientation of the vectors seen in Figure \[fig:fig1\_b1\_polarization\] (right) towards the bulk of the cloud (between B1-b N/S and SMM3) can be explained if B1 is part of a dense, slightly flattened cylindrical filament threaded perpendicularly by a large-scale magnetic field and viewed at an inclined angle to the line-of-sight [@Tomisaka2015]. While it may not be clear from Figure \[fig:fig1\_b1\_polarization\] alone, Perseus B1 is indeed part of a large filamentary structure extending towards the South-Western part of the map [@Chen2016]. Furthermore, magnetic field lines perpendicular to large-scale filaments have been hypothesized to funnel low density material into the striations (or sub-filaments) observed with *Herschel* in and around molecular clouds [@Andre2014]. Alternatively, if the cloud is collapsing gravitationally, then the apparent curving of the field lines West of SMM3 could be the sign of an emergent hourglass morphology [e.g., @Girart2006].
The largest discrepancy in the morphology of the large-scale magnetic field is seen towards the protostellar core B1-c, which is the source of a powerful molecular outflow viewed almost edge-on [@Matthews2006]. Indeed, the field turns more towards an East-West direction (or $\sim 120^{\circ}$ East of North) in the vicinity of B1-c, where it seems instead better aligned with the orientation of the protostellar outflow traced by the $^{12}$CO J=3-2 integrated intensity contour. In fact, the plane-of-sky component of the magnetic field towards B1-c is nearly parallel to the orientation of the outflow at $125^{\circ}$. In contrast, the local magnetic field direction is relatively well aligned with the mean field orientation in Perseus B1 ($\sim 165^{\circ}$) at the locations of the candidate first hydrostatic cores, and potentially less evolved, B1-bN ($\sim 155^{\circ}$) and B1-bS ($\sim 165^{\circ}$) objects [@Pezzuto2012; @Gerin2017], as well as at the previously identified young stellar objects associated with the submillimeter sources B1-a ($\sim 159^{\circ}$) and SMM3 ($\sim 158^{\circ}$), and to a lesser extent B1-d ($\sim 10^{\circ}$) and HH 789 ($\sim 180^{\circ}$) [@Bally2008]. This directional variation suggests that the magnetic field morphology is well ordered at large scales, but is potentially locally modified by the motion of the gas at smaller scales.
Perhaps the magnetic field orientation at B1-c originally followed the large-scale field of the molecular cloud, but was misaligned with the angular momentum of the initial prestellar core. As the core evolved, the magnetic field lines may have been “dragged” into a modified hourglass configuration [e.g., @Kataoka2012]. However, although hourglass structures have been seen toward some protostellar cores [e.g., @Girart2006; @Hull2017b], an alignment between magnetic field and outflow orientations does not appear to be a common occurrence [@Hull2014].
Alternatively, the orientation of the magnetic field at B1-c could be explained by more complex field models which have been shown to produce comparable polarization patterns [@Franzmann2017]. Indeed, recent ALMA observations of the protostellar core Ser-emb 8 in Serpens Main suggest that the magnetic field of that object, which is similarly misaligned with the large-scale field of the rest of the molecular cloud in which it is embedded, may not possess an hourglass morphology at all [@Hull2017]. However, the protostellar core Serpens SMM1 (also in Serpens Main) nevertheless shows evidence of having an hourglass field morphology while still being misaligned with the magnetic field at larger scales [@Hull2017b]. It would therefore be premature to assume that an observed misalignment in magnetic field orientations between core and cloud scales necessarily implies the absence of an hourglass field morphology.
Another peculiar property of B1-c is the orientation of the few polarization vectors found East from the protostellar core and along its outflow, as traced by the $^{12}$CO J=3-2 contour in Figure \[fig:fig1\_b1\_polarization\]. The inferred magnetic field orientation from the vectors found directly in the outflow’s path ($\sim 160^{\circ}$) is in better agreement with the large-scale field in B1 ($\sim 165^{\circ}$) than with the field orientation towards B1-c itself ($\sim 120^{\circ}$). Magnetic field orientations that are nearly perpendicular to outflows at large scales are not expected from ideal hourglass field morphologies.
An alternative explanation would be that elongated dust grains found in the vicinity of the outflow are aligned mechanically by the flow of gas instead of radiatively. In this case, the polarization vectors would be parallel (and the inferred magnetic field orientation perpendicular) to the outflow orientation, regardless of the field morphology [@Gold1952a; @Lazarian1997; @Lazarian2007_review], as is seen. This last scenario, however, has been shown to be unlikely even in the case of explosives outflows such as in Orion BN/KL [@Tang2010].
Indeed, the original mechanical alignment proposed by @Gold1952a requires supersonic flows to be efficient, and it is particularly inefficient for suprathermally rotating grains (see @Lazarian1997, @Das2016). Thus, although its polarization pattern seems to be consistent with the observed polarization map, it is rather difficult to explain the high polarization degree ($\sim~15$ per cent) shown in Figure \[fig:fig1\_b1\_polarization\]. On the other hand, the MechAnical Torque (MAT) alignment mechanism proposed by @Lazarian2007b and numerically demonstrated by @Hoang2018 predicts that the gas flow can efficiently align grains with the magnetic field. Specifically, the MAT mechanism predicts that the long-axis of the grains will be perpendicular either to the magnetic field or the gas flow. Therefore, the polarization vectors found along the outflow’s lobes may reveal that the magnetic field in the flow is not much different from the large-scale magnetic field in the rest of the molecular cloud.
Finally, there is the possibility that we are mainly measuring the polarization from dust grains found in the cavity walls of the B1-c outflow. Indeed, it has been suggested that strong irradiation of outflow cavity walls can enhance the polarized emission of the associated dust grains through radiative torques [e.g., @Maury2018]. This scenario is supported by ALMA observations of B1-c (or Per-emb-29) [@Cox2018] which provide evidence for significantly improved grain alignment (with $P > 5$ per cent) along outflow cavities near the protostar. Although previous ALMA studies have shown that the magnetic field along comparable outflow cavities tend to be parallel to the outflow orientation [@Hull2017b; @Maury2018; @Cox2018] instead of perpendicular as observed eastward from B1-c in Figure \[fig:fig1\_b1\_polarization\], their spatial resolutions were much smaller ($140$ au, $60$ au, and $100$ au respectively) than our resolution of $\sim 3500$ au. It could be that the dust grains with potentially enhanced polarized emission farther along the outflow cavity are instead tracing the large-scale field in the cloud, which would fit with the twisted field picture from @Kataoka2012 where the polarization signature becomes less affected by the outflow the farther away you look from the central source.
Magnetic and Turbulent Properties {#sub:magnetic_turbulence}
---------------------------------
In Section \[sub:dispersion\_results\], we derived the turbulent and magnetic properties of Perseus B1 from the angular dispersion analysis described by @Houde2009 (see Figure \[fig:fig7\_dcf\]). Specifically, we obtain a ratio of turbulent-to-total magnetic energy $\left\langle B_t^2 \right\rangle / \left\langle B^2 \right\rangle = 0.5 \pm 0.3$, which indicates that a large part of the magnetic energy in the cloud is found in the form of magnetized turbulence. This is larger than the ratio $\left\langle B_t^2 \right\rangle / \left\langle B^2 \right\rangle \sim 0.4$ found by @Levrier2018 for the galactic magnetic field using Planck data. As a comparison, a previous study utilizing the angular dispersion analysis presented in Section \[sub:dcf\] found ratios of turbulent-to-total magnetic energy $\left\langle B_t^2 \right\rangle / \left\langle B^2 \right\rangle$ of, respectively, 0.6, 0.7 and 0.7 for the high mass star-forming regions W3(OH), W3 Main and DR21(OH) [@Houde2016].
Since the ionized and neutral components of the gas in molecular clouds are typically well coupled, this magnetized turbulence is expected to be indistinguishable from the turbulence in the neutral gas as long as ambipolar diffusion remains negligible [e.g., @Krumholz2014]. Furthermore, the relatively large turbulent component of the magnetic field in B1 could be explained by the presence of at least five young stellar objects with confirmed molecular outflows (B1-a, B1-bS, B1-c, B1-d, and HH 789) in the main body of the cloud [@Hatchell2009]. Indeed, such outflows are among the most probable drivers of turbulence in molecular clouds [@Bally2008]. However, the signature of this protostellar feedback on the velocity dispersion of NH$_3$ does not appear to be as pronounced in B1 (GAS Consortium, in prep.) as it is in the more compact B59 in the Pipe nebula [see Figure 9 in @Redaelli2017], but a more detailed coherence analysis will be required to adequately investigate this effect.
The turbulent cells in B1 have a correlation length $\delta$ of $5.0 \pm 2.5$ arcsec, which for a distance of 295 pc represents a physical length of 1475 au. From Equation \[eq:turbulence\], we estimate that there are typically $\sim 30$ turbulent cells probed by the telescope’s beam along the depth of the cloud ($0.1$ pc). The number of turbulent cells along the line-of-sight could potentially be greater in higher density regions, such as towards pre-stellar cores. This larger number would explain the observed depolarization effect seen in Figure \[fig:fig2\_depolarization\] (top) as the Stokes $I$ intensity increases, which can be roughly understood as an increase in the dust column density. Indeed, an increased number of turbulent cells is expected to randomize dust orientations along the line-of-sight, and thus decrease the measured fraction of polarization $P$. Additionally, and perhaps counter-intuitively, numerical simulations by @Cho2016 have also shown that the averaging of a high number of turbulent cells along the line-of-sight could preserve the appearance of a well-ordered field morphology at large scales, which is an effect initially proposed by @Jones1992.
In Section \[sub:dispersion\_results\], we also find a plane-of-sky amplitude $B_{\text{pos}} = 120 \pm 60$ $\mu$G for the magnetic field, and a criticality criterion $\lambda_c = 3.0 \pm 1.5$. Although this magnetic field amplitude is relatively weak when compared to the fields found in high mass star-forming regions such as Orion A (where $B_{\text{pos}} \gtrsim 1.0$ mG) [e.g., @Houde2009; @Pattle2017] or in hub-filament structures such as IC 5146 (with $B_{\text{pos}} \sim 0.5$ mG) [e.g., @Wang2018arxiv], it is either comparable to or larger than the field strengths ($B_{\text{pos}} \lesssim 100$ $\mu$G) typically found in low-mass prestellar cores [e.g., @Crutcher2004; @Kirk2006; @Liu2019arxiv]. Above all, these results indicate that Perseus B1 is a supercritical molecular cloud (i.e., magnetic pressure alone cannot support the cloud against gravity). The criticality criterion $\lambda_c$ defined by Equation \[eq:crit\_ratio\], however, may be overestimated due to geometric effects. Indeed, @Crutcher2004 find that, on average, the effective criticality criterion is $\overline{\lambda_{c}} \approx \lambda_c / 3$. In the case of B1, this adjustment would lead to $\overline{\lambda_{c}} \approx 1.0$, which is the theoretical limit at which the cloud would be subcritical.
Since the inclination of the magnetic field in B1 can be calculated using published Zeeman line splitting measurements (see Section \[sub:dispersion\_results\]), we can better estimate the effect of geometry on the criticality criterion $\overline{\lambda_c}$. Assuming that the line-of-sight component obtained by @Goodman1989 ($27 \pm 4$ $\mu$G) is not an overestimation at large scales, we find an inclination $\theta = 12^{\circ}$ relative to the plane of the sky and an amplitude $B_{\text{tot}} \approx 125$ $\mu$G for the total magnetic field when combined with the plane-of-sky amplitude $B_{\text{pos}} = 120 \pm 60$ $\mu$G found in Section \[sub:dispersion\_results\]. If the cloud can also be approximated as a mostly prolate filament with a cylindrical symmetry, which is a reasonable assumption for a relatively weak magnetic field in a dense filament, then we get $\overline{\lambda_c} \approx \lambda_c$. We therefore find it likely that Perseus B1 is indeed supercritical by a factor $\sim 3$, although we cannot rule out if a combination of magnetic pressure and turbulence would be sufficient to significantly slow down the fall of additional material onto the central clump.
Polarization Fraction and Grain Alignment {#sub:depolarisaton}
-----------------------------------------
Fundamentally, the fraction $P$ of polarization can be understood as the alignment efficiency of a mixture of dust grains in the interstellar medium. Even though this fraction $P$ can be affected by purely environmental factors such as the number of integrated turbulent cells along the line-of-sight and complex magnetic field geometries, or even instrumental factors such as molecular contamination (see Appendix \[sub:contamination\]), it is intrinsically linked to the models of grain alignment.
Specifically, the contribution to the continuum emission of different grain sizes and compositions in the dust mixture could explain the apparent dependence of $P$ on the wavelength at far-infrared and submillimeter wavelengths [@Vaillancourt2012]. For example, grain growth in cold high density regions may lead to very large dust grains, with sizes $a \gtrsim 1.0$ $\mu$m [e.g., @Pagani2010], which align less efficiently through radiative torques than the typical grains ($a \sim 0.1$ $\mu$m) found in molecular clouds [@Hoang2009]. This scenario could potentially explain the apparent drop in polarization fraction $P$ seen in Figure \[fig:fig3\_depolarization\] above a visual extinction $A_V > 200$ mag, as well as towards B1-c, since there is significant evidence for grain growth across Perseus B1 [@Sadavoy2013; @Chen2016].
Furthermore, since the RAT theory of grain alignment depends on the stellar radiation field incident on the grains, the alignment efficiency is expected to be smaller towards regions with high dust opacities (e.g., dense prestellar cores) [@Andersson2015]. This effect would potentially explain the apparent minimum $P$ of $\sim$1 per cent seen both in Figure \[fig:fig2\_depolarization\] (top) and by @Matthews2002 for the highest opacity regions of the cloud, which in the case of Perseus B1 are associated with embedded young stellar objects such as the first hydrostatic core candidates B1-b N/S (see Figure \[fig:fig1\_b1\_polarization\]). This alignment efficiency, however, is expected to improve again if there is a significant source of radiation, such as a protostar, within the core itself. Such a scenario would explain the shallower than expected power index $\beta \sim -0.5$ given in Section \[sub:pol2\_perseusb1\] for the relation between the polarization fraction $P$ and the visual extinction $A_V$ in Perseus B1.
Nevertheless, B1-c, which is known to be a bright and warm protostellar core [@Sadavoy2013], also has among the lowest polarization fractions measured by POL-2 for B1. This behavior suggests that we may not be resolving the improved grain alignment efficiency seen by ALMA near the protostar [@Cox2018]. Indeed, @Jones2016 previously observed such an effect when comparing single-dish and interferometric polarization data of the protostellar core G034.43+00.24 MM1. Alternatively, it could be that factors other than alignment efficiency need to be taken into account to explain the polarization towards this object.
As an example, previous studies have found an inverse correlation between the polarization fraction $P$ and the local dispersion of magnetic field orientations at several scales in molecular clouds [@Planck2015XIX; @Planck2015XX; @Fissel2016; @Koch2018]. Such a measure towards B1-c would support the hypothesis of a complex but unresolved polarization structure, and higher resolution observations using interferometric facilities would provide further evidence to confirm or infirm this scenario. However, while there exist ALMA data of the linear polarization towards B1-c, only the most highly polarized emission is likely to have been recovered due to the short integration time ($8$ minutes) of these observations [@Cox2018]. A deeper ALMA polarization map of B1-c might therefore reveal a more complex magnetic field structure comparable to those observed in similar protostellar cores [e.g., @Hull2017; @Hull2017b; @Maury2018].
Conclusion {#sec:conclusion}
==========
We have observed the 850 $\mu$m linear polarization towards the B1 clump in the Perseus molecular cloud complex using the POL-2 polarimeter as part of the BISTRO survey at the JCMT. We have also compared the resulting polarization map with previously published SCUPOL observations of B1 from @Matthews2009 to illustrate the improvements brought by the increased sensitivity and reliability of POL-2 over its predecessor. From the POL-2 observations, we have inferred the plane-of-sky morphology of the magnetic field in Perseus B1 by rotating the 850 $\mu$m polarization vectors by 90$^\circ$ assuming the dust grains are aligned by radiative torques [e.g., @Andersson2015]. The plane-of-sky component of the magnetic field in most of the cloud is orientated in a North-South direction (or $\sim 165^{\circ}$ East of North), except towards the protostellar core B1-c where it turns more East-West in better agreement with the orientation of its associated molecular outflow.
We have also plotted the polarization fraction $P$ and the de-biased polarized intensity $I_P$ as a function of the Stokes $I$ total intensity. Specifically, we have fitted a power-law to the relationship between $P$ and $I$, and we find a power index $\alpha \sim -0.9$ in agreement with other BISTRO studies. There exists a clear trend in Perseus B1 of decreasing polarization fraction $P$ as a function of increasing Stokes $I$, although the polarized intensity $I_P$ itself appears to increase steadily. Such a behavior is likely linked to depolarization effects towards higher density regions, such as a complex field geometry, a low efficiency of grain alignment, or an increased number of turbulent cells along the line-of-sight.
Similarly, we have plotted the polarization fraction $P$ as a function of the visual extinction $A_V$ in Perseus B1, and fitted a power-law between the two parameters. We find a power index $\beta \sim -0.5$, which is a shallower value than those previously found in starless cores with comparable extinction measurements ($A_V>20$). This shallow power index $\beta \sim -0.5$ could therefore be explained by improved grain alignment due to the radiation from embedded young stellar objects in the cloud.
We have applied the angular dispersion analysis developed by @Houde2009 to the POL-2 850 $\mu$m polarization map of Perseus B1. By fitting the angular dispersion function, we have measured a turbulent magnetic correlation length $\delta$ of $5.0 \pm 2.5$ arcsec, which for a distance of 295 pc represents a physical length of $\sim 1500$ au, and a turbulent-to-total magnetic energy ratio of $0.5 \pm 0.3$ inside the cloud. Such a large ratio indicates that a significant part, if not most, of the magnetic energy in the cloud is found in the form of magnetized turbulence. Additionally, using an effective cloud depth of $\sim 0.1$ pc, we have evaluated that there are typically $\sim 30$ beam-integrated turbulent cells along the line-of-sight across B1.
With an updated version of the Davis-Chandrasekhar-Fermi method, we have evaluated the plane-of-sky amplitude of the magnetic field in Perseus B1 to be $B_{\text{pos}} = 120 \pm 60$ $\mu$G. From this amplitude, we have estimated the magnetic criticality criterion in this cloud to be $\lambda_c = 3.0 \pm 1.5$. We also found with measurements of OH Zeeman line splitting that the orientation of the magnetic field is nearly parallel to the plane of the sky, and thus this criticality criterion is unlikely to be overestimated due to geometric effects. Perseus B1 is therefore a magnetically supercritical molecular cloud.
Finally, our findings show that the angular dispersion analysis presented by @Houde2009 can be successfully applied to POL-2 observations of nearby star-forming regions. It will therefore be possible in future works to expand this analysis to a representative sample of molecular clouds in order to systematically quantify, and compare, their magnetic and turbulent properties. This illustrates how the BISTRO survey has the potential to provide us with unparalleled insight into the roles of magnetic fields and turbulence in the physical processes leading to the formation of stars and their planets.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the staff of the East Asian Observatory for their invaluable support in the completion of the BISTRO survey. We also wish to thank the people of Hawai’i for granting us access to the unique geographical site of the Maunakea observatory. Furthermore, we are grateful to the GAS Consortium for generously granting us access to their spectroscopic data. We also thank the anonymous reviewer for their helpful and detailed comments. Finally, we thank B.-G. Andersson, Kelvin Au, Jordan Guerra Aguilera, James Lane, Anna Ordog, Amélie Simon, Ian Stephens, and Julien Vandeportal for helpful discussions.
This research was conducted in part at the SOFIA Science Center, which is operated by the Universities Space Research Association under contract NNA17BF53C with the National Aeronautics and Space Administration.
The James Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of The National Astronomical Observatory of Japan; Academia Sinica Institute of Astronomy and Astrophysics; the Korea Astronomy and Space Science Institute; Center for Astronomical Mega-Science (as well as the National Key R&D Program of China with No. 2017YFA0402700). Additional funding support is provided by the Science and Technology Facilities Council of the United Kingdom and participating universities in the United Kingdom and Canada.
SCUBA-2 and POL-2 were built through grants from the Canada Foundation for Innovation. This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. This research has also made use of the <span style="font-variant:small-caps;">simbad</span> database and of NASA’s Astrophysics Data System Bibliographic Services. The Starlink software [@Currie2014] is currently supported by the East Asian Observatory.
Miju Kang was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2015R1C1A1A01052160). Woojin Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF-2016R1C1B2013642). C.W.L. was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2016R1A2B4012593). Keping Qiu is supported by National Key R&D Program of China No. 2017YFA0402600, and acknowledges the support from National Natural Science Foundation of China (NSFC) through grants U1731237, 11473011, 11629302, and 11590781.
Effect of Molecular Contamination {#sub:contamination}
=================================
Another effect which may influence the measured fractions of polarization is the contribution from molecular line emission at submillimeter wavelengths. The $^{12}$CO J=3-2 molecular line in particular has been shown in some special cases to be a significant source of contamination in SCUBA-2 continuum observations at 850 $\mu$m [@Drabek2012]. While relatively rare, high levels of $^{12}$CO J=3-2 line contamination ($>$10 per cent) in star-forming regions are usually associated with molecular outflows from young stellar objects [e.g., @Chen2016; @Coude2016]. This behavior occurs in SCUBA-2 observations of Perseus B1, where @Sadavoy2013 found $^{12}$CO J=3-2 line contamination levels of 90 per cent in the outflows of B1-c, 15 per cent in the central region of B1, and $<~1$ per cent in the rest of the cloud.
It is important to note that HARP, SCUBA-2, and POL-2 are not sensitive to the same spatial scales due to their different observing strategies. Specifically, SCUBA-2 observations for the JCMT Gould Belt Survey were taken using a PONG 1800 observing mode that is sensitive to larger spatial scales than the Daisy mode used for POL-2 [@Chapin2013; @Friberg2016]. We therefore expect contamination levels for POL-2 to be different than those previously measured for SCUBA-2 alone, but nonetheless still confined to molecular outflows if present. Similarly, HARP observations are sensitive to larger angular scales than those from SCUBA-2, and they had to be spatially filtered during data reduction to be subtracted accurately from the 850 $\mu$m maps of the JCMT Gould Belt Survey [e.g., @Mairs2016]. Such a subtraction procedure for $^{12}$CO J=3-2 molecular line contamination could potentially be adapted for future analyses of BISTRO observations.
The emission from the $^{12}$CO J=3-2 molecular line can be weakly linearly polarized by magnetic fields through the Goldreich-Kylafis effect [@Goldreich1981; @Goldreich1982]. Observational evidence, however, suggests that this polarization is only on the order of 1 per cent for single-dish observatories [e.g., @Greaves1999; @Forbrich2008]. Such a level of polarization would only be detectable by POL-2 in extreme cases of molecular contamination, such as the unlikely scenario of a $\sim 1.3$ Jy beam$^{-1}$ submillimeter source with a $^{12}$CO J=3-2 contamination level of 90 per cent (assuming a $3 \, \sigma$ detection threshold of $I_P \sim 12$ mJy beam$^{-1}$, and the maximum contamination fraction measured by @Sadavoy2013). If there is significant contamination from the $^{12}$CO J=3-2 molecular line in POL-2 observations at 850 $\mu$m, it is reasonable to assume that this additional contribution to the continuum flux is unpolarised. Therefore, the effect of contamination will be to overestimate the Stokes $I$ total intensity while the Stokes $Q$ and $U$ parameters remain unchanged.
In other words, molecular contamination from the $^{12}$CO J=3-2 molecular line will lead to an underestimation of the polarization fraction $P$, but the polarization angle $\Phi$ will be unaffected if the instrumental polarization is properly taken into consideration. This effect is thus unlikely to influence our characterization of the magnetic and turbulent properties of Perseus B1, although it could potentially affect the polarization fraction $P$ plotted in Figure \[fig:fig2\_depolarization\] (top). Such possible contamination may need to be taken into account for future, more detailed analysis of grain alignment efficiency using POL-2 data.
Finally, it is important to note that the Goldreich-Kylafis effect might nonetheless be important for polarimetric observations using interferometers such as the SMA. Indeed, @Ching2016 measured polarization fractions up to 20 $\%$ for the $^{12}$CO J=3-2 emission towards the IRAS 4A protostellar outflow. In such cases, continuum measurements of the Stokes $Q$ and $U$ parameters are likely to be affected by strong $^{12}$CO line contamination of the Stokes $I$ total intensity.
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address:
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Univ. Hamburg, Institut für Theor. Physik II\
22761 Hamburg, Germany, E-mail: kotikov@mail.desy.de
- |
Dep. de Física de Partículas, Univ. de Santiago de Compostela\
15706 Santiago de Compostela, Spain, E-mail: gonzalo@fpaxp1.usc.es
author:
- 'A. V. KOTIKOV'
- 'G. PARENTE'
title: SMALL $X$ BEHAVIOUR OF PARTON DISTRIBUTIONS
---
=cmr8
1.5pt
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
-0.5cm The measurements of the deep-inelastic scattering structure function $F_2$ in HERA [@H1] have permitted the access to a very interesting kinematical range for testing the theoretical ideas on the behavior of quarks and gluons carrying a very low fraction of momentum of the proton, the so-called small $x$ region. In this limit one expects that non-perturbative effects may give essential contributions. However, the resonable agreement between HERA data and the NLO approximation of perturbative QCD that has been observed for $Q^2 > 1 $GeV$^2$ (see the recent review in [@CoDeRo]) indicates that perturbative QCD could describe the evolution of structure functions up to very low $Q^2$ values, traditionally explained by soft processes.
Here we illustrate the results obtained recently in [@Q2evo]. These results are the extension to the NLO QCD approximation of previous LO studies [@Rujula]. The main ingredients are:
[**1.**]{} Both, the gluon and quark singlet densities are presented in terms of two components ($'+'$ and $'-'$) which are obtained from the analytical $Q^2$ dependent expressions of the corresponding ($'+'$ and $'-'$) parton distributions moments.
[**2.**]{} The $'-'$ component is constant at small $x$, whereas the $'+'$ component grows at $Q^2 \geq Q^2_0$ as $$\sim \exp{\left(2\sqrt{\left[
a_+\ln \left(
\frac{a_s(Q^2_0)}{a_s(Q^2)} \right) -
\left( b_+ + a_+ \frac{\beta_1}{\beta_0} \right)
\Bigl( a_s(Q^2_0) - a_s(Q^2) \Bigr) \right]
\ln \left( \frac{1}{x} \right)} \right)},$$ where the LO term $a_+ = 12/\beta_0$ and the NLO one $b_+ = 412f/(27\beta_0)$. Here the coupling constant $a_s=\alpha_s/(4\pi)$, $\beta_0$ and $\beta_1$ are the first two coefficients of QCD $\beta$-function and $f$ is the number of active flavors.
We have analyzed $F_2$ HERA data at small $x$ from the H1 coll.[@H1]. The initial scale of the parton distributions was fixed into the fits to $Q^2_0$ = 1 $GeV^2$, although later it was released to study the sensitivity of the fit to the variation of this parameter. The analyzed data region was restricted to $x<0.01$ to remain within the kinematical range where our results are accurate.
Fig. 1 shows $F_2$ calculated from the fit with Q$^2$ $>$ 1 GeV$^2$. Only the lower $Q^2$ bins are shown. One can observe that the NLO result (dot-dashed line) lies closer to the data than the LO curve (dashed line). The lack of agreement between data and lines observed at the lowest $x$ and $Q^2$ bins suggests that the initial flat behavior should occur at $Q^2$ lower than 1 GeV$^2$. In order to study this point we have done the analysis considering $Q_0^2$ as a free parameter. Comparing the results of the fits (see [@Q2evo]) one can notice the better agreement with the experiment of the NLO curve at fitted $Q^2_0=0.55 GeV^2$ (solid curve) which is more apparent at the lowest kinematical bins.\
-0.5cm
-0.3cm
-0.5cm
A.K. was supported by Alexander von Humboldt fellowship and by DIS2000 Orgcommittee. G.P. was supported in part by Xunta de Galicia (XUGA-20602B98) and CICYT (AEN96-1673).
[**References**]{}
[99]{} H1 Collab.: S. Aid et al., [*Nucl.Phys.*]{} B [**470**]{}, 3 (1996); ZEUS Collab.: M. Derrick et al., [*Zeit.Phys.*]{} C [**72**]{}, 399 (1996); A.M. Cooper-Sarkar et al., [*Int.J.Mod.Phys.*]{} A [**13**]{}, 3385 (1998). A.V. Kotikov and G. Parente, [*Nucl.Phys.*]{} B [**549**]{}, 242 (1999). A. De Rújula et al., [*Phys.Rev.*]{} [**D10**]{}, 1649 (1974); R.D. Ball and S. Forte, [*Phys.Lett.*]{} B [**336**]{}, 77 (1994); L. Mankiewicz, A. Saalfeld and T. Weigl, [*Phys.Lett.*]{} B [**393**]{}, 175 (1997).
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abstract: 'Braiding of Majorana fermions gives accurate topological quantum operations that are intrinsically robust to noise and imperfection, providing a natural method to realize fault-tolerant quantum information processing. Unfortunately, it is known that braiding of Majorana fermions is not sufficient for implementation of universal quantum computation. Here we show that topological manipulation of Majorana fermions provides the full set of operations required to generate random numbers by way of quantum mechanics and to certify its genuine randomness through violation of a multipartite Bell inequality. The result opens a new perspective to apply Majorana fermions for robust generation of certified random numbers, which has important applications in cryptography and other related areas.'
author:
- 'Dong-Ling Deng'
- 'Lu-Ming Duan'
title: Fault Tolerant Quantum Random Number Generator Certified by Majorana Fermions
---
The complex-valued solutions to the Dirac equation predict that every elementary particle should have a complex conjugate counterpart, namely an antiparticle. For example, an electron has a positron as its antiparticle. However, in $1937$ Ettore Majorana [@1937Majorana] showed that the complex Dirac equation can be modified to permit real wave-functions, leading to the possible existence of the so called *Majorana fermions* which are their own antiparticles [@2009Wilczek]. In condensed matter physics, Majorana fermions may appear as elementary qusi-particle excitations. To search for Majorana fermions, several proposals have been made in recent years, including $\nu =5/2$ fractional quantum Hall system [@2010Stern; @2008Nayak], topological insulator (TI)—superconductor (SC) interface [@2008Fu], interacting quantum spins [@2006Kitaev], chiral p-wave superconductors [@2000Read], spin-orbit coupled semiconductor thin film [@2010Sau] or quantum nanowire [@2010Lutchyn; @2010Oreg] in the proximity of an external s-wave superconductor. Based on these proposals, experimentalists have made great progress recently. For instance, Ref. [@2012Wang] reported an experimental observation of coexistence of the superconducting gap and the topological surface state in the $\mathtt{Bi}_{2}\mathtt{Se}_{3}$ thin film as a step towards realization of Majorana fermions. More recently, signature of Majorana fermions in hybrid superconductor-semiconductor nanowire device has been reported [@2012Mourik], which has raised strong interest in the community.
Majorana fermions are exotic particles classified as non-abelian anyons with fractional statistics, and braiding between them gives nontrivial quantum operations that are topological in nature. These topological quantum operations are intrinsically robust to noise and experimental imperfection, so they provide a natural solution to realization of fault-tolerant quantum gates. Application of Majorana fermions in implementation of fault-tolerant quantum computation has raised great interest [@2006Kitaev; @2008Nayak]. Unfortunately, braiding of Majorana fermions are not sufficient yet for realization of universal quantum computation [@2008Nayak], and we need assistance from additional non-topological quantum gates which are prone to influence of noise.
In this Letter, we show that topological manipulation of Majorana fermions alone can be used to realize a quantum random number generator in a fault tolerant fashion and to certifies its genuine randomness through violation of the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality [1993Belinskii,1992Ardehali,1990Mermin]{}. Random numbers have tremendous applications in science and engineering [2001Ackermann,1991Hultquist,2003Tu]{}. However, generation of genuine random numbers is a challenging task [@2010Pironio]. Any classical device does not generate genuine randomness as it allows a deterministic description in principle. Quantum mechanics is intrinsically random, and one can explore this feature to generate random numbers [1956Isida,1990Svozil,1994Rarity,2000Jennewein]{}. However, in real experiments, the intrinsic randomness of quantum mechanics is always mixed-up with an apparent randomness due to noise or imperfect control of the experiment [@2010Pironio]. The latter can be exploited by an adversary opponent and leads to security loopholes in various applications of randomness. Recently, a nice idea has been put forward to certify genuine randomness generated by a quantum device through test of violation of the Bell-CHSH (Clauser-Horn- Shimony-Holt [@1969Clauser]) inequality [2007Colbeck,2010Pironio]{}, and the idea has been demonstrated in a proof-of-principle experiment using remote entangled ions [@2010Pironio]. This implementation is not fault-tolerant yet as the remote entanglement is sensitive to noise and the quantum gates have limited precision which can all lead to security loopholes. We show here that all the operations for generation and certification of genuine randomness can be realized through topological manipulation of Majorana fermions. This implementation is inherently fault-tolerant and automatically closes security loopholes caused by influence of noise.
The implementation of certification of a quantum random number generator with Majorana fermions is tricky. First of all, one can not use the Bell-CHSH inequality anymore as proposed in Ref. [@2010Pironio], since it is impossible to violate this inequality through topological manipulation of Majorana fermions alone [@2009Brennen]. In fact, to observe violations of the CHSH inequality, measurements in the non-Clifford bases are required. However, topological operations on Majorana fermions can only give gates in the Clifford group, and thus not able to achieve the measurements required for the CHSH inequality violation for randomness certification. Consequently, we have to consider certification of randomness based on extension of the Bell inequalities in the multi-qubit case. For simplicity, here we use the MABK inequality for three logical qubits [@1993Belinskii; @1992Ardehali; @1990Mermin]. We show that first, this inequality can be used to certify randomness, and second, the inequality can be tested with topological manipulation of Majorana fermions alone. For the MABK inequality, we consider three qubits, each with two measurement settings. We denote the measurement settings for each qubit by the binary variables $x$, $y$, $z$, and the corresponding measurement outcomes by $a$, $b$, $c$, where $x,y,z,a,b,c=0,1$. The MABK inequality can be rewritten as [@1993Belinskii; @1992Ardehali; @1990Mermin] $$L\equiv \sum_{(x,y,z)\in \mathcal{S}}\tau (x,y,z)[P(\mathtt{even}|xyz)-P(\mathtt{odd}|xyz)]\leq 2, \label{MABK-Ineq}$$where $\mathcal{S}=\{(0,0,0),(0,1,1),(1,0,1),(1,1,0)\}$ and $\tau (x,y,z)$ is a sign function defined by $\tau (x,y,z)=(-1)^{(x+y+z)/2}$; $P(\mathtt{even}|xyz)$ ($P(\mathtt{even}|xyz)$) is the probability that $a+b+c$ is an even (odd) number when settings $(x,y,z)$ are chosen. The inequality ([MABK-Ineq]{}) is satisfied by all local hidden variable models. However, in quantum mechanics certain measurements performed on entangled states can violate this inequality. Experimentally, we can repeat the experiment $k$ times in succession to estimate the violation. For each trial, the measurement choices $(x,y,z)$ are generated by an independent identical probability distribution $P(xyz)$. Denote the input string as $\mathcal{I}=(x_{1},y_{1},z_{1};\cdots ;x_{k},y_{k},z_{k})$ and the corresponding output string as $\mathcal{O}=(a_{1},b_{1},c_{1};\cdots ;a_{k},b_{k},c_{k})$. The estimated violation of the MABK inequality can be obtained from the observed data as $$\hat{L}=\frac{1}{k}\sum_{(x,y,z)\in \mathcal{S}}\frac{\tau (x,y,z)}{P(xyz)}[N(\mathtt{even}|xyz)-N(\mathtt{odd}|xyz)],$$where $N(\mathtt{even}|xyz)$ ($N(\mathtt{odd}|xyz)$) denotes the number of trials that we get an even (odd) outcome $a+b+c$ after $k$ times of measurements with the measurement setting $(x,y,z)$.
We need to show that the output string $\mathcal{O}$ from the measurement outcomes contains genuine randomness by proving that it has a nonzero entropy. Let $\{\mathcal{L}_{m}:0\leq m\leq m_{max}\}$ be a series of violation thresholds with $\mathcal{L}_{0}=2$ and $\mathcal{L}_{m_{max}}=4$, corresponding respectively to the classical and quantum bound. Denote by $\mathcal{D}(m)$ the probability that the observed violation $\hat{L}$ lies in the interval $[\mathcal{L}_{m},\mathcal{L}_{m+1})$. We can use the min-entropy to quantify randomness of the output string $\mathcal{O}$ [2010Pironio,2012Pironio,2009Koenig]{}: $$E_{\infty }(\mathcal{O}|\mathcal{I},\mathcal{E},m)_{\mathcal{D}}\equiv -\mathtt{log}_{2}\sum_{\mathcal{I},\mathcal{E}}[\max_{\mathcal{O}}\mathcal{D}(\mathcal{O},\mathcal{I},\mathcal{E}|m)],$$where $\mathcal{E}$ represents the knowledge that a possible adversary has on the state of the device and the maximum is taken over all possible values of the output string $\mathcal{O}$. The probability distribution $\mathcal{D}(\mathcal{O},\mathcal{I},\mathcal{E}|m)$ is defined in the Supplemental Material. Based on a similar procedure as in Ref. [@2010Pironio], we can prove that if $\mathcal{D}(m)>\delta $, the min-entropy of the output string conditional on the input string and the adversary’s information has a lower bound (see derivation in the supplement), given by $$E_{\infty }(\mathcal{O}|\mathcal{I},\mathcal{E},m)_{\mathcal{D}}\geq kf(\mathcal{L}_{m}-\epsilon )-\log _{2}\left( 1/\delta \right) ,
\label{min-entropyBound}$$where the parameter $\epsilon \equiv \sqrt{-2(1+4r)^{2}(\ln \epsilon
^{\prime 2})}$ with $r=\min P(xyz)$, the smallest probability of the input pairs, and $\epsilon ^{\prime }$ is a given parameter that characterizes the closeness between the target distribution $\mathcal{D}(\mathcal{O},\mathcal{I},\mathcal{E})$ and the real distribution after $k$ successive measurements (see the supplement for an explicit definition). The function $f(\hat{L})$ can be obtained through numerical calculation based on semi-definite programming (SDP) [@1996Vandenberghe] and is shown in Fig. [Randomness-Show]{}. The minimum-entropy bound $kf(\mathcal{L}_{m}-\epsilon
)-\log _{2}\frac{1}{\delta }$ and the net entropy versus the number of trials $k$ are plotted in the insets (a) and (b) of Fig. [Randomness-Show]{}. Any observed quantum violation with $\hat{L}>2$ leads to a positive lower bound of the min-entropy, and a positive mini-entropy guarantees that genuine random numbers can be extracted from the string $\mathcal{O}$ of the measurement outcomes through the standard protocol of random number extractors [@1999Nisan]. As some amount of randomness needs to be consumed to prepare the input string according to the probability distribution $P(xyz)$, the scheme here actually realizes a randomness expansion device [@2007Colbeck; @2010Pironio]. Similar to Ref. [@2010Pironio], we can show that under a biased distribution $P(xyz)$ as shown in Fig. \[Randomness-Show\] we generate a much longer random output string of length $O(k)$ from a relatively small amount of random seeds of length $O(\sqrt{k}\log _{2}\sqrt{k})$ when $k$ is large.
![(Color online) Plot of the function $f(\hat{L})$ versus violation $\hat{L}$ of the MABK inequality. The function is calculated through optimization based on the semi-definite programming with the details shown in the Supplemental Material. The inset (a) shows the lower bound of the min-entropy $kf(\mathcal{L}_m-\protect\epsilon)-\log_2\frac{1}{\protect\delta}$ versus the number of trials $k$. Here we assume an observed MABK violation lies within the interval $3.9=\mathcal{L}_m\leq \hat{L}<\mathcal{L}_{max}=4$ with probability $\protect\delta$. The parameters are chosen as $\protect\delta=0.001$ and $\protect\epsilon^{\prime }=0.01$. The bound $kf(\mathcal{L}_m-\protect\epsilon)$ depends on the input probability distribution $P(xyz)$ through the parameter $r=\min_{xyz}P(xyz)$. The blue-square line represents the bound under a uniform distribution ($P(xyz)=1/4$ for all $(x,y,z)\in \mathcal{S}$), while the red-dotted line shows the bound under a biased probability distribution with $P(011)=P(101)=P(110)=\protect\alpha k^{-1/2}$ and $P(000)=1-3\protect\alpha k^{-1/2}$ with $\protect\alpha=10$. It consumes less randomness to generate a biased distribution for the input bits, so the net amount of randomness, defined as the number of output random bits minus that of the input, becomes positive when $k$ is large (typically $k$ needs to be of the order $10^5$). The inset (b) plots the net amount of randomness generated after $k$ trails under a biased distribution of the inputs. The parameters are the same as those in the inset (a). []{data-label="Randomness-Show"}](Fig1f.pdf){width="88mm"}
We now show how to generate and certify random numbers using Majorana fermions. The key step is to generate a three-qubit entangled state and find suitable measurements that lead to violation of the MABK inequality. Majorana fermions are non-Abelian anyons, and their braiding gives nontrivial quantum operations. However, this set of operations are very restricted. First, all the gates generated by topological manipulation of Majorana fermions belong to the Clifford group, and it is impossible to use such operations alone to violate the CHSH inequality [@2009Brennen]. We have to consider instead the multi-qubit MABK inequality. Second, it is not obvious that one can violate the MABK inequality as well using only topological operations. There are two ways to encode a qubit using Majorana fermions, using either two quasiparticles (Majorana fermions) or four quasiparticles (see the details in the supplement). In the two-quasiparticle encoding scheme, although the braiding gates exhaust the entire two-qubit Clifford group, they cannot span the whole Clifford group for more than two qubits [@2009Ahlbrecht]. Furthermore, braiding Majorana fermions within each qubit cannot change the topological charge of this qubit which fixes the measurement basis. Thus, no violation of the MABK inequality can be achieved using the topological operations alone in the two-quasiparticle encoding scheme. In the four-qusiparticle encoding scheme, it is not straightforward either as braidings in this scheme only allows certain single-qubit rotations and no entanglement can be obtained due to the no-entanglement rule proved already for this encoding scheme [2006Bravyi]{}.
Fortunately, we can overcome this difficulty by taking advantage of the non-destructive measurement of the anyon fusion, which can induce qubit entanglement [@2005Bravyi]. In a real physical device, the anyon fusion can be read out non-destructively through the anyon interferometry [2010Hassler]{}. In the four-qusiparticle encoding scheme: each qubit is encoded by four Majorana fermions, with the total topological charge $0$. The qubit basis-states are represented by $|0\rangle \equiv |((\bullet
,\bullet )_{\mathbf{I}},(\bullet ,\bullet )_{\mathbf{I}})_{\mathbf{I}}\rangle $ and $|1\rangle \equiv |((\bullet ,\bullet )_{\psi },(\bullet
,\bullet )_{\psi })_{\mathbf{I}}\rangle $. Here, each $\bullet $ represents a Majorana fermion; $\mathbf{I}$ and $\psi $ represent the two possible fusion channels of a pair of Majorana fermions, with $\mathbf{I}$ standing for the vacuum state and $\psi $ denoting a normal fermion. As explained in the Supplemental Material, a topologically protected two-qubit CNOT gate can be implemented using braidings together with non-destructive measurements of the anyon fusion [@2005Bravyi]. To certify randomness through the MABK inequality, we need to prepare a three-qubit entangled state. For this purpose, we need in total fourteen Majorana fermions, where twelve of them are used to encode three qubits and another ancillary pair is required for implementation of the effective CNOT gates through measurement of the anyon fusion. Initially, the logical state is $|\Phi \rangle
_{i}=|000\rangle $. We apply first a Hadamard gate on the qubit $1$, which can be implemented through a series of anyon braiding as shown in Fig. [Braidings]{}b, and then two effective CNOT gates on the logical qubits $1$, $2
$, and $2,$ $3$. The final state is the standard three-qubit maximally entangled state $|\Psi \rangle _{f}=(|000\rangle +|111\rangle )/\sqrt{2}$. After $|\Psi \rangle _{f}$ is generated, the three qubits can be separated and we need only local braiding and fusion of anyons within each qubit to perform the measurements in the appropriate bases to generate random numbers and certify them through test of the MABK inequality.
To perform the measurements, we read out each qubit according to the input string $\mathcal{I}$ through nondestructive detection of the anyon fusion. If the input is $0$, we first braid the Majorana fermions to implement a Hadamard gate $H$ on this qubit (as shown in Fig. \[Braidings\]b), and then measure the fusion of the first two Majorana fermions within each qubit. The measurement outcome is $0$ ($1$) if the fusion result is $\mathbf{I}$ ($\psi $). If the input is $1$, we first braid the Majorana fermions to implement a $B_{23}$ gate (see Fig. \[Braidings\]a) on this qubit before the same readout measurement. For instance, with the the input $(x,y,z)=(0,1,1)$, we apply a Hadamard gate to the first qubit and $B_{23}$ gates to the second and the third qubits, followed by the nondestructive measurement of fusion of the first two Majorana fermions in each qubit. Under the state $|\Psi \rangle _{f}$, the conditional probability of the measurement outcomes $(a,b,c)$ under the measurement setting $(x,y,z)$ for these three qubits is give by $$P(abc|xyz)=\left\vert \langle abc|(\mathtt{U}_{x}\mathtt{U}_{y}\mathtt{U}_{z})|\Psi \rangle _{f}\right\vert ^{2},$$where $\mathtt{U}_{0}=H$ and $\mathtt{U}_{1}=B_{23}$. With this conditional probability, we find the expected value of $\hat{L}$ defined in Eqs. (1,2) is $\hat{L}=4$, achieving the maximum quantum violation of the MABKinequality. All the steps for measurements and state preparation are based on the topologically protected operations such as anyon braiding or nondestructive detection of the anyon fusion, so the scheme here is intrinsically fault-tolerant and we should get the ideal value of $\hat{L}=4$ if the Majorana fermions can be manipulated at will in experiments. Such a large violation perfectly certifies genuine randomness of the measurement outcomes.
![(Color online) Illustration of the encoding scheme for a logic qubit using Majorana fermions and two single-qubit operations that can be implemented through anyon braiding. Each qubit is encoded by four Majorana fermions. (a)A counterclockwise braiding of Majorana fermions $2$ and $3$ implements a unitary gate $\mathtt{B}_{23}$ on the corresponding qubit. (b) Implementation of the Hadamard gate through composition of anyon braiding. In both (a) and (b), time flows from left to right and $\simeq$ means equal up to an irrelevant overall phase. []{data-label="Braidings"}](fig2ma.pdf){width="85mm"}
In summary, we have shown that genuine number numbers can be generated and certified through topologically manipulation of Majorana fermions, a kind of anyonic excitations in engineered materials. Such a protocol is intrinsically fault-tolerant. Given the rapid experimental progress on realization of Majorana fermions in real materials [@2012Mourik; @2012Wang], this protocol offers a promising prospective for application of these topological particles in an important direction of cryptography with broad implications in science and engineering.
We thank Y. H. Chan, J. X. Gong, and E. Lichko for discussions. This work was supported by the NBRPC (973 Program) 2011CBA00300 (2011CBA00302), the IARPA MUSIQC program, the ARO and the AFOSR MURI program.
Supplementary information: Fault Tolerant Quantum Random Number Generator Certified by Majorana Fermions
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This supplementary information gives more details about realization of fault-tolerant quantum random number generator through topological manipulation of Majorana fermions. In Sec. I, we give the detailed proof on how to certify genuine randomness through observation of violation of the MABK inequality. In Sec. II, we summarize the topological properties of Majorana fermions and show the implementation of the necessary topological quantum gates on the logic qubits encoded with these Majorana fermions.
Randomness certified by observation of violation of the MABK inequality
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In this section, we establish a link between randomness of the measurement outputs of a quantum system and violation of the MABK inequality. A link between randomness and violation of the Bell-CHSH inequality has been established in Ref. [@2010Pironio; @correction]. Here, we generalize the result from the two-qubit CHSH inequality to the three-qubit MABKinequality. Consider a quantum nonlocality test on three qubits. Each qubit has two settings of two-outcome measurements, denoted by $\left\{
x,y,z\right\} $, respectively for the three qubits. The measurement outputs $\left\{ a,b,c\right\} $ of this quantum system are characterized by the joint probability distribution $P=\{P(abc|xyz)\}$. Randomness of the outputs $\left\{ a,b,c\right\} $ are quantified by the min-entropy, defined as $E_{\infty }(ABC|XYZ)=-\mathtt{log}_{2}[\max_{abc}P(abc|xyz)]$. With an experimental observation of violation $\hat{L}$ of the MABK inequality, our aim is to find a lower bound on the min-entropy $$E_{\infty }(ABC|XYZ)\geq f(\hat{L}). \label{siglef}$$This is equivalent to solving of the following optimization problem [2010Pironio]{}: $$\begin{aligned}
P^{\ast }(abc|xyz)=\; &\max &\;P(abc|xyz) \notag \label{Min-entropy-Single}
\\
&\mathtt{subject}\text{ }\mathtt{to}&\;L=\hat{L} \\
&&P(abc|xyz)=\mathtt{Tr}(\rho M_{x}^{a}\otimes M_{y}^{b}\otimes M_{z}^{c})
\notag\end{aligned}$$where $L$ is defined in Eq.(2) of the main text and $(\rho
,M_{x}^{a},M_{y}^{b},M_{z}^{c})$ constitutes a quantum realization of the Bell scenario [@2011Acin]. Thus, the minimal value of $E_{\infty
}(ABC|XYZ)$ compatible with the MABK violation $\hat{L}$ and quantum theory is given by $E_{\infty }(ABC|XYZ)=-\mathtt{log}_{2}[\max_{abc}P^{\ast
}(abc|xyz)]$. Consequently, to obtain $f(\hat{L})$ we only need to solve (\[Min-entropy-Single\]) for all possible input and output triplets $(x,y,z)
$ and $(a,b,c)$. This can be effectively done by casting it to a *semi-definite program* (SDP) [@1996Vandenberghe]. An infinite hierarchy of conditions that need to be satisfied by all quantum correlations are introduced in Ref. [@2007Navascues; @2008Navascues; @2009Pironio]. All these conditions can be transformed to a SDP problem and the hierarchy is complete in the asymptotic limit, i.e., it guarantees existence of a quantum realization if all the conditions in the hierarchy are satisfied. Generally, conditions higher in the hierarchy are more constraining and thus better reflect the constraints in (\[Min-entropy-Single\]) and give a tighter lower bound. To obtain a lower bound of the min-entropy for a given MABK violation $\hat{L}$, we use the matlab toolbox SeDuMi [@Sturm-Sedumi] and solve the SDP corresponding to the certificates between order $1$ and order $2$ [@2007Navascues]. The result is plotted in Fig.1 in the main text. From the figure, $f(\hat{L})$ equals zero at the classical point $\hat{L}=2$ and increases monotonously as the MABK violation $\hat{L}$ increases. For the maximal violation $\hat{L}=4$, $P^{\ast }\approx 0.5003$, corresponding to $f(\hat{L})\simeq 0.9991$ bits.
Equation (4) in the main text can be derived using arguments similar to those in Ref. [@2010Pironio; @2012Pironio]. The difference is that the Bell scenario in Refs. [@2010Pironio] is based on the two-qubit CHSH inequality, which needs to be extended in our scheme with the three-qubit MABK inequality. Suppose we run the experiments $k$ times and denote the input and output string as $\mathcal{I}=(x_{1},y_{1},z_{1};\cdots ;x_{k},y_{k},z_{k})$ and $\mathcal{O}=(a_{1},b_{1},c_{1};\cdots ;a_{k},b_{k},c_{k})$, respectively. As in the main text, let $\{\mathcal{L}_{m}:0\leq m\leq m_{max}\}$ be a series of MABK violation thresholds, and denote $\mathcal{D}(m)$ the probability that the observed KCBS violation $\hat{L}$ lies in the interval $[\mathcal{L}_{m},\mathcal{L}_{m+1})$. Denote by $\mathcal{E}$ the possible classical side information an adversary may have. To derive Eq. (4) in the main text, let us first introduce the following theorem:
**Theorem 1**. Suppose the experiments are carried out $k$ times and each triplet of inputs $(x_{i},y_{i},z_{i})$ is generated independently with probability $P(xyz)$. Let $\delta $, $\epsilon ^{\prime }>0$ be two arbitrary parameters and $r=\min \{P(xyz))\}$, then the distribution $P(\mathcal{O}\mathcal{I}\mathcal{E})$ characterizing $k$ successive use of the devices is $\epsilon ^{\prime }$-close to a distribution $\mathcal{D}$ such that, either $\mathcal{D}(m)\leq \delta $ or $$E_{\infty }(\mathcal{O}|\mathcal{I},\mathcal{E},m)_{\mathcal{D}}\geq kf(\mathcal{L}_{m}-\epsilon )+\log _{2}\delta , \label{theorem1bound}$$where $\epsilon =(4+1/r)\sqrt{-2\ln \epsilon ^{\prime }/k}$.
Equation (\[theorem1bound\]) is equivalent to Eq. (4) in the main text. Theorem 1 tells us that the distribution $P$, which characterizes the output $\mathcal{O}$ of the device and its correlation with the input $\mathcal{I}$ and the adversary’s classical side information $\mathcal{E}$, is basically indistinguishable from a distribution $\mathcal{D}$ that will be defined below [@2012Pironio]. If we find that the observed MABK violation $\hat{L}$ lies in $[\mathcal{L}_{m},\mathcal{L}_{m+1})$ with a non-negligible probability, i.e., $\mathcal{D}(m)>\delta $, the entropy of the outputs $\mathcal{O}$ is guaranteed to have a positive lower bound $kf(\mathcal{L}_{m}-\epsilon )-\log _{2}\frac{1}{\delta }$, that is, the randomness of the outputs is guaranteed to be larger than $kf(\mathcal{L}_{m})$ up to epsilonic correction.
*Proof*. We use a procedure similar to those in Ref. [2012Pironio]{} to prove the above theorem. Let us define a function $\mathcal{G}(L)=2^{-f(L)}$, which is concave and monotonically decreasing given by the solution of the optimization problem in Eq. (\[Min-entropy-Single\]) (shown in Fig. 1 of the main text). Denote by $\mathcal{O}^{n}=(a_{i},b_{i},c_{i};\cdots ;a_{n},b_{n},c_{n})$ ($n\leq k$) the string of outputs before the $(n+1)$th round of experiment (similarly, $\mathcal{I}^{n}$ denotes the string of inputs). We introduce an indicator function $\chi (e)$ as: $\chi (e)=1$ if the event $e$ happens and $\chi (e)=0$ otherwise. Consider the following random variable $$\hat{L}_{i}=\sum_{abc;(x,y,z)\in \mathcal{S}}\tau (x,y,z)\Lambda (a,b,c)\frac{\chi (a_{i}=a,b_{i}=b,c_{i}=c;x_{i}=x,y_{i}=y,z_{i}=z)}{P(xyz)},
\label{randomvar}$$where $\mathcal{S}$ and $\tau (x,y,z)$ are defined in the main text, and $\Lambda (a,b,c)=1$ if $a+b+c$ is $\mathtt{even}$ and $\Lambda (a,b,c)=-1$ if $a+b+c$ is $\mathtt{odd}$. It is easy to check that Eq.(\[randomvar\]) reduces to the MABK expression (2) in the main text and the expectation value of $\hat{L}_{i}$ conditional on the past $W^{i}$ is equal to $L(W^{i})$, i.e., $\mathbb{E}(\hat{L}_{i}|W^{i})=L(W^{i})$. We use $\mathcal{W}^{i}\equiv (\mathcal{O}^{i-1}\mathcal{I}^{i-1}\mathcal{E})$ to denote all the events before the $i$th round of experiment and the possible adversary’s classical side information. The estimator of the MABK violation can be defined as: $\hat{L}=\frac{1}{k}\sum_{i=1}^{k}\hat{L}_{i}$. With these notations, first we introduce two lemmas for proof of the main theorem.
**Lemma 1**. For any given parameter $\epsilon^{\prime }>0$, let $\epsilon=(4+1/r)\sqrt{-2\ln\epsilon^{\prime }/k}$ and $S_{\epsilon}=\{(\mathcal{O},\mathcal{I},\mathcal{E})|\frac{1} {k}\sum_{i=1}^k\mathbb{E}(\hat{L}_i|W^i)\geq \hat{L}(\mathcal{O},\mathcal{I})-\epsilon\}$, then we have:
\(i) for any $(\mathcal{O},\mathcal{I},\mathcal{E})\in S_{\epsilon }$, $$P(\mathcal{O}|\mathcal{I}\mathcal{E})\leq \mathcal{G}^{k}(\hat{L}(\mathcal{O},\mathcal{I})-\epsilon ). \label{POIE}$$
\(ii) $$\mathtt{Pr}(S_{\epsilon })=\sum_{(\mathcal{O},\mathcal{I},\mathcal{E})\in
S_{\epsilon }}P(\mathcal{O},\mathcal{I},\mathcal{E})\geq 1-\epsilon ^{\prime
}. \label{PrT1}$$
*Proof*. According to the Bayes’ rule and the fact that the response of a system does not depend on the future inputs and outputs, we have:
$$\begin{aligned}
P(\mathcal{O}|\mathcal{I}\mathcal{E}) &=&\prod_{i=1}^{k}P(a_{i}b_{i}c_{i}|\mathcal{O}^{i-1}\mathcal{I}^{i}\mathcal{E}) \notag \label{IntroWi} \\
&=&\prod_{i=1}^{k}P(a_{i}b_{i}c_{i}|x_{i}y_{i}z_{i}\mathcal{W}^{i})\end{aligned}$$
From the solution to the optimization problem in Eq. ([Min-entropy-Single]{}), the probability $P(a_{i}b_{i}c_{i}|x_{i}y_{i}z_{i}\mathcal{W}^{i})$ is bounded by a function of the MABK violation $L(W^{i})$: $P(a_{i}b_{i}c_{i}|x_{i}y_{i}z_{i}\mathcal{W}^{i})\leq \mathcal{G}(L(W^{i}))$. Thus, we have: $$\begin{aligned}
P(\mathcal{O}|\mathcal{I}\mathcal{E}) &\leq &\prod_{i=1}^{k}\mathcal{G}(L(W^{i})) \notag \\
&\leq &\mathcal{G}^{k}(\frac{1}{k}\mathbb{E}(\hat{L}_{i}|W^{i})) \notag \\
&\leq &\mathcal{G}^{k}(\hat{L}(\mathcal{O},\mathcal{I})-\epsilon ).\end{aligned}$$Here, to obtain the second inequality, we have used the equality $\mathbb{E}(\hat{L}_{i}|W^{i})=L(W^{i})$ and the fact that $\mathcal{G}$ is logarithmically concave and monotonically decreasing. The third inequality is obtained from the definition of $S_{\epsilon }$ and the fact that $\mathcal{G}$ is decreasing. To get Eq. (\[PrT1\]), we can define another random variable $M^{q}=\sum_{i=1}^{q}(\hat{L}_{i}-\mathbb{E}(\hat{L}_{i}|W^{i}))$. Then it is easy to verify that (i) $|M^{q}|\leq 2q/r<\infty $, (ii) $|\hat{L}_{i}-L(W^{i})|\leq |\hat{L}_{i}|+|L(W^{i})|\leq \frac{1}{r}+4
$, and (iii) $\mathbb{E}(M^{q+1}|W^{q})=M^{q}$. Thus, the sequence $\{M^{q}:q\geq 1\}$ is a martingale process [@2001Grimmett-Book]. Applying the Azuma-Hoeffding inequality $P(M^{q}\geq k\epsilon )\leq \exp (-\frac{(k\epsilon )^{2}}{2k(1/r+4)^{2}})$ [1967Azuma,1960Hoeffding,2001Grimmett-Book]{}, we have $$P\left( \frac{1}{k}\sum_{i=1}^{k}\mathbb{E}(\hat{L}|W^{i})\leq \frac{1}{k}\sum_{i=1}^{k}\hat{L}_{i}-\epsilon \right) \leq \epsilon ^{\prime },
\label{MargEq1}$$where $\epsilon =(4+1/r)\sqrt{-2\ln \epsilon ^{\prime }/k}$. Equation ([MargEq1]{}) combined with the definition of $S_{\epsilon }$ gives Eq. ([PrT1]{}). Lemma 1 is thus proved.
In the above proof, we only considered the case that the random variable sequence $\mathcal{O}$ takes values in the output space $\mathbb{S}^{k}=\{-1,1\}^{k}$. As in Ref. [@2012Pironio], we can extend the range of $\mathcal{O}$ to include abort-output" $\bot $, and view $\mathcal{O}$ as an element of $\mathbb{S}^{k}\cup \bot $ with $P(\mathcal{O}|\mathcal{I}\mathcal{E})=0$ if $\mathcal{O}=\bot $. The physical meaning of $\bot $ is that when $\bot $ is produced by the device, then no MABK violation has been obtained and no randomness is certified.
**Lemma 2**. There exists a probability distribution $\mathcal{D}=\{\mathcal{D}(\mathcal{O},\mathcal{I},\mathcal{E})\}$, which is $\epsilon
^{\prime }$-close to $P=\{P(\mathcal{O},\mathcal{I},\mathcal{E})\}$, i.e., $d(\mathcal{D},P)=\frac{1}{2}\sum_{\mathcal{O},\mathcal{I},\mathcal{E}}|P(\mathcal{O},\mathcal{I},\mathcal{E})-\mathcal{D}(\mathcal{O},\mathcal{I},\mathcal{E})|\leq \epsilon ^{\prime }$, and satisfies the following condition $$\mathcal{D}(\mathcal{O}|\mathcal{I},\mathcal{E})\leq \mathcal{G}^{k}(\hat{L}(\mathcal{O},\mathcal{I})-\epsilon ), \label{Dcondition2}$$for all $(\mathcal{O},\mathcal{I},\mathcal{E})$ such that $\mathcal{O}\neq
\bot $.
*Proof*. We show how to construct a probability distribution satisfying the above two conditions. To this end, we introduce $\mathcal{D}(\mathcal{O},\mathcal{I},\mathcal{E})=P(\mathcal{I})P(\mathcal{E})\mathcal{D}(\mathcal{O}|\mathcal{I},\mathcal{E})$. $\mathcal{D}(\mathcal{O}|\mathcal{I},\mathcal{E})$ is defined as: $$\mathcal{D}(\mathcal{O}|\mathcal{I},\mathcal{E})=\left\{
\begin{array}{cc}
P(\mathcal{O}|\mathcal{I},\mathcal{E}), & \mathtt{if}\;(\mathcal{O},\mathcal{I},\mathcal{E})\in S_{\epsilon } \\
0, & \quad \quad \mathtt{if}\;\mathcal{O}\neq \bot \;\mathtt{and}\;(\mathcal{O},\mathcal{I},\mathcal{E})\notin S_{\epsilon } \\
1-\sum_{(\mathcal{O},\mathcal{I},\mathcal{E})\notin S_{\epsilon }}P(\mathcal{O}|\mathcal{I},\mathcal{E}) & \;\mathtt{otherwise}\end{array}\right.$$Then by Lemma 1, it is straightforward to get that the distribution $\mathcal{D}$ satisfies Eq. (\[Dcondition2\]) for all $(\mathcal{O},\mathcal{I},\mathcal{E})$ with $\mathcal{O}\neq \bot $. The distance between $P$ and $\mathcal{D}$ can be calculated as: $$\begin{aligned}
d(\mathcal{D},P) &=&\frac{1}{2}\sum_{\mathcal{O},\mathcal{I},\mathcal{E}}|P(\mathcal{O},\mathcal{I},\mathcal{E})-\mathcal{D}(\mathcal{O},\mathcal{I},\mathcal{E})| \notag \\
&=&\frac{1}{2}\sum_{\mathcal{I},\mathcal{E}}P(\mathcal{I},\mathcal{E})\sum_{\mathcal{O}}|P(\mathcal{O}|\mathcal{I},\mathcal{E})-\mathcal{D}(\mathcal{O}|\mathcal{I},\mathcal{E})| \notag \\
&=&\frac{1}{2}[\sum_{(\mathcal{O},\mathcal{I},\mathcal{E})\notin \mathcal{T}_{\epsilon }}P(\mathcal{O},\mathcal{I},\mathcal{E})+1-\sum_{(\mathcal{O},\mathcal{I},\mathcal{E})\in \mathcal{T}_{\epsilon }}P(\mathcal{O},\mathcal{I},\mathcal{E})] \\
&\leq &\epsilon ^{\prime }. \notag\end{aligned}$$This proves Lemma 2.
With Lemma 2, now the proof of Theorem 1 becomes straightforward. Define a subset of the outputs as $\mathcal{X}_{m}=\{\mathcal{O}|\mathcal{O}\neq \bot
\;\mathtt{and}\;\mathcal{L}_{m}\leq \hat{L}<\mathcal{L}_{m+1}\}$ and let $\mathcal{D}(\mathcal{O},\mathcal{I},
\mathcal{E}|m)$ denote the distribution of $\mathcal{O},\mathcal{I},
\mathcal{E}$ conditioned on a particular value of $m$, then we have: $$\begin{aligned}
E_{\infty }(\mathcal{O}|\mathcal{I},\mathcal{E},m)_{\mathcal{D}}&\equiv& -\mathtt{log}_{2}\sum_{\mathcal{I},\mathcal{E}}[\max_{\mathcal{O}}\mathcal{D}(\mathcal{O},\mathcal{I},\mathcal{E}|m)]\nonumber\\
&= &-\mathtt{log}_{2}\sum_{\mathcal{I},\mathcal{E}}\mathcal{D}(\mathcal{I},\mathcal{E}|m)[\max_{\mathcal{O}}\mathcal{D}(\mathcal{O}|\mathcal{I},\mathcal{E},m)] \notag \\
&=&-\mathtt{log}_{2}\sum_{\mathcal{I},\mathcal{E}}\mathcal{D}(\mathcal{I},\mathcal{E}|m)\frac{1}{\mathcal{D}(m|\mathcal{I},\mathcal{E})}\max_{\mathcal{O}\in \mathcal{X}_{m}}\mathcal{D}(\mathcal{O}|\mathcal{I},\mathcal{E})
\notag \\
&\geq &-\mathtt{log}_{2}\sum_{\mathcal{I},\mathcal{E}}\mathcal{D}(\mathcal{I},\mathcal{E}|m)\frac{\mathcal{G}^{k}(\mathcal{L}_{m}-\epsilon )}{\mathcal{D}(m|\mathcal{I},\mathcal{E})} \\
&=&-\mathtt{log}_{2}\sum_{\mathcal{I},\mathcal{E}}\frac{\mathcal{D}(\mathcal{I},\mathcal{E})}{\mathcal{D}(m)}\mathcal{G}^{k}(\mathcal{L}_{m}-\epsilon )
\notag \\
&=&kf(\mathcal{L}_{m}-\epsilon )-\log _{2}\frac{1}{\mathcal{D}(m)}. \notag\end{aligned}$$Here we have used the Bayes’ rule in the first, second and the fourth equalities and Eq. (\[Dcondition2\]) from Lemma 2 in the third inequality; for the last equality, the equation $f=-\log _{2}\mathcal{G}$ is used. The last equality immediately leads to the claim in Theorem 1. This concludes the proof.
It is worthwhile to clarify that in deriving Eq. (\[theorem1bound\]) we have made the following four assumptions [@2010Pironio; @2012Pironio]: (i) the system can be described by quantum theory; (ii) the inputs at the $j$th trial $(x_{j},y_{j},z_{j})$ are chosen randomly and their values are revealed to the systems only at step $j$; (iii) the three qubits are separated and non-interacting during each measurement step. (iv) the possible adversary has only classical side information. There are no constraints on the states, measurements, or the Hilbert space. Moreover, there is even no requirement that the system behaves identically and independently for each trial. In particular, the system could have an internal memory (classical or quantum) so that the results of the $j$th trial depend on the previous $j-1$ trials.
We also note that there is a significant difference between the two-qubit scenario in Ref. [@2010Pironio] and our three-qubit scenario here. In the two-qubit case, the randomness can be certified by the no-signalling conditions as well without the assumption of quantum mechanics. However, in our three-qubit scenario, the no-signalling conditions are not sufficient to certify randomness. Actually, we have numerically checked that even for the maximal possible MABK violation $\hat{L}_{max}=4$, $P^{\ast }(abc|xyz)$ can be equal to the unity for certain $(a,b,c)$ and $(x,y,z)$ if only the no-signalling conditions are imposed, which cannot certify any randomness. A possible reason for this difference is that the MABK inequality only contains four out of eight possible correlations. In other words, the input choices $\mathcal{S}$ is only a subset of $\{(x,y,z)|x,y,z=0,1\}$. As a result, the no-signalling constraints become less effective.
Encoding and operation of qubits by topological manipulation of Majorana fermions
----------------------------------------------------------------------------------
In this section, we discuss in detail how to control the logical qubits encoded with Majorana fermions. The fusion rule of Majorana fermions is of the Ising type: $\tau \times \tau \sim \mathbf{I}+\psi $, where $\tau $, $\mathbf{I}$, and $\psi $ stand for a Majorana fermion, the vacuum state, and a normal fermion, respectively. Generally, there are two encoding schemes. The first scheme encodes each logical qubit into a pair of Majorana fermions (two-quasiparticle encoding). When the pair fuse to a vacuum state $\mathbf{I}$, we say that the qubit is in state $|0\rangle $; and when they fuse to $\psi $, the state is $|1\rangle $. There is also an ancillary pair, which soak up the extra $\psi $ if necessary to maintain the constraint that the total topological charge must be $0$ for the entire system [2006Georgiev,2009Ahlbrecht]{}. In this encoding scheme, braiding operations of Majorana fermions exhaust the entire two-qubit Clifford group. However, for three or more qubits, not all Clifford gates could be implemented by braiding. The embedding of the two-qubit SWAP gate into a $n$-qubit system cannot be implemented by braiding [@2009Ahlbrecht]. In the two-quasiparticle encoding scheme, no violation of the MABK inequality can be obtained as we cannot change the measurement basis through local braiding of Majorana fermions within each logic qubit.
As we mentioned in the main text, we use the four-quasiparticle encoding scheme where the qubit basis-states are represented by $|0\rangle
=|((\bullet ,\bullet )_{\mathbf{I}},(\bullet ,\bullet )_{\mathbf{I}})_{\mathbf{I}}\rangle $ and $|1\rangle =|((\bullet ,\bullet )_{\psi },(\bullet
,\bullet )_{\psi })_{\mathbf{I}}\rangle $. Let us first consider braiding operations of Majorana fermions within each logic qubit. Consider four Majorana operators $c_{i}$ $(i=1,2,3,4)$ in one logic qubit, which satisfy $c_{i}^{\dagger }=c_{i}$, $c_{i}^{2}=1$ and the anti-commutation relation $\{c_{i},c_{j}\}=2\delta _{ij}$. The Pauli operators in the computational basis can be expressed as [@2010Hassler]: $$\sigma ^{x}=-ic_{2}c_{3},\quad \sigma ^{y}=-ic_{1}c_{3},\quad \sigma
^{z}=-ic_{1}c_{2}.$$Unitary operations can be implemented by counterclockwise exchange of two Majorana fermions $j<j^{\prime }$: $$\mathtt{B}_{jj^{\prime }}=e^{(i\pi /4)(ic_{j}c_{j^{\prime }})}.$$Specifically, we can write down the three basic braiding operators in the computational basis: $$\mathtt{B}_{12}=\mathtt{B}_{34}\simeq \left(
\begin{matrix}
1 & 0 \\
0 & i\end{matrix}\right) ,\;\mathtt{B}_{23}\simeq \frac{1}{\sqrt{2}}\left(
\begin{matrix}
1 & -i \\
-i & 1\end{matrix}\right) ,$$where $\simeq $ means that we ignore an unimportant overall phase. Using these basic braiding operators, a single-qubit Hadamard gate can be implemented as $\mathtt{H}=\frac{1}{\sqrt{2}}\left(
\begin{matrix}
1 & 1 \\
1 & -1\end{matrix}\right) \simeq \mathtt{B}_{23}^{2}\mathtt{B}_{12}^{-1}\mathtt{B}_{23}\mathtt{B}_{12}^{-1}\mathtt{B}_{23}^{2}$. The corresponding braidings are shown in Fig.2 of the main text. Note that the set of operations implemented through composition of $\mathtt{B}_{12}$ and $\mathtt{B}_{23}$ are still very limited, however, it is fortunate that $\mathtt{B}_{23}$ and $\mathtt{H}$ give all the gates that we need for change of the measurement bases in test of the MABK inequality. As shown in the main text, we actually get maximum quantum violation of the MABK inequality by randomly choosing either a $\mathtt{B}_{23}$ or an $\mathtt{H}$ gate on each logic qubit before measurement of the anyon fusion.
With only braiding operations of Majorana fermions, no entangling gate can be achieved for logic qubits in the four-quasiparticle encoding scheme due to the *no-entanglement rule* proved in Ref. [@2006Bravyi]. In order to overcome this problem, we need assistance from another kind of topological manipulation: nondestructive measurement of the anyon fusion, which can be implemented through the anyon interferometry as proposed in Ref. [@2010Hassler]. Suppose that we have eight Majorana modes $c_{1},c_{2},\ldots ,c_{8}$, where the first (last) four modes encode the control (target) qubit, respectively. As shown in Ref. [2002Bravyi,2005Bravyi]{}, a two-qubit controlled phase flip gate $\Lambda
(\sigma ^{z})$ can be implemented through the following identity: $$\Lambda (\sigma ^{z})=e^{-(\pi /4)c_{3}c_{4}}e^{-(\pi /4)c_{5}c_{6}}e^{(i\pi
/4)c_{4}c_{3}c_{5}c_{6}}e^{i\pi /4}. \label{conZ}$$ Note that the first two operations in Eq. (\[conZ\]) can be directly implemented by braiding operations. The key step is to implement the operation $e^{(i\pi /4)c_{4}c_{3}c_{5}c_{6}}$. To this end, we use another ancillary pair of Majorana fermions $c_{9}$ and $c_{10}$. We measure fusion of the four Majorana modes $c_{4}c_{3}c_{6}c_{9}$. The outcome is $\pm 1$, corresponding to either a vacuum state ($+1$) or a normal fermion ($-1$) . The corresponding projector is given by $\Pi _{\pm }^{(4)}=\frac{1}{2}(1\pm
c_{4}c_{3}c_{6}c_{9})$. Then, we similarly measure fusion of the Majorana modes (operator) $-ic_{5}c_{9}$, with the project denoted by $\Pi _{\pm
}^{(2)}=\frac{1}{2}(1\mp ic_{5}c_{9})$ corresponding to the measurement outcomes $\pm 1$. We have the following relation [@2002Bravyi; @2005Bravyi]:
$$\begin{aligned}
e^{(i\pi /4)c_{4}c_{3}c_{5}c_{6}}=2{\displaystyle\sum}\limits_{\eta ,\zeta =\pm }U_{\eta
\zeta }\Pi _{\eta }^{(2)}\Pi _{\zeta }^{(4)},\end{aligned}$$
where $U_{++}=e^{(\pi /4)c_{5}c_{10}}$, $U_{+-}=ie^{(\pi
/2)c_{4}c_{3}}e^{(\pi /2)c_{5}c_{6}}e^{(\pi /4)c_{5}c_{10}}$, $U_{-+}=ie^{(\pi /2)c_{4}c_{3}}e^{(\pi /2)c_{5}c_{6}}e^{-(\pi /4)c_{5}c_{10}}$, and $U_{--}=e^{-(\pi /4)c_{5}c_{10}}$. All the gates $U_{\eta \zeta }$ can be implemented through one or several braiding operations of Majorana fermions. So this identity shows that an effective controlled phase flip gate can be implemented on logic qubits through a combination of anyon braiding and measurement of anyon fusion. Depending on the measurement outcomes $(\zeta,\eta)$ of $c_{4}c_{3}c_{6}c_{9}$ and $-ic_{5}c_{9}$, one can always apply a suitable correction operator $U_{\eta \zeta }$ to obtain the desired operation $e^{(i\pi /4)c_{4}c_{3}c_{5}c_{6}}$. With controlled phase flip gates, one can easily realize quantum controlled-NOT (CNOT) gate with assistance from the Hadamard operations that can be implemented through the anyon braiding. With CNOT and Hadamard gates, we can then prepare the maximally entangled three-qubit state as required for test of quantum violation of the MABK inequality.
[99]{} E. Majorana, Nuovo Cimento **14**, 171 (1937).
F. Wilczek, Nat. Phys. **5**, 614 (2009).
A. Stern, Nature **464**, 187 (2010).
C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. **80**, 1083 (2008). L. Fu and C. L. Kane, Phys. Rev. Lett. **100**, 096407 (2008).
A. Kitaev, Ann. Phys. (N. Y.) **321**, 2 (2006).
N. Read and D. Green, Phys. Rev. B **61**, 10267 (2000).
J. D. Sau, R. M. Lutchyn, S. Tewari, and S. DasSarma, Phys. Rev. Lett. **104**, 040502 (2010). R. M. Lutchyn, J. D. Sau, and S. DasSarma, Phys. Rev. Lett. **105**, 077001 (2010).
Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. **105**, 177002 (2010).
M. X. Wang, *et al*. Science **336**, 52-55 (2012).
V. Mourik, *et al*., Science **336**, 1003 (2012).
N. D. Mermin, Phys. Rev. Lett. **65**, 1838 (1990).
M. Ardehali, Phys. Rev. A **46**, 5375 (1992).
A. V. Belinskii and D. N. Klyshko, Phys. Usp. **36**, 653 (1993).
J. Ackermann, *et al*. Comput. Phys. Commun. **140**, 293 (2001).
P. F. Hultquist, Simulation. **57**, 258 (1991).
S. J. Tu and E. Fischbach, Phys. Rev. E **67**, 016113 (2003). S. Pironio, *et al*. Nature **464**, 1021 (2010). M. Isida and Y. Ikeda, Ann. Inst. Stat. Math. **8**, 119 (1956).
K. Svozil, Phys. Lett. A **143**, 433 (1990).
J. G. Rarity, M. P. C. Owens, and P. R. Tapster, Journal of Modern Optics **41**, 2435 (1994).
T. Jennewein, *et al*. Rev. Sci. Instrum. **71**, 1675 (2000).
J. Clauser, F. M.Horne, A. A. Shimony, and R. A. Holt, Phys. Rev. Lett. **23**, 880 (1969).
Colbeck, R. Quantum and Relativistic Protocols for Secure Multi-Party Computation. PhD dissertation, Univ. Cambridge (2007).
G. K. Brennen, S. Iblisdir, J. K. Pachos, and J. K. Slingerland, New J. of Phys. **11**, 103023 (2009).
R. Koenig, R. Renner, and C. Schaffner, IEEE Trans. Inf. Theory **55**, 4337 (2009).
S. Pironio and S. Massar, arXiv: 1111.6056v4.
L. Vandenberghe and S. Boyd, SIAM Rev. **38**, 49 (1996).
N. Nisan and A. Ta-Shma, J. Comput. Syst. Sci. **58**, 148 (1999).
A. Ahlbrecht, L. S. Georgiev,and R. F. Werner, Phys. Rev. A **79**, 032311 (2009).
S. Bravyi, Phys. Rev. A **73**, 042313 (2006).
S. Bravyi and A. Y. Kitaev, Phys. Rev. A **71**, 022316 (2005).
F. Hassler, A. R. Akhmerov, C. Y. Hou, and C. W. J. Beenakker, New J. of Phys. **12** 125002 (2010). The theoretical results in Ref. [@2010Pironio] were improperly formulated, and the inaccuracies in formulation were corrected in the recent Refs. [@2012Pironio] and [@2012Fehr].
A. Acín, S. Massar, and S. Pironio, arXiv: 1107.2754.
M. Navascues, S. Pironio, and A. Acin, Phys. Rev. Lett. **98**, 010401 (2007).
M. Navascues, S. Pironio, and A. Acin, New J. Phys. **10**, 073013 (2008).
S. Pironio, M. Navascues, and A. Acin, arXiv: 0903.4368.
Sturm, J. SeDuMi, a MATLAB toolbox for optimization over symmetric cones. http://sedumi.mcmaster.ca.
G. Grimmett and D. Stirzaker, Probability and Random Processes (Oxford University Press, Oxford, 2001).
W. Hoeffding, Journal of the American Statistical Association **58**, 13 (1963).
K. Azuma, Tohoku Mathematical Journal **19**, 357(1967).
L. S. Georgiev, Phys. Rev. B **74**, 235112 (2006).
S. B. Bravyi and A. Y. Kitaev, Ann. Phys. **298** 210 (2002).
S. Fehr, R. Gelles, and C. Schaffner, arXiv: 1111.6052v3.
|
---
abstract: 'The 9 Myr old TW Hya Association (TWA) is the nearest group (typical distances of $\sim$50 pc) of pre-main-sequence (PMS) stars with ages less than 10 Myr and contains stars with both actively accreting disks and debris disks. We have studied the coronal X-ray emission from a group of low mass TWA common proper motion binaries using the [*[Chandra]{}*]{} and [*[Swift]{}*]{} satellites. Our aim is to understand better their coronal properties and how high energy photons affect the conditions around young stars and their role in photo-exciting atoms, molecules and dust grains in circumstellar disks and lower density circumstellar gas. Once planet formation is underway, this emission influences protoplanetary evolution and the atmospheric conditions of the newly-formed planets. The X-ray properties for 7 individual stars (TWA 13A, TWA 13B, TWA 9A, TWA 9B, TWA 8A, TWA 8B, and TWA 7) and 2 combined binary systems (TWA 3AB and TWA 2AB) have been measured. All the stars with sufficient signal require two-component fits to their CCD-resolution X-ray spectra, typically with a dominant hot ( 2 kev (25 MK)) component and a cooler component at 0.4 keV (4 MK). The brighter sources all show significant X-ray variability (at a level of 50-100% of quiescence) over the course of 5-15 ksec observations due to flares. We present the X-ray properties for each of the stars and find that the coronal emission is in the super-saturated rotational domain.'
author:
- 'Alexander Brown, Gregory J. Herczeg, Thomas R. Ayres, Kevin France, & Joanna M. Brown'
title: 'X-ray Emission from Young Stars in the TW Hya Association'
---
Introduction
============
Young stars are bright X-ray and ultraviolet (UV) emitters due to strong stellar magnetic activity fostered by rapid rotation. These high energy photons can greatly influence protoplanetary evolution and the atmospheric conditions of newly formed planets. Stellar X-ray/EUV photons are the major ionization source over most of a protoplanetary system [@alexander06]. Observation and modeling of representative samples of young stars can provide important insights into protoplanet evolution, because such data provides direct, unambiguous constraints for models. The evolutionary trend is from gas-dominated circumstellar physics to a solid-dominated structure where the gas component exists as icy surfaces on planetesimals and larger bodies or in the atmospheres of protoplanets.
[ccccccccc]{} TWA 13A \[NW\]&M1 V&11.5&5.1&55.6$^{+2.3}_{-2.1}$&5.56&14.57& 2.19$\pm$0.03 & 29.91$\pm$0.02\
TWA 13B \[SE\]&M1 V&12.0&&59.7$^{+2.8}_{-2.5}$&5.35 & & 2.80$\pm$0.03 & 30.08$\pm$0.03\
TWA 8A \[N\] &M3 V&12.2& 13 &46.9$^{+3.3}_{-2.9}$&4.65& 4.56& 3.30$\pm$0.06 & 29.94$\pm$0.04\
TWA 8B \[S\] &M5 V&15.3&&47.1$^{+3.4}_{-3.0}$&0.78 & & 0.16$\pm$0.01 & 28.64$\pm$0.06\
TWA 9A \[SE\]&K6 V&11.3& 5.8&46.7$^{+6.1}_{-4.9}$&5.10& 4.56& 2.13$\pm$0.04 & 29.74$\pm$0.07\
TWA 9B \[NW\]&M3 V&14.0&&50.3$^{+6.9}_{-5.4}$&3.98 & & 0.19$\pm$0.01 & 28.75$\pm$0.07\
TWA 7 & M3 V& 11.7 & ...&34.4$^{+2.8}_{-2.2}$& 5.05 & 4.69 & 3.33$\pm$0.13 & 29.67$\pm$0.05\
TWA 3AB &M4V+M4V&12.6;13.1& 1.5 &35.3$^{+2.2}_{-1.9}$ && 6.42 & 1.07$\pm$0.07 & 29.20$\pm$0.04\
TWA 2AB & M2V+M3V&11.1& 0.6 &46.6$^{+3.0}_{-2.7}$ & 4.86& 4.89 & 0.82$\pm$0.07 & 29.33$\pm$0.05\
The TW Hya association (TWA) is a nearby (distances $\sim$ 50 pc) group of 9 Myr old pre-main-sequence stars that samples a crucial phase of protoplanetary evolution. We have studied the high energy emission from a group of low mass, TWA common proper motion binaries, several of which can be spatially resolved by [*[Chandra]{}*]{} and thus permit measurement of the coronal properties for the individual stars. These stars possess extremely strong photospheric magnetic fields with typical field strengths of 3 kG [@yang08] and starspots large enough to show significant optical rotational modulation [@Lawson_Crause05]. These strong magnetic fields produce a high level of coronal heating and X-ray emission and most TWA members have been recognized initially as anomalously strong X-ray sources [@webb99]. The physical properties of the stars are listed in Table \[table1\].
Considerable effort has been devoted recently to obtaining better astrometry for TWA members and this has provided vastly improved knowledge of their distances, proper motions, and space motions. For our sample, astrometry is provided for TWA 13 and TWA 2 by @weinberger13, for TWA 8 by @riedel14, and for TWA 3 and TWA 7 by @ducourant14. The best astrometry source for TWA 9 is still the Hipparcos Catalogue. The distances that we have used are listed in Table \[table1\].
Observations and Data Analysis
==============================
We measured the X-ray emission from nine members of the TW Hya association using the [*[Chandra]{}*]{} ACIS-S3 detector (Obsids: 8569, 8570 – PI: Herczeg; 12389 – PI: Brown) and the [*[Swift]{}*]{} XRT (Obsids: 31981001, 90207001, 90410001 – PI:Brown), with the goal of a better detailed understanding of the coronal properties of the young, low-mass (K-M) dwarf stars in the TW Hya Association. CCD-resolution X-ray spectra with exposure times of $\sim$ 5-15 ksec were obtained The [*Chandra*]{} data were processed using CIAO Version 4.3 reduction recipes, while the [*Swift*]{} data were processed using the XTOOLS data commands outlined in the [*Swift XRT Data Reduction Guide*]{}. The X-ray fluxes and luminosities over the energy range 0.3-10 keV were measures (see Table \[table1\]). The source variability was investigated when sufficient counts were available (see Fig. \[fig1\]). All the resulting spectra were fitted using XSPEC Version 12.5.0 [@arnaud96; @dorman03] and typically required use of a two-temperature VAPEC model. At CCD-resolution the spectra are only sensitive to changes in a few elements, particularly Fe and Ne and with a weaker sensitivity to O.
![Chandra ACIS-S3 source variability for the three common-proper-motion binaries TWA 8AB, TWA 9AB, and TWA 13AB sampled in 500 second time-bins. TWA 8A, TWA 13A, and TWA 13B are clearly variable.\[fig1\]](TWAbinaries_lightcurves.ps){width="5.25in"}
Chandra/Swift Results
=====================
Our basic results can be summarized as follows:
- [The TWA stars were all readily detected as strong X-ray sources. Sufficient counts are collected to determine the X-ray sources positions accurately and these all agree with the expected proper-motion-corrected optical positions. ]{}
- [Our [*[Chandra]{}*]{} observations resolved the TWA 13, TWA 8 and TWA 9 common proper motion binaries and show that the lower mass but more rapidly rotating secondaries TWA 8B and TWA 9B are far less luminous.]{}
- [Stars in the TW Hya association have super-saturated coronae where increasing rotational velocity leads to a decrease in the X-ray luminosity (see @Jeffries11 for a general discussion). This has significant implications for how the X-ray radiation fields evolves as the stars contract and spin-up.]{}
- [The coronal emission is continuously varying (see Fig. \[fig1\]) due to magnetic flaring, with time-resolved spectral fitting showing higher temperatures corresponding to higher count rates. Similar time-scale FUV variations are seen in our contemporaneous [*[HST]{}*]{} COS spectra [@Loyd_France14].]{}
- [The higher luminosity (log L$_X$ = 29.5-30.1 ergs s$^{-1}$) stars have very hot (2 keV) coronal plasma, but the less active stars only show a cooler (0.5 keV) coronal component, based on 2-temperature XSPEC VAPEC spectral fitting (see Fig. \[fig2\]). This cooler component is present in the spectra of all the stars.]{}
Alexander, R. D., Clarke, C. J., & Pringle, J. E. 2006, MNRAS, 369, 229 Arnaud, K. A. 1996, Astronomical Data Analysis Software and Systems V, eds. G. Jacoby and J. Barnes, ASP Conf. Series Vol. 101, p.17 Dorman, B., Arnaud, K. A., & Gordon, C. A. 2003, BAAS, 35, 641 Ducourant, C., Teixeira, R., Galli, P. A. B., et al. 2014, A&A, 563, 121 Jeffries, R. D., Jackson, R. L., Briggs, K. R., Evans, P. A., & Pye, J. P. 2011, MNRAS, 411, 2099 Lawson, W. A. & Crause, L. A. 2005, MNRAS, 357, 1399 Loyd, R. O. Parke & France, K. 2014, ApJS, 211, 9 Riedel, A. R., Finch, C. T., Henry, T. J., et al, 2014, AJ, 147, 85 Webb, R. A., Zuckerman, B., Platais, I., et al. 1999, ApJ, 512, L63 Weinberger, A. J., Anglada-Escudé, G., & Boss, A. P. 2013, ApJ, 762, 118 Yang, H., Johns-Krull, C. M., & Valenti, J. A. 2008, AJ, 136, 2286
|
---
address: |
Universität Dortmund\
Institut für Physik, Otto-Hahn-Str. 4\
D–44221 Dortmund, Germany
author:
- Jürgen Baacke
title: 'Scalar O(N) Model at Finite Temperature — 2PI Effective Potential in Different Approximations'
---
Introduction
============
Inspired by earlier analyses out of equilibrium and in thermal equilibrium [@earlier] we are interested in the phase structure of the $O(N)$ linear sigma model in different approximations. Since renormalization is quite a task in 3+1 dimensions we first carry out this analysis in 1+1 dimensions. For all details the reader is referred to a more comprehensive analysis published recently [@Baacke:2004dp].
Basic Equations
===============
Classical action
----------------
We take the following classical action for the $O(N)$ linear sigma model with spontaneous symmetry breaking in 1+1 dimensions $$\label{eq:lagrange}
\mathcal{S}[\Phi]=\int {\mathrm d}^2x\
\frac{1}{2} (\partial_\mu \vec\Phi)^2 - \frac{\lambda}{4N}
\left( \vec\Phi^2 -N v^2 \right)^2
\quad\textrm{where}\quad
\vec\Phi=(\Phi_1,\dots,\Phi_N)\ .$$
In order to consider finite temperature we use the [Matsubara]{.nodecor} formalism where the [Minkowski]{.nodecor}an integral over momentum space is transformed to a sum and an integral over spatial momenta. We will use the short-hand notation $ T \sum_{\omega_n} \int\frac{\rm dp}{2\pi} \equiv {\ensuremath{ \int_{p} \! \! \! \! \! \! \Sigma}}$ . For the effective action we consider a homogeneous background condensate $\vec\phi = \langle \vec\Phi \rangle = (\phi,\phi,\dots,\phi)$ which can be $O(N)$-rotated such that it points only in the 1-direction $\vec\phi = \left( \sqrt{N} \phi, 0, \dots, 0 \right)$.
2PI effective action
--------------------
The two-particle irreducible (2PI) effective action [@CJT] reads $$\label{eq:effective action}
\Gamma[\phi,G] = \mathcal{S}\left[\sqrt{N}\phi\right]
+ {\frac{1}{2}}{\mathrm{Tr}\,}\ (i\mathcal{D}^{-1}G-1)
+ \frac{i}{2} \ln\det G^{-1}D_0 + \Gamma_2[G,\phi] \ .$$ Here the two-point function $G$ is a $N\times N$ matrix which is $O(N)$ symmetric, $ G = \mathrm{diag}\left[ G_\sigma, G_\pi, \dots, G_\pi \right] $, and $\mathcal{D}$ denotes the analogous matrix of classical Green functions with $
i \mathcal{D}_{\sigma,\pi}^{-1}(k) = k^2 - \lambda(f_{\sigma,\pi}\
\phi^2-v^2)$ and $f_{\sigma,\pi}=3,1$. The explicit form of all higher-order corrections denoted by $\Gamma_2[G,\phi]$ is related to the type of approximation used.
### Next-to-leading order of $1/N$ expansion
The $1/N$ expansion is a systematic expansion of the effective action in powers of $1/N$. The classical and one-loop part of the effective action is of leading order $\mathcal{O}(N)$ whereas all further contributions (except for the double-bubble graph) are of higher order. Here we will only take into account the contributions of next-to-leading order (NLO) of the 2PI-$1/N$ expansion[^1] Therefore, higher loop contributions are separated into three parts $$\Gamma_2[\phi,G] = \Gamma_2^\mathrm{db}[G] +
\Gamma_2^\textrm{pearls}[G] + \Gamma_2^\textrm{sunset}[\phi,G]
\ .$$ The first term (so called double-bubble) contributes both at leading and next-to-leading order of $1/N$ (cf. larger articles for details [@Baacke:2004dp; @Aarts:2002dj]) To sum up all contributions of next-to-leading order (NLO) we use the functional [@Aarts:2002dj] $$\label{eq:Gamma_2^pearls}
\Gamma_2^\textrm{pearls}[G] = \frac{i}{2} {\ensuremath{ \int_{p} \! \! \! \! \! \! \Sigma}}\
\left\{
\ln \left[1+ i \frac{\lambda}{N} {\mathrm{Tr}\,}\mathcal{F}(p) \right]
- i \frac{\lambda}{N} {\mathrm{Tr}\,}\mathcal{F}(p) \right\}
\ ,$$ where ${\mathrm{Tr}\,}\mathcal{F}$ is the trace of all fish graphs $$\label{eq:fish}
{\mathrm{Tr}\,}\mathcal{F}(p)= \mathcal{F}_\sigma(p) + (N-1) \mathcal{F}_\pi(p)
\quad\textrm{with}\quad
\mathcal{F}_*(p) = {\ensuremath{ \int_{p} \! \! \! \! \! \! \Sigma}}\ G_*(k)\, G_*(k+p) \ .$$ The subtracted graph in Eq. (\[eq:Gamma\_2\^pearls\]) is the NLO part of the double-bubble which is dealt with separately [@Baacke:2004dp].
The generalization of sunset diagrams in the effective action is achieved by cutting a sigma line of $\Gamma_2^\textrm{pearls}$ and pinning the two open legs to the background by multiplying by a factor of $N\phi^2$ $$\label{eq:Gamma_2^sunset}
\Gamma_2^\textrm{sunset}[\phi,G] =i \lambda\ \phi^2\ {\ensuremath{ \int_{p} \! \! \! \! \! \! \Sigma}}\
G_\sigma(p) \frac{ i \frac{\lambda}{N} {\mathrm{Tr}\,}\mathcal{F}(p)}
{1 + i\frac{\lambda}{N} {\mathrm{Tr}\,}\mathcal{F}(p)}\ .$$ Expanding this in powers of $\mathcal{F}$ one finds graphs of the generalized sunset kind (see again our larger article [@Baacke:2004dp] for figures and more details).
### “Loop” expansion for $N=1$
Since for $N=1$ the $1/N$ expansion is somewhat meaningless, one could have the idea to improve a loop expansion by summing up all pearls and generalized sunset graphs as in Eqs. and . We will call this “loop expansion” although this is not literally correct. We take into account the same graphs (topologically) as in the $1/N$ expansion but with a combinatorical factor that disregards their order of $1/N$. The respective expression for the resummation of pearls is (note the additional combinatorical factor) $$\label{eq:Gamma_2^pearlsloop}
\Gamma_2^\textrm{pearls}[G] = \frac{i}{2} {\ensuremath{ \int_{p} \! \! \! \! \! \! \Sigma}}\
\left\{
\ln \left[1+ 3i\lambda\mathcal{F}(p) \right]
- i3\lambda \mathcal{F}(p)
+ 3 \lambda^2 \left[ \mathcal{F}(p)\right]^2
\right\}
\ .$$ The subtracted graphs in Eq. (\[eq:Gamma\_2\^pearlsloop\]) have to be dealt with separately due to different combinatorical factors. The last term in Eq. (\[eq:Gamma\_2\^pearlsloop\]) is the basketball graph which has a different topology than the graphs with more than two vertices. The sum of generalized sunset graphs is obtained in an analogous way to the $1/N$ expression . It reads $$\label{eq:Gamma_2^sunsetloop}
\Gamma_2^\textrm{sunset}[\phi,G] =i\lambda\ \phi^2\ {\ensuremath{ \int_{p} \! \! \! \! \! \! \Sigma}}\
G_\sigma(p) \frac{ 3i\lambda \mathcal{F}(p)}
{1 + 3i\lambda\mathcal{F}(p)}
+ 6 \lambda^2 \mathcal{F}(p) G(p)
\ .$$
Dyson–Schwinger Equation
------------------------
In order to calculate the 1PI effective potential from the 2PI effective action we have to solve the [[Dyson]{}–[Schwinger]{} equation]{} $\delta \Gamma / \delta G = 0$ for the 2-point function. Using the convention $iG^{-1}_*(p) = i\mathcal{D}_*^{-1} - \Sigma_*(p)$, where $*=\sigma,\pi$, we can express the [[Dyson]{}–[Schwinger]{} equation]{} in terms of the self-energy $\Sigma(p)$
\[eq:Dyson-Schwinger\] $$\begin{aligned}
\Sigma_\sigma(p) &=& 3 \frac{\lambda}{N}\ \mathcal{B}_\sigma
+ (N-1) \frac{\lambda}{N} \mathcal{B}_\pi
- 2\, \frac{\delta}{\delta G_\sigma(p)}
\left( \Gamma_2^\textrm{pearls} + \Gamma_2^\textrm{sunset}
\right) \\
\Sigma_\pi(p) &=& \frac{\lambda}{N}\ \mathcal{B}_\sigma
+ (N+1) \frac{\lambda}{N} \mathcal{B}_\pi
- 2\, \frac{\delta}{\delta G_\pi(p)}
\left( \Gamma_2^\textrm{pearls} + \Gamma_2^\textrm{sunset}
\right)
\ .
\end{aligned}$$
The 1PI effective action is obtained by substituting a solution $G(\phi)$ of Eqs. into $\Gamma[\phi,G]$. We will plot the 1PI effective potential that differs from that only by a total factor of volume times temperature.
Numerical Results
=================
For a given temperature $T$ and different values of $\phi$ we numerically solve the [[Dyson]{}–[Schwinger]{} equation]{} by iteration.
Next-to-leading order of 2PI $1/N$ expansion
--------------------------------------------
We take here a value of $\lambda=0.5$ for the coupling constant and show results for $N=1$, $N=4$ and $N=10$.
Figure \[fig:1/N\] indicates that there is only one vacuum at $\phi=0$ for the considered temperatures and the cases $N=4$ and $N=10$. The potential for $N=1$ shows signs of a false vacuum which is actually not expected in a $1/N$ expansion. Though for $N=1$ this expansions is obviously not meaningful. For the sake of direct comparison we mention — without showing a plot — that for $N=1$ and $\lambda=0.5$ the $1/N$ potential is convex at temperatures $T=0.7$ and $T=0.8$, e.g., whereas it is not when using the “loop expansion” (see below and Fig. \[fig:loop\]).
“Loop expansion” for $N=1$
--------------------------
We display the effective potential in the resummed “loop expansion” (cf. Eqs. and ) in Fig. \[fig:loop\] at different temperatures and for three values of the coupling constant.
The potential exhibits a typical structure with a false vacuum — a clear sign of a first-order phase transition. For higher temperatures, e.g. $T=1.2$ and $\lambda=1$ or $T=0.8$ and $\lambda=0.5$, the false vacuum has disappeared but a “relic” consisting of two inflection points remains.
Conclusion and Outlook
======================
We have solved the [[Dyson]{}–[Schwinger]{} equation]{} to compute the effective potential of the $O(N)$ linear sigma model in 1+1 dimensions both in a “resummed loop” expansion for $N=1$ and at NLO of a $1/N$ expansion for arbitrary $N$. For $N=4$ and $N=10$ the effective potential is convex for all parameters we chose as expected from very old arguments [@Coleman:1973ci].
For $N=1$ we find (indications of) false vacua in both approximations. The $1/N$ expansion seems to be meaningless for $N=1$ concerning the shape of the effective potential. For the “resummed loop expansion” one has to admit that this approximation only serves as an example of a non-systematic expansion and therefore the effective potential has a non-physical shape. As stated above, further results can be found in a more comprehensive publication[@Baacke:2004dp].
Acknowledgments {#acknowledgments .unnumbered}
===============
S.M. thanks all the organizers of *SEWM 2004* for a wonderful meeting in Helsinki. S.M. was supported by *Deutsche Forschungsgemeinschaft* as a member of *Graduiertenkolleg 841*.
[99]{} <span style="font-variant:small-caps;">J. Baacke</span> and <span style="font-variant:small-caps;">S. Michalski,</span> Phys. Rev. D [**67**]{}, 085006 (2003) \[arXiv:hep-ph/0210060\]; Phys. Rev. D [**65**]{}, 065019 (2002) \[arXiv:hep-ph/0109137\]; <span style="font-variant:small-caps;">J. Baacke</span> and <span style="font-variant:small-caps;">A. Heinen,</span> Phys. Rev. D [**68**]{}, 127702 (2003) \[arXiv:hep-ph/0305220\]. Phys. Rev. D [**67**]{}, 105020 (2003) \[arXiv:hep-ph/0212312\].
<span style="font-variant:small-caps;">J. Baacke</span> and <span style="font-variant:small-caps;">S. Michalski</span>, arXiv:hep-ph/0407152 (to be published in PRD).
<span style="font-variant:small-caps;">G. Aarts, D. Ahrensmeier, R. Baier, J. Berges</span> and <span style="font-variant:small-caps;">J. Serreau</span>, Phys. Rev. D [**66**]{}, 045008 (2002) \[arXiv:hep-ph/0201308\]; <span style="font-variant:small-caps;">J. Berges</span>, Nucl. Phys. A [**699**]{}, 847 (2002) \[arXiv:hep-ph/0105311\].
see e.g. <span style="font-variant:small-caps;">J. M. Cornwall, R. Jackiw</span> and <span style="font-variant:small-caps;">E. Tomboulis,</span> Phys. Rev. D [**10**]{}, 2428 (1974). <span style="font-variant:small-caps;">S. R. Coleman</span>, *“There Are No Goldstone Bosons In Two-Dimensions”*, Commun. Math. Phys. [**31**]{}, 259 (1973).
[^1]: Due to resummation contributions of all powers of $1/N$ contribute as well. Though the order is determined from the 2PI graphs of the action.
|
---
abstract: 'This paper presents an analysis of the transient behavior of the Advanced LIGO suspensions used to seismically isolate the optics. We have characterized the transients in the longitudinal motion of the quadruple suspensions during Advanced LIGO’s first observing run. Propagation of transients between stages is consistent with modelled transfer functions, such that transient motion originating at the top of the suspension chain is significantly reduced in amplitude at the test mass. We find that there are transients seen by the longitudinal motion monitors of quadruple suspensions, but they are not significantly correlated with transient motion above the noise floor in the gravitational wave strain data, and therefore do not present a dominant source of background noise in the searches for transient gravitational wave signals.'
address: ' $^{1}$Louisiana State University, Baton Rouge, LA 70803, USA $^{2}$LIGO Livingston Observatory, Livingston, LA 70754, USA $^{3}$LIGO, California Institute of Technology, Pasadena, CA 91125, USA $^{4}$Syracuse University, Syracuse, NY 13244, USA $^{5}$LIGO Hanford Observatory, Richland, WA 99352, USA $^{6}$SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom $^{7}$LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA $^{8}$Columbia University, New York, NY 10027, USA $^{9}$University of Western Australia, Crawley, Western Australia 6009, Australia $^{10}$University of Florida, Gainesville, FL 32611, USA $^{11}$Leibniz Universität Hannover, D-30167 Hannover, Germany $^{12}$Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany $^{13}$The University of Sheffield, Sheffield S10 2TN, United Kingdom $^{14}$Stanford University, Stanford, CA 94305, USA $^{15}$University of Sannio at Benevento, I-82100 Benevento, Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy $^{16}$The University of Mississippi, University, MS 38677, USA $^{17}$University of Michigan, Ann Arbor, MI 48109, USA $^{18}$American University, Washington, D.C. 20016, USA $^{19}$University of Oregon, Eugene, OR 97403, USA $^{20}$University of Adelaide, Adelaide, South Australia 5005, Australia $^{21}$SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom $^{22}$Australian National University, Canberra, Australian Capital Territory 0200, Australia $^{23}$University of Minnesota, Minneapolis, MN 55455, USA $^{24}$Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain $^{25}$Hobart and William Smith Colleges, Geneva, NY 14456, USA $^{26}$Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia $^{27}$The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA $^{28}$MTA Eötvös University, “Lendulet” Astrophysics Research Group, Budapest 1117, Hungary $^{29}$SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom $^{30}$University of Maryland, College Park, MD 20742, USA $^{31}$California State University Fullerton, Fullerton, CA 92831, USA $^{32}$Monash University, Victoria 3800, Australia $^{33}$University of Birmingham, Birmingham B15 2TT, United Kingdom $^{34}$University of Washington, Seattle, WA 98195, USA '
author:
- 'M. Walker,$^{1}$ T. D. Abbott,$^{1}$ S. M. Aston,$^{2}$ G. Gonz[á]{}lez,$^{1}$ D. M. Macleod,$^{1}$ J. McIver,$^{3}$ B. P. Abbott,$^{3}$ R. Abbott,$^{3}$ C. Adams,$^{2}$ R. X. Adhikari,$^{3}$ S. B. Anderson,$^{3}$ A. Ananyeva,$^{3}$ S. Appert,$^{3}$ K. Arai,$^{3}$ S. W. Ballmer,$^{4}$ D. Barker,$^{5}$ B. Barr,$^{6}$ L. Barsotti,$^{7}$ J. Bartlett,$^{5}$ I. Bartos,$^{8}$ J. C. Batch,$^{5}$ A. S. Bell,$^{6}$ J. Betzwieser,$^{2}$ G. Billingsley,$^{3}$ J. Birch,$^{2}$ S. Biscans,$^{3,7}$ C. Biwer,$^{4}$ C. D. Blair,$^{9}$ R. Bork,$^{3}$ A. F. Brooks,$^{3}$ G. Ciani,$^{10}$ F. Clara,$^{5}$ S. T. Countryman,$^{8}$ M. J. Cowart,$^{2}$ D. C. Coyne,$^{3}$ A. Cumming,$^{6}$ L. Cunningham,$^{6}$ K. Danzmann,$^{11,12}$ C. F. Da Silva Costa,$^{10}$ E. J. Daw,$^{13}$ D. DeBra,$^{14}$ R. T. DeRosa,$^{2}$ R. DeSalvo,$^{15}$ K. L. Dooley,$^{16}$ S. Doravari,$^{2}$ J. C. Driggers,$^{5}$ S. E. Dwyer,$^{5}$ A. Effler,$^{2}$ T. Etzel,$^{3}$ M. Evans,$^{7}$ T. M. Evans,$^{2}$ M. Factourovich,$^{8}$ H. Fair,$^{4}$ A. Fernández Galiana,$^{7}$ R. P. Fisher,$^{4}$ P. Fritschel,$^{7}$ V. V. Frolov,$^{2}$ P. Fulda,$^{10}$ M. Fyffe,$^{2}$ J. A. Giaime,$^{1,2}$ K. D. Giardina,$^{2}$ E. Goetz,$^{12}$ R. Goetz,$^{10}$ S. Gras,$^{7}$ C. Gray,$^{5}$ H. Grote,$^{12}$ K. E. Gushwa,$^{3}$ E. K. Gustafson,$^{3}$ R. Gustafson,$^{17}$ E. D. Hall,$^{3}$ G. Hammond,$^{6}$ J. Hanks,$^{5}$ J. Hanson,$^{2}$ T. Hardwick,$^{1}$ G. M. Harry,$^{18}$ M. C. Heintze,$^{2}$ A. W. Heptonstall,$^{3}$ J. Hough,$^{6}$ K. Izumi,$^{5}$ R. Jones,$^{6}$ S. Kandhasamy,$^{16}$ S. Karki,$^{19}$ M. Kasprzack,$^{1}$ S. Kaufer,$^{11}$ K. Kawabe,$^{5}$ N. Kijbunchoo,$^{5}$ E. J. King,$^{20}$ P. J. King,$^{5}$ J. S. Kissel,$^{5}$ W. Z. Korth,$^{3}$ G. Kuehn,$^{12}$ M. Landry,$^{5}$ B. Lantz,$^{14}$ N. A. Lockerbie,$^{21}$ M. Lormand,$^{2}$ A. P. Lundgren,$^{12}$ M. MacInnis,$^{7}$ S. Márka,$^{8}$ Z. Márka,$^{8}$ A. S. Markosyan,$^{14}$ E. Maros,$^{3}$ I. W. Martin,$^{6}$ D. V. Martynov,$^{7}$ K. Mason,$^{7}$ T. J. Massinger,$^{4}$ F. Matichard,$^{3,7}$ N. Mavalvala,$^{7}$ R. McCarthy,$^{5}$ D. E. McClelland,$^{22}$ S. McCormick,$^{2}$ G. McIntyre,$^{3}$ G. Mendell,$^{5}$ E. L. Merilh,$^{5}$ P. M. Meyers,$^{23}$ J. Miller,$^{7}$ R. Mittleman,$^{7}$ G. Moreno,$^{5}$ G. Mueller,$^{10}$ A. Mullavey,$^{2}$ J. Munch,$^{20}$ L. K. Nuttall,$^{4}$ J. Oberling,$^{5}$ M. Oliver,$^{24}$ P. Oppermann,$^{12}$ Richard J. Oram,$^{2}$ B. O’Reilly,$^{2}$ D. J. Ottaway,$^{20}$ H. Overmier,$^{2}$ J. R. Palamos,$^{19}$ H. R. Paris,$^{14}$ W. Parker,$^{2}$ A. Pele,$^{2}$ S. Penn,$^{25}$ M. Phelps,$^{6}$ V. Pierro,$^{15}$ I. Pinto,$^{15}$ M. Principe,$^{15}$ L. G. Prokhorov,$^{26}$ O. Puncken,$^{12}$ V. Quetschke,$^{27}$ E. A. Quintero,$^{3}$ F. J. Raab,$^{5}$ H. Radkins,$^{5}$ P. Raffai,$^{28}$ S. Reid,$^{29}$ D. H. Reitze,$^{3,10}$ N. A. Robertson,$^{3,6}$ J. G. Rollins,$^{3}$ V. J. Roma,$^{19}$ J. H. Romie,$^{2}$ S. Rowan,$^{6}$ K. Ryan,$^{5}$ T. Sadecki,$^{5}$ E. J. Sanchez,$^{3}$ V. Sandberg,$^{5}$ R. L. Savage,$^{5}$ R. M. S. Schofield,$^{19}$ D. Sellers,$^{2}$ D. A. Shaddock,$^{22}$ T. J. Share,$^{5}$ B. Shapiro,$^{14}$ P. Shawhan,$^{30}$ D. H. Shoemaker,$^{7}$ D. Sigg,$^{5}$ B. J. J. Slagmolen,$^{22}$ B. Smith,$^{2}$ J. R. Smith,$^{31}$ B. Sorazu,$^{6}$ A. Staley,$^{8}$ K. A. Strain,$^{6}$ D. B. Tanner,$^{10}$ R. Taylor,$^{3}$ M. Thomas,$^{2}$ P. Thomas,$^{5}$ K. A. Thorne,$^{2}$ E. Thrane,$^{32}$ C. I. Torrie,$^{3}$ G. Traylor,$^{2}$ D. Tuyenbayev,$^{27}$ G. Vajente,$^{3}$ G. Valdes,$^{27}$ A. A. van Veggel,$^{6}$ A. Vecchio,$^{33}$ P. J. Veitch,$^{20}$ K. Venkateswara,$^{34}$ T. Vo,$^{4}$ C. Vorvick,$^{5}$ R. L. Ward,$^{22}$ J. Warner,$^{5}$ B. Weaver,$^{5}$ R. Weiss,$^{7}$ P. We[ß]{}els,$^{12}$ B. Willke,$^{11,12}$ C. C. Wipf,$^{3}$ J. Worden,$^{5}$ G. Wu,$^{2}$ H. Yamamoto,$^{3}$ C. C. Yancey,$^{30}$ Hang Yu,$^{7}$ Haocun Yu,$^{7}$ L. Zhang,$^{3}$ M. E. Zucker,$^{3,7}$ and J. Zweizig$^{3}$\'
bibliography:
- 'paper.bib'
title: Effects of transients in LIGO suspensions on searches for gravitational waves
---
Introduction
============
The Laser Interferometer Gravitational-wave Observatory (LIGO) was designed to detect gravitational waves from astrophysical sources. [@LIGO; @aLIGO] After six science runs over the course of several years, the detectors underwent major upgrades starting in 2010. In September 2015, the newly upgraded Advanced LIGO detectors began taking data for their first observational period (*O1*). Figure \[aligo\] shows the basic optical configuration of Advanced LIGO. With a sensitivity more than three times better than that of the previous generation, [@DenPaper] the detectors had the astrophysical reach to make the first direct observation of a gravitational wave signal, GW150914, from the merger of two black holes, [@GW150914detectionpaper] and later a second unambiguous detection, GW151226. [@GW151226] These observations have opened a new field of gravitational-wave astronomy, which will continue to grow brighter with further improvements to the detector network.
![Advanced LIGO optical configuration. [@aLIGO] The mirrors at the input and end of both arms (labeled ETM and ITM) are suspended from quadruple-stage pendulums, in addition to active seismic isolation systems.[]{data-label="aligo"}](figure1){width="\linewidth"}
The main low frequency noise source for LIGO is seismic activity. The LIGO detectors are affected by earthquakes from around the world, windy weather that shakes the buildings housing the interferometer instrumentation, microseismic vibrations from ocean waves crashing on the shores of the Pacific Ocean, Atlantic Ocean, and the Gulf of Mexico, and local anthropogenic activity. [@EfflerPEM; @GW150914detcharpaper] The study of the effects of seismic activity was especially important for improving the data quality of transient gravitational wave searches in the initial LIGO era.[@seisveto; @s6detchar] One of the key improvements from initial to Advanced LIGO is the implementation of a much more sophisticated seismic isolation system, which includes stages of active and passive isolation for all of the cavity optics.
It is important to check that this entirely new system provides the very high isolation expected, and that it does not introduce any new types of transient noise that could add to the noise background for searches of short duration gravitational waves, such as black hole mergers and supernovae. Here we present an investigation of the transient motion of the Livingston suspension systems as measured by local sensors on each suspension, specifically looking at the displacement of the quadruple stage pendulums in the longitudinal degree of freedom, which is the direction of the optical path used to sense spacetime strain induced by passing gravitational waves.
Advanced LIGO Suspensions
=========================
In Advanced LIGO, all optical cavities use optics suspended from multi-stage pendulums, in order to benefit from the lowpassing of seismic motion. The input and end mirrors (the optics whose motion most directly contributes to the gravitational-wave readout signal) are all hung from quadruple-stage suspensions, [@SuspensionsDesign] with each stage providing additional isolation at frequencies above the suspension resonances, which range from 0.4 Hz to 14 Hz. The quadruple pendulum is suspended at the top from maraging steel blade springs, with two further sets of springs incorporated into the top two masses, thus providing three stages of enhanced vertical isolation. The two lower masses of the quad are cylindrical silica masses connected by fused silica fibers to reduce thermal noise. Another similar quadruple suspension is hung next to the test mass suspension, so the actuation on lower stages can be done from a similarly isolated reaction chain. Figure \[quadschematic\] gives an overview of the design of the Advanced LIGO quadruple suspensions.
![Quadruple suspensions design. The left image shows the suspension systems with the blades, fibers, and reaction chain. On the right the whole structure is shown, with the four masses labeled. [@SuspensionsDesign][]{data-label="quadschematic"}](figure2a){width="\linewidth"}
![Quadruple suspensions design. The left image shows the suspension systems with the blades, fibers, and reaction chain. On the right the whole structure is shown, with the four masses labeled. [@SuspensionsDesign][]{data-label="quadschematic"}](figure2b){width="\linewidth"}
The local displacement of each stage of the suspensions is measured using Optical Sensor and ElectroMagnetic actuators, or OSEMs, which are electromagnetic sensors and actuators used for damping the suspensions’ resonances and controlling the mirrors to keep cavities aligned and locked. [@SuspensionsDesign; @sensors]
The OSEMs can sense suspension motion at low frequencies where the displacements are relatively large. At frequencies above 5 Hz the suspension motion has typically fallen below the sensitivity level of the OSEMs such that the resulting spectra are dominated by electronics noise. Multiple OSEMs on each stage allow the calculation of the mass’s motion in each degree of freedom using a linear combination of the sensors’ signals. The sensed displacement of the top stage is used in a feedback loop to actuate on that stage of the suspension in order to damp the mechanical resonances of the suspension. The sensors at lower stages are only used as witnesses of the optics’ displacement, for the purposes of diagnosing problems in the suspensions. The actuators on lower stages use interferometer and cavity signals to keep various degrees of the interferometer precisely on resonance.
Motion transients in suspensions
================================
It is important to understand the origins of non-astrophysical noise transients in the gravitational wave data in order to eliminate false positives from the gravitational wave searches. Each subsystem of the detector itself is therefore investigated in great detail to fully study all potential noise sources. [@GW150914detcharpaper]
In this article, we characterize transients in the displacement of the suspensions’ stages as measured by the OSEMs, the propagation of transients between different stages, and their effects on the gravitational wave strain data. Specifically, transients in the longitudinal degree of freedom were studied in the top three stages of the quadruple suspensions.
Motion transients seen by the local displacement sensors have a few potential sources. For example, they could be caused by motion that is intrinsic to the suspension systems themselves, from the crackling in the suspension wires or the steel blades. [@cracklingnoise; @cracklingnoise2] Transients could also come from excess seismic motion by propagating through each stage of active seismic isolation and then down through the suspension stages. Above the suspension’s resonance f$_0$, the seismic transients should decrease in amplitude by a factor of (f/f$_0)^{2}$ at each stage and be less likely to appear above the sensor noise at lower stages. Therefore any seismic transients that affect multiple stages should appear mostly at low frequencies. Another source of transients seen in the local sensors is the actuation on the suspensions from the feedback loops used to control the interferometer.
Suspension behavior in Advanced LIGO’s first observing run
----------------------------------------------------------
The typical spectrum of the suspension motion monitors is characterized by several peaks near the low frequency pendulum resonances between 0.4 to 5 Hz, and the flat noise above 5 Hz due to the sensors’ electronics noise. The main resonances in the longitudinal degree of freedom are modeled for the quadruple suspensions to be at 0.435 Hz, 0.997 Hz, 2.006 Hz, and 3.416, but coupling from other degrees of freedom and the active seismic isolation system creates additional peaks in the spectrum. Figure \[ITMY\_ASD\_AmpvsFreq\] shows a typical spectrum of the Y-arm input quadruple suspension (Input Test Mass Y, or *ITMY*) motion in O1 along with estimated sensor noise levels.
![Typical amplitude spectral density (ASD) for the ITMY longitudinal motion monitors from O1. Many of the low frequency features of the ASD correspond to the pendulum resonances of the suspension. The flat portion of the ASD above 5 Hz shows where the electronics noise of the OSEM dominates the spectrum. The noise in the penultimate stage at higher frequencies is slightly higher because it has a different kind of OSEM. [@sensors][]{data-label="ITMY_ASD_AmpvsFreq"}](figure3){width="\linewidth"}
![The panel on the left shows a five minute time series from the top stage of ITMY, with bandpass filters applied between 4 to 5 Hz and 10 to 11 Hz. The distribution of time series amplitude over the same time is shown on the right, with dashed lines to indicate a Gaussian distribution. While the Gaussian-distributed stationary sensor noise dominates the higher frequency band shown, the time series from 4 to 5 Hz exhibits large excursions from the average noise level. []{data-label="BandpassedTimeSeries_4to5_10to11"}](figure4a){width="\linewidth"}
![The panel on the left shows a five minute time series from the top stage of ITMY, with bandpass filters applied between 4 to 5 Hz and 10 to 11 Hz. The distribution of time series amplitude over the same time is shown on the right, with dashed lines to indicate a Gaussian distribution. While the Gaussian-distributed stationary sensor noise dominates the higher frequency band shown, the time series from 4 to 5 Hz exhibits large excursions from the average noise level. []{data-label="BandpassedTimeSeries_4to5_10to11"}](figure4b){width="\linewidth"}
Ideally, the noise would be stationary and the average spectrum would statistically characterize the noise level, but in actuality there are non-stationary disturbances at different frequencies. To demonstrate this, Figure \[BandpassedTimeSeries\_4to5\_10to11\] shows the time series of several minutes of data from the top stage of one suspension with two different bandpass filters applied to select for 4-5 Hz and 10-11 Hz. Above 10 Hz, the sensor noise dominates the signal and the resulting time series is Gaussian distributed, but the lower frequency shows large non-Gaussian transients.
Rather than visually inspecting time series, the Omicron algorithm is used to find transients in the data, producing *triggers* that indicate the time, frequency, amplitude, and signal-to-noise ratio of the transient noise. [@OmicronDocument; @OmicronArticle] Figure \[OmicronAmpvsFreq\_Nov1to8\] shows distribution in frequency and signal-to-noise ratio of Omicron triggers for the longitudinal degree of freedom, using data from the Y-arm input suspension over one week of the observing run. While stationary noise would produce a background of low SNR triggers across all frequencies, the actual data from the suspension monitors shows a varying structure in different frequency bands. This suggests the presence of non-stationary noise sources.
![Above, the Signal-to-Noise Ratio (SNR) and central frequencies of Omicron triggers of ITMY suspension during one week of O1. Below are histograms showing the SNR distributions of the penultimate stage triggers at three selected frequency ranges (note the different SNR scales). While the distribution of higher frequency triggers fall off much like Gaussian noise, the lower frequency ranges contain more outliers.[]{data-label="OmicronAmpvsFreq_Nov1to8"}](figure5a){width="\linewidth"}
Motion transient propagation
----------------------------
![Modelled transient response of a simple pendulum to a sine-Gaussian injection. Simulink was used to model a simple pendulum with a resonance at 2 Hz. A one-second 4 Hz sine-Gaussian signal (top panel) was used as the input to show the response of the system (middle panel) compared with the input signal multiplied by the transfer function of the system at 4 Hz. Local maxima and minima of the time series can be used to calculate the frequency for each half-cycle (bottom panel). This simulation shows that even for this simple model, the transient response of the system deviates from the steady state frequency response at 4 Hz.[]{data-label="SimpleModelResponse"}](figure6){width="\linewidth"}
To characterize the effects of short duration disturbances in the upper stages of the suspensions on the motion lower in the suspension chain, we need more than just the frequency domain models usually used to characterize the suspensions, due to the influence of the impulse response of the system. Simulink was used to model the response of a simple pendulum to a sine-Gaussian input signal. It is important to note that the input signal is not a pure sine wave at a single frequency, but rather a sine-Gaussian characterized by a peak frequency while also containing broader frequency content. Therefore, the pendulum response more strongly attenuates the higher frequencies of the input signal, and the peak frequency of the resulting motion is a mixture of the driving and the resonance frequencies, as demonstrated in Figure \[SimpleModelResponse\].
To examine the propagation of transients in Advanced LIGO suspensions and compare with expected behavior, sine-Gaussian waveforms were physically injected in the Y-end quadruple suspension (End Test Mass Y, or *ETMY*) in the longitudinal direction using the top mass actuators, with central frequencies ranging from 2 to 10 Hz. The Omicron algorithm was used to characterize the resulting transients caused in the top stage as well as in lower stages.
![Ratio of amplitudes of injection Omicron triggers at lower stages to the top stage, plotted against the peak frequency estimated by Omicron for the top stage motion, compared with the modeled transfer function. At lower frequencies the propagation of the transients is close to the model, but above a few Hz, the motion at the lower stages is smaller than the sensor noise, and the amplitude ratio is not as close to the model. There are fewer Omicron triggers at the penultimate stage, since the motion at that stage is at a lower amplitude and is not great enough at higher frequencies to be seen above the sensor noise.[]{data-label="AmpRatioInjections"}](figure7){width="\linewidth"}
Figure \[AmpRatioInjections\] shows the ratio of the Omicron trigger amplitudes of the lower suspension stages to the top stage for different frequencies, using the Omicron frequency estimate of the top stage trigger. The solid lines show the frequency response of the suspensions as predicted by the quadruple suspension models. One reason for apparent discrepancies from the model is variation of the transient motion frequency between stages, as well as the fact that the motion at each stage is not characterized by only a single frequency. To analyze this effect, time series of the injections were examined individually to characterize the frequencies and amplitudes of the signals at each stage, similar to the process used in analysis of the Simulink model shown in Figure \[SimpleModelResponse\].
Using a bandpass filter with a 1 Hz window around the injection frequency and finding the local maxima and minima of the resulting time series, the peak frequency of the induced transient motion was estimated with each cycle. Similar to the simulation performed in Simulink, the suspension’s response is not exactly at the peak sine-Gaussian frequency, and when the injection is finished the suspension’s motion begins to ring down with a frequency approaching the nearest resonance. As the motion propagates downwards, the pendulum filter response attenuates the signal more in the frequency range farther from the resonance, resulting in a slight frequency shift towards the resonance at the lower stage. Figure \[injtimeseries\] shows the period increasing in the bandpassed time series from one of the injections.
![Time series from one injection at 4.1 Hz, after application of a bandpass filter with a window of 1 Hz around the injection frequency. Similar to the simple pendulum model analysis, the frequency shifts throughout the time series. The period of the cycles in the top stage lengthens slightly, from 0.25 seconds (4.0 Hz) at the peak of the transient to 0.28 seconds (3.6 Hz) a few cycles later.[]{data-label="injtimeseries"}](figure8){width="\linewidth"}
The bandpass filter reduces the noise so that the time series cycles can be clearly determined. The frequency of the resulting motion at each stage was then estimated by taking the mean frequency, weighting each cycle by the amplitude of its maximum or minimum. The amplitude of motion was calculated using Omega, a multi-resolution technique for studying transients related to Omicron. [@OmicronDocument][@OmicronArticle] Using the weighted average frequency and the amplitudes calculated by Omega, ratio of motion transient amplitudes between suspension stages for each frequency can be better compared to the suspension model. Figure \[InjectionAmpRatiovsFreq\] displays this comparison for the propagation of the motion from the top stage to the second and third stages. Since the transient amplitude is much smaller at higher frequencies for each successive stage, the higher frequency injection measurements are farther from the model, due to the sensor noise at the lower stages. In both lower stages we see a shift in the frequency away from the frequency at the top stage, generally closer to the nearest suspension resonance, a pitch mode at 2.7 Hz.
Having understood the propagation of short transients in the suspension stages, we turn now to studying the effect of the actual suspension transients on the LIGO gravitational wave strain data during the first Advanced LIGO observing run.
![Amplitude ratios as calculated by the Omega algorithm, and frequency estimated using the maxima and minima of bandpassed time series. Errors are greater as sensor noise becomes dominant at higher frequencies. Red and yellow points represent the same amplitude ratios between stages, but red points show the frequency estimate at the top stage while yellow points show the frequency estimate at the lower stages. Error bars shown are the standard deviation of the measurement among the various injections of the same frequency, weighted by the amplitude of the injection at the top stage.[]{data-label="InjectionAmpRatiovsFreq"}](figure9a){width="\linewidth"}
![Amplitude ratios as calculated by the Omega algorithm, and frequency estimated using the maxima and minima of bandpassed time series. Errors are greater as sensor noise becomes dominant at higher frequencies. Red and yellow points represent the same amplitude ratios between stages, but red points show the frequency estimate at the top stage while yellow points show the frequency estimate at the lower stages. Error bars shown are the standard deviation of the measurement among the various injections of the same frequency, weighted by the amplitude of the injection at the top stage.[]{data-label="InjectionAmpRatiovsFreq"}](figure9b){width="\linewidth"}
### Correlations with gravitational wave strain data
Taking the data from the first week of November (the same week shown in Figure \[OmicronAmpvsFreq\_Nov1to8\]), Omicron was used to identify transients in the gravitational wave (GW) strain data in the same frequency range as used to produce the suspension motion triggers (0.1 to 60 Hz), as well as at higher frequencies to check for any nonlinear coupling. Figure \[DARMITMYROC\] shows the correlations between transients in the ITMY longitudinal displacement data and the gravitational wave strain data in both frequency ranges. The figures shown are Receiver Operator Characteristics (ROC) curves, which show the time coincidence rate between the two sets of triggers, with various coincidence windows from 0.1 to 10 seconds. This rate is compared with the number of time coincidences that would occur by chance (false alarm rate), using a number of time shifts between the two data sets. In both cases, the small number of coincidences between the sets of data are consistent with the number that would be expected by random chance. The observed transients in the ITMY suspension motion monitors did not show any significant correlation with GW strain noise transients, at any frequency.
![Receiver Operator Characteristic (ROC) curves showing the correlation between noise transients in the GW strain and ITMY suspension data from November 1 to 8, 2015. The lefthand plot shows the correlation with higher frequency GW strain triggers (above 60 Hz), while the plot on the right shows the correlation with GW strain triggers below 60 Hz. The y axis shows the fraction of triggers coincident between the two sets of data for varying time windows. The x axis represents the number of coincidences that would appear by chance, estimated by repeating the analysis at each time window with different time shifts between the two data sets. For both sets of GW strain triggers, the coincidence rate is approximately equal to the false alarm rate, whereas a significant correlation would have a much greater efficiency than false alarm rate.[]{data-label="DARMITMYROC"}](figure10a){width="\linewidth"}
![Receiver Operator Characteristic (ROC) curves showing the correlation between noise transients in the GW strain and ITMY suspension data from November 1 to 8, 2015. The lefthand plot shows the correlation with higher frequency GW strain triggers (above 60 Hz), while the plot on the right shows the correlation with GW strain triggers below 60 Hz. The y axis shows the fraction of triggers coincident between the two sets of data for varying time windows. The x axis represents the number of coincidences that would appear by chance, estimated by repeating the analysis at each time window with different time shifts between the two data sets. For both sets of GW strain triggers, the coincidence rate is approximately equal to the false alarm rate, whereas a significant correlation would have a much greater efficiency than false alarm rate.[]{data-label="DARMITMYROC"}](figure10b){width="\linewidth"}
We can now place upper limits on the level of noise that would be caused in GW strain from the observed transients in suspension monitors. The amplitude of the Omicron triggers from each of the upper stages of ITMY is multiplied by the suspension transfer function to estimate the amplitude of noise transients that would be caused in the test mass by a physical displacement of that amplitude. Figure \[DarmPredictions\] shows the resulting projections in equivalent GW strain amplitude, alongside the GW strain triggers from the same time. The sensor noise at the lower stages is much higher than the expected amplitude of motion at those stages, so the upper limit of motion at the lowest stage is above most of the GW strain triggers. The noise level predicted by the top stage triggers, however, is below most of the GW strain triggers up to 37 Hz, so if noise originating in that stage caused high amplitude transients in the GW data, it would be expected to also be seen by the top stage sensors.
![Livingston ITMY triggers from a week in the first observing run, multiplied by the transfer function to the lowest stage and divided by the arm length to convert the displacement into equivalent strain amplitude. The strain calibration of the gravitational wave data is only accurate above 10 Hz, at frequencies where the OSEM signals are dominated by sensor noise. Therefore, this calculation can only give us the upper limit of transient motion from each stage that could appear in the GW strain data without also appearing in the local displacement sensor. Where there are GW strain noise transients above one of these levels, we can rule out an origin in a particular stage of the suspension chain. A large number of the GW strain triggers are above the noise level from the top stage, eliminating the origin of the noise at the top of the suspension chain. However, only the very loudest GW strain triggers are above the level of the second stage, and no GW strain triggers are higher than the level of the third stage.[]{data-label="DarmPredictions"}](figure11){width="\linewidth"}
Since the top stage triggers are not statistically correlated with any of the GW strain triggers, we can conclude that transient noise originating at the top stage of the suspension is not a significant contribution to the transient noise in the interferometer. We cannot, however, rule out the possibility of GW strain noise transients caused by motion originating in the lower stages of the suspension, since there are a significant portion of GW strain triggers that fall below the level of transient noise caused by the local sensor noise.
Conclusions
===========
Using short duration hardware injections in the top stage of the suspension, we have studied the propagation of transient motion down the suspension chain. The difference of transient amplitudes at different stages is consistent with the models, although slight variations in frequency must be taken into account. The frequency of the transients shifts because the injected waveform is not a pure sine wave but a sine-Gaussian, and after the short duration injection, the suspension motion oscillates with a decreasing amplitude and frequency that shifts toward the closest mechanical resonance frequency. Transients at different stages of the suspension therefore show slightly different frequencies from the same initial sine-Gaussian injection.
Statistical comparisons of the times of transients in the OSEMs and in the GW strain data during O1 show that transients seen by the local displacement sensors of the suspensions are not a significant source of background transient noise in the interferometer. However, this does not rule out transient suspension motion that is below the local sensor noise as a possible source of background noise. Using the suspension models to propagate the sensor noise into the motion at the test mass, upper limits can be placed on the level of noise that could be caused in the GW strain data from transients in suspension motion at each stage. Most GW strain triggers are above the sensor noise level of the top stage of the suspension, but below the noise level of the third stage. Transient noise that originates in the lower stage of the suspension could therefore be a cause of noise in the GW data while not being loud enough to appear above the local sensor noise. [@aLIGO]
LSU authors acknowledge the support of the United States National Science Foundation (NSF) with grants PHY-1505779, 1205882, and 1104371.
The authors gratefully acknowledge the support of the NSF for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors also gratefully acknowledge the support of LSC related research by these agencies as well as by the Council of Scientific and Industrial Research of India, Department of Science and Technology, India, Science & Engineering Research Board (SERB), India, Ministry of Human Resource Development, India, the Istituto Nazionale di Fisica Nucleare of Italy, the Spanish Ministerio de Economía y Competitividad, the Vicepresidència i Conselleria d’Innovació, Recerca i Turisme and the Conselleria d’Educació i Universitat del Govern de les Illes Balears, the European Union, the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, and Innovation, International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), Russian Foundation for Basic Research, the Leverhulme Trust, the Research Corporation, Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS and the State of Niedersachsen/Germany for provision of computational resources.
References
==========
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abstract: 'In his recently discovered handwritten notes on “An alternate way to handle electrodynamics” dated on 1963, Richard P. Feynman speculated with the idea of getting the inhomogeneous Maxwell’s equations for the electric and magnetic fields from the wave equation for the vector potential. With the aim of implementing this pedagogically interesting idea, we develop in this paper the approach of introducing the scalar and vector potentials before the electric and magnetic fields. We consider the charge conservation expressed through the continuity equation as a basic axiom and make a heuristic handle of this equation to obtain the retarded scalar and vector potentials, whose wave equations yield the homogeneous and inhomogeneous Maxwell’s equations. We also show how this axiomatic-heuristic procedure to obtain Maxwell’s equations can be formulated covariantly in the Minkowski spacetime. “*He (Feynman) said that he would start with the vector and scalar potentials, then everything would be much simpler and more transparent.*”'
address: |
$^1$Instituto de Geofísica, Universidad Nacional Autónoma de México, Ciudad de México 04510, México. E-mail: herasgomez@gmail.com\
$^2$Department of Physics and Astronomy, University College London, London WC1E 6BT, UK. E-mail: ricardo.heras.13@ucl.ac.uk
author:
- 'José A. Heras$^1$ and Ricardo Heras$^2$'
---
{#section .unnumbered}
Searching through the historical Caltech archives, Gottlieb [@2] recently discovered five handwritten pages of notes dated on Dec. 13, 1963 in which Richard P. Feynman sketched some ideas on an alternate way to handle electrodynamics. More recently, De Luca et al. [@3] have presented their version of how a part of Feynman’s ideas may be implemented so that they may be used as a supplementary material to usual treatments on electrodynamics. Following Feynman’s ideas to a certain extent, they heuristically obtained the Lorentz force and the homogeneous Maxwell’s equations. Their procedure can be briefly outlined as follows.
- Following Feynman, De Luca et al. [@3] assume that the force on an electric charge $q$ moving with velocity $v_j$ is of the generic form $F_i=q(E_i+v_jB_{ij})$, where $E_i$ and $B_{ij}$ are functions of space and time to be determined (summation on repeated indices is understood). Next, this 3-force is assumed to be the spatial component of a 4-force. Considering the relativistic transformation of this 4-force, the form of the 3-force is found to be: $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$ where $\textbf{B}$ represents the independent components of $B_{ij}$ (they make $c\!=\!1$). Through this procedure the relativistic transformations of the vectors $\mathbf{E}$ and $\mathbf{B}$ may be identified with those of the electric and magnetic fields and this leads to the conclusion that $\mathbf{F}=q( \mathbf{E}+\mathbf{v}\times \mathbf{B})$ is the Lorentz force. We should emphasize that this procedure to obtain the Lorentz force was roughly sketched out by Feynman in his handwritten notes. In our opinion, however, Feynman’s route to the Lorentz force is criticisable: The hypothesis of a force linear in the velocity is not sufficiently well justified. But we must also recognize that the derivation of a Lorentz-like force from relativistic considerations and the assumption of a force depending linearly on velocity are conceptually interesting.
- De Luca et al. [@3] assume the relativistic action $S=\int_{t_1}^{t_2}[-m_0ds-qA_\mu dx^\mu]$, where $A_\mu$ is the 4-potential (they now use relativistic notation). They then vary this action to find the force $\mathbf{F}= q[-\nabla\Phi-\partial\mathbf{ A}/\partial t+\mathbf{v}\times (\nabla \times \mathbf{A})]$. Comparison of this force with the previously obtained Lorentz force yield the relations $\mathbf{E}=-\nabla\Phi-\partial \mathbf{A}/\partial t$ and $\mathbf{B}=\nabla\times \mathbf{A}$ which imply the homogeneous Maxwell’s equations $\nabla\cdot \textbf{B}=0$ and $\nabla \times \mathbf{E}=-
\,\partial \mathbf{B}/\partial t$. This procedure based on the least action principle, which starts with potentials and ends with the homogeneous Maxwell equations, was not drawn in Feynman’s handwritten notes. In getting the homogeneous Maxwell equations, De Luca et al. [@3] considered the Feynman’s Hughes Lectures [@4]. They justify their procedure by arguing that “It is conceivable that Feynman had something like this in mind in 1963, when he wrote his notes.”
Although the attempt of De Luca et al. [@3] to make useful Feynman’s alternate way to handle electrodynamics is valuable, it turns out to be incomplete because the inhomogeneous Maxwell equations: $\nabla\cdot \textbf{E}=\rho/\epsilon_0$ and $\nabla \times \mathbf{B}=\mu_0\textbf{ J} +\epsilon_0\mu_0\partial \mathbf{E}/\partial t$ were not inferred. De Luca et al. recognize this incompleteness but they make no attempt to address this problem. Interestingly, Feynman himself wasn’t sure how to get the inhomogeneous Maxwell’s equations as may be seen in the third point of his first handwritten page, which is partially reproduced in figure 1. With signs of doubt (he wrote: How!?) he speculated with the idea that such inhomogeneous equations could be obtained from the wave equation for the vector potential or from “other principle.” It is not surprising that Feynman was interested in following the unconventional route of starting with potentials before considering the inhomogeneous Maxwell’s equations. Feynman liked the idea that potentials and fields had the same level of reality. In the context of the Aharonov-Bohm effect and referring to the vector potential $\mathbf{A}$ and the magnetic field $\mathbf{B}$, he wrote [@5]: “$\mathbf{A}$ is as real as $\mathbf{B}$-realer, whatever that means.”
We think that the speculative idea raised by Feynman of introducing potentials before fields is pedagogically interesting and deserves to be explored. In this sense it is pertinent to say that in the traditional presentation of Maxwell’s equations appearing in textbooks, potentials are introduced using the homogeneous Maxwell’s equations. The electric and magnetic fields expressed in terms of the scalar and vector potentials are then used in the inhomogeneous Maxwell’s equations, obtaining explicit retarded forms of these potentials. The reversed idea of introducing first retarded potentials satisfying wave equations and then deriving the homogeneous Maxwell’s equations does not seem to have been explored so far, at least in the standard literature available to us. However, we believe that the idea exploring alternative presentations of Maxwell’s equations is important for pedagogical and conceptual reasons.
In this paper we suggest that the “other principle” to obtain the inhomogeneous Maxwell’s equations mentioned by Feynman may be the principle of local charge conservation represented by the continuity equation. We show how a heuristic procedure involving formal operations on the continuity equation evaluated at the retarded time leads to a first-order equation in which we identify the retarded scalar and vector potentials. We then apply the D’Alembertian operator to the retarded potentials, obtaining the wave equations they satisfy. In the final step, we use these wave equations to get not only the inhomogeneous Maxwell’s equations but also the homogeneous ones. Our approach is axiomatic in the sense that it starts with the continuity equation as the basic axiom but it is also heuristic in the sense that this equation is heuristically handled. We also show that this axiomatic-heuristic procedure to obtain the full set of Maxwell’s equations can be covariantly developed in the Minkowski spacetime. To put in context our axiomatic-heuristic procedure, it is pertinent to mention that in a series of papers [@6; @7; @8] which originated some comments [@9; @10] and their respective replies [@11; @12], one of us has developed the idea of getting Maxwell’s equations by starting with the continuity equation evaluated at the retarded time but without appealing to potentials as we now do in the present paper. In the cited papers it has been argued that charge conservation and causality, respectively represented by the local continuity equation and the retarded time are the cornerstones on which Maxwell’s equations are based and therefore they can be considered to be the two fundamental postulates for these equations. It is worth mentioning that although Maxwell’s equations are universally accepted, the question of what their fundamental physical postulates are remains a topic of discussion and debate [@13; @14; @15; @16; @17; @18; @19].
The derivation of Maxwell’s equations presented here in its three-dimensional and four-dimensional versions, which considers potentials as primary quantities and fields as derived quantities, may be useful to grasp the background of Maxwell’s theory and may be presented in undergraduate courses of electromagnetism.
{#section-1 .unnumbered}
The electric charge conservation can locally be expressed by the continuity equation $$\nabla\cdot\!\textbf{ J}+\frac{\partial\rho}{\partial t}=0,$$ where $\rho$ and $\textbf{J} $ are the *localised* charge and current densities which are functions of space and time. Our approach to obtain Maxwell’s equations involves two ingredients: The basic axiom expressed by the continuity equation and a heuristic handle of this equation which involves the concept of causality.
We then assume the existence of certain functions of space and time which are *causally* produced by these localised charge and current densities. Let us call these other unknown functions “the potentials.” We will justify this name later. We additionally assume that these potentials vanish sufficiently rapidly at spatial infinity so that the surface integrals containing these potentials vanish at infinity. Our first task will consist in finding the explicit form of these unknown potentials and the equations they satisfy. The causal connection between the expected potentials and their sources $\rho$ and $\textbf{J}$ means that the latter precede in time to the former, i.e., the potentials calculated at the field point $ \textbf{x}$ at the time $t$ are caused by the action of their sources $\rho$ and $\textbf{J}$ a distance $R=|\textbf{x}-\textbf{x}'|$ away at the source point $\textbf{x}'$ at the retarded time $t'\!=\!t -t_0$. It is clear that $t_0\!>\!0$ is the time required for the carrier of the charge-potential connection to travel the distance $R$ between the source point $\textbf{ x}'$ and the point $ \textbf{x}$. Consider now that the carrier of the interaction is the photon which moves in a straight line at the speed of light $c$ in vacuum. This implies $t_0=R/c$ and thus the retarded time takes the form: $t'\!=t -R/c$. Put differently, causality demands that the unknown potentials must be determined by their sources $\rho$ and $\textbf{J}$ evaluated at the retarded time. We then enclose the terms of the left of (1) in the retardation symbol $[\;\;]$ which indicates that the enclosed quantity is to be evaluated at the source point $\textbf{x}'$ at the retarded time $t'=t-R/c$, $$[\nabla'\cdot\!\textbf{ J}]+\bigg[\frac{\partial\rho}{\partial t'}\bigg]=0.$$ We now multiply the first term of (2) by the factor $\mu_0/(4\pi R)$ and the second term of (2) by the equivalent factor $1/(4\pi\epsilon_0 R c^2)$, where $\epsilon_0$ and $\mu_0$ are constants satisfying the relation $\epsilon_0\mu_0\!=\!1/c^2$, and integrate over all space, obtaining the equation $$\frac{\mu_0}{4\pi}\!\int\!\frac{[\nabla'\cdot\!\textbf{ J}]}{R}d^3x' + \frac{1}{c^2}\frac{1}{4\pi\epsilon_0}\!\int\! \frac{[\partial\rho/\partial t']}{R}d^3x'=0.$$ With the idea of taking out the derivative operators from the integrals in (3), we perform an integration by parts in the first term of (3), in which we use the result [@20]: $[\nabla'\cdot\!\textbf{ J}]/R=\nabla\cdot([\textbf{ J}]/R)+ \nabla'\cdot([\textbf{ J}]/R)$ and the fact that the surface integral arising from the term $\nabla'\cdot([\textbf{ J}]/R)$ vanishes at spatial infinity because $\textbf{J}$ is localised. Next we use the result [@6]: $[\partial\rho/\partial t']\!=\!\partial[\rho]/\partial t$ in the second term of (3). After performing the specified operations, the final result reads $$\nabla\cdot\Bigg\{\frac{\mu_0}{4\pi}\!\int\!\frac{[\textbf{ J}]}{R}d^3x'\Bigg\} + \frac{1}{c^2}\frac{\partial}{\partial t}\Bigg\{\frac{1}{4\pi\epsilon_0} \!\int\!\frac{[\rho]}{R}d^3x'\Bigg\} =0.$$ The terms within the curly braces $\{...\}$ are determined by the retarded values of the sources $\textbf{J}$ and $\rho$. We call these terms the retarded vector potential $\textbf{A}$ and the retarded scalar potential $\Phi$: $$\textbf{A}=\frac{\mu_0}{4\pi}\!\int\!\frac{[\textbf{ J}]}{R}d^3x', \quad \Phi=\frac{1}{4\pi\epsilon_0} \!\int\! \frac{[\rho]}{R}d^3x'.$$ These are the potentials we were looking for. Thus, equation (4) takes the compact form $$\nabla\cdot \textbf{A} + \frac{1}{c^2}\frac{\partial\Phi}{\partial t}=0.$$ In the standard presentation of Maxwell’s equations, the relation (6) is interpreted as a gauge condition, the so-called Lorenz condition. In our presentation, equation (6) should be rather interpreted as a field equation for potentials. At this stage we wonder what other field equations satisfy the potentials **A** and $\Phi$. We then apply the d’Alembert operator $\Box^2\!\equiv\!\nabla^2\!-\!(1/c^2)\partial^2/\partial t^2$ to the potentials in (5), use the result [@6]: $$\square^2\bigg\{\frac{[{\cal F}]}{R}\bigg\}\!=\!-4\pi[{\cal F}]\delta(\textbf{x}\!-\!\textbf{x}'),$$ where ${\cal F}$ is a function of space and time and $\delta$ is the Dirac delta function, and finally integrate the resulting expressions over all space. After this calculation, we get two wave equations $$\Box^2 \textbf{A}=-\mu_0\textbf{J},\quad \Box^2 \Phi=-\frac{\rho}{\epsilon_0}.$$ These are the second-order equations we were looking for. They imply expressions for the charge and current densities: $\textbf{J}\!=\!-\,\Box^2 \textbf{A}/\mu_0$ and $\rho\!=\!-\,\epsilon_0\Box^2\Phi$ that satisfy the continuity equation $$\nabla\cdot\!\textbf{ J}+\frac{\partial\rho}{\partial t}=-\frac{1}{\mu_0} \Box^2\Bigg(\nabla\cdot \textbf{A} + \frac{1}{c^2}\frac{\partial\Phi}{\partial t} \Bigg)=0,$$ because of (6). The retarded potentials $\textbf{A}$ and $\Phi$ given in (5) constitute the causal solution of the set formed by equations (6) and (8). This solution is shown to be unique [@21].
Equations (8) form a set of *second order* equations connecting the potentials $\textbf{A}$ and $\Phi$ with their sources $\textbf{J}$ and $\rho$. A question arises: could there be a set of *first order* field equations equivalent to the set of equations in (8)? Let us investigate this possibility. Using the identity $\nabla^2\textbf{A}\equiv \nabla (\nabla\cdot \textbf{A}) -\nabla \times (\nabla \times \textbf{A})$ and (6) and (8) we get the equivalent system of equations $$\begin{aligned}
\nabla\cdot\bigg\{\!-\nabla\Phi-\frac{\partial\textbf{A}}{\partial t}\bigg\} = \frac{\rho}{\epsilon_0}, \\
\nabla\times\big\{\nabla\times\textbf{A}\big\} -\frac{1}{c^2}\frac{\partial}{\partial t}\bigg\{\!-\nabla\Phi-\frac{\partial\textbf{A}}{\partial t}\bigg\} =\mu_0 \textbf{J}.\end{aligned}$$ We realise that the quantity $\{-\nabla\Phi-\partial\textbf{A}/\partial t\}$ appears in both (10) and (11), and this does not seem to be a fortuitous coincidence. This quantity together with its partner $\{\nabla\times\textbf{A}\}$ could be physically significant. Let us introduce the fields $\textbf{E}$ and $\textbf{B}$ through the equations $$\begin{aligned}
\textbf{E}=-\nabla\Phi-\frac{\partial\textbf{A}}{\partial t},\quad \textbf{B}= \nabla\times\textbf{A}.\end{aligned}$$ This justifies the name of potentials to the functions **A** and $\Phi$. According to (12) these potentials determine the fields $\textbf{E}$ and $\textbf{B}$. In terms of **E** and **B**, (10) and (11) take the compact form $$\begin{aligned}
\nabla\cdot \textbf{E} = \frac{\rho}{\epsilon_0}, \\
\nabla\times \textbf{B} -\frac{1}{c^2}\frac{\partial \textbf{E}}{\partial t} =\mu_0 \textbf{J}.\end{aligned}$$ We note that the divergence of (14) together with (13) yield (1) back. Clearly, we have inferred other equivalent expressions for $\textbf{J}$ and $\rho$, namely, $\textbf{J}=\nabla\times \textbf{B}/\mu_0 -\epsilon_0\partial \textbf{E}/\partial t $ and $\rho=\epsilon_0\nabla\cdot \textbf{E}$, which satisfy the continuity equation (1). Of course, (13) and (14) must be completed with other two equations that specify the quantities $\nabla\cdot \textbf{B}$ and $\nabla\times \textbf{E}$ as dictated by the Helmholtz theorem [@22]. These other equations are not difficult to find. We quickly note that the fields $\textbf{E}$ and $\textbf{B}$ given in (12) imply the other two field equations $$\begin{aligned}
\nabla\cdot \textbf{B}=0, \\
\nabla\times \textbf{E} +\frac{\partial \textbf{B}}{\partial t} =0.\end{aligned}$$ The set formed by the first order equations (13)-(16) is equivalent to the set formed by the second order equations (8) together with the equation (6). The set of equations (13)-(16) is uniquely determined whenever we adopt boundary conditions for the fields $\textbf{E}$ and $\textbf{B}$ that are consistent with those of the potentials $\textbf{A}$ and $\Phi$.
In order to find the significance of the fields $\textbf{E}$ and $\textbf{B}$ we use (5) and (12) and obtain the retarded solutions of (13)-(16), $$\begin{aligned}
\textbf{E}=-\nabla \!\int\! \frac{[\rho]}{4\pi\epsilon_0R}d^3x' -\frac{\partial}{\partial t}\!\int\!\!\frac{[\textbf{J}]}{4\pi\epsilon_0 c^2 R}d^3x',\\
\textbf{B}=\nabla\times\!\int\!\!\frac{[\textbf{J}]}{4\pi\epsilon_0 c^2 R}d^3x'.\end{aligned}$$ It becomes evident that $\textbf{E}$ and $\textbf{B}$ are retarded fields. The system formed by the *coupled* four first-order equations (13)-(16) imply a system formed by two *uncoupled* second-order equations. To find the latter system we apply the d’Alembertian operator $\Box^2$ to (17) and (18), use (7), and integrate the resulting expressions over all space to get the wave equations $$\Box^2 \textbf{E}=\frac{1}{\epsilon_0}\nabla\rho +\mu_0\frac{\partial\textbf{J}}{\partial t},\quad \Box^2\textbf{B} =-\mu_0\nabla\times\textbf{J}.$$ Our task will be complete if we identify $\epsilon_0$ and $\mu_0$ with the vacuum permittivity and the vacuum permeability. With this identification, the potentials $\Phi$ and **A** are the electromagnetic scalar and vector potentials and the fields $\textbf{E}$ and $\textbf{B}$ are the electric and magnetic fields.
We have obtained two equivalent versions of electromagnetic field equations. The first one is represented by equations (6) and (8) which are expressed in terms of the retarded scalar and vector potentials defined in (5) and the second one is represented by equations (13)-(16) which are expressed in terms of the retarded fields defined by equations (17) and (18). This second version of the equations is identified with the familiar Maxwell’s equations.
Let us emphasize that the fundamental elements of our axiomatic-heuristic approach to find the Maxwell equations were the principle of charge conservation expressed by the continuity equation (the basic axiom) and an heuristic handle of this equation which involved the principle of causality represented by the retarded time.
{#section-2 .unnumbered}
The preceding axiomatic-heuristic approach can also be used to obtain the Maxwell equations in the four-dimensional Minkowski spacetime. Let us introduce the corresponding notation. A point is denoted by $x=x^{\mu}=\{x^0, x^i\}=\{ct, \textbf{x}\}$ and the signature of the metric is $(+,-,-,-).$ Greek indices run from 0 to 3 and Latin indices run from 1 to 3. The summation convention on repeated indices is adopted.
The continuity equation in the four-space is elegantly simple $$\begin{aligned}
\partial_\nu J^\nu =0,\end{aligned}$$ where $J^\nu$ is the four-current which is assumed to be a localised function of spacetime and $\partial_\mu$ is the four gradient. Our basic axiom is now represented by the covariant form of the continuity equation. A heuristic manipulation of this equation will lead us to the manifestly covariant form of Maxwell’s equations.
Our first task consists in finding a four-potential which is *causally* connected with the four-current via a covariant equation. The causal connection will be now implemented through the retarded Green function $G\!=\!G(x,x')$ for the four-dimensional wave equation: $\partial_\mu\partial^\mu G\!=\!\delta^{(4)}(x\!-\!x'),$ where $\partial_\mu\partial^\mu\!=\!-\Box^2$ is the wave operator and $\delta^{(4)}(x\!-\!x')$ is the four-dimensional delta function. Integration of this wave equation yields the explicit form: $G\!=\!\delta\{t'\!-t+R/c\}/(4\pi R)$. The function $G$ satisfies the property $\partial^\mu G\!=\!-\partial'^\mu G.$ We now evaluate (20) at the source point $x'$ and multiply the resulting equation by $\mu_0G$ and integrate over all spacetime, obtaining $$\begin{aligned}
\int\! \mu_0G\partial'_\nu J^\nu d^4x'=0.\end{aligned}$$ After an integration by parts in (21), in which we use the relation $G\partial'_\nu J^\nu\!=\! \partial_\nu (G J^\nu) + \partial'_\nu (G J^\nu)$ and the fact that the surface integral originated by the term $\partial'_\nu (G J^\nu)$ vanishes at spatial infinity, we take out the operator $\partial_\nu$ from the integral in (21) and obtain $$\begin{aligned}
\partial_\nu\!\! \int\!\mu_0 G J^\nu d^4x'=0.\end{aligned}$$ The integral in (22) must have some significant interpretation, we call it the four-potential $$\begin{aligned}
A^\nu\!= \!\mu_0\!\int\!\!G J^\nu d^4x',\end{aligned}$$ in terms of which (22) becomes elegantly simple compact $$\begin{aligned}
\partial_\nu A^\nu=0.\end{aligned}$$ In the next step we take the wave operator $\partial_\mu\partial^\mu$ to (23), use the result $\partial_\mu\partial^\mu G\!=\!\delta^{(4)}(x-x')$ and integrate over all spacetime to obtain the wave equation $$\begin{aligned}
\partial_\mu\partial^\mu A^\nu=\mu_0J^\nu.\end{aligned}$$ This is the covariant equation we were looking for. It clearly provides an expression for the four-current $J^\nu=\partial_\mu\partial^\mu A^\nu/\mu_0$ that satisfies the continuity equation $$\begin{aligned}
\partial_\nu J^\nu=\frac{1}{\mu_0}\partial_\mu\partial^\mu \partial_\nu A^\nu=0,\end{aligned}$$ because of (24). Equation (25) is a second-order equation that causally connects the four-potential $A^\nu$ with the four-current $J^\nu$. Are there two first-order equations equivalent to the equation (25)? The answer is in the affirmative. We combine (24) and (25) to get the equation $$\begin{aligned}
\partial_\mu\big\{\partial^\mu A^\nu-\partial^\nu A^\mu\big\}=\mu_0J^\nu.\end{aligned}$$ We strongly suspect that the antisymmetric tensor $\partial^\mu A^\nu-\partial^\nu A^\mu$ could be physically significant. We find convenient to label this antisymmetric tensor as $$\begin{aligned}
F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu,\end{aligned}$$ in terms of which (27) takes the elegant form $$\begin{aligned}
\partial_\mu F^{\mu\nu} =\mu_0J^\nu.\end{aligned}$$ This provides us another expression for the four-current $J^\nu=\partial_\mu F^{\mu\nu}/\mu_0$ that satisfies the continuity equation $$\begin{aligned}
\partial_\nu J^\nu=\frac{1}{\mu_0}\partial_\mu\partial_\nu F^{\mu\nu}=0,\end{aligned}$$ because $\partial_\mu\partial_\nu F^{\mu\nu}\equiv 0$ since the operator $\partial_\mu\partial_\nu$ is symmetric in the indices $\mu$ and $\nu$ and the tensor $F^{\mu\nu}$ is antisymmetric in these indices. On the other hand, any antisymmetric tensor field $F^{\mu\nu}$ in the four-space has an associated a dual tensor defined by $^*\!{F}^{\mu\nu}\!=\!(1/2)\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$, where $\varepsilon^{\mu\nu\alpha\beta}$ is the four-dimensional Levi-Civita symbol with $\varepsilon^{0123}\!=\!1$. A generalised Helmholtz theorem [@22; @24] states that an antisymmetric tensor field is completely determined by specifying its divergence and the divergence of its dual. We can show that the dual of (28) is given by $ ^*\!{F}^{\mu\nu}\!=\!\varepsilon^{\mu\nu\alpha\beta}\partial_{\alpha}A_\beta$ and its divergence reads $\partial_\mu\,\!^*\!{F}^{\mu\nu}\!=\!\varepsilon^{\mu\nu\alpha\beta}\partial_\mu\partial_{\alpha}A_\beta,$ whose right-hand side identically vanishes because $\varepsilon^{\mu\nu\alpha\beta}$ is antisymmetric in the indices $\mu$ and $\alpha$ and the operator $\partial_\mu\partial_\alpha$ is symmetric in these indices. Therefore the additional required field equation is given by $$\begin{aligned}
\partial_\mu\,\!^*\!{F}^{\mu\nu}=0.\end{aligned}$$ The set formed by equations (24) and (25) is equivalent to the set formed by equations (29) and (31). Let us write (28) as $F^{\mu\nu} =(\delta^\nu_\lambda\partial^\mu-\delta^\mu_\lambda\partial^\nu) A^\lambda$, were $\delta^\nu_\lambda$ is the Kronecker delta. Using this expression for $F^{\mu\nu}$ together with (23) we obtain $$\begin{aligned}
F^{\mu\nu}=\mu_0 (\delta^\nu_\lambda\partial^\mu-\delta^\mu_\lambda\partial^\nu) \!\int\!\!G J^\lambda d^4x'.\end{aligned}$$ We now take the wave operator $\partial_\alpha\partial^\alpha$ to (32), use $\partial_\mu\partial^\mu G\!=\!\delta^{(4)}(x-x')$ and integrate over all spacetime, obtaining the wave equation $$\begin{aligned}
\partial_\alpha\partial^\alpha F^{\mu\nu}=\mu_0 (\partial^\mu J^\nu-\partial^\nu J^\mu).\end{aligned}$$ Our task will be complete if we appropriately specify the components of the four-current $J^{\mu}$, the four-potential $A^{\mu}$, the electromagnetic field $F^{\mu\nu}$ and its dual $^*\!{F}^{\mu\nu}$. The four-gradient is defined by $\partial_\mu=\big\{(1/c)\partial/\partial t, \nabla\big\}$. Therefore, if we write $$\begin{aligned}
J^\nu\!=\!\{c\rho, \textbf{J}\}, \quad A^\nu\!=\!\{\Phi/c, \textbf{A}\},\end{aligned}$$ then (23)-(25) reproduce (5), (6) and (8) respectively. Similarly, if we write $$\begin{aligned}
F^{i0}\!=\!(\textbf{E})^i/c,\quad F^{ij}\!=\!-\varepsilon^{ijk}(\textbf{B})_k,\quad ^*\!{F}^{i 0}\!=\!(\textbf{B})^i,\quad ^*\!{F}^{i j}\!=\!\varepsilon^{ijk}( \textbf{E})_k/c.\end{aligned}$$ where $(\textbf{E})^i$ and $(\textbf{B})_k$ are the Cartesian components of the fields **E** and **B**, then (29), (31) and (32) reproduce (13)-(18).
We have obtained two equivalent covariant versions of the electromagnetic field equations in the Minkowski spacetime. The first one is represented by equations (24) and (25) which are expressed in terms of the retarded four-potential defined in (23). The second one is represented by equations (29) and (31) which are expressed in terms of the retarded electromagnetic field (32) and its dual. This second version of the equations is identified with the covariant form of Maxwell’s equations. The basic physical ingredients of our axiomatic-heuristic procedure to find these equations were charge conservation mathematically represented by the covariant form of the continuity equation and a heuristic handling of this equation involving the retarded Green function of the wave equation.
{#section-3 .unnumbered}
How should we interpret the procedure proposed here to *obtain* Maxwell’s equations? Have we really made a *derivation* of these equations or just a *construction* of them?
Following the traditional procedure starting with Maxwell’s equations, one introduces potentials and derives their wave equations (by adopting the Lorenz condition). By assuming appropriate boundary conditions the solutions of these wave equations yield the retarded potentials which are then differentiated to get the corresponding retarded electric and magnetic fields. This conventional procedure is logically well-structured and then one can conclude that if Maxwell’s equations are postulated from the beginning then one can derive the retarded potentials and hence their corresponding fields. End of the story.
On the other hand, the reverse procedure starting with the retarded potentials and ending with Maxwell’s equations does not seem to be simple at first sight. Suppose that by some means (which of course does not involve the Maxwell equations) we have found the retarded potentials (5). Differentiating these potentials one obtains their wave equations (8) and equation (6). Combining (6) and (8) one infers equations (10) and (11) which are then identified with the inhomogeneous Maxwell’s equations whenever the electric and magnetic fields are defined as (12). In the final step, one uses these definitions of fields to obtain the homogeneous Maxwell’s equations. This reversed procedure is conceptual and pedagogically significative as long as one can convincingly justify the existence of the retarded potentials without explicitly appealing to Maxwell’s equations. This is the most difficult problem to solve.
But there is a conceptual disadvantage in the traditional procedure. If one *postulates* Maxwell’s equations from the beginning then the task of identifying the basic postulates of these equations *loses* its meaning. On the contrary, the reversed procedure starting with retarded potentials can help to elucidate the nature of these postulates. In the task of finding these potentials, we have argued that charge conservation should be considered the fundamental axiom underlying Maxwell’s equations.
Clearly, the interest sketched by Feynman in his handwritten notes was how to obtain Maxwell’s equations by starting with potentials and using physical principles like relativity and charge conservation. In this aim we think the recourse of heuristic arguments is unavoidable. Put differently, the procedure followed by De Luca et al. [@3] to arrive at the Lorenz force and the homogeneous Maxwell’s equation as well as our procedure to arrive at the inhomogeneous and homogeneous Maxwell’s equations could be interpreted as *constructive* procedures. *In this kind of procedures one makes use of heuristic arguments to show the *existence* of a mathematical object by providing a method for creating the object.* Of course, one generally has knowledge of this object by other means. In this perspective, our procedure to obtain Maxwell’s equations could be considered as a constructive method to demonstrate the existence of retarded potentials which leads to the electric and magnetic fields satisfying Maxwell’s equations. In other words, from a conceptual point of view our procedure could (and should!) be formulated as an existence theorem. Let us enunciate this theorem. *Existence Theorem*. Let $\cJ(\textbf{x},t)$ and $\mathscr{G}(\textbf{x},t)$ be vector and scalar functions which are spatially localised and satisfy the continuity equation $$\nabla\cdot \cJ+\frac{\partial\mathscr{G}}{\partial t}=0.$$ If this equation is evaluated at the source point $\textbf{x}'$ at the retarded time $t'=t-R/\mathscr{C}$ with $\mathscr{C}$ being a constant with units of velocity, then *there exist* the retarded scalar and vector functions: ${\mA}(\textbf{x},t)$ and $\mathscr{P}(\textbf{x},t)$ defined by $$\mA=\frac{1}{4\pi}\!\int\!\frac{[\cJ]}{R}d^3x', \quad \mathscr{P}=\frac{1}{4\pi}\!\int\! \frac{[\mathscr{G}]}{R}d^3x',$$ that satisfy the equation $$\nabla\cdot \mA +\frac{\partial \mathscr{P}}{\partial t}=0,$$ where the retardation symbol $[\;\;]$ indicate that the enclosed quantity is to be evaluated at the source point at the retarded time.
*Corollary 1.* The functions $\mathscr{P}$ and ${\mA}$ in (37) satisfy the wave equations $$\Box^2\mathscr{P} =-\mathscr{G}, \quad \Box^2 \mA=-\cJ,$$ where $\Box^2\!\equiv\!\nabla^2\!-\!(1/\mathscr{C}^2)\partial^2/\partial t^2$. *Corollary 2.* There exist retarded fields: ${\cE}(\textbf{x},t)$ and ${\cB}(\textbf{x},t)$ defined by $$\begin{aligned}
{\cE}=-\nabla\mathscr{P}-\frac{1}{\mathscr{C}^2}\frac{\partial \mA}{\partial t},\quad {\cB}= \nabla\times \mA,\end{aligned}$$ that satisfy the field equations $$\begin{aligned}
\nabla\cdot {\cE} =\mathscr{G} , \quad \nabla\times {\cE} +\frac{1}{\mathscr{C}^2}\frac{\partial {\cB}}{\partial t} =0,\\
\nabla\cdot {\cB}=0, \quad \nabla\times {\cB} -\frac{\partial {\cal E}}{\partial t} =\cJ.\end{aligned}$$ The proof of this general theorem and the proof of its corollaries are entirely similar to those given in the section 2 for the particular case of electromagnetic expressions in SI units. Furthermore, if we make the particular specifications $$\begin{aligned}
\mathscr{C}=c,\;\cJ=\textbf{J},\; \mathscr{G}=\rho,\; \mA=\textbf{A}/\mu_0, \; \mathscr{P}=\epsilon_0\Phi,\; {\cB}=\textbf{B}/\mu_0,\;
{\cE}=\epsilon_0\textbf{E},\end{aligned}$$ in the general theorem and its corollaries then we obtain the corresponding electromagnetic expressions in SI units. In the Minkowski spacetime the existence theorem is indeed elegant: *Existence Theorem*. Let ${\cal J}^\nu$ a localised four-vector that satisfies the continuity equation $\partial_\nu {\cal J}^\nu =0$ then there exists a four-vector ${\cal A}^\nu$ defined as $$\begin{aligned}
{\cal A}^\nu =\int {\cal G} {\cal J}^\nu d^4x',\end{aligned}$$ that satisfies the field equation $\partial_\nu {\cal A}^\nu =0$, where the Green function is defined by ${\cal G}=\delta\{t'\!-t+R/\mathscr{C}\}/(4\pi R)$ with $\mathscr{C}$ being a constant with units of velocity. *Corollary 1.* The four-vector ${\cal A}^\nu$ satisfies the wave equation $\partial_\mu\partial^\mu {\cal A}^\nu={\cal J}^\nu$, where $\partial_\mu\partial^\mu\!=-\nabla^2\!+\!(1/\mathscr{C}^2)\partial^2/\partial t^2$. *Corollary 2.* There exists the antisymmetric tensor ${\cal F}^{\mu\nu}= \partial^\mu {\cal A}^\nu-\partial^\nu {\cal A}^\mu$ that satisfies the field equations $\partial_\mu {\cal F}^{\mu\nu} ={\cal J}^\nu$ and $\partial_\mu\,\!^*{\cal F}^{\mu\nu}=0$, where $^*\!{\cal F}^{\mu\nu}\!=\!\varepsilon^{\mu\nu\alpha\beta}\partial_{\alpha}{\cal A}_\beta$. The proof of this covariant form of the theorem and the proof of its corollaries are entirely similar to those given in the section 3 for the case of electromagnetic expressions in SI units. If $\mathscr{C}=c$ then ${\cal G}=G$. If in this case we make ${\cal J}^\nu= J^\nu$ and ${\cal A}^\nu= A^\nu/\mu_0$ with $A^\nu=(\Phi/c,\mathbf{A})$ then (44) becomes (23) and $A^\nu$ is the electromagnetic four-potential in SI units.
It is possible consider a different heuristic handle of the continuity equation (the basic axiom) to formulate a theorem that is equivalent to the previously considered existence theorem. For example, we can formulate the following existence theorem [@6]: Given the localised sources $\rho(\textbf{x},t)$ and $\textbf{J}(\v x,t)$ satisfying the continuity equation $\nabla\cdot \textbf{J}+\partial\rho/\partial t=0$ there exist the retarded fields $\textbf{F}(\textbf{x},t)$ and $ \textbf{G}(\textbf{x},t)$ defined by $$\begin{aligned}
\textbf{F} =
\frac{\alpha}{4\pi}\int\bigg(\frac{\hat{\textbf{R}}}{R^2}[\rho]+\frac{\hat{\textbf{R}}}{Rc}\left[\frac{\partial \rho}{\partial t}\right]
-\frac{1}{Rc^2}\left[\frac{\partial \textbf{J}}{\partial t}\right]\bigg)\, d^3x',\\
\textbf{G}=
\frac{\beta}{4\pi}\int \bigg([\textbf{J}]\times\frac{\hat{\textbf{R}}}{R^2 }+\bigg[\frac{\partial \v J}{\partial t}\bigg]\times\frac{\hat{\textbf{R}}}{R c}\bigg)\, d^3x'.\end{aligned}$$ that satisfy the following field equations: $\nabla\cdot \textbf{F}=\alpha\rho,\,\nabla\cdot \textbf{G}=0,\,
\nabla\times\textbf{F}+\chi \partial \textbf{G}/\partial t=0$ and $
\nabla\times \textbf{G}-(\beta/\alpha)\partial \textbf{F}/\partial t
=\beta \textbf{J}.$ Here $\hat{\textbf{R}}={\textbf{R}}/R =(\textbf{x}-\textbf{x}')/|\textbf{x}- \textbf{x}'|$ and equations (45) and (46) are in the $``\alpha\beta\chi$” system defined by $\alpha=\beta\chi c^2$. In this case the axiomatic-heuristic approach shows the existence of the electric and magnetic fields in the generalized form of Coulomb and Biot-Savart laws given by Jefimenko [@6] which satisfy Maxwell’s equations.
Similarly, an alternate heuristic manipulation of the continuity equation in the Minkowski spacetime leads to the existence of an electromagnetic tensor satisfying the covariant form of Maxwell’s equations. This is a consequence the following existence theorem [@7]: Given the localized four-vector ${\cal J}^{\mu}$ satisfying the continuity equation $\partial_\mu {\cal J}^\mu=0$ there exists the antisymmetric tensor field $${\cal F}^{\mu\nu}=
\int {\cal G}(\partial'^\mu {\cal J}^\nu-\partial'^\nu {\cal J}^\mu)\,d^4x',$$ that satisfies the field equations: $\partial_{\mu} {\cal F}^{\mu\nu} = {\cal J}^\nu$ and $
\partial_{\mu}{\cal^*F}^{\mu\nu} = 0,$ where $^*{\cal F}^{\mu\nu}=(1/2)\varepsilon^{\mu\nu\alpha\beta} {\cal F}_{\alpha\beta}$ is the dual of ${\cal F}^{\mu\nu}$ and ${\cal G}=\delta\{t'\!-t+R/\mathscr{C}\}/(4\pi R)$ with $\mathscr{C}$ being a constant whose units are of velocity. If we make the identification $\mathscr{C}=c$ then ${\cal G}=G$ and if in addition we make ${\cal J}^\mu=\mu_0J^\nu$ with $J^\nu=(c\rho,\textbf{J})$ then ${\cal F}^{\mu\nu}={F}^{\mu\nu}$ is the electromagnetic field tensor in SI units.
The point to remark is that in the proof of an existence theorem of an object, one is generally free to use all heuristic devices that allows one to exhibit the explicit form of such an object. This is the more essential aspect in a constructive approach.
{#section-4 .unnumbered}
Most authors agree that the continuity equation is a *consequence* of Maxwell’s equations [@26]. Other authors state that it is an *integrability condition* of these equations [@27; @28]. Some other authors are more cautious and claim that Maxwell’s equations are *consistent* with the continuity equation [@23; @29]. Although Maxwell’s equations formally imply the continuity equation, the idea that the latter is a consequence of the former is in a sense questionable. The fact is that the continuity equation has its own existence *independent* of Maxwell’s equations. This can be illustrated by the fact that there are field equations of different electromagnetic theories that are also consistent with the continuity equation. For example, one of these theories arises when the Faraday induction term of Maxwell’s equations is eliminated, obtaining the field equations of a Galilean-invariant instantaneous electrodynamics [@30; @31]. Other examples are the Proca equations of the massive electrodynamics [@32] and the field equations of an electrodynamics in an Euclidean four-space [@33; @34]. Therefore, one should interpret the continuity equation as a formal representation of the principle of charge conservation, but having always in mind that this principle is not exclusive of Maxwell’s theory.
Accordingly, we can equally use the continuity equation to formulate other existence theorems for potentials or fields which can be applied to the aforementioned alternative electromagnetic theories. Here we have evaluated this equation at the retarded time to obtain Maxwell’s equations. But we can equally evaluate this equation at present time, for example, and following a similar heuristic procedure we will obtain the field equations of a Galilean-invariant instantaneous electrodynamics in Gaussian units [@30; @31]: $\nabla\cdot \textbf{E}=4\pi \rho,\nabla\cdot \textbf{B}=\!0, \nabla\times\textbf{E}=0$ and $
\nabla\times \textbf{B}-(1/c)\partial \textbf{E}/\partial t
=(4\pi/c) \textbf{J}.$ However, this does not prevent us to consider that the continuity equation is the cornerstone on which Maxwell’s equations can be constructed. It is in this sense that we claim that charge conservation must be unavoidable considered as one of the basic postulates of Maxwell’s equations. It has been argued that the other basic postulate may be the principle of causality [@6; @7; @8] represented by the retarded time or by the retarded Green function of the wave equation. Of course, we can integrate these two postulates in a single fundamental postulate which would state that *the continuity equation is valid at all times*. Therefore, evaluating this equation at a particular time is not a new postulate but only one special case of the fundamental postulate.
The alert reader might argue that if charge conservation is really the fundamental physical principle underlying Maxwell’s equations then one should be able to obtain these equations using only the continuity equation without making any further assumptions. In our opinion this demand is very hard to satisfy, at least at the level in which we call basic postulates in physics. Furthermore, as already pointed out, the continuity equation may imply other fields equations depending on the “further assumptions.” Let us give an example to illustrate our point. Most physicists would agree that the basic postulates used to derive the Lorentz transformations are the principle of relativity (the first postulate), which states that physical laws must exhibit the same form in inertial frames, and the constancy of the speed of light (the second postulate), which states that the speed of light is the same in inertial frames. What is not well-known is that in 1887, Voigt [@35] used these same two postulates and derived a set of spacetime transformations different from the Lorentz transformations [@36]. In other words, the same postulates may lead to distinct space-time theories! The explanation is simple, the basic postulates are the same but there are different additional assumptions (implicit or explicit) underlying in the derivation of Lorentz and Voigt transformations. We think such additional assumptions are important but they do not qualify to be fundamental postulates.
Similarly, charge conservation can be seen as a basic postulate which requires of some additional considerations to imply Maxwell’s equations. One of these additional assumptions is, for example, the retarded time or the retarded Green function of the wave equation. Nevertheless, we should point out that this assumption is sufficient but not necessary since we could equally assume the advanced time $(t'=t+R/c)$ or the advanced Green function of the wave equation $(G\!=\!\delta\{t'\!-t-R/c\}/(4\pi R))$ and obtain Maxwell’s equations as well. Put differently, charge conservation is a basic postulate (fully justified by experimental considerations) and causality (represented by the retarded time or the retarded Green function of the wave equation) is a sufficient but not a necessary assumption which –we think– does not qualify to be a basic postulate but rather as a complementary assumption. Under this wisdom, the idea of considering that charge conservation is the basic postulate of Maxwell’s equations is similar to the idea of considering that the principle of relativity and the constancy of the speed of light are the basic postulates of special relativity.
{#section-5 .unnumbered}
We have evidence that Feynman attempted to find a different derivation of Maxwell’s equations in at least two periods of his life. The first attempt was around 1948, year in which Feynman showed Dyson an unusual proof of the homogeneous Maxwell’s equations [@37]. Dyson reconstructed Feynman’s proof as an existence theorem: If a non-relativistic particle satisfies Newton’s law of motion and the commutation relations between its position and velocity then *there exist* two fields that satisfy the Lorentz force and the homogeneous Maxwell’s equations. The inhomogeneous Maxwell’s equations were merely assumed to be the definitions of charge and current densities. The second attempt was at the end of 1963 as may be seen in the Feynman’s handwritten notes recently discovered by Gottlieb [@2] and discussed by De Luca et al [@3]. In this second attempt, the Lorentz force was inferred by assuming that the force that acts on a charge is linear in its velocity and is the spatial component of a four-force of special relativity. The homogeneous Maxwell’s equations were obtained via the well-known principle of least action. There is a certain parallelism between these two attempts: both were unpublished and both fail to obtain the inhomogeneous Maxwell’s equations. In the first attempt these equations were defined but not derived and in the second attempt they were not inferred. Charge conservation represented by the continuity equation was not considered in both attempts. Perhaps we may never know what Feynman had in mind in 1966 when he said that he had “cooked up a much better way of presenting the electrodynamics, a much more original and much more powerful way than is in the book,” but it is intriguing that in his first handwritten page wrote in 1963 (see figure 1) he clearly wrote charge conservation and not charge invariance. Was this an error or an unconscious desire?
Here we have pointed out that charge conservation expressed by the continuity equation is the key to obtain the Maxwell equations. We have shown that if the continuity equation evaluated at the retarded time is heuristically handled then we can show that there exist defined retarded potentials that imply not only the inhomogeneous Maxwell’s equations but also the homogeneous ones. In the search for this alternative presentation of Maxwell’s equations in which potentials are introduced before fields, we have been motivated by Feynman’s words that [@38]: “... there is a pleasure in recognising old things from a new point of view. Also, there are problems for which the new point of view offers a distinct advantage.”
{#section-6 .unnumbered}
We dedicate this paper to the memory of Richard P. Feynmanon the occasion of its 101st anniversary.
{#section-7 .unnumbered}
[37]{} R. P. Feynman, R. B. Leighton and M. Sands. The Feynman Lectures on Physics. Addison-Wesley (1963).
Available from the online *Feynman’s Lectures on Physiscs* ([www.feynmanlectures.caltech.edu](http://www.feynmanlectures.caltech.edu/)). See: <http://www.feynmanlectures.caltech.edu/info/other/Alternate_Way_to_Handle_Electrodynamics.html>
De Luca R, Di Mauro M, Esposito S, and Naddeo A 2019 Feynman’s different approach to electromagnetism *Eur. J. Phys.* [**40**, 065205](https://doi.org/10.1088/1361-6404/ab423a)
Feynman R P Lectures on Electrostatics, Electrodynamics, Matter-Waves Interacting, Relativity. Lectures at the Hughes Aircraft Company; notes taken and transcribed by John T. Neer (1967-8)\
[http://www.thehugheslectures.info/wp-content/uploads/lectures/FeynmanHughesLecturesVol2.pdf](http://www.thehugheslectures.info/wp-content/uploads/lectures/FeynmanHughesLectures_Vol2.pdf)
Goodstein D and Goodstein J 2000 Richard Feynman and the History of Superconductivity *Phys. Perspect.* [**2** 3-47](https://doi.org/10.1007/s000160050035)
Heras J A 2007 Can Maxwell’s equations be obtained from the continuity equation? *Am. J. Phys.* [**75** 652-56](http://dx.doi.org/10.1119/1.2739570)
Heras J A 2009 How to obtain the covariant form of Maxwell’s equations from the continuity equation *Eur. J. Phys.* [ **30**, 845-54](http://dx.doi.org/10.1088/0143-0807/30/4/017)
Heras J A 2016 An axiomatic approach to Maxwell’s equations *Eur. J. Phys.* [**37**, 055204](http://dx.doi.org/10.1088/0143-0807/37/5/055204)
Jefimenko O D 2008 Causal equations for electric and magnetic fields and Maxwell’s equations: Comment on a paper by Heras *Am. J. Phys.* [**76**, 101](http://dx.doi.org/10.1119/1.2825390)
Kapuscik E 2009 Comment on ‘Can Maxwell’s equations be obtained from the continuity equation?’ by Heras J A \[Am. J. Phys. **75** 652-56 (2007)\] *Am. J. Phys.* [ **77**, 754](http://dx.doi.org/10.1119/1.3039030)
Heras J A 2008 Author’s response *Am. J. Phys.* [ **76**, 101-2](http://dx.doi.org/10.1119/1.2826656)
Heras J A 2009 Reply to “Comment on ‘Can Maxwell’s equations be obtained from the continuity equation?’" by E Kapuscik \[Am. J. Phys. **77**, 754 (2009)\] *Am. J. Phys.* [**77**, 755-56](http://dx.doi.org/10.1119/1.3039031)
Hehl F W and Obukhov Y N 2003 *Foundations of Classical Electrodynamics: Charge, Flux, and Metric* (MA: Birkhäuser)
Diener G, Weissbarth J, Grossmann F and Schmidt R Obtaining Maxwell’s equations heuristically *Am. J. Phys.* [**81** 120$-$23](http://dx.doi.org/10.1119/1.4768196).
Kosyakov B P 2014 The pedagogical value of the four-dimensional picture: II. Another way of looking at the electromagnetic field *Eur. J. Phys.* [**35** 025013](http://dx.doi.org/10.1088/0143-0807/35/2/025013)
Sobouti Y 2015 Lorentz covariance almost implies electromagnetism and more *Eur. J. Phys.* [ **36**, 065036](http://dx.doi.org/10.1088/0143-0807/36/6/065036)
Hanno E and Nordmark A B 2016 Relativistic version of the Feynman$-$Dyson$-$Hughes derivation of the Lorentz force law and Maxwell’s homogeneous equations *Eur. J. Phys.* [**37**, 05520](http://dx.doi.org/10.1088/0143-0807/37/5/055201)
Heras R 2017 Alternative routes to the retarded potentials *Eur. J. Phys.* [ **38** 055203](https://doi.org/10.1088/1361-6404/aa7f18)
Kosyakov B 2007 *Introduction to the Classical Theory of Particles and Fields* (Springer-Verlag Berlin Heidelberg) Jefimenko O D 1989 *Electricity and Magnetism* 2nd edn (Star City, WV: Electrect Scientific) Anderson J L 1967 *Principles of Relativity Physics* (New York: Academic Press)
Heras R 2016 The Helmholtz theorem and retarded fields, *Eur. J. Phys.* [**37** 065204](http://dx.doi.org/10.1088/0143-0807/37/6/065204)
Jackson J D 1999 *Classical Electrodynamics* 3rd edn (New York: Wiley)
Heras J A 1990 A short proof of the generalized Helmholtz theorem *Am. J. Phys.* [ **58**, 154-55 ](http://dx.doi.org/10.1119/1.16225).
Heras J A and Báez G 2009 The covariant formulation of Maxwell’s equations expressed in a form independent of specific units [*Eur. J. Phys.*]{} [[**30**]{} 23-33](http://dx.doi.org/10.1088/0143-0807/30/1/003)
Griffiths D J 1999 *Introduction to Electrodynamics* 3rd edn (Upper Saddle River, NJ: Prentice-Hall).
Burke W L 1985 *Applied Differential Geometry* (Cambridge: Cambridge University Press).
Betounes D 1998 *Partial Differential Equations for Computational Science* (Springer-Verlag New York).
Zangwill A 2012 *Modern Electrodynamics* (Cambridge: Cambridge University Press).
Jammer M and Stachel J 1980 If Maxwell had worked between Ampére and Faraday: An historical fable with a pedagogical moral *Am. J. Phys.* [**48** 5-7](https://doi.org/10.1119/1.12239)
Heras J A 2005 Instantaneous fields in classical electrodynamics *Eur. Phys. Lett.* [**69** 1-7](http://dx.doi.org/10.1209/epl/i2004-10318-y)
Goldhaber A S and Nieto M M 2010 Photon and Graviton Mass Limits [Rev. Mod. Phys. **82** 939-979](https://doi.org/10.1103/RevModPhys.82.939)
Heras J A 1994 Euclidean electromagnetism in four space: A discussion between God and the Devil *Am. J. Phys.* [**62** 914–16](https://doi.org/10.1119/1.17681)
Heras J A 2006 The Kirchhoff gauge *Ann. Phys.* [**321** 1265–73](https://doi.org/10.1016/j.aop.2005.12.001)
Voigt W 1887 Über das Doppler’sche *Princip Nachr. Ges. Wiss.* Göttingen **8** 41-51
Heras R A review of Voigt’s transformations in the framework of special relativity arXiv:[1411.2559](https://arxiv.org/abs/1411.2559)
Dyson F J 1990 Feynman’s proof of the Maxwell equations [*Am. J. Phys.*]{} [](http://dx.doi.org/10.1119/1.16188)
Feynman R P 1948 Space-time approach to non-relativistic quantum mechanics *Rev. Mod. Phys.* [**20** 367–87](https://doi.org/10.1103/RevModPhys.20.367)
|
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abstract: 'Star formation depends on the available gaseous “fuel” as well as galactic environment, with higher specific star formation rates where gas is predominantly molecular and where stellar (and dark matter) densities are higher. The partition of gas into different thermal components must itself depend on the star formation rate, since a steady state distribution requires a balance between heating (largely from stellar UV for the atomic component) and cooling. In this presentation, I discuss a simple thermal and dynamical equilibrium model for the star formation rate in disk galaxies, where the basic inputs are the total surface density of gas and the volume density of stars and dark matter, averaged over $\sim {{\;\rm kpc}}$ scales. Galactic environment is important because the vertical gravity of the stars and dark matter compress gas toward the midplane, helping to establish the pressure, and hence the cooling rate. In equilibrium, the star formation rate must evolve until the gas heating rate is high enough to balance this cooling rate and maintain the pressure imposed by the local gravitational field. In addition to discussing the formulation of this equilibrium model, I review the current status of numerical simulations of multiphase disks, focusing on measurements of quantities that characterize the mean properties of the diffuse ISM. Based on simulations, turbulence levels in the diffuse ISM appear relatively insensitive to local disk conditions and energetic driving rates, consistent with observations. It remains to be determined, both from observations and simulations, how mass exchange processes control the ratio of cold-to-warm gas in the atomic ISM.'
title: |
Star Formation and Gas Dynamics in Galactic Disks:\
Physical Processes and Numerical Models
---
Introduction
============
Disk galaxies are gas-rich systems, with a multi-phase, highly structured interstellar medium (ISM). Within the ISM, star formation takes place in giant molecular clouds (GMCs), sometimes concentrated in spiral arms. The rate and character of star formation are influenced by physical processes from sub-pc to multi-kpc scales . In spite of the complexity of the ISM and star formation at small scales, there are nevertheless clear correlations between the large-scale rate at which stars are born, and the properties of the ISM and (intra-)galactic environment on large ($\sim$ kpc) scales.
As discussed by Frank Bigiel at this meeting (see also @Bigiel08, and references therein), in regions of galaxies where the gaseous surface density $\Sigma {\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}100 {{\;\rm\,M_\odot}}{{\;\rm\,pc}}^{-2}$, the star formation rate closely follows the surface density of molecular gas. This can be understood in terms of having an essentially constant star formation timescale, ${t_{\rm SF}}\sim 2\times 10^9{{\;\rm yr}}$, within molecular gas (which is observed to be in organized in gravitationally bound clouds with properties that are similar in different galaxies). As a consequence, ${\Sigma_{\rm SFR}}\propto \Sigma$ in regions where the molecular gas dominates the atomic gas. For regions where atomic gas dominates (primarily in the outer parts of galaxies), ${\Sigma_{\rm SFR}}$ instead varies as a steeper power of $\Sigma$. In addition to this superlinear behavior, there is considerable scatter in the relation between ${\Sigma_{\rm SFR}}$ vs. $\Sigma$ at low surface density, suggesting that one or more other parameters, in addition to $\Sigma$, controls the star formation rate.
Indeed, recent examination of the correlation of ${\Sigma_{\rm SFR}}$ with “non-interstellar” galactic environmental properties has revealed interesting dependences, indicating that in the outer parts of galaxies, both the specific star formation rate and the ratio of molecular-to-atomic gas increase roughly linearly with the *stellar* surface density $\Sigma_s$ [@Ler08]. Previously, @BR06 found an approximately linear increase of the molecular content with the estimated dynamic pressure of the ISM, and this is evident in the sample analyzed by @Ler08 as well. The physical reason for the relationship between molecular content (and star formation) and pressure has not, however, been clear from these empirical studies.
Observations of star formation pose a number of challenges: Why is there an increase in the slope of ${\Sigma_{\rm SFR}}\propto \Sigma^{1+p}$ in going from molecular- to atomic-dominated regions? What is the physical reason for the empirical relation between ISM pressure and star formation; more generally, how do galactic parameters such as $\Sigma_s$, the velocity dispersions of stars and of gas, and spiral structure affect ${\Sigma_{\rm SFR}}$? Is it possible to explain the observed behavior of ${\Sigma_{\rm SFR}}$ using simplified theoretical models, and what is required in numerical simulations in order to reproduce observed star formation relationships? Recent theoretical work has taken on these challenges with increasing success; a key to these advances has been a more sophisticated treatment of both the ISM and the galactic environment. For example, @KO09a found, using numerical simulations of the ISM and a cooling function allowing multiple phases, star formation rates and proportions between self-gravitating and diffuse gas similar to the observations of @BR06 and @Ler08 provided that turbulent driving is included; for non-turbulent models, the proportion of self-gravitating gas was found to be much too high.
A thermal/dynamical equilibrium model for ${\Sigma_{\rm SFR}}$
==============================================================
Motivated by recent observations as well as simulations and earlier theory, @2010ApJ...OML (hereafter OML) have developed a simple model for star formation regulation in multiphase, turbulent ISM disks. In essence, the OML model combines three basic principles: (1) the diffuse (atomic) component of the ISM is in approximate thermal equilibrium, with a density (and pressure) proportional to the heating rate; (2) the diffuse component of the ISM is in approximate dynamical equilibrium, with the pressure at any height above the galactic midplane given by the weight of the overlying gas; (3) UV from young stars provides most of the heating for the atomic component of the ISM, with star formation taking place only within the gravitationally-bound component of the ISM. These principles have been individually established and extensively studied (over several decades) in the astrophysical literature. @1969ApJ...155L.149F combined (1) and (2) to conclude that the diffuse atomic gas in the local Milky Way must consist of a two-phase cloud/intercloud medium. In this and subsequent treatments of thermal and dynamical equilibrium, the heating rate has generally been treated as an independent (empirical) parameter. But, by including (3) together with (1) and (2), OML obtained a closed system representing a local patch of a disk galaxy. For this closed system, the partition of gas into phases and the star formation rate are obtained self-consistently.
In the OML model, the (simplified) ISM is treated as having two components, one consisting of diffuse gas (including both high-density cold atomic cloudlets and a low-density warm atomic intercloud medium), and the other consisting of gravitationally-bound clouds (GBCs). Although hot gas is also present in the ISM, it is a tiny fraction of the mass, and fills ${\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}20\%$ of the volume [@2001ASPC..231..294H] (OML describe how to correct for this effect). For galaxies with normal metallicity, the GBCs would represent giant molecular clouds, including their atomic shielding layers. Averaged over $\sim {{\;\rm kpc}}$ scales (which may contain many or few individual GBCs), the total surface density of the GBC component is ${\Sigma_{\rm GBC}}$, and the total surface density of the diffuse component is ${\Sigma_{\rm diff}}$.
The diffuse component is assumed to be in vertical dynamical equilibrium (as has been verified by numerical simulations; e.g. @PO07 [@KO09b]), with the vertical gravity (from the diffuse gas, the GBC component, the stellar disk, and the dark matter halo) balanced by the difference between midplane and external values of thermal pressure ${P_{\rm th}}$, turbulent pressure $\rho v_z^2$, and magnetic stresses $(8\pi)^{-1}(B^2 - 2 B_z^2)$. Because cooling times are short compared to other timescales, the diffuse gas is assumed to be in thermal equilibrium, with the additional provision that both warm and cold phases are present. This allows a range of pressures between $P_{\rm min, cold}$ and $P_{\rm max, warm}$; for the model of OML, it is assumed that the pressure is equal to the geometric mean of these limits, $P_{\rm two-phase}\equiv (P_{\rm min, cold} P_{\rm max,
warm})^{1/2}$. For atomic gas, heating is generally dominated by the UV and cooling by collisionally-excited lines [@Wol03], which yields $P_{\rm two-phase} \propto J_{\rm UV}$. (Note that other heating – e.g. cosmic rays and shocks – can be more important for very dense, shielded cores and very hot gas, respectively.) Finally, the OML model assumes that the rate of star formation is proportional to the total surface density ${\Sigma_{\rm GBC}}$ of gas in the GBC component, ${\Sigma_{\rm SFR}}={\Sigma_{\rm GBC}}/{t_{\rm SF}}=(\Sigma -{\Sigma_{\rm diff}})/{t_{\rm SF}}$.
Vertical dynamical equilibrium within the diffuse layer is expressed as $P_{\rm tot} \equiv \alpha P_{\rm th} = \Sigma_{\rm diff} \langle g_z \rangle /2$, where the mean vertical gravity is $$\langle g_z \rangle
\approx \pi G ({\Sigma_{\rm diff}}+ 2{\Sigma_{\rm GBC}}) +2 (2 G \rho_{\rm sd})^{1/2} \sigma_z;
\label{gz-eq}$$ $\rho_{\rm sd}$ is the midplane density of stars plus dark matter, $\sigma_z$ is the total vertical velocity dispersion of the diffuse gas, and the total pressure is larger than the thermal pressure by a factor $\alpha$ (see below). The GBC component contributes more strongly (per unit mass) to the gravity because its scale height is smaller than that of the diffuse gas.
If $n^2\Lambda(T)$ is the cooling rate per unit volume and $n\Gamma$ is the heating rate per unit volume, then the two-phase pressure is given by $$\begin{aligned}
\frac{P_{\rm two-phase}}{k}
&\equiv& \left(n_{\rm min, cold} T_{\rm min, cold} n_{\rm max,warm} T_{\rm max, warm}
\right)^{1/2}\cr
&=&\Gamma
\frac{\left( T_{\rm min, cold} T_{\rm max, warm}\right)^{1/2} }
{\left[\Lambda(T_{\rm min, cold })\Lambda(T_{\rm max,warm})\right]^{1/2}},\end{aligned}$$ where we have used the equilibrium condition $\Gamma = n \Lambda$ for both phases. Cooling of the cold atomic medium is dominated by metals (in particular, C II) so that $\Lambda \propto
Z_{\rm gas}$, while heating is dominated by the photoelectric effect with $\Gamma \propto Z_{\rm dust} J_{\rm UV}$; since $T_{\rm min, cold
}$ and $T_{\rm max,warm}$ are relatively independent of the heating rate [@Wol95], this yields $P_{\rm two-phase} \propto J_{\rm UV}
$ if $Z_{\rm dust}/Z_{\rm gas} = const$. The terms $Z_{\rm gas}$ and $Z_{\rm dust}$ represent the ratios of metals and dust to hydrogen, respectively. The mean UV intensity is affected by radiative transfer through the diffuse gas, but for modest optical depth in the diffuse gas the relation $J_{\rm UV} \propto {\Sigma_{\rm SFR}}$ is expected to hold. In addition, a larger fraction of the UV escapes from GBCs if $Z_d$ is very sub-Solar, which increases $J_{\rm UV}$ for a given ${\Sigma_{\rm SFR}}$ (this effect is quite uncertain, but might increase $J_{\rm UV}$ by a factor $\sim 2$). In thermal equilibrium with ${P_{\rm th}}\sim P_{\rm two-phase}$, the midplane pressure is therefore expected to vary roughly as ${P_{\rm th}}\propto {\Sigma_{\rm SFR}}$, with a somewhat larger coefficient for very low-metallicity regions.
Combining the thermal equilibrium relation ${P_{\rm th}}= P_{\rm th,0}
{\Sigma_{\rm SFR}}/\Sigma_{\rm SFR, 0}$ (normalized using the Solar neighborhood thermal pressure $P_{\rm th,0}$ and star formation rate $\Sigma_{\rm SFR, 0}$) with the dynamical equilibrium relation ${P_{\rm th}}= \Sigma_{\rm diff} \langle g_z \rangle /(2\alpha)$ and the star formation relation ${\Sigma_{\rm SFR}}={\Sigma_{\rm GBC}}/{t_{\rm SF}}$, we obtain $$\begin{aligned}
\frac{{\Sigma_{\rm GBC}}}{{\Sigma_{\rm diff}}}&=&\frac{\langle g_z \rangle}{g_*}
\propto \pi G ({\Sigma_{\rm diff}}+ 2{\Sigma_{\rm GBC}}) +2 (2 G \rho_{\rm sd})^{1/2} \sigma_z.
\label{ratio-eq}\end{aligned}$$ Here, $g_* = 2\alpha P_{\rm th,0}/(\Sigma_{\rm SFR, 0} {t_{\rm SF}})$; for fiducial parameters, this acceleration is $g_* \sim {{\;\rm\,pc}}\ {\rm Myr}^{-2}$.
It is interesting to compare outer and inner disks. In outer disks (similar to the Solar neighborhood and beyond, in galaxies like the Milky-Way), diffuse gas dominates the total so that ${\Sigma_{\rm GBC}}\ll
{\Sigma_{\rm diff}}\approx \Sigma$; in addition, the term depending on $\rho_{\rm sd}$ dominates the gravity $g_z$. In this regime, the relation ${\Sigma_{\rm SFR}}\propto {\Sigma_{\rm GBC}}\propto \Sigma \sqrt{\rho_{\rm sd}}$ is therefore expected to hold. Physically, this regime may be thought of as the result of star formation increasing until the heating it provides is sufficient to balance cooling at the (dynamically-imposed) midplane pressure. If there is too little gas in the GBC component, the star formation rate would be extremely low, and the UV field would be very weak. A very low heating rate could not maintain a warm medium at the pressure imposed by the local gravitational field, so that (a portion of the) warm gas would condense out and become cold clouds. These cold clouds would collect to create more GBCs, which would then initiate star formation, raising the local UV radiation field until heating balances cooling. Given the low gravity and pressure of outer disks, cooling rates are moderate, and relatively low levels of star formation are needed to produce enough UV that heating balances cooling.
For outer disks where the stars and dark matter dominate gravity, the vertical oscillation time is $t_{\rm osc}=\pi^{1/2}/(G\rho_{\rm
sd})^{1/2}$; a dense cloud settles to the midplane in $\sim t_{\rm osc}/4$. In this regime, the conversion time from gas to stars, $t_{\rm con}\equiv \Sigma/{\Sigma_{\rm SFR}}$, is given by $$t_{\rm con}= t_{\rm osc}\frac{\sigma_z P_{\rm th, 0} }
{(2 \pi)^{1/2} \langle v_{\rm th}^2\rangle \Sigma_{\rm SFR,0}},$$ where $\langle v_{\rm th}^2 \rangle \equiv \tilde f_w c_w^2\approx c_w^2
M_{\rm diff, warm}/M_{\rm diff, total}$ is the mean thermal dispersion in the diffuse medium (here $c_w\sim 8 {{\;\rm km\; s^{-1}}}$ is the thermal speed in the warm ISM). Using $P_{\rm th, 0}\sim \langle v_{\rm th}^2 \rangle
P_{\rm gas, 0}/\sigma_z^2$ and defining a star formation energy conversion efficiency $\varepsilon_{\rm rad} \equiv 4 \pi J_{\rm rad,
0}/(c^2 \Sigma_{\rm SFR, 0})$ for $P_{\rm rad, 0} = 4 \pi J_{\rm rad,
0}/(3c)$, $$t_{\rm con}= t_{\rm osc} \frac{c }{3(2 \pi)^{1/2} \sigma_z }
\frac{P_{\rm gas, 0} }{P_{\rm rad, 0}}
\varepsilon_{\rm rad}.$$ That is, the gas conversion time (or depletion time) is set by the time for gas to settle to the midplane, scaled by factors for the ratio of gas-to-radiation pressure in the Solar neighborhood, the mass-to-energy conversion efficiency, and $c/\sigma_z$.
In inner disks, unlike outer disks, we have $ {\Sigma_{\rm diff}}\ll {\Sigma_{\rm GBC}}\approx \Sigma$, so that ${\Sigma_{\rm SFR}}\propto \Sigma$. In inner disks, it is straightforward to show that there is an upper limit on the diffuse gas surface density ${\Sigma_{\rm diff}}$. Physically, the reason for this limit is that the diffuse-gas cooling rate per particle increases with higher density and pressure in the inner parts of disks at least as $n \Lambda\propto
{\Sigma_{\rm diff}}{\Sigma_{\rm GBC}}$ (since $n \Lambda \propto {\Sigma_{\rm diff}}/H \propto
{\Sigma_{\rm diff}}g_z /\sigma_z^2 \propto {\Sigma_{\rm diff}}{\Sigma_{\rm GBC}}[1 + g_{\rm
sd}/g_{\rm GBC}]/\sigma_z^2$), whereas the heating rate per particle varies as $\Gamma \propto {\Sigma_{\rm SFR}}\propto {\Sigma_{\rm GBC}}$. Thus, cooling will exceed heating (causing mass to drop out of the diffuse component) unless ${\Sigma_{\rm diff}}$ is sufficiently low. Enhanced cooling and mass “dropout” is likely responsible at least in part for the “saturation” of HI surface densities at ${\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}10 {{\;\rm\,M_\odot}}{{\;\rm\,pc}}^{-2}$ that has been observed in the inner parts of galaxies [@Bigiel08].
Based on the relations described above, the star formation law is expected to steepen from ${\Sigma_{\rm SFR}}\propto \Sigma$ in inner disks to ${\Sigma_{\rm SFR}}\propto
\Sigma \sqrt{\rho_{\rm sd}}$ in outer disks. A reduction of the specific star formation rate ${\Sigma_{\rm SFR}}/\Sigma$ is indeed observed in galaxies starting at $\Sigma {\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}10 {{\;\rm\,M_\odot}}{{\;\rm\,pc}}^{-2}$ [@Bigiel08; @Ler08]. For some galaxies, a further power-law relation $\rho_{\rm sd} \propto \Sigma^{2p}$ may hold such that ${\Sigma_{\rm SFR}}\propto \Sigma^{1+p}$ in outer disks, but this need not be the case in general – that is, integrated “Schmidt”-type relations need not hold.
In OML, the full solution for ${\Sigma_{\rm SFR}}$ is obtained as a function of $\Sigma$, $\rho_{\rm sd}$, and the parameters $\alpha
\equiv P_{\rm tot}/{P_{\rm th}}$ and $\tilde f_w \equiv \langle v_{\rm th}^2
\rangle/c_w^2$, under the assumptions of thermal and dynamical equilibrium described above. It is also shown that the theoretical solution for ${\Sigma_{\rm SFR}}$ agrees well overall with a sample of disk galaxies analyzed in @Ler08, with especially close correspondence for the large flocculent galaxies NGC 7331 and NGC 5055. Figure 1 shows an example of the comparison between the model and data, for NGC 5055.
![Comparison between annular averages of the data ([*squares*]{}) for NGC 5055 [@Ler08], and the thermal/dynamical equilibrium model ([*triangles*]{}) developed in OML. Both the star formation rates as a function of radius in the galaxy (panel d), and star formation rates as a function of gas and stellar density (panels a and b) agree with the model predictions. ](ostriker_fig1.eps){width="\hsize"}
Given the promising comparisons between the analytic theory and observations, it will be quite interesting to develop numerical simulations that fully test the assumptions and results of the thermal/dynamical equilibrium model. Encouragingly, the poster presented by C.-G. Kim at this meeting shows that initial numerical tests support the assumptions of thermal and dynamical equilibrium adopted in the analysis of OML. As discussed above, the OML theory contains parameters that must be set from either observations or detailed simulations. In the remainder of this contribution, we review what is known in this regard based on previous numerical work, and what measurements will be needed from future modeling efforts.
Numerical evaluation of parameters
==================================
From equations (\[gz-eq\]) and (\[ratio-eq\]), the star formation rate in outer-disk regions is expected to vary as ${\Sigma_{\rm SFR}}\propto \Sigma \sqrt{2 G \rho_{\rm sd}} \sigma_z/\alpha$, where $\alpha \equiv \sigma_z^2/v_{\rm th}^2$ and $\sigma_z^2=v_{\rm th}^2 + v_{\rm turb}^2 + (1/2)\Delta(v_A^2-2v_{A,z}^2)$, with $v_{\rm th}^2$, $ v_{\rm turb}^2$, and $v_A^2$ the (mass-weighted) mean thermal, turbulent, and Alfvén speeds in the diffuse gas (we now omit angle brackets denoting averaging). The coefficient $\sigma_z/\alpha$ can also be written as $v_{\rm th}^2/\sigma_z = c_w^2 \tilde f_w /\sigma_z$. Thus, the star formation rate is expected to depend on the total velocity dispersion $\sigma_z$ (or the ratio $\sigma_z/c_w$, where $c_w$ is fixed by atomic physics), and on the fraction of diffuse gas in the warm phase $\approx \tilde f_w$
The ratios $\sigma_z/c_w$ and $\tilde f_w \approx M_{\rm diff,
warm}/M_{\rm diff, total}$ depend on the details of gas dynamics in the diffuse ISM. Important effects include warm and cold phase exchange via thermal instability; turbulence (with the associated shock heating and adiabatic temperature changes, as well as turbulent mixing); conversion of diffuse gas to GBCs via midplane settling, self-gravity, and turbulence-induced cloudlet collisions; return of GBC gas to the diffuse phase by photodissociation and by “mechanical” destruction processes (including expanding HII regions, winds, SNe, and radiation pressure). Turbulence in the diffuse gas can be driven by stellar energetic inputs as well as spiral shocks, the magnetorotational instability, large-scale gravitational instabilities in the disk, and cosmic infall.
Numerical studies to understand the various effects involved are very much a work in progress, but some consensus is already beginning to emerge on a number of points:
- For a medium with a bistable cooling curve, the midplane thermal pressure tends to evolve, by exchange of mass between cold and warm components of the diffuse phase, such that the mean value is comparable to, or slightly below, $P_{\rm two-phase}$ [@PO05; @PO07]. Since out-of equilibrium effects depend on the heating time from shocks compared to the cooling time, the mean value of the thermal pressure, as well as the breadth of the pressure distribution, must in general be affected by the scale and the amplitude of turbulence (see ). Realistic numerical evaluations of the mean thermal pressure (for a given radiative heating rate) therefore will require numerical simulations in which the vertical box size is comparable to the true scale height of the diffuse ISM, and in which the turbulent amplitude is $\sim 5-10{{\;\rm km\; s^{-1}}}$.
- Magnetic fields in differentially-rotating multiphase disks are amplified by the magnetorotational instability until the magnetic pressure becomes comparable to the thermal gas pressure, with $B_z^2 \ll B^2$ [@PO05; @PO07; @2009ApJ...696...96W]. Supernova-driven turbulence also contributes to amplifying the magnetic field .
- The energy input from supernovae yield ISM velocity dispersions $\sim 5-10 {{\;\rm km\; s^{-1}}}$ for models with a wide range of supernova driving rates and disk properties (e.g. ). These values are comparable to those observed in the HI gas. Simulations have also shown that the turbulent amplitudes decrease at smaller scales and for higher densities. With this range of turbulent velocity dispersions, the turbulent pressure in simulations of the diffuse ISM is comparable to the thermal pressure.
- The interaction between self-gravity and rotational shear also drives turbulence at significant levels (${\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}10 {{\;\rm km\; s^{-1}}}$) in galactic disks [@KO01; @KO07; @2002ApJ...577..197W; @2008ApJ...684..978S; @2009ApJ...700..358T; @2009MNRAS.392..294A; @2010ApJ...719.1230A; @2010arXiv1007.2566B]. However, the turbulent power is much larger at the large ($\sim {{\;\rm kpc}}$) scales that dominate the swing amplifier than at scales below the disk scale height, and in-plane velocities (which do not contribute to vertical support of the disk) are much larger than vertical velocities. Thus, turbulence driven by instabilities on large scales is likely of limited importance in regulating the effective midplane pressure (for a given local gas surface density $\Sigma$), and hence the star formation rate. (Gravitational instabilities would, however, enhance $\Sigma$ and thus ${\Sigma_{\rm SFR}}$ locally.) Flapping associated with non-steady spiral shocks also drives turbulence in the diffuse ISM [@2006ApJ...649L..13K; @2010arXiv1006.4691K], but again, vertical motions are small compared to horizontal motions.
Although numerical results have shown that the total turbulent velocity dispersion $\sigma_z$ is relatively insensitive to the disk properties and the supernova driving rate (consistent with observations), it is much less certain how the warm fraction, or $v_{\rm th}^2=\tilde f_w c_w^2\approx c_w^2 M_{\rm diff, warm}/M_{\rm
diff, total}$, depends on disk conditions and/or the star formation rate. Assessing this dependence, including a full exploration of parameter space, is an important task for future numerical studies. The fraction of diffuse atomic gas in different phases is not well known empirically, either, although observations of C II with [*Herschel*]{} potentially afford a means to separate cold and warm components of the atomic medium (which both contribute to 21 cm emission).
Finally, it remains important to understand more fully how spiral structure develops, and in particular, whether it is possible to characterize in a simple way the fraction of gas in a given annulus that is found in “arm” vs. “interarm” conditions, and what the compression factor is for the gas surface density. Numerical simulations have begun to marry spiral structure with an increasingly realistic treatment of the ISM (including multiple phases, turbulence, and magnetic fields); much more, however, remains to be done on this front. It also remains to be determined how well models like that of OML apply locally for galaxies with strong spiral structure. More generally, it is important to assess which equilibria (thermal, dynamical, star formation) still apply locally even in galaxies with large-scale transient structure in the ISM (due to spiral arms, tidal interactions, mergers, cosmic inflows, etc.).
Conclusion
==========
Gas is the raw material for star formation, but the detailed state of the ISM, which depends in turn on the internal galactic environment, determines the rate at which this material is processed to create new stars. Recent observations have begun to explore the correlation between gas content and star formation at increasingly high spatial resolution, revealing changes in star formation “laws” between inner and outer disks; other environmental dependences of star formation have also been explored, including intriguing correlations between molecular and stellar content of galactic disks.
Although the simplest recipes for star formation (such as a rate that depends inversely on the free-fall time at the mean ISM density) have difficulty matching the data, models that account for feedback and the multiphase character of the ISM are more successful. In particular, recent work suggests that the empirical correlation between molecular content and estimated midplane pressure can be understood as reflecting a state of simultaneous thermal and dynamical equilibrium in the diffuse ISM. The thermal/dynamical equilibrium model of OML develops the idea that UV from OB stars provides a feedback loop that regulates the star formation rate: the proportions of diffuse and self-gravitating gas adjust themselves so that the heating rate (proportional to the mass of self-gravitating gas) matches the cooling rate (proportional to the mass of diffuse gas and to the vertical gravitational field). The model formulated in OML is promising in terms of explaining star-forming behavior in observed systems. With numerical simulations, it will be possible to appraise – and potentially revise – the simplifying assumptions and parameterizations adopted by this equilibrium model. Time-dependent simulations will also lead to a much clearer understanding of how GBCs form and disperse, and how their properties and the formation/destruction timescales relate to galactic environment. This will aid in defining limits for applying equilibrium relations, while also pointing the way towards non-equilibrium theories of star formation.
: This work was supported by grant AST-0908185 from the National Science Foundation, and by a fellowship from the John Simon Guggenheim Foundation. The author thanks the referee for a helpful report.
[28]{} natexlab\#1[\#1]{}
, O., [Lake]{}, G., [Teyssier]{}, R., [Moore]{}, B., [Mayer]{}, L., & [Romeo]{}, A. B. 2009, [*MNRAS*]{}, 392, 294
, E. & [Hennebelle]{}, P. 2005, [*A&A*]{}, 433, 1
—. 2010, [*A&A*]{}, 511, A76+
Aumer, M., Burkert, A., Johansson, P. H., & Genzel, R. 2010, [*ApJ*]{}, 719, 1230
Bournaud, F., Elmegreen, B. G., Teyssier, R., Block, D. L., & Puerari, I. 2010, arXiv:1007.2566
, F., [Leroy]{}, A., [Walter]{}, F., [Brinks]{}, E., [de Blok]{}, W. J. G., [Madore]{}, B., & [Thornley]{}, M. D. 2008, [*AJ*]{}, 136, 2846
, L. & [Rosolowsky]{}, E. 2006, [*ApJ*]{}, 650, 933
, M. A. & [Breitschwerdt]{}, D. 2005, [*A&A*]{}, 436, 585
, S., [Bell]{}, E., & [Burkert]{}, A. 2006, [*ApJ*]{}, 638, 797
Field, G. B., Goldsmith, D. W., & Habing, H. J. 1969, [*ApJL*]{}, 155, L149
, A., [Luis]{}, L., & [Kim]{}, J. 2009, [*ApJ*]{}, 693, 656
, A., [V[á]{}zquez-Semadeni]{}, E., & [Kim]{}, J. 2005, [*ApJ*]{}, 630, 911
Heiles, C. 2001, Tetons 4: Galactic Structure, Stars and the Interstellar Medium, 231, 294
, P. & [Audit]{}, E. 2007, [*A&A*]{}, 465, 431
, M. K. R. & [Mac Low]{}, M. 2006, [*ApJ*]{}, 653, 1266
, M. R., [Mac Low]{}, M., & [Bryan]{}, G. L. 2009, [*ApJ*]{}, 704, 137
Kim, C.-G., Kim, W.-T., & Ostriker, E. C. 2006, [*ApJL*]{}, 649, L13
, C., [Kim]{}, W., & [Ostriker]{}, E. C. 2010, ArXiv e-prints
, J. 2004, Journal of Korean Astronomical Society, 37, 237
, W. & [Ostriker]{}, E. C. 2007, [*ApJ*]{}, 660, 1232
, W.-T. & [Ostriker]{}, E. C. 2001, [*ApJ*]{}, 559, 70
, H. & [Ostriker]{}, E. C. 2009, [*ApJ*]{}, 693, 1316
—. 2009, [*ApJ*]{}, 693, 1346
, A. K., [Walter]{}, F., [Brinks]{}, E., [Bigiel]{}, F., [de Blok]{}, W. J. G., [Madore]{}, B., & [Thornley]{}, M. D. 2008, [*AJ*]{}, 136, 2782
, M., [Balsara]{}, D. S., [Kim]{}, J., & [de Avillez]{}, M. A. 2005, [*ApJ*]{}, 626, 864
McKee, C. F., & Ostriker, E. C. 2007, [*ARAA*]{}, 45, 565
Ostriker, E. C., McKee, C. F., & Leroy, A. K. 2010, [*ApJ*]{}, 721, 975 (OML)
, R. A. & [Ostriker]{}, E. C. 2005, [*ApJ*]{}, 629, 849
—. 2007, [*ApJ*]{}, 663, 183
, R. & [Ostriker]{}, E. C. 2008, [*ApJ*]{}, 684, 978
, E. J. & [Tan]{}, J. C. 2009, [*ApJ*]{}, 700, 358
, K., [Meurer]{}, G., & [Norman]{}, C. A. 2002, [*ApJ*]{}, 577, 197
, P. & [Abel]{}, T. 2009, [*ApJ*]{}, 696, 96
, M. G., [Hollenbach]{}, D., [McKee]{}, C. F., [Tielens]{}, A. G. G. M., & [Bakes]{}, E. L. O. 1995, [*ApJ*]{}, 443, 152
, M. G., [McKee]{}, C. F., [Hollenbach]{}, D., & [Tielens]{}, A. G. G. M. 2003, [*ApJ*]{}, 587, 278
|
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abstract: 'Granular solid hydrodynamics, constructed to describe quasi-elastic and plastic motion of granular solid, is shown also capable of accounting for the rheology of granular dense flow. This makes it a unified, though still qualitative, hydrodynamic description, enabling one to tackle fluidization and jamming, the hysteretic transition between elasto-plastic motion and uniform dense flow.'
author:
- Stefan Mahle
- Yimin Jiang
- Mario Liu
title: |
Granular Solid Hydrodynamics:\
Dense Flow, Fluidization and Jamming
---
A potentially catastrophic feature of granular media is their variable capability to sustain external stresses. As the mechanical stability of any structure relies on this capability, it is important to have a thorough understanding when and why it weakens, and how it recovers. The transition from a solid-like response to a liquid-like one is [*fluidization*]{}; and [*jamming*]{} denotes increasingly often the reverse change. After a mud slide starts, after it fluidizes, jamming is when it stops again – at what stress and density, and whether the village downhill will be reached, are then questions of considerable interests.
The two limiting states of the transition may be referred to as [*granular solid*]{} and [*uniform dense flow*]{}. In the first, the grains are deformed and at rest, with all energy being elastic. In the second, they jiggle, rattle, move macroscopic distances, and a significant portion of the energy is kinetic. The transition from the dominance of one energy to the other may be gradual or abrupt, and has two possible intermediate states: the uniform, ideally plastic [*critical state*]{} [@CSSM; @PSM], or [*shear band*]{}, the nonuniform path. A unified theory of these limiting and intermediate states does not as yet exist, though [gsh]{} (for granular solid hydrodynamics) seems close. It was originally constructed to account for granular solid and its elasto-plastic motion. Here, we demonstrate its applicability to dense flow: [gsh]{} displays broad agreement with three existing theories on various aspects of dense flow, and accounts for the data of Savage and Sayed [@Savage].
The first of the three existing theories is by Pouliquen et al. [@pouliquen1]. Starting from the insight that granular rheology in dense flows is controlled by a dimensionless parameter $\sim\dot\gamma/\sqrt P$ (where $\dot\gamma$ is the shear rate, $P$ the pressure), they distilled two locally applicable constitutive relations from experiments and simulations, for the density $\rho$ and the friction angle $\sigma_s/P$ (with $\sigma_s$ the shear stress), $$\begin{aligned}
\label{sb1} 1-\rho_{r}\sim \dot\gamma/\sqrt P, \quad
\frac{\sigma_s}{P}=\frac{\mu_1+\mu_2\,(\dot\gamma/\sqrt
P)^n}{1+(\dot\gamma/\sqrt P)^n},\end{aligned}$$ where, with $\rho_{cp}$ the closed-packed density, $\rho_r\equiv\rho/\rho_{cp}$ is the relative one. $\mu_1,\mu_2$ denote, respectively, the friction angle for $\dot\gamma\to0,\infty$. The authors took $n=1$, though the difference to $n=2$ is subtle, as both describe a gentle change from $\mu_1$ to $\mu_2$ with $\dot\gamma$.
Earlier, Boquet et al. [@Bocquet] developed a continuum theory to account for their experiment. Starting from the results of the kinetic theory for inelastic hard spheres, they modified the density dependence of the pressure $P$, viscosity $\eta$ and relaxation rate $\gamma$ to accommodate the higher density in their system. They employ the Cauchy stress $\sigma_{ij}$, and a balance equation for the granular temperature $T_G$: $$\begin{aligned}
\label{sb2}
\sigma_{ij}=P-\eta v^0_{ij},\quad \partial
T_G/\partial t\sim \eta v_s^2-\gamma \, T_G,
\quad\text{with}
\\\label{sb3}
P\sim \frac{T_G}{1-\rho_r}, \quad \eta\sim
\frac{\sqrt{T_G}}{(1-\rho_r)^\beta},\quad \gamma\sim
\frac{\sqrt{T_G}}{(1-\rho_r)^\alpha},\end{aligned}$$ where $\alpha=1$, $\beta$ between 1 and 2.5. $v_{ij}\equiv\frac12(\nabla_iv_j+\nabla_jv_i)$ denotes the strain rate, with $v^0_{ij}$ its traceless part, and $v_s^2\equiv{v^0_{ij}v^0_{ij}}$ the scalarized shear rate. (The notation $^0$ and $_s$ are used also for other tensors below, such as the strain and stress.) Solving both equations for a Couette cell, the solution was found to agree well with their data. This theory II does not consider any elastic contributions.
Considering shallow flows on an inclined plane and rotating drums, Aranson and Tsimring identified the hysteresis of transition, or the delay between jamming and fluidization, as a key feature of granular behavior [@Aranson2]. Their theory III treats the Cauchy stress as the sum of two parts, a solid-like, possibly elastic contribution $\hat\varrho\sigma^s_{ij}$, and a rate-dependent fluid one. A crucial variable is an order parameter $\hat\varrho$ that is 1 for granular solid, and 0 for dense flow. The authors take the friction angle $\phi$, differently than above, as the ratio of the solid stress components, and postulate a free energy $f(\hat\varrho)$ such that granular solid, $\hat\varrho=1$, is unstable for large shear stresses, $\phi>\phi_1$; while dense flow, $\hat\varrho=0$, is unstable for small ones, $\phi<\phi_0$. Both are stable in the intermediate region, $\phi_1>\phi>\phi_0$. This theory does not consider variations in the density $\rho$, or in $T_G$, and takes $\sigma^s_{ij}$ as an input from some other theory. But its success provides a pivotal insight: The viability, even appropriateness, of using a partially bistable energy to account for the hysteresis.
[gsh]{} starts from the basic fact that grains with enduring contacts are elastically deformed. Its essential idea is that this deformation is slowly lost when grains jiggle, as they briefly loose contact with one another. Granular solid’s complex elasto-plastic behavior was shown to be a result of this simple physics, assuming the dominance of the elastic energy. Kinetic energy, or granular heat, is what underlies the behavior of granular gas. So it seems obvious that the behavior of dense flow results when both energies are comparable, when the contribution to the stress from granular temperature becomes equally important as that from deformation.
[gsh]{} was first employed to calculate static stress distribution for various geometries, including sand piles, silos, and point load, achieving results in agreement with observation [@ge1]. It was then employed to consider slowly strained granular solid, and found to yield response envelopes similar to those from modern hypoplastic theory [@granL3]. Recently, the critical state – generally considered a hallmark of granular behavior – was identified as a steady-state, elastic solution of [gsh]{} [@critState]: Although given as a simple analytic expression, the solution realistically renders the critical state and the approach to it, including dilatancy and contractancy. Finally, the velocity of elastic waves were calculated as a function of the stress [@elaWave], and found to agree well with experiments [@Jia2009].
[gsh]{} consists of conservation laws for the energy $w$, mass $\rho$, and momentum $\rho
v_i$, an evolution equation for the elastic strain $u_{ij}$, and balance equations for two entropy densities, $s$ and $s_g$. Two entropies are necessary, because granular media display a [*two-stage irreversibility*]{}: Macroscopic energy, kinetic and elastic, dissipates into mesoscopic, inter-granular degrees of freedom, mainly granular jiggling and the collision-induced, fluctuating elastic deformation. After a characteristic time, the energy degrades further into microscopic, inner-granular degrees of freedom, especially phonons. The granular and the true entropy, $s_g, s$, account respectively for the energy of the meso- and microscopic degrees of freedom. The elastic strain $u_{ij}$ is the portion of the total strain $\varepsilon_{ij}$ that deforms the grains and leads to reversible storage of elastic energy. The rest-frame energy density $w_0$ is a function of $s_g,s,\rho,u_{ij}$ (though we shall neglect $s$, as we are not interested in effects such as thermal expansion at present). The conjugate variables are: Granular temperature $T_g\equiv{\partial
w_0}/{\partial s_g}$, chemical potential $\mu\equiv{\partial w_0}/{\partial \rho}$, elastic stress, $\pi_{ij}\equiv -{\partial w_0}/{\partial
u_{ij}}$, and the gaseous pressure $P_T\equiv\rho^2\left.{\partial(w_0/\rho)}
/{\partial\rho}\right|_{s_g/\rho,\dots}$. The elastic stress $\pi_{ij}$ derives from granular deformation, while $P_T$ is generated by granular temperature – similar to the temperature generated pressure in a gas. All conjugate variables: $(T_g, \mu,
\pi_{ij},P_T)$ are given once $w_0$ is.
In [gsh]{}, the Cauchy stress $\sigma_{ij}$ \[given by momentum conservation, $\partial(\rho v_i)/\partial
t+\nabla_j(\sigma_{ij}+\rho v_iv_j)=0$\] and the balance equation for $s_g$ are given as $$\begin{aligned}
\label{sb4}
\sigma_{ij}=(1-\alpha)\pi_{ij}+P_T\delta_{ij} -\eta_g
v^0_{ij},
\\\label{sb5} \partial s_g/\partial t=
(\eta_gv_s^2 -\gamma T_g^2)/T_g.\end{aligned}$$ Although a result of general principles, the expression for $\sigma_{ij}$ is, remarkably, a simple sum of the elastic stress, the gaseous pressure, and the viscous stress, with $\eta_g$ the shear viscosity. (Compressional flow is usually negligible. If not, one needs to include the bulk viscosity.) For elasto-plastic motion, only $(1-\alpha)\pi_{ij}$ is important; granular gas is well accounted for by $P_T\delta_{ij} -\eta_g v^0_{ij}$; dense flow needs all three terms. $\alpha\approx0.8$ is a softening coefficient that remains constant for all shear rates considered in the present context. (It becomes smaller only for ultra low shear rates, in ratcheting or elastic waves). Note the similarity of Eq (\[sb4\]) to the above cited theories, with the difference that theory II ignores $\pi_{ij}$, and theory III takes it as given. In Eq (\[sb5\]), $\gamma$ is the relaxation rate of $s_g$, accounting for the inelastic collisions that occur when grains jiggle. The positive term $\eta_g v_s^2\equiv\eta_g v^0_{ij}v^0_{ij}$ describes how grains, being sheared past one another, start to jiggle in the process, leading to an increase of $s_g$. From a more general point of view, this term describes how the kinetic energy dissipates into granular heat. In the stationary limit, for $\partial
s_g/\partial t=0$, we have $T_g=v_s\sqrt{\eta_g/\gamma}$. (Only a uniform $T_g$ is considered – more terms exist otherwise.) See [@granR2] for derivation and detailed explanation.
Eqs (\[sb4\],\[sb5\]) hold for the given set of variables, independent of the granular material, or the specific form of $w_0$. Material-specific properties are encoded in $w_0(\rho,u_{ij},s_g)$, also the transport coefficients: $\eta_g,\gamma$. We obtain them from qualitative consideration, also comparison to experiments and existing theories. (More puristically, one would of course like to obtain them from simulation or microscopic calculations.) For dry sand and glass beads, a simple energy expression, the sum of the elastic energy $w_1(u_{ij},\rho)$ and granular heat $w_2(s_g,\rho)$: $w_0=w_1+w_2$, has turned out to be quite adequate as a first approximation. Then $\pi_{ij}=-{\partial
w_1}/{\partial u_{ij}}$, $P_T\approx\rho^2{\partial(w_2/\rho)} /{\partial\rho}$ (if one assumes $u_{ij}\ll1$, see [@granR2]), which is why stress and energy contributions are simply linked: If $w_1$ dominates, only $\pi_{ij}$ is important; while $P_T$ hinges on a sufficiently large $w_2$. We have $w_1={\cal B}(\rho) \sqrt\Delta\,
[\Delta^2+u_s^2/\xi]$, with $\Delta\equiv
-u_{\ell\ell}$, $u_s^2=u^0_{ij}u^0_{ij}$. For a granular system at rest, $w_1$ is the only energy, and the elastic stress is the total stress. The mentioned calculation of static stress distributions was carried out using $w_1$ [@granR1]. Granular heat $w_2$ is the lowest order expansion in $s_g$, $$\label{sb6}
w_2=\frac{s_g^2}{2\rho b},\quad T_g=\frac{s_g}{\rho
b}, \quad P_T=-\frac{T_g^2\rho^2}2\frac{\partial
b(\rho)}{\partial\rho}.$$ The linear term vanishes because granular jiggling dissipates and decreases toward zero, implying $w_2(s_g)$ is minimal for $s_g=0$. Expanding also the transport coefficients, $$\label{sb7}
\eta_g=\eta_0+\eta_1 T_g,\,\, \gamma=\gamma_0+\gamma_1
T_g, \,\, T_g=v_s\sqrt{\eta_1/\gamma_1},$$ we take $\eta_0\ll\eta_1 T_0$, $\gamma_0\ll\gamma_1
T_g$, as is appropriate for any $T_g$ typical of elasto-plastic motion and dense flow, see [@granR2]. Then the last of Eqs (\[sb7\]) holds, for $\partial s_g/\partial t=0$.
In theory II, $T_{\rm G}$ is the energy per degree of freedom, $w_2\sim T_{\rm G}$, while $w_2\sim s_g^2\sim
T_g^2$. We therefore identify $T_g\sim\sqrt{T_G}$, and note the perfect agreement between Eqs (\[sb2\],\[sb3\]) and Eqs (\[sb4\],\[sb5\],\[sb6\],\[sb7\]). This is important, because the $T_g$-dependence (in contrast to the density dependence discussed below) is rather fixed – it is the result of the kinetic theory on one hand, and the general consideration rendered above on the other. (Note taking $w_2\sim T_g$ would disregard the fact that $T_g$ dissipates, and $w_2$ is minimal for $s_g,T_g=0$ in an adiabatic system. It is more appropriate for ideal gas than the granular one.)
To consider the density dependence, we focus on $1-\rho_r$ $\equiv1-\rho/\rho_{cp}$, which represents a stronger dependency than $\rho$ if the sand is dense, $\rho_r\approx1$. We take $$\label{sb8}
P_T=\frac{ab\rho_{cp}\rho_r^2\,T_g^2}{2(1-\rho_r)},\,\,
\eta_1=\frac{h_1\rho_{cp}}{({1-\rho_r})^{\beta}},\,
\gamma_1=\frac{g_1\rho_{cp}}{({1-\rho_r})^\alpha},$$ with $\alpha=\frac12$, $\beta=\frac32$ to fit the experimental results of [@Savage] with respect to the polystyrene beads. The expression for $P_T$ derives from $b=b_0(1-\rho_r)^a$, cf Eq (\[sb6\]). Taking $b\sim\ln(1-\rho_r)$ would yield $P_T\sim
1/(1-\rho_r)$ exactly, as in Eqs (\[sb3\]), but also leads to a divergent $s_g$, see [@granR2]. Assuming $a\approx0.1$ approximates the result, yet avoids the problem. The coefficients $b_0,h_1,g_1$ are (material dependent) numbers. Combining Eq (\[sb4\],\[sb6\],\[sb7\],\[sb8\]) and denoting the shear rate as $\dot\gamma\equiv\nabla_xv_y$ (hence $\dot\gamma=\sqrt2\,v_s$ in simple-shear geometry), we arrive at the final expressions for the pressure $P\equiv\sigma_{\ell\ell}/3$ and the shear stress $\sigma_s\equiv
%\sqrt
({\sigma^0_{ij}\sigma^0_{ij}})^{1/2}$, $$\begin{aligned}
\label{sb9}
P=P_c+C_1\frac{\dot\gamma^2}{(1-\rho_r)^2},\quad%
\sigma_s=\Pi_c +C\frac{\dot\gamma^2}{(1-\rho_r)^2},\end{aligned}$$ where $C_1=\frac14ab\rho_{cp}\rho_r^2\,{h_1}/{g_1}$, $C=\frac12{\rho_{cp}}\sqrt{{h_1^3}/{2g_1}}$. $P_c,\Pi_c$ denote the rate-independent, elastic contributions, with $\Pi_c/P_c$ independent of $1-\rho_r$, see the explanation below.
The first of Eq (\[sb9\]) may be written as $1-\rho_{r}\sim
\dot\gamma/\sqrt{P-P_c}=\dot\gamma/\sqrt{P_T}$. It is the same as Eq (\[sb1\]) if $P_c$ is neglected – an understandable mismatch, because the consideration of theory I involves inertia and confining pressure, but neglects elasticity. The friction angle $\sigma_s/P$ is given by $\Pi_c/P_c$ for $\dot\gamma\to0$, and $C/C_1$ for $\dot\gamma\to\infty$, which we may respectively identify as $\mu_1,\mu_2$. The number $n$ of Eq (\[sb1\]) is 2 in [gsh]{}. As mentioned, the difference to $n=1$ is subtle for the friction angle – but less so if one look at the pressure or shear stress individually. That both grow with $\dot\gamma^2$ is in fact a behavior that already Bagnold observed [@Bagnold]. Finally, a note on volume versus pressure control: Yielding $P, \sigma_s$ for given $\dot\gamma, 1-\rho_r$, Eqs (\[sb9\]) are directly appropriate for experiments performed under constant volume. If $P$ is fixed, one uses the first calculate $\rho$, and rewrite the second as $\sigma_s-\Pi_c=(P-P_c)C/C_1$, with a coefficient $C/C_1$ that does not depend on $1-\rho_r$ – though still on $\rho$, a weaker function of $P$ and $\dot\gamma$.
![Comparison of [gsh]{} to the polystyrene data of Savage and Sayed [@Savage], with Pressure $P$, shear stress $\sigma_s$, and $\sigma_s/P$ given as functions of the shear rate $\dot\gamma$. The first two figures show the $\dot\gamma^2$-dependence, and the third the convergence onto the weakly density-dependent, high-rate limit $\mu_2=C/C_1$. Diamonds, squares and circles are the experimental points at the specified densities $\rho$. (Data for the largest $\rho$ are not used, because the authors believe they may be plagued by “finite-particle-size effects.") The curves render Eqs (\[sb9\]), with $h_1=3.1\cdot10^{-4}\sqrt{ab_0}$, $g_1=121.7\sqrt{ab_0}^{3}$, $a=0.1$, $\rho_{cp}=0.64\rho_{bulk}$.[]{data-label="fig1"}](1)
Returning to the exponent of $\gamma,\eta_1$, we note $\alpha+\beta=2$ if $(\sigma_s-\Pi_c)/(P-P_c)$ is to be independent of $1-\rho_r$; and $\beta-\alpha=1$, if $P_T\sim(1-\rho_r)^{-2}$. Together, they imply $\alpha=\frac12, \beta=\frac32$, as given above, see Fig \[fig1\]. However, for glass beads of the same experiment [@Savage], $P_T\sim\dot\gamma^2/(1-\rho_r)$, or $\beta=\alpha=1$ is more appropriate. In addition, the friction angle decreases for increasing $\dot\gamma$ here, implying $\Pi_c/P_c>C/C_1$, without contradicting any general principle.
Next we discuss the elastic contributions: $P_c,\Pi_c$. Applying a constant shear rate $v_s$ to an elastic body, the shear stress will monotonically increase – until the point of breakage. Sand is different and can maintain a constant stress, $\sigma^c_{ij}$. This is the famous critical state [@CSSM; @PSM] that has, for given density, a unique, rate-independent stress value. Employing [gsh]{}, this is easy to understand: Because elastic deformation $u_{ij}$ is slowly lost if the grains jiggle, and because grains indeed jiggle when forced to shear past one another, a shear rate $v_s$ not only increases $u_{ij}$, as in any elastic medium, but also decreases it. The critical state is the steady state in which both processes balance each other, such that the elastic deformation remains constant over time, in spite of a finite $v_s$. As shown in [@critState], the stationary solution $u^c_{ij}$ depends on the density, but not on $v_s$. The associated elastic stress $\pi^c_{ij}\equiv\pi_{ij}(u^c_{k\ell},\rho)
=\pi^c_{\ell\ell}\,\delta_{ij}/3
-\pi^c_s\,v^0_{ij}/v_s$, characterized by two scalars, $\pi^c_{\ell\ell}$ and $\pi^c_s$, is also independent of $v_s$. Its contributions in Eq (\[sb9\]) are: $P_c=\frac13(1-\alpha)\pi^c_{\ell\ell}$, $\Pi_c=(1-\alpha)\pi^c_s$. Although both $P_c,\Pi_c$ depend on $1-\rho_r$, the ratio $\Pi_c/P_c=\pi^c_s/\pi^c_{\ell\ell}$ does not.
We did not find any independent data on the critical state of polystyrene beads, though that from [@Savage] indicate $P_c\approx50$ Pa, $\sigma_c/P_c\approx0.25$, implying that the softer polystyrene beads have a ${\cal B}\approx10^5$ Pa, while the other coefficients retain their orders of magnitude as given in [@critState]. (Note: $\pi_{ij}\sim{\cal B}$, and ${\cal
B}\approx5\times10^9$ Pa for sand.)
At lower shear rates, say for $v_s\lesssim$ 10 s$^{-1}$, the rate-dependent terms of $\sigma_{ij}$ are quadratically small, $P_T\sim T_g^2\sim v_s^2$, $\eta_1T_g v^0_{ij}\sim v_s^2$, and may be neglected. This is the reason the total stress is given by the rate-independent critical state, $\sigma_{ij}=(1-\alpha)\pi^c_{ij}$, for a fairly broad range of shear rates, and why soil mechanic textbooks emphasize the rate-independence of granular behavior.
We note that fluidization, as considered above, is uniform and continuous, without anything resembling “failure" or “yield." Starting from a state of isotropic stress, a sheared granular system will approach the critical state, in the continuous way as calculated in [@critState]. The end state is, more generally, given by Eqs (\[sb9\]), though the difference to the critical state is evident only at higher shear rates. There is an alternative path that goes through an energetic instability, eg. the Coulomb yield contained in $w_1(u_{ij})$, see [@granR2], which sets in when the ratio $\pi_s/ \pi_{\ell\ell}$ becomes too large. This transition is discontinuous, non-uniform, and shear bands necessarily appear. We shall consider it in a forthcoming paper.
Jamming, the reverse transition – a drop of the shear rate $v_s$ from a finite value to zero at given stress – is necessarily discontinuous. In contrast to the authors of theory III, however, we do not believe this instability is marked by a lower bound of $\pi_s/
\pi_{\ell\ell}$, as elastic solutions are perfectly stable at isotropic stresses, $\pi_s=0$. Rather, jamming seems an instability that sets in when the density is too high to enable a shear flow $v_s$. Although $v_s$ is not a state variable, $T_g$ is, and we have $T_g\sim v_s$ \[cf. Eq (\[sb7\])\] for any processes slow enough for Eq (\[sb5\]) to have reached its stationary limit. The appropriate instability must therefore be in $w_2(s_g,\rho)={s_g^2}/{2\rho b}$. If we substitute $\hat b$ for $b$, we have $\hat P_T$ instead of $P_T$, $$\label{sb10}
%\!\!
\frac{\hat b}{b}=
\left[1+\frac{b_1}{1-\rho_r}\right],\quad \frac{\hat
P_T}{P_T}=1-\frac{(1-a)\,b_1}{a(1-\rho_r)}.$$ With $b_1$ small, we may neglect the correction term as long as $\rho_r$ is away from 1, and all results above remain valid. For $\rho$ equal to $$\label{sb11}
\rho_{jam}=\rho_{cp}(1-2b_1/a),$$ however, the convexity of $w_2$ with respect to $\rho$ is lost, and no finite value of $T_g\sim v_s$ is stable. $\rho_{jam}$ is obtained from the condition: $\partial^2 w_2/\partial\rho^2|_{s_g}=0$, or equivalently, from $\partial^2
f_2/\partial\rho^2|_{T_g}\sim
\frac\partial{\partial\rho}(\rho^2\frac\partial
{\partial\rho}b)|_{T_g}=0$, where $f_2\equiv
w_2-T_gs_g=-\rho bT_g^2/2$. Eq (\[sb11\]) is the result to lowest order in $b_1$ and $a$. Note $\rho<\rho_{jam}$ imples a lower bound for $v_s$ if the pressure is given instead of the density, as a smaller $v_s$ will imply a larger $\rho$, see the first of Eq (\[sb9\]). On a plane inclined by the angle $\varphi$, the friction angle is $\tan\varphi=\sigma_s/P$, with the angle of repose given by $\varphi_{r}=\varphi(\rho_{jam})$. Since the two terms $\sim\dot\gamma^2$ are negligible by then, the angle of repose is given by the critical angle at $\rho_{jam}$: $\tan\varphi_{r}=\Pi_c(\rho_{jam})/P_c(\rho_{jam})$. This is consistent with observation, because the critical angle is necessarily smaller than the angle at which Coulomb yield sets in. All these statements are independent of the specific form of $\hat b$, which may possibly prove inappropriate – though the case for an instability in $b(\rho)$ seems watertight.
Summary: Because [gsh]{} is capable of accounting for elasto-plastic motion, including the critical state, and also for dense flow, fluidization and jamming, we believe that this hydrodynamic theory, conventionally based on conservation laws and thermodynamics, is a viable candidate for a unified theory of granular media.
[99]{}
A. Schofield and P. Wroth. . McGraw-Hill, London, 1968.
G. Gudehus. [*Physical Soil Mechanics*]{}. Springer SPIN, 2010
S.B. Savage, M. Sayed. , 142:391, 1984.
P. Jop, Y. Forterre, O. Pouliquen. , 441:727, 2006. GDR MiDi. , E14(4):341, 2004. Y. Forterre, O. Pouliquen. , 40(1):1, 2008.
L. Bocquet, W. Losert, D. Schalk, T. C. Lubensky, and J. P. Gollub. , 65(1):011307, 2001.
I. S. Aranson and L. S. Tsimring. , 65:061303, 2002. , 78:641, 2006.
D. O. Krimer, M. Pfitzner, K. Bräuer, Y. Jiang, and M. Liu. , 74(6):061310, 2006.
Y. Jiang, M. Liu. , 99(10):105501, 2007.
S. Mahle, Y.M. Jiang, and M. Liu. arXiv:1006.5131v1 M. Mayer, M. Liu. , 82:042301, 2010.
Y. Khidas, X. Jia., 81:021303, 2010.
Y. Jiang and M. Liu. , 11:139, 2009. In D. Kolymbas and G. Viggiani, editors, [*Mechanics of Natural Solids*]{}, pages 27–46. Springer, 2009. Y. Jiang and M. Liu. , E22(3):255, 2007.
R.A.Bagnold. , 225(1160):49, 1954.
|
---
abstract: 'In the Colonel Blotto game, two players with a fixed budget simultaneously allocate their resources across $n$ battlefields to maximize the aggregate value gained from the battlefields where they have the higher allocation. Despite its long-standing history and important applicability, the Colonel Blotto game still lacks a complete Nash equilibrium characterization in its most general form—the non-constant-sum version with asymmetric players and heterogeneous battlefields. In this work, we propose a simply-constructed class of strategies—the independently uniform strategies—and we prove them to be approximate equilibria of the non-constant-sum Colonel Blotto game; moreover, we also characterize the approximation error according to the game’s parameters. We also introduce an extension called the Lottery Blotto game, with stochastic winner-determination rules allowing more flexibility in modeling practical contexts. We prove that the proposed strategies are also approximate equilibria of the Lottery Blotto game.'
address:
- 'Nokia Bell Labs, Nokia Paris-Saclay, Route de Villejust, 91620 Nozay, France'
- 'Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LIG, 38000 Grenoble, France'
- 'Max Planck Institute for Software Systems (MPI-SWS), Campus E1 5, D-66123, Saarbrücken, Germany'
- 'Safran Tech, Signal and Information Technologies, 78117 Châteaufort, France'
author:
- Dong Quan Vu
- Patrick Loiseau
- Alonso Silva
bibliography:
- 'mybibfile.bib'
title: 'Approximate Equilibria in Non-constant-sum Colonel Blotto and Lottery Blotto Games with Large Numbers of Battlefields'
---
resource allocation games ,epsilon-equilibrium ,Colonel Blotto game ,Lottery Blotto game ,contest success function
INTRODUCTION {#sec:Intro}
============
The *Colonel Blotto game* (henceforth, CB game) is one of the most well-known resource allocation games. Its description is very simple: two players, each having a fixed amount of resources (called budget), compete over a finite number of battlefields. Each battlefield is evaluated by the players with a certain value. Players simultaneously allocate their resources toward the battlefields and each player’s payoff is her aggregate gains from all the battlefields. In each battlefield, the winner, who is simply the one that has the higher allocation, gains the corresponding value and the loser gains zero—this is called the winner-takes-all rule—; in battlefields with tie allocations, the value is shared between the players with a predetermined tie-breaking rule, e.g., sharing equally between them. Throughout its long-standing history since its first introduction by [@borel1921], the CB game has attracted interest from different research communities for its potential to elegantly model a large range of practical situations. One of its original applications is military logistics, see e.g., [@gross1950; @grosswagner]; but it is also used to model problems in politics (where political parties distribute their budgets to compete over voters), see e.g., [@myerson1993; @kovenock2012; @roberson2006]; in cybersecurity (where effort is distributed to attack/defend targets), see e.g., [@chia2012; @schwartz2014]; in online advertising (where marketing campaigns allocate the time to broadcast ads to attract web users), see e.g., [@masucci2014; @masucci2015]; in telecommunication (where network service providers distribute and lease their spectrum to the users), see e.g., [@hajimirsaadeghi2017dynamic].
In this paper, we consider the most general version of the *non-constant-sum CB game*, where the evaluations of the battlefields’ values can be heterogeneous across battlefields and different between the two players; moreover, players’ budgets can be asymmetric. Despite the long-standing history of the CB game, the characterization of the Nash equilibrium in this most general version remains an open question—even the existence of an equilibrium has not been proved or disproved. Our study also examines the Nash equilibrium but we take a different angle: instead of looking for an exact equilibrium, *our first contribution* is to propose a class of approximate equilibria of the non-constant-sum CB game, called the ${{\rm IU}^{{\gamma^*}}}$ strategies.[^1] Importantly, we characterize the approximation error of this solution according to the games’ parameters and show that it is negligible when the number of battlefields is sufficiently large (it quickly decreases as the number of battlefields increases). Note also that it is simple and efficient to construct ${{\rm IU}^{{\gamma^*}}}$ strategies even in large-scale problems. Our work extends the state-of-the-art where the only known results regarding the equilibria of the non-constant-sum CB game[^2] are given by [@kovenock2015]. They provide a set of univariate marginal distributions (one per battlefield) that are the equilibrium marginals if they can be achieved with the budget constraints. They then indicate a sufficient condition for this to hold[^3]—which is identical to that of [@schwartz2014] for the constant-sum case—, that only covers a restricted range of games; and they also show a necessary condition where there is no equilibrium satisfying such a set of marginals.
The constant-sum CB game, where both players assign the same value to each battlefield, has been studied profoundly in the literature; however, even in this simpler version the equilibrium characterization is still not completely solved. When players have symmetric budgets, the equilibria are constructed by [@borel1938] in the game involving three battlefields and by [@grosswagner; @gross1950] in the game containing any number of battlefields (see also [@laslier2002; @thomas2017] for a modern presentation of this solution). For the constant-sum CB game with asymmetric budgets, equilibria characterization remains an open question in general; the exceptions are the following restricted cases: the games with only two battlefields ([@macdonell2015]), the games with any number of battlefields but homogeneous values ([@roberson2006]), and the games where there exists a sufficient number of battlefields of each possible value ([@schwartz2014]). In our model of the non-constant-sum CB game, we make no assumption on the players’ symmetry nor on the battlefields homogeneity; therefore, our results for the non-constant-sum CB game can be trivially adapted to the constant-sum game with the most general configuration of parameters, i.e., the ${{\rm IU}^{{\gamma^*}}}$ strategy is also an approximate equilibrium. Moreover, we show an additional result that the ${{\rm IU}^{{\gamma^*}}}$ strategy is also an approximate max-min strategy of the constant-sum CB game (with the same approximation error). It is also worth mentioning that a few works have considered extensions to non-constant-sum of the CB game, though with a significantly different flavor. In particular, [@Hortala2012] consider the discrete CB game and identify conditions under which a pure Nash equilibrium exists; and [@Kvasov07a; @roberson2012non] consider the relaxation of the use-it-or-lose-it rule that changes the payoffs.
In practice, there exists situations where the winner-takes-all rule of the CB game is too restrictive. In order to model these situations with more flexibility, in this work, we introduce and study an extension of the non-constant-sum CB game, called the *Lottery Blotto game* (henceforth, LB game),[^4] where each player only gains a part of her value in each battlefield. Alternatively, one can interpret the LB game as a version of the CB game in which each player wins a battlefields’ value with a certain probability depending on players’ allocations on that battlefield and this probability can be non-zero even for the player with smaller allocation. Some examples where the LB game model may prove to be useful are online advertising competitions, political contests for voters’ attention, research and development activities, radio-wave transmission with noises, etc. We formulate the LB game by presenting the players’ payoffs based on the concept of contest success function (henceforth, CSF). CSFs, studied profoundly in the rent-seeking literature—see e.g., [@skaperdas1996; @corchon2007theory]—, are functions that take the players’ allocations as inputs and output the probability of winning a battlefield. The definition of CSF that we adopt (see Section \[sec:LotteryFormulation\]) also includes the winner-takes-all rule as a special case so that the CB game is a particular case of the LB game. Similar to the non-constant-sum CB game, the equilibrium characterization of the LB game is an open question, except for several particular instances. *Our second main contribution* is to prove that the ${{\rm IU}^{{\gamma^*}}}$ strategy is also an approximate equilibrium of the LB game with an approximation error that decreases quickly as the number of battlefields increases and the corresponding CSFs converge pointwise to that of the CB game.
The LB game that we propose as an extension of the CB game with a general CSF, has not been formally defined in previous works; but several particular instances have been considered. [@friedman1958] investigated the pure equilibrium of the (constant-sum) LB game where players’ gains in each battlefield follows the Tullock function (termed by [@tullock1980]).[^5][^6] [@osorio2013] studied an extension of this model with a generalization of the Tullock function (coincidentally, it is also called there by the Lottery Blotto game); however, only numerically computed approximate-results of the equilibrium are proposed and no tractable close-form solution is provided in the general cases where battlefields’ values are asymmetric across players. The CSFs considered in these works belong to one specific class that we call the ratio-form CSFs (see Section \[sec:LotteryFormulation\] for a formal definition). Note that our result works for any LB game with any general CSF; therefore, they can be applied to the LB game with ratio-form CSFs. As an illustration, we analyze our ${{\rm IU}^{{\gamma^*}}}$ strategy in the LB games with two of the most well-known cases of ratio-form CSFs, the power form and the logit form (see Section \[sec:LotteryFormulation\] for more details)—in this case we obtain more precise results on the convergence of the error.
We note finally that a strategy construction similar to the ${{\rm IU}^{{\gamma^*}}}$ strategy can be found in [@Vu18a] for the (constant-sum) discrete CB game (i.e., where the budgets and every allocation are required to be integers) with asymmetric budgets and heterogeneous battlefields. Due to the discrete condition, their analysis has essential differences to our work; particularly, their asymptotic results involve a double limits of the number of battlefields and the ratio of players’ budgets; moreover, the convergence of the players’ payoffs in their work does not have the difficulties of continuous allocation encountered in our work. Finally, they do not consider the non-constant-sum CB game and the extension to the LB game.
The remainder of this paper is organized as follows. Section \[sec:GamesFormulation\] introduces the formulations of the non-constant-sum CB game and the LB game. Although the LB game model is essentially more general; we first focus on the CB game due to the fact that the CB game is a more classical game and our analysis for the LB game also depends on our results for the CB game. After providing some preliminary results for the CB game in Section \[sec:preliminaries\], we propose the ${{\rm IU}^{{\gamma^*}}}$ strategy in Section \[sec:ApproximateBlotto\] and state the result that any ${{\rm IU}^{{\gamma^*}}}$ strategy is an approximate equilibrium of the non-constant-sum CB game. In Section \[sec:LotteryApproximation\], we claim the results that the ${{\rm IU}^{{\gamma^*}}}$ strategy is also an approximate equilibrium of the LB game. Finally, the detailed proofs of all lemmas and theorems are given in Appendix.
Throughout the paper, we use bold symbols (e.g., $\boldsymbol{x}$) to denote vectors and subscript indices to denote its elements (e.g., $\boldsymbol{x} = (x_1, x_2, \ldots , x_n)$). The notation $[n]$ denotes the set $\{1,2, \ldots , n\}$, for any $n \in \mathbb{N} \backslash \{0\}$. We often use the letter ${p}$ to denote a player and use $-{p}$ to indicate her opponent in the games; $R^n_{\ge 0}$ denotes the set of all $n$-tuples whose elements are non-negative ($R_{\ge 0}:=R^1_{\ge 0}$). We denote the Euler’s number by ${e}$. For any random variable $X$, we use $F_{X}$ to denote its corresponding cumulative density function (abbreviated by CDF). Finally, we use ${\mathbb{P}}(E)$ to denote the probability that an event $E$ happens and ${\mathbb{E}}X$ to denote the expectation of a random variable $X$. A table of notations that are used in this work (Table \[table:notation\]) is given in \[sec:appen\_preliminary\].
GAMES FORMULATION {#sec:GamesFormulation}
=================
In this section, we define the two games that are our main focus: in Section \[sec:BlottoFormulation\], we introduce the non-constant-sum Colonel Blotto game; in Section \[sec:LotteryFormulation\], we present the Lottery Blotto game, as an extension of the Colonel Blotto game.
The Colonel Blotto game {#sec:BlottoFormulation}
-----------------------
We consider the following one-shot, complete information game between two players A and B. Each player has a fixed amount of resources (called the *budgets*), denoted $X_A$ and $X_B$, respectively. Without loss of generality, we assume that . Players simultaneously allocate their resources across $n$ *battlefields* . Each battlefield $i \in [n]$ is embedded with two parameters $w^A_i, w^B_i > 0$, corresponding to the *values* at which player A and player B respectively assess this battlefield. A *pure strategy* of player ${{p}} \in \left\{ A, B \right\}$ is a vector $\boldsymbol{x}^{{p}} = {\left( {x_i^{p}} \right)_{i \in [n]}} \in \mathbb{R}_ {\ge 0} ^n$ that satisfies the budget constraint $\sum\nolimits_{i = 1}^n {x_i^{{p}} \le {X_{{p}}}}$. In each battlefield $i$, when player ${{p}}$ allocates strictly more than her opponent, she gains completely her embedded values $w^{{p}}_i$ while the opponent gains $0$. In case of a tie, i.e., if $x^A_i = x^B_i$, then player A receives $\alpha w^A_{i}$ and player B receives $(1-\alpha) w^B_{i}$, where is a fixed parameter. Each player’s payoff is the summation of values she gains from all battlefields; formally, for any pure strategy profile $(\boldsymbol{x}^A, \boldsymbol{x}^B)$, the payoffs of players A and B are and respectively; here, $\beta_A$ and $\beta_B$ (henceforth, we called them the Blotto functions) are functions defined as follows: $$\beta_A\left( {x,y} \right) = \left\{ \begin{array}{l}
1 \text{ , if } x >y\\
\alpha \text{ , if } x =y \\
0 \text{ , if } x < y
\end{array} \right. \quad
\textrm{ and } \quad
\beta_B\left( {x,y} \right) = \left\{ \begin{array}{l}
1 \text{ , if } y > x\\
1-\alpha \text{ , if } y = x \\
0 \text{ , if } y < x
\end{array} \right., \textrm{ for all } x, y \in \mathbb{R}_ {\ge 0}. \label{eq:betafunction}$$
\[def:BlottoGame\] **A non-constant-sum Colonel Blotto game**, denoted by ${{\mathcal{CB}_n}}$, is the game defined above; in particular, the action set of player ${p}\in \{A,B\}$ is $\{\boldsymbol{x}^{p}\in \mathbb{R}^n_{\ge 0}: \sum \nolimits_{i=1}^n{x^{{p}}_i \le X_{{p}} } \}$ and her payoff is $\Pi^{{p}} (\boldsymbol{x}^A, \boldsymbol{x}^B)$ when players A and B play the pure strategies $\boldsymbol{x}^A$ and $\boldsymbol{x}^B$ respectively.
To lighten the notation, we only include the subscript $n$—the number of battlefields—in the notation ${{\mathcal{CB}_n}}$ and omit the other parameters; in particular the values $X_A, X_B$, $\alpha$ and $w^A_i, w^B_i$ for $i \in [n]$. Hereinafter, in places with no ambiguity, we drop the term non-constant-sum and simply address the game ${\mathcal{CB}_n}$ as the Colonel Blotto game. In this game, a *mixed strategy* is a joint distribution on the allocations of all battlefields, such that any drawn pure strategy of a player is an $n$-tuple that satisfies her budget constraint. We reuse the notations $\Pi^A\left(s_A, s_B \right)$ and $\Pi^B\left(s_A, s_B \right)$ to denote the payoffs of players A and B when they play the mixed strategies $s_A$ and $s_B$, respectively. Note that the definition of ${\mathcal{CB}_n}$ above allows asymmetry in players’ budgets and heterogeneity in battlefields values; moreover, it allows battlefield values to differ between the two players. Furthermore, the defined payoff functions can be understood as if we randomly break the tie (if it happens) such that player A wins battlefield $i$ with probability $\alpha$ while player B wins it with probability $(1-\alpha)$. This includes all the classical tie-breaking rules considered in the literature; for instance, the rule of giving the whole value to player B used by [@roberson2006; @schwartz2014] corresponds to $\alpha=0$; the 50-50 rule used by [@kovenock2015; @Ahmadinejad16a; @Behnezhad17a] corresponds to .
In this paper, we also often work with the *normalized values* of the battlefields defined as and , where and for $i \in [n]$. We trivially observe that $v^{{p}}_i \in \left[0, 1\right]$ for all $i$ and that $\sum \nolimits_{j=1}^n {v^{{p}}_j} = 1$. Most of our analysis relies on an additional assumption that the battlefields’ values are bounded away from zero and infinity (see the Assumption $(A0)$ below). This is a fairly mild assumption that is satisfied in most of (if not all) practical applications. $$\llap{$(A0)$ \hspace{100pt}} \exists {\underaccent{\bar}{w}}, {\bar{w}}>0: {\underaccent{\bar}{w}}\leq w^p_i \leq {\bar{w}}, \forall i \in [n], \forall p \in \{A, B\}$$ As a direct consequence, the normalized values satisfy $$\frac{{\underaccent{\bar}{w}}}{n {\bar{w}}} \le v^{{p}}_i \le \frac{{\bar{w}}}{n {\underaccent{\bar}{w}}}, \quad \forall i \in [n], \forall p \in \{A, B\}. \label{eq:bound_v^p_i}$$
Finally, we note that most works in the literature (the only exception, in our knowledge, being the work of [@kovenock2015]) focus only on the *constant-sum* Colonel Blotto game where players have the same evaluations on battlefields’ values. The non-constant-sum game ${\mathcal{CB}_n}$ given in Definition \[def:BlottoGame\] is more general; hence all our results for ${\mathcal{CB}_n}$ can be straightforwardly applied to this constant-sum version as well. However, for the purpose of comparing with the literature and because we can show stronger results in this special case, it is useful to also formally define the constant-sum game variant as follows.
\[def:constantsumGame\] **A constant-sum Colonel Blotto game**, denoted by ${\mathcal{CB}_n}^c$, is a game that has the same formulation as the game ${\mathcal{CB}_n}$ but with the additional condition that $w^A_i= w^B_i, \forall i \in [n]$.
As a trivial corollary of this additional condition, in ${\mathcal{CB}_n}^c$, players also have common normalized valuation on battlefields, i.e., for all $i \in [n]$ and the players’ maximum payoffs are equal, i.e., .
The contest success functions and the Lottery Blotto game {#sec:LotteryFormulation}
---------------------------------------------------------
In this section, we present a new game, the Lottery Blotto game, that extends the model of the Colonel Blotto game. This new game is based on the notion of contest success functions (CSFs), that we introduce below before defining the game model.
Contest success functions (CSFs) are functions that quantify the winning probability in *contests* (also called *rent-seeking* competitions) where several players compete for a single prize by exerting resources/efforts. CSFs can be defined for any number of players (see e.g., a general definition by [@skaperdas1996]), but in this work, we focus only on the case of two players.
\[def:CSF\_general\] and is a pair of contest success functions (**CSF**s) if and only if the following two conditions are satisfied:
- $\zeta_A(x,y),\zeta_B(x,y) \ge 0$ and $\zeta_A(x,y) + \zeta_B(x,y)= 1$, $\forall x,y\ge 0$.
- $\zeta_{A}(x, y)$ (resp. $\zeta_{B}(x, y)$) is non-decreasing in $x$ (resp. in $y$) and non-increasing in $y$ (resp. in $x$).
Intuitively, the function $\zeta_A$ (resp. $\zeta_B$) maps any pair of players’ invested resources to the probability that player A (resp. player B) wins the prize. Condition $(C1)$ indicates that the outputs of any pair of the CSFs always satisfy the condition of a probability distribution. On the other hand, Condition $(C2)$ states that a player’s winning probability increases (or at least stays the same) when she increases her effort and decreases (or at least stays the same) when her opponent increases her effort. We note that Definition \[def:CSF\_general\] allows a more general definitions of the CSFs (in two-player cases) compared to the definition given by [@skaperdas1996; @hirshleifer1989conflict; @clark1998contest] that contains other assumptions.[^7] While many of the CSFs considered in the literature are continuous functions, we do not include continuity in Definition \[def:CSF\_general\] to keep the generality. Importantly, the Blotto functions $\beta_A, \beta_B$ of the game ${\mathcal{CB}_n}$ (i.e., the winner-takes-all rule) satisfy Conditions $(C1)$ and $(C2)$, hence $\beta_A, \beta_B$ are CSFs. Besides these functions, some examples of other CSFs considered in the literature are:
(a) $\zeta_A(x,y) = x/(x+y)$ and $\zeta_B(x,y) = y/(x+y)$, proposed by [@tullock1980];
(b) $\zeta_A(x,y) = \max\left\{ \min \left\{\frac{1}{2} \! +\! C(x\!-\!y),1 \right\},0 \right\} $ and $\zeta_B(x,y) = 1- \zeta_A(x,y)$, proposed by [@che2000difference], where $C>0$ is a fixed parameter;
(c) $\zeta_A(x,y) = \frac{1}{2} - \frac{y-x}{2y}$ if $x \le y$ and $\zeta_A(x,y) = \frac{1}{2} + \frac{x-y}{2x}$ if $x \ge y$; and $\zeta_B(x,y)=1-\zeta_A(x,y)$, proposed by [@alcalde2007tullock].
Building on the notion of CSFs and the Colonel Blotto game, we now define a new game model based on the following idea: in a game ${\mathcal{CB}_n}$, we view each battlefield as a contest between players where the prize is the battlefield’s value and players’ effort correspond to their allocations; by doing this, each pair of CSFs defines an instance of a new game where the probability of winning a battlefield follows them accordingly.
\[def:LotteryGame\] Let $\zeta = (\zeta_A,\zeta_B)$ be a pair of CSFs. **A Lottery Blotto game** with $n$ battlefields, denoted ${\mathcal{LB}_n}(\zeta)$, is the game with the same players A and B and the same strategy sets as in ${\mathcal{CB}_n}$; but where payoffs are given, for any pure strategy profile $(\boldsymbol{x}^A,\boldsymbol{x}^B)$, by $$\Pi^A_{\zeta}(\boldsymbol{x}^A, \boldsymbol{x}^B) = \sum \nolimits _{i=1}^n {w^A_i\cdot \zeta_A \left( {x_i^A,x_i^B} \right)} \qquad \textrm{and } \qquad \Pi^B_{\zeta}(\boldsymbol{x}^A, \boldsymbol{x}^B) = \sum \nolimits _{i=1}^n {w^B_i \cdot \zeta_B \left( {x_i^A,x_i^B} \right)}.$$
The Lottery Blotto game model is more flexible than that of the Colonel Blotto game, as it allows choosing the CSFs that define the players’ payoffs for each specific practical situation. Intuitively, the players’ payoffs in a Lottery Blotto game can be seen as the expected payoffs in the Colonel Blotto game with respect to the following random process determining the winner in any battlefield $i$: player A wins with probability $\zeta_A(x^A_i,x^B_i)$ and player B wins with probability $\zeta_B(x^A_i,x^B_i)$ if they allocate $x^A_i$ and $x^B_i$ respectively. Similar to the game ${\mathcal{CB}_n}$, players’ payoffs in the ${\mathcal{LB}_n}$ game are also monotonic with respect to the allocations in a battlefield (due to Condition $(C2)$).
Besides the Lottery Blotto game with the generally defined CSFs, we additionally consider the games corresponding to the CSFs that belong to a special class called the *ratio-form* CSFs. These are the CSFs that are studied the most profoundly in the literature. We will use the games with these ratio-form CSFs to illustrate the results obtained in the Lottery Blotto game.
\[def:CSF\_ratioform\] CSFs are called **ratio-form CSFs** if they have the form: $$\zeta_A(x,y) = \frac{\eta(x)}{\eta(x)+ \kappa(y)} \quad \textrm{and} \quad \zeta_B(x,y) = \frac{\kappa(y)}{\eta(x)+ \kappa(y)},$$ where $\eta, \kappa: \mathbb{R}_{\ge 0} \to \mathbb{R}$ are non-negative functions such that $\zeta_A$ and $\zeta_B$ satisfy Conditions $(C1)$ and $(C2)$.
Two classical ratio-form CSFs in the literature (see e.g., [@hillman1989; @corchon2010foundations]) are the power form where and the logit form where , where $R>0$ is a parameter chosen a priori. These functions yield the sharing 50-50 tie-breaking rule, i.e., $\zeta_A(x,y) = \zeta_B(x,y) =1/2$ if $x=y$. We define in Table \[table:CSF\] the generalized versions of these ratio-form CSFs using the parameter $\alpha\in (0,1)$ that leads to the general tie-breaking rule as in the Colonel Blotto game ${\mathcal{CB}_n}$.[^8] Henceforth, we use the terms power and logit form to indicate the CSFs $\mu^R$ and $\nu^R$ with this generalization. It is trivial to verify that both pairs $(\mu^R_A, \mu^R_B)$ and $(\nu^R_A, \nu^R_B)$ satisfy the Conditions $(C1)$ and $(C2)$. An important remark is that both the power and logit form CSFs converge pointwise toward the Blotto functions $\beta_A, \beta_B$ as $R$ tends to infinity (see Section \[sec:Approx\_Lottery\_ratio\] for more details). This convergence can be observed in Figure \[fig1\] that illustrates several instances of the ratio-form CSFs in comparison with the Blotto functions.
Notation [If $x^2+ y^2 >0$]{} [If $x=y=0$]{}
------------ -------------------------------- ---------------------- ----------------
Power form $\mu^R:=\! (\mu^R_A,\mu^R_B)$
Logit form $\nu^R := \!(\nu^R_A,\nu^R_B)$
: Power and logit form CSFs with generalized tie-breaking rule ($\alpha \in (0,1)$).[]{data-label="table:CSF"}
[90]{} [ $\mu^R_A(x, 4)$]{}
[90]{} [ $\nu^R_A(x,4)$]{}
Throughout the paper, to refer to a Colonel Blotto game ${\mathcal{CB}_n}$ that has the same parameters $n,X_A,X_B, w^A_i$, $w^B_i, \forall i \in [n]$ as a Lottery Blotto game ${\mathcal{LB}_n}$, we call ${\mathcal{CB}_n}$ the *corresponding game* of ${\mathcal{LB}_n}$ and vice versa. Note that, to derive our results for ${\mathcal{LB}_n}$, we will also use Assumption $(A0)$ introduced above.
PRELIMINARIES {#sec:preliminaries}
=============
In this section, we briefly review some results from the literature that are useful for our analyses of the Colonel Blotto games and the Lottery Blotto games; and we show new bounds on the involved parameters, based on Assumption $(A0)$, that are essential for the asymptotic analysis in the next sections.
The Nash equilibrium characterization of the non-constant-sum Colonel Blotto game still remains an open question. However, under certain assumptions, the set of *univariate marginal distributions* of players in an equilibrium of the game ${\mathcal{CB}_n}$ is well-known. To see this, observe that we can break down the problem of finding the best-response of a player against a fixed strategy of her opponent into solving $n$ all-pay auctions involving the Lagrange multipliers corresponding to the budget constraints (see e.g., [@kovenock2015; @roberson2006; @schwartz2014]). The equilibrium of two-player all-pay auctions is well-known and can be expressed as uniform-type distributions (see e.g., [@hillman1989; @baye1996all]). In other words, these uniform-type distributions are the set of Nash equilibrium univariate marginals of the Colonel Blotto game. We have a Nash equilibrium if we can construct a joint distribution with these univariate marginals such that its realizations always satisfy the budget constraints. However, as mentioned in Section \[sec:Intro\], the existence of such a construction is known only for some special cases and remains unknown in the general setting of ${\mathcal{CB}_n}$. Note that if we consider a relaxation of the game that only requires the budget constraints to be hold in expectation (this relaxation is called the General Lotto game by [@kovenock2015]), an equilibrium is to independently draw allocations from the uniform-type distributions.
Although in this work we do not attempt to solve the open question of the equilibria characterization of the non-constant-sum Colonel Blotto game, we still use several preliminary results from this approach to construct an approximate equilibrium of the games. We present these results below, using a notation similar to [@kovenock2015].
For each instance of the game ${\mathcal{CB}_n}$ (and of the game ${\mathcal{LB}_n}$), for any $\gamma \in (0,\infty)$, we define $$\Omega_A(\gamma):= \left\{ i \in [n] : {v^A_i}/{v^B_i} > \gamma \right\},$$ and consider the following equation with the variable $\gamma$ (other coefficients are the parameters of ${\mathcal{CB}_n}$ and ${\mathcal{LB}_n}$): $$\frac{X_B \gamma}{X_A} = \frac{\gamma^2 \sum\nolimits_{i \in \Omega_{A}(\gamma)}{\frac{(v^B_i)^2}{v^A_i}} + \sum\nolimits_{i \notin \Omega_{A}(\gamma)}{v^A_i}} {\sum\nolimits_{i \in \Omega_{A}(\gamma)}{v^B_i} + \frac{1}{\gamma^2} \sum\nolimits_{i \notin \Omega_{A}(\gamma)}{\frac{(v^A_i)^2}{v^B_i}} }.
\label{eq:Equagamma}$$
Let us denote by ${\mathcal{S}_n}$ *the set containing all positive solutions* of Equation corresponding to the game ${\mathcal{CB}_n}$ (or ${\mathcal{LB}_n}$).[^9] Based on Brouwer’s fixed-point theorem, the following lemma is proved by [@kovenock2015].
\[lem:positivegamma\] For any game ${\mathcal{CB}_n}$ (or ${\mathcal{LB}_n}$), Equation has at least one positive solution; i.e., ${\mathcal{S}_n}\neq \emptyset$.
Equation may have more than one solution and it can be solved in ${\mathcal{O}}(n \ln(n))$ time.[^10] Now, corresponding to each positive solution ${\gamma^*}\in {\mathcal{S}_n}$, we define two constants,[^11] namely ${\lambda^*_A}$ and ${\lambda^*_B}$ as follows: $$\begin{aligned}
& {\lambda _A^*}: = \frac{({{\gamma^*}})^2}{2{X_B}}\sum\limits_{i \in {\Omega _A}({{\gamma^*}})} {\frac{{{{\left( {v_i^B} \right)}^2}}}{{v_i^A}}} + \frac{1}{2{X_B}}\sum\limits_{i \notin {\Omega _A}({{\gamma^*}})} {v_i^A} , \label{eq:lambdaA} \\
& {\lambda _B^*}: = \frac{1}{2{X_A}}\sum\limits_{i \in {\Omega _A}({{\gamma^*}})} {v_i^B} + \frac{1}{2({\gamma^*}) ^2 {X_A}}\sum\limits_{i \notin {\Omega _A}({{\gamma^*}})} {\frac{{{{\left( {v_i^A} \right)}^2}}}{{v_i^B}}}.\label{eq:lambdaB}\end{aligned}$$ Note importantly that we have ${\gamma^*}= {\lambda^*_A}/ {\lambda^*_B}$ (see Lemma \[lem:Preliminary\] in \[sec:appen\_preliminary\] for a proof). We now use these constants ${\lambda^*_A}$ and ${\lambda^*_B}$ to define several important distributions.
\[def:UnifromDistributions\] Given a game ${\mathcal{CB}_n}$ (or ${\mathcal{LB}_n}$), for any ${\gamma^*}\in {\mathcal{S}_n}$ and the corresponding constants ${\lambda^*_A}, {\lambda^*_B}$, we define the following random variables and distributions,[^12] for each $i \in [n]$:
(a) If $i \in {\Omega_A({\gamma^*})}$ (i.e., $\frac{v^A_i}{{\lambda^*_A}} \! > \! \frac{v^B_i}{ {\lambda^*_B}}$), we define ${ A^S_{{\gamma^*},i}}$ and ${ B^W_{{\gamma^*},i}}$ as the random variables whose distributions are $$\begin{aligned}
& F_{{ A^S_{{\gamma^*},i}}}\left( x \right) := \frac{x {\lambda^*_B}}{v^B_i}, \forall x \in \left[0, \frac{v_i^B}{{\lambda^*_B}}\right], \label{As}
\\
& {F_{{ B^W_{{\gamma^*},i}}}}\left( x \right) := \frac{\frac{v^A_i}{{\lambda^*_A}} - \frac{v^B_i}{{\lambda^*_B}}}{\frac{v^A_i}{{\lambda^*_A}}} + \frac{x {\lambda^*_A}}{v^A_i}, \forall x \in \left[0, \frac{v_i^B}{{\lambda^*_B}}\right]. \label{Bw}
\end{aligned}$$
(b) If $i \notin {\Omega_A({\gamma^*})}$ (i.e., $\frac{v^A_i}{{\lambda^*_A}} \! \le \! \frac{v^B_i}{{\lambda^*_B}}$), we define ${ A^W_{{\gamma^*},i}}$ and ${ B^S_{{\gamma^*},i}}$ as the random variables whose distributions are $$\begin{aligned}
& {F_{{ A^W_{{\gamma^*},i}}}}\left( x \right) := \frac{\frac{v^B_i}{{\lambda^*_B}} - \frac{v^A_i}{{\lambda^*_A}}}{\frac{v^B_i}{{\lambda^*_B}}} + \frac{x {\lambda^*_B}}{v^B_i}, \forall x \in \left[0, \frac{v_i^A}{{\lambda^*_A}}\right],\label{Aw}
\\
& F_{{{ B^S_{{\gamma^*},i}}}}\left( x \right) := \frac{x {\lambda^*_A}}{v^A_i}, \forall x \in \left[0, \frac{v_i^A}{{\lambda^*_A}}\right]. \label{Bs}
\end{aligned}$$
To lighten the notation, hereinafter, we often commonly denote these random variables as follows (the corresponding distributions are denoted by ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$): $$\begin{aligned}
& A^*_i := \left\{
\begin{array}{ll}
{ A^S_{{\gamma^*},i}}& \mbox{if } i \in {\Omega_A({\gamma^*})}\\
{ A^W_{{\gamma^*},i}}& \mbox{if } i \notin {\Omega_A({\gamma^*})}\end{array}
\right.
\textrm{ and }
\hspace{-2.5cm}& B^*_i := \left\{
\begin{array}{ll}
{ B^S_{{\gamma^*},i}}& \mbox{if } i \notin {\Omega_A({\gamma^*})}\\
{ B^W_{{\gamma^*},i}}& \mbox{if } i \in {\Omega_A({\gamma^*})}\end{array}
\right.. \label{A*B*}\end{aligned}$$
We term these distributions the *uniform-type distributions*: $F_{{ A^S_{{\gamma^*},i}}}\left( x \right)$ is the continuous uniform distribution on $\left[ 0, {v^B_i}/{{\lambda^*_B}} \right]$ and $F_{{ B^W_{{\gamma^*},i}}}\left( x \right)$ is the distribution placing a positive mass $\left( \frac{v^A_i}{{\lambda^*_A}}\!-\! \frac{v^B_i}{{\lambda^*_B}}\right)\! \Big/ \! \frac{v^A_i}{{\lambda^*_A}}$ at $0$ and uniformly distributing the remaining mass on $\left( 0, {v^B_i}/{{\lambda^*_B}} \right]$; similarly, $F_{{ B^S_{{\gamma^*},i}}}$ is the uniform distribution on $[0, v^A_i / {\lambda^*_A}]$ and $F_{{ B^W_{{\gamma^*},i}}}$ is uniform on $(0,v^A_i/{\lambda^*_A}]$ with a positive mass at $0$.
If player A can construct and plays a mixed strategy such that her sampled allocation to any battlefield $i \in [n]$ follows the distribution ${ F_{A^*_i}}$, it is optimal for player B to play such that her allocation to $i$ follows ${ F_{B^*_i}}$ (if it is possible) and vice versa. We will revisit this result (with more details) in Section \[sec:ApproximateBlotto\] and in Lemma \[lem:best\_response\] in \[sec:Appen\_Proof\_TheoBlotto\]. Therefore, under the condition that player A and player B can respectively construct joint distributions of and such that their sampled allocations satisfy the budget constraint, these mixed strategies yield an equilibrium of the game ${\mathcal{CB}_n}$. However, in general, that condition does not always hold. For instance, although $A^*_i$ and $B^*_i$ have finite upper-bounds,[^13] we note that among these random variables, some may (with strictly positive probability) exceed the budgets $X_A, X_B$ for certain parameters’ configuration of the game; therefore, allocating according to ${ F_{A^*_i}}, { F_{B^*_i}}$ may violate the budget constraints and it is then trivial that there exists no equilibrium yielding ${ F_{A^*_i}}, { F_{B^*_i}}, \forall i \in [n]$ as marginals. On the other hand, given fixed $X_A, X_B$, if $n$ is large enough, we can guarantee that $A^*_i, B^*_i$ do not exceed the budgets for each $i$; however, even in this case, we still do not have guarantees on the summation of allocations sampled from all $A^*_i, B^*_i, i \in [n]$, i.e., it is still unknown if there exists an equilibrium yielding ${ F_{A^*_i}},{ F_{B^*_i}}, i \in [n]$ as marginals. Note importantly that the budget-constraints violation of $A^*_i, B^*_i$ does not affect our work and our results hold for any parameters’ configuration of the games.
Finally, under Assumption $(A0)$, we obtain a novel result, presented below as Proposition \[Prop:BoundLambda\], stating that the parameters ${\gamma^*}, {\lambda^*_A}$ and ${\lambda^*_B}$ are all bounded. The main results of this work are based on asymptotic analyses in terms of the number of battlefields of the games; therefore, it is noteworthy that the bounds of these parameters do not depend on $n$. The proof of this proposition is given in \[sec:appen\_preliminary\]. From the proof of Proposition \[Prop:BoundLambda\], we observe that as the ratios ${\bar{w}}/{\underaccent{\bar}{w}}$ and (or) $X_B/X_A$ increase, the ranges in which ${\gamma^*}$ and ${\lambda^*_A},{\lambda^*_B}$ belong to also become larger (i.e., the ratios ${\bar{\gamma}}/{\underaccent{\bar}{\gamma}}$ and ${\bar{\lambda}}/{\underaccent{\bar}{\lambda}}$ also increase).
[proposition]{}[boundpropo]{} \[Prop:BoundLambda\] Under Assumption $(A0)$, for any game ${\mathcal{CB}_n}$ (or ${\mathcal{LB}_n}$), there exist constants ${\underaccent{\bar}{\gamma}}, {\bar{\gamma}}, {\underaccent{\bar}{\lambda}}, {\bar{\lambda}}\! >\!0$, that do not depend on $n$, such that for any ${\gamma^*}\! \in\! {\mathcal{S}_n}$ and its corresponding ${\lambda^*_A}, {\lambda^*_B}$, we have and .
APPROXIMATE EQUILIBRIA OF THE COLONEL BLOTTO GAME {#sec:ApproximateBlotto}
=================================================
In this section, we propose a class of strategies in the Colonel Blotto game ${\mathcal{CB}_n}$, called the independently uniform strategies, and we show that it is an approximate Nash equilibrium (and an approximate min-max strategy in the constant-sum case). Note that the independently uniform strategies are also approximate equilibria of the Lottery Blotto game ${\mathcal{LB}_n}$, we analyze that in Section \[sec:LotteryApproximation\].
We begin by recalling the concept of approximate Nash equilibria (see e.g., [@myerson1991game; @Nisan07]) in the context of our games: *for any $\varepsilon \ge 0$, an ** of the game ${\mathcal{CB}_n}$ is any strategy profile $\left(s^{*},t^{*} \right)$ such that and for any strategy $s$ and $t$ of players A and B.* Replacing $\Pi^A$ and $\Pi^B$ by $\Pi^A_{\zeta}$ and $\Pi^B_{\zeta}$, we have the definition of of the Lottery Blotto games ${\mathcal{LB}_n}(\zeta)$. Hereinafter, we use the generic term *approximate equilibrium* whenever the approximation error $\varepsilon$ need not be emphasized.
The Independently Uniform strategies {#sec:IU_Strategy}
------------------------------------
Given a game ${\mathcal{CB}_n}$ (or ${\mathcal{LB}_n}$), consider the corresponding Equation and set ${\mathcal{S}_n}$. For any ${\gamma^*}\in {\mathcal{S}_n}$, we define in Definition \[def:IU\_strategy\] a mixed strategy via an algorithm, called Algorithm \[alg:IU\_strategy\]. We term this strategy as the *independently uniform* strategy (or ${{\rm IU}^{{\gamma^*}}}$ strategy), parameterized by ${\gamma^*}$. Intuitively, this strategy is constructed by a simple procedure: players start by *independently* drawing $n$ numbers from the *uniform-type* distributions defined in Definition \[def:UnifromDistributions\], then they re-scale these numbers to guarantee the budget constraints.
\[def:IU\_strategy\] For any ${\gamma^*}\in {\mathcal{S}_n}$ and any player $p \in \{ A, B\}$, ${{\rm IU}^{{\gamma^*}}}_p$ is the **mixed** strategy of player $p$ where her allocation $\boldsymbol{x}^p$ is randomly generated from Algorithm \[alg:IU\_strategy\].
Draw $a_i \sim { F_{A^*_i}}, b_i \sim { F_{B^*_i}}, \forall i \in [n]$ independently
Henceforth, we use the term ${{\rm IU}^{{\gamma^*}}}$ strategy to denote the strategy profile $( {{{\rm IU}^{{\gamma^*}}}_A}, {{{\rm IU}^{{\gamma^*}}}_B} )$. We also simply use the notation ${{\rm IU}^{{\gamma^*}}}$ in some places to commonly address either ${{\rm IU}^{{\gamma^*}}}_A$ or ${{\rm IU}^{{\gamma^*}}}_B$ strategy in case the name of the player need not be specified. Observe that for any player $p \in \{A,B\}$, any output $\boldsymbol{x}^p$ from Algorithm \[alg:IU\_strategy\] is an $n$-tuple that satisfies her budget constraint. In other words, ${{\rm IU}^{{\gamma^*}}}_p$ is a mixed strategy that is implicitly defined by Algorithm \[alg:IU\_strategy\] and each run of this algorithm outputs a feasible pure strategy sampled from ${{\rm IU}^{{\gamma^*}}}_p$. Note that the marginals of the ${{\rm IU}^{{\gamma^*}}}$ strategy are *not* the uniform-type distributions ${ F_{A^*_i}}, { F_{B^*_i}}, i \in [n]$ defined in Section \[sec:preliminaries\]. In terms of computational complexity, Algorithm \[alg:IU\_strategy\] terminates in ${\mathcal{O}}(n)$ time. Below we discuss the specificity of the outputs of Algorithm \[alg:IU\_strategy\] in the cases where $\sum_{j \in [n]} a_j =0$ or $\sum_{j \in [n]} b_j =0$.
If $\sum_{j \in [n]} a_j =0$ or $\sum_{j \in [n]} b_j =0$, the ${{\rm IU}^{{\gamma^*}}}_p$ strategy allocates zero resource to all battlefields for the corresponding player (line $3$ and line $7$ of Algorithm \[alg:IU\_strategy\]). It may seem more natural that, if , player A allocates equally on all battlefields, i.e., set in line $3$ of Algorithm \[alg:IU\_strategy\] (and similarly for player B). In reality though, these assignments can be chosen to be any arbitrary $n$-tuple $\boldsymbol{x}^p$ in $\mathbb{R}^n_{\ge 0}$ as long as without affecting the results in our work. This comes from the fact that in most cases, the conditions in line $2$ and $6$ hold with probability zero. They can happen with a positive probability only when one player is the “weak player" and the other is the “strong player" on all of the battlefields (i.e., either ${\Omega_A({\gamma^*})}= \emptyset$ or ${\Omega_A({\gamma^*})}= [n]$), e.g., in a constant-sum game ${\mathcal{CB}_n}^c$. Yet, even in this case, this probability decreases exponentially as the number of battlefields increases (see in \[sec:Appen\_Proof\_TheoBlotto\]). The asymptotic order of the approximation error in all of our results is larger than this probability; therefore, it does not matter which assignment we choose in lines $3$ and $7$ of Algorithm \[alg:IU\_strategy\]. Here, we choose to assign $x^A_i = 0, \forall i$ and $x^B_i = 0, \forall i$ to ease the notation in the proofs of the results in the following sections; in particular, it avoids creating a discontinuity outside $0$ in the CDF of the effective allocation in each battlefield (see also Lemma \[lem:continuity\_Ani\_and\_Bni\] in \[sec:Appen\_Proof\_TheoBlotto\]).
Approximate equilibria of the non-constant-sum Colonel Blotto game ${\mathcal{CB}_n}$ {#sec:Approx_NonConstantSum}
-------------------------------------------------------------------------------------
We now present our main result stating that the ${{\rm IU}^{{\gamma^*}}}$ strategy is an approximate equilibrium with an error that is only a negligible fraction of the maximum payoffs that the players can achieve, quickly decreasing as $n$ increases. In the following results, note that since we focus on the setting of games with a large number of battlefields, we now focus on characterizing the approximation error according to $n$ and treat other parameters of the ${\mathcal{CB}_n}$ games, including $X_A, X_B, {\underaccent{\bar}{w}}, {\bar{w}}$ and $\alpha$, as constants (but not the values $w^p_i, v^p_i, \forall i \in [n], p \in \{A,B\}$). Using the notation $\tilde{{\mathcal{O}}}$—a variant of the big-${\mathcal{O}}$ notation that ignores the logarithmic factors—, we have the first result as follows.
[theorem]{}[TheoMainBlotto]{} \[TheoMainBlotto\]
- In any game ${\mathcal{CB}_n}$, there exists a positive number such that for any ${\gamma^*}\in {\mathcal{S}_n}$, the following inequalities hold for any pure strategy $\boldsymbol{x}^A$ and $\boldsymbol{x}^B$ of players A and B: $$\begin{aligned}
& \Pi^A(\boldsymbol{x}^A,{{{\rm IU}^{{\gamma^*}}}_B}) \le \Pi^A({{{\rm IU}^{{\gamma^*}}}_A},{{{\rm IU}^{{\gamma^*}}}_B}) + \varepsilon W_A,\label{eq:MainTheo_A}\\
& \Pi^B({{{\rm IU}^{{\gamma^*}}}_A},\boldsymbol{x}^B) \le \Pi^B({{{\rm IU}^{{\gamma^*}}}_A},{{{\rm IU}^{{\gamma^*}}}_B}) + \varepsilon W_B. \label{eq:MainTheo_B}
\end{aligned}$$
- For any $\varepsilon\! \in (0, 1]$, there exists $C^*>0$ (that does not depend on $\varepsilon$) such that in any game ${\mathcal{CB}_n}$ with $n\!\ge C^* \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, and hold for any ${\gamma^*}\in {\mathcal{S}_n}$, any pure strategy , $\boldsymbol{x}^B$ of players A and B.
A proof of this theorem is presented in \[sec:Appen\_Proof\_TheoBlotto\]. The two results stated in Theorem \[TheoMainBlotto\] are two equivalent statements that can be interpreted from different perspectives as follows. Result $(i)$ states that given a priori a game ${\mathcal{CB}_n}$, there exists no unilateral deviation from the ${{\rm IU}^{{\gamma^*}}}$ strategy that can profit any player $p \in \{A,B\}$ more than a small portion of her maximum payoff $W_p$. As a trivial corollary, the ${{\rm IU}^{{\gamma^*}}}$ strategy is an approximate equilibrium of the game ${\mathcal{CB}_n}$ with a bounded approximation error (depending on $n$); this is formally stated as follows:
\[Corol:Blotto\_Approx\_Equi\] In any game ${\mathcal{CB}_n}$, there exists a positive number such that for any ${\gamma^*}\in {\mathcal{S}_n}$, the ${{\rm IU}^{{\gamma^*}}}$ strategy is an where $W:= \max\{W_A, W_B\}$.
The bound $\tilde{{\mathcal{O}}} (n^{-1/2})$ tells us the order of and how fast the level of error $\varepsilon$ decreases if we consider games with larger and larger numbers of battlefields. Moreover, note that this upper-bound on $\varepsilon$ also depends on other parameters of the game ${\mathcal{CB}_n}$, including $X_A, X_B, {\underaccent{\bar}{w}}, {\bar{w}}$ and $\alpha$.[^14] We can extract from the proof of Theorem \[TheoMainBlotto\] that as ${\bar{w}}/ {\underaccent{\bar}{w}}$ and/or $X_B/X_A$ increases, $\varepsilon$ also increases, i.e., for games with higher heterogeneity of the battlefields values and/or higher asymmetry in players’ budgets, the ${{\rm IU}^{{\gamma^*}}}$ strategy yields higher errors. Additionally, we note that to keep the generality, Result $(i)$ is presented such that the approximation error $\varepsilon$ is commonly addressed for any ${{\rm IU}^{{\gamma^*}}}$ strategy with any ${\gamma^*}\in {\mathcal{S}_n}$. For each specific solution ${\gamma^*}$ of Equation (implying ${\lambda^*_A}$ and ${\lambda^*_B}$), the corresponding ${{\rm IU}^{{\gamma^*}}}$ strategy is an approximate equilibrium of ${\mathcal{CB}_n}$ with an approximation error that is at most (and it might be strictly smaller than) $\varepsilon$.
On the other hand, Result $(ii)$ of Theorem \[TheoMainBlotto\] indicates the number of battlefields that a Colonel Blotto game needs to contain in order to guarantee a desired level of the approximation error by using the ${{\rm IU}^{{\gamma^*}}}$ strategy as an approximate equilibrium. Hence, in practical situations involving large instances of the Colonel Blotto game, the ${{\rm IU}^{{\gamma^*}}}$ strategy (simply and efficiently constructed by Algorithm \[alg:IU\_strategy\]) can be used as a safe replacement for any Nash equilibrium whose construction may be unknown or too complicated. Now, let us introduce an important notation:
Corresponding to the players’ allocations toward each battlefield $i \in [n]$, let ${ F_{A^n_i}}$ and ${ F_{B^n_i}}$ denote the univariate marginal distributions of the ${{\rm IU}^{{\gamma^*}}}_A$ and ${{\rm IU}^{{\gamma^*}}}_B$ strategies (see and in \[sec:Appen\_Proof\_TheoBlotto\] for a more explicit formulation of ${ F_{A^n_i}}$ and ${ F_{B^n_i}}$).
Intuitively, Result $(ii)$ can be proved by showing the two following results: *(a)* when player B’s allocation to the battlefield $i \in [n]$ follows ${ F_{B^*_i}}$, the best response of player A is to play such that her allocation to $i$ follows the distribution ${ F_{A^*_i}}$ (and vice versa); *(b)* as $n$—the number of battlefields—increases, ${ F_{A^n_i}}$ and ${ F_{B^n_i}}$ uniformly converge toward the distributions ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$, i.e., the marginal distributions of the ${{\rm IU}^{{\gamma^*}}}$ strategy approximate the distributions ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$. This convergence can be proved by applying concentration inequalities on the random variables $\sum \nolimits_{j \in [n]} A^*_j$ and $\sum \nolimits_{j \in [n]} B^*_j$ (see Lemma \[lem:convergence\] in \[sec:Appen\_Proof\_TheoBlotto\]); moreover, the relation between $\varepsilon$ and $n$ in the results of Theorem \[TheoMainBlotto\] depends directly on the rate of this convergence. In this work, we use the Hoeffding’s inequality ([@hoeffding1963probability]) that yields a better convergence rate than working with other types of concentration inequalities (e.g., Chebyshev’s inequality). To complete the proof of Result $(ii)$, we finally show that as $n$ increases, when player $-p \in \{A,B\}$ plays the ${{\rm IU}^{{\gamma^*}}}_{-p}$ strategy, the ${{\rm IU}^{{\gamma^*}}}_{p}$’s payoff of player $p$ converges toward her best-response payoff. Note that these payoffs can be written as expectations with respect to different measures (see , and Lemma \[lem:SufCon\] in \[sec:Appen\_Proof\_TheoBlotto\]). To prove the convergence of payoffs, we use a variant of the portmanteau theorem (see Lemma \[lem:portmanteau\] in \[sec:Appen\_Proof\_TheoBlotto\]) regarding the equivalent definitions of the weak convergence of a sequence of measures. Note importantly that a direct application of the portmanteau theorem leads to a slow convergence rate (notably, and only hold when $n = \Omega(\varepsilon^{-4}))$. This is due to the fact that the players’ payoffs involve the bounded Lipschitz functions ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$ and that these functions depend on $n$, particularly, their Lipschitz constants (that are either ${\lambda^*_A}/v^A_i$ or ${\lambda^*_B}/v^B_i$) increase as $n$ increases. In order to obtain the convergence rate as indicated in Theorem \[TheoMainBlotto\], we exploit the special relation between ${ F_{A^n_i}}$ and ${ F_{A^*_i}}$, and between ${ F_{B^n_i}}$ and ${ F_{B^*_i}}$; then we apply a telescoping-sum trick allowing us to avoid the need of using the Lipschitz properties (for more details, see the proof of Lemma \[lem:portmanteau\] in \[sec:appen\_proof\_lem\_portmanteau\]).
Approximate equilibria of the constant-sum Colonel Blotto game ${\mathcal{CB}_n}^c$ {#sec:Approx_ConstantSum}
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In this section, we discuss the constant-sum variant ${\mathcal{CB}_n}^c$ of the Colonel Blotto game, defined in Definition \[def:constantsumGame\]. As an instance of the non-constant-sum game ${\mathcal{CB}_n}$, the game ${\mathcal{CB}_n}^c$ satisfies all results presented in Sections \[sec:IU\_Strategy\] and \[sec:Approx\_NonConstantSum\]. Additionally, we show that any ${{\rm IU}^{{\gamma^*}}}$ strategy is an approximate max-min strategy of the game ${\mathcal{CB}_n}^c$.
In any game ${\mathcal{CB}_n}^c$, Equation has a unique solution ; this ${\gamma^*}$ uniquely induces and . Moreover, in ${\mathcal{CB}_n}^c$, ; therefore, we have ; intuitively, player A is the “weak player” (and B the “strong player”) in *all* battlefields. Recall the notation $W:= \max \{W_A,W_B \}$, in the constant-sum game ${\mathcal{CB}_n}^n$, we have $W = W_A = W_B$. Applying Theorem \[TheoMainBlotto\], we obtain the following result.
\[corol:constant\_sum\_Equi\] In any game ${\mathcal{CB}_n}^c$, there exists a positive number such that the ${\rm IU} ^{{\gamma}^*}$ strategy is an with ${\gamma}^* \in {\mathcal{S}_n}=\left\{ {X_B}/{X_A} \right\}$.
Note that if a Nash equilibrium exists in ${\mathcal{CB}_n}^c$, then the set of equilibrium univariate marginal distributions is unique (see e.g., Corollary 1 of [@kovenock2015]) and they correspond to the distributions $F_{{ A^W_{{\gamma^*},i}}}$ and $F_{{ B^S_{{\gamma^*},i}}}$, defined in and , where ${\lambda^*_A}$ and ${\lambda^*_B}$ are respectively replaced by ${1}/(2X_B)$ and ${X_A}/{(2{X_B}^2)}$. The marginals of the ${{\rm IU}^{{\gamma^*}}}$ strategy with ${\gamma^*}= X_B/X_A$ converge toward these unique equilibrium marginals. Finally, we also deduce that the ${{\rm IU}^{{\gamma^*}}}$ strategy is an approximate $\max$-$\min$ strategy of the game ${\mathcal{CB}_n}^c$; formally stated as follows.
\[corol:max\_minConstantSum\] In any game ${\mathcal{CB}_n}^c$, there exists a positive number $\varepsilon \le \tilde{O}(n^{-1/2})$ such that the following inequalities hold for ${\gamma}^* \in {\mathcal{S}_n}=\left\{ {X_B}/{X_A} \right\}$ and any strategy $\tilde{s}$ and $\tilde{t}$ of players A and B: $$\begin{aligned}
& \min \limits_{t}{\Pi^A(\tilde{s}, t)} \le \min \limits_{t} {\Pi^A({{\rm IU}^{{\gamma^*}}}_A, t)} + \varepsilon {W},\\
& \min \limits_{s}{\Pi^B(s, \tilde{t})} \le \min \limits_{s} {\Pi^B(s,{{\rm IU}^{{\gamma^*}}}_B)} + \varepsilon {W}.
\end{aligned}$$
Intuitively, if player ${p}\in \{A,B \}$ plays the ${{\rm IU}^{{\gamma^*}}}_{p}$ strategy, she guarantees a near-optimal payoff even in the worst-case scenario when her opponent $-{p}$ plays strategies that minimize ${p}$’s payoff (no matters how it affects $-{p}$’s payoff). The proofs of Corollary \[corol:constant\_sum\_Equi\] and Corollary \[corol:max\_minConstantSum\] can be trivially deduced by applying specifically Theorem \[TheoMainBlotto\] to the constant-sum Colonel Blotto games and thus are omitted in this work.
APPROXIMATE EQUILIBRIA OF THE LOTTERY BLOTTO GAME {#sec:LotteryApproximation}
=================================================
In this section, we present the results regarding the ${{\rm IU}^{{\gamma^*}}}$ strategy in the Lottery Blotto games. In Section \[sec:Approx\_Lottery\_LB\_n\], we analyze the game ${\mathcal{LB}_n}(\zeta)$ with an arbitrary pair of CSFs $\zeta=(\zeta_A,\zeta_B)$ and show that the ${{\rm IU}^{{\gamma^*}}}$ strategy is an approximate equilibrium of ${\mathcal{LB}_n}(\zeta)$ with an error depending on the number of battlefields as well as the dissimilarity between $\zeta_A$ and $\beta_A$ (and between $\zeta_B$ and $\beta_B$). In Section \[sec:Approx\_Lottery\_ratio\], we illustrate this result in two particular instances, the games ${\mathcal{LB}_n}(\mu^R)$ and ${\mathcal{LB}_n}(\nu^R)$, belonging to the class of ratio-form Lottery Blotto games. We characterize the approximation error of the ${{\rm IU}^{{\gamma^*}}}$ strategy in these games according to $n$ and the parameter $R$ of these CSFs.
Approximate equilibria of Lottery Blotto games ${\mathcal{LB}_n}(\zeta)$ with general CSFs {#sec:Approx_Lottery_LB_n}
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We start by defining a parameter that expresses the dissimilarity between a given pair of CSFs and the Blotto functions $\beta_A, \beta_B$ (defined in ). First recall that for any $n$ and $i \in [n]$, the random variables $A^*_i, B^*_i$ are upper-bounded by $2 X_B$ (see Lemma \[lem:Preliminary\] in \[sec:appen\_preliminary\]) and by definition, the variables $A^n_i, B^n_i$ are trivially upper-bounded by $X_A,X_B$ (and thus by $2X_B$). Then, given any $\varepsilon>0$, for any $x^*\in [0, 2 X_B]$ and $y^* \in [0, 2X_B]$ (i.e., any number that can be sampled from ${ F_{A^*_i}},{ F_{B^*_i}}, { F_{A^n_i}}$ or ${ F_{B^n_i}}$), we introduce the following sets: $$\begin{aligned}
& {\mathcal{X}_{\zeta}}(y^*,\varepsilon):= \left\{x \in [0,2 X_B]: |\zeta_A(x,y^*) - \beta_A(x,y^*)| \ge \varepsilon \right\}, \label{eq:Xset} \\
& {\mathcal{Y}_{\zeta}}(x^*,\varepsilon):= \left\{y \in [0,2 X_B]: |\zeta_B(x^*,y) - \beta_B(x^*,y)| \ge \varepsilon \right\}.\label{eq:Yset}
$$
\[def:delta\] For any pair of CSFs $\zeta=(\zeta_A, \zeta_B)$, $\varepsilon>0$ and ${\gamma^*}\in {\mathcal{S}_n}$, we define the following set[^15] $${\Delta_{{\gamma^*}}(\zeta, \varepsilon)}:= \left\{ \delta\in [0,1] : \max \limits_{i \in [n]} \max \limits_{y^* \in [0,2X_B]}{\int_{{\mathcal{X}_{\zeta}}(y^*,\varepsilon)} { \!\!\!\!\!{{\rm d}}{ F_{A^*_i}}(x)}} \le \delta, \;\; \textrm{ and } \;\;
\max \limits_{i \in [n]} \max \limits_{x^* \in [0,2X_B]}{\int_{{\mathcal{Y}_{\zeta}}(x^*,\varepsilon)} { \!\!\!\!\!{{\rm d}}{ F_{B^*_i}}(y)}} \le \delta
\right\}.$$
Intuitively, the set ${\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ contains all numbers $\delta \in [0,1]$ such that for any allocation $y^*$ of player B toward an arbitrary battlefield $i$, if player A draws an allocation $x$ from the distribution ${ F_{A^*_i}}$, it only happens with probability at most $\delta$ that the value of the CSF $\zeta_A$ at $(x, y^*)$ is significantly different (i.e., $\varepsilon$-away) from that of the Blotto function $\beta_A$; and we have a similar statement for the distribution ${ F_{B^*_i}}$ of player B and any allocation $x^*$ of player A. Note that the set ${\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ depends on ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$, thus it depends on ${\gamma^*}$. We can trivially see that ${\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ is an interval with the form $[\delta_0, 1]$ since if $\delta_0 \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ then $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ for any $\delta \ge \delta_0$.
Based on the convergence of ${ F_{A^n_i}}$ and ${ F_{B^n_i}}$ toward ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$ (see Lemma \[lem:convergence\] in \[sec:Appen\_Proof\_TheoBlotto\]), we can prove the following lemma (a formal proof is given in \[sec:appen\_proof\_lem\_delta\]):
[lemma]{}[deltalemma]{} \[lem:deltalemma\] For any $\varepsilon \in (0,1]$, there exists a constant $L_0>0$ (that does not depend on $\varepsilon$), such that for any $n \ge L_0 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, for any game ${\mathcal{LB}_n}(\zeta)$, $\gamma^* \in {\mathcal{S}_n}$, $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ and $i \in [n]$, we have: $$\max \left\{ \sup_{y^* \in [0,2X_B]}{\int \nolimits_{{\mathcal{X}_{\zeta}}(y^*,\varepsilon)} { {{\rm d}}{ F_{A^n_i}}(x)}}, \sup_{x^* \in [0,2X_B]}{\int \nolimits_{{\mathcal{Y}_{\zeta}}(x^*,\varepsilon)} { {{\rm d}}{ F_{B^n_i}}(y)}} \right\} \le \delta + \varepsilon. \label{eq:lemma_Fn_LB}$$
Intuitively, this lemma provides an upper-bound for the probability of the value of the CSFs $\zeta$ being $\varepsilon$-away from the Blotto functions when player A (resp. player B) plays such that her allocation to battlefields $i$ follows ${ F_{A^n_i}}$ (resp. ${ F_{B^n_i}}$), i.e., when she plays the ${{\rm IU}^{{\gamma^*}}}$ strategy.
Based on the definition of ${\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ and Lemma \[lem:deltalemma\], we can now show the following result regarding the ${{\rm IU}^{{\gamma^*}}}$ strategy in Lottery Blotto games.
[theorem]{}[LotteTheo]{}**(Approximate equilibria of the Lottery Blotto game).** \[theo:Lottery\_generic\_approx\]
- In any game ${\mathcal{LB}_n}(\zeta)$, there exists a positive number such that for any ${\gamma^*}\in {\mathcal{S}_n}$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, the following inequalities hold for any pure strategy $\boldsymbol{x}^A$ and $\boldsymbol{x}^B$ of players A and B: $$\begin{aligned}
& \Pi_{\zeta}^A(\boldsymbol{x}^A,{{{\rm IU}^{{\gamma^*}}}_B}) \le \Pi_{\zeta}^A({{{\rm IU}^{{\gamma^*}}}_A},{{{\rm IU}^{{\gamma^*}}}_B}) + \left(8\delta + 13{\varepsilon} \right) W_A,\label{eq:lottery_theo_A}\\
& \Pi_{\zeta}^B({{{\rm IU}^{{\gamma^*}}}_A},\boldsymbol{x}^B) \le \Pi_{\zeta}^B({{{\rm IU}^{{\gamma^*}}}_A},{{{\rm IU}^{{\gamma^*}}}_B}) + \left(8\delta + 13{\varepsilon} \right) W_B. \label{eq:lottery_theo_B}
\end{aligned}$$
- For any $\varepsilon \in (0,1]$, there exists a constant $L^* >0 $ (that does not depend on $\varepsilon$) such that in any game ${\mathcal{LB}_n}(\zeta)$ where $ n \ge L^* \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, and hold for any ${\gamma^*}\in {\mathcal{S}_n}$, $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ and any pure strategy $\boldsymbol{x}^A, \boldsymbol{x}^B$ of players A and B.
The proof of this theorem is given in \[sec:appen\_proof\_theoLottery\]. The main idea to prove these results is that we can approximate the players’ payoffs in the game ${\mathcal{LB}_n}(\zeta)$ when they play the ${{\rm IU}^{{\gamma^*}}}$ strategies by that in the corresponding game ${\mathcal{CB}_n}$ (the difference between these payoffs is controlled by the parameter $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$); and then use the results from Section \[sec:ApproximateBlotto\] for the game ${\mathcal{CB}_n}$ (involving the error $\varepsilon$) to prove and . The coefficients (8 and 13) in front of the parameters $\delta$ and $\varepsilon$ come from the application of several triangle inequalities to connect these approximate results. Note that if the CSFs $\zeta_A$ and $\zeta_B$ are Lipschitz continuous on $[0,2X_B] \times [0,2X_B]$, we can avoid the need to approximate several terms involved in the analysis of using the ${{\rm IU}^{{\gamma^*}}}$ strategy in the game ${\mathcal{LB}_n}(\zeta)$ via the corresponding terms in the game ${\mathcal{CB}_n}$; thus, we can improve the results in Theorem \[theo:Lottery\_generic\_approx\] to obtain an approximation error of $2\delta + 5 \varepsilon$ instead of $8\delta + 13 \varepsilon $ (see Remark \[remark\_conti\_CSF\] in \[sec:appen\_remark\_conti\_CSF\] for more details). Here, to keep the generality, we do not include the continuity assumption of the CSFs in Theorem \[theo:Lottery\_generic\_approx\] (recall that our definition of a CSF allows for discontinuity). Intuitively, Result $(i)$ of Theorem \[theo:Lottery\_generic\_approx\] determines the order of the approximattion error while using ${{\rm IU}^{{\gamma^*}}}$ in any given game ${\mathcal{LB}_n}(\zeta)$. Straightforwardly, we can deduce that the ${{\rm IU}^{{\gamma^*}}}$ strategy is an approximate equilibrium of the game ${\mathcal{LB}_n}(\zeta)$, formally stated as follows.
\[corol:Lottery\_LB\_n\] In any game ${\mathcal{LB}_n}(\zeta)$, there exists a positive number such that for any ${\gamma^*}\in {\mathcal{S}_n}$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, the ${{\rm IU}^{{\gamma^*}}}$ strategy is an $\left(8\delta + 13{\varepsilon} \right)W$-equilibrium where $W:= \max\{W_A,W_B \}$.
We observe that the error bound in Theorem \[theo:Lottery\_generic\_approx\] (and in Corollary \[corol:Lottery\_LB\_n\]) is valid for any $\delta$ of the set ${\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$. Naturally, it is the tightest for $\delta_0 = \min \{\delta: \delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}\}$; but this quantity is not always easy to compute; for instances, in the Lottery Blotto games with the power and logit form CSFs (i.e., $\mu^R$ and $\nu^R$). Still, in Section \[sec:Approx\_Lottery\_ratio\], we show that there exists an element of ${\Delta_{{\gamma^*}}(\mu^R, \varepsilon)}$ and ${\Delta_{{\gamma^*}}(\nu^R, \varepsilon)}$ that is negligibly small, given appropriate parameter’s configurations of the games; in other words, we can still obtain a good error’s upper-bound for the ${{\rm IU}^{{\gamma^*}}}$ strategy in these games. Note that, on the other hand, the Colonel Blotto game ${\mathcal{CB}_n}$ can be considered as an instance of the game ${\mathcal{LB}_n}(\zeta)$ where the CSFs are $\zeta_A= \beta_A$ and $\zeta_B = \beta_B$; therefore, it also satisfies Theorem \[theo:Lottery\_generic\_approx\]. In ${\mathcal{CB}_n}$, we trivially have ${\mathcal{X}_{\zeta}}(y^*, \varepsilon) = {\mathcal{Y}_{\zeta}}(x^*, \varepsilon) = \emptyset$ for any $x^*, y^*$; thus ${\Delta_{{\gamma^*}}(\zeta, \varepsilon)}= [0,1]$ for any $\varepsilon>0$ and $\min \{\delta: \delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}\} =0$.[^16] This is consistent with results obtained in Theorem \[TheoMainBlotto\] in Section \[sec:ApproximateBlotto\]. In Theorem \[theo:Lottery\_generic\_approx\], Result $(ii)$ is an equivalent statement of Result $(i)$. It indicates the number of battlefields needed to guarantee a certain level of approximation error when using the ${{\rm IU}^{{\gamma^*}}}$ strategy in the game ${\mathcal{LB}_n}(\zeta)$. For instance, to obtain an approximate equilibrium of the game ${\mathcal{LB}_n}(\zeta)$ where the level of error is less than a certain number $\bar \varepsilon$, one needs $\varepsilon \le \bar{\varepsilon}$ such that we can find a $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ satisfying $8 \delta + 13 \varepsilon \le \bar{\varepsilon}$; from these parameters, by Result $(ii)$, one can deduce the sufficient number of battlefields needed for an ${\mathcal{LB}_n}$ game to yield that desired level of error.
Finally, in the constant-sum variant of the Lottery Blotto game denoted by ${\mathcal{LB}_n}^c(\zeta)$ (i.e., when , ), we can easily deduce from Theorem \[theo:Lottery\_generic\_approx\] that the ${{\rm IU}^{{\gamma^*}}}$ strategy is also an approximate max-min strategy:
\[corol:max\_minLB\] In any game ${\mathcal{LB}_n}^c(\zeta)$, there exists such that for any and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, the following inequalities hold for any strategy $\tilde{s}$ and $\tilde{t}$ of players A and B:[^17] $$\begin{aligned}
& \min \limits_{t}{\Pi^A_\zeta(\tilde{s}, t)} \le \min \limits_{t} {\Pi^A_\zeta({{\rm IU}^{{\gamma^*}}}_A, t)} + (8 \delta + 13\varepsilon) {W}, \\
& \min \limits_{s}{\Pi^B_\zeta(s, \tilde{t})} \le \min \limits_{s} {\Pi^B_\zeta(s,{{\rm IU}^{{\gamma^*}}}_B)} + (8 \delta + 13\varepsilon){W}.
\end{aligned}$$
Approximate equilibria of the ratio-form Lottery Blotto games ${\mathcal{LB}_n}(\mu^R)$ and ${\mathcal{LB}_n}(\nu^R)$ {#sec:Approx_Lottery_ratio}
---------------------------------------------------------------------------------------------------------------------
We now consider the ratio-form Lottery Blotto games ${\mathcal{LB}_n}(\mu^R)$ and ${\mathcal{LB}_n}(\nu^R)$. Recall that the corresponding CSFs are defined in Table \[table:CSF\] and that for those CSFs, we do not consider the degenerate cases where $\alpha=0$ or $\alpha =1$ in which trivial equilibria exist. The games ${\mathcal{LB}_n}(\mu^R)$ and ${\mathcal{LB}_n}(\nu^R)$ are instances of the game ${\mathcal{LB}_n}(\zeta)$ studied in Section \[sec:Approx\_Lottery\_LB\_n\]; therefore, by Theorem \[theo:Lottery\_generic\_approx\] (and Corollary \[corol:Lottery\_LB\_n\]), the ${{\rm IU}^{{\gamma^*}}}$ strategy is also an approximate equilibrium of them. In this section, we focus on characterizing the approximation error of the ${{\rm IU}^{{\gamma^*}}}$ strategy in these games according to $n$ (the number of battlefields) and $R$ (the corresponding parameter of the CSFs). We will show that this error quickly tends to zero as $n$ and $R$ increase under appropriate conditions. To do this, we first notice that although it is non-trivial to analyze the closed form of the sets ${\Delta_{{\gamma^*}}(\mu^R, \varepsilon)}$ and ${\Delta_{{\gamma^*}}(\nu^R, \varepsilon)}$ and find their minimum, we can find small elements of theses sets.
[lemma]{}[lemmamunu]{} \[lem:delta\_mu\_nu\] Fix $n \ge 2 $, $R>0$ and $\alpha \in (0,1)$, for any $\varepsilon <
\min \{\alpha, 1 - \alpha\}$, we have:[^18]
- In any game ${\mathcal{LB}_n}(\mu^R)$ with $\alpha$ as the tie-breaking parameter, there exists such that for any ${\gamma^*}\in {\mathcal{S}_n}$.
- In any game ${\mathcal{LB}_n}(\nu^R)$ with $\alpha$ as the tie-breaking parameter, there exists such that for any ${\gamma^*}\in {\mathcal{S}_n}$.
The proof of Lemma \[lem:delta\_mu\_nu\] is given in \[sec:appen\_proof\_ratio-form\]. Note that for the sake of generality, the parameters $\delta_{\mu}$ and $\delta_{\nu}$ are indicated in this lemma in such a way that they do not depend on ${\gamma^*}$, but for each ${\gamma^*}\in {\mathcal{S}_n}$, we can find smaller elements of the corresponding sets ${\Delta_{{\gamma^*}}(\mu^R, \varepsilon)}$ and ${\Delta_{{\gamma^*}}(\nu^R, \varepsilon)}$. More importantly, for a fixed $n$, the numbers $\delta_\mu$ and $\delta_\nu$ decrease as $R$ increases; but $\delta_\mu$ and $\delta_\nu$ increase as $\varepsilon$ decreases. While the lemma is valid for any parameter values, since $1$ is a trivial element of ${\Delta_{{\gamma^*}}(\mu^R, \varepsilon)}$ and ${\Delta_{{\gamma^*}}(\nu^R, \varepsilon)}$, it is useful only if $\delta_\mu,\delta_\nu < 1$; this is guaranteed whenever $R \ge {\mathcal{O}}\left( n \ln(\varepsilon^{-1}) \right)$. Note finally that the condition $\varepsilon < \min\{\alpha, 1- \alpha\}$ in the statement of Lemma \[lem:delta\_mu\_nu\] does not limit its use since our goal is to obtain asymptotic results on the ${{\rm IU}^{{\gamma^*}}}$ strategy when $\varepsilon$ tends to $0$. Moreover, in the games ${\mathcal{LB}_n}(\mu^R)$ and ${\mathcal{LB}_n}(\nu^R)$ where $\alpha$ is either very close to $0$ or $1$, one player has a very high advantage and always obtains large gains from all battlefields (where her allocation is strictly positive) while her opponent gains very little regardless of her allocations; therefore, there exists (many) trivial approximate equilibria with small errors.
Combining the results of Corollary \[corol:Lottery\_LB\_n\] and Lemma \[lem:delta\_mu\_nu\], we can deduce directly that in any game ${\mathcal{LB}_n}(\mu^R)$ (resp. ${\mathcal{LB}_n}(\nu^R)$), there exists $\varepsilon \le \tilde{{\mathcal{O}}}(n^{-1/2})$ such that for any ${\gamma^*}\in {\mathcal{S}_n}$, the ${{\rm IU}^{{\gamma^*}}}$ strategy is an $(8\varepsilon + 13\delta_{\mu}) W$-equilibrium (resp. $(8\varepsilon + 13\delta_{\nu}) W$-equilibrium). Next, we look for the asymptotic relation between these error terms and the parameters $n, R$ of the games. First, as $n$ increases, the error level $\varepsilon$ decreases; on the other hand, from Lemma \[lem:delta\_mu\_nu\], the number $\delta_\mu$ (and $\delta_\nu$) decreases if $R$ increases with a faster rate than $\tilde{{\mathcal{O}}}(n)$. However, there is a trade-off between $\varepsilon$ and $\delta_\mu$ (or $\delta_\nu$): as $\varepsilon$ decreases, $\delta_\mu$ (and $\delta_\nu$) increases and vice versa. To handle this trade-off between $\delta_\mu$ and $\varepsilon$ (resp. $\delta_\nu$ and $\varepsilon$), we can first find a condition on $n$ that generates a small error $\varepsilon$, and then find a condition on $R$ (with respect to $n$) such that the error $\delta_\mu$ (resp. $\delta_\nu$) is of the same order as $\varepsilon$. Formally, we state the result that the ${{\rm IU}^{{\gamma^*}}}$ strategy yields an approximate equilibrium of the games ${\mathcal{LB}_n}(\mu^R)$ and ${\mathcal{LB}_n}(\nu^R)$ with any arbitrary small error in the next theorem.
[theorem]{}[theoratioform]{}**(Approximate equilibria of the ratio-form Lottery Blotto games)** \[theoratioform\] For any and $\alpha \in (0,1)$ such that ${\bar{\varepsilon}}< \min\{\alpha, 1- \alpha\}$, there exists $\tilde{L}>0$ such that for any $ n \ge \tilde{L} {\bar{\varepsilon}}^{-2} {\ln\left(\frac{1}{\min\{{\bar{\varepsilon}},1/{e}\}}\right)}$, $R \ge {\mathcal{O}}\left(\frac{n}{{\bar{\varepsilon}}} \ln\left( \frac{1}{{\bar{\varepsilon}}}\right) \right)$ and , the ${{\rm IU}^{{\gamma^*}}}$ strategy is an ${\bar{\varepsilon}}W$-equilibrium of any game ${\mathcal{LB}_n}(\mu^R)$ and ${\mathcal{LB}_n}(\nu^R)$ having $\alpha$ as the tie-breaking-rule parameter.
The proof of this theorem is based on Theorem \[theo:Lottery\_generic\_approx\] and Lemma \[lem:delta\_mu\_nu\] (see \[sec:appen\_proof\_theoratioLB\] for more details). Theorem \[theoratioform\] involves a double limit in $R$ and $n$. Intuitively, if $n$ and $R$ increase but $R$ increases with a slower rate, then $\varepsilon$ decreases but the corresponding $\delta_\mu$ and $\delta_\nu$ do not decrease; thus, the total error is not guaranteed to decrease.
CONCLUSION {#Conclu}
==========
In this work, we consider the most general variant of the Colonel Blotto game—the non-constant-sum variant with heterogeneous battlefields and asymmetric players. While most of (if not all) works in the literature attempt (but do not completely succeed) to construct an exact equilibrium of the CB game by looking for a joint distribution with the uniform-type marginals that satisfies the budget constraints, we take a different angle. We propose a class of strategies called the ${{\rm IU}^{{\gamma^*}}}$ strategies that is simply constructed by an efficient algorithm; the ${{\rm IU}^{{\gamma^*}}}$ strategies guarantee the budget constraints but their marginals are not the uniform-type distributions. Yet, we prove the ${{\rm IU}^{{\gamma^*}}}$ strategy to be an approximate equilibrium of the CB games. We also define an extended game called the Lottery Blotto game and obtain similar results. We characterize the approximate error in our results in terms of the number of battlefields of the games. Our work extends the scope of applications of the CB games and its variants.
Throughout the paper, we emphasized the dependence of the approximation error on the number of battlefields $n$. Yet, although the dependence on other parameters of the games is not explicitly emphasized, it can be extracted from our analysis and the proofs of the stated results. It is also interesting to note that although the notion of approximate equilibrium is defined in terms of payoffs (the payoffs when players play the ${{\rm IU}^{{\gamma^*}}}$ strategy are close to optimal), the ${{\rm IU}^{{\gamma^*}}}$ strategy also approximates the equilibrium marginals (if an equilibrium exists)—that is, it is also an approximate equilibrium in terms of strategies.
Our approximation results are valid even in the case where no equilibrium exists (and we do not include the assumption that requires the existence of the equilibrium). Particularly in the cases of the CB game where it is known that there exists no equilibrium yielding the uniform-type marginals, the ${{\rm IU}^{{\gamma^*}}}$ strategy is still an approximate equilibrium, yet we suspect that in those cases the approximation error might be large. On the other hand, our work does not solve the question of the existence of an exact Nash equilibrium. In particular, we leave as future work the investigation of possible conditions under which a Nash equilibrium exists, for instance for a large-enough number of battlefields. We also finally note that, in the non-constant-sum version, the existence of multiple solutions ${\gamma^*}$ of Equation leads to problems of equilibrium selection (in practical contexts involving a social welfare measurement) among the ${{\rm IU}^{{\gamma^*}}}$ strategies with different ${\gamma^*}\in {\mathcal{S}_n}$, which we also leave as future work.
ACKNOWLEDGEMENT {#acknowledgement .unnumbered}
===============
This work was supported by the French National Research Agency through the “Investissements d’avenir” program (ANR-15-IDEX-02) and through grant ANR-16-TERC0012; and by the Alexander von Humboldt Foundation.
REFERENCES {#references .unnumbered}
==========
NOMENCLATURES AND PRELIMINARIES {#sec:appen_preliminary}
===============================
[r c p[10cm]{} ]{}\
CB (${\mathcal{CB}_n}$) & $\triangleq$ & non-constant-sum Colonel Blotto game (with $n$ battlefields)\
LB (${\mathcal{LB}_n}$) & $\triangleq$ & non-constant-sum Lottery Blotto game (with $n$ battlefields)\
${\mathcal{CB}_n}^c, {\mathcal{LB}_n}^c$ & $\triangleq$ & the constant-sum versions of ${\mathcal{CB}_n}$ and ${\mathcal{LB}_n}$ games\
CSF & $\triangleq$ & contest success function.\
${{\rm IU}^{{\gamma^*}}}$ ($=({{\rm IU}^{{\gamma^*}}}_A, {{\rm IU}^{{\gamma^*}}}_B)$) & $\triangleq$ & independent uniform strategy (corresponding to $\gamma^*$)\
\
$X_A, X_B$ & $\triangleq$ & budgets of player A and B respectively\
$n$ & $\triangleq$ & number of battlefields\
$w^A_i, w^B_i$ & $\triangleq$ & values of battlefield $i$ assessed by player A and B respectively\
${\underaccent{\bar}{w}}, {\bar{w}}$ & $\triangleq$ & lower and upper bounds of battlefields’ values\
$W_A, W_B$ & $\triangleq$ & sums of battlefields’ values, $W_A:= \sum_{i=1}^n w^A_i$, $W_B:= \sum_{i=1}^n w^B_i$\
$W$ & $\triangleq$ & $\max\{W_A, W_B\}$\
$v^A_i, v^B_i$ & $\triangleq$ & normalized values of battlefield $i$ assessed by player A and B\
$x^A_i, x^B_i$ & $\triangleq$ & the allocation to battlefield $i$ of player A and B respectively\
$ \Pi^A(s,t), \Pi^B(s,t)$ & $\triangleq$ & players’ payoffs in CB games when playing the strategies $s$ and $t$\
$\alpha$ & $\triangleq$ & the tie-breaking parameter\
$\beta_A, \beta_B$ & $\triangleq$ & Blotto functions (see )\
$\zeta$ & $\triangleq$ & $(\zeta_A, \zeta_B)$—the generic CSFs\
${\mathcal{LB}_n}(\zeta)$ & $\triangleq$ & the LB game with CSFs $\zeta_A, \zeta_B$\
$\mu^R$ & $\triangleq$ & $(\mu_A^R, \mu^R_B)$—the power form CSFs with parameter $R$ (see Table \[table:CSF\])\
$\nu^R$ & $\triangleq$ & $(\nu_A^R, \nu^R_B)$—the logit form CSFs with parameter $R$ (see Table \[table:CSF\])\
$ \Pi_{\zeta}^A(s,t), \Pi_{\zeta}^B(s,t)$ & $\triangleq$ & players’ payoffs in ${\mathcal{LB}_n}(\zeta)$ games when playing the strategies $s$, $t$\
${\mathcal{X}_{\zeta}}(y^*, \varepsilon), {\mathcal{Y}_{\zeta}}(x^*, \varepsilon) $ & $\triangleq$ & the sets characterizing the dissimilarity between $(\beta_A,\beta_B)$ and $(\zeta_A,\zeta_B)$ (see , )\
${\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ & $\triangleq$ & see Definition \[def:delta\]\
\
$\gamma^*$ & $\triangleq$ & a positive solution of Equation \
${\mathcal{S}_n}$ & $\triangleq$ & the set of positive solutions of Equation w.r.t. ${\mathcal{CB}_n}$ (or ${\mathcal{LB}_n}$)\
${\underaccent{\bar}{\gamma}}, {\bar{\gamma}}$ & $\triangleq$ & lower and upper bounds of any $\gamma^* \in {\mathcal{S}_n}$ (see Proposition \[Prop:BoundLambda\])\
$\Omega_A(\gamma^*)$ & $\triangleq$ & $ \left\{i \in [n]: v^A_i/v^B_i > \gamma^* \right\}$\
${\lambda^*_A}, {\lambda^*_B}$ & $\triangleq$ & Lagrange multipliers corresponding to $\gamma^*$ (see , )\
${\underaccent{\bar}{\lambda}}, {\bar{\lambda}}$ & $\triangleq$ & lower and upper bounds of ${\lambda^*_A}, {\lambda^*_B}$ (see Proposition \[Prop:BoundLambda\])\
${ F_{A^*_i}},{ F_{B^*_i}}$& $\triangleq$ & uniform-type distributions (see -\
${ A^S_{{\gamma^*},i}},{ A^W_{{\gamma^*},i}},A^n_i$ & $\triangleq$ & random variables defined in -\
$ { B^S_{{\gamma^*},i}},{ B^W_{{\gamma^*},i}},B^n_i$ & &\
${ F_{A^n_i}}, { F_{B^n_i}}$ & $\triangleq$ & the marginals corresponding to battlefield $i$ of the ${{\rm IU}^{{\gamma^*}}}$ strategy\
$A_{=0}, A_{> 0}$ & $\triangleq$ & the events $\{\sum_{i \in [n]} A^*_i = 0\}$ and $\{\sum_{i \in [n]} A^*_i > 0\}$, respectively\
$B_{=0}, B_{> 0}$ & $\triangleq$ & the events $\{\sum_{i \in [n]} B^*_i = 0\}$ and $\{\sum_{i \in [n]} B^*_i > 0\}$, respectively\
\[table:notation\]
\[lem:Preliminary\] Given a game ${\mathcal{CB}_n}$ (or ${\mathcal{LB}_n}$), for any ${\gamma^*}\in {\mathcal{S}_n}$, we have:
1. ${\lambda^*_A}, {\lambda^*_B}>0$ and ${\gamma^*}= {{\lambda^*_A}}/{{\lambda^*_B}}$.
2. For any $i \in [n]$, ${\mathbb{E}}[{ A^S_{{\gamma^*},i}}]= \frac{1}{2}\frac{v^B_i}{{\lambda^*_B}}$, ${\mathbb{E}}[{ A^W_{{\gamma^*},i}}]= \left(\frac{v^A_i}{{\lambda^*_A}}\right)^2 \frac{{\lambda^*_B}}{2 v^B_i}$, ${\mathbb{E}}[{ B^S_{{\gamma^*},i}}]= \frac{1}{2}\frac{v^A_i}{{\lambda^*_A}}$ and ${\mathbb{E}}[{ B^W_{{\gamma^*},i}}]= \left(\frac{v^B_i}{{\lambda^*_B}}\right)^2 \frac{{\lambda^*_A}}{2 v^A_i}$.
3. $X_A = \sum\nolimits_{i \in [n]}{{\mathbb{E}}[A^*_i]}$ and $X_B = \sum\nolimits_{i \in [n]}{{\mathbb{E}}[B^*_i]}$.
4. For any $i \in [n]$, $A^*_i$ and $B^*_i$ have a constant upper-bound; particularly, .
$ $
- The positivity of ${\lambda^*_A}$ and ${\lambda^*_B}$ follows from the positivity of ${\gamma^*}$ and the definitions of ${\lambda^*_A}$ and ${\lambda^*_B}$ in and . By dividing by and combining with , we trivially have that ${\gamma^*}= {{\lambda^*_A}}/{{\lambda^*_B}}$.
- These results come directly from the definitions of the distributions $F_{{ A^S_{{\gamma^*},i}}}$, $F_{{ A^W_{{\gamma^*},i}}}$ $F_{{ B^S_{{\gamma^*},i}}}$ and $F_{{ B^W_{{\gamma^*},i}}}$.
- We multiply both sides of by $X_A/{\lambda^*_B}$ and both sides of by $X_B/{\lambda^*_A}$ then using the fact that ${\gamma^*}= {\lambda^*_A}/{\lambda^*_B}$ to obtain the following: $$\begin{aligned}
X_A & = \sum_{j \in {\Omega_A({\gamma^*})}}{\frac{1}{2} \frac{v^B_j}{{\lambda^*_B}}} + \sum_{j \notin {\Omega_A({\gamma^*})}}{\left(\frac{v^A_j}{{\lambda^*_A}}\right)^2 \frac{{\lambda^*_B}}{2 v^B_j}}, \label{eq:prelimi_XA}\\
X_B & = \sum_{j \in {\Omega_A({\gamma^*})}}{\left(\frac{v^B_j}{{\lambda^*_B}}\right)^2 \frac{{\lambda^*_A}}{2 v^A_j}} + \sum_{j \notin {\Omega_A({\gamma^*})}}{\frac{1}{2}\frac{v^A_j}{{\lambda^*_A}}}. \label{eq:prelimi_XB}
\end{aligned}$$ Combining with $(ii)$, we deduce that $X_A = \sum\nolimits_{i \in [n]}{{\mathbb{E}}[A^*_i]}$ and $X_B = \sum\nolimits_{i \in [n]}{{\mathbb{E}}[B^*_i]}$.
- If $i \in {\Omega_A({\gamma^*})}$, we have $A^*_i = { A^S_{{\gamma^*},i}}$ and $B^*_i = { B^W_{{\gamma^*},i}}$. Recalling Definition \[def:UnifromDistributions\], we have that and . On the other hand, from , we deduce $$X_B \ge X_A \ge \sum_{j \in {\Omega_A({\gamma^*})}}{\frac{v^B_j}{2{\lambda^*_B}}} \ge \frac{v^B_i}{2 {\lambda^*_B}}.$$ Therefore, $ {\mathbb{P}}( A^S_i \! \le\! 2X_B) \!\ge \!{\mathbb{P}}\left(A^S_i \! \le \! {v^B_i}/{{\lambda^*_B}} \right) \! = \! 1 $ and ${\mathbb{P}}(B^W_i \le 2X_B) \ge {\mathbb{P}}(B^W_i \le {v^B_i}/{{\lambda^*_B}}) = 1$. We conclude that for any $i \in {\Omega_A({\gamma^*})}$, $A^*_i, B^*_i$ are bounded by $2X_B$.
If $i \notin {\Omega_A({\gamma^*})}$, we have $A^*_i = { A^W_{{\gamma^*},i}}$ and $B^*_i = { B^S_{{\gamma^*},i}}$. Recalling Definition \[def:UnifromDistributions\], we have that and . On the other hand, from , we deduce $$X_B \ge \sum_{j \notin {\Omega_A({\gamma^*})}}{\frac{v^A_j}{2{\lambda^*_A}}} \ge \frac{v^A_i}{2 {\lambda^*_A}}.$$ Therefore, $ {\mathbb{P}}( A^W_i \le 2X_B) \ge {\mathbb{P}}\left(A^W_i \le {v^A_i}/{{\lambda^*_A}} \right) =1$ and ${\mathbb{P}}(B^S_i \le 2X_B) \ge {\mathbb{P}}(B^S_i \le {v^A_i}/{{\lambda^*_A}})=1$. We conclude that for $i \notin {\Omega_A({\gamma^*})}$, $A^*_i, B^*_i$ are also bounded by $2X_B$.
Let ${\gamma^*}\in {\mathcal{S}_n}$, we consider the following cases:
#### Case 1: If $0< {\gamma^*}< \min \limits_{i \in [n]} \left\{ \frac{v^A_i}{v^B_i} \right\}$.
In this case, ${\Omega_A({\gamma^*})}= [n]$, and since ${\gamma^*}$ is a solution of , we deduce: $${\gamma^*}= \frac{X_B}{X_A} \frac{\sum\nolimits_{i=1}^n{v^B_i}} {\sum\nolimits_{i=1}^n{\frac{(v^B_i)^2}{v^A_i}}} \ge \frac{X_B}{X_A} \frac{n \frac{{\underaccent{\bar}{w}}}{n {\bar{w}}}}{n\frac{\left(\frac{{\bar{w}}}{n {\underaccent{\bar}{w}}} \right)^2 }{\frac{{\underaccent{\bar}{w}}}{n {\bar{w}}}}} = \frac{X_B}{X_A} \left(\frac{{\underaccent{\bar}{w}}}{{\bar{w}}} \right)^4.$$
Here, the inequality comes directly from .
#### Case 2: If ${\gamma^*}\ge \max \limits_{i \in [n]} \left\{ \frac{v^A_i}{v^B_i} \right\}$.
In this case, ${\Omega_A({\gamma^*})}= \emptyset$, and since ${\gamma^*}$ is a solution of , we deduce: $${\gamma^*}= \frac{X_B}{X_A} \frac{\sum\nolimits_{i=1}^n{\frac{(v^A_i)^2}{v^B_i}}} {\sum\nolimits_{i=1}^n{v^A_i}} \le \frac{X_B}{X_A} \left(\frac{{\bar{w}}}{{\underaccent{\bar}{w}}} \right)^4.$$
#### Case 3: If $\exists i, j : \frac{v^A_i}{v^B_i} \le {\gamma^*}< \frac{v^A_j}{v^B_j}$.
In this case, trivially from , we have ${\gamma^*}\in \left[ {\left( \frac{{\underaccent{\bar}{w}}}{{\bar{w}}} \right)^2} , {\left(\frac{{\bar{w}}}{{\underaccent{\bar}{w}}} \right)^2} \right]$.
In conclusion, by denoting and , we have the conclusion on the bounds of ${\gamma^*}$.
On the other hand, from the definition of ${\lambda^*_A}$ in , we deduce $$\begin{aligned}
{\lambda^*_A}&\ge \frac{({{\gamma^*}})^2}{2{X_B}}\sum\nolimits_{i \in {\Omega _A}({{\gamma^*}})} { \left(\frac{{\underaccent{\bar}{w}}}{n {\bar{w}}}\right)^2 \frac{1}{\frac{{\bar{w}}}{n {\underaccent{\bar}{w}}}} } + \frac{1}{{2{X_B}}}\sum\nolimits_{i \notin {\Omega _A}({{\gamma^*}})} {\frac{{\underaccent{\bar}{w}}}{n {\bar{w}}}} \\
& \ge \min \left\{\frac{({{\gamma^*}})^2}{2{X_B}},\frac{1}{2{X_B}} \right\} \cdot \sum \nolimits_{i \in [n]} {\frac{1}{n} \left(\frac{{\underaccent{\bar}{w}}}{ {\bar{w}}}\right)^3}\\
& \ge \min \left\{\frac{({{\gamma^*}})^2}{2{X_B}},\frac{1}{2{X_B}} \right\} \cdot \left(\frac{{\underaccent{\bar}{w}}}{ {\bar{w}}}\right)^3.
\end{aligned}$$ Similarly, we have the upper-bound $${\lambda^*_A}\le \max \left\{\frac{({{\gamma^*}})^2}{2{X_B}},\frac{1}{2{X_B}} \right\} \cdot \left[ \sum\limits_{i \in {\Omega _A}({{\gamma^*}})} { \frac{1}{n} \left(\frac{{\bar{w}}}{ {\underaccent{\bar}{w}}}\right)^3 } + \sum\limits_{i \notin {\Omega _A}({{\gamma^*}})} { \frac{1}{n} \left(\frac{{\bar{w}}}{ {\underaccent{\bar}{w}}}\right)^3 }\right] = \max \left\{\frac{({{\gamma^*}})^2}{2{X_B}},\frac{1}{2{X_B}} \right\} \cdot\left(\frac{{\bar{w}}}{ {\underaccent{\bar}{w}}}\right)^3.$$ Similarly, we can prove that $\min\left\{\frac{1}{2{X_A}}, \frac{1}{2{{\gamma^*}} ^2 {X_A}}\right\} \left(\frac{{\underaccent{\bar}{w}}}{ {\bar{w}}}\right)^3 \!\le\! {\lambda^*_B}\le \max \left\{\frac{1}{2{X_A}}, \frac{1}{2{{\gamma^*}} ^2 {X_A}} \right\} \left(\frac{{\bar{w}}}{ {\underaccent{\bar}{w}}}\right)^3 $; therefore, $$\min \left\{ \frac{{{\gamma^*}}^2}{2{X_B}},\frac{1}{2{X_B}} , \frac{1}{2{X_A}}, \frac{1}{2({\gamma^*}) ^2 {X_A}} \right\}\left(\frac{{\underaccent{\bar}{w}}}{ {\bar{w}}}\right)^3 \le {\lambda^*_A}, {\lambda^*_B}\le \max \left\{\frac{{{\gamma^*}}^2}{2{X_B}},\frac{1}{2{X_B}}, \frac{1}{2{X_A}}, \frac{1}{2({\gamma^*}) ^2 {X_A}} \right\} \left(\frac{{\bar{w}}}{ {\underaccent{\bar}{w}}}\right)^3.$$ Since ${\gamma^*}\in [{\underaccent{\bar}{\gamma}}, {\bar{\gamma}}]$, ${\lambda^*_A}$ and ${\lambda^*_B}$ are bounded in $[{\underaccent{\bar}{\lambda}}, {\bar{\lambda}}]$, where $$\begin{aligned}
& {\underaccent{\bar}{\lambda}}:= \min \left\{ \frac{{{\underaccent{\bar}{\gamma}}}^2}{2{X_B}},\frac{1}{2{X_B}} , \frac{1}{2{X_A}}, \frac{1}{2 {\bar{\gamma}}^2 {X_A}} \right\}\left(\frac{{\underaccent{\bar}{w}}}{ {\bar{w}}}\right)^3, \\
& {\bar{\lambda}}:= \max \left\{\frac{{{\bar{\gamma}}}^2}{2{X_B}},\frac{1}{2{X_B}}, \frac{1}{2{X_A}}, \frac{1}{2 {\underaccent{\bar}{\gamma}}^2 {X_A}} \right\} \left(\frac{{\bar{w}}}{ {\underaccent{\bar}{w}}}\right)^3.\end{aligned}$$
Finally, we prove a trivial result that will be used quite often in the remainder of this work.
\[lem:log\_pre\] For any ${\hat{\varepsilon}}>0$ and ${\hat{C}}\ge 1$, we have that .
If ${\hat{\varepsilon}}< 1/{e}$. In this case, we have $\ln(1/{\hat{\varepsilon}}) > 1$; therefore, $$(\ln({\hat{C}}) + 1) \ln \left( \frac{1}{\min\{{\hat{\varepsilon}}, 1/{e}\}} \right) = (\ln({\hat{C}}) + 1) \ln \left( \frac{1}{{\hat{\varepsilon}}} \right) = \ln({\hat{C}})\ln \left( \frac{1}{{\hat{\varepsilon}}} \right) + \ln \left( \frac{1}{{\hat{\varepsilon}}} \right) > \ln({\hat{C}}) + \ln \left( \frac{1}{{\hat{\varepsilon}}} \right) = \ln \left( \frac{{\hat{C}}}{{\hat{\varepsilon}}} \right).$$
If ${\hat{\varepsilon}}\ge 1/{e}$. We have $\ln(1/{\hat{\varepsilon}}) \le 1$; therefore, $$(\ln({\hat{C}}) + 1) \ln \left( \frac{1}{\min\{{\hat{\varepsilon}}, 1/{e}\}} \right) = (\ln({\hat{C}}) + 1) \ln \left( \frac{1}{1/{e}} \right) = \ln({\hat{C}}) + 1 \ge \ln({\hat{C}}) + \ln \left( \frac{1}{{\hat{\varepsilon}}} \right) = \ln \left( \frac{{\hat{C}}}{{\hat{\varepsilon}}} \right).$$
PROOF OF THEOREM \[TheoMainBlotto\] {#sec:Appen_Proof_TheoBlotto}
===================================
First note that in the remainders of the paper, for any bounded, non-negative random variable $Z$ (i.e., ${\mathbb{P}}(Z\in[0,C])=1$), any measurable function $g$ on $\mathbb{R}$, we write $\int \nolimits_0^{\infty} \! {g(x) {{\rm d}}{F_{Z}(x)}}$ instead of $\int \nolimits_0^{C} \! {g(x) {{\rm d}}{F_{Z}(x)}}$ if there is no need to emphasize the bounds of $Z$. For the sake of notation, we also denote by $A_{=0}$ the event $\left\{ \sum \nolimits_{j \in [n]}{A^*_j} = 0 \right\}$ and by $A_{>0}$ its complement event, that is $\left\{ \sum \nolimits_{j \in [n]}{A^*_j} > 0 \right\}$. Similarly, we denote by $B_{=0}$ the event $\left\{ \sum \nolimits_{j \in [n]}{B^*_j} = 0 \right\}$ and by $B_{>0}$ the event $\left\{ \sum \nolimits_{j \in [n]}{B^*_j} > 0 \right\}$.
Recall the notation ${ F_{A^n_i}}$ and ${ F_{B^n_i}}$ as the univariate marginal distributions corresponding to battlefield $i \in [n]$ of the ${{\rm IU}^{{\gamma^*}}}_A$ and ${{\rm IU}^{{\gamma^*}}}_B$ strategies (the corresponding random variables are denoted $A^n_i$ and $B^n_i$). Due to the definition of the ${{\rm IU}^{{\gamma^*}}}$ strategy (via Algorithm \[alg:IU\_strategy\]), for any $x\ge 0$ and $i \in [n]$, we have: $$\begin{aligned}
{ F_{A^n_i}}(x) &= {\mathbb{P}}\left(\left\{A^n_i \le x\right\} \bigcap A_{=0} \right) + {\mathbb{P}}\left( \left\{A^n_i \le x \right\} \bigcap A_{>0} \right) \nonumber \\
& = {\mathbb{P}}\left( A_{=0}\right) + \! {\mathbb{P}}\left( \left\{ \frac{A^*_i \cdot X_A}{\sum \nolimits_{j\in[n]} A^*_j} \!\le\! x \right\} \bigcap A_{>0} \right). \label{eq:A^n_Def}\end{aligned}$$ Here, we have used the fact that if $\sum \nolimits_{j\in[n]} A^*_j = 0$ (i.e., when $A_{=0}$ happens), then $A^n_i = 0$ by definition and thus, ${\mathbb{P}}(A^n_i \le x) =1$ and ${\mathbb{P}}\left(\left\{A^n_i \le x\right\} \bigcap A_{=0} \right) = {\mathbb{P}}(A_{=0})$. Similarly to , for any $x\ge 0$ and $i \in [n]$, $${ F_{B^n_i}}(x) = {\mathbb{P}}\left( B_{=0}\right) + \! {\mathbb{P}}\left( \left\{ \frac{B^*_i \cdot X_B}{\sum \nolimits_{j\in[n]} B^*_j} \!\le\! x \right\} \bigcap B_{>0} \right). \label{eq:B^n_Def}$$ Regarding the random variables $A^n_i$ and $B^n_i$ ($i \in [n]$), we prepare a lemma stating several useful results as follows (its proof is given in \[sec:appen\_proof\_continuity\]).
\[lem:continuity\_Ani\_and\_Bni\] For any $n$ and $i \in [n]$, we have
- ${\mathbb{P}}(A^n_i =0) = {\mathbb{P}}(A^*_i = 0)$ and ${\mathbb{P}}(B^n_i =0) = {\mathbb{P}}(B^*_i =0 )$.
- ${\mathbb{P}}(A^n_i = x) = {\mathbb{P}}(B^n_i =y) = 0$ for any $x \in (0,\infty) \backslash \{X_A\}$ and $y \in (0,\infty) \backslash \{X_B\}$.
- ${\mathbb{P}}(A^n_i = X_A) \le \left( 1 -\frac{{\underaccent{\bar}{\lambda}}}{{\bar{\lambda}}} \frac{{\underaccent{\bar}{w}}^2}{{\bar{w}}^2} \right)^{n-1}$ and ${\mathbb{P}}(B^n_i = X_B) \le \left( 1 -\frac{{\underaccent{\bar}{\lambda}}}{{\bar{\lambda}}} \frac{{\underaccent{\bar}{w}}^2}{{\bar{w}}^2} \right)^{n-1}$.
Intuitively, Result $(ii)$ states that the function ${ F_{A^n_i}}$ (resp. ${ F_{B^n_i}}$) is continuous on $(0, X_A)$ (resp. $(0,X_B)$). The discontinuity of ${ F_{A^n_i}}$ (resp. ${ F_{B^n_i}}$) at $X_A$ (resp. at $X_B$) is due to the normalization step involved in the definition of the ${{\rm IU}^{{\gamma^*}}}$ strategy; note that the probability that $A^n_i = X_A$ (resp. $B^n_i = X_B$) quickly tends to zero when $n$ increases as has been shown in Result $(iii)$. Finally, Result $(i)$ shows that in some cases, ${ F_{A^n_i}}$ and ${ F_{B^n_i}}$ may be discontinuous at $0$. This is due to the fact that the functions ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$ may be discontinuous at $0$. Moreover, recall that we chose the assignments of the outputs in line 3 and 7 of Algorithm \[alg:IU\_strategy\] to be allocating zero to every battlefield, i.e., the mass at $0$ of ${ F_{A^n_i}}$ and ${ F_{B^n_i}}$ is added by a (negligibly small) positive probability. While other assignments do not affect our results, they make ${ F_{A^n_i}}$ (resp. ${ F_{B^n_i}}$) be discontinuous at some points differing from $0$ and $X_A$ (resp. $X_B$), e.g., if in line 3 of Algorithm \[alg:IU\_strategy\], we assign $x^A_i = X_A/n$, the distribution ${ F_{A^n_i}}$ would also be discontinuous at the point $X_A/n$. Our choice of assignments provides more convenience in our analysis since we have to consider their discontinuity at $0$ in any case.
Finally, with all the preparation steps mentioned above, we are ready to prove Theorem \[TheoMainBlotto\].
In this section, we first give a proof of Result $(ii)$ of Theorem \[TheoMainBlotto\]. Result $(i)$ will be deduced from $(ii)$. We first look for the condition on $n$ such that holds for any pure strategy $\boldsymbol{x}^A$ of player A. The proof that holds for any pure strategy of player B under the same condition can be done similarly and thus is omitted.
First, we write explicitly the payoffs of player A when player B plays the ${{\rm IU}^{{\gamma^*}}}_B$ strategy and player A plays either the pure strategy $\boldsymbol{x}^A$ or the ${{\rm IU}^{{\gamma^*}}}_A$ strategy: $$\begin{aligned}
&& \Pi^A(\boldsymbol{x}^A, {{\rm IU}^{{\gamma^*}}}_B) & = \alpha \sum \limits_{i=1}^{n}w^A_i {\mathbb{P}}(B^n_i = x^A_i) + \sum \limits_{i=1}^{n}w^A_i {\mathbb{P}}(B^n_i < x^A_i), \label{eq:Pi_A_pure}\\
&& \Pi^A({{\rm IU}^{{\gamma^*}}}_A, {{\rm IU}^{{\gamma^*}}}_B) & = \alpha \sum \limits_{i=1}^{n}w^A_i {\mathbb{P}}(B^n_i = A^n_i) + \sum \limits_{i=1}^{n}w^A_i {\mathbb{P}}(B^n_i< A^n_i) \nonumber \\
&& & = \alpha \sum\limits_{i = 1}^n {\int_0^\infty {w_i^A{{\mathbb{P}}(B^n_i = x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } + \sum\limits_{i = 1}^n {\int_0^\infty {w_i^A{{\mathbb{P}}(B^n_i < x)} {{\rm d}}{{ F_{A^n_i}}(x)}} }. \label{eq: Pi_A_IU}
$$ We then prepare a useful lemma, its proof is given in \[sec:appen\_proof\_lem:SufCon\]. Intuitively, this lemma shows that as $n$ is large enough, we can prove without the need of analyzing separately the case where players get tie allocations (that is our results hold regardless of the tie-breaking-rule parameter $\alpha$).
[lemma]{}[lem:SufCon]{} \[lem:SufCon\] Given $\varepsilon \in (0,1]$, there exists a constant $C^*_0>0$ (that does not depend on $\varepsilon$) such that for any $n\ge C^*_0 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, for any game ${\mathcal{CB}_n}$ and ${\gamma^*}\in {\mathcal{S}_n}$ the following inequality is a sufficient condition of : $$\sum \limits_{i=1}^{n}v^A_i { F_{B^n_i}}\left( x^A_i\right) \le \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } + \frac{\varepsilon}{2}. \label{eq:lem_ignore_tie}$$
In the remainders of the proof, we focus on and look for the condition of $n$ such that it holds; this will be done in the following five steps. After that, from Lemma \[lem:SufCon\], we can conclude that also holds with the corresponding condition on $n$.
#### Prove that $\{{ F_{A^*_i}}\}_i$ is optimal against $\{{ F_{B^*_i}}\}_i$.
[lemma]{}[lemdevia]{} \[lem:best\_response\] In any game ${\mathcal{CB}_n}$, for any pure strategy $\boldsymbol{x}^A$ of player A and ${\gamma^*}\in {\mathcal{S}_n}$, we have $$\sum\limits_{i=1}^n {v_i^A{{ F_{B^*_i}}\left( {x_i^A} \right)} } \le \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^*_i}}(x)} {{\rm d}}{{ F_{A^*_i}}(x)}} } . \label{eq:lem_best_res}$$
The proof of Lemma \[lem:best\_response\] is given in \[sec:appen\_proof\_best\_respond\]. This lemma can be interpreted as follows: if the allocation of player B to battlefield $i$ follows the distribution ${ F_{B^*_i}}$, then it is optimal for player A to play such that her allocation at this battlefield follows ${ F_{A^*_i}}$ (we do not know if it is possible to construct a mixed strategy such that player A’s allocation at battlefield $i$ follows ${ F_{A^*_i}}$ for all $i \in [n]$; however, this does not affect our results in this work). Using this lemma, we will analyze the validity of by proving that, as $n \rightarrow \infty$, the terms in respectively converge toward the terms in . To do this, we consider the next step.
#### Prove that ${ F_{A^n_i}}$ and ${ F_{B^n_i}}$ uniformly converge toward ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$ as $n$ increases.
[lemma]{}[lemmaconverge]{} \[lem:convergence\] For any $ \varepsilon_1 \in (0, 1]$, there exists $C_1>0$ (that does not depend on $\varepsilon_1$) such that for any $n \ge C_1 {\varepsilon_1^{-2}} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$ and $ i \in [n]$, $$\mathop {\sup }\limits_{ x \in [0, \infty) } \left|{ F_{A^n_i}}(x) - { F_{A^*_i}}(x)\right| \le \varepsilon_1 \hspace{0.3cm} \text{ and } \hspace{0.3cm} \mathop {\sup }\limits_{ x \in [0, \infty) } \left|{ F_{B^n_i}}(x) - { F_{B^*_i}}(x)\right| \le \varepsilon_1. \label{ineqconver}$$
A proof of this lemma is given in \[sec:appen\_proof\_lem\_convergence\]. The main intuition of this result comes from the fact that $A^n_i$ (resp. $B^n_i$) is the normalization of $A^*_i, i \in [n]$ (except for the special cases of the events $A_{=0}$ and $B_{=0}$) and the use of concentration inequalities on the random variables $\sum_{j \in [n]} A^*_j$ (and $\sum_{j \in [n]} B^*_j$). In this work, we apply the Hoeffding’s inequality (Theorem 2, [@hoeffding1963probability]) to obtain the rate of convergence indicated here in Lemma \[lem:convergence\].
#### Prove that the left-hand-side of converges toward the left-hand-side of .
Take $C_1$ as indicated in Lemma \[lem:convergence\], we define and deduce that .[^19] Therefore, take $\varepsilon_1:= \varepsilon/4$, for any $n \ge C^*_1 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $n \ge C_1 \varepsilon_1^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$; apply Lemma \[lem:convergence\], for any pure strategy $\boldsymbol{x}^A$ of player A, we have $$\begin{aligned}
\left| \sum \limits_{i=1}^{n}v^A_i { F_{B^n_i}}\left( x^A_i\right) - \sum \limits_{i=1}^{n}v^A_i { F_{B^*_i}}\left( x^A_i\right) \right|
\le & \sum \limits_{i=1}^n{v^A_i \mathop {\sup }\limits_{ x \in [0, \infty) } \left| {{{ F_{B^n_i}}}\left( {x} \right)}-{{{ F_{B^*_i}}}\left( {x} \right)} \right|} \nonumber \\
\le & \sum \limits_{i=1}^n{v^A_i \frac{\varepsilon}{4}} = \frac{\varepsilon}{4} \label{lhs}.\end{aligned}$$
#### Prove that the right-hand-side of converges toward the right-hand-side of .
We consider the difference of the involved terms as follows. $$\begin{aligned}
& \left|\sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } - \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^*_i}}(x)} {{\rm d}}{{ F_{A^*_i}}(x)}} } \right| \nonumber \\
\le & \sum\limits_{i = 1}^n v^A_i \int_0^\infty {\left|{{ F_{B^n_i}}(x) - { F_{B^*_i}}(x)} \right| {{\rm d}}{{ F_{A^n_i}}(x)}} + \sum\limits_{i = 1}^n v^A_i \left| {\int_0^\infty {{{ F_{B^*_i}}}\left( x \right)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) - \int_0^\infty {{{ F_{B^*_i}}}\left( x \right)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \label{rhs_1}.
$$ Let us define $C^*_2:= C_1 \cdot 64 (\ln(8)+1)$ (again, $C_1$ is the constant indicated in Lemma \[lem:convergence\]), we have that $C^*_2 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}\ge C_1 \left( \frac{8}{\varepsilon} \right)^{2} \ln \left(\frac{1}{\min\{\frac{\varepsilon}{8}, \frac{1}{{e}} \}} \right)$.[^20] Therefore, take $\varepsilon_1:= \varepsilon/8$, for any $n \ge C^*_2 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $n \ge \varepsilon_1^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$ and by Lemma \[lem:convergence\], we have $$\sum\limits_{i = 1}^n v^A_i \int_0^\infty {\left|{{ F_{B^n_i}}(x) - { F_{B^*_i}}(x)} \right| {{\rm d}}{{ F_{A^n_i}}(x)}} \le \sum\limits_{i = 1}^n v^A_i \int_0^\infty { \frac{\varepsilon}{8} {{\rm d}}{{ F_{A^n_i}}(x)}} = \sum\limits_{i = 1}^n v^A_i \frac{\varepsilon}{8} . \label{bla1}$$ Now, we need to find an upper-bound of the second term in the right-hand-side of. To do this, we present a lemma, called Lemma \[lem:portmanteau\] (stated below), that is based on the portmanteau lemma (see, e.g., [@van2000asymptotic]) regarding the weak convergence of a sequence of measures. Note importantly that by a direct application of the portmanteau lemma (since ${ F_{B^*_i}}$ is Lipschitz continuous and from Lemma \[lem:convergence\], ${ F_{A^n_i}}$ uniformly converges to ${ F_{A^*_i}}$), we can prove that $\int_0^\infty { { F_{B^*_i}}\left( x \right)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right)$ converges toward $\int_0^\infty {{ F_{B^*_i}}\left( x \right)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)$ as $n \rightarrow \infty$; however, note that the convergence rate obtained by doing this is large due to the fact that the Lipschitz constant of ${ F_{B^*_i}}$ (that is ${\lambda^*_A}/ v^A_i$) increases as $n$ increases. To obtain a better convergence rate as indicated in Lemma \[lem:portmanteau\], we exploit the properties of the involved functions that allow us to use the telescoping sum trick (see \[sec:appen\_proof\_lem\_portmanteau\] for more details).
\[lem:portmanteau\] For any $\varepsilon_2\in (0, 1]$, there exists a constant $C_2>0$ (that does not depend on $\varepsilon_2$) such that for any $n \ge C_2 \cdot {\varepsilon_2^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_2, 1/{e}\}}\right)}}$ and $ i \in [n]$, we have $$\label{eq:portmanteau_main}
\left| {\int_0^\infty { { F_{B^*_i}}\left( x \right)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) - \int_0^\infty {{ F_{B^*_i}}\left( x \right)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \le \varepsilon_2.$$
The proof of Lemma \[lem:portmanteau\] is given in \[sec:appen\_proof\_lem\_portmanteau\]. Based on this constant $C_2$, we define . Now, take $\varepsilon_2:= \varepsilon/8$, we have that $C^*_3 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}\ge C_2 \varepsilon_2^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_2, 1/{e}\}}\right)}$;[^21] therefore, for any $n \ge C^*_3 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $n \ge C_2 \varepsilon_2^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_2, 1/{e}\}}\right)}$ and by Lemma \[lem:portmanteau\], we deduce $$\left| {\int_0^\infty { { F_{B^*_i}}\left( x \right)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) - \int_0^\infty {{ F_{B^*_i}}\left( x \right)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \le \varepsilon/8.$$
Combine this with and , for any $n = \max\{C^*_2, C^*_3 \} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $$\left|\sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } - \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^*_i}}(x)} {{\rm d}}{{ F_{A^*_i}}(x)}} } \right| \le \sum\limits_{i = 1}^n v^A_i \varepsilon/8 + \sum\limits_{i = 1}^n v^A_i \varepsilon/8 = \frac{\varepsilon}{4}. \label{rhs}$$
#### Conclusion.
For any $n \ge \max\{C^*_1, C^*_2, C^*_3\} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$ and any pure strategy $\boldsymbol{x}^A$ of player A, we conclude that $$\begin{aligned}
\sum \limits_{i=1}^{n}v^A_i { F_{B^n_i}}\left( x^A_i\right) & \le \sum \limits_{i=1}^{n}v^A_i { F_{B^*_i}}\left( x^A_i\right) + \frac{\varepsilon}{4} & \textrm{(from \eqref{lhs})} \\
& \le \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^*_i}}(x)} {{\rm d}}{{ F_{A^*_i}}(x)}} } +\frac{\varepsilon}{4} & \textrm{(from \eqref{eq:lem_best_res})} \\
& \le \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } + \frac{\varepsilon}{4} + \frac{\varepsilon}{4} & \textrm{(from \eqref{rhs})} \\
& = \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } + \frac{\varepsilon}{2}.\end{aligned}$$ This is exactly ; therefore, applying Lemma \[lem:SufCon\] (involving $C^*_0$), denote $C^*_{\eqref{eq:MainTheo_A}}:= \max\{C^*_0, C^*_1, C^*_2, C^*_3\}$, we have proved that holds for any $n\ge C^*_{\eqref{eq:MainTheo_A}} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$. Similarly, we can prove that there exists a constant $C^*_{\eqref{eq:MainTheo_B}}$ such that holds for any $n \ge C^*_{\eqref{eq:MainTheo_B}} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$. Finally, define , we conclude the proof for Result $(ii)$.
Now, to obtain Result $(i)$, we prove that Result $(ii)$ implies Result $(i)$. Note that the constant $C^*$ found in the Result $(ii)$ does not depend on neither $n$ nor $\varepsilon$. Moreover, the function $$\begin{aligned}
\xi \colon (0,\infty) &\to (0, \infty)\\
{\tilde{\varepsilon}}&\mapsto C^* {\tilde{\varepsilon}}^{-2} \ln\left( \frac{1}{\min\{ {\tilde{\varepsilon}}, 1/{e}\}} \right).
\end{aligned}$$ is continuous and increases to infinity when $ \varepsilon $ tends to zero. Therefore, for any $n \ge 1$, there exists an such that $n = C^* \varepsilon ^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$. Now, apply Result $(ii)$, and hold in the game ${\mathcal{CB}_n}$ for any ${\gamma^*}\in {\mathcal{S}_n}$ and pure strategies $\boldsymbol{x}^A, \boldsymbol{x}^B$. We conclude the proof by notice that if $ \varepsilon \ge 1/{e}$, we have $n=C^* \varepsilon ^{-2}$ and thus $ \varepsilon = \sqrt{n/C^*} = {\mathcal{O}}(n^{-1/2})$; on the other hand, if $ \varepsilon < \frac{1}{{e}} $, we have $\ln\left(\frac{1}{\varepsilon} \right) >1$ that induces , thus, $\frac{1}{ \varepsilon } \le \frac{n}{C^*}$. We deduce that .
Proof of Lemma \[lem:continuity\_Ani\_and\_Bni\] {#sec:appen_proof_continuity}
------------------------------------------------
- Assuming $A^*_i =0$, if $\sum_{j \neq i}A^*_j =0$ then $A^n_i = 0$ (due to line 3 of Algorithm \[alg:IU\_strategy\]) and if $\sum_{j \neq i} A^*_j > 0$ then $A^n_i = A^*_i \big/ \sum_{j \in [n]} A^*_j = 0$. Reversely, assuming $A^n_i = 0$, then regardless whether $\sum_{j \in [n]} A^*_j=0$ or $\sum_{j \in [n]} A^*_j >0$, we have $A^*_i =0$. Therefore, $A^n_i = 0 \Leftrightarrow A^*_i =0$ for any $n$ and $i \in [n]$. Similarly, we can prove that $B^n_i = 0 \Leftrightarrow B^*_i =0$.
- The results are trivial in cases where $x > X_A$ and $y>X_B$ due to the definition of $A^n_i$ and $B^n_i$ (that guarantees that with probability $1$, $A^n_i \le X_A$ and $B^n_i \le X_B$). In the following, we consider the case where $x\in (0,X_A)$. For any $n$, $i \in [n]$, we denote $Z_i:= \sum_{j \neq i}A^*_j$ and obtain: $$\begin{aligned}
& {\mathbb{P}}(A^n_i = x) \\
=& {\mathbb{P}}\left( \{A^n_i =x \} \bigcap A_{>0} \right) & \textrm{(since } x>0) \\
=& {\mathbb{P}}\left( \left\{A^*_i =\frac{x}{X_A} \sum \nolimits_{j \in [n]}A^*_j \right\} \bigcap A_{>0} \right) \\
= & {\mathbb{P}}\left( \left\{A^*_i \left(1- \frac{x}{X_A} \right) =\frac{x}{X_A} \sum \nolimits_{j \neq i}A^*_j \right\} \bigcap A_{>0} \right)\\
= & {\mathbb{P}}\left( \left\{A^*_i =\frac{Z_i \cdot x}{X_A -x} \right\} \bigcap A_{>0} \right) & \textrm{(note that } X_A-x >0) \\
\le & {\mathbb{P}}(\{A^*_i = Z_i =0 \} \cap A_{>0}) + \int_{z>0}{\mathbb{P}}\left(A^*_i = \frac{z \cdot x}{X_A - x} \right) {{\rm d}}F_{Z_i}(z) \\
\le & {\mathbb{P}}(A_{=0} \cap A_{>0}) + \int_{z>0}{0~ {{\rm d}}F_{Z_i}(z) }
\\
= & 0.
\end{aligned}$$ Here, the second-to-last inequality comes from the fact that $\frac{z x}{X_A-x} >0, \forall z >0, \forall x \in (0,X_A)$ and ${\mathbb{P}}(A^*_i = a) =0 $ for any $a >0$. Similarly, we can prove that ${\mathbb{P}}(B^n_i = y) = 0$ for any $y \in (0,X_B)$.
- We have $$\begin{aligned}
{\mathbb{P}}(A^n_i = X_A) & = {\mathbb{P}}\left( \left\{A^*_i = \sum_{j \in [n]}A^*_j \right\} \bigcap A_{>0} \right) \\
& \le {\mathbb{P}}\left( \sum_{j \neq i} A^*_j =0 \right)\\
& = \prod_{j \neq i} {\mathbb{P}}\left( A^*_j = 0 \right) & \textrm{(since } A^*_j, j \in [n] \textrm{ are non-negative and independent)}.\end{aligned}$$ Now, if there exists $j \neq i $ such that $ j \in {\Omega_A({\gamma^*})}$, then ${\mathbb{P}}(A^*_j = 0) =0 $ due to the fact that $A^*_j = A^S_{{\gamma^*},j}$ and the definition of $ A^S_{{\gamma^*},j}$ (see ). In this case, $ \prod_{j \neq i} {\mathbb{P}}\left( A^*_j = 0 \right)=0$. On the other hand, if $j \notin {\Omega_A({\gamma^*})}$ for any $j \neq i$, then $A^*_j= A^W_{{\gamma^*}, j}$ for $j \neq i$; therefore, $$\prod_{j \neq i} {\mathbb{P}}\left( A^*_j = 0 \right) = \prod_{j \neq i}\left[{\left(\frac{v^B_j}{{\lambda^*_B}} - \frac{v^A_j}{{\lambda^*_A}} \right)} \middle/ {\frac{v^B_j}{{\lambda^*_B}}}\right] = \prod_{j\neq i} \left(1 - \frac{v^A_j}{v^B_j}\frac{{\lambda^*_B}}{{\lambda^*_A}} \right) \le \left(1-\frac{{\underaccent{\bar}{\lambda}}}{{\bar{\lambda}}} \frac{\frac{{\underaccent{\bar}{w}}}{n{\bar{w}}}}{\frac{{\bar{w}}}{n{\underaccent{\bar}{w}}}} \right)^{n-1} = \left(1-\frac{{\underaccent{\bar}{\lambda}}}{{\bar{\lambda}}} \frac{{\underaccent{\bar}{w}}^2}{{\bar{w}}^2} \right)^{n-1}.$$ Here, to obtain the last equality, we use for the bounds of $v^A_j, v^B_j$ and Proposition \[Prop:BoundLambda\] for the bounds of ${\lambda^*_A}, {\lambda^*_B}$.
Similarly, we can obtain ${\mathbb{P}}(B^n_i = X_B) \le \left(1-\frac{{\underaccent{\bar}{\lambda}}}{{\bar{\lambda}}} \frac{{\underaccent{\bar}{w}}^2}{{\bar{w}}^2} \right)^{n-1}$.
Proof of Lemma \[lem:SufCon\] {#sec:appen_proof_lem:SufCon}
-----------------------------
Fix $\varepsilon \in (0, 1]$ and assume that is satisfied, we prove that also holds by comparing the terms in with the terms in . First, due to the fact that $\alpha \le 1$, we can find a lower bound of the left-hand side of as follows: $$\begin{aligned}
\sum \limits_{i=1}^{n}v^A_i { F_{B^n_i}}\left( x^A_i\right) & = \sum \limits_{i=1}^{n}v^A_i {\mathbb{P}}(B^n_i = x^A_i) + \sum \limits_{i=1}^{n}v^A_i {\mathbb{P}}(B^n_i < x^A_i) \nonumber \\
& \ge \alpha \sum \limits_{i=1}^{n}v^A_i {\mathbb{P}}(B^n_i = x^A_i) + \sum \limits_{i=1}^{n}v^A_i {\mathbb{P}}(B^n_i < x^A_i) \nonumber \\
& = \Pi^A(\boldsymbol{x}^A, {{\rm IU}^{{\gamma^*}}}_B)/W_A. \label{eq:B2_lhs}\end{aligned}$$
Now, we turn our focus to the right-hand-side of , we can rewrite the involved term as follows. $$\begin{aligned}
\sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } & = \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{\mathbb{P}}(B^n_i = x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } + \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{\mathbb{P}}(B^n_i < x)} {{\rm d}}{{ F_{A^n_i}}(x)}} }.\end{aligned}$$ We observe that $\sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } $ is very similar to the expression of $\Pi^A(\boldsymbol{x}^A, {{\rm IU}^{{\gamma^*}}}_B)$ stated in . The main difference lies at the coefficient of the term related to the tie cases that is the tie-breaking parameter $\alpha$. Therefore, we consider the following two cases of $\alpha$:
*Case 1:* $\alpha = 1$. For any $n$, divide two sides of (with $\alpha =1$) by $W_A$ and recall that $v^A_i:= w^A_i/W_A, \forall i $, we trivially have .
*Case 2:* $\alpha < 1$. Due to Results $(ii)$ and $(iii)$ of Lemma \[lem:continuity\_Ani\_and\_Bni\], for any $x >0$, we have ${\mathbb{P}}(B^n_i = x) \le D^{n-1}$ where we define $D:= \left(1-\frac{{\underaccent{\bar}{\lambda}}}{{\bar{\lambda}}} \frac{{\underaccent{\bar}{w}}^2}{{\bar{w}}^2} \right) < 1$. We consider two cases of $\alpha$ as follows.
- If $2(1-\alpha) \le 1$, define $\hat{C}_{1} := \frac{1}{\ln(1/D)} + 1>0$, we have that[^22] ; therefore, for any $n \ge \hat{C}_1 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we obtain and $$D^{n-1} \le D^{\log_D {\varepsilon}} = \varepsilon \le \frac{\varepsilon}{2(1-\alpha)} \qquad (\textrm{note that } D < 1 \textrm{ and in this case } 2(1-\alpha) \le 1).$$
- If $2(1\!-\!\alpha)\! > \!1$, define $\hat{C}_{2}\! := \! \frac{1}{\ln(1/D)} + \frac{\ln(2\!-\!2\alpha)}{\ln(1/D)} \!+ \! 1 \!> \!0$; we have .[^23] We conclude that for any $n \ge \hat{C}_2 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we obtain and $$D^{n-1} \le D^{\log_D {\frac{\varepsilon}{2(1- \alpha)}}} = \frac{\varepsilon}{2(1-\alpha)}.$$
Let us define $C^*_0 = \max\{\hat{C}_1, \hat{C}_2\} > 0$, we conclude that for any $\alpha < 1$, $n\ge C^*_0 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, $i \in [n]$ and $x >0$, we have $$\label{eq:probAn=Bn}
{\mathbb{P}}(B^n_i =x) \le D^{n-1} \le \frac{\varepsilon}{2(1- \alpha)}.$$ Note also that ,[^24] we conclude that when $\alpha < 1$, for any $n \ge C^*_0 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $$\begin{aligned}
& \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } \\
= & \left[ \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{\mathbb{P}}(B^n_i < x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } + \alpha \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{\mathbb{P}}(B^n_i = x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } \right] + (1-\alpha)\sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{\mathbb{P}}(B^n_i = x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } \\
= & \Pi^A({{\rm IU}^{{\gamma^*}}}_A, {{\rm IU}^{{\gamma^*}}}_B)/W_A + (1-\alpha) \sum\limits_{i = 1}^n v_i^A {\int_{(0, \infty)} { \frac{\varepsilon}{2(1-\alpha)} {{\rm d}}{{ F_{A^n_i}}(x)}} } + (1-\alpha) \sum_{i=1}^n v^A_i {\mathbb{P}}(A^n_i = B^n_i =0) \\
= & \Pi^A({{\rm IU}^{{\gamma^*}}}_A, {{\rm IU}^{{\gamma^*}}}_B)/W_A + (1-\alpha) \sum\limits_{i = 1}^n v_i^A {\int_{(0,\infty)} { \frac{\varepsilon}{2(1-\alpha)} {{\rm d}}{{ F_{A^n_i}}(x)}} } + 0 \\
\le & \Pi^A({{\rm IU}^{{\gamma^*}}}_A, {{\rm IU}^{{\gamma^*}}}_B)/W_A + (1- \alpha)\frac{\varepsilon}{2(1-\alpha)} \\
= & \Pi^A({{\rm IU}^{{\gamma^*}}}_A, {{\rm IU}^{{\gamma^*}}}_B)/W_A + {\varepsilon}/{2}. \end{aligned}$$
In conclusion, regardless of the value of $\alpha$, for any $n \ge C^*_0 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $$\label{eq:B2_rhs}
\sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} } \le \Pi^A({{\rm IU}^{{\gamma^*}}}_A, {{\rm IU}^{{\gamma^*}}}_B)/W_A + {\varepsilon}/{2}.$$
Combine , and the assumption that holds, for any $n \ge C^*_0 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $$\frac{\Pi^A(\boldsymbol{x}^A, {{\rm IU}^{{\gamma^*}}}_B)}{W_A} \stackrel{\eqref{eq:B2_lhs}}{\le} \sum \limits_{i=1}^{n}v^A_i { F_{B^n_i}}\left( x^A_i\right) \stackrel{\eqref{eq:lem_ignore_tie}}{\le} \sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^n_i}}(x)} {{\rm d}}{{ F_{A^n_i}}(x)}} }\! +\! \varepsilon/2 \stackrel{\eqref{eq:B2_rhs}}{\le} \frac{\Pi^A({{\rm IU}^{{\gamma^*}}}_A, {{\rm IU}^{{\gamma^*}}}_B)}{W_A} \!+\! \varepsilon.$$ By multiplying both sides of this inequality by $W_A$, we obtain .
Proof of Lemma \[lem:best\_response\] {#sec:appen_proof_best_respond}
-------------------------------------
We compute the right-hand-side of based on the definition of ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$ (see Definition \[def:UnifromDistributions\]). $$\begin{aligned}
\sum\limits_{i = 1}^n {\int_0^\infty {v_i^A{{ F_{B^*_i}}(x)} {{\rm d}}{{ F_{A^*_i}}(x)}} } & = \sum\limits_{i \in {\Omega _A}\left( {{{\gamma^*}}} \right)} {\int_0^\infty {v_i^A{F_{B_{{{\gamma^*}},i}^W}(x)} {{\rm d}}{F_{A_{{{\gamma^*}},i}^S}(x)}} } + \sum\limits_{i \notin {\Omega _A}\left( {{{\gamma^*}}} \right)} {\int_0^\infty {v_i^A{F_{B_{{{\gamma^*}},i}^S}(x)} {{\rm d}}{F_{A_{{{\gamma^*}},i}^W}(x)}} } \nonumber \\
&= \sum\limits_{i \in {\Omega _A}\left( {{{\gamma^*}}} \right)} {\int_0^{\frac{{v_i^B}}{{\lambda _B^*}}} {v_i^A\left( {\frac{{\frac{{v_i^A}}{{\lambda _A^*}} - \frac{{v_i^B}}{{\lambda _B^*}}}}{{\frac{{v_i^A}}{{\lambda _A^*}}}} + \frac{x}{{\frac{{v_i^A}}{{\lambda _A^*}}}}} \right)\frac{1}{{\frac{{v_i^B}}{{\lambda _B^*}}}} {{\rm d}}x} } + \sum\limits_{i \notin {\Omega _A}\left( {{{\gamma^*}}} \right)} {\int_0^{\frac{{v_i^A}}{{\lambda _A^*}}} {v_i^A\frac{x}{{\frac{{v_i^A}}{{\lambda _A^*}}}}\frac{1}{{\frac{{v_i^B}}{{\lambda _B^*}}}} {{\rm d}}x} } \nonumber \\
& = \sum\limits_{i \in {\Omega _A}\left( {{{\gamma^*}}} \right)} v_i^A\left( 1 - \frac{{v_i^B {\gamma^*}}}{2v^A_i} \right) + \sum\limits_{i \notin {\Omega _A}\left( {{{\gamma^*}}} \right)} {(v_i^A)^2\frac{1}{{2{\gamma^*}v^B_i}}}. \label{eq:proof_best_res_1}\end{aligned}$$ On the other hand, for any pure strategy $\boldsymbol{x}^A$ of player A, we have: $$\begin{aligned}
\sum\limits_{i=1}^n {v_i^A{{ F_{B^*_i}}\left( {x_i^A} \right)} } = & \sum\limits_{i \in {\Omega _A}\left( {{{\gamma^*}}} \right)} v_i^A{F_{B_{{\gamma^*},i}^W}}\left( {x_i^A} \right) + \sum\limits_{i \notin {\Omega _A}\left( {{{\gamma^*}}} \right)} v_i^A{F_{B_{{\gamma^*},i}^S}}\left( {x_i^A} \right) \nonumber \\
\le & \sum\limits_{i \in {\Omega _A}\left( {{{\gamma^*}}} \right)} v^A_i \left(\frac{\frac{v^A_i}{{\lambda^*_A}} - \frac{v^B_i}{{\lambda^*_B}}}{\frac{v^A_i}{{\lambda^*_A}}} + \frac{x^A_i {\lambda^*_A}}{v^A_i}\right) + \sum\limits_{i \notin {\Omega _A}\left( {{{\gamma^*}}} \right)} v_i^A \left( \frac{x^A_i {\lambda^*_A}}{v^A_i} \right) \nonumber \\
\le & \sum\limits_{i \in {\Omega _A}\left( {{{\gamma^*}}} \right)} {\left( {\frac{{v_i^A}}{{\lambda _A^*}} - \frac{{v_i^B}}{{\lambda _B^*}}} \right)\lambda _A^*} + \lambda _A^* X_A
&& \textrm{ (since } \sum \nolimits_{i=1}^n{ x^A_i} \le X_A \textrm{)} \nonumber \\
=& \sum\limits_{i \in {\Omega _A}\left( {{{\gamma^*}}} \right)} v_i^A\left( 1 - \frac{{v_i^B {\gamma^*}}}{2v^A_i} \right) + \sum\limits_{i \notin {\Omega _A}\left( {{{\gamma^*}}} \right)} {(v_i^A)^2\frac{1}{{2{\gamma^*}v^B_i}}} . \label{eq:proof_best_res_2}\end{aligned}$$ Here, to obtain the last equality, we use to rewrite $X_A$. Finally, from and , we conclude that holds for any $\boldsymbol{x}^A$ and ${\gamma^*}$.
Proof of Lemma \[lem:convergence\] {#sec:appen_proof_lem_convergence}
----------------------------------
Since the definition of ${ F_{A^n_i}}$ involves ${\mathbb{P}}(A_{=0})$ (see ), we first look for an upper-bound of ${\mathbb{P}}(A_{=0})$. For any $n$ and ${\gamma^*}\in {\mathcal{S}_n}$, if ${\Omega_A({\gamma^*})}\neq \emptyset$, i.e., there exists $i$ such that $A^*_i = { A^S_{{\gamma^*},i}}$, then ${\mathbb{P}}(A^*_i = 0) =0 $ due to the definition of $ A^S_{{\gamma^*},i}$ (see ); in this case, ${\mathbb{P}}(A_{=0}) = \prod_{j \in [n]} {\mathbb{P}}\left( A^*_j = 0 \right)=0$. On the other hand, if ${\Omega_A({\gamma^*})}= \emptyset $, then $A^*_j= A^W_{{\gamma^*}, j}$ for any $j \in [n]$; therefore, $${\mathbb{P}}(A_{=0}) = \prod_{j \in [n]} {\mathbb{P}}\left( A^*_j = 0 \right) = \prod_{j \in [n]}\left[{\left(\frac{v^B_j}{{\lambda^*_B}} - \frac{v^A_j}{{\lambda^*_A}} \right)} \middle/ {\frac{v^B_j}{{\lambda^*_B}}}\right] = \prod_{j \in [n]} \left(1 - \frac{v^A_j}{v^B_j}\frac{{\lambda^*_B}}{{\lambda^*_A}} \right) \le \left(1-\frac{{\underaccent{\bar}{\lambda}}}{{\bar{\lambda}}} \frac{{\underaccent{\bar}{w}}^2}{{\bar{w}}^2} \right)^n . \label{eq:Prop_all_zero}$$ Here, the last inequality comes directly from and Proposition \[Prop:BoundLambda\]. Recall the notation and define $\tilde{C}_{0}\! :=\! \frac{\ln(4)+1}{\ln(1/D)}\!>\!0$, we have .[^25] Therefore, for any we have $n \ge \log_D{(\varepsilon_1/4)}$ and since $D<1$ we have: $$\label{eq:proofB4_A=0}
{\mathbb{P}}(A_{=0}) \le D^n \le D^{\log_D(\varepsilon_1 /4)} = \varepsilon_1/ 4.$$
Now, we look for an upper-bound of ${\mathbb{P}}(A_{>0})$. For any $n$, define the constants $\epsilon_n:= \frac{\varepsilon_1 }{4} \frac{ {\underaccent{\bar}{w}}}{n {\bar{w}}{\bar{\lambda}}}$ and , we consider the following term for any $i \in [n]$: $$\begin{aligned}
& {\mathbb{P}}\left( \left\{ A^*_i - \frac{A^*_i }{\sum \nolimits_{j \in [n]}{A^*_j}}X_A > \epsilon_n \right\} \bigcap A_{>0} \right) \nonumber \\
& \le {\mathbb{P}}\left( \left\{ \left| A^*_i - \frac{A^*_i }{\sum \nolimits_{j \in [n]}{A^*_j}}X_A \right| > \epsilon_n \right\} \bigcap A_{>0} \right) \nonumber \\
& \le {\mathbb{P}}\left( A^*_i \left|\sum \nolimits _{j \in [n]}{A^*_j} - X_A \right| \!>\! \epsilon_n \sum \nolimits _{j \in [n]}{A^*_j} \right) \nonumber \\
& = {\mathbb{P}}\left( A^*_i \left|\sum \nolimits _{j \in [n]}{A^*_j} - X_A \right| \!>\! \epsilon_n X_A \! -\! \epsilon_n \left(X_A \! -\! \sum \nolimits _{j \in [n]}{A^*_j} \right) \right) \nonumber \\
& \le {\mathbb{P}}\left( A^*_i \left|\sum \nolimits _{j \in [n]}{A^*_j} - X_A \right| \!>\! \epsilon_n X_A \! -\! \epsilon_n \left|\sum \nolimits _{j \in [n]}{A^*_j}- X_A \right| \right) \nonumber \\
& = {\mathbb{P}}\left( \left|\sum \nolimits_{j \in [n]}{A^*_j} - X_A \right| \!>\! \frac{\epsilon_n X_A}{ A^*_i \! + \!\epsilon_n}\right) \nonumber \\
& \le {\mathbb{P}}\left( \left| \sum \nolimits_{j \in [n]}{A^*_j} - X_A \right| \!>\! \frac{\epsilon_n X_A}{\frac{{\bar{w}}}{n {\underaccent{\bar}{w}}{\underaccent{\bar}{\lambda}}} + \epsilon_n}\right) \nonumber \\
& = {\mathbb{P}}\left( \left| \sum \nolimits_{j \in [n]}{A^*_j} - X_A \right| \!>\! \frac{1}{\tau} \right) \label{1.1}.\end{aligned}$$ Here, the second-to-last inequality comes from the fact that for any $i \in [n]$, $A^*_i$ is upper-bounded by either ${v^A_i}/{{\lambda^*_A}}$ or ${v^B_i}/{{\lambda^*_B}}$ (see and ), thus, it is bounded by ${{\bar{w}}}/({n {\underaccent{\bar}{w}}{\underaccent{\bar}{\lambda}}})$ (due to and Proposition \[Prop:BoundLambda\]).
Recall that $X_A= \mathbb{E}\left[\sum \limits _{j=1}^n{A^*_j}\right]$ (see Lemma \[lem:Preliminary\]-$(iii)$), we use the Hoeffding’s inequality (see e.g., Theorem 2, [@hoeffding1963probability]) on the random variables $A^*_i, i \in [n]$ (bounded in $\left[ 0, {{\bar{w}}}/(n {\underaccent{\bar}{w}}{\underaccent{\bar}{\lambda}})\right]$) to obtain $$\begin{aligned}
P\left( \left| \sum \limits _{j \in [n]}{A^*_j} - X_A \right| > \frac{1}{\tau} \right)
\le & 2 \exp \left(\frac{-2\frac{1}{\tau^2}}{\sum\limits_{j \in [n]} {\left( \frac{{\bar{w}}}{n {\underaccent{\bar}{w}}{\underaccent{\bar}{\lambda}}} \right)^2}} \right) \nonumber \\
= & 2 \exp \left[ \frac{-2n}{\tau ^2} \left(\frac{ {\underaccent{\bar}{\lambda}}{\underaccent{\bar}{w}}}{{\bar{w}}} \right)^2\right].\label{eq:from_Hoeffiding}\end{aligned}$$ Now, we define ; due to the definition of $\tau$, we have that[^26] ; therefore, for any $n \ge \tilde{C}_1 {\varepsilon_1^{-2}} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$, we can deduce that and thus, $$2 \exp \left[ \frac{-2n}{\tau^2} \left(\frac{ {\underaccent{\bar}{\lambda}}{\underaccent{\bar}{w}}}{{\bar{w}}} \right)^2\right] \le 2 \exp \left[-\ln\left(\frac{8}{\varepsilon_1} \right) \right] = \frac{\varepsilon_1 }{4}. \label{inequN1}$$ Combining , and , we deduce that $${\mathbb{P}}\left( \left\{ A^*_i - \frac{A^*_i }{\sum \nolimits_{j \in [n]}{A^*_j}}X_A > \epsilon_n \right\} \bigcap A_{>0} \right) \le \frac{\varepsilon_1}{4}, \forall n \ge \tilde{C}_1 {\varepsilon_1^{-2}} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}. \label{eq:bound_E_non_0}$$
Finally, note that for any $n$, $i \in [n]$ and $x\ge 0$, we also have $$\begin{aligned}
& {\mathbb{P}}\left( \left\{ \frac{A^*_i \cdot X_A }{\sum \nolimits_{j\in[n]} A^*_j} \le x \right\} \bigcap A_{>0} \right) \nonumber \\
= & {\mathbb{P}}\left( \left\{ \frac{A^*_i X_A}{\sum \limits_{j\in[n]} A^*_j} \! \le \! x \right\} \bigcap \left\{ A_i^*\!- \! \frac{A^*_i X_A}{\sum \limits_{j\in[n]} A^*_j} \! \le \! \epsilon_n \right\} \bigcap A_{>0} \right) \!+\! {\mathbb{P}}\left( \left\{ \frac{A^*_i \!\cdot\! X_A}{\sum \limits_{j\in[n]} A^*_j} \le x \right\} \bigcap \left\{ A_i^* \! - \! \frac{A^*_i X_A}{\sum \limits_{j\in[n]} A^*_j} \! > \! \epsilon_n \right\} \bigcap A_{>0} \right) \nonumber \\
\le & {\mathbb{P}}\left(\{A^*_i \le x\! +\! \epsilon_n \} \right) + {\mathbb{P}}\left( \left\{ A_i^* \! - \! \frac{A^*_i X_A}{\sum \limits_{j\in[n]} A^*_j} \! > \! \epsilon_n \right\} \bigcap A_{>0} \right). \label{eq:proof_converge_inter} \end{aligned}$$
Therefore, define $C_1 := \max\{\tilde{C_0}, \tilde{C_1} \}$, for any $n \ge C_1 \varepsilon_1^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$, from , we have $$\begin{aligned}
& {{ F_{A^n_i}}}\left( x \right) - {{ F_{A^*_i}}}\left( x \right) \nonumber\\
= & {\mathbb{P}}\left( A_{=0} \right) + {\mathbb{P}}\left( \left\{ \frac{A^*_i \cdot X_A }{\sum \nolimits_{j\in[n]} A^*_j} \le x \right\} \bigcap A_{>0} \right) - {{ F_{A^*_i}}}\left( x \right) \nonumber\\
\le & \frac{\varepsilon_1}{4} + {\mathbb{P}}\left(\{A^*_i \le x\! +\! \epsilon_n \} \right) + {\mathbb{P}}\left( \left\{ A_i^* \! - \! \frac{A^*_i X_A}{\sum \limits_{j\in[n]} A^*_j} \! > \! \epsilon_n \right\} \bigcap A_{>0} \right) - {{ F_{A^*_i}}}\left( x \right) && (\textrm{due to } \eqref{eq:proofB4_A=0} \textrm{ and } \eqref{eq:proof_converge_inter}) \nonumber\\
\le & \frac{\varepsilon_1}{4} + { F_{A^*_i}}(x+ \epsilon_n) + \frac{\varepsilon_1}{4} - { F_{A^*_i}}(x) && (\textrm{due to } \eqref{eq:bound_E_non_0}) . \label{eq:final_lem_converge}\end{aligned}$$
The final step is to bound the term ${ F_{A^*_i}}(x + \epsilon_n) - { F_{A^*_i}}(x)$; we present this as the following lemma.
\[remark\_prepare\] For any $\epsilon >0$, $n>0$, $i \in [n]$ and $x \in [0,\infty)$, we have ${{ F_{A^*_i}}}\left( {x + {\epsilon}} \right) - {{ F_{A^*_i}}}\left( x \right) \le \frac{\epsilon {\lambda^*_B}}{v^B_i}$.
If $i\in {\Omega_A({\gamma^*})}$, then $A^*_i = { A^S_{{\gamma^*},i}}$ and $$\label{inequaFAS}
F_{{ A^S_{{\gamma^*},i}}}(x+ \epsilon) - F_{{ A^S_{{\gamma^*},i}}}(x) = \left\{ \begin{array}{l}
\frac{(x+\epsilon){\lambda^*_B}}{v^B_i} - \frac{x {\lambda^*_B}}{v^B_i} = \frac{\epsilon {\lambda^*_B}}{v^B_i}, \textrm{ if } 0 \le x < \frac{v^B_i}{{\lambda^*_B}} - \epsilon \\
1 - \frac{x{\lambda^*_B}}{v^B_i} \le \frac{\epsilon {\lambda^*_B}}{v^B_i}, \qquad \quad \textrm{ if } \frac{v^B_i}{{\lambda^*_B}} - \epsilon \le x \le \frac{v^B_i}{{\lambda^*_B}}\\
1 - 1 \le \frac{\epsilon v^B_i}{{\lambda^*_B}}, \qquad \qquad \quad \textrm{ if } x > \frac{v^B_i}{{\lambda^*_B}}\\
\end{array} \right..$$ On the other hand, if $i\notin {\Omega_A({\gamma^*})}$, then $A^*_i = { A^W_{{\gamma^*},i}}$ and we have $$\label{inequaFAW}
F_{{ A^W_{{\gamma^*},i}}}(x+ \epsilon) - F_{{ A^W_{{\gamma^*},i}}}(x) = \left\{ \begin{array}{l}
\frac{(x+\epsilon){\lambda^*_B}}{v^B_i} - \frac{x {\lambda^*_B}}{v^B_i} = \frac{\epsilon {\lambda^*_B}}{v^B_i}, \quad \textrm{ if } 0 \le x < \frac{v^A_i}{{\lambda^*_A}} - \epsilon\\
1 - \frac{\frac{v^B_i}{{\lambda^*_B}} - \frac{v^A_i}{{\lambda^*_A}}}{ \frac{v^B_i}{{\lambda^*_B}}}-\frac{x{\lambda^*_B}}{v^B_i} \le \frac{\epsilon {\lambda^*_B}}{v^B_i}, \textrm{ if } \frac{v^A_i}{{\lambda^*_A}} - \epsilon \le x \le \frac{v^A_i}{{\lambda^*_A}}\\
1 - 1 \le \frac{\epsilon v^B_i}{{\lambda^*_B}}, \quad \qquad \qquad \quad \textrm{ if } x > \frac{v^A_i}{{\lambda^*_A}}\\
\end{array} \right..$$
Combine this lemma with and recall the definition of $\epsilon_n$ (which induces that $\epsilon_n {\lambda^*_B}/ v^B_i \le \varepsilon_1 /2$), we conclude that ${ F_{A^n_i}}(x)\! - \! { F_{A^*_i}}(x) \le \varepsilon_1$ for any $n \ge C_1 \varepsilon_1^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$. Similarly, for and $i \in [n]$, we can deduce that for any $x \in [0, \infty)$. We conclude that for any $n \ge C_1 \varepsilon_1^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$, . The inequality can be proved in a similar way.
Proof of Lemma \[lem:portmanteau\] {#sec:appen_proof_lem_portmanteau}
----------------------------------
In this proof, we will use the notation ${\mathbb{E}}f(X):= \int_{0}^\infty {f(z) {{\rm d}}F_Z(x)} $ and ${\mathbb{E}}_{\mathcal{I}} f(X):= \int_{\mathcal{I}} {f(z) {{\rm d}}F_Z(x)}$ for any function $f$, random variable $Z$ and interval $\mathcal{I}$. To simplify the notation, let us define $M:= \frac{{\bar{\lambda}}}{{\underaccent{\bar}{\lambda}}} \frac{{\bar{w}}^2}{{\underaccent{\bar}{w}}^2}$ and we denote by $\mathcal{I}_i$ the interval $\left[ 0,{v^B_i}/{{\lambda^*_B}} \right]$. For any $\varepsilon_2 \in (0,1]$, we define $\delta_2:=\frac{\varepsilon_2}{6 + 2M}$. We first consider the case where $i \in {\Omega_A({\gamma^*})}$, i.e., $B^*_i = { B^W_{{\gamma^*},i}}$. Note that for any $x \ge 2X_B$ (see Lemma \[lem:Preliminary\]-$(iv)$); the left-hand-side of can be rewritten as follows. $$\begin{aligned}
& \left|{\mathbb{E}}{ F_{B^*_i}}(A^n_i) - {\mathbb{E}}{ F_{B^*_i}}(A^*_i) \right| \nonumber \\
= & \left| {\int_{[0,2X_B]} { { F_{B^*_i}}\left( x \right)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) - \int_{[0,2X_B]} {{ F_{B^*_i}}\left( x \right)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \nonumber \\
\le & \left| {\mathbb{E}}_{[0, v^B_i/ {\lambda^*_B}]} F_{{ B^W_{{\gamma^*},i}}} (A^n_i) - {\mathbb{E}}_{[0, v^B_i/ {\lambda^*_B}]} F_{{ B^W_{{\gamma^*},i}}} (A^*_i) \right| + \left| {\int \limits_{\left({v^B_i}/{{\lambda^*_B}}, 2X_B \right]} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) \! -\! \int \limits_{\left({v^B_i}/{{\lambda^*_B}}, 2X_B \right]} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \nonumber \\
= & \left| {\mathbb{E}}_{\mathcal{I}_i} F_{{ B^W_{{\gamma^*},i}}} (A^n_i) - {\mathbb{E}}_{\mathcal{I}_i} F_{{ B^W_{{\gamma^*},i}}} (A^*_i) \right| + \left|{ F_{A^n_i}}(2X_B) - { F_{A^n_i}}(v^B_i/{\lambda^*_B}) - { F_{A^*_i}}(2X_B) + { F_{A^*_i}}(v^B_i/{\lambda^*_B}) \right| \nonumber \\
\le & \left| {\mathbb{E}}_{\mathcal{I}_i} F_{{ B^W_{{\gamma^*},i}}} (A^n_i) - {\mathbb{E}}_{\mathcal{I}_i} F_{{ B^W_{{\gamma^*},i}}} (A^*_i) \right| \! + 2 \sup \limits_{x \in [0, \infty)} \left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right|. \label{eq:portmanteau_case1}\end{aligned}$$ We now focus on bounding the first term in . Let us define $K := \big \lceil \frac{M}{\delta_2} \big \rceil$ and $K + 1$ points ${x}_j$ such that ${x_0} := 0$ and ${x}_j := {x}_{j-1} + \frac{v^B_i }{{\lambda^*_B}K}, \forall j \in [K]$. In other words, we have the partitions where we denote by $P_1$ the interval $[{x}_0,{x}_1]$ and by $P_j$ the interval $({x}_{j-1}, {x}_{j}]$ for $j = 2, \ldots, K$. For any $ x, x^{\prime} \in P_j, \forall j \in [K]$, from the definition of ${ B^W_{{\gamma^*},i}}$, we have $$|F_{{ B^W_{{\gamma^*},i}}}(x)- F_{{ B^W_{{\gamma^*},i}}}(x^{\prime}) | = \frac{{\lambda^*_A}}{v^A_i}|x - x^{\prime}| \le \frac{{\lambda^*_A}}{v^A_i} \cdot \frac{v^B_i}{{\lambda^*_B}} \cdot \frac{1}{K} \le \frac{{\bar{\lambda}}n {\bar{w}}}{{\underaccent{\bar}{w}}} \cdot \frac{{\bar{w}}}{n {\underaccent{\bar}{w}}{\underaccent{\bar}{\lambda}}} \cdot \frac{1}{K} = \frac{M}{K} \le \delta_2 . \label{eq:port_delta_2}$$ Now, we define the function $g(x)\!:=\! \sum \limits_{j=1}^{K} {F_{{ B^W_{{\gamma^*},i}}}(x_j) \boldsymbol{1}_{P_j} (x)}$. Here, $\boldsymbol{1}_{P_j}$ is the indicator function of the set ${P_j}$. From this definition and Inequality , we trivially have , $\forall x \in \mathcal{I}_i$. Therefore, $$\begin{aligned}
& \left| {\mathbb{E}}_{\mathcal{I}_i} F_{{ B^W_{{\gamma^*},i}}} (A^n_i) - {\mathbb{E}}_{\mathcal{I}_i} g (A^n_i) \right| \le \int _{I_i} \left| F_{{ B^W_{{\gamma^*},i}}}(x) - g(x) \right| {{\rm d}}{ F_{A^n_i}}(x) \le \int_{\mathcal{I}_i} \delta_2 {{\rm d}}{ F_{A^n_i}}(x) \le \delta_2, \label{eq:triangle1}\\
& \left| {\mathbb{E}}_{\mathcal{I}_i} F_{{ B^W_{{\gamma^*},i}}} (A^*_i) - {\mathbb{E}}_{\mathcal{I}_i} g (A^*_i) \right| \le \int _{I_i} \left| F_{{ B^W_{{\gamma^*},i}}}(x) - g(x) \right| {{\rm d}}{ F_{A^*_i}}(x) \le \int_{\mathcal{I}_i} \delta_2 {{\rm d}}{ F_{A^*_i}}(x) \le \delta_2 .\label{eq:triangle2}\end{aligned}$$ Now, we note that for any $j \in [K]$, $F_{{ B^W_{{\gamma^*},i}}}(x_j) = \sum \limits_{m=0}^{j} \left[{F_{{ B^W_{{\gamma^*},i}}}(x_m) - F_{{ B^W_{{\gamma^*},i}}}(x_{m-1})} \right]$; here, for the sake of notation, we denote by $x_{-1}$ an arbitrary negative number (that is $F_{{ B^W_{{\gamma^*},i}}}(x_{-1}) = 0$). Using this, we have: $$\begin{aligned}
&\left|{\mathbb{E}}_{\mathcal{I}_i} g \left(A^n_i \right) - {\mathbb{E}}_{\mathcal{I}_i} g \left(A^*_i \right) \right|\nonumber \\
= & \left| \sum \limits_{j=1}^K F_{{ B^W_{{\gamma^*},i}}}(x_j) \left[{ {\mathbb{E}}_{\mathcal{I}_i} \boldsymbol{1} _{P_j}\left( A^n_i\right)} - {{\mathbb{E}}_{\mathcal{I}_i} \boldsymbol{1} _{P_j}\left( A^*_i\right)} \right] \right| \nonumber \\ = & \left| \sum \limits_{j=1}^K F_{{ B^W_{{\gamma^*},i}}}(x_j) \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right| \nonumber \\
= & \left| \sum \limits_{j=1}^K \left( \sum \limits_{m=0}^{j}
\left[ {F_{{ B^W_{{\gamma^*},i}}}(x_m) - F_{{ B^W_{{\gamma^*},i}}}(x_{m-1})} \right] \left[{\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right) \right| \nonumber \\
\le & \left| \left[ F_{{ B^W_{{\gamma^*},i}}}(x_0) - F_{{ B^W_{{\gamma^*},i}}}(x_{-1}) \right] \sum \limits_{j=1}^K \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right| \nonumber \\
& \qquad + \left| \sum \limits_{m=1}^K \left( \left[ F_{{ B^W_{{\gamma^*},i}}}(x_m) - F_{{ B^W_{{\gamma^*},i}}}(x_{m-1}) \right] \sum \limits_{j=m}^K \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right) \right|. \label{eq:portmanteau_bridge}\end{aligned}$$ Note that .[^27] Moreover, due to the fact that and $F_{{ B^W_{{\gamma^*},i}}}(x_{-1}) = 0$, we can rewrite the first term in as follows: $$\begin{aligned}
& \left| \left[ F_{{ B^W_{{\gamma^*},i}}}(x_0) - F_{{ B^W_{{\gamma^*},i}}}(x_{-1}) \right] \sum \limits_{j=1}^K \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right| \nonumber \\
= & \left| F_{{ B^W_{{\gamma^*},i}}}(0) \cdot \left[ \sum \limits_{j=1}^K \left({ F_{A^n_i}}(x_j) - { F_{A^n_i}}(x_{j-1}) - { F_{A^*_i}}(x_j) + { F_{A^*_i}}(x_{j-1}) \right) \right] \right| \nonumber \\
= & \left| F_{{ B^W_{{\gamma^*},i}}}(0) \cdot \left[{ F_{A^n_i}}(x_K) - { F_{A^n_i}}(x_0) - { F_{A^*_i}}(x_K) + { F_{A^*_i}}(x_{0}) \right] \right| \nonumber \\
\le & F_{{ B^W_{{\gamma^*},i}}}(0) \cdot 2 \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| } \nonumber \\
\le & 2 \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| } . \label{eq:portmanteau_final1}\end{aligned}$$ Now, recall that $x_m = x_{m-1} + {v^B_i}/{({\lambda^*_B}\cdot K)}, \forall m \in [K]$, by the definition of $F_{{ B^W_{{\gamma^*},i}}}$, we deduce that . Therefore, the second term in is $$\begin{aligned}
&\left| \sum \limits_{m=1}^K \left( \left[ F_{{ B^W_{{\gamma^*},i}}}(x_m) - F_{{ B^W_{{\gamma^*},i}}}(x_{m-1}) \right] \sum \limits_{j=m}^K \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right) \right| \nonumber \\
= & \left| \sum \limits_{m=1}^K \left( \left[ F_{{ B^W_{{\gamma^*},i}}}(x_m) - F_{{ B^W_{{\gamma^*},i}}}(x_{m-1}) \right] \sum \limits_{j=m}^K \left[{ F_{A^n_i}}(x_j) - { F_{A^n_i}}(x_{j-1}) - { F_{A^*_i}}(x_j) + { F_{A^*_i}}(x_{j-1}) \right] \right) \right| \nonumber \\
= & \left| \sum \limits_{m=1}^K \left( \left[ F_{{ B^W_{{\gamma^*},i}}}(x_m) - F_{{ B^W_{{\gamma^*},i}}}(x_{m-1}) \right] \left[{ F_{A^n_i}}(x_K) - { F_{A^n_i}}(x_{m-1}) - { F_{A^*_i}}(x_K) + { F_{A^*_i}}(x_{m-1}) \right] \right) \right| \nonumber \\
\le & \sum \limits_{m=1}^K \left( F_{{ B^W_{{\gamma^*},i}}}(x_m) - F_{{ B^W_{{\gamma^*},i}}}(x_{m-1}) \right) \cdot 2 \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| } \nonumber \\
\le & \sum \limits_{m=1}^K \frac{M}{K} \cdot 2 \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| } \nonumber \\
= & 2 M \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| }. \label{eq:portmanteau_final2}\end{aligned}$$ Inject and into , we obtain that $$\left|{\mathbb{E}}_{\mathcal{I}_i} g \left(A^n_i \right) - {\mathbb{E}}_{\mathcal{I}_i} g \left(A^*_i \right) \right| \le \left( 2 + 2 M \right) \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| }. \label{eq:triangle3}$$ Apply the triangle inequality and combine , , , we have that: $$\left| {\mathbb{E}}_{\mathcal{I}_i} F_{{ B^W_{{\gamma^*},i}}} (A^n_i) - {\mathbb{E}}_{\mathcal{I}_i} F_{{ B^W_{{\gamma^*},i}}} (A^*_i) \right| \le 2 \delta_2 + \left( 2 + 2M \right) \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| }.$$ From this and , we obtain that $$|{\mathbb{E}}{ F_{B^*_i}}(A^n_i) - {\mathbb{E}}{ F_{B^*_i}}(A^*_i) | \le 2 \delta_2 + \left( 4 + 2M \right) \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| }. \label{eq:portmanteau_the_end}$$ Recall the constant $C_1$ indicated in Lemma \[lem:convergence\], we define (note that $C_2$ does not depend on $n$ nor $\varepsilon_2$) and deduce that $C_2 \varepsilon_2^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_2, 1/{e}\}}\right)}\ge C_1 \delta_2^{-2} \ln\left(\frac{1}{\min\{\delta_2, 1/{e}\}} \right)$.[^28] Take $\varepsilon_1:= \delta_2$, for any $n \ge C_2 \varepsilon_2^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_2, 1/{e}\}}\right)}$, we have $n \ge C_1 \varepsilon_1^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$ and by applying Lemma \[lem:convergence\], we obtain that $\sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| } \le \varepsilon_1 = \delta_2 $ and thus by , we have: $$|{\mathbb{E}}{ F_{B^*_i}}(A^n_i) - {\mathbb{E}}{ F_{B^*_i}}(A^*_i) | \le 2 \delta_2 + \left( 4 + 2M \right) \delta_2 = (6 + 2M) \delta_2 = \varepsilon_2.$$ This is exactly . We can have a similar result in the case where $i \notin {\Omega_A({\gamma^*})}$ (its proof is omitted here) and we conclude the proof of this lemma.
PROOF OF RESULTS IN SECTION \[sec:LotteryApproximation\] {#sec:Appendix_Lotte}
========================================================
Proof of Lemma \[lem:deltalemma\] {#sec:appen_proof_lem_delta}
---------------------------------
Fix $y^* \in [0,2X_B]$, we look for the condition on $n$ such that holds. The condition corresponding to the inequality $\int_{{\mathcal{Y}_{\zeta}}(x^*, \varepsilon)} {{\rm d}}{ F_{B^n_i}}(x) \le \varepsilon + \delta$ with $x^* \in [0,2X_B]$ can be proved similarly and thus is omitted in this section.
First, we note that if ${\mathcal{X}_{\zeta}}(y^*, \varepsilon)$ is empty, $\int_{{\mathcal{X}_{\zeta}}(y^*, \varepsilon)} {{\rm d}}{ F_{A^n_i}}(x)=0$ and the result trivially holds. Now, let us assume that ${\mathcal{X}_{\zeta}}(y^*, \varepsilon) \neq \emptyset$, we can write ${\mathcal{X}_{\zeta}}(y^*, \varepsilon) = I_1 \bigcup I_2 \bigcup I_3$ with[^29] $$\begin{aligned}
I_1 &:= \{x \in [0,2X_B]: x=y^*, |\zeta_A(x,y^*) - \alpha| \ge \varepsilon \}, \\
I_2 &:= \{x \in [0,2X_B]: x < y^*, \zeta_A(x,y^*) \ge \varepsilon \}, \\
I_3 &:= \{x \in [0,2X_B]: x>y^*, 1 -\zeta_A(x,y^*) \ge \varepsilon \}.\end{aligned}$$ It is trivial that $I_1$ is either an empty set or a singleton; on the other hand, due to the monotonicity of the CSF $\zeta_A$ (see $(C2)$, Definition \[def:CSF\_general\]), $I_2$ and $I_3$ are either empty sets or half intervals. Moreover, for any arbitrary distribution $F$, we have that
$$\int_{ x \in I^{\prime}} {{\rm d}}F(x) = \left\{ \begin{array}{l}
0 \text{ , if } I^{\prime} = \emptyset, \\
F(a) \text{ , if } I^{\prime} = \{a\}, \textrm{i.e., } I^{\prime} \textrm{ is a singleton},\\
F(b) - F(a) \text{ , if } I^{\prime} = (a,b], \textrm{i.e., } I^{\prime} \textrm{ is a half interval}.
\end{array} \right. \quad
$$
Therefore, we can deduce that
$$\int_{{\mathcal{X}_{\zeta}}(y^*, \varepsilon)} {{\rm d}}{ F_{A^n_i}}(x) - \int_{{\mathcal{X}_{\zeta}}(y^*, \varepsilon)} {{\rm d}}{ F_{A^*_i}}(x) = \sum_{j=1}^3 \left( \int_{I_j} {{\rm d}}{ F_{A^n_i}}(x) - \int_{I_j} {{\rm d}}{ F_{A^*_i}}(x) \right) \le 5 \sup_{x \in [0,\infty)}{|{ F_{A^n_i}}(x) - { F_{A^*_i}}(x)| }$$
Recall the constant $C_1$ indicated in Lemma \[lem:convergence\], we define $L_0:= C_1 5^2 (\ln(5) + 1)$. Note that $L_0$ does not depend on the choice of $y^*$. Take $\varepsilon_1:= \varepsilon /5$, we can deduce that .[^30] Therefore, for any $n \ge L_0 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $n \ge C_1 \varepsilon_1^2 {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$ and by Lemma \[lem:convergence\], $\sup \limits_{x \in [0,\infty)}{|{ F_{A^n_i}}(x) - { F_{A^*_i}}(x)| } \le \varepsilon_1 = \varepsilon/5 $. Hence, for any and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, $$\int_{{\mathcal{X}_{\zeta}}(y^*, \varepsilon)} {{\rm d}}{ F_{A^n_i}}(x) \le \int_{{\mathcal{X}_{\zeta}}(y^*, \varepsilon)} {{\rm d}}{ F_{A^*_i}}(x) + 5 \cdot \varepsilon/5 \le \delta + \varepsilon.$$
Proof of Theorem \[theo:Lottery\_generic\_approx\] {#sec:appen_proof_theoLottery}
--------------------------------------------------
We first give the proof of Result $(ii)$. For the sake of brevity, we only focus on . The proof that holds under the same condition can be done similarly and thus is omitted. Note that in this proof, we often use the Fubini’s Theorem to exchange the order of the double integrals.
Recall that $\boldsymbol{x}^A = (x^A_i)_{i \in [n]}$, by the definition of the payoff functions in ${\mathcal{LB}_n}(\zeta)$, can be rewritten as $$\sum\limits_{i = 1}^n {\left( {{v^A_i}\int\limits_0^\infty {\zeta_A\left( {x_i^A,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} } \right)} - \sum\limits_{i = 1}^n {\left( {{v^A_i}\int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{A^n_i}}}\left( x \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} } } \right)} \le 8\delta + 13 \varepsilon. \label{eq:lottery_proof_rewrite_A}$$ We now prove that holds under appropriate parameters values. To do this, we prepare two useful lemmas as follows.
\[lem:proof\_lottery\_LBn\_prepare1\] For any pair of CSFs $\zeta = (\zeta_A, \zeta_B)$, any $\varepsilon \in (0,1]$ and $x^* \in [0,2X_B]$, the following results hold:
- For any $n$, $i \in [n]$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, $$\left| \int\limits_0^\infty {\zeta_A\left( {x^*,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} - \int\limits_0^\infty {\beta_A\left( {x^*,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} \right| \le \delta + \varepsilon \label{eq:lottery_delta_prepare1*}.$$
- There exists a constant $L_1 >0$ such that for any $n \ge L_1 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, $i \in [n]$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, $$\left| \int\limits_0^\infty {\zeta_A\left( {x^*,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} - \int\limits_0^\infty {\beta_A\left( {x^*,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} \right| \le \delta + 2\varepsilon. \label{eq:lottery_delta_prepare1n}$$
\[lem:proof\_lottery\_LBn\_prepare2\] For any $\varepsilon \in (0,1]$, there exists a constant $L_2>0$ such that for any , any game ${\mathcal{LB}_n}(\zeta)$, any $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ and , we have: $$\begin{aligned}
& \left|{\int\limits_0^\infty {\zeta_A\left( {x ,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} - \int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} } \right| \le 2\delta + 4\varepsilon , \forall x \ge 0, \label{eq:propo_appen_lottery_1} \\
&\left|\int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} - \int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} \right| \le 2\delta + 3\varepsilon, \label{eq:propo_appen_lottery_2} \\
& \left|\int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} - \int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right)} \right| \le 2\delta + 4\varepsilon. \label{eq:propo_appen_lottery_3}\end{aligned}$$
Lemma \[lem:proof\_lottery\_LBn\_prepare1\] states the relation between the first term appearing in the left-hand-side of and the corresponding terms when we replace the CSF $\zeta$ by the Blotto functions $\beta$ and replace ${ F_{B^n_i}}$ by ${ F_{B^*_i}}$. These relations are useful to connect the statement we want to prove and the results obtained in Section \[sec:ApproximateBlotto\]. A proof of Lemma \[lem:proof\_lottery\_LBn\_prepare1\] is given in \[sec:lemma\_proof\_lottery\_LBn\_prepare1\]. On the other hand, Lemma \[lem:proof\_lottery\_LBn\_prepare2\] indicates several useful inequalities involving the players’ payoffs in the game ${\mathcal{LB}_n}$ (when they play according to the ${{\rm IU}^{{\gamma^*}}}$ strategy or playing such that the marginals are ${ F_{A^*_i}}, { F_{B^*_i}}$). Its proof is given in \[sec:lem:proof\_lottery\_LBn\_prepare2\] that is based on Lemma \[lem:proof\_lottery\_LBn\_prepare1\] and the convergence of the distributions ${ F_{A^n_i}},{ F_{B^n_i}}$ toward ${ F_{A^*_i}},{ F_{B^*_i}}$ (i.e., Lemma \[lem:convergence\]).
We have another remark: for any $n$ and $i \in [n]$, $${\mathbb{P}}(A^*_i = B^*_i = x) = 0, \forall x \ge 0. \label{eq:A^*=B^*}$$ This can be trivially proved as follows: first, ${\mathbb{P}}(A^*_i = B^*_i = x) = {\mathbb{P}}(A^*_i = x) {\mathbb{P}}(B^*_i =x)$ since they are independent; now, if $x > 0$, both ${ F_{A^*_i}}$ and ${ F_{B^*_i}}$ are continuous at $x$ and thus ${\mathbb{P}}(A^*_i = x) = {\mathbb{P}}(B^*_i =x) = 0$; on the other hand, if $x =0$, in the case where $i \in {\Omega_A({\gamma^*})}$, since $A^*_i = { A^S_{{\gamma^*},i}}$, we have ${\mathbb{P}}(A^*_i = x) = 0 $, in the case where $i \notin {\Omega_A({\gamma^*})}$, since $B^*_i = { B^S_{{\gamma^*},i}}$, we have ${\mathbb{P}}(B^*_i = x) = 0 $.
Finally, use Lemma \[lem:proof\_lottery\_LBn\_prepare1\] and \[lem:proof\_lottery\_LBn\_prepare2\] and take $L^* = \max\{L_1, L_2\}$, for any $n\ge L^* \varepsilon^{-2 } {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, and any pure strategy $\boldsymbol{x}^A$ of player A, we have: $$\begin{aligned}
& \sum\limits_{i = 1}^n {\left( {{v^A_i}\int\limits_0^\infty {\zeta_A\left( {x_i^A,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} } \right)} \nonumber\\
\le & \sum\limits_{i = 1}^n {\left( {{v^A_i}\int\limits_0^\infty {\zeta_A\left( {x_i^A,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} } \right)} + \sum\limits_{i = 1}^n { v^A_i (2\delta + 4\varepsilon)} & \text{ (due to \eqref{eq:propo_appen_lottery_1}) } \nonumber\\
= & \sum\limits_{i = 1}^n {\left( {{v^A_i}\int\limits_0^\infty {\zeta_A\left( {x_i^A,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} } \right)} + 2\delta + 4\varepsilon & (\textrm{note that } \sum\nolimits_{i=1}^n{v^A_i} = 1 )\nonumber\\
\le & \sum\limits_{i = 1}^n {\left( {{v^A_i}\int\limits_0^\infty {\beta_A\left( {x_i^A,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} } \right)} + 3\delta + 5\varepsilon & ( \text{due to } \eqref{eq:lottery_delta_prepare1*})\nonumber \\
= & \sum\limits_{i = 1}^n {\left[ {v^A_i} \left(\alpha {\mathbb{P}}(B^*_i =x^A_i) + {\mathbb{P}}(B^*_i < x^A_i) \right) \right]} + 3\delta + 5\varepsilon \nonumber \\
\le & \sum\limits_{i = 1}^n {{v_i^A}{{ F_{B^*_i}}}\left( {x_i^A} \right)} + 3\delta + 5\varepsilon & (\textrm{since } \alpha \le 1) \nonumber\\
\le & \sum\limits_{i = 1}^n {\left( {{v_i^A}\int\limits_0^\infty {{{ F_{B^*_i}}}(x) {{\rm d}}{{ F_{A^*_i}}}(x)} } \right)} + 3\delta + 5\varepsilon & (\text{due to Lemma~\ref{lem:best_response}}) \nonumber \\
= & \sum\limits_{i = 1}^n {\left( {v_i^A} \int\limits_0^\infty {\mathbb{P}}(B^*_i < x) {{\rm d}}{{ F_{A^*_i}}}(x) \right)} + 3\delta + 5\varepsilon & \text{ (due to } \eqref{eq:A^*=B^*}) \nonumber \\
\le & \sum\limits_{i = 1}^n {\left( {{v^A_i}\int\limits_0^\infty {\int\limits_0^\infty {{\beta_A}\left( {x,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} } } \right)} + 3\delta + 5\varepsilon \nonumber \\
\le & \sum\limits_{i = 1}^n {\left( {{v^A_i}\int\limits_0^\infty {\int\limits_0^\infty {{\zeta_A}\left( {x,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} } } \right)} + 4\delta + 6\varepsilon & \text{ (due to } \eqref{eq:lottery_delta_prepare1*}) \nonumber \\
\le & \sum\limits_{i = 1}^n {\left( {v^A_i}\int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} \right)} + 6\delta + 9\varepsilon & \text{ (due to \eqref{eq:propo_appen_lottery_2}) } \nonumber \\
\le & \sum\limits_{i = 1}^n {\left( {v^A_i}\int\limits_0^\infty {\int\limits_0^\infty {\zeta_A \left( {x,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right)} \right)}+ 8\delta + 13\varepsilon & \text{ (due to \eqref{eq:propo_appen_lottery_3})}. \nonumber\end{aligned}$$ Hence, we conclude that for $n \ge L^* \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$ holds and thus, also holds.
To prove that Result $(ii)$ implies Result $(i)$, we can proceed similarly to the proof that Theorem \[TheoMainBlotto\]-$(ii)$ implies Theorem \[TheoMainBlotto\]-$(i)$ (see \[sec:Appen\_Proof\_TheoBlotto\]). We conclude this proof.
Proof of Lemma \[lem:proof\_lottery\_LBn\_prepare1\] {#sec:lemma_proof_lottery_LBn_prepare1}
----------------------------------------------------
First, we prove . Note that ${ F_{B^*_i}}(y)=1, \forall y > 2X_B$ (see Lemma \[lem:Preliminary\]-$(iv))$, for any $n$, $i \in [n]$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, we have $$\begin{aligned}
& \left| \int\limits_0^\infty {\zeta_A\left( {x^*,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} - \int\limits_0^\infty {\beta_A\left( {x^*,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} \right| \nonumber\\
\le & \int\nolimits_{{\mathcal{Y}_{\zeta}}(x^*, \varepsilon )}{\left|\zeta_A(x^*,y) - \beta_A(x^*,y)\right| {{\rm d}}{{ F_{B^*_i}}}\left( y \right) } + \int\nolimits_{[0,\infty) \backslash {\mathcal{Y}_{\zeta}}(x^*,\varepsilon)}{\left|\zeta_A(x^*,y) - \beta_A(x^*,y)\right| {{\rm d}}{{ F_{B^*_i}}}\left( y \right) } \nonumber\\
= & \int\nolimits_{{\mathcal{Y}_{\zeta}}(x^*, \varepsilon )}{\left|1 - \zeta_B(x^*,y) - 1 + \beta_B(x^*,y)\right| {{\rm d}}{{ F_{B^*_i}}}\left( y \right) } + \int\nolimits_{[0,2X_B] \backslash {\mathcal{Y}_{\zeta}}(x^*,\varepsilon)}{\left|1 - \zeta_B(x^*,y) - 1 + \beta_B(x^*,y)\right| {{\rm d}}{{ F_{B^*_i}}}\left( y \right) } \nonumber\\
= & \int\nolimits_{{\mathcal{Y}_{\zeta}}(x^*, \varepsilon )}{\left|\zeta_B(x^*,y) - \beta_B(x^*,y)\right| {{\rm d}}{{ F_{B^*_i}}}\left( y \right) } + \int\nolimits_{[0,2X_B] \backslash {\mathcal{Y}_{\zeta}}(x^*,\varepsilon)}{\left|\zeta_B(x^*,y) - \beta_B(x^*,y)\right| {{\rm d}}{{ F_{B^*_i}}}\left( y \right) } \nonumber\\
\le & \int\nolimits_{{\mathcal{Y}_{\zeta}}(x^*, \varepsilon )}{ {{\rm d}}{{ F_{B^*_i}}}\left( y \right) } + \int\nolimits_{[0,2X_B] \backslash {\mathcal{Y}_{\zeta}}(x^*,\varepsilon)}{\varepsilon {{\rm d}}{{ F_{B^*_i}}}\left( y \right) } \nonumber \\
\le & \delta + \varepsilon. \label{eq:prepare1_first}\end{aligned}$$ Here, the second-to-last inequality comes from the fact that $0 \le \zeta_B(x,y), \beta_B(x,y) \le 1$ for any $x,y$ and the definition of ${\mathcal{Y}_{\zeta}}(x^*, \varepsilon)$ while the last inequality is due to the definition of ${\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$.
Now, in order to prove , we proceed similarly as in to show that $$\begin{aligned}
& \left| \int\limits_0^\infty {\zeta_A\left( {x^*,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} - \int\limits_0^\infty {\beta_A\left( {x^*,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} \right| \nonumber\\
\le & \int\nolimits_{{\mathcal{Y}_{\zeta}}(x^*, \varepsilon )}{ {{\rm d}}{{ F_{B^n_i}}}\left( y \right) } + \int\nolimits_{[0,2X_B] \backslash {\mathcal{Y}_{\zeta}}(x^*,\varepsilon)}{\varepsilon {{\rm d}}{{ F_{B^n_i}}}\left( y \right) } \nonumber \\
\le & \int\nolimits_{{\mathcal{Y}_{\zeta}}(x^*, \varepsilon )}{ {{\rm d}}{{ F_{B^n_i}}}\left( y \right) } + \varepsilon. \label{eq:LemC1_2}\end{aligned}$$ Finally, by Lemma \[lem:deltalemma\], for any $n \ge L_0 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$ and $\delta \!\in\! {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, we have . Combine this with , we conclude that holds for any $n \ge L_0 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$. Take $L_1:= L_0$, we conclude the proof.
Proof of Lemma \[lem:proof\_lottery\_LBn\_prepare2\] {#sec:lem:proof_lottery_LBn_prepare2}
----------------------------------------------------
In this proof, we use the notation ${\mathbb{E}}h(X,y):= \int \nolimits_{0}^{\infty}h(x,y) {{\rm d}}F_{X}(x)$ and where $X, Y$ are arbitrary non-negative random variables and $h$ is any function.
For any $i \in [n]$ and $x \ge 0$, we have $$\begin{aligned}
&\left| {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} - \int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} }\right| \nonumber \\
\le &\left| {\mathbb{E}}\zeta_A(x,B^n_i) \! -\! {\mathbb{E}}\beta_A(x,B^n_i)\right|\! +\! \left|{\mathbb{E}}\beta_A(x,B^n_i) \!-\! {\mathbb{E}}\beta_A(x,B^*_i)\right|\! +\! \left|{\mathbb{E}}\beta_A(x,B^*_i) \!-\! {\mathbb{E}}\zeta_A(x,B^*_i) \right|. \label{eq:proof_propo_LBn_1}\end{aligned}$$ We notice that upper-bounds of the first and third terms in the right-hand-side of are given by and from Lemma \[lem:proof\_lottery\_LBn\_prepare1\]. We focus on finding an upper-bound of the second term of ; to do this, we rewrite this term as follows. $$\begin{aligned}
& {\mathbb{E}}\beta_A(x,B^n_i) = \int_{y<x} {{\rm d}}{ F_{B^n_i}}(y) + \alpha {\mathbb{P}}(B^n_i = x) = { F_{B^n_i}}(x ) - (1-\alpha){\mathbb{P}}(B^n_i = x),\label{eq:Ex_betaBn}\\
\textrm{and~} & {\mathbb{E}}\beta_A(x,B^*_i) = \int_{y<x} {{\rm d}}{ F_{B^*_i}}(y) + \alpha {\mathbb{P}}(B^*_i = x) = { F_{B^*_i}}(x) - (1-\alpha){\mathbb{P}}(B^*_i = x).\label{eq:Ex_betaB*}\end{aligned}$$ If $\alpha =1$, we trivially have $ \left| {\mathbb{E}}\beta_A(x,B^n_i) - {\mathbb{E}}\beta_A(x,B^*_i) \right| = \left| { F_{B^n_i}}(x) - { F_{B^*_i}}(x) \right| $. In the following, we assume that $\alpha < 1$ and consider three cases:
*Case 1:* If $x =0$. From Lemma \[lem:continuity\_Ani\_and\_Bni\]-$(i)$, we have ${\mathbb{P}}(B^n_i = 0) = {\mathbb{P}}(B^*_i = 0)$ and thus $$\begin{aligned}
\left| {\mathbb{E}}\beta_A(0,B^n_i) - {\mathbb{E}}\beta_A(0,B^*_i) \right| = \left|\int_{y<0} {{\rm d}}{ F_{B^n_i}}(y) - \int_{y<0} {{\rm d}}{ F_{B^*_i}}(y)+ \alpha {\mathbb{P}}(B^n_i = 0) - \alpha {\mathbb{P}}(B^*_i =0)\right| = 0 .\end{aligned}$$
*Case 2:* If $x>0$, ${\mathbb{P}}(B^*_i = x) =0$ by definition. On the other hand, from Results $(ii)$ and $(iii)$ of Lemma \[lem:continuity\_Ani\_and\_Bni\], we have where we define $D:=\left(1 - \frac{{\underaccent{\bar}{\lambda}}}{{\bar{\lambda}}} \frac{{\underaccent{\bar}{w}}^2}{{\bar{w}}^2} \right)$. Following , for any $n \ge C_0 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$ (here, $C_0$ is defined as in \[sec:appen\_proof\_lem:SufCon\]), we have $D^{n-1 } \le \frac{\varepsilon}{2(1-\alpha)}$. Therefore, for any , we have $$\begin{aligned}
& \left| {\mathbb{E}}\beta_A(x,B^n_i) - {\mathbb{E}}\beta_A(x,B^*_i) \right| \\
\le & |{ F_{B^n_i}}(x) - { F_{B^*_i}}(x)| + (1-\alpha)\left|{\mathbb{P}}(B^n_i = x)\right| \qquad (\textrm{due to }\eqref{eq:Ex_betaBn}-\eqref{eq:Ex_betaB*})\\
\le & \sup \limits_{x \in [0,\infty)}{|{ F_{B^n_i}}(x) - { F_{B^*_i}}(x)| } + (1-\alpha)\frac{\varepsilon}{2 (1-\alpha)} \\
= & \sup \limits_{x \in [0,\infty)}{|{ F_{B^n_i}}(x) - { F_{B^*_i}}(x)| } + \frac{\varepsilon}{2}.\end{aligned}$$
In conclusion, for any $x \ge 0$, and . Now, let us define $C^{\prime}_1 = C_1 \cdot 4 (\ln(2) + 1)$ (where $C_1$ is indicated in Lemma \[lem:convergence\]); take $\varepsilon_1:= \varepsilon/2$, we have . Therefore, for any $n \ge C^{\prime}_1 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $n \ge C_1 \varepsilon_1^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$ and apply Lemma \[lem:convergence\], we have $\sup \limits_{x \in [0,\infty)}{|{ F_{B^n_i}}(x) - { F_{B^*_i}}(x)| } \le \varepsilon_1 = \varepsilon/2$.
We deduce that for any $x \ge 0$, for any $n \ge \max\{C_0, C^{\prime}_1\} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$ and $i \in [n]$, we have: $$\left| {\mathbb{E}}\beta_A(x,B^n_i) - {\mathbb{E}}\beta_A(x,B^*_i) \right| \le \varepsilon/2 + \varepsilon/2 = \varepsilon. \label{eq:proof_C5}$$
Finally, apply Lemma \[lem:proof\_lottery\_LBn\_prepare1\] to to bounds the first and third term of its right-hand-side, use to bound its second-term and take $L_{\eqref{eq:propo_appen_lottery_1}} = \max\{L_1, C_0, C^{\prime}_1\}$, we deduce that for any $n \ge L_{\eqref{eq:propo_appen_lottery_1}}\varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, $$\left| {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} - \int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} }\right| \le (\delta+ 2\varepsilon) + \varepsilon + (\delta + \varepsilon) = 2\delta + 4 \varepsilon.$$ To prove this inequality, we note that similar to the proof of in Lemma \[lem:proof\_lottery\_LBn\_prepare1\] (by replacing ${ F_{B^*_i}}$ by ${ F_{A^*_i}}$ and replacing $\zeta_A(x^*,y), \beta_A(x^*,y)$ by $\zeta_A(x,y^*), \beta_A(x,y^*)$), we can prove that for any $n$, $i \in [n]$, $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ and $y^* \in [0,2X_B]$, the following inequality holds $$\left| \int\limits_0^\infty {\zeta_A\left( {x,y^*} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} - \int\limits_0^\infty {\beta_A\left( {x,y^*} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \le \delta + \varepsilon.\label{eq:zeta_A*}$$ Using this, we have $$\begin{aligned}
& \left|\int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} - \int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} \right| \\
\le & \int \limits_{0}^{\infty} \left| \int \limits_0^\infty \zeta_A(x,y) {{\rm d}}{ F_{A^*_i}}(x) \!- \! \int \limits_0^\infty \beta_A(x,y) {{\rm d}}{ F_{A^*_i}}(x) \right| {{\rm d}}{ F_{B^*_i}}(y) \!\\
& \qquad \qquad +\! \left| \int \limits_{0}^{\infty} \int \limits_0^\infty \beta_A(x,y) {{\rm d}}{ F_{A^*_i}}(x) {{\rm d}}{ F_{B^*_i}}(y)\! -\! \int \limits_{0}^{\infty} \int\limits_0^\infty \beta_A(x,y) {{\rm d}}{ F_{A^*_i}}(x) {{\rm d}}{ F_{B^n_i}}(y) \right|\\
& \qquad \qquad \qquad \qquad + \int\limits \limits_{0}^{\infty} \left| \int \limits_0^\infty \beta_A(x,y) {{\rm d}}{ F_{A^*_i}}(x) - \int \limits_0^\infty \zeta_A(x,y) {{\rm d}}{ F_{A^*_i}}(x)\right| {{\rm d}}{ F_{B^n_i}}(y) \\
\le & \int \nolimits_{0}^{\infty} (\delta \!+\! \varepsilon) {{\rm d}}{ F_{B^*_i}}(y) \!+\! \left| \int \nolimits_{0}^{\infty} {\mathbb{E}}\beta_A(x,B^*_i) {{\rm d}}{ F_{A^*_i}}(x) \! -\! \int \nolimits_{0}^{\infty} {\mathbb{E}}\beta_A(x,B^n_i) {{\rm d}}{ F_{A^*_i}}(x) \right| \! +\! \int \nolimits_{0}^{\infty} (\delta \! +\! \varepsilon) {{\rm d}}{ F_{B^n_i}}(y) \\
\le & 2\delta \!+\! 2\varepsilon \!+\! \int \nolimits_{0}^{\infty} \left|{\mathbb{E}}\beta_A(x,B^n_i)- {\mathbb{E}}\beta_A(x,B^*_i) \right|{{\rm d}}{ F_{A^*_i}}(x).\end{aligned}$$ Finally, take $L_\eqref{eq:propo_appen_lottery_2} = \max\{C_0, C^{\prime}_1\}$ and apply , we deduce that for any , holds.
To prove this inequality, we note that similar to the proof of in Lemma \[lem:proof\_lottery\_LBn\_prepare1\] (by replacing ${ F_{B^n_i}}$ by ${ F_{A^n_i}}$ and replacing $\zeta_A(x^*,y), \beta_A(x^*,y)$ by $\zeta_A(x,y^*), \beta_A(x,y^*)$), we can prove that for , $i \in [n]$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, $$\label{eq:last}
\left| \int\limits_0^\infty {\zeta_A\left( {x,y^*} \right){{\rm d}}{{ F_{A^n_i}}}\left( x \right)} - \int\limits_0^\infty {\beta_A\left( {x,y^*} \right) {{\rm d}}{{ F_{A^n_i}}}\left( x \right)} \right| \le \delta + 2\varepsilon.$$ Now, similar to the proof leading to , we can prove that for any , $i \in [n]$ and $y \ge 0$, we have $$\label{eq:mid}
\left| {\mathbb{E}}\beta_A(A^*_i,y) - {\mathbb{E}}\beta_A(A^n_i,y) \right| \le \varepsilon.$$
Finally, take $L_\eqref{eq:propo_appen_lottery_3} = \max\{L_1, C_0, C^{\prime}_1\}$, for any $n \ge L_\eqref{eq:propo_appen_lottery_3} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, $i \in [n]$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, we have $$\begin{aligned}
& \left|\int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} - \int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right)} \right| \\
\le & \int \limits_{0}^{\infty} \left| \int \limits_0^\infty \zeta_A(x,y) {{\rm d}}{ F_{A^*_i}}(x) \!- \! \int \limits_0^\infty \beta_A(x,y) {{\rm d}}{ F_{A^*_i}}(x) \right| {{\rm d}}{ F_{B^n_i}}(y) \!\\
& \qquad \qquad +\! \left| \int \limits_{0}^{\infty} \int \limits_0^\infty \beta_A(x,y) {{\rm d}}{ F_{A^*_i}}(x) {{\rm d}}{ F_{B^n_i}}(y) \! -\! \int \limits_{0}^{\infty} \int\limits_0^\infty \beta_A(x,y) {{\rm d}}{ F_{A^n_i}}(x) {{\rm d}}{ F_{B^n_i}}(y)\right| \\
& \qquad \qquad \qquad \qquad + \int\limits \limits_{0}^{\infty} \left| \int \limits_0^\infty \beta_A(x,y) {{\rm d}}{ F_{A^n_i}}(x) - \int \limits_0^\infty \zeta_A(x,y) {{\rm d}}{ F_{A^n_i}}(x)\right| {{\rm d}}{ F_{B^n_i}}(y) \\
\le & \int \limits_{0}^{\infty} (\delta + \varepsilon) {{\rm d}}{ F_{A^*_i}}(x) + \int \limits_0^\infty \left| {\mathbb{E}}\beta_A(A^*_i,y) - {\mathbb{E}}\beta_A(A^n_i,y) \right| {{\rm d}}{{ F_{B^n_i}}}(y) \!+ \! \int \limits_{0}^{\infty} (\delta + 2\varepsilon) {{\rm d}}{ F_{A^n_i}}(x) & \textrm{(due to } \eqref{eq:zeta_A*} \textrm{ and } \eqref{eq:last} )\\
\le & 2\delta + 4 \varepsilon & \textrm{(due to } \eqref{eq:mid} ).\end{aligned}$$ In conclusion, take $L_2:= \max\{L_{\eqref{eq:propo_appen_lottery_1}},L_{\eqref{eq:propo_appen_lottery_2}},L_{\eqref{eq:propo_appen_lottery_3}} \}$, we conclude the proof of this lemma.
Remark on the Lottery Blotto games with continuous CSFs {#sec:appen_remark_conti_CSF}
-------------------------------------------------------
In this section, we present and prove the remark stating that under the additional assumption that the CSFs $\zeta_A$ and $\zeta_B$ are Lipschitz continuous on $[0,2X_B] \times [0,2X_B]$, the statements in Theorem \[theo:Lottery\_generic\_approx\] also hold with and (see below) in places of and . For the sake of completeness, we formally state this result as follows.
\[remark\_conti\_CSF\] For any CSF $\zeta_A$ and $\zeta_B$ that are Lipschitz continuous on $[0,2X_B] \times [0,2X_B]$, the following results hold (here, we denote $\zeta:=(\zeta_A, \zeta_B)$):
- In any game ${\mathcal{LB}_n}(\zeta)$, there exists a positive number such that for any ${\gamma^*}\in {\mathcal{S}_n}$ and $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$, the following inequalities hold for any pure strategy $\boldsymbol{x}^A$ and $\boldsymbol{x}^B$ of players A and B: $$\begin{aligned}
& \Pi_{\zeta}^A(\boldsymbol{x}^A,{{{\rm IU}^{{\gamma^*}}}_B}) \le \Pi_{\zeta}^A({{{\rm IU}^{{\gamma^*}}}_A},{{{\rm IU}^{{\gamma^*}}}_B}) + \left(2\delta + 5{\varepsilon} \right) W_A,\label{eq:lottery_theo_A_improve}\\
& \Pi_{\zeta}^B({{{\rm IU}^{{\gamma^*}}}_A},\boldsymbol{x}^B) \le \Pi_{\zeta}^B({{{\rm IU}^{{\gamma^*}}}_A},{{{\rm IU}^{{\gamma^*}}}_B}) + \left(2\delta + 5{\varepsilon} \right) W_B. \label{eq:lottery_theo_B_improve}
\end{aligned}$$
- For any $\varepsilon \in (0,1]$, there exists a constant $L_{\zeta} >0 $ (that depends on $\zeta$ but does not depend on $\varepsilon$) such that in any game ${\mathcal{LB}_n}(\zeta)$ where $ n \ge L_{\zeta} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, and hold for any ${\gamma^*}\in {\mathcal{S}_n}$, $\delta \in {\Delta_{{\gamma^*}}(\zeta, \varepsilon)}$ and any pure strategy $\boldsymbol{x}^A, \boldsymbol{x}^B$ of players A and B.
We define the Lipschitz constant of $\zeta_A, \zeta_B$ respectively by $\mathcal{L}_{\zeta_A}, \mathcal{L}_{\zeta_B}$ and let ${\mathcal{L}_{\zeta}}:= \max\{\mathcal{L}_{\zeta_A}, \mathcal{L}_{\zeta_B}\}$. We focus on proving Result $(ii)$ of this Remark; Result $(i)$ can be deduced from Result $(ii)$ and thus is omitted.
: We prove that for any $x^*, y^* \in [0,2X_B]$, there exists a constant ${C_{\zeta}}$ (that does not depend on $\varepsilon$ nor $x^*,y^*$) such that for any $n \ge C_{\zeta} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, the following inequalities hold:
$$\begin{aligned}
&\left|{\int\limits_0^\infty {\zeta_A\left( {x ,y^*} \right) {{\rm d}}{{ F_{A^n_i}}}\left( x \right)} - \int\limits_0^\infty {\zeta_A\left( {x,y^*} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} } \right| \le \varepsilon, \label{eq:CSF_intergal_1} \\
& \left|{\int\limits_0^\infty {\zeta_A\left( {x^* ,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} - \int\limits_0^\infty {\zeta_A\left( {x^*,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} } \right| \le \varepsilon \label{eq:CSF_intergal_2}.\end{aligned}$$
The proof of this statement is quite similar to the proof of Lemma \[lem:portmanteau\] (see \[sec:appen\_proof\_lem\_portmanteau\]). We present here the proof of ; the proof of can be done similarly.
Fix $y^* \in [0,2X_B]$; to simplify the notation, we define $f(x):= \zeta_A(x,y^*)$ and ${\tilde{\varepsilon}}_1:=\frac{\varepsilon}{4 + 4X_B {\mathcal{L}_{\zeta}}}$. From Lemma \[lem:Preliminary\], ${ F_{A^n_i}}(x) = { F_{A^*_i}}(x)=1, \forall x > 2X_B$; therefore, the left-hand-side of can be rewritten as follows. $$\left|{\int\limits_0^\infty {\zeta_A\left( {x ,y^*} \right) {{\rm d}}{{ F_{A^n_i}}}\left( x \right)} - \int\limits_0^\infty {\zeta_A\left( {x,y^*} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} } \right| = \left| {\int_{0}^{2X_B} {f(x)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) - \int_{0}^{2X_B} {f(x)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right|. \label{eq:CSF_integral_first}$$ Let us define $K := \big \lceil \frac{2 X_B {\mathcal{L}_{\zeta}}}{{\tilde{\varepsilon}}_1} \big \rceil$ and $K + 1$ points ${x}_j$ such that ${x_0} := 0$ and ${x}_j := {x}_{j-1} + \frac{2X_B}{K}, \forall j \in [K]$. In other words, we have the partitions where we denote by $P_1$ the interval $[{x}_0,{x}_1]$ and by $P_j$ the interval $({x}_{j-1}, {x}_{j}]$ for $j = 2, \ldots, K$. For any $ x, x^{\prime} \in P_j, \forall j \in [K]$, since $f$ is Lipschitz continuous, we have $$|f(x)- f(x^{\prime}) | \le {\mathcal{L}_{\zeta}}|x - x^{\prime}| \le {\mathcal{L}_{\zeta}}\frac{2X_B}{K} \le {\tilde{\varepsilon}}_1 . \label{eq:CSF_intergral_lipsc}$$ Now, we define the function $g(x)\!:=\! \sum \limits_{j=1}^{K} {f(x_j) \boldsymbol{1}_{P_j} (x)}$. Here, $\boldsymbol{1}_{P_j}$ is the indicator function of the set ${P_j}$. From this definition and Inequality , we have , $\forall x \in [0,2X_B]$. Therefore, $$\begin{aligned}
& \left| {\int_{0}^{2X_B} {f(x)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) - \int_{0}^{2X_B} {g(x)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right)} \right| \le \int_{0}^{2X_B} {\tilde{\varepsilon}}_1 {{\rm d}}{ F_{A^n_i}}(x) \le {\tilde{\varepsilon}}_1, \label{eq:CSF_intergral_triangle1}\\
& \left| {\int_{0}^{2X_B} {f(x)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right) - \int_{0}^{2X_B} {g(x)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \le \int_{0}^{2X_B} {\tilde{\varepsilon}}_1 {{\rm d}}{ F_{A^*_i}}(x) \le {\tilde{\varepsilon}}_1 .\label{eq:CSF_integral_triangle2}\end{aligned}$$ Now, we note that for any $j \in [K]$, $f(x_j) = \sum \limits_{m=0}^{j} \left[{f(x_m) - f(x_{m-1})} \right]$; here, by convention, we denote by $x_{-1}$ an arbitrary negative number and set $f(x_{-1}) = 0$. Using this, we have: $$\begin{aligned}
& \left| {\int_{0}^{2X_B} {g(x)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) - \int_{0}^{2X_B} {g(x)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \nonumber \\
= & \left| \sum \limits_{j=1}^K f(x_j) \left[{ \int_{0}^{2X_B} \boldsymbol{1} _{P_j}\left( x\right)} {{\rm d}}{ F_{A^n_i}}(x) - {\int_{0}^{2X_B} \boldsymbol{1} _{P_j}\left(x\right) {{\rm d}}{ F_{A^*_i}}(x) } \right] \right| \nonumber \\ = & \left| \sum \limits_{j=1}^K f(x_j) \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right| \nonumber \\
= & \left| \sum \limits_{j=1}^K \left( \sum \limits_{m=0}^{j}
\left[ {f(x_m) - f(x_{m-1})} \right] \left[{\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right) \right| \nonumber \\
\le & \left| \left[ f(x_0) - f(x_{-1}) \right] \sum \limits_{j=1}^K \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right| \nonumber \\
& \qquad + \left| \sum \limits_{m=1}^K \left( \left[ f(x_m) - f(x_{m-1}) \right] \sum \limits_{j=m}^K \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right) \right|. \label{eq:CSF_intergral_bridge}\end{aligned}$$ Note that .[^31] Now, we can rewrite the first term in as follows. $$\begin{aligned}
& \left| \left[ f(x_0) - f(x_{-1}) \right] \sum \limits_{j=1}^K \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right| \nonumber \\
= & \left| f(0) \cdot \left[ \sum \limits_{j=1}^K \left({ F_{A^n_i}}(x_j) - { F_{A^n_i}}(x_{j-1}) - { F_{A^*_i}}(x_j) + { F_{A^*_i}}(x_{j-1}) \right) \right] \right| \nonumber \\
= & \left| f(0) \cdot \left[{ F_{A^n_i}}(x_K) - { F_{A^n_i}}(x_0) - { F_{A^*_i}}(x_K) + { F_{A^*_i}}(x_{0}) \right] \right| \nonumber \\
\le &2 \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| } \label{eq:CSF_intergral_final1}.\end{aligned}$$ Here, the last inequality comes from the fact that $f(x) \le 1, \forall x \in [0,2X_B]$ (since it is a CSF).
Now, we recall that for any $m \in [K]$, . Therefore, the second term in is $$\begin{aligned}
&\left| \sum \limits_{m=1}^K \left( \left[ f(x_m) - f(x_{m-1}) \right] \sum \limits_{j=m}^K \left[ {\mathbb{P}}\left(A^n_i \in P_j \right)\!- \! {\mathbb{P}}\left(A^*_i \in P_j \right) \right] \right) \right| \nonumber \\
= & \left| \sum \limits_{m=1}^K \left( \left[ f(x_m) - f(x_{m-1}) \right] \sum \limits_{j=m}^K \left[{ F_{A^n_i}}(x_j) - { F_{A^n_i}}(x_{j-1}) - { F_{A^*_i}}(x_j) + { F_{A^*_i}}(x_{j-1}) \right] \right) \right| \nonumber \\
= & \left| \sum \limits_{m=1}^K \left( \left[ f(x_m) - f(x_{m-1}) \right] \left[{ F_{A^n_i}}(x_K) - { F_{A^n_i}}(x_{m-1}) - { F_{A^*_i}}(x_K) + { F_{A^*_i}}(x_{m-1}) \right] \right) \right| \nonumber \\
\le & \sum \limits_{m=1}^K \frac{2X_B {\mathcal{L}_{\zeta}}}{K} \cdot 2 \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| } \nonumber \\
= & 4 X_B {\mathcal{L}_{\zeta}}\sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| }. \label{eq:CSF_intergral_final2}\end{aligned}$$ Inject and into , we obtain that $$\left| {\int_{0}^{2X_B} {g(x)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) - \int_{0}^{2X_B} {g(x)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \le \left( 2 + 4 X_B {\mathcal{L}_{\zeta}}\right) \sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| }. \label{eq:CSF_intergral_triangle3}$$ Apply the triangle inequality and combine , , , we have that: $$\left| {\int_{0}^{2X_B} {f(x)} {{\rm d}}{{ F_{A^n_i}}}\left( x \right) - \int_{0}^{2X_B} {f(x)} {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} \right| \le 2 {\tilde{\varepsilon}}_1 + (2+ 4 X_B {\mathcal{L}_{\zeta}})\sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right|}. \label{eq:CSF_intergral_the_end}$$ Recall the constant $C_1$ indicated in Lemma \[lem:convergence\], we define (note that $C_{\zeta}$ does not depend on $n$ nor $\varepsilon$) and deduce that $C_{\zeta} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}\ge C_1 {\tilde{\varepsilon}}_1^{-2} \ln\left(\frac{1}{\min\{{\tilde{\varepsilon}}_1, 1/{e}\}} \right)$.[^32] Take $\varepsilon_1:= {\tilde{\varepsilon}}_1$, for any $n \ge C_{\zeta} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$, we have $n \ge C_1 \varepsilon_1^{-2} {\ln\left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}}\right)}$ and by applying Lemma \[lem:convergence\], we obtain that $\sup \limits_{x \in [0, \infty)} {\left| { F_{A^n_i}}(x) - { F_{A^*_i}}(x) \right| } \le \varepsilon_1 = {\tilde{\varepsilon}}_1 $ and thus by and , we have: $$\left|{\int\limits_0^\infty {\zeta_A\left( {x ,y^*} \right) {{\rm d}}{{ F_{A^n_i}}}\left( x \right)} - \int\limits_0^\infty {\zeta_A\left( {x,y^*} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} } \right| \le 2 {\tilde{\varepsilon}}_1 + \left( 2 + 4 X_B {\mathcal{L}_{\zeta}}\right) {\tilde{\varepsilon}}_1 = (4 + 4 X_B {\mathcal{L}_{\zeta}}) {\tilde{\varepsilon}}_1 = \varepsilon.$$ This is exactly .
: Based on and , we can trivially deduce that the following inequalities hold for any $n \ge C_{\zeta} \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$ and $i \in [n]$: $$\begin{aligned}
& \left|{\int\limits_0^\infty {\zeta_A\left( {x ,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} - \int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right)} } \right| \le \varepsilon , \forall x \ge 0, \label{eq:1} \\
&\left|\int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^*_i}}}\left( y \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} } - \int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right)} } \right| \le \varepsilon, \label{eq:2} \\
& \left|\int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{A^*_i}}}\left( x \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} } - \int\limits_0^\infty {\int\limits_0^\infty {\zeta_A\left( {x,y} \right) {{\rm d}}{{ F_{A^n_i}}}\left( x \right) {{\rm d}}{{ F_{B^n_i}}}\left( y \right)} } \right| \le \varepsilon. \label{eq:3}\end{aligned}$$ We notice that the left-hand-sides of these inequalities are exactly the terms considered in Lemma \[lem:proof\_lottery\_LBn\_prepare2\]; moreover, the upper-bounds given in, and are smaller than that in , and of Lemma \[lem:proof\_lottery\_LBn\_prepare2\].
: To complete the proof of Remark \[remark\_conti\_CSF\], we follow the proof of Theorem \[theo:Lottery\_generic\_approx\] where we use , and instead of , and . By doing this, we obtain and .
PROOF OF LEMMA \[lem:delta\_mu\_nu\] AND THEOREM \[theoratioform\]
==================================================================
Proof of Lemma \[lem:delta\_mu\_nu\] {#sec:appen_proof_ratio-form}
------------------------------------
*$(i)$ We first consider the games ${\mathcal{LB}_n}(\mu^R)$.*
*We want to prove that there exists $\delta_0 = {\mathcal{O}}(\varepsilon^{-1/R} - 1)$ such that* for any $y^* \in [0,2X_B]$. Note that this is trivial if ${\mathcal{X}_{\mu^R}}(y^*, \varepsilon) = \emptyset$. In the following, we consider the case where ${\mathcal{X}_{\mu^R}}(y^*, \varepsilon) \neq \emptyset$. We denote by $f: [0, 2X_B] \times [0,2X_B] \rightarrow [0,1]$ the function: $$f(x, y^*) := | \mu^R_A(x,y^*) - \beta_A(x, y^*)| = \left\{ \begin{array}{l}
\frac{\alpha x^R}{\alpha x^R + (1-\alpha ) (y^*)^R }, \text{ if } x < y^* \\
0 ,\text{ if } x = y^* \\
1 - \frac{\alpha x^R}{\alpha x^R + (1-\alpha ) (y^*)^R }, \text{ if } x > y^*
\end{array} \right..$$ Trivially, $y^* \notin {\mathcal{X}_{\mu^R}}(y^*, \varepsilon)$. Take an arbitrary $x \in {\mathcal{X}_{\mu^R}}(y^*, \varepsilon)$. If $ x < y^*$, we have $$f(x,y^*) \ge \varepsilon \Rightarrow \frac{\alpha x^R}{\alpha x^R + (1-\alpha ) (y^*)^R } \ge \varepsilon \Rightarrow \frac{x}{y^*} \ge \left(\frac{\varepsilon}{1-\varepsilon} \frac{1- \alpha}{\alpha}\right)^{1/R}.$$ Therefore, $0< y^*- x \le y^* \left[ 1 -\left(\frac{\varepsilon}{1-\varepsilon} \frac{1- \alpha}{\alpha}\right)^{1/R} \right]$. Here, we note that the right-hand side is positive (due to the condition $\varepsilon < \alpha$); moreover, it is upper-bounded by ${\mathcal{O}}(1 - \varepsilon^{1/R} ) \le {\mathcal{O}}(\varepsilon^{-1/R} - 1) $.
On the other hand, if $x > y^*$, we have: $$f(x,y^*) \ge \varepsilon \Rightarrow 1 - \frac{\alpha x^R}{\alpha x^R + (1-\alpha ) (y^*)^R } \ge \varepsilon \Rightarrow \frac{x}{y^*} \le \left(\frac{1-\varepsilon}{\varepsilon} \frac{1- \alpha}{\alpha}\right)^{1/R}.$$ Therefore we have $0< x - y^* \le y^* \left[\left(\frac{1-\varepsilon}{\varepsilon} \frac{1- \alpha}{\alpha}\right)^{1/R} - 1 \right]$. Here the right-hand side is positive (due to the condition $\alpha+\varepsilon < 1$) and is upper-bounded by ${\mathcal{O}}(\varepsilon^{-1/R} - 1)$.
In conclusion, for any $\varepsilon < \min\{\alpha, 1\!- \!\alpha\}$, there exists $\delta_0 \!= \!{\mathcal{O}}(\varepsilon^{-1/R}\! -\! 1)$ such that . Note that a similar proof can be done to prove that there exists $\hat{\delta}_0 = {\mathcal{O}}(\varepsilon^{-1/R} - 1) $ such that for any $x^* \in [0, 2X_B]$, .
####
For any $y^* \in [0,2X_B]$ and $\delta_0 \ge 0$, let us define the set $I_0 (y^*) := [y^* - \delta_0, y^* + \delta_0] \bigcap [0, 2 X_B] $; we want to show that $\int_{x \in I_0 (y^*)} {{\rm d}}{ F_{A^*_i}}(x) \le \frac{2 n {\bar{\lambda}}\delta_0 {\bar{w}}}{{\underaccent{\bar}{w}}}, \forall i \in [n]$.
*[Case 1:]{}* For $i \in {\Omega_A({\gamma^*})}$, then $A^*_i { A^S_{{\gamma^*},i}}$, we have that $$\begin{aligned}
\int_{x \in I_0 (y^*)} {{\rm d}}{ F_{A^*_i}}(x) \le & {F_{ { A^S_{{\gamma^*},i}}}}\left( {y^* + \delta_0 } \right) - {F_{ { A^S_{{\gamma^*},i}}}}\left( {y^* - \delta_0 } \right) \\
= & \left\{ \begin{array}{l}
\frac{(y^* + \delta_0 ) {\lambda^*_B}}{v^B_i} \le \frac{2\delta_0 {\lambda^*_B}}{v^B_i} ,\text{~if~}0 \le y^* \le \delta_0 \\
\frac{(y^* + \delta_0 ) {\lambda^*_B}}{v^B_i} - \frac{(y^* - \delta_0 ) {\lambda^*_B}}{v^B_i} = \frac{ 2\delta_0 {\lambda^*_B}}{v^B_i} , \text{~if~} \delta_0 \le y^* < \frac{v^B_i}{{\lambda^*_B}} - \delta_0 \\
1 - \frac{(y^* - \delta_0 ) {\lambda^*_B}}{v^B_i} = \frac{v^B_i - y^* {\lambda^*_B}+ \delta_0 {\lambda^*_B}}{v^B_i}\le \frac{ 2\delta_0 {\lambda^*_B}}{v^B_i} ,\text{~if~}\frac{v^B_i}{{\lambda^*_B}} - \delta_0 \le y^* \le \frac{v^B_i}{{\lambda^*_B}} + \delta_0 \\
1 - 1 = 0 , \text{~otherwise}
\end{array} \right.\\
\le &\frac{2 n {\bar{\lambda}}\delta_0 {\bar{w}}}{{\underaccent{\bar}{w}}}.
\end{aligned}$$
*[Case 2:]{}* For $i \notin {\Omega_A({\gamma^*})}$, then $A^{*}_i= { A^W_{{\gamma^*},i}}$. We have $$\begin{aligned}
\int_{x \in I_0 (y^*)} {{\rm d}}{ F_{A^*_i}}(x) \le & {F_{ { A^W_{{\gamma^*},i}}}}\left( {y^* + \delta_0 } \right) - {F_{ { A^W_{{\gamma^*},i}}}}\left( {y^* - \delta_0 } \right) \\
= & \left\{ \begin{array}{l}
\frac{(y^* + \delta_0) {\lambda^*_B}}{v^B_i} \le \frac{2 n {\bar{\lambda}}\delta_0 {\bar{w}}}{{\underaccent{\bar}{w}}} ,\text{~if~}0 \le y^* \le \delta_0
\\
\frac{(y^* + \delta_0) {\lambda^*_B}}{v^B_i} - \frac{(y^* - \delta_0) {\lambda^*_B}}{v^B_i} = \frac{ 2\delta_0{\lambda^*_B}}{v^B_i} , \text{~if~} \delta_0 < y^* < \frac{v^A_i}{{\lambda^*_A}} -\delta_0\\
1 - \frac{\frac{v^B_i}{{\lambda^*_B}} - \frac{v^A_i}{{\lambda^*_A}}}{\frac{v^B_i}{{\lambda^*_B}}} -\frac{(y^* - \delta_0) {\lambda^*_B}}{v^B_i} = \frac{v^A_i \frac{{\lambda^*_B}}{{\lambda^*_A}} - y^* {\lambda^*_B}+ \delta_0 {\lambda^*_B}}{v^B_i}\le \frac{ 2\delta_0 {\lambda^*_B}}{v^B_i} ,\text{~if~}\frac{v^A_i}{{\lambda^*_A}} - \delta_0 \le y^* \le \frac{v^A_i}{{\lambda^*_A}} + \delta_0\\
1 - 1 = 0 , \text{~otherwise}
\end{array} \right.\\
\le &\frac{2 n {\bar{\lambda}}\delta_0 {\bar{w}}}{{\underaccent{\bar}{w}}}.
\end{aligned}$$ Note that we also can similarly prove that for any $ x^* \in [0, 2X_B] $ and $\delta_0 \ge 0$, for any $i \in [n]$, we also have $\int_{y \in I_0 (x^*)} {{\rm d}}{ F_{B^*_i}}(y) \le \frac{2 n {\bar{\lambda}}\delta_0 {\bar{w}}}{{\underaccent{\bar}{w}}}$.
#### Conclusion.
We note that all random variable $ A^*_i, B^*_i, i \in [n]$ are bounded in $[0, 2X_B]$; therefore, for any $x^*, y^* \in [0, 2X_B]$ and $\delta_0 \ge 0$, we have: $$\begin{aligned}
\int_{x \in [y^* - \delta, y^* + \delta_0]} {{\rm d}}{ F_{A^*_i}}(x) = \int_{x \in I_0 (y^*)} {{\rm d}}{ F_{A^*_i}}(x) & \textrm{ and } & \int_{y \in [x^* - \delta, x^* + \delta_0]} {{\rm d}}{ F_{B^*_i}}(y) = \int_{y \in I_0 (x^*)} {{\rm d}}{ F_{B^*_i}}(x).\end{aligned}$$
Let us define $\delta_\mu: = min\{ 1, \frac{2 n {\bar{\lambda}}\delta_0 {\bar{w}}}{{\underaccent{\bar}{w}}} \}= {\mathcal{O}}\left(n (\varepsilon^{-1/R}-1) \right)$ and we conclude that: $$\max \left\{ \max_{y^* \in [0,2X_B]}{\int \nolimits_{{\mathcal{X}_{\mu^R}}(y^*,\varepsilon)} { {{\rm d}}{ F_{A^*_i}}(x)}}, \max_{x^* \in [0,2X_B]}{\int \nolimits_{{\mathcal{Y}_{\mu^R}}(x^*,\varepsilon)} { {{\rm d}}{ F_{B^*_i}}(y)}} \right\} \le \delta_\mu.$$ This implies that $\delta_\mu \in {\Delta_{{\gamma^*}}(\mu^R, \varepsilon)}$.
#### $(ii)$ We now turn our focus on the games ${\mathcal{LB}_n}(\nu^R)$.
We first prove the existence of $\delta_1>0$ such that ${\mathcal{X}_{\nu^R}}(y^*, \varepsilon) \subset [y^* - \delta_1, y^* + \delta_1]$ for any $y^* \in [0,2X_B]$. Similar to step 1 in the above analysis for the game ${\mathcal{LB}_n}(\mu^R)$, we denote by $g: [0, 2X_B] \times [0,2X_B] \rightarrow [0,1]$ the function: $$g(x, y^*) := | \nu^R_A(x,y^*) - \beta_A(x, y^*)| = \left\{ \begin{array}{l}
\frac{\alpha e^{xR}}{\alpha e^{xR} + (1-\alpha ) e^{y^* R}} \text{ , if } x < y^*, \\
0 \text{ , if } x = y^*, \\
1 - \frac{\alpha e^{xR}}{\alpha e^{xR} + (1-\alpha ) e^{y^* R}} \quad \text{ , if } x > y^*.
\end{array} \right.$$ Trivially, $y^* \notin {\mathcal{X}_{\nu^R}}(y^*, \varepsilon)$. Take an arbitrary $x \in {\mathcal{X}_{\mu^R}}(y^*, \varepsilon)$. If $ x < y^*$, we have $$g(x,y^*) \ge \varepsilon \Rightarrow \frac{\alpha e^{xR}}{\alpha e^{xR} + (1-\alpha ) e^{y^* R} } \ge \varepsilon.$$ Therefore, $0< y^*- x \le \frac{1}{R} \ln \left(\frac{1-\varepsilon}{\varepsilon} \frac{\alpha}{1-\alpha}\right)$. Here, we note that the right-hand side is positive (due to the condition $\varepsilon < \alpha$).
On the other hand, if $x > y^*$, we have: $$g(x,y^*) \ge \varepsilon \Rightarrow 1 - \frac{\alpha e^{xR}}{\alpha e^{xR} + (1-\alpha ) e^{y^* R} } \ge \varepsilon.$$ Therefore, $0< x - y^* \le \frac{1}{R} \ln \left( \frac{1-\varepsilon}{\varepsilon} \frac{1-\alpha}{ \alpha}\right)$. Here, the right-hand side is positive (due to the condition $\alpha+\varepsilon < 1$). In conclusion, let us denote $\delta_1 = \mathcal{O} (R^{-1} \ln(\varepsilon^{-1}))$, we have proved that ${\mathcal{X}_{\nu^R}}(y^*,\varepsilon) \subset [y^*- \delta_1, y^* + \delta_1]$ for any $ y^* \in [0,2X_B]$. Now, we define $I_1 (y^*) := [y^* - \delta_1, y^* + \delta_1] \bigcap [0, 2 X_B]$. Similar to step 2 of the above analysis regarding the game ${\mathcal{LB}_n}(\mu^R)$, we can prove that $\int \nolimits_{I_1(y^*)} {{\rm d}}{ F_{A^*_i}}(x) \le 2 n {\bar{\lambda}}\delta_1 {\bar{w}}/{\underaccent{\bar}{w}}$ for any $y^* \in [0,2X_B]$. Therefore, $$\max \left\{ \max_{y^* \in [0,2X_B]}{\int \nolimits_{{\mathcal{X}_{\nu^R}}(y^*,\varepsilon)} { {{\rm d}}{ F_{A^*_i}}(x)}}, \max_{x^* \in [0,2X_B]}{\int \nolimits_{{\mathcal{Y}_{\nu^R}}(x^*,\varepsilon)} { {{\rm d}}{ F_{B^*_i}}(y)}} \right\} \le \delta_\nu,$$ where $\delta_\nu := \min\{1,\frac{2 n {\bar{\lambda}}\delta_1 {\bar{w}}}{{\underaccent{\bar}{w}}}\} = {\mathcal{O}}\left(n R^{-1} \ln(\varepsilon^{-1}) \right)$ and $\delta_{\nu} \in {\Delta_{{\gamma^*}}(\nu^R, \varepsilon)}$.
Proof of Theorem \[theoratioform\] {#sec:appen_proof_theoratioLB}
----------------------------------
Take $\varepsilon = {\bar{\varepsilon}}/ 21$ and $\tilde{L} = L^* 21^2 (\ln(21)+1)$ (where $L^*$ is indicated in Theorem \[theo:Lottery\_generic\_approx\]). We note that ;[^33] therefore, for any $n \ge \tilde{L} {\bar{\varepsilon}}^{-2} {\ln\left(\frac{1}{\min\{{\bar{\varepsilon}},1/{e}\}}\right)}$, we have and thus, apply Theorem \[theo:Lottery\_generic\_approx\]-$(ii)$, for any $R>0$, the ${{\rm IU}^{{\gamma^*}}}$ strategy is an -equilibrium of the game ${\mathcal{LB}_n}(\mu^R)$ (recall that $W:= \max \{W_A, W_B \}$). Similarly, the ${{\rm IU}^{{\gamma^*}}}$ strategy is an $(8\delta_\nu + 13\varepsilon) W$-equilibrium of the game ${\mathcal{LB}_n}(\nu^R)$).
We first consider the game ${\mathcal{LB}_n}(\mu^R)$. Apply Lemma \[lem:delta\_mu\_nu\], for any and ${\gamma^*}\in {\mathcal{S}_n}$, we have $\delta_\mu \le \varepsilon$. Therefore, for any $n \ge \tilde{L} {\bar{\varepsilon}}^{-2} {\ln\left(\frac{1}{\min\{{\bar{\varepsilon}},1/{e}\}}\right)}$, , the ${{\rm IU}^{{\gamma^*}}}$ strategy is an -equilibrium (i.e., -equilibrium) of the game $LB(\mu^R)$.
Similarly, apply Lemma \[lem:delta\_mu\_nu\], for any ${\gamma^*}\in {\mathcal{S}_n}$ and $R\ge {\mathcal{O}}\left( \frac{n}{{\bar{\varepsilon}}}\ln\left( \frac{1}{{\bar{\varepsilon}}} \right) \right)$, we have $\delta_\nu \le \varepsilon$. Therefore, for any $n \ge \tilde{L} {\bar{\varepsilon}}^{-2} {\ln\left(\frac{1}{\min\{{\bar{\varepsilon}},1/{e}\}}\right)}$, $R \ge {\mathcal{O}}\left(\frac{n}{{\bar{\varepsilon}}} \ln\left( \frac{1}{{\bar{\varepsilon}}} \right) \right) $, the ${{\rm IU}^{{\gamma^*}}}$ strategy is an -equilibrium (i.e., -equilibrium) of the game $LB(\nu^R)$.
[^1]: We explain the name in Section \[sec:IU\_Strategy\].
[^2]: It is called the generalized Colonel Blotto game by [@kovenock2015].
[^3]: The set of battlefields are partitioned such that two battlefields are in the same partition if they have the same (normalized) values; the sufficient condition on the attainability of equilibria requires a sufficient number of battlefields in each partition.
[^4]: Note that the LB game is also a non-constant-sum game; however, to lighten the notation, hereinafter, we do not highlight this and only call it the LB game in places with no ambiguity.
[^5]: That is the LB game where two players, called A and B, commonly evaluate each battlefield $i$ with a value $w_i$; if players allocates $x^A_i, x^B_i$ to battlefield $i$ then player A gains $x^A_i w_i/ (x^A_i + x^B_i)$ and player B gains $x^B_i w_i/ (x^A_i + x^B_i)$ from this battlefield.
[^6]: A similar function to define the winning probability is also used by [@rinott2012] to study a variant of the CB game involving sequential tournaments.
[^7]: For example, [@skaperdas1996] defines $\zeta_A$, $\zeta_B$ with an axiom of anonymity; they also require that any player who puts a strictly positive amount of resources has a strictly positive probability of winning the prize; [@clark1998contest] considers the CSFs additionally satisfying the Choice Axiom. These are technical conditions needed for proving their results and we omit them here lest they unnecessarily limit our scope of study.
[^8]: When $\alpha=1/2$, the CSFs $\mu^R$ and $\nu^R$ match the classical power form and logit form CSFs. Note that we exclude the cases where $\alpha =0$ or $\alpha =1$ since these are the trivial cases: in the corresponding Lottery Blotto game, a player, say $p \in \{A,B\}$, always has the payoff $W_p$ while player $-p$’s payoff is always zero regardless how they allocate their resources.
[^9]: Note that and ${\mathcal{S}_n}$ also depend on other parameters of the game ${\mathcal{CB}_n}$ but we use the notation with only the subscript $n$ and omit other parameters to lighten the notation.
[^10]: To solve this equation, we first sort out all ratios ${v^A_i}/{v^B_i}$ in a non-decreasing order (which can be done in ${\mathcal{O}}(n \ln(n))$), then there are three possible cases: ${\gamma^*}< \min\{v^A_i/ v^B_i, i \in [n]\}$ or ${\gamma^*}\ge \max\{v^A_i/ v^B_i, i \in [n]\}$ or $\exists j: {\gamma^*}\in \left[ {v^A_{j}}/{v^B_j},{v^A_{j+1}}/{v^B_{j+1}} \right)$. In all of these cases, Equation becomes a cubic equation; therefore, it can be solved algebraically.
[^11]: These constants are the Lagrange multipliers corresponding to the budget constraints in finding players’ best-response; see [@kovenock2015] for more details.
[^12]: Here, the superscripts $S$ and $W$, standing for strong and weak, are used to emphasize the intuition on players’ incentive to play according to these distributions in the CB games: if , player A has a “stronger" incentive to win battlefield $i$ and player B has a “weaker" incentive; if , the roles of players are exchanged.
[^13]: Trivially from Proposition \[Prop:BoundLambda\], the random variables $A^*_i, B^*_i, \forall n , \forall i \in [n]$ are upper-bounded by ${\bar{w}}/ ({\underaccent{\bar}{w}}n {\underaccent{\bar}{\lambda}})$. In the remainders of the paper, we sometimes need an upper-bound of these random variables that does not depend on $n$: we can prove that they are bounded by $2 X_B$ (see Lemma \[lem:Preliminary\] in \[sec:appen\_preliminary\]).
[^14]: This dependency is implicitly presented in the asymptotic notation $\tilde{{\mathcal{O}}}$ in Result $(i)$ and the constant $C^*$ in Result $(ii)$.
[^15]: Note that ${ F_{A^*_i}}, { F_{B^*_i}}$ are continuous, bounded functions on $[0,2X_B]$; therefore, they attain a maximum on this interval.
[^16]: Note also that for the case of the game ${\mathcal{CB}_n}$, the left-hand side in equals zero for any $n$ and $i \in [n]$.
[^17]: Recall that in the constant-sum variant, $W:= \max\{W_A, W_B\} = W_A =W_B$.
[^18]: The asymptotic notations are taken w.r.t. when $\varepsilon \rightarrow 0$.
[^19]: This is due to $C^*_1 \cdot \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}= C_1 \left(\frac{4}{\varepsilon} \right)^2 \cdot (\ln(4)+1){\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}\ge C_1 \cdot \left(\frac{4}{\varepsilon} \right)^2 \ln \left(\frac{4}{\varepsilon} \right)$; here, we have applied Lemma \[lem:log\_pre\] with ${\hat{\varepsilon}}:= \varepsilon$ and ${\hat{C}}:= 4$; moreover, $\frac{\varepsilon}{4} = \min\{\frac{\varepsilon}{4}, \frac{1}{e}\}$ since $\varepsilon \le 1$; thus, we can rewrite $\ln\left(\frac{4}{\varepsilon} \right) = \ln \left(\frac{1}{\min\{\varepsilon/4, 1/{e}\}} \right)$.
[^20]: This is due to $C^*_2 \cdot \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}= C_1 \left(\frac{8}{\varepsilon} \right)^2 \cdot (\ln(8)+1){\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}\ge C_1 \cdot \left(\frac{8}{\varepsilon} \right)^2 \ln \left(\frac{8}{\varepsilon} \right)$; here, we have applied Lemma \[lem:log\_pre\] with ${\hat{\varepsilon}}:= \varepsilon$ and ${\hat{C}}:= 8$; moreover, $\frac{\varepsilon}{8} = \min\{\frac{\varepsilon}{8}, \frac{1}{e}\}$ since $\varepsilon \le 1$; thus, we can rewrite $\ln\left(\frac{8}{\varepsilon} \right) = \ln \left(\frac{1}{\min\{\varepsilon/8, 1/{e}\}} \right)$.
[^21]: Once again, apply Lemma \[lem:log\_pre\], $C^*_3 \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}\! = \!C_2 \left( \frac{8}{\varepsilon}\right)^2 (\ln(8)\! +\! 1) {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}\ge C_2 \left( \frac{8}{\varepsilon}\right)^2 \ln\left(\frac{8}{\varepsilon} \right)$; moreover, we have .
[^22]: If $\varepsilon < 1/ {e}$, then $\ln(1/\varepsilon) > 1$ and $\hat{C}_1 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}= \frac{\ln(1/\varepsilon)}{\ln(1/D)} + \ln(1/\varepsilon) > \log_{D}{\varepsilon} +1 $; otherwise, if $\varepsilon \ge 1/e$, we have $\ln(1/\varepsilon) \le 1 $ and $\hat{C}_1 {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}= \frac{1}{\ln(1/D)} + 1 \ge \frac{\ln(1/\varepsilon)}{\ln(1/D)} + 1 = \log_{D}{\varepsilon} +1$ (note that $\ln(1/D)>0$ since $D<1$).
[^23]: If $\varepsilon< 1/e$, we have ; otherwise, if $\varepsilon \!\ge\! 1/{e}$, we have (since ).
[^24]: Note that if then , if $i \notin {\Omega_A({\gamma^*})}$ then $ {\mathbb{P}}(B^*_i = 0 ) = 0$ (see -); therefore, .
[^25]: This is due to the fact that ; here, we have applied Lemma \[lem:log\_pre\] (see \[sec:appen\_preliminary\]) for ${\hat{\varepsilon}}:= \varepsilon_1$ and ${\hat{C}}:= 4$.
[^26]: This is due to ; here, we have used Lemma \[lem:log\_pre\] with ${\hat{\varepsilon}}:= \varepsilon_1$ and ${\hat{C}}:=8$ and the fact that $1/\varepsilon \ge 1$.
[^27]: For any $j \! \ge \! 2$, this is trivially since $P_j \!:= \! (x_{j-1}, x_j]$. For $P_1 = [0,x_1]$, we have that ; moreover, due to Lemma \[lem:continuity\_Ani\_and\_Bni\]-$(i)$, we also note that .
[^28]: Apply Lemma \[lem:log\_pre\], we have . Moreover, since $\frac{\varepsilon_2}{6+2M} = \min\left\{\frac{\varepsilon_2}{6+2M}, \frac{1}{{e}} \right\} = \min\left\{\delta_2, \frac{1}{{e}} \right\} $ (due to the fact that $\delta_2 = \varepsilon_2 /(6+2M) < 1/{e}$). Therefore, we have .
[^29]: Recall that by definition, $\beta_A(x,y^*) = \alpha$ if $x=y^*$, $\beta_A(x,y^*) = 0$ if $x<y^*$ and $\beta_A(x,y^*) = 1$ if $x>y^*$
[^30]: Note that $\varepsilon_1 = \frac{\varepsilon}{5}$ and apply Lemma \[lem:log\_pre\], $L_0 \cdot \varepsilon^{-2} {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}= C_1 \left(\frac{5}{\varepsilon} \right)^2 \cdot (\ln(5)+1){\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}\ge C_1 \cdot \left(\frac{5}{\varepsilon} \right)^2 \ln \left(\frac{5}{\varepsilon} \right)$; moreover, $\frac{\varepsilon}{5} = \min\{\frac{\varepsilon}{5}, \frac{1}{e}\}$ since $\varepsilon \le 1$; thus, we can rewrite $\ln\left(\frac{5}{\varepsilon} \right) = \ln \left(\frac{1}{\min\{\varepsilon/5, 1/{e}\}} \right) = \ln \left(\frac{1}{\min\{\varepsilon_1, 1/{e}\}} \right)$.
[^31]: For any $j \! \ge \! 2$, this is trivially since $P_j \!:= \! (x_{j-1}, x_j]$. For $P_1 = [0,x_1]$, we have that ; moreover, due to Lemma \[lem:continuity\_Ani\_and\_Bni\]-$(i)$, we also note that .
[^32]: Apply Lemma \[lem:log\_pre\], . Moreover, since $\frac{\varepsilon}{4+4X_B {\mathcal{L}_{\zeta}}} = \min\left\{\frac{\varepsilon}{4+4X_B {\mathcal{L}_{\zeta}}}, \frac{1}{{e}} \right\} = \min\left\{{\tilde{\varepsilon}}_1, \frac{1}{{e}} \right\} $ (due to the fact that ${\tilde{\varepsilon}}_1 = \frac{\varepsilon}{4+4X_B {\mathcal{L}_{\zeta}}} < \frac{1}{{e}}$).
[^33]: Note that $\varepsilon = {\bar{\varepsilon}}/ 21$ and apply Lemma \[lem:log\_pre\] to have that $\tilde{L} {\bar{\varepsilon}}^{-2} {\ln\left(\frac{1}{\min\{{\bar{\varepsilon}},1/{e}\}}\right)}\ge L^* \left( \frac{21}{{\bar{\varepsilon}}} \right)^2 \ln \left( \frac{21}{{\bar{\varepsilon}}}\right)$; moreover, we recall that $\frac{{\bar{\varepsilon}}}{21} < \frac{1}{{e}}$; therefore, $\ln\left(\frac{21}{{\bar{\varepsilon}}} \right) = \ln \left( \frac{1}{\min\{{\bar{\varepsilon}}/21, 1/{e}\}} \right) = {\ln\left(\frac{1}{\min\{\varepsilon, 1/{e}\}}\right)}$.
|
---
abstract: 'The emergence of online open source repositories in the recent years has led to an explosion in the volume of openly available source code, coupled with metadata that relate to a variety of software development activities. As an effect, in line with recent advances in machine learning research, software maintenance activities are switching from symbolic formal methods to data–driven methods. In this context, the rich semantics hidden in source code identifiers provide opportunities for building semantic representations of code which can assist tasks of code search and reuse. To this end, we deliver in the form of pretrained vector space models, distributed code representations for six popular programming languages, namely, Java, Python, PHP, C, C++, and C\#. The models are produced using *fastText*, a state–of–the–art library for learning word representations. Each model is trained on data from a single programming language; the code mined for producing all models amounts to over 13.000 repositories. We indicate dissimilarities between natural language and source code, as well as variations in coding conventions in between the different programming languages we processed. We describe how these heterogeneities guided the data preprocessing decisions we took and the selection of the training parameters in the released models. Finally, we propose potential applications of the models and discuss limitations of the models.'
author:
-
title: |
Semantic Source Code Models\
Using Identifier Embeddings
---
fastText, Code Semantics, Vector Space Models, Semantic Similarity
Introduction {#sec:intro}
============
The emergence of online open source repositories, such as GitHub, in the recent years has drastically increased the volume of archived software artifacts that are openly available to the community. Such artifacts include source code, combined with an assortment of meta data related to various stages of the development lifecycle. This large–scale mass of data, often referred to as “Big Code” [@Allamanis2018] encompasses rich information related to documentation, maintenance events, and authorship of software. An increasing research interest focuses on leveraging the wealth of this data and extracting actionable results for automating related activities.
Data–driven methods have attracted substantial attention, following recent advances in machine learning research and foreseeing the practical potential with the availability of computational resources that can nowadays afford data–intensive tasks. In this context, statistical regularities observed in source code have revealed the repetitive and predictable nature of programming languages, which has been compared to that of natural languages [@Hindle2012; @Ernst2017]. Consequently, research on problems of automation in natural language processing, such as identification of semantic similarity between texts, translation, text summarisation, word prediction and language generation has inspired parallel lines of research regarding the automation of software development tasks. Relevant problems in software development include clone detection [@White2016; @Wei2017], deobfuscation [@Vasilescu2017], language migration [@Nguyen2013], source code summarisation [@Iyer2016; @Allamanis2016], auto–correction [@Pu2016; @Gupta2017], auto–completion [@Foster2012], code generation [@Oda2015; @Ling2016; @Yin2017], and comprehension [@alexandru2017]. The perceived similarity between natural language and source code has largely driven the practice of mining source code, with relevant problems being addressed through the latest state–of–the–art natural language processing methods [@Allamanis2013; @Palomba2016; @Iyer2016; @Vasilescu2017; @Yin2017].
Besides similarities, there also exist major differences that need to be taken into consideration when designing such studies. State–of–the art text mining techniques produce impressive results when given sufficient amounts of data expressed in natural language [@halevy2009unreasonable]. Substantial volumes of data expressed in a programming language however do not necessarily yield comparable results in equivalent tasks. Especially, when it comes to extracting semantic topics from source code, results are poor. This has been attributed to data sparsity issues [@Mahmoud2017] as semantically rich elements in source code tend to amount only to a small fragment of the overall data. As a solution to addressing these gaps, we propose the use of pretrained source code embeddings.
The emergence of word embeddings [@mikolov2013efficient] — *i. e.*, representations of words in the continuous vector space — has revolutionized information retrieval in natural language processing. The method relies on the idea that shared textual context implies semantic relatedness, which is in turn reflected as topological proximity in the vector space. At a practical level, this information is delivered through portable models that have been pretrained over large–scale textual data. We claim that on a par with natural language, source code demonstrates similar qualities through the information encoded in source code identifiers. Following the natural language paradigm, we deliver a set of general–purpose models, pretrained over large amounts of code, which can be used to assist a number of information retrieval tasks.
Continuous Vector Space Models {#sec:embeddings_general}
==============================
Word embeddings are based on the distributional hypothesis proposed by Harris [@harris1954distributional], which states that words that occur in the same contexts tend to have similar meanings. Traditional approaches of distributional similarity have treated words as atomic units, represented in a discrete manner as indices in a vocabulary. Sparse, high dimensional vectors for encoding this information, however, suffer from scalability issues. Continuous, low dimensional dense vectors provide an alternative representation that overcomes these issues. Continuous representations of words, capture distributional similarity by encoding words into dense vectors where each word is associated with a point in continuous vector space, and semantically related words tend to share context in the vector space. The seminal work by Mikolov et al. [@mikolov2013efficient] with the Word2Vec model brought continuous vector space models into play, with an efficient implementation of an unsupervised algorithm for learning word representations. Follow–up work resulted to the implementation of the *fastText* library [@Fasttext] by Facebook research which outperforms Word2Vec and, most importantly, builds representations at character-level granularity. This key feature allows the representations of synthetic words that do not appear in the training corpus, and builds models for highly diverse languages that contain many rare words [@bojanowski2017enriching].
The success of word embeddings, relies to a certain extent on the fact that pretrained readily available models are easy to access and further exploit by communities with no particular machine learning expertise. Mikolov et al. demonstrated the potential of the method through a Word2Vec model pretrained over $100$ billion words of Google news data. The model was released in a portable binary format along with its implementation [@Word2Vec], bringing the method into the mainstream. Similar approaches followed this paradigm [@pennington2014glove], releasing toolkits and readily available pretrained models. Ever since, substantial work is constantly under development, oriented towards releasing pretrained general–purpose models [@mikolov2018advances] in a variety of languages [@Grave2018], as well as domain–specific embeddings for disambiguating words to their specialized context [@taghipour2015semi].
In the software engineering community, embeddings have been trained over small datasets pertinent to ad–hoc tasks [@xu2016predicting; @fu2017easy], but also released as general–purpose domain specific–knowledge [@Efstathiou2018]. In both cases, these models are trained over natural language artifacts related to software development. In this work we provide a collection of models trained on source code in six different programming languages. To the best of our knowledge this is the first set of pretrained source code models to be released for general–purpose use.
Source Code Embeddings
======================
Following current trends in natural language processing, with readily available pretrained models being released as exploitable resources of general, common sense knowledge, we propose the release of general–purpose, pre–trained source code models. We motivate towards this idea and describe the implementation steps, from data selection criteria to training the models. We discuss results and demonstrate the potential of the models through a simple code similarity example.
Motivation
----------
Good coding practices dictate that source code identifiers be given meaningful, descriptive names. As a result, source code identifiers tend to encompass distinctive semantics that render them useful for communicating information across developers [@Allamanis2013]. Furthermore, by considering the fact that code comes in self–contained units of relevant functionality, we postulate that, to a certain extent, contextual distributional semantics in code are captured in ways comparable to those in natural language. The availability of high quality repositories provides the grounds for investigations towards this direction.
Data Selection
--------------
We selected GitHub public repositories where the primary programming language was one of Java, Python, PHP, C, C++, C\#. We chose these languages due to their popularity and diversity in application domains, spanning from web programming to systems programming, and general application programming. In addition, we were interested in training models for languages of varying verbosity, with Python at one end being concise and Java and C\# at the other end being more verbose. All six languages are listed within the top 10 most popular programming languages according to Tiobe’s index as of January 2019 [@Tiobe] and are supported by the framework proposed by Munaiah et al. [@Munaiah2017] for quality assessment.
We consulted the list of repositories already analyzed by the RepoReapers tool [@Reapers], and for each language separately we sorted the related repositories in decreasing order of GitHub stars. We chose repositories with over 100 stars and filtered out of these, few cases of repositories that have not been classified as engineered projects by any of the implemented classifiers [@Munaiah2017]. The resulting lists of repositories of a total of $13,144$ repositories that match our selection criteria can be found in our repository [@ScodeEmb].
Data Collection and Preprocessing
---------------------------------
We used a number of shell scripts for compiling and transforming the data in the appropriate format. The complete toolkit is available on our GitHub repository [@ScodeEmb]. We followed the preprocessing steps described below.
### Tokenization
After fetching the selected repositories, we selected source code files with extensions that matched each of the six programming languages of choice, i.e., { .java, .py, .cpp, .php, .c, .cpp, .cs}. We used Tokenizer [@Tokenizer], an open-source tool that provides, among others, functionality for tokenizing source code elements into string tokens. For each programming language we tokenized the content of source code files and stored them in a single file by maintaining their original order. We further preprocessed the tokenized files by filtering out some elements as described in the next section.
### Data Cleansing {#sec:cleaning}
It is a common practice for studies that employ text mining techniques with software artifacts to religiously follow the guidelines akin to natural language processing tasks. Efforts towards adapting to the needs of the task in hand mainly focus on fine–tuning training parameters [@Agrawal2018; @Panichella2013; @Mahmoud2017]. The importance of the decisions taken at data preprocessing level is a parameter rarely stressed, despite the fact that the quality of the produced models depends heavily on the features expressed through the representational strength of the data provided. In this study, we performed trials with variations of preprocessed data and decided to follow some of the standard text mining preprocessing steps and to omit others, as described in our rationale below.\
[*Text Normalization:*]{} Lemmatization and stemming are standard normalization techniques employed in order to mitigate the noise produced by grammatical inflections in a variety of natural language processing tasks. However, in continuous vector space models, this type of normalization could lead to information loss as inflections may capture relational analogies, e.g., nominal plural analogies, such as “dog is to dogs what horse is to horses" [@Finley2017]. Inflection phenomena are not equally pervasive in programming scripts; still source code identifiers do incorporate aspects of inflection, e.g., a class named “Node” versus a collection which is named “nodes” and holds instances of “Node” objects. We maintained the inflected forms of source code identifiers as these originally appear in the scripts. Furthermore, we did not split composite name signatures into their counterparts as we observed that the dictionary of the individual words that compose the highly synthetic vocabulary of a source code document such a class is limited and repetitive. Hence, flattening compositionality of identifiers led to repetition of identical terms limiting the representational strength of the data. Similarly, we maintained typesetting aspects that naming conventions of the different programming languages dictate.\
*Conversion to Lowercase:* Capital case in English language is sparse. Typically words that start with a capital letter are found in the beginning of a sentence, in which case capital case does not assign special semantics to words. The only exception is proper nouns *i.e.*, instances of entities (e.g., “Bob” is an instance of a person). Due to the relative sparsity of named entities, in order to avoid the noise occurring from typesetting diversities it is common practice in text mining to uniformly convert text to lowercase. On the contrary, capital letters used thoroughly in source code text as naming conventions dictate. Naming conventions imply underlying functional semantics of code identifiers. Particularly in object–oriented languages conventions function conversely, with identifiers starting with a capital letter denoting higher–level entities such as classes and interfaces. In order to maintain such features in the data, we maintained the source code text in the original form found in scripts without converting case.\
[*Stop Word Removal:*]{} Stop words are short function words that commonly occur in language, and carry limited semantic content (e.g., “the”, “and”, “this”). Because their presence in a text does not contribute in distinguishing concrete semantics, such words are considered as noise and are often removed at preprocessing in text mining. In analogy to natural language stop words, for each of the programming languages that we mined, we compiled a corresponding list of reserved keywords. The lists of keywords that we filtered out of the data are available in our repository [@ScodeEmb].\
[*Punctuation Removal:*]{} Heavy use of punctuation is ubiquitous throughout the source code in all six programming languages that we processed, inducing considerable noise. Thus, we decided to remove punctuation symbols in all six languages, except for “\_” which is regularly used for compound identifier labels. In addition, we maintained “\$” in the PHP dataset due to its use for denoting variables.\
[*Other Noise Removal:*]{} We found substantial noise in the data in the form of single characters that occur from a variety of statements (e.g., “e” from “catch Exception e”), numeric values and hexadecimal numbers. We cleared the data from these types of tokens.
The final data set, after cleansing, totals up to nearly one billion tokens. Even though the number is significant, it seems disproportionately low with respect to the overall 2.4 billion lines of code these tokens were obtained from. This implies that a substantial content of the initial data amounts to noise, corroborating the evidence of sparsity of useful information within source code. Table \[tab:data\] summarizes key metrics of the data used for training the models.
**Pr. Language** **\# Repos** **\# Files** **\# LOC** **\# Clean Tokens**
------------------ -------------- -------------- --------------- ---------------------
Java 2,963 2,456,267 589,043,498 258,011,215
Python 3,862 374,225 76,756,824 106,245,311
PHP 2,394 563,258 96,287,040 82,082,221
C 1,826 2,093,090 749,520,681 238,358,382
C++ 1,335 2,691,489 822,175,363 167,149,674
C\# 764 390,919 69,006,942 92,620,757
**Total** 13,144 8,569,248 2,402,790,348 944,467,560
: A Summary of Analyzed Repositories
\[tab:data\]
Training
--------
We used the fastText library [@Fasttext] for training the models. With fastText word vectors are built from vectors of character substrings contained in a word [@bojanowski2017enriching]. This feature allows for the representation of made–up words, hence we found it to be the most appropriate for dealing with artificial languages such as the programming languages under consideration. We used each of the six language–specific consolidated preprocessed files for training the models. We chose the skip-gram model over the cbow-model for training as the former has been observed to be more efficient with subword information [@bojanowski2017enriching]. Skip-gram predicts the target word by using a random close-by word within a context window of determined width. We set this to be equal to 5 for all languages besides Python, where we set the context window to be equal to 4 due to the the concise style of the language. For subwords, we set the minimum length of character n-gram to range between 3 and 6. We set the dimensionality of vectors to be equal to 100 and trained the models in 20 epochs. The .bin files of the resulting models are archived on Zenodo. [^1] Table \[tab:models\] summarizes key metrics of the trained models.
**Pr. Language** **Vocabulary Size** **.bin File Size (GBs)**
------------------ --------------------- --------------------------
Java 2,480,481 $2.8$
Python 1,005,902 $1.6$
PHP 715,760 $1.4$
C 2,734,020 $3.1$
C++ 2,223,393 $2.6$
C\# 990,330 $1.6$
: A Summary of the Trained Models
\[tab:models\]
Results {#sec:results}
-------
In natural language processing, pretrained models are typically evaluated against established benchmarks. Evaluating source code embeddings is not as straightforward. We empirically assessed the models by using the nearest neighbor functionality of *fastText* which, given a query, returns its closest words in a trained model. We produced several versions of models by changing i) formats of the data, and ii) training parameters. Interestingly, variations in data formatting produced substantial differences with extensively preprocessed data (composite identifier labels split and all tokens lowercased) resulting to poor representations. As discussed in section \[sec:cleaning\], we decided to keep preprocessing minimal for the delivered models. In terms of variations of the training parameters, we experimented with models that ignored subword information and found the produced representations inadequate. Variations in training windows (5–10) and reduction on dimensionality (80–100) of the models did not change the results to our queries dramatically. It is worth noting that examples from the software engineering literature [@Mahmoud2017] were on agreement with queries on the Java model where, for instance, the top $10$ nearest words to the word “FullScreen” included, among others, terms such as “toggleFullScreen” in accordance with the ad-hoc topic models presented by the authors.
In order to obtain a more clear perspective on the value offered by the models and at the same time demonstrate a potential application, we performed a small case study for repository similarity assessment. We used the Word Movers Distance (WMD) [@Kusner2015], a metric proposed for assessing document similarity by considering their embedded word representations in a trained model. We used as documents the tokenized versions of the Java logging libraries SLF4j and Log4j and a similarly–sized general purpose spatial Java library, Spatial4j. By applying pairwise WMD in between the three libraries we found SLF4j and Log4j located closely together with a distance of $0.59$ whereas their distances with Spatial4j were equal to $2.39$ and $1.99$ respectively. Thus, even though the model is trained on a wide range of Java repositories, it incorporates condensed knowledge that renders it capable of drawing out similarity details.
Discussion
==========
The breakout of word embeddings is recent, hence their potential for empowering other processes such as recommendation and classification is currently under development [@barkan2016item2vec], [@Kusner2015], [@fu2014learning], [@rekabsaz2017toward]. We propose potential applications of source code embeddings and discuss challenges and limitations that we observed in training and using the models.
Opportunities
-------------
Combining semantic models of source code together with semantic models of software documentation provides grounds for addressing a variety of problems in software engineering. We mention some indicative examples below.\
[*Identifying Semantic Errors:*]{} Semantic errors refer to compilable code which does something other than what is intended to do [@Ernst2017]. Source code embeddings can contribute in inferring semantic inconsistencies and assist tasks such as semantic bug localization and recommendations for semantic bug fix.\
[*Robust Topic Modeling:*]{} Agrawal et al. [@Agrawal2018] present a comprehensive review of topic modeling studies in software engineering. Following the paradigm of natural language word embeddings, pretrained source code embeddings provide background knowledge that can further enhance existing methods.\
*Coupling With Other Artifacts:* The models can be used for facilitating tasks that require the association of source code with related artifacts in natural language, e.g., assessment of relevance between proposed changes in code and code review comments, recommendation of reviewers, prediction of programming comments.\
*Auto Completion:* The official *fastText* documentation stresses the value of the subword information captured by such models for auto–correction of misspellings. We observed that in a code–writing setting this feature could prove useful for auto-completion with the combined spelling and meaning that identifiers share in the model (e g., identifiers located in the nearest neighborhood of “isFullScreen” include “useFullscreen”, “isFullScreenAllowed”, “toggleFullscreen”, “behindFullscreen”).
Challenges and Limitations
--------------------------
The main challenge in producing embeddings for source code identifiers was deciding the appropriate input format for optimizing the representational strength of the models. [*fastText*]{} is being used extensively for training word representations for natural languages, there exist however lexical details specific to source code that led to counterintuitive preprocessing decisions as discussed in sections \[sec:cleaning\] and \[sec:results\].
In addition to challenges, there are limitations that ensue when models for artificial languages are trained using techniques originally designed for natural languages. Besides their representational strengths, source code embeddings also suffer from weaknesses when compared to their natural language counterparts. *fastText* provides the character n-gram prediction granularity that makes it more suitable for source code than other models, such as Word2Vec. Still, the feature of relational analogy is not prominent in the models. Further work needs to be done at the algorithmic level in order to capture intricacies implicit in source code, which do not apply in the case of natural language. Research towards this direction is just being initiated [@Alon2019]. In the meantime, we believe that semantic representations of source code, embedded in dense vector space models, can drive a variety of information retrieval tasks in software engineering.
Acknowledgment {#acknowledgment .unnumbered}
==============
The research described has been carried out as part of the CROSSMINER Project, which has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement No. 732223.
[00]{} M. Allamanis, E. T. Barr, P. Devanbu, and C. Sutton, “A survey of machine learning for big code and naturalness,” *ACM Computing Surveys (CSUR)*, vol. 51, no. 4, p. 81, 2018.
A. Hindle, E. T. Barr, Z. Su, M. Gabel, and P. Devanbu, “On the naturalness of software,” in *Software Engineering (ICSE), 2012 34th International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2012.
M. D. Ernst, “Natural language is a programming language: Applying natural language processing to software development,” in *LIPIcs-Leibniz International Proceedings in Informatics*, vol. 71.1em plus 0.5em minus 0.4emSchloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017.
M. White, M. Tufano, C. Vendome, and D. Poshyvanyk, “Deep learning code fragments for code clone detection,” in *Proceedings of the 31st IEEE/ACM International Conference on Automated Software Engineering*. 1em plus 0.5em minus 0.4emACM, 2016.
H. Wei and M. Li, “Supervised deep features for software functional clone detection by exploiting lexical and syntactical information in source code.” in *IJCAI*, 2017.
B. Vasilescu, C. Casalnuovo, and P. Devanbu, “Recovering clear, natural identifiers from obfuscated js names,” in *Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering*.1em plus 0.5em minus 0.4emACM, 2017, pp. 683–693.
A. T. Nguyen, T. T. Nguyen, and T. N. Nguyen, “Lexical statistical machine translation for language migration,” in *Proceedings of the 2013 9th Joint Meeting on Foundations of Software Engineering*.1em plus 0.5em minus 0.4emACM, 2013.
S. Iyer, I. Konstas, A. Cheung, and L. Zettlemoyer, “Summarizing source code using a neural attention model,” in *Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics*, vol. 1, 2016, pp. 2073–2083.
M. Allamanis, H. Peng, and C. Sutton, “A convolutional attention network for extreme summarization of source code,” in *International Conference on Machine Learning*, 2016.
Y. Pu, K. Narasimhan, A. Solar-Lezama, and R. Barzilay, “sk\_p: a neural program corrector for moocs,” in *Companion Proceedings of the 2016 ACM SIGPLAN International Conference on Systems, Programming, Languages and Applications: Software for Humanity*.1em plus 0.5em minus 0.4em ACM, 2016.
R. Gupta, S. Pal, A. Kanade, and S. Shevade, “Deepfix: Fixing common c language errors by deep learning.” in *AAAI*, 2017.
S. R. Foster, W. G. Griswold, and S. Lerner, “Witchdoctor: Ide support for real-time auto-completion of refactorings,” in *Software Engineering (ICSE), 2012 34th International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2012.
Y. Oda, H. Fudaba, G. Neubig, H. Hata, S. Sakti, T. Toda, and S. Nakamura, “Learning to generate pseudo-code from source code using statistical machine translation (t),” in *Automated Software Engineering (ASE), 2015 30th IEEE/ACM International Conference on*.1em plus 0.5em minus 0.4em IEEE, 2015.
W. Ling, P. Blunsom, E. Grefenstette, K. M. Hermann, T. Ko[č]{}isk[y]{}, F. Wang, and A. Senior, “Latent predictor networks for code generation,” in *Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics*, vol. 1, 2016.
P. Yin and G. Neubig, “A syntactic neural model for general-purpose code generation,” in *Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics*, vol. 1, 2017.
C. V. Alexandru, S. Panichella, and H. C. Gall, “Replicating parser behavior using neural machine translation,” in *Proceedings of the 25th International Conference on Program Comprehension*.1em plus 0.5em minus 0.4emIEEE Press, 2017.
M. Allamanis and C. Sutton, “Mining source code repositories at massive scale using language modeling,” in *Proceedings of the 10th Working Conference on Mining Software Repositories*.1em plus 0.5em minus 0.4emIEEE Press, 2013.
F. Palomba, A. Panichella, A. De Lucia, R. Oliveto, and A. Zaidman, “[A textual-based technique for Smell Detection]{},” in *2016 IEEE 24th International Conference on Program Comprehension (ICPC)*, Universita di Salerno, Salerno, Italy.1em plus 0.5em minus 0.4emIEEE, 2016.
A. Halevy, P. Norvig, and F. Pereira, “The unreasonable effectiveness of data,” *IEEE Intelligent Systems*, vol. 24, no. 2, pp. 8–12, 2009.
A. Mahmoud and G. Bradshaw, “Semantic topic models for source code analysis,” *Empirical Software Engineering*, vol. 22, no. 4, pp. 1965–2000, 2017.
T. Mikolov, K. Chen, G. Corrado, and J. Dean, “Efficient estimation of word representations in vector space,” *arXiv preprint arXiv:1301.3781*, 2013.
Z. S. Harris, “Distributional structure,” *Word*, vol. 10, no. 2-3, pp. 146–162, 1954.
“fasttext - library for efficient text classification and representation learning,” <https://fasttext.cc/>, \[last accessed 5-Feb-2019\].
P. Bojanowski, E. Grave, A. Joulin, and T. Mikolov, “Enriching word vectors with subword information,” *Transactions of the Association for Computational Linguistics*, vol. 5, pp. 135–146, 2017.
“word2vec - tool for computing continuous distributed representations of words.” <https://code.google.com/archive/p/word2vec/>, \[last accessed 5-Feb-2019\].
J. Pennington, R. Socher, and C. D. Manning, “Glove: Global vectors for word representation,” in *Empirical Methods in Natural Language Processing (EMNLP)*, 2014. \[Online\]. Available: <http://www.aclweb.org/anthology/D14-1162>
T. Mikolov, E. Grave, P. Bojanowski, C. Puhrsch, and A. Joulin, “Advances in pre-training distributed word representations,” in *Proceedings of the International Conference on Language Resources and Evaluation (LREC 2018)*, 2018.
E. Grave, P. Bojanowski, P. Gupta, A. Joulin, and T. Mikolov, “Learning word vectors for 157 languages,” in *Proceedings of the International Conference on Language Resources and Evaluation (LREC 2018)*, 2018.
K. Taghipour and H. T. Ng, “Semi-supervised word sense disambiguation using word embeddings in general and specific domains,” in *Proceedings of the 2015 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies*, 2015.
B. Xu, D. Ye, Z. Xing, X. Xia, G. Chen, and S. Li, “Predicting semantically linkable knowledge in developer online forums via convolutional neural network,” in *Proceedings of the 31st IEEE/ACM International Conference on Automated Software Engineering*.1em plus 0.5em minus 0.4em ACM, 2016.
W. Fu and T. Menzies, “Easy over hard: a case study on deep learning,” in *Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering*.1em plus 0.5em minus 0.4emACM, 2017.
V. Efstathiou, C. Chatzilenas, and D. Spinellis, “Word embeddings for the software engineering domain,” in *2018 IEEE/ACM 15th International Conference on Mining Software Repositories (MSR)*.1em plus 0.5em minus 0.4emIEEE, 2018.
“Tiobe - the software quality company,” <https://www.tiobe.com/tiobe-index/>, 2016, \[last accessed 5-Feb-2019\].
N. Munaiah, S. Kroh, C. Cabrey, and M. Nagappan, “Curating github for engineered software projects,” *Empirical Software Engineering*, vol. 22, no. 6, pp. 3219–3253, 2017.
“Home of the reporeapers,” <https://reporeapers.github.io/results/1.html>, \[last accessed 5-Feb-2019\].
<https://github.com/vefstathiou/scode-ft-embeddings>, 2019, \[last accessed 5-Feb-2019\].
D. Spinellis, “dspinellis/tokenizer: Version 1.1,” Feb. 2019. \[Online\]. Available: <https://doi.org/10.5281/zenodo.2558420>
A. Agrawal, W. Fu, and T. Menzies, “What is wrong with topic modeling?(and how to fix it using search-based software engineering),” *Empirical Software Engineering*, vol. 4, p. 2, 2018.
A. Panichella, B. Dit, R. Oliveto, M. Di Penta, D. Poshyvanyk, and A. De Lucia, “How to effectively use topic models for software engineering tasks? an approach based on genetic algorithms,” in *Proceedings of the 2013 International Conference on Software Engineering*.1em plus 0.5em minus 0.4emIEEE Press, 2013.
G. Finley, S. Farmer, and S. Pakhomov, “What analogies reveal about word vectors and their compositionality,” in *Proceedings of the 6th Joint Conference on Lexical and Computational Semantics (\* SEM 2017)*, 2017.
M. Kusner, Y. Sun, N. Kolkin, and K. Weinberger, “From word embeddings to document distances,” in *International Conference on Machine Learning*, 2015.
O. Barkan and N. Koenigstein, “Item2vec: neural item embedding for collaborative filtering,” in *Machine Learning for Signal Processing (MLSP), 2016 IEEE 26th International Workshop on*.1em plus 0.5em minus 0.4emIEEE, 2016.
R. Fu, J. Guo, B. Qin, W. Che, H. Wang, and T. Liu, “Learning semantic hierarchies via word embeddings,” in *Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics*, vol. 1, 2014.
N. Rekabsaz, B. Mitra, M. Lupu, and A. Hanbury, “Toward incorporation of relevant documents in word2vec,” *arXiv preprint arXiv:1707.06598*, 2017.
U. Alon, M. Zilberstein, O. Levy, and E. Yahav, “code2vec: Learning distributed representations of code,” *Proceedings of the ACM on Programming Languages*, vol. 3, no. POPL, 2019.
[^1]: <https://doi.org/10.5281/zenodo.2558730>
|
---
abstract: 'We present the first three-dimensional (3D), hydrodynamic simulations of the core convection zone (CZ) and extended radiative zone spanning from 1% to 90% of the stellar radius of an intermediate mass ($3\,{\ensuremath{\mathrm{M}_\odot}}$) star. This allows us to self-consistently follow the generation of internal gravity waves (IGWs) at the convective boundary and their propagation to the surface. We find that convection in the core is dominated by plumes. The frequency spectrum in the CZ and that of IGW generation is a double power law as seen in previous two-dimensional (2D) simulations. The spectrum is significantly flatter than theoretical predictions using excitation through Reynolds stresses induced by convective eddies alone. It is compatible with excitation through plume penetration. An empirically determined distribution of plume frequencies generally matches the one necessary to explain a large part of the observed spectrum. We observe waves propagating in the radiation zone and excited standing modes, which can be identified as gravity and fundamental modes. They show similar frequencies and node patterns to those predicted by the stellar oscillation code GYRE. The continuous part of the spectrum fulfills the IGW dispersion relation. A spectrum of tangential velocity and temperature fluctuations close to the surface is extracted, which are directly related to observable brightness variations in stars. Unlike 2D simulations we do not see the high frequencies associated with wave breaking, likely because these 3D simulations are more heavily damped.'
author:
- 'P. V. F. Edelmann'
- 'R. P. Ratnasingam'
- 'M. G. Pedersen'
- 'D. M. Bowman'
- 'V. Prat'
- 'T. M. Rogers'
bibliography:
- 'igw3d.bib'
title: 'Three-Dimensional Simulations of Massive Stars: I. Wave Generation and Propagation'
---
Introduction {#sec:intro}
============
In addition to sound waves, fluid dynamical systems can have other wave-like solutions for which the restoring force is not pressure but buoyancy. These waves are commonly referred to as internal gravity waves (IGWs) to distinguish them from surface gravity waves. They occur in many stratified systems, such as atmospheres and oceans, in many of which they have an important impact on the large scale dynamics. IGWs excited by equatorial convection were found to be crucial in driving the quasi-biennial oscillation (QBO) in the Earth’s equatorial stratosphere [@baldwin2001a]. In the oceans, IGWs excited through the surface wind or tides cause turbulent mixing when they break [@munk1998a].
In stars IGWs have been suggested to play an important role in angular momentum transport and chemical mixing in radiative regions, where other mechanisms are not efficient. @press1981a suggested that IGWs in the sun can cause mixing in the convectively stable interior and affect the effective radiative opacity by a factor of two or more. IGW mixing was also suggested as the cause of lithium depletion in F stars [@garcia-lopez1991a] and in the sun [@schatzman1993a; @montalban1994a; @talon2005a].
IGWs are candidates for being the cause of some of the observed properties of stars that are poorly explained by current stellar models, such as the internal rotation structure of stars [@beck2012a; @aerts2017b], stellar cores counter-rotating to their envelopes [@triana2015a; @rogers2015a], or the enhanced mass loss needed to explain certain classes of core-collapse supernovae [@quataert2012a]. Photometric observations suggest the presence of convectively generated IGWs in, at least, some massive stars since the observed velocity spectrum at the surface compares well to that obtained using numerical simulations of IGWs [@aerts2015a; @bowman2019a].
To understand the role IGWs play in all these physical situations it is important to know what spectrum of waves in frequency and wave number space is excited by convection. Theoretical work characterizing these spectra mostly focuses on two mechanisms, excitation through the Reynolds stresses of convective eddies or through penetration of plumes. The former approach was taken by @lighthill1952a, @goldreich1990a, @kumar1999a, and later by @lecoanet2013a. All these studies found a power law dependence in frequency, i.e. proportional to $f^{-\alpha}$, with wave frequency $f$ and exponent $\alpha$. The exact value of the exponent depended on the profile of the [Brunt–Väisälä frequency]{} at the convective boundary (CB). The spectrum generated by plume penetration was first studied by @townsend1966a in a terrestrial context and later extended to stars by @montalban2000a. A recent semianalytical model for the IGW flux caused by plumes at the base of a convection zone, as is the case in the sun, has been developed by @pincon2016a. The predicted spectrum takes a very similar functional form in all these plume-driven cases, which is proportional to $\exp[-(f/f_\text{b})^2]$, with wave frequency $f$ and the plume frequency $f_\text{b}$.
Multidimensional hydrodynamic simulations generally do not impose a specific IGW generation mechanism, and are able to follow convection, IGW generation and propagation directly from the basic equations. Yet numerical limitations and the extreme scales within stellar interiors often restrict them to a more dissipative regime than is realistic in stars. Nevertheless, careful choice of parameters and interpretation of the results allow us to assess theoretical predictions. Simulations showed that the Li depletion in the sun cannot be explained by IGWs [@rogers2006a]. Similarly the uniform rotation of the sun’s radiative interior is not completely caused by IGWs [@rogers2005b; @denissenkov2008a]. @rogers2013a performed two-dimensional (2D) hydrodynamic simulations of IGW generation at the boundary of convective cores of massive stars. They found that the IGW generation spectrum is generally much shallower than theoretical predictions. It shows two frequency regimes with different slopes, suggesting different excitation mechanisms at work. Recent research on breaking of IGWs in the radiative envelopes of massive stars affirmed the importance of the shape of the wave generation spectrum [@ratnasingam2019a].
@browning2004a performed simulations of the inner 30% in radius of a $2\,{\ensuremath{\mathrm{M}_\odot}}$ star with methods very similar to the ones used in this work. Their work focused on convective motions in the core, overshooting, and the influence of rotation. They do not study IGWs in detail, but mention their excitation at the convective boundary. Later work by the same group included magnetic fields and specifically studied the dynamo in the convective core [@brun2005a]. In contrast, our work specifically studies IGW excitation and propagation and therefore includes a much larger part of the radiation zone (up to 90% in radius).
In their work on IGWs in solar-like stars, @alvan2014a performed a detailed analysis of wave excitation and propagation, similar to the one carried out in our work. The main difference is their work is based on solar-like stars with a convective envelope and radiative core, while the opposite is the case in our $3\,{\ensuremath{\mathrm{M}_\odot}}$ star. Propagation through a radiative envelope along a falling density gradient causes wave amplification, which makes nonlinear behavior more likely in intermediate-mass and massive stars.
The remaining parts of this paper are structured as follows: Section \[sec:method\] describes the hydrodynamic equations solved and their pseudo-spectral discretization. Section \[sec:simulations\] discusses the stellar models used as the background state of the simulations and assumptions on heating and dissipation needed for numerical reasons. The general properties of three-dimensional (3D) convection in the core are presented in Sect. \[sec:convection-zone\]. Frequency spectra of core convection and their implications for the generation of IGWs are discussed in Sect. \[sec:conv-spectra\]. The properties of the overshooting region is the subject of Sect. \[sec:overshoot\]. Section \[sec:radiation-zone\] treats IGW propagation and the modes excited in the radiation zone, while Sect. \[sec:igw-surface\] discusses the signature they are expected to leave on the surface, which could be observed by photometry and spectroscopy. We conclude in Sect. \[sec:conclusions\].
Computational Method {#sec:method}
====================
The simulations presented here are a logical continuation of those of @rogers2013a. One caveat of their work is the restriction to 2D geometry, which is expected to yield significantly different behavior of turbulence and also altered wave propagation to some degree due to the difference between 2D annulus geometry and a 3D sphere. We extend their method to 3D by using the same type of anelastic approximation, but discretizing the horizontal part of the equation in terms of spherical harmonics instead of $\sin$ and $\cos$ functions.
We solve the following equations for the deviation from the reference state (indicated by a bar, e.g., $\overline{\rho}$) given by a hydrostatic stellar evolution model, $$\begin{aligned}
\label{eq:anelastic-rho}
\nabla \cdot \overline{\rho} {\mathitbf}{v} = 0,\end{aligned}$$ $$\begin{aligned}
\label{eq:anelastic-v}
{\frac{\partial {\mathitbf}{v}}{\partial t}}&= - ({\mathitbf}{v} \cdot \nabla) {\mathitbf}{v}
- \nabla P - C \overline{g} {\mathitbf}{\hat{r}} + 2({\mathitbf}{v} \times {\mathitbf}{\hat{z}} \Omega) \\
\nonumber
& + \overline{\nu} \left( \nabla^2 {\mathitbf}{v} + \frac{1}{3} \nabla (\nabla \cdot {\mathitbf}{v}) \right),\\
\label{eq:anelastic-T}
{\frac{\partial T}{\partial t}}&= - ({\mathitbf}{v} \cdot \nabla) T + (\gamma - 1) T h_\rho v_r\\
\nonumber
&- v_r \left( {\frac{\partial \overline{T}}{\partial r}} - (\gamma - 1) \overline{T} h_\rho \right) + \frac{\overline{Q}}{c_v \overline{\rho}}\\
\nonumber
& + \frac{1}{c_v\overline{\rho}} \nabla \cdot (c_p \overline{\kappa}\overline{\rho}\nabla T) + \frac{1}{c_v\overline{\rho}} \nabla \cdot (c_p \overline{\kappa}\overline{\rho}\nabla \overline{T}).\end{aligned}$$ Here, ${\mathitbf}{v}$ is the 3D fluid velocity, $v_r$ its radial component, $\overline{\rho}$ is the background density, $\gamma$ is the adiabatic index of the ideal gas equation of state, $\overline{T}$ and $T$ are the temperature background and fluctuation, $\overline{\kappa}$ and $\overline{\nu}$ are the thermal and viscous diffusivities, $\overline{Q}$ is the energy release rate, $c_v$ is the specific heat at constant density, $\overline{g}$ is gravitational acceleration, and $h_\rho=\partial \ln \overline{\rho} / \partial r$ is the negative inverse of the density scale height. We use a standard spherical coordinate system with radius $r$, colatitude $\theta$, and azimuthal angle $\phi$. The unit vector ${\mathitbf}{\hat{r}}$ points in radial direction. Rotating stars are set up using a rotating frame of reference with an angular velocity $\Omega$ and rotation axis along ${\mathitbf}{\hat{z}}$ in direction of the pole at $\theta=0$.
This formulation of the anelastic equations includes self-gravity perturbations $\Phi$ to the reference state gravitational potential $\overline{\Phi}$ by introducing the reduced pressure $P=p/\overline{\rho}+\Phi$ and co-density $C$ [@braginsky1995a; @rogers2005b]. This introduces no additional computational effort as long as the thermodynamic pressure is not calculated. The co-density takes the form $$\label{eq:codensity}
C=-\frac{1}{\overline{T}}\left(T + \frac{1}{\overline{g} \overline{\rho}}{\frac{\partial \overline{T}}{\partial z}}p \right).$$
In their comparison of different variants of the anelastic approximation @brown2012a also investigated this variant of the anelastic equations[^1]. They find that the equations do not conserve energy for non-isothermal stratifications and suggest the removal of the term proportional to $p$ in Eq. to ensure energy conservation. While all simulations used for the analysis in Sect. \[sec:results\] did not contain the @brown2012a modification, we performed several test calculations including it. The modes we find in the radiation zone are not affected by the inclusion or exclusion of this factor.
@brown2012a also predict that IGW frequencies are larger by a factor of $\sqrt{\gamma}$. When comparing the frequencies generated in our simulations to those generated with the 1D pulsation code GYRE [@townsend2013a], we find small deviations, initially of the order of a few ($\sim$ 2% relative deviation) getting larger at higher wavenumbers, but this $\sqrt{\gamma}$ factor does not explain the differences. Hence, we are unsure how this factor manifests itself in our simulations or how that work extends to these fully nonlinear simulations.
The numerical solution method we choose is similar to the approach taken by @glatzmaier1984a and in the ASH code [@clune1999a] with some different choices adapted to the application at hand. To implicitly fulfill Eq. we replace the mass flux $\overline{\rho} {\mathitbf}{v}$ by its decomposition into a poloidal ($W$) and toroidal stream function ($Z$). These are related to the mass flux by $$\label{eq:pol-tor-decomp}
\overline{\rho} {\mathitbf}{v} = \nabla \times \nabla \times W {\mathitbf}{\hat{r}} + \nabla \times Z {\mathitbf}{\hat{r}}.$$ Resulting purely from a curl of a vector this is naturally divergence free. Together with temperature $T$ and reduced pressure $P$ these form the four unknown quantities we are solving for. They are expressed as a linear combination of spherical harmonics $Y_{l,m}$ and radius-dependent, complex coefficients. For temperature this is $$T(r, \theta, \phi, t) = \sum_{m=-m_\mathrm{max}}^{m_\mathrm{max}} \sum_{l=|m|}^{l_\mathrm{max}} T_{l,m}(r,t) Y_{l,m}(\theta,\phi),$$ and its equivalent for the other quantities. This allows us to compute horizontal derivatives via computationally inexpensive recursion relations and it avoids the singularities at the poles. As the coefficients of real-valued quantities fulfill $$T_{l,-m} = (-1)^m T_{l,m}^*,$$ only the components with $m\geq0$ need to be stored. The choice of *triangular truncation* ($l_\mathrm{max} = m_\mathrm{max}$) results in uniform angular resolution. The method we use is pseudo-spectral, i.e. the linear terms are computed in spectral space and the nonlinear terms are computed in grid space. This approach makes it necessary to chose the number of grid points in latitudinal and longitudinal direction, $N_\theta$ and $N_\phi$, corresponding to the number of spectral modes. To avoid aliasing errors we set $N_\theta = (3l_\mathrm{max} + 1)/2$ and $N_\phi = 2 N_\theta$. Details on this kind of spectral discretization can be found in @glatzmaier2013a. We do not use a spectral basis in the radial direction to be able to easily adjust the grid to the underlying stellar model. In the present case we have an increased radial resolution in the convection zone. Radial derivatives are computed using second-order finite differences accounting for the nonuniform grid spacing.
We use the implicit Crank–Nicolson method for the linear diffusion terms to avoid the strict CFL condition that depends quadratically on the step size associated with explicit time-stepping. The nonlinear terms are calculated using the explicit Adams–Bashforth linear multistep method, which makes the method second-order accurate in time. We choose a constant time step of , which is well below the CFL condition of the explicit terms and makes later Fourier analysis of the time series easier.
![\[fig:scaling-openmp\]Strong scaling on the NASA NAS Pleiades system using Ivy Bridge CPUs. The reference for measuring speed-up is the case of 100 MPI tasks with 1 thread/task. The best efficiency at 1500 cores is 77% using 5 OpenMP threads per MPI task.](figures/scaling-openmp.pdf)
The code is parallelized using a domain decomposition in the radial coordinate only. The communication involves halo updates for computing finite differences in the radial direction and all-to-all communication for solving the linear equations involved in implicit time-stepping. It is implemented using the message passing interface (MPI). To alleviate the problem that domains become small when using many cores we additionally implement thread-based parallelization using OpenMP, which starts to be more efficient than pure MPI when there are less than 3 radial points per task (see Fig. \[fig:scaling-openmp\]). The achieved scaling efficiency from 100 to 1500 cores is 77% on the NASA Pleiades system.
Simulations {#sec:simulations}
===========
The equations discussed in Sect. \[sec:method\] rely on a spherically symmetric reference state for the thermodynamic variables on top of which the evolution of small perturbations is calculated. We use the MESA (Modules for Experiments in Stellar Astrophysics) stellar evolution code[^2] [@paxton2011a; @paxton2013a; @paxton2015a; @paxton2018a] to produce the reference state. We use the default settings to generate a nonrotating, $3\,{\ensuremath{\mathrm{M}_\odot}}$ zero-age main-sequence (ZAMS) star of metallicity $Z=10^{-2}$. The exact code configuration (inlists) and MESA profiles can be obtained at this URL[^3]. No convective overshooting was used. The values of density, temperature, and gravity are adopted unchanged from the model and interpolated onto a grid with 400 cells in the convection zone and 1100 cells in the radiation zone. The total radius of the star is $R_\star = \SI{1.42e11}{cm} = 2.05\,{\ensuremath{\mathrm{R}_\odot}}$.
![\[fig:rho-t-mesa\]Background stratification of density $\rho$ (*solid blue line*) and temperature $T$ (*dashed red line*) used in the 3D anelastic simulations. The vertical dotted lines show the extent of the simulation domain. The radius coordinate is scaled to the total radius of the star $R$. The vertical solid line indicates the convective–radiative boundary.](figures/mesa-model.pdf)
![\[fig:brunt-mesa\]Square of [Brunt–Väisälä frequency]{} $N^2$ and Lamb frequencies $S^2_l$ (dashed) for the background stratification from Fig. \[fig:rho-t-mesa\]. The vertical solid line indicates the outer boundary of the core convection zone and the vertical dashed lines are the boundaries of the computational domain of the 3D simulations.](figures/mesa-brunt.pdf)
Figure \[fig:rho-t-mesa\] shows the density and temperature profile of the stellar model. The radial extent of the 3D simulation domain indicated by vertical, dashed lines is limited at 1% of the stellar radius to avoid the coordinate singularity at the core and at 90%, where density drops below $10^{-4}\,\mathrm{g\,cm^{-3}}$, covering nearly six orders of magnitude in density.[^4] The [Brunt–Väisälä frequency]{} profile, which governs the propagation of IGWs, is plotted in Fig. \[fig:brunt-mesa\]. It shows the convective radiative boundary at 13% of the total radius.
[c|ccccccccc]{} H6R5 & 3741 & $8$ (rising) & $10^5 \kappa_\star$ & $10^6 \varepsilon_\star$ & $5$ & & 126 & 60 – & $38.7$\
H6R10 & 3741 & $8$ (rising) & $10^5 \kappa_\star$ & $10^6 \varepsilon_\star$ & $10$ & & 126 & 60 – & $59.3$\
H5 & 3741 & $1$ & $10^5 \kappa_\star$ & $10^5 \varepsilon_\star$ & 0 & & 468 & 7 – & $13.3$\
H6E & 3741 & $10$ & $10^5 \kappa_\star$ – $50 \kappa_\star$ & $\approx 10^6 \varepsilon_\star$ (exp.) & 0 & & 101 & 100 – 2 & $61.7$\
H6LD & 3741 & $10$ & $10^5 \kappa_\star$ – $50 \kappa_\star$ & $10^6 \varepsilon_\star$ & 0 & & 100 & 100 – 2 & $58.8$\
H6LD-HR& 14706 & $10$ & $10^5 \kappa_\star$ – $50 \kappa_\star$ & $10^6 \varepsilon_\star$ & 0 & & 100 & 100 – 2 & $6.5$\
H7E & 3741 & $8$ (rising) & $\SI{5e13}{cm^2.s^{-1}}$ & $\approx 10^7 \varepsilon_\star$ (exp.) & 0 & & 272 & 1 – 39 & $16.4$\
H7E-HR & 14706 & $8$ (rising) & $\SI{5e13}{cm^2.s^{-1}}$ & $\approx 10^7 \varepsilon_\star$ (exp.) & 0 & & 272 & 1 – 39 & $33.2$\
For numerical stability we need to increase the thermal diffusivity $\kappa$ and kinematic viscosity $\nu$ beyond their physical values in the star, $\kappa_\star$ and $\nu_\star$, respectively. In the $3\,{\ensuremath{\mathrm{M}_\odot}}$ MESA model $\kappa_\star$ ranges from in the CZ to at the top of the simulated region ($r=0.9\,R_\star$), and $\nu_\star$ ranges from to . As increased diffusivity and viscosity would damp convection too strongly in order to reach a somewhat turbulent state, we increase the luminosity of the star by a similar factor to balance the increased damping. In a series of models we explore the effect of increased forcing and that of using different profiles for viscosity. These are summarized in Table \[tab:models\]. To put this into context we compare several characteristic nondimensional numbers. The Rayleigh number, $$\label{eq:rayleigh}
{\ensuremath{\mathrm{Ra}}}= \frac{g \overline{Q} D^5}{c_v \kappa^2 \nu \overline{T}},$$ with a typical length scale $D$ (chosen to be the size of the convective core in this case), controls the details of convection and determines if energy transport is mostly through radiation or convection. This particular form of ${\ensuremath{\mathrm{Ra}}}$ is also called a flux Rayleigh number. The stellar value is , which is more than six orders of magnitude higher than the values reached in the simulations. This is the rationale for increasing the energy release. If we had used the original value of $\overline{Q}$, [$\mathrm{Ra}$]{} would be approximately , which might even be subcritical. The actual convection in the star is likely even more vigorous and plume dominated than that observed in the simulations.
Flows with a high Reynolds number, $$\label{eq:reynolds}
{\ensuremath{\mathrm{Re}}}= \frac{v_\text{rms} D}{\nu},$$ develop turbulence, while low values of [$\mathrm{Re}$]{} normally result in laminar flow. Due to the extreme length scales $D\approx 14\%\,R_\star$ and velocities, [$\mathrm{Re}$]{} is typically extremely large in stellar environments, in the current case ${\ensuremath{\mathrm{Re}}}\approx\num{e12}$. These parameters are not currently possible in numerical simulations. As can be seen in Table \[tab:models\] we can only reach values of approximately in the CZ. However, it is expected that as long as a part of the inertial range of the turbulent cascade is numerically resolved, the energy dissipation rate will not change significantly at higher [$\mathrm{Re}$]{} [e.g., @frisch1995a Chapter 5]. Yet the small scale velocity field will definitely show differences, which is a caveat of the presented simulations. The Reynolds number is another reason for using an increased convective forcing, as using the original value would result in velocities corresponding to ${\ensuremath{\mathrm{Re}}}\approx 1$.
The Prandtl number, $$\label{eq:prandtl}
{\ensuremath{\mathrm{Pr}}}= \frac{\nu}{\kappa},$$ is the ratio of viscous to thermal diffusion. In stars it is typically extremely low, ranging from in the core to at the surface. The only way to reach these values in our numerical simulations would be to increase $\kappa$ to very high values, which would damp the waves too much. As a compromise we settle on [$\mathrm{Pr}$]{} around 1 in the envelope and around 100 in the core in most models (see Tab. \[tab:models\]). In a few models [$\mathrm{Pr}$]{} reaches much lower values of or even in the envelope, but these are subject to excessive damping due to too much thermal diffusion.
In most models we increase luminosity by setting the heating function $\overline{Q}$ to the nuclear energy generation rate from MESA multiplied by a constant factor. These models are referred to with a name starting with “H$X$” for $10^X$ times the stellar luminosity $L_\star$. For example, “H6” corresponds to a luminosity of $10^6\,L_\star$. In a few models we used an exponential heating function, $$\overline{Q} = A c_v \overline{\rho} \exp(-r/r_\mathrm{min}) (r-r_\mathrm{min})/R_\star,$$ with a scaling factor $A$, which is used to adjust it to a boosted stellar luminosity. These are labeled with “H$X$E” for an exponential heating profile corresponding to a luminosity of $10^X$ the stellar value, and $r_\mathrm{min}$ the innermost radius of the simulation domain.
The thermal diffusivity is treated in a similar way by multiplying the stellar value with a constant factor, which was the lowest value that did not show stability problems. As the increased diffusivity is mainly needed in the convection zone, we also tried a different approach where just the CZ is subject to a value of $10^5 \kappa_\star$, while diffusivity in the radiation zone can be reduced to $50\kappa_\star$. Both regions are blended using a hyperbolic tangent function with a width of (3.5% of the stellar radius and 26% of the size of the convection zone). This was used in models H6E and H6LD. Model H6LD is a combination of the low diffusivity of Model H6E with the boosted MESA energy release of model H6.
Important conclusions in this paper are drawn from spectra. To clarify their interpretation we give an exact definition here. For a real function $E(t)$ sampled in an interval $[t_a, t_b]$ the Fourier transform is $$\label{eq:FT}
\hat{E}(f) = \frac{1}{t_b - t_a} \int_{t_a}^{t_b} E(t) e^{- 2 \pi i f t} dt.$$ By normalizing with the length of the interval the units of $\hat{E}$ are the same as those of $E$, which makes it easier to interpret the magnitude of components of the spectra.
As data from the simulations is sampled at discrete times $t_0,\ldots,t_{n-1}$ with equidistant spacing $\Delta t$, we approximate Eq. with a discrete Fourier transform (DFT), $$\label{eq:DFT}
\hat{E}(f_j) = \frac{1}{n} \sum_{m=0}^{n-1} E(t_m) e^{-2 \pi i \frac{{m j}}{n}},$$ where $j$ takes values from 0 to $\lfloor\frac{n}{2}\rfloor$. Higher values of $j$ are redundant due the real input data. The corresponding frequencies are $f_j = \frac{j}{n\Delta t}$.
Results {#sec:results}
=======
Convection Zone {#sec:convection-zone}
---------------
![\[fig:convvel-compare\]Root mean square velocity as a function of radius in the 3D simulations for different luminosity boosting factors. The dashed line is the velocity estimate according to mixing-length theory returned from MESA. The surface convection zone is visible at $r\approx R_\star$ in the MESA data. The region shaded in blue marks the position of the convection zones in MESA. The vertical dashed line is the radius at which the spectra from Fig. \[fig:CZ-spec-all\] are computed.](figures/convvel)
![\[fig:L-vconv\]Relation of increased stellar luminosity and rms velocity in the convection zone. The dotted line is a power law fit to data from the 3D hydrodynamics simulations. The MESA value computed from a volume average of the MLT velocity is plotted for comparison.](figures/L-vconv)
As we have to increase the heating term $\overline{Q}$ (equivalent to an increase in luminosity $L$), thermal diffusivity $\kappa$, and kinematic viscosity $\nu$ for numerical reasons, the convective velocities are higher than those predicted by mixing-length theory (MLT) using quantities from the stellar evolution model. Figure \[fig:convvel-compare\] compares the angular average of velocities of the different models to the MLT value. In the convection zone all our simulations have velocities one to two orders of magnitude higher than the MLT value. This causes waves at the convective–radiative boundary (CB) to be excited at higher amplitudes, which is intended to offset the increased dissipation within the RZ with the hope of surface amplitudes being more realistic. The rise of velocity close to the largest radii is related to the outer boundary condition.
The scaling of convective velocities with changing luminosity has been subject of previous studies. Other hydrodynamic simulations of convection zones in stars [@porter2000a; @viallet2013a; @jones2017a] find that, $$\label{eq:L-vconv}
L \propto v_\text{rms}^3.$$ This is also the result found using MLT [e.g., @kippenhahn2012a]. The scaling relation agrees perfectly with the observations in our simulations, which fit $v_\text{rms} \propto L^{0.34}$ (see Fig. \[fig:L-vconv\]).
As expected from the stratification of the 1D reference state, convection immediately starts to develop in the core. From early times convection is dominated by large plumes. These plumes often rise until they reach the CB, but are sometimes dissolved by interacting with large eddies. Their disintegration at the convective boundary perturbs the stably stratified radiation zone directly above. This process can be seen in the time series in Fig. \[fig:H6R10-T-series\].
Figure \[fig:H6R10-slice\] illustrates the correlation of positive radial velocity and temperatures higher than the horizontal average. It shows that in model H6R10 plumes reach typical velocities of . Scaling this down to the rms convective velocities of the actual star using Eq. yields a rising speed of . Figure \[fig:3d-temp\] shows a 3D view of the whole star using the same model.
The series of meridional slices in Fig. \[fig:H6R10-T-series\] show an example of several plumes hitting the convective boundary and triggering wave motion in the region above. Between $t=0$ and $t=\SI{2.8}{h}$ the large plume in the bottom part of the slice splits into two parts, which subsequently cause small-scale disturbances in the previously much more uniform temperature field of that region. At $t=\SI{5.6}{h}$ a larger plume hits the boundary in the upper left corner of the convection zone. It spreads out at the boundary over more than half a hemisphere and causes Kelvin–Helmholtz-like vortices on its inner side. These seem to be the cause of many of the small-scale eddies at the interface, which can themselves drive waves in the RZ.
Turbulent kinetic energy in the CZ shows a typical cascade behavior, where most energy is present at low wavenumbers, i.e. large length scales. Figure \[fig:k-spectrum\] shows the kinetic energy spectrum of several models as a function of $l$ mode. The energy contained in a single $l$ mode is computed from the poloidal ($W$) and toroidal ($Z$) decomposition (see Eq. ) with the expression [e.g., @glatzmaier2013a Sect. 10.6.6], $$\begin{aligned}
\label{eq:wave-energy-l}
\nonumber
E_l(r) &= \sideset{}{'}\sum^l_{m=0} \frac{l(l+1)}{4 \pi r^2 \overline{\rho}}\\
& \times \left(\frac{l(l+1)}{r^2}|W_l^m|^2 + \left|{\frac{\partial W_l^m}{\partial r}}\right|^2 + |Z_l^m|^2 \right),\end{aligned}$$ where the primed sum means that the $m=0$ terms are multiplied by $1/2$. In all cases most energy is contained in the low-order modes ($l\lesssim 5$), although the actual peak of the spectrum varies between $l=1$ and $l=3$ for the different parameters.
Although numerical diffusivity limits the inertial range in these spectra, we can still obtain a power law slope for each of the models. The slope becomes negatively steeper with increased convective forcing. In Fig. \[fig:k-spectrum\] we fit the inertial range of each model with power laws. In the strongly forced models H6LD and H6R10, in which we see a strong influence of rising plumes (see Fig. \[fig:H6R10-T-series\]), the kinetic energy spectrum drops with $l^{-2.1}$ or $l^{-2.3}$, respectively. This approaches the value predicted by Bolgiano–Obukhov scaling of $l^{-2.2}$ for buoyancy-driven turbulence [@obukhov1959a; @bolgiano1959a]. The more strongly forced models H7E and H7E-HR, show an even steeper slope in the inertial range, following $l^{-3.0}$. This is significantly steeper than the $l^{-5/3}$ relation predicted by @kolmogorov1941a, which forms the basis for theoretical spectra using the eddy excitation mechanism. This might explain why our simulations show a different slope in the frequency spectra.
The deviation from theoretically predicated slopes might be due to the relatively low Reynolds numbers reached in the simulations (see Tab. \[tab:models\]). On the other hand, the case of heating concentrated in a small spherical region is quite different from the plane-parallel, Boussinesq convection underlying some theoretical models and the velocity field is not necessarily isotropic in this case.
Comparing with previous hydrodynamic simulations we find that Model H6R10 agrees well with a comparable model from @rogers2013a, who find a broken power law fit with exponents of and and a break at $l\approx 10$ in a singular-value decomposition of the frequency and wavenumber spectrum. @augustson2016a obtain a qualitatively similar spectrum in their simulations of magnetic turbulence, with a low-wavenumber exponent of approximately and a steeper power law for higher wavenumbers. Their simulations also have a peak in the kinetic energy spectrum at low spherical harmonic degree, in the range from $l=1$ to $l=10$. We note that the models of @augustson2016a do not have enhanced forcing and yet show a similar spectrum to those in this work. This indicates that the spectrum is more dependent on the regime that nondimensional numbers like [$\mathrm{Ra}$]{} and [$\mathrm{Re}$]{} are in than the actual value of convective forcing, as expected.
The inset in Fig. \[fig:k-spectrum\] shows the comparison of the high-wavenumber tail for the simulations H7E and H7E-HR, which are run with identical parameters except for the number of spectral modes being used. Their spectra are almost identical apart from a small bend at the highest $l$ values. This suggests that enough of the inertial subrange of the turbulent cascade is resolved to get the correct energy dissipation and that the simulations do not suffer from severe anomalous behavior at the smallest resolved length scales. The same is true for H6LD and its high-resolution counterpart H6LD-HR.
![\[fig:H6LD-HR-spec\]Frequency spectra of radial velocity in simulations H6LD ($128\times256$ angular resolution) and H6LD-HR ($256\times512$ angular resolution). The spectrum is shown in the radiation zone at $r=0.74\,R_\star$. The number of time samples in H6LD was reduced to match H6LD-HR, which was run for a shorter time.](figures/H6LD-HR-spec)
As a main concern of this paper are the IGW spectra, we also assess the impact of angular resolution on these. Figure \[fig:H6LD-HR-spec\] shows the frequency spectrum of radial velocity in the radiation zone for the two simulations H6LD and H6LD-HR, where both are identical except for latter having twice the angular resolution. We see that both simulations are very similar, including the magnitude and position of the modes between and , the continuous signal between and , and the low frequency drop due to radiative damping at .
IGW generation {#sec:conv-spectra}
--------------
![\[fig:CZ-spec-all\]Frequency spectra of kinetic energy just above the CZ ($r=\SI{2e10}{cm}=0.14\,R_\star$). The vertical dotted lines represent an estimate for the convective turnover frequency from Eq. . Panel (a) shows the Fourier transform according to Eq. . Panel (b) is the same multiplied by frequency to account for integration over a logarithmic coordinate.](figures/CZ-spec-all)
![\[fig:CZ-spec-l\]Absolute value of the Fourier transform of kinetic energy in particular $l$ modes according to Eq. . The data are taken from simulation H6LD. The logarithmic data are fitted with a broken power law (*orange line*). Its slopes and the position of the break are indicated next to the fit. The vertical dashed lines are estimates for the position of the break from Eq. .](figures/CZ-spec-l.pdf)
It is controversial which physical mechanism is most important for the excitation of IGWs at the CB. The two common candidates are bulk Reynolds stresses produced by convective eddies [@lighthill1952a; @goldreich1990a] and plume overshoot [@townsend1966a; @zahn1991a]. Most theories about the effect of IGWs in stellar interiors [e.g., @talon2005a; @fuller2014a] employ the spectrum of IGWs derived from convective eddies [@kumar1999a; @lecoanet2013a]. Therefore our analysis focuses on this spectrum, but the plume spectrum is considered later.
To study the spectrum of waves generated, we first investigate the spectrum of motions generated at the CB. We analyze our 3D data by computing the spectrum of kinetic energy density at a radius of 0.07 $H_P$[^5] above the top of the convection zone (as defined by the Schwarzschild criterion). This spectrum is given by, $$\label{eq:ekin-spec}
\hat{E}_\mathrm{kin} = \frac{1}{2} \bar{\rho} \left(\hat{v}_r^2 + \hat{v}_\theta^2 + \hat{v}_\phi^2 \right),$$ with the Fourier transforms of the individual velocity components, $\hat{v}_r$, $\hat{v}_\theta$, $\hat{v}_\phi$, according to Eq. . Figure \[fig:CZ-spec-all\] shows the spectra for different models. For guidance we show an estimate of the convective turnover frequency given by, $$\label{eq:turnover}
f_\text{TO} = \frac{v_\text{rms}}{\pi r_\text{CZ}},$$ which assumes the largest eddy extends from the center of the star to the radius of the convection zone $r_\text{CZ}$ and it turns at the rms velocity. Panel (b) shows the spectra multiplied by $f$ to account for integration over $d \log f$, which makes it easier to see the regions containing most energy in the logarithmic plot. We see that, while the peak is not too far from $f_\text{TO}$, the distribution is almost flat in the low frequency regime.
Clearly, this is a spectrum of motions at this radius and is not necessarily waves (although see Sect. \[sec:igw-identification\]). However, this motion is what drives the waves and if it has a high-frequency component then high-frequency waves can be efficiently driven.
At this radius the integrated (i.e. including all harmonic degrees $l$) frequency spectrum is nearly flat with a transition to a more steeply declining power law at higher frequencies ($f\gtrsim \SI{20}{\micro Hz}$). The spectrum is not dominated by values at $f_\text{TO}$ and indeed it is hard to make out this frequency in the spectrum. However, if we look at the frequency spectrum at particular length scales, by selecting individual values of $l$, we start to see a sharp transition between the power laws at low and high frequency as evidenced in Fig. \[fig:CZ-spec-l\].
In this scale-dependent spectrum the break point between the two power laws depends mostly linearly on angular degree $l$ and can be approximated with, $$\label{eq:turnover-l}
s = \SI{4.0}{\micro Hz} \cdot l.$$ The slope is not too far from the estimate for the convective turnover frequency $f_\text{TO}=\SI{7.2}{\micro Hz}$ for this model, considering the uncertainty in the estimate of $f_\text{TO}$ in Eq. . This fits the conjecture by @rogers2013a that the eddy mechanism efficiently generates waves below this frequency.
It is worth noting that, even in this scale-dependent spectrum in which the break between power laws corresponds to the scale-dependent turnover frequency, the energy is *not* concentrated at that frequency. This is in stark contrast to the theoretical predictions which posit that the frequency spectrum (within the CZ) is strongly peaked at the convective turnover frequency.
![\[fig:conv-slopes\]Exponents of a broken power law fit to the kinetic energy spectrum on top of the convection zone of model H6E as a function of $l$ mode in the spherical harmonic decomposition.](figures/conv-slopes.pdf)
![\[fig:conv-break\]Position of the break $s$ of the broken power law fit to the kinetic energy on top of the convection zone of model H6LD as a function of $l$ mode in the spherical harmonic decomposition. The dashed line is the turnover frequency for the particular $l$ mode, as estimated in Fig. \[fig:CZ-spec-l\].](figures/conv-break.pdf)
In a more systematic analysis of the broken power law fits to the frequency spectrum at the CB in Fig. \[fig:conv-slopes\], we notice that the exponent of the low frequency regime stays relatively constant for $l>3$ at a value of $\num{-0.46}\pm\num{0.07}$. The high frequency component covers a wider range of exponents from at $l=2$ to at $l=33$. Both exponents show very little change at higher values of $l$. The position of the frequency break point $s$ in the power law in Fig. \[fig:conv-break\] on the other hand is rising with $l$ roughly following the estimate for the convective turnover frequency from Eq. multiplied by $l$ to account for the smaller length scales at higher spherical harmonic degree. For $l\gtrsim 31$ we observe a rise in the exponent of the high frequency range, which is due to the difficulty in fitting an increasingly smaller part of the curve. This makes the determination of $s$ less certain as well. The value of $s$ lies within the range between 12 and in our simulations, which is a bit higher than the range of 10 to in the 2D simulations of @rogers2013a. This is understandable if the position of the break really depends on $f_\text{TO}$, and in turn $v_\text{rms}$, because the 2D simulations show a lower convective velocity.
![\[fig:CZ-spec-integrated\]Cumulative spectrum of kinetic energy density just above the CZ ($r=\SI{2e10}{cm}=0.14\,R_\star$) normalized to the value integrated over all frequencies. The vertical dotted lines represent an estimate for the convective turnover frequency from Eq. . The shaded area shows the frequency span centered around $f_\text{TO}$ that contains 40% of kinetic energy for model H6LD.](figures/CZ-spec-integrated)
The logarithmic scaling of Fig. \[fig:CZ-spec-all\] makes it hard to see, which frequencies contribute most to kinetic energy at the top of the convection zone. To analyze this we plot the cumulative energy distribution, i.e. the function of energy contained below a certain frequency. It is shown normalized to the full energy of the particular model in Fig. \[fig:CZ-spec-integrated\]. We see that for models with a heating rate increased by a factor of $10^6$, roughly 50% of the energy is below $f_\text{TO}$ and 50% is above. While the steepest increase is around $f_\text{TO}$, the distribution of energy is widely spread in frequency. The shaded area shows the frequency range around $f_\text{TO}$ that contains 40% of energy. In model H7E with a heating rate of $10^7$ times the stellar value, 80% of the energy is located below $f_\text{TO}$, while model $H5$ has almost all energy far above its value of $f_\text{TO}$ at . As discussed earlier we believe model H6LD is the best trade-off between an increased heating rate and increased diffusivity.
Comparing these numerical spectra to theoretical spectra is not straightforward as clearly the former include wave motion as well as overshooting motion (although see Sect. \[sec:igw-identification\]). However, one can trace differences between theoretical IGW spectra to differences in the assumed convective spectra. While the theoretical wavenumber spectrum has some observational basis, the frequency spectra supposed in the theoretical analysis of @kumar1999a and @lecoanet2013a – based on the assumption of Kolmogorov scaling of eddy sizes and their corresponding turnover times – do not. Yet it is this frequency spectrum in the CZ that determines the frequency spectra of excited IGWs.
For example, theoretical spectra do not efficiently generate high-frequency waves because of the assumption that most of the convective energy is concentrated at the convective turnover frequency. If the energy of convection itself is not limited to a narrow band around the convective turnover frequency, there is no reason to suppose that the IGW frequency spectrum would be. Moreover, if the CZ has high frequencies then it can efficiently generate waves of high frequency. Therefore, based on comparisons of CZ spectra one can see two important issues arise between theoretical and numerical results that would affect the IGW spectra: (1) for an integrated spectrum, energy is not concentrated at the convective turnover frequency, but is spread among a wide range of frequencies; and (2) frequencies higher than $f_\mathrm{TO}$ are clearly present with significant energy within the convection zone. We also note that while Kolmogorov scaling may be the appropriate description for isotropic turbulence in a Boussinesq box, it is wholly unclear that it is appropriate for spherical configurations with a centrally peaked heating term, such as stars.
@kumar1999a mention that they deliberately ignore wave excitation by plumes due to limited information on their properties. As the flow pattern we observe in the simulations is obviously dominated by large plumes, it is natural to compare the spectra to theory of IGW excitation by plume penetration. @montalban2000a developed expressions for the IGW spectrum generated by plume penetration at the bottom of the solar convection zone. It is based on the plume model by @rieutord1995a. They explicitly caution against its use at the top of a convective envelope because of the typical importance of radiative cooling there, characterized by a low Péclet number (${\ensuremath{\mathrm{Pe}}}\approx 1$). This argument does not apply here, where ${\ensuremath{\mathrm{Pe}}}\gtrsim \num{e4}$. For comparison, the stellar value in the core is ${\ensuremath{\mathrm{Pe}}}\gtrsim \num{e6}$. Therefore, for lack of a dedicated theory, we apply their model to our spectra.
The frequency dependence of the energy spectrum is determined by the plume timescale $t_\text{b}$, which is often approximated by the ratio of plume velocity $v_\text{pl}$ and plume incursion depth $\Delta_\text{p}$. The corresponding frequency is then, $$\label{eq:plume-freq}
f_\text{b} \sim \frac{v_\text{pl}}{\Delta_\text{p}}.$$ The predicted energy spectrum takes the form [@montalban2000a], $$\label{eq:montalban-spectrum}
E(f) \propto \exp\left(-(f / f_\text{b})^2\right),$$ where we absorbed all factors depending on radius and wavenumber into the proportionality constant. We choose to work directly with the expression for kinetic energy instead of wave flux because the spectrum is taken right at the top of the CZ, where a conversion to flux is not straightforward.
![\[fig:plume-spec\]Kinetic energy spectrum on top of the convection zone of model H6LD. The dashed lines show the theoretical spectrum for plume excitation from Eq. . The red line is the case of a single plume frequency $f_\text{b}$. The orange line is a combination of three different frequencies. The cyan line is using a plume frequency distribution following Eq. .](figures/plume-spec)
A combination of plumes with different timescales can be fit to the simulation spectra. Figure \[fig:plume-spec\] shows fits with one and three values of $f_\text{b}$. This shows that the plume spectrum as described by Eq. generally fits the shallow power law in the low-frequency regime very well, even with a single plume frequency (red line). With just three plume timescales (orange line) it is possible to fit most of the spectrum. Assuming plumes are distributed so that their frequencies follow an exponential function with a low frequency cut-off, allows us to fit the whole spectrum apart from the high frequency turnoff (cyan line). This heuristic expression has the form, $$\label{eq:plume-fit}
A\int_c^\infty e^{-f_\text{b}/{\alpha}} e^{-(f/f_\text{b})^2} df_\text{b}$$ where the shape of the exponential is given by $\alpha=\SI{11.0}{\micro Hz}$ and the low-frequency cut-off is $c=\SI{1}{\micro Hz}$. It should be noted that similar fits can be obtained with other steeply declining functions (e.g., a power law with a negative exponent) for the plume frequency distribution.
To acquire an estimate for typical plume length and time scales from our simulations, we study the process of plume incursion in more detail. Figure \[fig:plume-rect\] shows a Cartesian projection of temperature perturbation at the CB in two different cases. In the left panel a large plume hits the boundary exciting waves at a large range of phase angles, including very steep angles. A representation of excitation through eddies is shown in the right panel. It results in much smaller phase angles. Both cases are remarkably similar to 2D simulations [@rogers2013a Fig. 4].
The large plume of Fig. \[fig:plume-rect\] is studied in greater detail in Fig. \[fig:plume-detail\] to extract its size, velocity, and penetration depth. The [Brunt–Väisälä frequency]{} $N^2$ (Panel (d)) is significantly reduced in the overshooting region above the original convective boundary. This coincides almost perfectly with the penetration depth of the plume, which can be identified by $v_r$ approaching 0 (Panel (b)) and a discontinuity in $T$ (Panel (c)). The penetration depth is $\Delta_\text{p}=0.4\,H_P$ from the original convective boundary. The maximum plume velocity is $v_\text{pl}=\SI{7.8}{km.s^{-1}}$. This allows us to estimate a plume incursion time $t_\text{b} \sim \frac{\Delta_\text{p}}{v_\text{pl}}=\SI{2.2}{h}$, which corresponds to a frequency $f_\text{b}$ of . The lateral extent of the plume $b$ can be defined as the region of positive $v_r$ in Panel (e). It has a value of $b=\SI{1.4e10}{cm}$ in this case. This is also sometimes used to compute the plume timescale, which results in a value of here. As it is easier to extract from simulations in a systematic way, we stay with the first definition using $\Delta_\text{p}$ in the following analysis.
![\[fig:plume-freq-pdf\]Probability density function (PDF) of plume frequencies. It was extracted from simulation H6LD and computed using Eq. . The red, dashed line is an exponential fit to the high-frequency ($f_\text{b}>\SI{210}{\micro Hz}$) end of the PDF. The inset shows a zoom on the peak of the PDF.](figures/plume-freq-pdf)
We apply this estimate of the plume frequency statistically to all plumes in simulation H6LD. For each longitude and latitude, and each output snapshot (every ) we determine if there is a rising plume and in that case compute a plume incursion depth and plume velocity. The criterion for a plume is that $v_r$ is positive at the position of the convective boundary. The incursion radius $r_\text{p}$ is then defined as the radius at which $v_r$ first becomes negative along a line at this particular angle. The penetration depth is calculated as the distance to the convective boundary $\Delta_\text{p}=r_\text{p}-r_\text{conv}$. The plume velocity $v_\text{pl}$ is the highest value of $v_r$ between the top of the convective boundary and $r_p$. Using the estimate for the plume frequency from Eq. , we compute the probability density function (PDF) of $f_\text{b}$ throughout the simulation (Fig. \[fig:plume-freq-pdf\]). It rises sharply to its maximum at and then drops roughly following an exponential distribution. An exponential fit to the data does not perfectly match the values found when fitting Eq. . The parameter $\alpha$ is too high by a factor of 3. Yet considering the simplistic definition of $f_\text{b}$, this still makes a strong argument for an exponential distribution of plume frequencies as the explanation of a large part of the kinetic energy spectrum at the top of the convection zone and hence, the IGW frequency spectrum.
To understand the effect of increased forcing and diffusivity we follow the discussion of plume lifetimes of @pincon2016a. They argue that plume velocity scales with luminosity as $v_\text{pl}\propto L^{1/3}$ which is consistent with the scaling of the convective velocities from Eq. \[eq:L-vconv\]. A luminosity increased by a factor of $10^6$ would thus result in $v_\text{pl}$ increased by a factor of 100. The penetration depth $\Delta_\text{p}$ is not expected to be strongly affected by the change in forcing (see end of Sect. \[sec:overshoot\] for an estimate). The effect of radiative thermalization, while strongly increased due to the higher value of $\kappa$ in the simulations, is still negligible as the timescale $t_\text{rad}\sim \Delta_\text{p}/\kappa$ is of the order of two years, much longer than any observed plume lifetime. The turbulent timescale inside the plume $t_\text{turb} \sim b / v_\text{pl}$, with the lateral plume size $b$. Assuming that $b$ is not strongly affected by increased forcing, similar to $\Delta_\text{p}$, this means that plume frequency $f_\text{b}=1/t_\text{b}$ scales like $v_\text{pl}\propto L^{1/3}$. In model H6LD ($L = 10^6 L_\star$) this results in $f_\text{b}$ being too high by a factor of 100.
Convective overshoot {#sec:overshoot}
--------------------
The treatment of convective–radiative boundaries (CB) in 1D stellar evolution codes is a long-standing problem. It can have a significant impact on the evolution and nucleosynthetic signature of stars by mixing of species beyond convective regions. Hydrodynamic simulations in two or three dimensions promise insight based on first principles and have been subject of previous work [e.g., @freytag1996a; @rogers2006a; @meakin2007a; @jones2017a; @cristini2017a]. There is no single accepted definition of the overshooting depth in terms of angular averages of 3D quantities. For better comparability between different stellar parameters the overshooting depth is usually stated in multiples of the pressure scale height $H_P$ above the convective boundary as defined by the Schwarzschild or Ledoux criterion. Both criteria are equivalent in the case studied here because the star is chemically homogeneous.
![\[fig:depth-pdf\]Probability density function (PDF) of plume incursion depth computed from simulation H6LD. The distribution peaks at $0.54\,H_P$ above the convective boundary. The region after the peak was fitted with a power low with an exponent of .](figures/depth-pdf)
The statistics of velocities and penetration depth $\Delta_\text{p}$ from Sect. \[sec:conv-spectra\] can also be used to make statements on the size of the overshooting region. Figure \[fig:depth-pdf\] shows the PDF of plume penetration depth in model H6LD. The distribution at low $\Delta_\text{p}$ is relatively flat until it peaks at $0.54\,H_P$. Beyond that it drops following a power law with exponent . This is consistent with the picture in Panel (d) of Fig. \[fig:plume-detail\], where $N^2$ is affected by penetration up to a value of approximately $0.5\,H_P$. 95% of plumes penetrate no further than $0.695\,H_P$, which is the value we will use as the boundary of the overshooting region in Sect. \[sec:igw-identification\].
![\[fig:vr-pdf\]Probability density function (PDF) of $v_r$ $0.42\,H_P$ above the top of the convection zone in model H6LD (blue line). The blue shaded area signifies standard deviation over all time steps. A Lorentzian (*green line*) was fitted to the central part of the distribution. A power law (*yellow line*) and an exponential function (*red line*) were fitted in the regions of positive $v_r$. The inset plot shows a log-log plot of the same data.](figures/vr-pdf)
Figure \[fig:vr-pdf\] shows the distribution of updrafts and downdrafts in the overshooting region ($0.42\,H_P$ above $r_\text{conv}$). The PDF is peaked in Lorentzian shape at . The inward velocities are distributed in a smaller range, at most, than the outward velocities, which extend up to .
Our use of an increased convective forcing and thermal diffusivity raises the question of the validity of these results for the actual stellar values. In his study of convective penetration in stellar interiors @zahn1991a found a simple scaling law for the size of the penetrative region [see also, @rogers2006a], $$\label{eq:zahn-Lp}
\Delta_\text{p}^2 = \frac{3}{5} H_P H_\kappa f \frac{\rho v_\text{pl}^3}{F_\text{tot}},$$ with the scale height of thermal diffusivity $H_\kappa = - d \ln r / d \ln \kappa$, plume filling factor $f$, and total energy flux $F_\text{tot}$. As $\kappa$ is only multiplied by a constant in the radiation zone, $H_\kappa$ is identical to the stellar value. The same is true for $H_P$ and $\rho$. While the simulations have an increased $F_\text{tot}$, we found the scaling $F_\text{tot} \propto v^3$ (Eq. and Fig. \[fig:L-vconv\]). This means the penetration depth in the simulations and in the star only vary by a factor of $\sqrt{f_\text{sim}/f_\star}$, which we expect to be a number not too far from unity.
IGW propagation {#sec:radiation-zone}
---------------
![\[fig:vr-spec-pcolor-full\]Frequency spectrum of radial velocity at the equator of model H6LD for all radii. The values were computed by sampling 8 points at different longitudes and averaging over the absolute value of the Fourier transform.](figures/spec-r-heat-1e6-rot-1e-5.pdf)
![\[fig:H6LD-spec-rads\]Frequency spectrum of $v_r$ in simulation H6LD at different radii integrated over all $l$. These are line plots of the spectra shown in Fig. \[fig:vr-spec-pcolor-full\] at several radii. The black dashed line is a theoretical prediction for the frequency dependence of $v_r$ from [@lecoanet2013a]. The light blue dashed line is a power-law fit to the simulation data in the range at $r=0.87\,R_\star$.](figures/H6LD-spec-rads)
\
![\[fig:amp-l-2\]Amplitude variation for different standing modes ($l=2$) from model H6LD computed for a single point at the equator. This corresponds to a vertical slice through the top panel of Fig. \[fig:vr-spec-pcolor-l-2-4\]. The blue shaded area is the convection zone according to the Schwarzschild criterion. The modes where identified by the number of nodes in the RZ.](figures/spec-amp-l-2)
Convective motions in the core generate IGWs at the CB, which propagate through the cavity of positive $N^2$ in the radiation zone. To visualize the excited frequencies and the change of the wave spectrum with radius, we compute the frequency spectrum of $v_r$ sampled at several longitudes around the stellar equator at all times for all radii and show it as a heat map in Fig. \[fig:vr-spec-pcolor-full\] for model H6LD. In the convection zone ($r\lesssim \SI{2e10}{cm}=0.14\,R_\star$) we note the presence of all frequencies with a clear dominance of the range below . This is reflective of the large range of timescales of convective motion (see Sect. \[sec:conv-spectra\]). In the radiation zone frequencies up to are excited. Low-frequency waves are strongly damped and only frequencies above reach the top of the simulation domain. This is qualitatively in agreement with linear theory, which predicts that lower frequency waves experience stronger damping [e.g., @kumar1999a]. The exact position of this lower cut-off depends on the value of $\kappa$ as well, which is why the present simulations cannot predict it quantitatively. Figure \[fig:H6LD-spec-rads\] shows line plots of the same spectrum at different radii. It shows that the spectrum in the RZ at low frequencies ($\lesssim \SI{20}{\micro Hz}$) does not reach the numerical noise level, as would be expected by the excessive numerical diffusion at this frequency, but turns flat at a higher value. The figure also indicates the expected frequency dependence of $v_r$ from theoretical work by @lecoanet2013a, which is $f^{-3.25}$ for the radial velocity of waves excited at a discontinuous $N$ profile. We see that the simulated spectrum is much flatter than this prediction, following $f^{0.8}$. The steep drop around for $r>0.8\,R_\star$ is due to the limit imposed by the [Brunt–Väisälä frequency]{} at these radii.
Strong vertical features are visible in Fig. \[fig:vr-spec-pcolor-full\]. These are peaks in the spectrum which are present at the same frequency at all radii in the radiation zone. This identifies them as standing waves. Their frequencies are determined by the cavity they resonate in and can be computed numerically using the stellar oscillation code GYRE [@townsend2013a].
It is hard to disentangle individual modes because the contributions of several wave numbers overlap, but due to the horizontal discretization of the simulations using spherical harmonics it is simple to extract a spectrum for particular $l$ and $m$ modes. The panels in Fig. \[fig:vr-spec-pcolor-l-2-4\] show the frequency spectrum for the modes $l=2$ and $l=4$. Here, the radial order of the individual standing modes can be clearly identified by the number of radial nodes. The strong mode at without any nodes is a *fundamental mode* or f mode. The other visible modes show an increasing number of nodes with decreasing frequency. This identifies them as g modes [e.g., @aerts2010a Sect. 3.5]. We computed expected mode frequencies with GYRE[^6] for comparison. They are labeled in the figure using the Eckart-Osaki-Scuflaire-Takata scheme [e.g., @aerts2010a], where negative numbers indicate g modes, 0 is the f mode, and positive numbers are p modes. In the case of $l=2$ we find quite good agreement for the g modes (at least up to $\text{g}_4$) and the f mode, especially considering that our 3D simulation has a slightly different resonant cavity due to the different equation of state and outer boundary compared to the 1D MESA model. As expected there are no p modes as the chosen set of equations (see Sect. \[sec:method\]) does not include the physics of sound waves. For $l=4$ (lower panel of Fig. \[fig:vr-spec-pcolor-l-2-4\]) the $\text{g}_1$, $\text{g}_2$, and $\text{g}_3$ modes match very well, while the identified f mode is within a few of the expected frequency of the $\text{p}_1$ mode according to GYRE. This is probably coincidental as the discrepancy between 3D hydrodynamics and GYRE gets even larger at higher wave numbers. The identification of modes by counting the number of nodes in the RZ is illustrated in Figure \[fig:amp-l-2\], which shows the radial change of $v_r$ amplitude of particular frequency components corresponding to the standing waves. The amplitudes were computed by projection on a complex phase angle of the Fourier transform at the radius with the maximum absolute value in the RZ. The $\mathrm{g_2}$ and $\mathrm{g_3}$ show nodes at the top of the overshooting region, which is ignored for the mode identification.
![\[fig:waveamp-l\]Wave amplitude in $v_r$ for the $l=6$ mode at three different frequencies from simulation H6LD. The dashed lines are the theoretical prediction using radiative damping and pseudomomentum conservation from Eq. . This curve uses the enhanced values of thermal diffusivity $\kappa$ from the simulation instead of the stellar values.](figures/waveamp-l-6)
Linear theory predicts amplification of waves moving along a decreasing density profile through pseudomomentum conservation [e.g., @buehler2009a]. At the same time thermal diffusivity damps the wave. @ratnasingam2019a give an expression for the linear wave amplitude based on @press1981a and @kumar1999a. The amplitude of the radial velocity follows $$\label{eq:wave-ampl}
v_r \propto \left(\frac{r_0}{r}\right)^{3/2} \sqrt{\frac{\rho_0}{\rho}} \left(\frac{N^2-\omega^2}{N_0^2 - \omega^2}\right)^{1/4} \exp(-\tau/2),$$ with $$\tau = \int_{r_0}^r dr \frac{\kappa \left[l(l+1)\right]^{3/2} N^3}{r^3 \omega^4} \sqrt{1-\frac{\omega^2}{N^2}},$$ using $\omega = 2 \pi f$ and the starting radius of wave propagation $r_0$ with its corresponding density $\rho_0$ and [Brunt–Väisälä frequency]{} $N_0^2$. We extract the amplitude of $v_r$ at several frequencies for a particular $l$ mode and show it together with the theoretical prediction from Eq. in Fig. \[fig:waveamp-l\]. We see that the waves generally follow amplification through the $\sqrt{\rho_0/\rho}$ term and are hardly affected by radiative damping, except for the low frequency case, as expected.
Generally the match between the GYRE predictions and data extracted from 3D hydrodynamics is quite promising, considering that both approaches make slightly different assumptions about the physics. Even with the high thermal diffusivity needed for the simulation we can see wave amplification. We might be able to observe wave breaking in future simulations which extend to regions closer to the surface at much lower densities.
Nature of the signal in the radiation zone {#sec:igw-identification}
------------------------------------------
![\[fig:wave-brunt\]High-frequency regime of the spectrum of $v_r$ in simulation H6E at all radii integrated over all $l$ components. The white line is the [Brunt–Väisälä frequency]{} $N/2\pi$. We see that the signal in the radiation zone is approximately limited to the region where $2\pi f<N$, with the notable exception of f modes (strong vertical features) going beyond that limit.](figures/wave-brunt)
Although a visualization of the temperature field in the radiation zone such as in Fig. \[fig:3d-temp\] suggests a wave nature of the flow field, a more rigorous analysis is needed to prove the motions are indeed IGWs excited close to the convective boundary or by nonlinear interaction in the RZ. IGWs are naturally limited to frequencies below the [Brunt–Väisälä frequency]{}, i.e. $\omega < N$ with $\omega=2\pi f$. In Fig. \[fig:wave-brunt\] we show the high-frequency part of the spectrum of $v_r$ at all radii. The white line in the figure indicates the local [Brunt–Väisälä frequency]{}. We see that the bulk of the signal in the RZ is constrained to the region $\omega < N$. Beyond this frequency there is a sharp drop in the amplitude which is consistent with IGW nature. A notable exception are the strong f modes (e.g. at ) going beyond that limit, which does not contradict this interpretation because these modes are not subject to the frequency limit.
In the locally Boussinesq but globally anelastic approximation IGWs follow the dispersion relation [e.g., @press1981a], $$\label{eq:dispersion-igw}
\frac{k_\perp}{k} = \frac{\omega}{N}.$$ Here, $k_\perp$ is the horizontal wave number, $k_r$ the radial wave number, and $k=\sqrt{k_\perp^2 + k_r^2}$ the magnitude of the total wave vector. We verify this relation for individual values of angular degree $l$ and frequencies because the resulting velocity field is a combination of many individual waves. The horizontal wave vector can easily be computed for a given $l$ by $$k_\perp = \frac{\sqrt{l(l+1)}}{r}.$$ The radial wave number is not as straightforward to derive because the wave length changes with radius as the [Brunt–Väisälä frequency]{} varies. An additional complication is that the wavelength becomes comparable to the stellar radius above $r \sim 0.5\,R_\star$, which makes an accurate determination very hard.
We determine the radial wavelength $\lambda_r$ for each individual frequency by finding the peaks of $v_r$ along a ray in the radial direction and calculating the difference between them. For this we employ the routine `signal.find_peaks` from the scipy Python package [@scipy], which finds isolated local extrema and is resilient to small numerical noise. As there are only very few wave cycles along the total radius of the star, we use a cubic spline to interpolate the wavelengths at every radial coordinate. Several other methods proved unsuccessful in this particular case: using a radial Fourier transform with a sliding window is inaccurate, as there are only few wave cycles per window; calculating the radial derivative of the phase of the spectrum works well except for regions where the phase is poorly defined when the amplitude is close to 0. This makes this method inapplicable to determining the wavelength in standing modes. The simple method of measuring the distance between peaks and interpolating the found wavelengths is the most robust.
 \
 
Using this method we calculate the radial wave number $k_r=2\pi/\lambda_r$ for every frequency at every radius to check how closely Eq. is fulfilled. The two panels in Fig. \[fig:dispersion\] show this for $l=2,4,10,20$. White regions indicate a match of the dispersion relation, red regions have a too large $k_\perp$, blue regions have a too small $k_\perp$. All components show no match in the very low frequency range ($\lesssim \SI{10}{\micro Hz}$), which is expected to be totally dominated by diffusion at all radii. Just above the overshooting region at $0.2\,R_\star$ we find excellent agreement at higher frequencies, which we interpret as waves being emitted from the convection zone over a large range of frequencies. Due to the increased thermal diffusivity needed for numerical reasons, low-frequency waves cannot propagate far into the RZ. This is evidenced by the increasing size of the non-IGW (red) region at the low-frequency end.
The higher $l$ values show a remarkable phenomenon. At $r\gtrsim 0.4 \, R_\star$ in the low-frequency region which should be completely dominated by damping ($f\lesssim\SI{70}{\micro Hz}$ for $l=10$) a signal appears which matches the dispersion relation. This cannot be explained by waves originating from the convective boundary because there are no waves of these frequencies present at lower radii. A plausible explanation is that these are generated by nonlinear interaction of low $l$ waves in the middle of the RZ. These secondary waves reach frequencies from .
The ability of the discretization to resolve IGWs is checked in this context as well. For given values of $k_\perp$ and $N$ we can calculate a frequency below which the radial IGW wavelength would not be resolved by at least 10 grid points in the radial direction. This frequency forms the lower limit for resolving IGWs at a given radius in this simulation. The limit is indicated as a magenta line in Fig. \[fig:dispersion\]. The lower limit is highest close to the convective boundary, where $N$ is lowest.
Another limit on wave resolution is imposed by diffusion. As a rough estimate for the minimum wave length of waves not dissipated by diffusion and viscosity we use, $$\label{eq:kappa-flim}
\max(\overline{\kappa}, \overline{\nu}) \sim \frac{\lambda^2}{\tau_\mathrm{d}} = \frac{(2\pi)^5 r^2 f^4}{N^3 l (l+1)},$$ with the IGW wavelength $\lambda=2\pi/k$ and the diffusion time $\tau_\mathrm{d} = \lambda_\perp/v_\mathrm{g}$. This uses the magnitude of the group velocity, $$v_g = \frac{\partial\omega}{\partial k} = \frac{r \omega^2}{N\sqrt{l(l+1)}}.$$ Solving this equation for $f$ leads to the yellow colored line in Fig. \[fig:dispersion\]. As expected, we can see that motions below this frequency largely do not fulfill the dispersion relation. This proves that the radial resolution in our simulations is sufficient to resolve waves with frequencies above in the most energy bearing wave numbers ($l \lesssim 5$).
![\[fig:dispersion-filter\]Fraction of energy in IGW motions in simulation H6LD after applying the filtering process from Eq. at different radii for angular degrees $l \le 10$. The energy computed from the filtered velocities $E_\mathrm{IGW}$ is compared to the original kinetic energy $E_\mathrm{kin}$ for these values of $l$. We do not include frequencies below the limit definitely dominated by diffusion (Eq. ). The line colors represent different error margins $C_\mathrm{cut}$ around the expected dispersion relation. The vertical dotted line is the boundary of the overshooting region determined in Sect. \[sec:conv-spectra\] at $0.695\,H_P$ beyond the original convection zone.](figures/dispersion-match-H6LD)
While this analysis gives a good overview of the frequencies, wave numbers, and radii where the dispersion relation is fulfilled well, it is also important to see what fraction of the kinetic energy is actually contained in the wave motions. To compute this we filter the Fourier transform of the velocity components $\hat{v}_r$, $\hat{v}_\theta$, and $\hat{v}_\phi$ to only include values at frequencies and radii, where $k_\perp/k$ is close to the IGW dispersion relation $\omega/N$. The filtered velocities are, $$\label{eq:dispersion-filter}
\hat{v}_{r, \theta, \phi}^\mathrm{filt} =
\begin{cases}
\hat{v}_{r, \theta, \phi} &\text{if $\frac{\omega}{N} \frac{1}{C_\mathrm{cut}} < \frac{k_\perp}{k} < \frac{\omega}{N} {C_\mathrm{cut}}$,}\\
0 &\text{otherwise.}
\end{cases}$$ The kinetic energy computed from these velocities is then identified as the energy in IGW motions $E_\mathrm{IGW}$. Figure \[fig:dispersion-filter\] shows the ratio of this energy to the unfiltered kinetic energy integrated over angular degrees $l \le 10$. We do not include frequencies below the limit definitely dominated by diffusion (Eq. ) in this analysis. As expected, we see almost no energy is in IGWs from the center up to the top of the overshooting region at $r=0.2\,R_\star$, which matches the previously determined position from Sect. \[sec:conv-spectra\] as indicated by the vertical dotted line. In the RZ it rises to 90% when applying the error margin $C_\mathrm{cut}=1.3$. The fraction of kinetic energy in IGWs drops beyond $r=0.4\,R_\star$. The main cause of this is the uncertainty in determining $\lambda_r$ at large radii, where $\lambda_r$ approaches $R_\star$ and our method of measuring the distance between peaks breaks down. Another reason is the growth of the low-frequency, red regions in Fig. \[fig:dispersion\], which arises because of the limited range of IGWs due to high numerical diffusion. At even higher radii, $r \gtrsim 0.7\,R_\star$, we notice an increase in the IGW energy fraction, which is likely due to the increased fraction of secondary waves, as discussed earlier.
This analysis makes us confident that the motions in the RZ are indeed of IGW nature to a significant fraction, with the exception of very low frequency motions dominated by numerical diffusion. We see evidence for secondary generation of waves within the RZ.
IGW surface signature {#sec:igw-surface}
---------------------
![\[fig:vr-spec-H6LD-surface\]Frequency spectrum of tangential velocity at the equator of model H6E close to the outer boundary of the simulation domain ($r=0.89R_\star$). Just the velocity in $\theta$ direction (i.e., aligned in southern direction at the equator) is used, as the $\phi$ velocity is subject to boundary artifacts. The 2D spectrum from [@rogers2013a] was scaled to match the 3D spectrum. It was started with an initially uniform rotation of using a different, but similar $3\,{\ensuremath{\mathrm{M}_\odot}}$ reference state. The spectrum of brightness variations of HD46966 is from CoRoT observations [@blomme2011a; @aerts2015a]. The empirical conversion factor between velocity and brightness variations is [@decat2002a; @aerts2015a]. ](figures/spec-H6LD-surface)
The observed brightness variations in O stars have been suggested as signatures of convectively excited IGWs [@aerts2015a]. Their spectrum is likely linked to that of tangential velocity close to the surface of the star [@decat2002a; @tkachenko2014a]. Figure \[fig:vr-spec-H6LD-surface\] shows a spectrum of latitudinal velocities from model H6LD (*orange line*). These are less affected by numerical influence from the boundary condition than the azimuthal velocities. This is compared to a spectrum obtained from 2D simulations of a $3\,{\ensuremath{\mathrm{M}_\odot}}$ star from @rogers2013a (*blue line*). We see the same low-frequency power excess in range from and a similar drop in amplitude below as in the 3D simulations. We also plot photometric observations [@blomme2011a] with an amplitude ratio of [@decat2002a; @aerts2015a] for comparison.
![\[fig:spec-H6LD-surface-T\]Frequency spectrum of temperature fluctuations $T$ from the reference state at the equator of model H6LD close to the outer boundary of the simulation domain ($r=0.89\;R_\star$). The different panels show different values of angular degree ($l$) as indicated.](figures/spec-H6LD-surface-T)
The same low-frequency power excess shows in the spectra of temperature fluctuations in Fig. \[fig:spec-H6LD-surface-T\]. It is expected to be the dominant cause of photometric variability in observations. In the spectrum integrated over all $l$ values (top panel) this excess makes it hard to distinguish individual excited mode frequencies, expect for one mode at (), which is part of the $l=2$ component. In the spectra for individual $l$ values, several modes can be identified, corresponding to those in Fig. \[fig:vr-spec-pcolor-l-2-4\]. This decomposition also makes it clear that the low-frequency power excess is a combination of the power excesses in different $l$ components, each contributing to a small frequency range. The lack of signal at low-frequencies in numerical simulations is due to the high numerical diffusivity and not expected to be physical. The amplitude of the waves is expected to increase as they propagate from $r=0.89\;R_\star$, where the spectrum was computed, to the surface. According to pseudomomentum conservation it should increase by a factor of 380.
The low-frequency power excess is also found in observations of stars with a convective core [@bowman2019a]. In the simulations it is caused by the high density of high radial order, low-$l$ g modes and because most energy in the CZ is at low $l$ values (see Fig. \[fig:k-spectrum\]). The drop in amplitude below on the other hand is in disagreement with observed photometry [@blomme2011a; @aerts2017a; @aerts2018a; @bowman2019a]. This disagreement is likely caused by the increased thermal diffusivity in the simulations (both in 2D and 3D) which damps low-frequency waves more strongly than in stellar interiors. Another possibility is the lack of differential rotation in our 3D model. @rogers2013a found that differential rotation between core and envelope introduces a significant low frequency component in the spectrum. At higher frequencies above the 2D simulations drop more slowly than the 3D simulations and show many excited modes. This is possibly due to the lack of wave breaking brought about by the high thermal diffusivity and viscosity needed in our present set of 3D simulations.
Conclusions and Outlook {#sec:conclusions}
=======================
We showed the first 3D simulations of convection in the core of an intermediate-mass star, with a convective core and radiative envelope, that also include a large part of the radiation zone (RZ). The simulations using the anelastic equations (i.e., removing the physics of sound waves) and a spectral discretization using spherical harmonics were run using a realistic reference state from the stellar evolution code MESA. For numerical reasons the simulations were run with increased thermal and viscous diffusivity. To compensate for the increased wave damping this produces we increased the luminosity of the star causing higher velocities in the convective core. We do this in the hope that wave velocities at the surface of the star are more realistic.
We see wave patterns in the RZ, which are identified to be standing g and f modes with frequencies similar to those predicted by the oscillation code GYRE. Although there are differences, they are not of the sort predicted in @brown2012a and are dependent on the $l$ and $m$ values of the spherical harmonics. These differences are likely due to slightly different physics (e.g., equation of state, outer boundary condition) and changes in the temperature profile at the top of the convection zone due to overshooting.
Apart from the standing modes the simulations also show a continuous signal in the RZ between frequencies of approximately and . An analysis of the dispersion relation (see Fig. \[fig:dispersion\]) identifies the physical mechanism as IGWs. The decline of this continuous spectrum with frequency is markedly smaller than theoretically predicted values for excitation purely due to convective eddies (see Fig. \[fig:H6LD-spec-rads\]).
An analysis of the kinetic energy distribution over spherical harmonic degree $l$ shows a spectrum which peaks at a low value of $l$ and then declines with a power law with an exponent in the range from in the inertial range. This is closer to the theoretical value for Bolgiano–Obukhov scaling ($-2.2$) of buoyancy dominated convection than to the Kolmogorov value ($-1.6$) of isotropic turbulence. We do not have enough information to get a conclusive answer on the realized scaling in convective stellar cores. This should be studied further in detailed simulations of just the core. These slopes are measured at the top of the CZ, which is subject to convective overshooting, and therefore do not directly match the IGW spectrum. Yet they show what energy is available for wave excitation at a given frequency. The slope in the inertial range is similar to that observed in other 2D or 3D simulations of core convection [@rogers2013a; @augustson2016a].
The broken power law structure of the frequency spectra of kinetic energy above the convection zone is similar to those obtained in the 2D simulations of @rogers2013a, suggesting the mechanism driving the bulk of this spectrum does not fundamentally change with dimensionality. It is likely that bulk Reynolds stresses induced by convective eddies contribute more in higher Reynolds number flows, but this would still only affect the low frequencies ($f<f_\text{TO}$) and hence, have little impact on angular momentum transport or mixing within the bulk of the RZ [@shiode2013a; @kumar1999a; @lecoanet2013a].
Excitation by plume penetration is obviously involved as can be seen in the temperature and velocity fields (see Fig. \[fig:H6R10-slice\]). It can explain the excitation of higher frequency waves and the extracted distribution of plume frequencies fits a large part of the simulation spectrum. One may argue that the plume penetration depths, and hence, frequencies generated are too large. However, at least in the theory by @zahn1991a, this penetration depth scales like the velocities cubed divided by the total flux, a number which is the same in the simulations as it is in the star. The production of high-frequency waves is extremely important for explaining the photometrically observed brightness variations at high frequencies (see Fig. \[fig:vr-spec-H6LD-surface\]). They are likely underestimated in our simulations due to high dissipation preventing wave breaking.
Stochastic brightness variations caused by velocity and temperature fluctuations at the stellar surface have been inferred to be caused by IGWs in massive stars [@aerts2015a; @aerts2017b; @aerts2018a; @bowman2019a]. We extracted frequency spectra of these quantities from the simulation close to the stellar surface. General features are a low-frequency power excess and the presence of standing modes at low $l$ harmonics. This is in agreement with the findings of 2D simulations [@rogers2013a], which match observations in the power bearing range, but lack both amplitudes at very low frequencies (due to excessive radiative damping) and high frequencies (possibly due to lack of wave breaking from overdamped waves).
The simulations presented in this article show the feasibility of hydrodynamic modeling of convectively excited IGWs and their propagation through a large part of the radiative zone using a consistent numerical treatment. In future work employing more computational resources the limitations forcing us to use unphysically high diffusivities and luminosities can hopefully be overcome to achieve more realistic wave amplitudes throughout the interior and at the surface, and hence more realistic angular momentum transport. More realistic physical parameters in the simulations combined with coverage of a wider range of stellar models will also allow us to make quantitative predictions of the expected signature of IGWs in asteroseismological observations.
The general similarity of our results with those of previous 2D simulations encourage us to consider those results with less reservations due to their dimensionality and to use 2D simulations as a tool in the future to quickly cover a wider parameter range in models than is possible with 3D simulations.
Support for this research was provided by STFC grant ST/L005549/1 and NASA grant NNX17AB92G. MGP and DMB received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No670519: MAMSIE). VP acknowledges support from the European Research Council through ERC grant SPIRE 647383. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. This research made use of the Rocket High Performance Computing service at Newcastle University. The authors thank C. Pin[ç]{}on and M. Rieutord for helpful comments.
[^1]: They call this set of equations the RG equations.
[^2]: The MESA version used was SVN revision number 10000.
[^3]: <https://www.mas.ncl.ac.uk/~npe27/igw3d/>
[^4]: Models H7E and H7E-HR were run with an earlier version of the stellar model, which reached 95% of stellar radius. There is no qualitative change in the wave spectra of these models.
[^5]: The pressure scale height is defined as $H_P=-\partial r / \partial \ln P$.
[^6]: We used version 5.1 from the GYRE web page.
|
---
address: |
University of Library and Information Science\
1-2 Kasuga Tsukuba 305-8550, JAPAN\
[{fujii, ishikawa}@ulis.ac.jp]{}
title: 'A Novelty-based Evaluation Method for Information Retrieval'
---
Introduction {#sec:introduction}
============
In information retrieval (IR) research, the notion of precision and recall have commonly been used to evaluate the empirical performance of systems [@keen:ipm-92; @salton:ipm-92]. Precision is the ratio of the number of relevant documents retrieved by a system under evaluation, compared to the total number of documents retrieved by the system. On the other hand, recall is the ratio of the number of relevant documents retrieved by the system, compared to the total relevant documents in a given benchmark test collection.
In other words, the precision/recall-based evaluation method regards all the relevant documents as equally important or informative for the user, and thus highly values systems that retrieve as many relevant documents as possible, with little noise.
However, in the real world, where a number of IR systems are available, for example, on the World Wide Web, it is often the case that the user has already read some of relevant documents using other systems. Thus, systems that always retrieve relevant documents similar to those retrieved by ubiquitous systems have little practical utility. In addition, meta search systems, which integrate document sets retrieved by more than one system, are less effective, in the case where individual systems retrieve similar documents.
In view of these problems, our proposed IR evaluation method favors systems that retrieve more [*novel*]{} documents, that is, relevant documents which cannot be retrieved by other existing systems.
From a different perspective, our evaluation method is also effective in producing test collections. The pooling method [@voorhees:sigir-98], which has commonly been used to produce test collections, requires a variety of participating systems. However, in the case where most participating systems adopt similar techniques, it is not feasible to collect a sufficient “pool” (i.e., a set of candidates for relevant documents). Our evaluation method is expected to promote a development of IR systems with various concepts, and therefore resolve the above problem.
Section \[sec:measure\] formalizes the evaluation measure based on the novelty of documents, and Section \[sec:case\_study\] applies this measure to evaluate IR systems that participated in the IREX workshop [@sekine:irex-99].
Formalizing the Measure {#sec:measure}
=======================
Instead of the notion of precision and recall, we propose as a new evaluation measure the utility of system $x$ with respect to relevant document $d$, . This measure denotes the extent to which $x$ contributes to providing the user with $d$, for a given query. Note that in this paper, $d$ generally refers to a [*relevant*]{} document.
From an information theoretical point of view, we calculate as the ratio of the probability that the user reads document $d$ by using system $x$, , compared to the probability that the user reads $d$ by using another system (i.e., even without using $x$), , as shown in Equation [(\[eq:udx\])]{}. $$\label{eq:udx}
U_{d}(x) = \log\frac{\textstyle P(D=d|S=x)}{\textstyle P(D=d)}$$ In the case where system $x$ adopts a ubiquitous retrieval technique, the value of becomes similar to that of , and thus the utility of $x$ becomes small. On the other hand, the utility of $x$ becomes greater as the number of [*novel* ]{} relevant documents provided by $x$ increases.
We then calculate the [*total*]{} utility of $x$, $U(x)$, by summing up $U_{d}(x)$’s of all the relevant documents for the query, as shown in Equation [(\[eq:ux\])]{}. $$\label{eq:ux}
U(x) = \sum_{d} U_{d}(x)$$ To sum up, our evaluation method favors systems with greater .
In Equation [(\[eq:udx\])]{}, is the summation of ’s for existing systems, averaged by the probability that the user utilizes system $y$, . Thus, given a set of existing system excluding $x$, $E$, we calculate as in Equation [(\[eq:pd\])]{}. $$\begin{aligned}
\label{eq:pd}
\begin{array}{lll}
P(D=d) & = & {\displaystyle \sum_{y\in E}P(D=d|S=y)\cdot P(S=y)} \\
\noalign{\vskip 2ex}
& \approx & {\displaystyle \sum_{y\in
E}P(D=d|S=y)\cdot\frac{\textstyle 1}{\textstyle |E|}}
\end{array}\end{aligned}$$ Here, note that we assume uniformity with respect to .
Finally, the crucial content is the way to estimate , i.e., the probability that the user reads document $d$ by using system $x$. It can safely be assumed that the user always reads the top document, $d_1$, and thus $P(D=d_{1}|S=x)$ always takes 1. However, the probability that the user reads remaining documents becomes smaller according to their ranking.
Given $N$ documents sorted according to their relevance degree, in descending order, the user can choose a threshold for the ranking (i.e., the boundary until which he/she continues to read) out of $N$ choices. Consequently, documents ranked lower than the threshold will be discarded.
In other words, we can calculate as the probability that the user chooses a threshold equal to or greater than the ranking of $d$, as in Equation [(\[eq:pdx\])]{}. $$\label{eq:pdx}
\begin{array}{lll}
P(D=d|S=x) & = & {\displaystyle \sum_{i = r_{x,d}}^{N}
\frac{\textstyle 1}{\textstyle N}} \\
\noalign{\vskip 2ex}
& = & \frac{\textstyle N - r_{x,d} + 1}{\textstyle N}
\end{array}$$ Here, $r_{x,d}$ is the ranking of document $d$ determined by system $x$.
A Case Study using the IREX Collection {#sec:case_study}
======================================
Our concern in this section is to investigate the characteristic of our evaluation method. For this purpose, we targeted IR systems participated in the IREX workshop [@sekine:irex-99], and compared the result obtained based on our newly proposed evaluation method, with that based on the precision/recall. We also investigated reasons behind the difference between those two results, if any.
Overview of the IREX Collection {#subsec:irex}
-------------------------------
The IREX collection was produced through the IREX workshop [@sekine:irex-99], which consists of TREC-style IR and MUC-style named entity (NE) tasks for Japanese.[^1] Hereafter, the IREX collection/workshop refers solely to that related to the IR task.
The IREX collection consists of 30 queries, 211,853 articles collected from two years worth of “Mainichi Shimbun” newspaper articles [@mainichi:94-95],[^2] relevance assessment for each query, retrieval results of 22 participating systems, and technical details of each system.
Each query consists of the ID, description and narrative. While descriptions are usually phrases to briefly express the topic, narratives consist of several sentences and synonyms associated with the topic. Figure \[fig:query\] shows an example query in the SGML form (translated into English by one of the organizers of the IREX workshop).
> <TOPIC>\
> <TOPIC-ID>1001</TOPIC-ID>\
> <DESCRIPTION>Corporate merging</DESCRIPTION>\
> <NARRATIVE>The article describes a corporate merging and in the article, the name of companies have to be identifiable. Information including the field and the purpose of the merging have to be identifiable. Corporate merging includes corporate acquisition, corporate unifications and corporate buying.</NARRATIVE>\
> </TOPIC>
Relevance assessment was performed based on the pooling method [@voorhees:sigir-98]. That is, candidates for relevant documents were first pooled using the 22 participating systems. Thereafter, for each candidate document, human experts assigned one of three ranks of relevance, i.e., “relevant”, “partially relevant” and “irrelevant”. The average number of documents pooled for each query is 2,105, among which the number of relevant and partially relevant documents are 68 and 116, respectively.
Each retrieval result consists of the top 300 articles submitted in the same form as used in the TREC.[^3] For each of the 22 results, the TREC evaluation software was used to investigate the performance (e.g., non-interpolated average precision). Figure \[fig:trec\] shows a fragment of the retrieval result obtained with one of the participating systems, which consists of the query ID, dummy field, article ID, ranking of the article, relevance degree computed by the system, and system ID.
------ --- ----------- --- ---------- ------
1007 0 940228106 1 0.306856 1106
1007 0 940110130 2 0.246505 1106
1007 0 950106119 3 0.237173 1106
1007 0 940131126 4 0.236115 1106
1007 0 940614009 5 0.223313 1106
1007 0 940614002 6 0.222998 1106
1007 0 941107114 7 0.217324 1106
1007 0 940428222 8 0.215979 1106
------ --- ----------- --- ---------- ------
[Question]{} [Answers]{}
---------------------------- --------------------------------------------------------------------------------
query information used only description (8), description+narrative (14)
indexing method word (9), n-gram (3), word+character (2), character (1), syntactic phrase (1),
statistical phrase (1)
proper noun identification yes (5)
query expansion local feedback (2), use of a thesaurus (2)
retrieval method vector space model (13), probabilistic model (4), latent semantic indexing (1)
It should be noted that using relevance assessment and retrieval results for each system, we can easily calculate in Equation [(\[eq:pdx\])]{}, which is the central issue in estimating our evaluation measure.
Technical details of participating systems were collected from questionnaires answered by each participant, where questions ranged from retrieval algorithms used to execution time. Although several questions are relatively vague, a number of questions are effective to characterize each system.
Table \[tab:spec\] shows representative questions in terms of retrieval accuracy. In this table, the number of answers are indicated in parentheses. However, answers classified as “no”, “unknown” and “etc.” are not shown. Roughly speaking, most systems adopted the word-based indexing and vector space model combined with TF$\cdot$IDF term weighting.
On the other hand, note that in the IREX workshop, the correspondence between system IDs and participants is not available to the public. Additionally, several participants did not have oral presentations and papers in the proceedings. Consequently, for some systems it is difficult to obtain sufficient technical details.
For example, although most participants answered “TF$\cdot$IDF” for the question about term weighting method, it is not possible to identify the exact formula used, out of a number of variants [@salton:ipm-88; @zobel:sigir-forum-98], for several systems.
Experimentation {#subsec:experiment}
---------------
As explained in Section \[subsec:irex\], the 22 IREX participating systems have already been ranked based on the conventional precision/recall, using the TREC evaluation software.
Thus, we re-evaluated the 22 systems based on our evaluation method, and compared results derived from different evaluation methods. To put it more precisely, we conducted 22 trials in each of which a different system was under evaluation and the rest were regarded as existing systems. That is, the former and latter correspond to $x$ and $E$ in Section \[sec:measure\], respectively.
Note that in this evaluation, we did not regard “partially relevant” documents as relevant ones, because interpretation of “partially relevant” is not fully clear to the authors.
Table \[tab:all\_A\] compares rankings obtained based on non-interpolated average precision and the utility factor we proposed in this paper. Table \[tab:qbq\_A\] compares rankings obtained with two evaluation methods on a query-by-query basis, where we show solely the difference of rankings for enhanced readability. Since in the IREX collection, every query ID consists of four digits stating with “10”, we simply show the remaining two digits in Table \[tab:qbq\_A\].
System ID [Avg. Precision]{} [Utility]{} [Difference]{}
----------- -------------------- ------------- ----------------
1144b 2 1 +1
1135a 3 2 +1
1144a 1 3 -2
1135b 4 4 0
1103b 5 5 0
1106 17 6 +11
1145b 16 7 +9
1122b 7 8 -1
1103a 10 9 +1
1128b 9 10 -1
1142 6 11 -5
1122a 8 12 -4
1110 11 13 -2
1133a 19 14 +5
1133b 18 15 +3
1128a 12 16 -4
1120 14 17 -3
1145a 13 18 -5
1112 15 19 -4
1146 20 20 0
1132 22 21 +1
1126 21 22 -1
: Comparison of rankings obtained based on non-interpolated average precision and utility factor.[]{data-label="tab:all_A"}
----------- ---- ---- ----- ---- ----- ----- ----- ----- ----- ---- ---- ---- ---- ---- ---- ----- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
System ID 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
1103a 8 -7 14 0 8 3 3 -14 1 13 5 -3 0 -4 -2 3 -6 -3 6 1 -2 13 2 14 -3 -5 -7 -2 -3 3
1103b -2 -5 6 4 -1 -3 -6 -9 4 -5 -1 1 -3 -2 -1 8 0 -2 1 -2 -1 7 1 -3 -5 -1 -6 -3 -2 5
1106 8 -4 -9 -2 9 -2 7 11 5 -1 -2 -4 5 4 0 -3 -3 2 0 0 -1 -1 1 2 1 2 0 2 17 0
1110 6 -1 -4 4 -1 9 -4 -10 -1 0 4 -2 -5 -1 0 3 0 -2 -1 0 0 16 13 -1 -3 -3 8 1 3 -2
1112 -2 -5 0 0 -5 3 -3 1 -11 0 5 -5 12 -2 -1 5 -3 -4 -3 -1 -1 -4 -6 -4 3 1 -4 -2 0 0
1120 1 -2 -2 -1 0 -3 4 -8 -1 0 5 -2 7 1 0 5 0 2 0 2 0 -3 -1 -1 2 2 6 5 -1 0
1122a -2 2 -2 -7 -5 5 -5 -11 -1 -5 1 8 -1 -6 -2 -8 1 1 0 -1 4 -4 1 -1 -3 -1 3 -2 -3 -1
1122b -5 0 -8 1 0 -8 1 -5 -9 -5 0 -2 -3 -6 1 -4 4 0 -2 1 7 -3 -2 -4 -4 0 6 0 -1 -2
1126 0 4 -10 0 0 -2 0 3 -1 -1 -1 1 -1 0 0 0 0 0 0 1 1 0 -2 -3 0 0 -3 -1 0 0
1128a -1 -1 4 -2 -3 0 3 -6 -8 -1 -3 4 2 9 1 -13 0 6 2 -1 0 -2 1 0 -1 1 4 -4 0 4
1128b -2 14 -4 -4 -7 -5 11 9 -2 -2 -5 4 -1 3 -2 -13 -1 1 2 2 0 1 0 -5 1 -1 0 -4 0 -1
1132 0 16 -9 2 0 0 0 12 21 0 0 10 0 8 15 0 -4 0 0 0 0 0 2 0 0 -1 0 13 0 0
1133a -2 -2 -4 0 3 2 3 15 11 1 -5 -1 1 7 -1 3 4 1 4 1 0 -2 -1 1 4 7 -1 0 0 1
1133b -3 -2 -4 2 3 1 11 15 3 0 -4 2 0 5 1 6 5 0 3 1 0 -3 -5 -1 10 3 -2 -2 1 -1
1135a -1 -2 9 -2 4 -11 -6 4 9 2 -6 -4 -1 -1 -1 -2 -3 -1 -1 -1 0 -2 -2 0 1 -1 -1 0 -1 -3
1135b 2 0 6 -1 -12 -13 -6 1 2 0 -3 1 -5 -6 -3 -1 -3 -2 0 -1 -4 -7 -2 0 0 -2 -1 -7 -2 0
1142 -4 -1 10 0 -5 -1 -7 -14 -7 -3 -2 -3 -4 -7 -5 -2 4 -3 -3 -1 -2 -2 -2 -5 2 -6 -7 -6 -1 -4
1144a -2 -1 -1 3 -1 5 -16 -9 -3 5 1 -6 -1 -2 0 6 -1 -2 -2 -3 0 0 -2 -1 0 -4 7 2 -1 -1
1144b -2 3 -1 2 -2 5 -16 -5 -2 5 2 -5 2 -2 1 5 -3 1 1 -1 0 0 -5 -2 0 1 4 2 -1 2
1145a 0 -4 -7 -4 -5 -1 5 11 -2 -1 -1 -3 -1 -1 -1 1 8 -3 -5 5 -1 -4 5 6 -2 2 -4 -3 1 -3
1145b 3 -3 -5 5 13 7 12 13 -5 -1 -2 8 -3 4 0 2 1 1 -2 0 -1 0 5 6 -2 7 0 13 -5 0
1146 0 1 21 0 7 9 9 -4 -3 -1 12 1 0 -1 0 -1 0 7 0 -2 1 0 -1 2 -1 -1 -2 -2 -1 3
----------- ---- ---- ----- ---- ----- ----- ----- ----- ----- ---- ---- ---- ---- ---- ---- ----- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
Discussion {#subsec:discussion}
----------
Looking at Table \[tab:all\_A\], one may notice that rankings of systems “1106”, “1145b”, “1133a” and “1133b” were significantly improved within our evaluation method. Thus, we investigated properties that characterize each of those four systems, in a comparison with other systems.
First, we found that “1106” adopted a relatively simple implementation, while most systems used more elaborate ones. To put it more precisely, morphological analysis was performed, and nouns/verbs were extracted for a word-based indexing. For term weighting, a TF$\cdot$IDF formula as in Equation [(\[eq:tf\_idf\])]{} was used, while most systems used different methods, such as the logarithmic TF formulation as in Equation [(\[eq:log\_tf\_idf\])]{} and one proposed by Robertson and Walker . $$\label{eq:tf_idf}
f_{t,d}\cdot\log\frac{\textstyle N}{\textstyle n_{t}} \\$$ $$\label{eq:log_tf_idf}
(1 + \log f_{t,d})\cdot\log\frac{\textstyle N}{\textstyle n_{t}}$$ Here, $f_{t,d}$ denotes the frequency that term $t$ appears in document $d$, and $n_{t}$ denotes the number of documents containing term $t$. $N$ is the total number of documents in the collection.
Second, “1145b” conducted a query expansion [@qiu:sigir-93], while a few systems used query expansion (e.g., one based on a thesaurus). In addition, a term weighing method based on mutual information between two terms was introduced. Possible rationales behind this method include that two terms frequently co-occur are effective to characterize the domain of documents, and are thus assigned with greater term weights.
Third, “1133a” and “1133b” also used domain knowledge for term weighting. However, unlike the case of “1145b”, they regarded pages of news articles as domain. In practice, a greater weight is assigned to terms whose distribution varies more strongly depending on the page, because they are expected to characterize the domain. On the other hand, terms commonly appear in more pages are assigned with a lesser weight.
To sum up, our novelty-based evaluation revealed the effectiveness of those properties above, specifically term weighting methods introduced in “1145b”, “1133a” and “1133b”, which were overshadowed or underestimated within the precision/recall-based evaluation.
We devote a little space to consider Table \[tab:qbq\_A\] for further investigation. We arbitrarily regarded improvements above seven as significant, and focused solely on systems with relatively many significant improvements, that is, “1103a” and “1132”. Although “1145b” is associated with the same number of significant improvements as “1132”, we previously discussed system “1145b” above.
We found that “1103a” is one of five systems that conducts a proper noun identification, and that five of six queries where “1103a” achieved significant improvements are directly or indirectly associated with proper nouns.
Samples of query descriptions directly and indirectly related to proper nouns include “1016: Nick Price (a golfer)” and “1011: arrest of suspects of robbery in the [*Kanto*]{} region”, respectively. Note that in the latter (indirect) case, Japanese prefectures within the “[*Kanto*]{}” region, which are not explicitly described in the query (e.g., “[*Tokyo*]{}” and “[*Kanagawa*]{}”), must be identified in news articles.
Finally, “1132” is the only system that used Latent Semantic Indexing (LSI), which is an extension of the vector space model, so as to retrieve relevant documents including no common terms in a given query. While as shown in Table \[tab:all\_A\], “1132” had the lowest ranking in terms of the average precision, our evaluation method indicated that in many cases (queries) an LSI-based method is expected to retrieve relevant documents that other types of methods fail to retrieve.
Conclusion {#sec:conclusion}
==========
Evaluation methods based on precision and recall have long been used in information retrieval (IR) research, where systems that retrieve as many relevant documents as possible are usually highly valued.
However, given the fact that a number of retrieval systems resembling one another are available to the public (not only in laboratories), it is valuable to retrieve relevant documents that can never be retrieved by those existing systems. This notion is also true in various contexts that require a variety of IR systems, such as meta search systems and the pooling method in producing IR test collections.
In consideration of these factors, we proposed a new evaluation method for IR, which favors systems that retrieve more novel documents, i.e., relevant documents that many systems fail to retrieve. To realize this notion, we estimated the utility of a system in question by comparing the probability that the user reads relevant documents by using the system, and the probability that the user can read those documents even without using the system.
We also applied our evaluation method to the 22 systems that participated in the IREX workshop, and identified several effective techniques that have been underestimated in the conventional precision/recall-based evaluation method.
E. Michael Keen. 1992. Presenting results of experimental retrieval comparisons. , 28(4):491–502.
. 1994-1995. Mainichi shimbun [CD-ROM]{} ’94-’95. (In Japanese).
Y. Qiu and H. Frei. 1993. Concept based query expansion. In [*Proceedings of the 16th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval*]{}, pages 160–169.
S. E. Robertson and S. Walker. 1994. Some simple effective approximations to the 2-poisson model for probabilistic weighted retrieval. In [*Proceedings of the 17th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval*]{}, pages 232–241.
Gerard Salton and Christopher Buckley. 1988. Term-weighting approaches in automatic text retrieval. , 24(5):513–523.
Gerard Salton. 1992. The state of retrieval system evaluation. , 28(4):441–449.
Satoshi Sekine and Hitoshi Isahara. 1999. project overview. In [*Proceedings of the IREX Workshop*]{}, pages 7–12.
Ellen M. Voorhees. 1998. Variations in relevance judgments and the measurement of retrieval effectiveness. In [*Proceedings of the 21st Annual International ACM SIGIR Conference on Research and Development in Information Retrieval*]{}, pages 315–323.
Justin Zobel and Alistair Moffat. 1998. Exploring the similarity space. , 32(1):18–34.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank organizers and participants of the IREX workshop for their support with the IREX collection.
[^1]: [http://cs.nyu.edu/cs/projects/proteus/irex/\
index-e.html]{}
[^2]: Practically speaking, the IREX collection provides only article IDs, which corresponds to articles in Mainichi Shimbun newspaper CD-ROM’94-’95. Participants must get a copy of the CD-ROMs themselves.
[^3]: [http://trec.nist.gov/pubs.html]{}
|
---
abstract: 'We introduce the concept of *sos-convex* Lyapunov functions for stability analysis of both linear and nonlinear difference inclusions (also known as discrete-time switched systems). These are polynomial Lyapunov functions that have an algebraic certificate of convexity and that can be efficiently found via semidefinite programming. We prove that sos-convex Lyapunov functions are universal (i.e., necessary and sufficient) for stability analysis of switched linear systems. We show via an explicit example however that the minimum degree of a convex polynomial Lyapunov function can be arbitrarily higher than a non-convex polynomial Lyapunov function. In the case of switched *nonlinear* systems, we prove that existence of a common non-convex Lyapunov function does *not* imply stability, but existence of a common convex Lyapunov function does. We then provide a semidefinite programming-based procedure for computing a full-dimensional subset of the region of attraction of equilibrium points of switched polynomial systems, under the condition that their linearization be stable. We conclude by showing that our semidefinite program can be extended to search for Lyapunov functions that are pointwise maxima of sos-convex polynomials.'
author:
- 'Amir Ali Ahmadi and Raphaël M. Jungers [^1]'
bibliography:
- 'pablo\_amirali.bib'
title: |
**SOS-Convex Lyapunov Functions and\
Stability of Difference Inclusions**
---
Difference inclusions, switched systems, nonlinear dynamics, convex Lyapunov functions, algebraic methods in optimization, semidefinite programming.
Introduction
============
The most commonly used Lyapunov functions in control theory, namely the quadratic ones, are convex functions. This convexity property is not always purposefully sought after; it is simply an artifact of the nonnegativity requirement of Lyapunov functions, which for quadratic forms coincides with convexity. If one however seeks Lyapunov functions that are polynomial functions of degree larger than two (for instance, for improving some sort of performance metric), then convexity is no longer implied by the nonnegativity requirement of the Lyapunov function (consider, e.g., the polynomial $x_1^2x_2^2$). In this paper we ask the following question: what is there to gain (or to lose) by requiring that a polynomial Lyapunov function be convex? We also present a computational methodology, based on semidefinite programming, for automatically searching for convex polynomial Lyapunov functions.
Our study of this question is motivated by, and for the purposes of this paper exclusively focused on, the stability problem for difference inclusions, also known as discrete time switched systems. We are concerned with an uncertain and time-varying map $$x_{k+1}=\tilde{f_k}(x_k), \label{eq:switched.nonlinear.system}$$ where $$\tilde{f_k}(x_k)\in conv\{f_1(x_k),\ldots,f_m(x_k)\}.\label{eq:ftilda=conv}$$ Here, $f_1,\ldots,f_m: \mathbb{R}^n\rightarrow\mathbb{R}^n$ are $m$ different (possibly nonlinear) continuous maps with $f_i(0)=0$, and $conv$ denotes the convex hull operation. The question of interest is (local or global) *asymptotic stability under arbitrary switching*. This means that we would like to know whether the origin is stable in the sense of Lyapunov (see [@Khalil:3rd.Ed] for a definition) and attracts all initial conditions (either in a neighborhood or globally) for *all* possible values that $\tilde{f}_k$ can take at each time step $k$.
The special case of this problem where the maps $f_1,\ldots,f_m$ are *linear* has been and continues to be the subject of intense study in the control community, as well as in the mathematics and computer science communities [@jsr-toolbox; @morris-ergodic; @cfbousch; @Shorten05stabilitycriteria; @jungers_lncis; @LeeD06; @bcv12; @liberzon-cdc10]. A switched linear system in this setting is given by $$\label{eq:switched.linear.system}
x_{k+1}\in conv\{A_ix_k\}, \quad i=1,\ldots,m,$$ where $A_1,\ldots,A_m$ are $m$ real $n\times n$ matrices. Local (or equivalently global) asymptotic stability under arbitrary switching of this system is equivalent to the *joint spectral radius* of these matrices being strictly less than one.
The [*joint spectral radius*]{} of a set of matrices ${{\cal{M}}}$ is defined as $$\label{eq:JSR}
\rho({{\cal{M}}})=\lim_{k\rightarrow \infty} \
\max_{A_1,\dots,A_k\in {{\cal{M}}}}||A_1\dots A_k||^{1/k},$$ where $\|\cdot\|$ is any matrix norm on $\mathbb{{{\mathbb R}}}^{n \times n}.$
Deciding whether $\rho<1$ is notoriously difficult. No finite time procedure for this purpose is known to date, and the related problems of testing whether $\rho \leq 1$ or whether the trajectories of (\[eq:switched.linear.system\]) are bounded under arbitrary switching are known to be undecidable [@BlTi3]. On the positive side however, a large number of sufficient conditions for stability of such systems are known. Most of these conditions are based on the numerical construction of special classes of Lyapunov functions, a subset of which enjoy theoretical guarantees in terms of their quality of approximation of the joint spectral radius [@GP11; @Ando98; @Pablo_Jadbabaie_JSR_journal; @protasov-jungers-blondel09; @jungersguglielmicicone12].
It is well known that if the switched linear system (\[eq:switched.linear.system\]) is stable[^2], then it admits a common *convex* Lyapunov function, in fact a norm [@jungers_lncis]. It is also known that stable switched linear systems admit a common *polynomial* Lyapunov function [@Pablo_Jadbabaie_JSR_journal]. It is therefore natural to ask whether existence of a common *convex polynomial* Lyapunov function is also necessary for stability. One would in addition want to know how the degree of such convex polynomial Lyapunov function compares with the degree of a non-convex polynomial Lyapunov function. We address both of these questions in this paper.
It is not difficult to show (see [@jungers_lncis Proposition 1.8]) that stability of the linear inclusion (\[eq:switched.linear.system\]) is equivalent to stability of its “corners”; i.e. to stability of a switched system that at each time step applies one of the $m$ matrices $A_1,\ldots,A_m$, but never a matrix strictly inside their convex hull. This statement is no longer true for the switched nonlinear system in (\[eq:switched.nonlinear.system\])-(\[eq:ftilda=conv\]); see Example \[ex:nonconvex.fails\] in Section \[subsec:nl.global\] of this paper. It turns out, however, that one can still prove switched stability of the entire convex hull by finding a common *convex* Lyapunov function for the corner systems $f_1,\ldots,f_m$. This is demonstrated in our Proposition \[prop:convex.lyap\] and Example \[ex:convex.lyap\], where we demonstrate that convexity of the Lyapunov function is important in such a setting.
Such considerations motivate us to seek efficient algorithms that can automatically search over all candidate convex polynomial Lyapunov functions of a given degree. This task, however, is unfortunately intractable even when one restricts attention to quartic (i.e., degree-four) Lyapunov functions and switched linear systems. See our discussion in Section \[section:sos-convex\]. In order to cope with this issue, we introduce the class of *sos-convex* Lyapunov functions (see Definition \[def:sos-convex\]). Roughly speaking, these Lyapunov functions constitute a subset of convex polynomial Lyapunov functions whose convexity is certified through an explicit algebraic identity. One can search over sos-convex Lyapunov functions of a given degree by solving a single semidefinite program whose size is polynomial in the description size of the input dynamical system. The methodology can directly handle the linear switched system in (\[eq:switched.linear.system\]) or its nonlinear counterpart in (\[eq:switched.nonlinear.system\])-(\[eq:ftilda=conv\]), if the maps $f_1,\ldots,f_m$ are *polynomial functions*.[^3]
We will review some results from the thesis of the first author which show that for certain dimensions and degrees, the set of convex and sos-convex Lyapunov functions coincide. In fact, in relatively low dimensions and degrees, it is quite challenging to find convex polynomials that are not sos-convex [@AAA_PP_not_sos_convex_journal]. This is evidence for the strength of this semidefinite relaxation and is encouraging from an application viewpoint. Nevertheless, since sos-convex polynomials are in general a strict subset of the convex ones, a more refined (and perhaps more computationally relevant) converse Lyapunov question for switched linear systems is to see whether their stability guarantees existence of an sos-convex Lyapunov function. This question is also addressed in this paper.
We shall remark that there are other classes of convex Lyapunov functions whose construction is amenable to convex optimization. The main examples include polytopic Lyapunov functions, and piecewise quadratic Lyapunov functions that are a geometric combination of several quadratics [@gwz05; @protasov2; @jungersprotasov09; @protasov-jungers-blondel09; @BlNes05; @dual_LMI_diff_inclusions; @LeeD06]. These Lyapunov functions are mostly studied for the case of linear switched systems, where they are known to be necessary and sufficient for stability. The extension of their applicability to polynomial switched systems should be possible via the sum of squares relaxation. Our focus in this paper however is solely on studying the power of sos-convex polynomial Lyapunov functions. Only in our last section, do we briefly comment on extensions to piecewise sos-convex Lyapunov functions.
Related work
------------
The literature on stability of switched systems is too extensive for us to review. We simply refer the interested reader to [@Shorten05stabilitycriteria; @gst-book; @jungers_lncis] and the references therein. Closer to the specific focus of this paper is the work of Mason et al. [@mason-boscain-chitour06], where the authors prove existence of polynomial Lyapunov functions for switched linear systems in *continuous time*. Our proof of the analogous statement in discrete time closely follows theirs. In [@ahmadi2017sum], Ahmadi and Parrilo show that in the continuous time case, existence of the Lyapunov function of Mason et al. further implies existence of a Lyapunov function that can be found with sum of squares techniques. In [@Pablo_Jadbabaie_JSR_journal], Parrilo and Jadbabaie prove that stable switched linear systems in discrete time always admit a (not necessarily convex) polynomial Lyapunov function which can be found with sum of squares techniques. Blanchini and Franco show in [@blanchini_no_convex_Lyap] that in contrast to the case of uncontrolled switching (our setting), controlled linear switched systems, both in discrete and continuous time, can be stabilized by means of a suitable switching strategy without necessarily admitting a convex Lyapunov function.
In [@Chesi_Hung_journal], [@Chesi_Hung_conf], Chesi and Hung motivate several interesting applications of working with convex Lyapunov functions or Lyapunov functions with convex sublevel sets. These include establishing more regular behavior of the trajectories, ease of optimization over sublevel sets of the Lyapunov function, stability of recurrent neural networks, etc. The authors in fact propose sum of squares based conditions for imposing convexity of polynomials. However, it is shown in [@AAA_PP_CDC10_algeb_convex Sect. 4] that these conditions lead to semidefinite programs of larger size than those of sos-convexity, while at the same time being at least as conservative. Moreover, the works in [@Chesi_Hung_journal], [@Chesi_Hung_conf] do not offer an analysis of the performance (existence) of convex Lyapunov functions.
On the optimization side, the reader interested in knowing more about sos-convex polynomials, their role in convex algebraic geometry and polynomial optimization, and their applications outside of control is referred to the works by Ahmadi and Parrilo [@AAA_PP_not_sos_convex_journal], [@AAA_PP_table_sos-convexity], Helton and Nie [@Helton_Nie_SDP_repres_2], and Magnani et al. [@convex_fitting], or to Section 3.3.3 of the edited volume [@Convex_Alg_Geom_BOOK]. Finally, we note that a shorter version of the current paper with some preliminary results appears in [@aaa_raph_sosconvex_cdc] as a conference paper.
Organization and contributions of the paper
-------------------------------------------
The paper is organized as follows. In Section \[section:sos-convex\], we present the mathematical and algorithmic machinery for working with sos-convex Lyapunov functions and explain its connection to semidefinite programming. In Section \[section:linear\], we study switched linear systems. We show that given any homogeneous Lyapunov function, the Minkowski norm defined by the convex hull of its sublevel set is also a valid (convex) Lyapunov function (Proposition \[prop:Minkowski\]). We then show that any stable switched linear system admits a convex polynomial Lyapunov function (Theorem \[thm:existence-convex-poly\]). Furthermore, we give algebraic arguments to strengthen this result and prove existence of an sos-convex Lyapunov function (Theorem \[thm:existence-sos-convex-poly\]). While existence of a convex polynomial Lyapunov functions is always guaranteed, we prove that in worst case, the degree of such a Lyapunov function can be arbitrarily higher than that of a non-convex polynomial Lyapunov function (Theorem \[thm:degree.higher\]).
In Section \[section:nonlinear\], we study nonlinear switched systems. We show that stability of these systems cannot be inferred from the existence of a common Lyapunov function for the corner systems (Example \[ex:nonconvex.fails\]). However, we prove that this conclusion can be made if the common Lyapunov function is convex (Proposition \[prop:convex.lyap\]). We also give a lemma that shows that the radial unboundedness requirement of a Lyapunov function is implied by its convexity (Lemma \[lem:convex.coercive\]). We then provide an algorithm based on semidefinite programming that under mild conditions finds a full-dimensional inner approximation to the region of attraction of a locally stable equilibrium point of a polynomial switched system (Theorem \[thm:nl.beta.guarantee\]). This algorithm is based on a search for an sos-convex polynomial whose sublevel set is proven to be in the region of attraction via a sum of squares certificate coming from Stengle’s Positivstellensatz. Some examples are provided in Section \[subsec:examples\].
Finally, in Section \[sec:future\], we briefly describe some future directions and extensions of our framework to a broader class of convex Lyapunov functions that are constructed from combining several sos-convex polynomials. These extensions are still amenable to semidefinite programming and have connections to the theory of path-complete graph Lyapunov functions proposed in [@ajpr-sicon].
Sos-convex polynomials {#section:sos-convex}
======================
A multivariate polynomial $p(x)\mathrel{\mathop:}=p(x_1,\ldots,x_n)$ is said to be *nonnegative* or *positive semidefinite* (psd) if $p(x)\geq0$ for all $x\in\mathbb{R}^n$. We say that $p$ is a *sum of squares* (sos) if it can be written as $p=\sum_i q_i^2$, where each $q_i$ is a polynomial. It is well known that if $p$ is of even degree four or larger, then testing nonnegativity is NP-hard, while testing existence of a sum of squares decomposition, which provides a sufficient condition and an algebraic certificate for nonnegativity, can be done by solving a polynomially-sized semidefinite program [@PhD:Parrilo],[@sdprelax].
A polynomial $p\mathrel{\mathop:}=p(x)$ is *convex* if its Hessian $\nabla^2p(x)$ (i.e., the $n\times n$ polynomial matrix of the second derivatives) is positive semidefinite matrix for all $x\in\mathbb{R}^n$. This is equivalent to the scalar-valued polynomial $y^T\nabla^2p(x)y$ in $2n$ variables $(x_1,\ldots,x_n,y_1,\ldots,y_n)$ being nonnegative. It has been shown in [@NPhard_Convexity] that testing if a polynomial of degree four is convex is NP-hard in the strong sense. This motivates the algebraic notion of *sos-convexity*, which can be checked with semidefinite programming and provides a sufficient condition for convexity.
\[def:sos-convex\] A polynomial $p\mathrel{\mathop:}=p(x)$ is *sos-convex* if its Hessian $\nabla^2p(x)$ can be factored as $$\nabla^2p(x)=M^T(x)M(x),$$ where $M(x)$ is a polynomial matrix; i.e., a matrix with polynomial entries.
Polynomial matrices which admit a decomposition as above are called *sos matrices*. The term *sos-convex* was coined in a seminal paper of Helton and Nie [@Helton_Nie_SDP_repres_2]. The following theorem is an algebraic analogue of a classical theorem in convex analysis and provides equivalent characterizations of sos-convexity.
\[Ahmadi and Parrilo [@AAA_PP_table_sos-convexity]\] \[thm:sos.convexity.3.equivalent.defs\] Let $p\mathrel{\mathop:}=p(x)$ be a polynomial of degree $d$ in $n$ variables with its gradient and Hessian denoted respectively by $\nabla p\mathrel{\mathop:}=\nabla p(x) $ and $\nabla^2p\mathrel{\mathop:}=\nabla^2 p(x)$. Let $g_{\lambda}$, $g_\nabla$, and $g_{\nabla^2}$ be defined as $$\label{eq:defn.g_lambda.g_grad.g_grad2}
\begin{array}{lll}
g_{\lambda}(x,y)&=&(1-\lambda)p(x)+\lambda p(y)-p((1-\lambda)
x+\lambda y),\\
g_\nabla(x,y)&=&p(y)-p(x)-\nabla p(x)^T(y-x), \\
g_{\nabla^2}(x,y)&=&y^{T}\nabla^2p(x)y.
\end{array}$$ Then the following are equivalent to sos-convexity of $p$:
**(a)** $g_{\frac{1}{2}}(x,y)$ is sos[^4].
**(b)** $g_\nabla(x,y)$ is sos.
**(c)** $g_{\nabla^2}(x,y)$ is sos.
The above theorem is reassuring in the sense that it demonstrates the invariance of the definition of sos-convexity with respect to the characterization of convexity that one may choose to apply the sos relaxation to. Since existence of an sos decomposition can be checked via semidefinite programming (SDP), any of the three equivalent conditions above, and hence sos-convexity of a polynomial, can also be checked by SDP. Even though the polynomials $g_{\frac{1}{2}}$, $g_\nabla$, $g_{\nabla^2}$ above are all in $2n$ variables and have degree $d$, the structure of the polynomial $g_{\nabla^2}$ allows for much smaller SDPs (see [@AAA_PP_CDC10_algeb_convex] for details).
In general, finding examples of convex polynomials that are not sos-convex seems to be a nontrivial task, though a number of such constructions are known [@AAA_PP_not_sos_convex_journal]. A complete characterization of the dimensions and the degrees for which the notions of convexity and sos-convexity coincide is available in [@AAA_PP_table_sos-convexity]. Crucial for our purposes is the fact that semidefinite programming allows us to not only check if a given polynomial is sos-convex, but also search and optimize over the set of sos-convex polynomials of a given degree. This feature enables an automated search over a subset of convex polynomial Lyapunov functions. Of course, a Lyapunov function also needs to satisfy other constraints, namely positivity and monotonic decrease along trajectories. Following the standard approach, we replace the inequalities underlying these constraints with their sum of squares counterparts as well.
Throughout this paper, what we mean by an *sos-convex Lyapunov function* is a polynomial function which is sos-convex and satisfies all other required Lyapunov inequalities with sos certificates.[^5] When the Lyapunov function can be taken to be homogeneous—as is the case when the dynamics are homogeneous [@HomogHomog]—then the following lemma establishes that the convexity requirement of the polynomial automatically meets its nonnegativity requirement.
Recall that a *homogeneous polynomial* (or a *form*) is a polynomial whose monomials all have the same degree.
Convex forms are nonnegative and sos-convex forms are sos.
See [@Helton_Nie_SDP_repres_2 Lemma 8] or [@AAA_PP_table_sos-convexity Lemma 3.2].
For stability analysis of the switched linear system in (\[eq:switched.linear.system\]), the requirements of a (common) sos-convex Lyapunov function $V$ are therefore the following: $$\label{eq:sos-convex.Lyap.requirements}
\begin{array}{ll}
V(x) & \mbox{sos-convex}\\
V(x)-V(A_ix) & \mbox{sos for}\ i=1,\ldots,m.
\end{array}$$
Given a set of matrices $\{A_1,\ldots,A_m\}$ with rational entries, the search for the coefficients of a fixed-degree polynomial $V$ satisfying the above conditions amounts to solving an SDP whose size is polynomial in the bit size of the matrices. If this SDP is (strictly) feasible, the switched system in (\[eq:switched.linear.system\]) is stable under arbitrary switching. We remark that the same implication is true if the sos-convexity requirement of $V$ is replaced with the requirement that $V$ is sos; see [@Pablo_Jadbabaie_JSR_journal Thm. 2.2]. (This statement fails to hold for switched nonlinear systems.)
In the next section, we study the converse question of existence of a Lyapunov function satisfying the semidefinite conditions in (\[eq:sos-convex.Lyap.requirements\]).
Sos-convex Lyapunov functions and switched linear systems {#section:linear}
=========================================================
As remarked in the introduction, it is known that asymptotic stability of a switched linear system under arbitrary switching implies existence of a common convex Lyapunov function, as well as existence of a common polynomial Lyapunov function. In this section, we show that this stability property in fact implies existence of a common Lyapunov function that is both convex and polynomial (cf. Subsection \[subsec:existence.convex\]). Moreover, we strengthen this result to show existence of a common sos-convex Lyapunov function (cf. Subsection \[subsec:sos-convex\]).
Before we prove these results, we state a related proposition which shows that in the particular case of switched linear systems, any common Lyapunov function (e.g. a non-convex polynomial) can be turned into a common *convex* Lyapunov function, although not necessarily an efficiently computable one. We believe that this statement must be known, but since we could not pinpoint a reference, we include a proof here.
\[prop:Minkowski\] Consider the switched linear system in (\[eq:switched.linear.system\]). Suppose $V$ is a common homogeneous and continuous Lyapunov function for (\[eq:switched.linear.system\]); i.e. satisfies Let $$\mathcal{S}\mathrel{\mathop:}=\{x\in\mathbb{R}^n|~\ V(x)\leq 1\}.$$ Then, the Minkowski (a.k.a. gauge) norm defined by the set $\mathcal{S}$, i.e. the function $$W(x)\mathrel{\mathop:}=\inf \{t>0|\ \frac{x}{t}\in conv(\mathcal{S})\},$$ is a convex common Lyapunov function for (\[eq:switched.linear.system\]).
Since under the assumptions of the proposition the set $conv(\mathcal{S})$ is compact, origin symmetric, and has nonempty interior, the function $W$ is a norm (see e.g. [@BoydBook p. 119]) and hence convexity and positivity of $W$ are already established. It remains to show that for any $i\in\{1,\ldots,m\}$ and $x\neq 0$ we have $$\nonumber
\begin{array}{lll}
W(A_ix)&=& \inf \{t>0|\ \frac{A_ix}{t}\in conv(\mathcal{S})\}\\
\ &<& \inf \{t>0|\ \frac{x}{t}\in conv(\mathcal{S})\}\\
\ &=& W(x).
\end{array}$$ To see the inequality, first note that because $V$ is a common Lyapunov function, there must exist a constant $\gamma\in(0,1)$ such that if $z\in\mathcal{S}$, then $A_iz\in\gamma \mathcal{S}$. Now observe that if for some $t>0$ we have $\frac{x}{t}\in conv(\mathcal{S}),$ then by definition $\frac{x}{t}=\sum_j \lambda_j y_j$ for some $y_j\in\mathcal{S}$ and $\lambda_j\geq 0$ with $\sum_j \lambda_j =1.$ Hence, $$\frac{A_i x}{t}=\sum_j \lambda_j A_iy_j=\sum_j \lambda_j \gamma w_j,$$ for some $w_j\in\mathcal{S}$. But this means that $\frac{A_i x}{\gamma t}\in conv(\mathcal{S}).$
Existence of convex polynomial Lyapunov functions {#subsec:existence.convex}
-------------------------------------------------
The goal of this subsection is to prove the following theorem.
\[thm:existence-convex-poly\] If the switched linear system (\[eq:switched.linear.system\]) is asymptotically stable under arbitrary switching, then there exists a convex positive definite homogeneous polynomial $p$ that satisfies $p(A_ix)<p(x)$ for all $x\neq 0$ and all $i\in\{1,\ldots,m\}.$
Our proof is inspired by [@mason-boscain-chitour06], which proves the existence of a convex polynomial Lyapunov function for continuous time switched systems, but we are not aware of an equivalent statement in discrete time. We also need the following classical result, which to the best of our knowledge first appears in [@RoSt60].
\[thm-rotastrang\] Consider a set of matrices ${{\cal{M}}}\subset {{\mathbb R}}^{n\times n}$ with JSR $\rho$. For all $\epsilon>0$, there exists a vector norm $|\cdot|_\epsilon$ in ${{\mathbb R}}^n$ such that for any $A$ in ${{\cal{M}}},$ $$|x|_\epsilon \leq 1 \quad \Rightarrow \quad |Ax|_\epsilon \leq \rho + \epsilon.$$
(of Theorem \[thm:existence-convex-poly\].) Let ${{\cal{M}}}\mathrel{\mathop:}=\{A_1,\ldots,A_m\}$ and denote the JSR of ${{\cal{M}}}$ by $\rho$. By assumption we have $\rho<1$ and by Theorem \[thm-rotastrang\], there exists a norm, which from here on we simply denote by $|\cdot|$, such that $\forall i\in\{1,\ldots,m\},$ $$|x| \leq 1 \Rightarrow \quad |A_ix| \leq \rho + \frac12 (1-\rho).$$ We denote the unit ball of this norm by $B$ and use the notation $${{\cal{M}}}B\mathrel{\mathop:}=\{A_ix|~A_i\in{{\cal{M}}}\mbox{ and } x\in B\}.$$ Hence, we have ${{\cal{M}}}B \subseteq (\rho + \frac12 (1-\rho)) B . $ The goal is to construct a convex positive definite homogeneous polynomial $p_d$ of some degree $2d$, such that its 1-sublevel set $S_d$ satisfies $$(\rho + \frac12(1-\rho)) B \subseteq int(S_d) \subset S_d \subseteq B.$$ As $S_d \subseteq B$ and ${{\cal{M}}}B \subseteq (\rho + \frac12 (1-\rho)) B$, it would follow that $${{\cal{M}}}S_d \subseteq {{\cal{M}}}B \subseteq int(S_d).$$ This would imply that $p_d(A_ix)<p(x),\forall x \in \partial S_d,$ and for $i=1,\ldots,m.$ By homogeneity of $p_d$, we get the claim in the statement of the theorem.
To construct $p_d$, we proceed in the following way. Let $$C=\{x\in \mathbb{R}^n |~|x|\leq \rho + \frac{3}{4}(1-\rho)\}.$$ To any $x\in \partial C,$ we associate a (nonzero) dual vector $v(x)$ orthogonal to a supporting hyperplane of $C$ at $x$. This means that $\forall y \in C,\ v(x)^Ty\leq v(x)^Tx$. Since $x\in {{{\rm int} }}{B},$ the set $$S(x)=\{y\in \mathbb{R}^n|~v(x)^Ty>v(x)^Tx \mbox{ and }|y|=1\}$$ is a relatively open nonempty subset of the boundary $\partial B$ of our unit ball. Moreover, $\frac{x}{|x|}\in S(x).$ Now, the family of sets $S(x)$ is an open covering of $\partial B,$ and hence we can extract a set of points $x_1,\dots,x_N$ such that the union of the sets $S(x_i)$ covers $\partial B.$ Let $v_i\mathrel{\mathop{:}}=v(x_i).$ For any natural number $d$, we define [^6] $$p_d(y)=\sum_{i=1}^N \left(\frac{v_i^Ty}{v_i^Tx_i}\right)^{2d} \text{ and } S_d=\{y \in \mathbb{R}^n|~ p_d(y)\leq 1\}.$$ Note that $p_d$ is convex as the sum of even powers of linear forms and homogeneous. We first show that $$(\rho + \frac12(1-\rho)) B \subseteq int(S_d).$$ As $(\rho + \frac12(1-\rho)) B \subset int(C)$, for all $y \in (\rho + \frac12(1-\rho)) B$ and for all $i=1,\ldots,N$, we have $v_i^Ty < v_i^Tx_i$. Hence there exists a positive integer $d$ such that $$\left(\max_i \max_{y \in (\rho +\frac12 (1-\rho))B} \frac{v_i^Ty}{v_i^Tx_i}\right)^{2d}<\frac{1}{N}$$ and so $p_d(y)<1$ for all $y \in (\rho + \frac12(1-\rho)) B$.
We now show that $S_d \subseteq B.$ Let $y \in S_d$, and so $p_d(y)\leq 1.$ This implies that $$\frac{v_i^Ty}{v_i^Tx_i} \leq 1, \forall i=1,\ldots,N.$$ From this, we deduce that $y \notin \partial B.$ Indeed if $y \in \partial B,$ there exists $i \in \{1,\ldots,N\}$ such that $y \in S(x_i)$, which implies that $v_i^T y>v_i^Tx_i$ and contradicts the previous statement. Hence $\partial B \cap S_d =\emptyset$. As both $B$ and $S_d$ contain the zero vector, we conclude that $S_d \subseteq int(B) \subseteq B$. Note that this guarantees positive definiteness of $p_d$ as $p_d$ is homogeneous and its 1-sublevel set is bounded.
Existence of sos-convex polynomial Lyapunov functions {#subsec:sos-convex}
-----------------------------------------------------
We now strengthen the converse result of the previous subsection by showing that asymptotically stable switched linear systems admit an *sos-convex* Lyapunov function. This in particular implies that such a Lyapunov function can be found with semidefinite programming.
Recall that a homogeneous polynomial $h$ is said to be *positive definite (pd)* if $h(x)>0$ for all $x\neq 0$.
\[thm:existence-sos-convex-poly\] If the switched linear system (\[eq:switched.linear.system\]) is asymptotically stable under arbitrary switching, then there exists a homogeneous polynomial $q$ that satisfies the sum of squares constraints $$\nonumber
\begin{array}{ll}
q(x) & \mbox{sos-convex},\\
q(x)-q(A_jx) & \mbox{sos for}\ j=1,\ldots,m.
\end{array}$$ Moreover, this polynomial $q$ is positive definite and is such that the $m$ polynomials $q(x)-q(A_jx)$ are also positive definite.
Our proof will make crucial use of the following Positivstellensatz due to Scheiderer.
\[thm:claus\] Given any two positive definite homogeneous polynomials $h$ and $g$, there exists a positive integer $N$ such that $hg^N$ is sos.
(of Theorem \[thm:existence-sos-convex-poly\]). We have already shown in the proof of Theorem \[thm:existence-convex-poly\] that under our assumptions, there exist vectors $a_1,\ldots,a_N\in\mathbb{R}^n$ and a positive integer $d$ such that the convex form $p(x)=\sum_{i=1}^N (a_i^Tx)^{2d}$ is positive definite and makes the $m$ forms $p(x)-p(A_jx)$ also positive definite. Note also that as a sum of powers of linear forms, $p$ is already sos-convex and sos. Let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$ and define $$\alpha_j\mathrel{\mathop{:}}=\frac{1}{2} \min_{x\in S^{n-1}} p(x)-p(A_jx).$$ By definition of $\alpha_j$, we have that is positive definite. Furthermore, as $\alpha_j>0$ (this is a consequence of $p(x)-p(A_jx)$ being positive definite) and as $p(A_jx)$ is sos, we get that $p(A_jx)+\alpha_j (\sum_i x_i^2)^d$ is positive definite. there exist an integer $K$ such that $$\begin{aligned}
\label{eq:claus.p}
\left(p(x)-p(A_jx)-\alpha_j (\sum_i x_i^2)^d\right)\cdot p(x)^K
\end{aligned}$$ is sos and an integer $K'$ such that $$\begin{aligned}
\label{eq:claus.pAj}
\left(p(x)-p(A_jx)-\alpha_j (\sum_i x_i^2)^d\right) \cdot \left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^{K'}
\end{aligned}$$ is sos.
Take $k\mathrel{\mathop{:}}=\max\{2K+1,2K'+1\}$ and define $q(x)=p(x)^k.$ It is easy to see that $q$ is positive definite as $p$ is positive definite. We first show that $q$ is sos-convex. We have $$\nabla^2 q(x)=k(k-1)p(x)^{k-2}\nabla p(x)\nabla p(x)^T+kp(x)^{k-1}\nabla^2 p(x).$$ As $p$ is sos, any power of it is also sos. Furthermore, we have $$\nabla^2 p(x)=\sum_{i=1}^N 2d(2d-1)(a_i^Tx)^{2d-2}a_ia_i^T,$$ which implies that there exists a polynomial matrix $V(x)$ such that $\nabla^2p(x)=V(x)V(x)^T$. As a consequence, we see that $$y^T\nabla^2 q(x)y=k(k-1)p(x)^{k-2} (\nabla p(x)^Ty)^2+kp(x)^{k-1}(V(x)^Ty)^2$$ is a sum of squares and hence $q$ is sos-convex.
We now show that for $j=1,\ldots,m$, the form $q(x)-q(A_jx)$ is positive definite and sos. For positive definiteness, simply note that as $p(x)>p(A_jx)$ for any $x \neq 0$ and $p$ is nonnegative, we get $p^k(x)>p^k(A_jx)$ for any $k\geq 1$ and $x\neq 0.$ To show that $q(x)-q(A_jx)$ is sos, we make use of the following identity: $$\begin{aligned}
\label{eq:identity}
a^k-b^k=(a-b)\sum_{l=0}^{k-1} a^{k-1-l}b^l.
\end{aligned}$$ Applying (\[eq:identity\]), we have $$\begin{aligned}
&p^k(x)-\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^k \nonumber \\ &=\left(p(x)-p(A_jx)-\alpha_j (\sum_i x_i^2)^d\right)\cdot \sum_{l=0}^{k-1}p(x)^{k-1-l}\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^l\nonumber\\
&=\sum_{l=0}^{k-1}\left(p(x)-p(A_jx)-\alpha_j (\sum_i x_i^2)^d\right)\cdot p(x)^{k-1-l}\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^l. \label{eq:sum}
\end{aligned}$$ For any $l \in \{0,\ldots,k-1\}$, either $k-1-l\geq \frac{k-1}{2}$ or $l \geq \frac{k-1}{2}$. Suppose that the index $l$ is such that $k-1-l \geq \frac{k-1}{2}$: by definition of $k$, this implies that $k-1-l\geq K$. Since the polynomial in (\[eq:claus.p\]) sos and since $(p(A_jx)+\alpha_j (\sum_i x_i^2)^d)^l$ is sos, we get that the term $$\left(p(x)-p(A_jx)-\alpha_j (\sum_i x_i^2)^d\right)\cdot p(x)^{k-1-l}\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^l$$ in the sum (\[eq:sum\]) is sos. Similarly, if the index $l$ is such that $l \geq \frac{k-1}{2}$, we have that $l \geq K'$ by definition of $k.$ Sine the polynomial in (\[eq:claus.pAj\]) is sos, we come to the conclusion that the term $$\left(p(x)-p(A_jx)-\alpha_j (\sum_i x_i^2)^d\right)\cdot p(x)^{k-1-l}\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^l$$ in the sum (\[eq:sum\]) is sos. When summing over all possible $l \in \{0,\ldots,k-1\}$, as each term in the sum is sos, we conclude that the sum $$p^k(x)-\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^k$$ itself is sos. Now note that we can write $$\begin{aligned}
&p^k(x)-\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^k=p(x)^k-p(A_j)^k -\sum_{s=1}^k \binom{k}{s} \alpha_j^s (\sum_i x_i^2)^{ds}\cdot p(A_jx)^{k-s},
\end{aligned}$$ which enables us to conclude that $$q(x)-q(A_jx)=p^k(x)-\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^k+\sum_{s=1}^k \binom{k}{s} \alpha_j^s (\sum_i x_i^2)^{ds}\cdot p(A_jx)^{k-s}$$ is sos as the sum of sos polynomials.
Non-existence of a uniform bound on the degree of convex polynomial Lyapunov functions
--------------------------------------------------------------------------------------
It is known that there are families of $n\times n$ matrices ${{\cal{M}}}=\{A_1,\ldots,A_m\}$ for which the switched linear system (\[eq:switched.linear.system\]) is asymptotically stable under arbitrary switching, but such that the minimum degree of a common polynomial Lyapunov function is arbitrarily large [@aaa_raph_lower_bounds]. (In fact, this is the case already when $m=n=2$.) In the case where ${{\cal{M}}}$ admits a common polynomial Lyapunov function of degree $d$, it is natural to ask whether one can expect ${{\cal{M}}}$ to also admit a common *convex* polynomial Lyapunov function of some degree $\hat{d}$, where $\hat{d}$ is a function of $d,n,m$ only? In this subsection, we answer this question in the negative.
Consider the set of matrices $\mathcal{A}=\{A_1,A_2\},$ with $$\label{eq:A1,A2.ando.shi}
A_{1}=\left[
\begin{array}
[c]{cc}1 & 0\\
1 & 0
\end{array}
\right] ,\text{ }A_{2}=\left[
\begin{array}
[c]{cr}0 & 1\\
0 & -1
\end{array}
\right].$$ This is a benchmark set of matrices that has been studied in [@Ando98], [@Pablo_Jadbabaie_JSR_journal] mainly because it provides a “worst-case” example for the method of common quadratic Lyapunov functions. Indeed, it is easy to show that $\rho(\mathcal{A})=1$, but a common quadratic Lyapunov function can only produce an upper bound of $\sqrt{2}$ on the JSR. In [@Pablo_Jadbabaie_JSR_journal], Parrilo and Jadbabaie give a simple degree-4 (non-convex) common polynomial Lyapunov function that proves stability of the switched linear system defined by the matices $\{\gamma A_1, \gamma A_2\},$ for any $\gamma<1$. In sharp contrast, we show the following:
\[thm:degree.higher\] Let $A_1,A_2$ be as in (\[eq:A1,A2.ando.shi\]) and consider the sets of matrices ${{\cal{M}}}_\gamma=\{\gamma A_1, \gamma A_2\}$ parameterized by a scalar $\gamma <1$. As $\gamma\rightarrow 1$, the minimum degree of a common convex polynomial Lyapunov function for ${{\cal{M}}}_\gamma$ goes to infinity.
It is sufficient to prove that the set $\{A_1,A_2\}$ has no convex invariant set defined as the sublevel set of a polynomial. Indeed, if there were a uniform bound $D$ on the degree of a convex polynomial Lyapunov function this would imply the existence of an invariant set—which is the sublevel set of a convex polynomial function of degree $D$—\
We prove our claim by contradiction. In fact, we will prove the slightly stronger fact that for these matrices, the only convex invariant set is the unit square $$S=\{(x,y)\in\mathbb{R}^2|\ ||(x,y)||_\infty\leq 1\},$$ or, of course, a scaling of it.
Let $\mathcal{A}=\{A_1,A_2\}$ and let $\mathcal{A}^*$ denote the set of all matrix products out of $\mathcal{A}$. Suppose for the sake of contradiction that there was a convex bivariate polynomial $p$ whose unit level set was the boundary of an invariant set for the switched system defined by $\mathcal{A}$. More precisely, suppose we had $$\label{eqn-contrex-lyap}\forall x \in {{\mathbb R}}^2,\ \forall A\in \mathcal{A}, \quad p(Ax) \leq p(x).$$ Let $x^*\in\mathbb{R}$ be such that It is easy to check that the following matrices can be obtained as products of matrices in $\mathcal{A}$: $$\label{eqn-ando-semigroup}
\left \{
\begin{pmatrix}
0 & 1\\
0 & -1
\end{pmatrix},\ \begin{pmatrix}
0 & 1\\
0 & 1
\end{pmatrix}
,\
\begin{pmatrix}
0 & -1\\
0 & 1
\end{pmatrix},\right \}\quad \subset {{\cal{A}}}^*.$$ This implies that $$\begin{aligned}
\nonumber p({\bf x})&=&1 \\
\nonumber \mbox{for } {\bf x}&\in&\{(x^*,-x^*),(-x^*,-x^*),(-x^*,x^*)\}\end{aligned}$$ as well, because these points can all be mapped onto each other with matrices from (\[eqn-ando-semigroup\]).
Suppose that there is an $x>x^*,$ $-x^*<y<x^*,$ such that Then we reach a contradiction because (\[eqn-ando-semigroup\]) implies that $(x,y)$ can be mapped on $(x,x),$ which contradicts (\[eqn-contrex-lyap\]) because $x>x^*.$ This implies that $p(x^*,y)\geq 1, \forall y\in (-x^*,x^*)$. However, convexity of $p$ implies that $p(x^*,y)\leq 1, \forall y\in (-x^*,x^*)$. Thus, we have proved that $p(x^*,y)=1, \forall y\in (-x^*,x^*).$ The same is true for $p(-x^*,y)$ by symmetry.
In the same vein, if there is a $y>x^*,$ $-x^*<x<x^*$ such that $p(x,y)=1,$ this point can be mapped on $(-y,-y),$ which again leads to a contradiction, because $p(-x^*,-x^*)=1.$ Hence, $p(x,x^*)=p(x,-x^*)=1, \forall x\in(-x^*,x^*),$ which concludes the proof.
SOS-convex Lyapunov functions and switched nonlinear systems {#section:nonlinear}
============================================================
In this section, we turn our attention to stability analysis of switched nonlinear systems
$$\begin{aligned}
\label{eq:nl.systems.again}
x_{k+1}&=&\tilde{f}(x_k), \label{eq:switched.nonlinear.system.ex}\\ \nonumber
\tilde{f}(x_k)&\in & conv\{f_1(x_k),\dots, f_m(x_k)\},\end{aligned}$$
where $f_1,\ldots,f_m: \mathbb{R}^n\rightarrow\mathbb{R}^n$ are continuous and satisfy $f_i(0)=0.$ We start by demonstrating the significance of convexity of Lyapunov functions in this setting. We then consider the case where $f_1,\ldots,f_m$ are polynomials and devise algorithms that under mild conditions find algebraic certificates of local asymptotic stability under arbitrary switching. These algorithms produce a full-dimensional inner approximation to the region of attraction of the origin, which comes in the form of a sublevel set of an sos-convex polynomial.
The significance of convexity of the Lyapunov function {#subsec:nl.global}
------------------------------------------------------
The following example demonstrates that unlike the case of switched linear systems, one *cannot* simply resort to a common Lyapunov function for the maps $f_1,\ldots, f_m$ to infer a proof of stability of a nonlinear difference inclusion.
\[ex:nonconvex.fails\] Consider the nonlinear switched system (\[eq:switched.nonlinear.system.ex\]) with $m=n=2$ and $$\begin{aligned}
f_1(x)&=&(x_1x_2,0)^T,\label{eq:ex.nonconvex.fails}\\\nonumber
f_2(x)&=&(0,x_1x_2)^T.\end{aligned}$$ function $$\label{eq:Lyap.nonconvex}V(x)=x_1^2x_2^2+(x_1^2+x_2^2)$$ is a common Lyapunov function for both $f_1$ and $f_2$, but nevertheless the system in (\[eq:switched.nonlinear.system.ex\]) is unstable.
To see this, note that $$V(f_i(x))=x_1^2x_2^2<V(x)=x_1^2x_2^2+(x_1^2+x_2^2)$$ for $i=1,2,$ and for all $x\neq 0.$\
On the other hand, (\[eq:switched.nonlinear.system.ex\]) is unstable since in particular the dynamics $x_{k+1}=f(x_k)$ with $$f(x)=\left(\frac{x_1x_2}{2},\frac{x_1x_2}{2}\right)\in conv\{f_1(x),f_2(x)\}$$ is obviously unstable.
Note that the Lyapunov function in (\[eq:Lyap.nonconvex\]) was not convex. Proposition \[prop:convex.lyap\] below shows that a convexity requirement on the Lyapunov function gets around the problem that arose above. To prove this proposition, we first give a lemma which is potentially of independent interest for global stability analysis. Recall that Lyapunov’s theorem for global asymptotic stability commonly requires that the Lyapunov function $V$ be radial unbounded (i.e., satisfy $||x||\rightarrow\infty\implies V(x)\rightarrow\infty$). Our lemma shows that convexity brings this property for free.[^7]
\[lem:convex.coercive\] Suppose a function $V:\mathbb{R}^n\rightarrow\mathbb{R}$ satisfies $V(0)=0$ and $V(x)>0$ for all $x\neq 0$. If $V$ is convex, then it is radially unbounded.
We proceed by contradiction. Suppose that $V$ is not radially unbounded. This implies that there exists a scalar $s>0$ for which the sublevel set $$S\mathrel{\mathop:}=\{x\in\mathbb{R}^n|\ V(x)\leq s \}$$ of $V$ is unbounded. As $V$ is convex, $S$ is convex, and as any nonempty sublevel set of $V$ contains the origin, $S$ contains the origin. We claim that $S$ must in fact contain an entire ray originating from the origin.
Indeed, as $S$ is unbounded, there exists a sequence of points $\{x_k\}$ such that $\lim_{k\rightarrow \infty}||x_k||=\infty$ and such that $V(x_k)\leq s$ for all $k \in \mathbb{N}.$ Consider now the sequence $\{x_k/||x_k||\}$: this is a bounded sequence and hence has a subsequence that converges. Let $\hat{x}$ be the limit of this subsequence. We argue that the ray $\{c\hat{x}|\ c\geq 0 \}$ is contained in $S$. Suppose that it was not: then $V(\alpha \hat{x})>s$ for some fixed $\alpha>0$, and since $S$ is closed (as a sublevel set of a continuous function), there exists a scalar $\epsilon>0$ such that for all $y\in\mathbb{R}^n$ with $||y-\alpha\hat{x}||\leq \epsilon$, we have $V(y)>s.$ As $\lim_{k\rightarrow \infty} ||x_k||=\infty$ and a subsequence of $\{x_k/||x_k||\}$ converges to $\hat{x}$, there must exist an integer $k_0$ such that $$||x_{k_0}||> \alpha \text{ and } ||\hat{x}-\frac{x_{k_0}}{||x_{k_0}||}||\leq \epsilon/ \alpha.$$ Note that $$||\alpha \hat{x} -\alpha \frac{x_{k_0}}{||x_{k_0}||}|| \leq \epsilon,$$ which implies that $V(\alpha x_{n_0}/||x_{k_0}||)>s$ and hence $\alpha x_{k_0}/||x_{k_0}||$ does not belong to $S$. But this contradicts convexity of $S$ as $$\frac{\alpha x_{k_0}}{||x_{k_0}||}=\frac{\alpha}{||x_{k_0}||} \cdot x_{k_0}+(1-\frac{\alpha}{||x_{k_0}||})\cdot 0$$ and $x_{k_0}$ and $0$ are in $S$.
We now consider the restriction of $V$ to this ray, which we denote by $g(z)=V(z\hat{x})$, where $z\geq 0.$ We remark that as a univariate function, $g$ is convex, and positive everywhere except at zero where it is equal to zero. By convexity of $g$, we have the inequality $$\frac{1}{w} g(w)+\left(1-\frac{1}{w}\right)g(0)\geq g \left( \frac{1}{w}\cdot w+(1-1/w) \cdot 0 \right)$$ for all $w \in \mathbb{N}$. This is equivalent to $$\label{eq:g.univariate.inequality}
\frac{g(w)}{w} \geq g(1).$$ Note that $g(1)>0$, but $g(w)\leq s~\forall w \in \mathbb{N}$ since $g$ is a restriction of $V$ to a ray contained in $S$. This contradicts the inequality in (\[eq:g.univariate.inequality\]) when $w$ is large. Hence, $S$ cannot be unbounded and it follows that $V$ must be radially unbounded.
\[prop:convex.lyap\] Consider the nonlinear switched system in (\[eq:switched.nonlinear.system.ex\]).
(i) If there exists a convex function $V:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfies $V(0)=0$, $V(x)>0$ for all $x\neq 0,$ and $$\label{eq:corner.decrease}
V(f_i(x))<V(x), \quad \forall x\neq 0, \forall i\in\{1,\ldots,m\},$$ then the origin is globally asymptotically stable under arbitrary switching.
(ii) If there exist a scalar $\beta>0$ and a convex function $V:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfies $V(0)=0$, $V(x)>0$ for all $x\neq 0,$ and $$\label{eq:corner.decrease.local}
V(f_i(x))<V(x), \quad \forall x\neq 0 \ \mbox{with}\ V(x)\leq \beta, \mbox{and}\ \forall i\in\{1,\ldots,m\},$$ then the origin is locally asymptotically stable under arbitrary switching and the set $\{x\in \mathbb{R}^n|~V(x)\leq \beta \}$ is a subset of the region of attraction of the origin.
The proof of this proposition is similar to the standard proofs of Lyapunov’s theorem except for the parts where convexity intervenes. Hence we only prove part (i) and leave the very analogous proof of part (ii) to the reader.
Suppose the assumptions of (i) hold. Then, for all $x_k \neq 0$ we have $$\label{eq:Lyap.decrease}
\begin{aligned} V(x_{k+1})-V(x_k)&=V\left(\sum_{i=1}^{m}{\lambda_i(k) f_i(x_k)}\right)-V(x_k) \\ &\leq \sum_{i=1}^{m}{\lambda_i(k) V(f_i(x_k))}-V(x_k)\\ &= \sum_{i=1}^{m}{\lambda_i(k) \left(V(f_i(x_k))-V(x_k)\right)}\\ &<0,
\end{aligned}$$ where the first inequality follows from convexity of $V,$ and the second from (\[eq:corner.decrease\]) and the fact that $\sum_{i=1}^{m}\lambda_i(k)=1$. Hence, our Lyapunov function decreases in each iteration independent of the realization of the uncertain and time-varying map $\tilde{f}$ in (\[eq:switched.nonlinear.system.ex\]).
To show that the origin is stable in the sense of Lyapunov, consider an arbitrary scalar $\delta>0$ and the ball $B(0,\delta)\mathrel{\mathop:}=\{x\in\mathbb{R}^n|\ ||x||\leq \delta \}.$ Recall that as a consequence of Lemma \[lem:convex.coercive\], all sublevel sets of $V$ are bounded. Let $\hat{\delta}>0$ be the radius of a ball that is contained in a (full-dimensional) sublevel set of $V$ which itself is contained in $B(0,\delta)$. Then, from (\[eq:Lyap.decrease\]), we get that $x_0\in B(0,\hat{\delta})\implies x_k\in B(0,\delta), \forall k.$
To show that the origin attracts all initial conditions, consider an arbitrary nonzero point $x_0\in\mathbb{R}^n$ and denote by $\{x_k\}$ any sequence that this initial condition can generate under the iterations of (\[eq:switched.nonlinear.system.ex\]). We know the sequence $\{V(x_k)\}$ is positive and decreasing (unless $x_k$ in finite time lands on the origin, in which case the proof is finished). It follows that $\{V(x_k)\}\rightarrow c$ for some scalar $c\geq 0.$ We claim that $c=0$, in which case we must have $x_k\rightarrow 0$ as $k\rightarrow\infty$ which is the desired statement. Suppose for the sake of contradiction that we had $c>0.$ Then, we must have $$x_k\in\Omega\mathrel{\mathop:}=\{x\in\mathbb{R}^n| \ c\leq V(x)\leq V(x_0) \}, \forall k.$$
Note that the set $\Omega$ is closed and bounded as $V$, being a convex function, is continuous and by Lemma \[lem:convex.coercive\] also radially unbounded. Let $\Delta_m$ denote the unit simplex in $\mathbb{R}^m$ and let $$\eta= \sup_{x\in\Omega,\lambda\in\Delta_m} V\left(\sum_{i=1}^{m}{\lambda_i f_i(x)}\right)-V(x).$$
We claim that $\eta<0$. This is because of (\[eq:Lyap.decrease\]) and the fact that the above supremum is achieved as the objective functions is continuous and the feasible set is compact. Hence, the sequence $\{V(x_k)\}$ decreases in each step by at least $|\eta|$ and hence must go to $-\infty$. This however contradicts positivity of $V$ on $\Omega$.
Computing regions of attraction for switched nonlinear systems
--------------------------------------------------------------
In this section, we consider the switched nonlinear system in (\[eq:nl.systems.again\]), where $f_1,\ldots,f_m$ are polynomials. It is quite common in this case for the system to not be globally stable but yet to have a locally attractive equilibrium point. Under the assumption that $\rho(A_1,\ldots,A_m)<1$, where $A_1,\ldots,A_m$ are the matrices associated with the linearizations of $f_1,\ldots,f_m$ around the origin, we design an algorithm based on semidefinite programming that provably finds a full-dimensional inner approximation to the region of attraction of the nonlinear switched system. Note that if the origin of (\[eq:nl.systems.again\]) is locally asymptotically stable, then we must have $\rho(A_1,\ldots,A_m) \leq 1.$ The only case to remain is the boundary case $\rho(A_1,\ldots,A_m)=1,$ which is left for our future work.
Our procedure for finding the region of attraction will have two steps:
(i) Use SDP to find a common sos-convex Lyapunov function $V$ for the linearizations of $f_1,\ldots, f_m$ around the origin; i.e., find a positive definite sos-convex form $V(x)$ such that $V(x)-V(A_i x)$ is sos and positive definite for $i=1,\ldots,m.$ Existence of such a function is guaranteed by Theorem \[thm:existence-sos-convex-poly\], which was the main result of Section \[section:linear\].
(ii) Find a scalar $\beta>0$ such that $$\forall x \neq 0, V(x)\leq \beta \Rightarrow V(f_i(x))<V(x), \text{ for } i=1,\ldots,m.$$ We will prove that semidefinite programming can find such a $\beta$ in finite time and certify the above implication algebraically.
Once this procedure is carried out, the set $\{x\in\mathbb{R}^n|~V(x)\leq \beta\}$ is guaranteed to be a subset of the region of attraction. Implementation of step (ii) requires the reader to be reminded of the following fundamental theorem in algebraic geometry. Recall that a basic semialgebraic set is a set defined by a finite number of polynomial inequalities and equations.
\[thm:stengle\] The basic semialgebraic set $$S=\{x\in \mathbb{R}^n |~ g_1(x)\geq 0,\ldots, g_m(x)\geq 0, h_1(x)=0,\ldots, h_s(x)=0\}$$ is empty if and only if there exist polynomials $t_1(x),\ldots,t_s(x)$ and sum of squares polynomials $\{\sigma_{a_1\ldots a_m}|~ (a_1,\ldots,a_m)\in \{0,1\}^{m}\}$ such that $$\begin{aligned}
-1=\sum_{j=1}^s t_j(x) h_j(x)+\sum_{a_1,\ldots,a_m \in \{0,1\}^m} \sigma_{a_1 \ldots a_m}(x) \prod_{i=1}^m g_i^{a_i}(x).
\end{aligned}$$
\[thm:nl.beta.guarantee\] Consider the switched nonlinear system in (\[eq:nl.systems.again\]), where $f_1,\ldots,f_m: \mathbb{R}^n \rightarrow \mathbb{R}^n$ are polynomials. Let $$f_1^l(x)=A_1x,\ldots,f_m^l(x)=A_mx$$ be the linearizations of $f_1,\ldots,f_m$ around zero, and suppose $\rho(A_1,\ldots,A_m)<1$. Let $y \in \mathbb{R}$ be a new variable. Then, there exist an sos-convex positive definite form $V(x)$, a scalar $\beta>0$, a polynomial $t$, and sum of squares polynomials $$\{\sigma_{a_0\ldots a_m}|~ (a_0,\ldots,a_m)\in \{0,1\}^{m+1}\}$$ such that Conversely, if (\[eq:stengle.lyap\]) holds, then the switched nonlinear system in (\[eq:nl.systems.again\]) is locally asymptotically stable under arbitrary switching and the set $$\{x \in \mathbb{R}^n|~ V(x)\leq \beta\}$$ is a subset of the region of attraction.
We start with the converse as it is the easier direction to prove. Note that if (\[eq:stengle.lyap\]) holds for some sos-convex positive definite form $V(x)$ and some scalar $\beta>0$, then the set $$\begin{aligned}
\label{eq:empty.set}
\{(x,y)\in\mathbb{R}^{n+1}|~ V(x) \leq \beta, (\sum_{i=1}^n x_i^2)\cdot y=1, V(f_i(x))-V(x)\geq 0, i=1,\ldots,m \}
\end{aligned}$$ is empty. Indeed, if there was a point $(\hat{x},\hat{y})$ in this set, then plugging it into (\[eq:stengle.lyap\]) would give a contradiction as the right hand side would evaluate to a nonnegative real number. We observe that emptiness of (\[eq:empty.set\]) is equivalent to emptiness of $$\begin{aligned}
\label{eq:empty.set.2}
\{x \in \mathbb{R}^n|~ V(x) \leq \beta, x \neq 0, V(x)-V(f_i(x))\leq 0, i=1,\ldots,m\},
\end{aligned}$$ which in turn implies that $$\forall x \neq 0, V(x)\leq \beta \Rightarrow V(f_i(x))<V(x), \text{ for } i=1,\ldots,m.$$ From Proposition \[prop:convex.lyap\] (part (ii)), it follows that the switched nonlinear system in (\[eq:nl.systems.again\]) is locally asymptotically stable under arbitrary switching and that the set $$\{ x \in \mathbb{R}^n|~ V(x)\leq \beta\}$$ is a subset of the region of attraction.
We now show the opposite direction. Since $\rho(A_1,\ldots,A_m)<1$, we know from Theorem \[thm:existence-sos-convex-poly\] that there exists a positive definite sos-convex form $V$ of some even degree $r$ such that $$\begin{aligned}
\label{eq-proof-roa}
V(A_ix)<V(x),~\forall x\neq 0, \text{ and } i=1,\ldots,m.
\end{aligned}$$ We prove that there exists a scalar $\beta>0$ such that $$\begin{aligned}
\label{eq:roa.proof}
\forall x \neq 0, V(x)\leq \beta \Rightarrow V(f_i(x))<V(x), \text{ for } i=1,\ldots,m,
\end{aligned}$$ by considering the Taylor expansion of $V$ around the origin. As the maps $f_i,~ i=1,\ldots,m, $ are twice differentiable, we have ote that $V(A_ix)-V(x)$ is a degree-$r$ form which is negative definite. Hence, if we define $$\lambda_i \mathrel{\mathop{:}}=-\frac{1}{2}\min_{||x||=1} (V(A_ix)-V(x)),$$ we have $\lambda_i>0$ and $$\begin{aligned}
\label{eq:neg.def}
V(A_ix)-V(x)<-\lambda_i ||x||^r.
\end{aligned}$$ Let $\epsilon_i=\min(\delta_i, \frac{\lambda_i}{K_i})$ and note that $\epsilon_i>0.$ For any nonzero $x$ such that $||x||\leq \epsilon_i$, we have $$\begin{aligned}
V(f_i(x))-V(x)&< -\lambda_i||x||^r+O(||x||^{r+1})\\
&\leq -\lambda_i ||x||^r+K_i||x||^{r+1}\\
&\leq 0,
\end{aligned}$$ where the first inequality follows from (\[eq:V.f\]) and (\[eq:neg.def\]), the second from (\[eq:defn.big.O\]) as $||x||\leq \delta$, and the third from the fact that $||x||\leq \frac{\lambda_i}{K_i}.$ By compactness of the sublevel sets of $V$ and homogeneity of $V$, there exists $\beta_i>0$ such that $V(x)\leq \beta_i \Rightarrow ||x||<\epsilon_i$. Taking $\beta=\min_{i=1,\ldots,m} \beta_i$ concludes the proof of (\[eq:roa.proof\]).
Now observe that the statement in (\[eq:roa.proof\]) implies that the set in (\[eq:empty.set.2\]) is empty. This is equivalent to the set in (\[eq:empty.set\]) being empty as noted previously. From Theorem \[thm:stengle\], this implies that there exist a polynomial $t$, and sum of squares polynomials $\{\sigma_{a_0\ldots a_m}|~ (a_0,\ldots,a_m)\in \{0,1\}^{m+1}\}$ such that the algebraic identity in (\[eq:stengle.lyap\]) holds.
Theorem \[thm:nl.beta.guarantee\] gives rise to a hierarchy of semidefinite programs whose $r^{th}$ level involves searching for a polynomial $t$ and sum of squares polynomials $$\{\sigma_{a_0\ldots a_m}|~ (a_0,\ldots,a_m) \in \{0,1\}^{m+1} \}$$ of degree less than or equal to $2r$ that satisfy the algebraic identity in (\[eq:stengle.lyap\]) (note that $V$ is fixed here). For fixed $r$, one can obtain the largest $\beta$ for which (\[eq:stengle.lyap\]) is feasible by doing bisection on $\beta$. If the number $m$ of maps and the level $r$ of the hierarchy are fixed, one can check that the size of the resulting SDP is polynomial in the number of variables $n.$
We also remark that this SDP-based procedure terminates in finite time with a full-dimensional estimate of the ROA. Indeed, one can bound the degrees of the polynomials $t_j, j=1,\ldots,r$ and $\sigma_{a_1\ldots a_m}, a_1,\ldots,a_m \in \{0,1\}^m$ in Theorem \[thm:stengle\] by quantities that only depend on the degree of the polynomials $h_i$ and $g_i$, $m$, $n$, and $s$ (see [@lombardi2014elementary] for the precise bound). So in theory, if we fix the degree of the polynomials $t$ and $\sigma_{a_1\ldots a_m}, a_1,\ldots,a_m \in \{0,1\}^m$ in (\[eq:stengle.lyap\]) to that bound, start with any $\beta>0$, and halve $\beta$ when the SDP is infeasible, then the procedure will terminate in finite time with a positive $\beta$ for which the SDP is feasible. The bounds in [@lombardi2014elementary] are too large however to be practical and hence our remark here is of theoretical interest only. In practice, we observe that the first few levels of the hierarchy are sufficient to obtain a full-dimensional estimate of the ROA.
Examples: ROA computation for nonlinear switched systems {#subsec:examples}
--------------------------------------------------------
We give two examples of the ideas we have seen so far for local stability analysis.
\[ex:convex.lyap\] Let us revisit the system (\[eq:ex.nonconvex.fails\]) of Example \[ex:nonconvex.fails\]. We claim that the function $$W(x)=x_1^2+x_2^2,$$ which is convex, is a common Lyapunov function for $f_1,f_2$ on the set $$S=\{x \in \mathbb{R}^n|~||(x_1,x_2)^T||_\infty\leq 1\}.$$ Indeed, for $i=1,2,$ and nonzero $x\in\mathcal{S},$ $$\begin{aligned}
\nonumber W(f_i(x))&=&x_1^2x_2^2 \\ \nonumber &< & x_1^2+x_2^2\\\nonumber &=& W(x).\end{aligned}$$ Moreover, $S$ is an invariant set for $f_1$ and $f_2$. Hence, for the system (\[eq:ex.nonconvex.fails\]), the set $S$ is part of the region of attraction of the origin under arbitrary switching.
We now give an example where quadratic Lyapunov functions do not suffice for a proof of local stability and our SDP procedure is carried out in full.
\[ex:nonlin.roa\]
\[h\]
Consider the dynamical system in (\[eq:nl.systems.again\]), with $m=n=2$ and $$\begin{aligned}
f_1(x_1,x_2)&=\begin{pmatrix}-\frac14 x_1-\frac14 x_2+\frac15 x_1^2\\ -x_1+\frac{1}{10}x_1x_2 \end{pmatrix},\\
f_2(x_1,x_2)&=\begin{pmatrix} \frac34 x_1+\frac34 x_2-\frac{1}{10}x_1x_2\\ -\frac12 x_1+\frac14 x_2 \end{pmatrix}.\end{aligned}$$ The linearizations of $f_1$ and $f_2$ at $(x_1,x_2)=0$ are given by $f_{1}^l(x)=A_1x$ and $f_2^l (x)=A_2x$, where $$A_1=\begin{pmatrix} -1/4&-1/4\\ -1 & 0\end{pmatrix} \text{ and }
A_2=\begin{pmatrix}3/4 & 3/4 \\ -1/2 & 1/4 \end{pmatrix}.$$ One can check that these matrices do not admit a common quadratic Lyapunov function. We will consequently be searching for polynomials of higher order. In this case, imposing convexity becomes essential as it is no longer implied by nonnegativity of the polynomial. Using the parser YALMIP[@yalmip] and the SDP solver MOSEK[@mosek2015], we look for a quartic form $V$ satisfying the sos conditions of Theorem \[thm:existence-sos-convex-poly\]. Our SDP solver returns the sos-convex form $$\begin{aligned}
\label{eq:V.ex}
V(x_1,x_2)=19.14x_1^4+10.57x_1^3x_2+47.88x_1^2x_2^2+16.47x_1x_2^3+10.49x_2^4.\end{aligned}$$ This implies that $\rho(A_1,A_2)<1.$ By solving a second SDP, one can find a polynomial $t$ of degree $\leq 4$ and sos polynomials $\sigma_0$, $\sigma_1$, $\sigma_2$, $\sigma_3$, $\sigma_{12}, \sigma_{23},\sigma_{13}$ and $\sigma_{123}$ of degree $\leq 4$ that satisfy (\[eq:stengle.lyap\]) with $\beta=1.$ From the “easy” direction of Theorem \[thm:nl.beta.guarantee\], this implies that the set $$\{x \in \mathbb{R}^n|~V(x)\leq 1\}$$ is part of the region of attraction of the nonlinear switched system given by $f_1$ and $f_2$. This is illustrated in Figure \[fig:traj\], where we have plotted the 1-level set of $V$, and three possible trajectories of our switched dynamical system. These trajectories are generated by the dynamics $x_{k+1}=\lambda f_1(x_k)+(1-\lambda)f_2(x_k),$ where $x_0=(0.2,0.4)$ for all three trajectories and $\lambda$ is picked uniformly at random in $[0,1]$ at each iteration. As can be seen, all trajectories flow to the origin as predicted by the theory.
Conclusions and extensions to multiple Lyapunov functions {#sec:future}
=========================================================
In this paper, we introduced the concept of sos-convex Lyapunov functions for stability analysis of switched linear and nonlinear systems. For switched linear systems, we proved a converse Lyapunov theorem on guaranteed existence of sos-convex Lyapunov functions. We further showed that the degree of a convex polynomial Lyapunov function can be arbitrarily higher than the degree of a non-convex one. For switched nonlinear systems, we showed that sos-convex Lyapunov functions allow for computation of regions of attraction under arbitrary switching, while non-convex Lyapunov functions in general do not.
Our work can be extended in at least two different directions. The first direction concerns the computation of the region of attraction of the nonlinear switched system in (\[eq:nl.systems.again\]) when the joint spectral radius of the matrices associated to the linearizations of $f_1\ldots,f_m$ is exactly equal to one. In this scenario, the assumption of Theorem \[thm:nl.beta.guarantee\] is violated. Nevertheless, one can directly search for an sos-convex polynomial $V$, a scalar $\beta>0$, a polynomial $t$, and sos polynomials $\sigma_{a_0\ldots a_m}$ satisfying (\[eq:stengle.lyap\]) to have a certificate that the $\beta$-sublevel set of $V$ is in the ROA of the origin. The problem with this approach however is that the coefficients of $V$ and $\sigma_{a_0\ldots a_m}$ are all decision variables and their multiplication leads to a nonconvex constraint. A principled way of getting around this issue with convex relaxations is left for our future work.
The second direction is motivated by scalability issues encountered when solving semidefinite programs arising from sos constraints on high-degree polynomials. In general, it is more efficient to work with *multiple* low-degree sos-convex Lyapunov functions as opposed to a single one of high degree. This is because the underlying semidefinite program will end up having semidefinite constraints on much smaller matrices (though possibly a higher number of them). Nevertheless this trade-off is almost always computationally beneficial for interior point solvers.
A systematic approach for searching for multiple Lyapunov functions that together imply stability of a switched linear system has been proposed in [@ajpr-sicon]. If the switched system is defined by $x_{k+1}=A_i x_k, ~i=1,\ldots,m,$ and our candidate Lyapunov functions are $V_1,\ldots,V_r,$ the works in [@ajpr-sicon] and [@jungers2017characterization] completely characterize all collections of Lyapunov inequalities of the type $$\{V_j(A_ix) < V_k(x)\}$$ that prove stability. This characterization is based on the concept of *path-complete graphs* (see [@ajpr-sicon Definition 2.2]), which is a notion that relates to the theory of finite automata and languages. In our future work, we would like to extend this theory to cover nonlinear switched systems. In this setting, the property of convexity needs to be carefully incorporated, as the current paper has demonstrated. More precisely, we would like to understand which path-complete paths give rise to a common *convex* Lyapunov function, assuming that the nodes of the graph are all associated with convex Lyapunov functions. The proposition below provides a large family of such graphs, though we suspect that there must be others. In the reader’s interest, we present the proposition in a self-contained fashion with no mention to the terminology of path-complete graphs. The common convex Lyapunov function obtained here will be a pointwise maximum of convex functions. A complete study of the more general question above would likely need to extend the ideas in [@philippe2017path Section III] and [@jungers2017characterization Section IV].
For simplicity, we state the proposition below for global asymptotic stability. The analogous statement for local asymptotic stability is simple to derive (similarly to what was done in Proposition \[prop:convex.lyap\]).
\[cor:max.of.quadratics\] Consider the nonlinear switched system in (\[eq:switched.nonlinear.system.ex\]) defined by continuous maps $f_1,\ldots,f_m:\mathbb{R}^n \rightarrow \mathbb{R}^n$. If there exist $K$ convex Lyapunov functions $V_1,\ldots,V_K:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfy $V_i(0)=0$, $V_i(x)>0$ for all $x\neq 0$, $\forall i\in \{1,\ldots,K\},$ and $$\begin{aligned}
\label{eq:max.quadratics.LMIs}
\forall (i,j) \in\{1,\ldots,m\}\times \{1,\ldots,K\},\ \exists k\in\{1,\ldots,K\}\
\nonumber
\\
\mbox{such that}\quad V_j(f_i(x))< V_k(x), \quad \forall x \neq 0,\end{aligned}$$ then the origin is globally asymptotically stable under arbitrary switching. Moreover, if these constraints are satisfied, then the convex function $$W(x)\mathrel{\mathop{:}}=\max \{V_1(x),\ldots ,V_K(x)\}$$ is a common Lyapunov function for $f_1,\ldots, f_m$.
It suffices to show the latter claim because the former would then follow from Proposition \[prop:convex.lyap\], part (i), as it is clear that $W$ so constructed is positive definite and convex. Let $i \in \{1,\ldots,m\}$ be fixed. From (\[eq:max.quadratics.LMIs\]), for any $j \in \{1,\ldots,K\},$ there exists $k \in \{1,\ldots,K\}$ such that $$V_j(f_i(x))<V_k(x), \forall x\neq 0.$$ As $W$ is the pointwise maximum of $V_k, k=1,\ldots,K$, it follows that $$V_j(f_i(x))<W(x), \forall x\neq 0 \text{ and } \forall j\in \{1,\ldots,K\}.$$ Hence $W(f_i(x))< W(x), \forall x\neq 0.$
In the case where $f_1,\ldots,f_m$ are polynomials, and $V_1,\ldots,V_k$ are parametrized as sos-convex polynomials, the search for $W$ can be carried out by semidefinite programming after replacing the inequalities in (\[eq:max.quadratics.LMIs\]) with their sos counterparts. Note that the above proposition does not give just one way of formulating such an SDP, but rather $K^{m^2}$ of them. Indeed, for any fixed pair $(i,j)$, there are $K$ choices for the index $k$. In the language of [@ajpr-sicon], each of these SDPs corresponds to a particular path-complete graph and its feasibility provides a proof of stability.
Acknowledgments {#acknowledgments .unnumbered}
===============
The thankful to Alexandre Megretski for insightful discussions around convex Lyapunov functions.
[^1]: Amir Ali Ahmadi is with the Department of Operations Research and Financial Engineering at Princeton University (email: `a_a_a@princeton.edu`). His research has been partially supported by the DARPA Young Faculty Award, the Young Investigator Award of the AFOSR, the CAREER Award of the NSF, the Google Faculty Award, and the Sloan Fellowship. Raphaël Jungers is an F.R.S.-FNRS Research Associate at the ICTEAM Institute, Université catholique de Louvain (email: `raphael.jungers@uclouvain.be`).
[^2]: Throughout this paper, by the word “stable” we mean asymptotically stable under arbitrary switching.
[^3]: While polynomial dynamical systems are already a broad and significant class of nonlinear dynamical systems, certain extensions are possible. For example, our methodology extends in a straightforward fashion to the case where the functions $f_i$ are rational functions with sign-definite denominators. Extensions to trigonometric dynamical systems may also be possible using the ideas in [@megretski_trig].
[^4]: The constant $\frac{1}{2}$ in $g_{\frac{1}{2}}(x,y)$ of condition **(a)** is arbitrary and chosen for convenience. One can show that $g_{\frac{1}{2}}$ being sos implies that $g_{\lambda}$ is sos for any fixed $\lambda\in[0,1]$. Conversely, if $g_{\lambda}$ is sos for some $\lambda\in(0,1)$, then $g_{\frac{1}{2}}$ is sos.
[^5]: Even though an sos decomposition in general merely guarantees , sos decompositions obtained numerically from interior point methods generically provide proofs of *positivity*; see the discussion in [@AAA_MS_Thesis p.41]. In this paper, whenever we are concerned with asymptotic stability and prove a result about existence of a Lyapunov function satisfying certain sos conditions, we make sure that the resulting inequalities are strict (cf. Theorem \[thm:existence-sos-convex-poly\]).
[^6]: Note that $v_i^Tx_i \neq 0$. In fact, we have $v_i^Tx_i>0$, $\forall i$. Indeed, there exists $\alpha_i>0$ such that $\alpha_iv_i \in C$ and hence $v_i^Tx_i \geq \alpha_i ||v_i||_2^2>0.$
[^7]: We remind the reader that radial unboundedness is not equivalent to radial unboundedness along restrictions to all lines, hence the need for the subtleties in this proof.
|
---
abstract: 'In this paper, Arnold diffusion is proved to be generic phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom: $$H(x,y)=h(y)+\epsilon P(x,y), \qquad x\in\mathbb{T}^3,\ y\in\mathbb{R}^3.$$ Under typical perturbation $\epsilon P$, the system admits “connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided $E$ is bigger than the minimum of the average action, namely, $E>\min\alpha$.'
address: |
Department of Mathematics\
Nanjing University\
Nanjing 210093, China\
author:
- 'Chong-Qing CHENG'
title: |
Arnold diffusion in nearly integrable\
Hamiltonian systems
---
Introduction
============
For nearly integrable Hamiltonian systems, the set of KAM tori has a relatively large Lebesgue measure in phase space. For systems with two degrees of freedom, it implies the dynamical stability: all orbits are stable, the variation of actions stays small for all the time as each 2-dimensional KAM torus separates the 3-dimensional energy level. However, this is a special property of lower-dimensional space, KAM torus of $n$-dimension does not separate $(2n-1)$-dimensional energy level if $n>2$. It is conceivable that the complement of all $n$-dimensional invariant tori forms dense and connected set in phase space. This would mean that by arbitrary small changes of the initial states one would find orbits along which the action variables ultimately escape. The underlying phenomenon is now called “Arnold diffusion".
[**Conjecture**]{} ([@Ar2; @AKN]): [*The typical case in a higher-dimensional problems is topological instability: through an arbitrarily small neighborhood of any point there passes a phase trajectories along which the slow variables drift away from the initial value by a quantity of order 1.*]{}
Since the celebrated example of Arnold [@Ar1] was published half a century ago, there are many works for the study of this problem. In recent years, it has become clear that diffusion is a typical phenomenon in so called [*a priori*]{} unstable systems, refer to [@Be3; @CY1; @CY2; @DLS; @LC; @Tr]. The [*a priori*]{} unstable condition guarantees the existence of normally hyperbolic cylinder, from which one derives certain regularity of the barrier functions. The genericity of the diffusion is obtained by using the regularity [@CY1; @CY2]. There are also many works for the study of the problem, for instance, see [@Bs; @BCV; @Fo; @DH1; @DH2; @GL; @GR1; @GR2; @KL1; @KL2; @X; @Zha].
General perturbation of integrable Hamiltonian is usually called [*a priori*]{} stable system. A bit away from strong complete resonance in such systems, some pieces of normally hyperbolic cylinder still exist and the method for [*a priori*]{} unstable system can also be applied [@BKZ; @Be4]. For [*a priori*]{} stable systems with three degrees of freedom, a notable difficulty occurs at the point of double resonance, around which the cylinder for prescribed single resonance may disappear. The averaged system has two homoclinic orbits associated with different classes in $H_1(\mathbb{T}^2,\mathbb{Z})$. As the energy decreases, the periodic orbit on the cylinder approaches these two homoclinic orbits simultaneously. Thus, the transition chain in $H^1(\mathbb{T}^2,\mathbb{R})$ for the single resonance may break. To solve this difficulty, Mather suggested a path in $H^1(\mathbb{T}^2,\mathbb{R})$ to cross double resonance, along which one moves the cohomology class in the channel determined by the prescribed homology class and switches it to the channel determined by one of these two classes when it is getting close to the double resonance [@Ma6].
In this paper, the path we choose to construct transition chain is different from that suggested by Mather. We find an annulus surrounding the flat of double resonance in $H^1(\mathbb{T}^2,\mathbb{R})$, which has a foliation of circles. Each of these circles is actually a transition chains, possibly, of incomplete intersection. Although the annulus is not so thick, each single resonance path extends into. It allows us to use one of these circles connecting one single resonance path to another. In this way, we find a path of transition chain along which the diffusion orbits are constructed by variational method.
Statement of the main result
----------------------------
We consider nearly integrable Hamiltonian systems with 3 degrees of freedom: $$\label{introeq1}
H(x,y)=h(y)+\epsilon P(x,y), \qquad (x,y)\in\mathbb{T}^3\times\mathbb{R}^3,$$ where $h$ is assumed to strictly convex, namely, the Hessian matrix $\partial^2h/\partial y^2$ is positive definite. It is also assumed that $\min h=0$, both $h$ and $P$ are $C^r$-function with $r\ge 8$.
For $E>0$, let $H^{-1}(E)=\{(x,y):H(x,y)=E\}$ denote the energy level set, $B\subset\mathbb{R}^3$ denote a ball in $\mathbb{R}^3$ such that $\bigcup_{E'\le E+1}h^{-1}(E')\subset B$. Let $\mathfrak{S}_a,\mathfrak{B}_a\subset C^r(\mathbb{T}^3\times B)$ denote a sphere and a ball with radius $a>0$ respectively: $F\in\mathfrak{S}_a$ if and only $\|F\|_{C^r}=a$ and $F\in\mathfrak{B}_a$ if and only $\|F\|_{C^r}\le a$. They inherit the topology from $C^r(\mathbb{T}^3\times B)$.
For perturbation $P$ independent of $y$ (classical mechanical system) we use the same notation $\mathfrak{S}_a,\mathfrak{B}_a\subset C^r(\mathbb{T}^3)$ to denote a sphere and a ball with radius $a>0$
Let $\mathfrak{R}_a$ be a set residual in $\mathfrak{S}_a$, each $P\in\mathfrak{R}_a$ is associated with a set $R_P$ residual in the interval $[0,a_P]$ with $a_P\le a$. A set $\mathfrak{C}_a$ is said cusp-residual in $\mathfrak{B}_a$ if $$\mathfrak{C}_a=\{\lambda P:P\in\mathfrak{R}_a,\lambda\in R_P\}.$$
Let $\Phi_H^t$ denote the Hamiltonian flow determined by $H$. Given an initial value $(x,y)$, $\Phi_H^t(x,y)$ generates an orbit of the Hamiltonian flow $(x(t),y(t))$. An orbit $(x(t),y(t))$ is said to visit $B_{\delta}(y_0)\subset\mathbb{R}^3$ if there exists $t\in\mathbb{R}$ such that $y(t)\in B_{\delta}(y_0)$ a ball centered at $y_0$ with radius $\delta$.
\[mainthm\] Given any two balls $B_{\delta}(x_0,y_0),B_{\delta}(x_k,y_k)\subset\mathbb{T}^3\times\mathbb{R}^3$ and finitely many small balls $B_{\delta}(y_i)\subset \mathbb{R}^3$ $(i=0,1,\cdots,k)$, where $y_i\in h^{-1}(E)$ with $E>0$ and $\delta>0$ is small, there exists a cusp-residual set $\mathfrak{C}_{\epsilon_0}$ such that for each $\epsilon P\in\mathfrak{C}_{\epsilon_0}$, the Hamiltonian flow $\Phi_H^t$ admits orbits which, on the way between passing through $B_{\delta}(x_0,y_0)$ and $B_{\delta}(x_k,y_k)$, visit the balls $B_{\delta}(y_i)$ in any prescribed order.
This is the main result of the paper. It is generic not only in usual sense, but also in the sense of Mañé, namely, it is a typical phenomenon when the system is perturbed by potential.
The same result for time-periodic systems can be proved by using the same method. The statement of the result is: for typical time-periodic perturbations of integrable Hamiltonian with 2-degrees of freedom, the Hamiltonian flow admits orbits passing through any prescribed two balls $B_{\delta}(x_0,y_0)$ and $B_{\delta}(x_k,y_k)$ in the phase space and finitely many small balls $B_{\delta}(y_i)\subset \mathbb{R}^2$ $(i=0,1,\cdots,k)$ in the action variable space. The proof is easier from technical point of view, one can see it in the proof.
Similar result was announced by Mather earlier [@Ma4]. Recently, two other groups (Kaloshin and Zhang, Marco [@Mac]) announced, using different approach, a similar result for time-periodic systems with two degrees of freedom.
The result obtained here is stronger than what was formulated in [@Ar2]. By dropping the requirement that orbit passes two prescribed balls in the phase space and using the same construction, one can get an orbit that visits these balls $B_{\delta}(y_i)\subset \mathbb{R}^3$ $(i=0,1,\cdots,k)$ infinitely many times with any prescribed order, as it was announced in [@Ma4]. Indeed, as finitely many balls are given, there exists a path with finite length passing through finitely many resonance layers and connecting any two of these balls directly. It does not damage the cusp-residual property.
Brief introduction of Mather theory
-----------------------------------
We use variational method to prove the result, which is based on Mather theory. This theory is established for Tonelli Lagrangian.
Let $M$ be a closed manifold. A $C^2$-function $L$: $TM\times\mathbb{T}\to\mathbb{R}$ is called Tonelli Lagrangian if it satisfies the following conditions:
[Positive definiteness]{}. For each $(x,t)\in M\times\mathbb{T}$, the Lagrangian function is strictly convex in velocity: the Hessian $\partial_{\dot x\dot x}L$ is positive definite.
[Super-linear growth]{}. We assume that $L$ has fiber-wise superlinear growth: for each $(x,t)\in M\times\mathbb{T}$, we have $L/\|\dot x\|\to\infty$ as $\|\dot x\|\to \infty$.
[Completeness]{}. All solutions of the Lagrangian equations are well defined for the whole $t\in\mathbb{R}$.
For autonomous systems, the completeness is automatically satisfied, since each orbit entirely stays in certain compact energy level set.
Let $\eta_c(x)$ denote a closed 1-form $\langle\eta_c(x),dx\rangle$ evaluated at $x$, with its first co-homology class $[\langle\eta_c(x),dx\rangle]=c\in H^1(M,\mathbb{R})$. We introduce a Lagrange multiplier $\eta_c=\langle\eta_c(x),\dot x\rangle$. Without danger of confusion, we call it closed 1-form also.
For each $C^1$ curve $\gamma$: $\mathbb{R}\to M$ with period $k$, there is unique probability measure $\mu_{\gamma}$ on $TM\times\mathbb{T}$ so that the following holds $$\int_{TM\times\mathbb{T}}fd\mu_{\gamma}=\frac 1k\int_0^kf(d\gamma(s),s)ds$$ for each $f\in C^0(TM\times\mathbb{T},\mathbb{R})$, where we use the notation $d\gamma=(\gamma,\dot\gamma)$. Let $$\mathfrak{H}^*=\{\mu_{\gamma} |\ \gamma\in C^1(\mathbb{R},M)\ \text{\rm is periodic of}\ k\}.$$ The set $\mathfrak{H}$ of holonomic probability measures is the closure of $\mathfrak{H}^*$ in the vector space of continuous linear functionals. One can see that $\mathfrak{H}$ is convex.
For each $\nu\in\mathfrak{H}$ the action $A_c(\nu)$ is defined as follows $$A_c(\nu)=\int (L-\eta_c)d\nu.$$ It is proved in [@Ma1; @Me] that for each co-homology class $c$ there exists at least one invariant probability measure $\mu_c$ minimizing the action over $\mathfrak{H}$ $$A_c(\mu_c)=\inf_{\nu\in\mathfrak{H}}\int (L-\eta_c)d\nu,$$ called $c$-minimal measure. Let $\mathfrak{H}_c\subset\mathfrak{H}$ be the set of $c$-minimal measures, the Mather set $\tilde{\mathcal {M}}(c)$ is defined as $$\tilde{\mathcal{M}}(c)=\bigcup_{\mu_c\in\mathfrak{H}_c}\text{\rm supp} \mu_c.$$ The $\alpha$-function is defined as $\alpha(c)=-A_c(\mu_c): H^1(M,\mathbb{R})\to\mathbb{R}$, it is convex, finite everywhere with super-linear growth. Its Legendre transformation $\beta: H_1(M,\mathbb{R})\to\mathbb{R}$ is called $\beta$-function $$\beta(\omega)=\max_c (\langle\omega,c\rangle -\alpha(c)).$$ It is also convex, finite everywhere with super-linear growth (see [@Ma1]).
Note that $\int\lambda d\mu_{\gamma}=0$ for each exact 1-form $\lambda$ and each $\mu_{\gamma} \in\mathfrak{H}^*$. Therefore, for each measure $\mu\in\mathfrak{H}$ one can define its rotation vector $\omega(\mu)\in H_1(M,\mathbb{R})$ such that $$\langle[\lambda],\omega(\mu)\rangle=\int\lambda d\mu,$$ for every closed 1-form $\lambda$ on $M$. For a closed curve $\gamma$: $[0,k]\to M$ its rotation vector is defined as $$[\gamma]=\frac{\bar{\gamma}(k)-\bar{\gamma}(0)}k,$$ where $\bar{\gamma}$ stands for a curve in the lift of $\gamma$ to the universal covering $\mathbb{R}^n$. Let $\gamma_k$: $[0,k]\to M$ be a closed curve such that $[\gamma_k]=\omega_k$ and $$\frac 1kA(\gamma_k)=\inf_{[\gamma]=\omega_k}\frac 1k\int_0^kL(d\gamma(t),t)dt.$$ The curve determines a periodic orbit $(\gamma_k,\dot\gamma_k)$ of $\phi^t_L$, the Lagrange flow determined by the Lagrangian $L$, it supports an invariant measure $\mu_k$ whose rotation vector is $\omega_k$. The measure $\mu_k$ is not necessarily minimal for $\beta(\omega_k)$. Nevertheless, if we choose a sub-sequence of closed curves $\{\gamma_{k_i}\}$ such that $$\lim_{k_i\to\infty}\frac 1{k_i}A(\gamma_{k_i})=\liminf_{k\to\infty}\inf_{[\gamma]=\omega_k}\frac
1k\int_0^kL(d\gamma(t),t)dt,$$ and if $[\gamma_{k_i}]\to\omega$, then $$\lim_{k_i\to\infty}\frac 1{k_i}A(\gamma_{k_i})=\beta(\omega).$$ Clearly, there is at least one invariant measure $\mu$ such that $\mu_{k_i}\rightharpoonup\mu$, and $\mu$ is a holonomic probability measure with the prescribed rotation vector $\omega(\mu)=\omega$. According to the definition of holonomic measure, and due to the work in [@Me], we have $$\beta(\omega)=\inf _{\nu\in\mathfrak{H}_{\omega}}\int\ell d\nu$$ where $\mathfrak{H}_{\omega}$ is a set of holonomic probability measures with the given rotation vector $\omega$, not necessarily invariant for $\phi_L^t$.
The Fenchel-Legendre transformation $\mathscr{L}_{\beta}$: $H_1(M,\mathbb{R})\to H^1(M,\mathbb{R})$ is defined by the following relation $$c\in\mathscr{L}_{\beta}(\rho)\ \ \iff \ \ \alpha(c)+\beta(\rho)=\langle c,\rho\rangle.$$
The concept of semi-static curves is introduced by Mather and Mañé (cf. [@Ma2; @Me]). A curve $\gamma$: $\mathbb{R}\to M$ is called $c$-semi-static if in time-1-periodic case we have $$[A_c(\gamma)|_{[t,t']}]=F_c((\gamma(t),t),(\gamma(t'),t'))$$ where $$\label{introeq2}
[A_c(\gamma)|_{[t,t']}]=\int_{t}^{t'}\Big (L(d\gamma(t),t)-\eta_c(d\gamma(t))\Big ) dt+\alpha(c)(t'-t),\notag$$ $$F_c((x,t),(x',t'))=\inf_{\stackrel {\tau=t\ \text{\rm mod}\, 1}{\scriptscriptstyle \tau '=t'\text{\rm mod}\,1}} h_c((x,\tau),(x',\tau')),$$ in which $$\label{introeq3}
h_c((x,\tau),(x',\tau'))=\inf_{\stackrel{\stackrel {\xi\in C^1}{\scriptscriptstyle \xi(\tau)=x}}{\scriptscriptstyle \xi(\tau')=x'}}[A_c(\xi)|_{[\tau,\tau']}].\notag$$ In autonomous case, the period can be considered as any positive number. Consequently, the notation of semi-static curve in this case is somehow simpler $$[A_c(\gamma)|_{(t,t')}]=F_c(\gamma(t),\gamma(t')),$$ where $$\label{introeq4}
F_c(x,x')=\inf_{\tau>0}h_c((x,0),(x',\tau)).\notag$$
[**Convention**]{}: Let $I\subseteq\mathbb{R}$ be an interval $($either bounded or unbounded$)$. A continuous map $\gamma$: $I\to M$ is called curve. If it is differentiable, the map $d\gamma=(\gamma,\dot\gamma)$: $I\to TM$ is called orbit. When the implication is clear without danger of confusion, we use the same symbol to denote the graph, $\gamma:=\cup_{t\in I}(\gamma(t),t)$ is called curve and $d\gamma:=\cup_{t\in I}(\gamma(t),\dot\gamma(t),t)$ is called orbit. In autonomous system, the terminology also applies to the image: $\gamma:=\cup_{t\in I}\gamma(t)$ is called curve and $d\gamma:=\cup_{t\in I}(\gamma(t),\dot\gamma(t))$ is called orbit.
A semi-static curve $\gamma\in C^1(\mathbb{R},M)$ is called $c$-static if, in addition, the relation $$[A_c(\gamma)|_{(t,t')}]=-F_c((\gamma(t'),\tau'),(\gamma(t),\tau))$$ holds in time-1-periodic case and $$[A_c(\gamma)|_{(t,t')}]=-F_c(\gamma(t'),\gamma(t))$$ holds in autonomous case. An orbit $X(t)=(d\gamma(t), t\, \text{\rm mod}\ 2\pi)$ is called $c$-static (semi-static) if $\gamma$ is $c$-static (semi-static). We call the Mañé set $\tilde{\mathcal {N}}(c)$ the union of $c$-semi-static orbits $$\tilde{\mathcal{N}}(c)=\bigcup\{d\gamma:\gamma\ \text{\rm is}\ c\text{\rm -semi static}\}$$ and call the Aubry set $\tilde{\mathcal {A}}(c)$ the union of $c$-static orbits $$\tilde{\mathcal{A}}(c)=\bigcup\{d\gamma :\gamma\ \text{\rm is}\ c\text{\rm -static}\}.$$
We use $\mathcal {M}(c)$, $\mathcal {A}(c)$ and $\mathcal {N}(c)$ to denote the standard projection of $\tilde{\mathcal {M}}(c)$, $\tilde{\mathcal {A}}(c)$ and $\tilde{\mathcal {N}}(c)$ from $TM\times\mathbb{T}$ to $M\times\mathbb{T}$ respectively. They satisfy the inclusion relation $$\tilde{\mathcal{M}}(c)\subseteq\tilde{\mathcal{A}}(c)\subseteq\tilde{\mathcal{N}}(c).$$ It is showed in [@Ma1; @Ma2] that the inverse of the projection is Lipschitz when it is restricted to $\mathcal {A}(c)$ as well as to $\mathcal {M}(c)$. By adding subscript $s$ to $\mathcal{N}$, i.e. $\mathcal{N}_s$ we denote its time-$s$-section. This principle also applies to $\tilde{\mathcal{N}}(c)$, $\tilde{\mathcal{A}}(c)$, $\tilde{\mathcal{M}}(c)$, $\mathcal{A}(c)$ and $\mathcal{M}(c)$ to denote their time-$s$-section respectively. For autonomous systems, these sets are defined without the time component.
On the time-1-section of Aubry set a pseudo-metric $d_c$ is introduced by Mather in [@Ma2], its definition relies on the quantity $h_c^{\infty}$. Let $$\label{introeq5}
h_c^{\infty}((x,s),(x',s'))=\liminf_{\stackrel{\stackrel{s=t\ \text{\rm mod}\ 1}{\scriptscriptstyle t'=s'\ \text{\rm mod}\ 1}}{\scriptscriptstyle t'-t\to\infty}}h_c((x,t),(x',t')),\notag$$ $$\label{introeq6}
h^{\infty}_c(x,x')=\liminf_{k\to\infty}h_c((x,0),(x',k)).\notag$$ The pseudo-metric $d_c$ on Aubry set is defined as $$d_c((x,t),(x',t'))=h_c^{\infty}((x,t),(x',t'))+h_c^{\infty}((x',t'),(x,t)).$$ With the pseudo-metric $d_c$ one defines equivalence class in Aubry set. The equivalence $(x,t)\sim (x',t')$ implies $d_c((x,t),(x',t'))=0$, with which one can define quotient Aubry set $\mathcal{A}(c)/\sim$. Its element is called Aubry class, denoted by $\mathcal{A}_i(c)$, its lift to $TM\times\mathbb{T}$ is denoted by $\tilde{\mathcal{A}}_i(c)$. Thus, $$\mathcal{A}(c)= \bigcup_{i\in\Lambda}\mathcal{A}_i(c), \qquad \tilde{\mathcal{A}}(c)=\bigcup_{i\in\Lambda}\tilde{\mathcal{A}}_i(c).$$ In [@Ma5] Mather constructed an example with some quotient Aubry set homeomorphic to an interval. However, it is proved generic in [@BC] that, for system with $n$ degrees of freedom, each $c$-minimal measure contains not more than $n+1$ ergodic components. In this case, each Aubry set contains at most $n+1$ classes.
The definition of semi-static curve as well as of Mañé set depends on which configuration manifold under our consideration. Let $\pi:\bar M\to M$ be a finite covering, a curve $\gamma$: $\mathbb{R}\to M$ is said semi-static in $\bar M$ if each curve in its lift $\bar\gamma$ is semi-static in $\bar M$. Accordingly, we define $\tilde{\mathcal{N}}(c,\bar M)$ ($\mathcal{N}(c,\bar M)$) as the set containing all $c$-semi-static orbits (curves) in $\bar M$. We use the symbol $\mathcal{N}(c)$ when $M$ is defaulted as the configuration manifold.
It is possible that $\pi\mathcal{N}(c,\bar M)\supsetneq\mathcal{N}(c,M)$. For instance, if $N\subset M$ is a open region such that $H_1(M,N,\mathbb{Z})\neq H_1(M,\mathbb{Z})$, $\mathcal{N}\subset N$ and the lift of $N$ in $\bar M$ has more than one connected component, then this phenomenon takes place. But we have
Let $\pi:\bar M\to M$ be a finite covering space, then $$\pi\mathcal{A}(c,\bar M)=\mathcal{A} (c,M).$$
Pick up any $\bar x\in\pi\mathcal{A}(c,\bar M)$ and any small $\delta>0$, by definition, there exists sufficiently large $T>0$ as well as a curve $\bar\xi:[0,T]\to\bar M$ such that $\bar\xi(0)=\bar\xi(T)=\bar x$ and $[A_c(\bar\xi)]<\delta$. Let $\xi$ denote the project of $\bar\xi$ down to $M$, clearly, we have $[A_c(\xi)]=[A_c(\bar\xi)]<\delta$. Let $x=\pi\bar x$, clearly, $x\in\mathcal{A}(c)$.
Outline of the proof
--------------------
We use variational method to prove the result. Since the work of Mather [@Ma2; @Ma3], the variational method has become a powerful tool for the study of dynamical instability in positive definite Lagrange systems with multiple degrees of freedom.
In the study of Arnold diffusion in [*a priori*]{} stable systems with three degrees of freedom, mainly due to the work of Mather [@Ma6], it has been widely known that the main difficulty takes place near double resonance. To describe what puzzled us and to explain the strategy of our proof, let us recall the example of Arnold and previous study on [*a priori*]{} unstable systems.
In the example of Arnold, there exists a 2-dimensional cylinder in the phase space, which is invariant and normally hyperbolic for the time-1-map determined by the Hamiltonian flow. This cylinder is foliated into a family of invariant circles, each of them has stable and unstable manifold which intersect each other transversally. Consequently, the unstable manifold of some circle intersects the stable manifold of other circles nearby, it implies the existence of a sequence of successively connected heteroclinic orbits. This structure is called transition chain by Arnold. Diffusion orbits are then constructed shadowing these heteroclinic orbits.
Such argument heavily depends on the geometric structure and variational method turns out to have wider range of application. Let us interpret the proof by variational language. Each invariant circle is the Aubry set for certain cohomology class, the stable as well as the unstable manifold is actually the graph of the differential of the weak KAM solution. Expressed as the difference of backward and forward weak KAM, the barrier function reaches its minimum at primary intersection points of these two manifolds. These homoclinic orbits and the Aubry set constitute the Mañé set in certain finite covering space. For positive definite systems, the transversal intersection implies the minimality of the homoclinic orbits as well as local heteroclinic orbits, along which the Lagrange action reaches the minimum among all those curves with the same boundary conditions. The diffusion orbits are obtained by searching for global minimizers which generate orbits shadowing a sequence of local heteroclinic orbits.
The variational arguments still work even if there does not exist such nice geometric structure, provided the following conditions are satisfied for each cohomology class
[1, *the Aubry set is lower dimensional: $H_1(M,\mathcal{A}(c),\mathbb{Z})\neq 0$*]{};
[2, *the stable “manifold" intersects the unstable “manifold" transversally*]{}.
However, it turns out very difficult to verify whether the stable “manifold" intersects the unstable “manifold" transversally for each cohomology class along a path of first cohomology class. As uncountably many stable and unstable “manifolds" have to be considered, one can not verify the genericity of the transversal intersection by taking the intersection of countably many open-dense sets. One possible way is to study some regularity of barrier functions with respect to some parameter, with which one obtains the finiteness of Hausdorff dimensions of the set of barrier functions. As such regularity is obtained in the case when a normally hyperbolic cylinder exists we are succeeded in solving the problem in [*a priori*]{} unstable case [@CY1; @CY2; @LC]. In fact, the regularity was obtained in [@CY1] only for those barrier functions for which the minimal measure is supported on an invariant circle. It was extended in [@Zho1] to all other barrier functions, which allows us to construct diffusion orbits of which the picture looks like what was constructed by Arnold, while in our previous work, the constructed orbits keep close to the cylinder when they pass through strong resonance (Birkhoff instability region). The “gap" problem was then solved.
Intuitively, diffusion orbits in [*a priori*]{} stable systems may be constructed along some resonant path. In terms of rotation vector (first homology class), each point on this curve satisfies at least one resonant condition for the system with $3$ degrees of freedom. In integrable systems, each resonant path corresponds to an invariant cylinders without any hyperbolicity. Under generic perturbations, it breaks into many pieces of normally hyperbolic cylinder, but may disappear around double resonant points. It implies a bad consequence: we lost a handhold to get certain regularity of barrier functions in suitable parameter. It then becomes unclear whether there is a transition chain near double resonance.
In terms of first cohomology, strong double resonance corresponds to a convex disc with size $O(\sqrt{\epsilon})$ if the perturbation is of order $O(\epsilon)$, each piece of normally hyperbolic cylinder corresponds to a channel which extends to a small neighborhood of the disc. But it is unclear whether these channels are connected to the disc of double resonance.
The method we use to overcome this difficulty bases on following discoveries:
First, each double resonant disc is surrounded by an annulus foliated into a family of paths, along each of these paths, there is another invariant (a coordinate component of the cohomology class) besides the average action. For each class in this annulus, the intersection of stable “manifold" with unstable “manifold" is nontrivial although it may not be transversal. This annulus has width of order $O(\epsilon)$.
Next, incomplete intersection of the stable and unstable “manifold" of an Aubry set does not implies that it can not be connected to any other Aubry set nearby. It does if they are $c$-equivalent.
Finally, the channels of normally hyperbolic cylinder reach to somewhere $\epsilon^{1+\delta}$-close to double resonant disc ($\delta>0$), i.e. it has overlap with the annulus. For each class in the channel, the relative homology of the Aubry set is non-trivial.
Therefore, we are able to find a path close to prescribed one, for each class on the path, the Aubry set is connected to another one nearby if the class is also on the path. All of these connecting orbits are minimal in local sense. The diffusion orbits are constructed shadowing these successively connected orbits.
We organize the proof in following way. Section 2 is used to establish the concept of elementary weak KAM, with which one has simpler expression of barrier function. It thus becomes easier to study genericity of transition chain. Section 3, 4 and 5 are devoted to study the structure of Mañé set and of Aubry set. Since these sets are symplectic invariants [@Be2], we do it by studying the normal form which is put into the appendix. The truncated Hamiltonian of the normal form is a system with two degrees of freedom. In Section 3, we study the dynamics around the double resonance, and the modulus continuity of the period on energy (average action). With these preliminary works, normally hyperbolic cylinder is shown to get very close to the double resonance in Section 4, the existence of annulus of $c$-equivalence is established in Section 5. Section 6 is devoted to establish two types of local connecting orbits. The local minimality of these local connecting orbits are naturally given, it enables us to construct global connecting orbit shadowing these local connecting orbits. It is obtained by searching for the minimizer of certain modified Lagrangian, which is done in Section 7. Finally in Section 8, we verify the cusp-residual property of the transition chain in nearly integrable Hamiltonian systems with three degrees of freedom. Consequently, the main result of this paper is proved.
Elementary Weak KAM and Barrier
===============================
The concept of elementary weak KAM solution is introduced for $c$-minimal measure with finitely many ergodic components, this condition has been shown to be generic in [@BC]. Each ergodic component $\mu^i_c$ determines a pair of elementary weak KAM $u^{\pm}_{c,i}$, with which we introduce a barrier function $$B_{c,i,j}=u^-_{c,i}-u^+_{c,j}.$$ With this formula, it is easier to show that, generically, the set $\arg\min B_{c,i,j}\backslash(\mathcal{A}(c)+\delta)$ is totally disconnected. This property is crucial for the construction of diffusion orbits.
Elementary weak KAM
-------------------
The concept of $c$-semi-static curves can be extended to the curves only defined on $\mathbb{R}^{\pm}$, which are called forward or backward $c$-semi-static curves respectively. Usually one uses $\gamma^-_c(t,x,\tau)$: $(-\infty,\tau]\to M$ to denote backward $c$-semi-static curve such that $\gamma^-_c(\tau)=x$, and uses $\gamma^+_c(t,x,\tau)$: $[\tau,\infty)\to M$ to denote forward $c$-semi-static curve such that $\gamma^+_c(\tau)=x$. In autonomous case, one uses the notation $\gamma^{\pm}_c(t,x)$ such that $\gamma^{\pm}(0,x)=x$. Let $$\begin{aligned}
\tilde{\mathcal{N}}^+(c)&=\{(x,\dot x,\tau)\in TM\times\mathbb{T}: \pi_x\phi^t_L (z,\tau)|_{[\tau,+\infty)}\ \text{\rm is \it c\rm-semi-static}\},\notag\\
\tilde{\mathcal{N}}^-(c)&=\{(x,\dot x,\tau)\in TM\times\mathbb{T}: \pi_x\phi^t_L (z,\tau)|_{(-\infty,\tau]}\ \text{\rm is \it
c\rm-semi-static}\},\notag\end{aligned}$$ where $0\le\tau<1$, $\pi_x(x,\dot x)=x$ denotes the standard projection along the tangent fiber and $\phi^t_L(x,\dot x,\tau)$ denotes the orbit of the Lagrangian flow with the initial value $(x,\dot x)$ at the time $\tau$. The corrsponding orbits are called forward (backward) semi-static orbit set respectively. These two sets are upper semi-continuous for the cohomology class.
\[weakpro1\] If the Lagrangian $L$ is of Tonelli type, for each point $(x,\tau)\in M\times\mathbb{T}$, there is at least one $\gamma^{\pm}_c(t,x,\tau)$ which is forward $($backward$)$ semi-static curve.
As both the $\omega$-limit set of $d\gamma_c^+$ and the $\alpha$-limit set of $d\gamma_c^-$ are in the Aubry set one can define $$W^{\pm}_c=\bigcup_{(x,\tau)\in M\times\mathbb{T}}
\left\{x,\tau,\frac{d\gamma^{\pm}_c(\tau,x,\tau)}{dt}\right\},$$ and call $W^+_c$ the stable set, $W^-_c$ the unstable set of the $c$-minimal measure respectively. If $\dot\gamma^-(\tau,x,\tau)=\dot\gamma^+(\tau,x,\tau)$ holds for some $(x,\tau)\in M\times\mathbb{T}$, passing through the point $(x,\tau,\dot\gamma^-_c(\tau,x,\tau))$ the orbit is either in the Aubry set or homoclinic to this Aubry set.
When the Aubry set contains only one class, the stable as well as the unstable set has its own generating function $u_c^{\pm}$ such that $W^{\pm}_c=\text{\rm Graph}(du_c^{\pm})$ holds almost everywhere [@Fa1; @E]. These functions are weak KAM solutions, which are the fixed points of so called Lax-Oleinik operator. We use $u_c^{\pm}$ to denote the weak KAM solution for the Lagrangian $L-\eta_c$, where $\eta_c$ is a closed form with $[\eta_c]=c$. These functions are Lipschitz, thus differentiable almost everywhere. At each differentiable point $(x,\tau)$, $(x,\tau,\partial_xu^-(x,\tau))$ uniquely determines backward $c$-semi static curve $\gamma^-_x$: $(-\infty, \tau]\to M$ such that $\gamma^-_x(\tau)=x$, $\dot\gamma^-_x(\tau)=\partial_yH(x,\tau,\partial_xu^-(x,\tau))$. Similarly, $(x,\tau,\partial_xu^+(x,\tau))$ uniquely determines forward $c$-semi static curve $\gamma^-_x$: $[\tau,\infty)\to M$ such that $\gamma^+_x(\tau)=x$, $\dot\gamma^+_x(\tau)=\partial_yH(x,\tau,\partial_xu^+(x,\tau))$.
Given a class $c\in H^1(M,\mathbb{R})$, we use $\{\tilde{\mathcal{A}}_c^i\}_{i\in\Lambda} \subset TM$ to denote the set of Aubry classes, use $\{\mathcal{A}^i_c\} _{i\in\Lambda}\subset M$ to denote the projected set along the tangent fibers, where $\Lambda$ is the subscript set: $\tilde{\mathcal{A}}(c)=\cup_{i\in\Lambda}\tilde{\mathcal{A}}_c^i$. We also use the notation $\tilde{\mathcal{M}}_c^i=\text{\rm supp}\mu_c^i$ where $\mu_c^i$ is an ergodic component of the $c$-minimal measure $\mu_c$. Let $\mathcal{M}_c^i=\pi\tilde{\mathcal{M}}_c^i$.
\[weakpro2\] [([@Fa2])]{} Let $u^{\pm}_c$ and $u'^{\pm}_c$ be two weak-KAM solutions for $c$. Their difference keeps constant when they are restricted on an Aubry class $(u^{\pm}_c-u'^{\pm}_c)|_{\mathcal{A}_c^i}=\text{constant}$.
Recall the definition of $h_c^{\infty}$ in the introduction. We use the symbol $h^{\infty}_L$ to denote the quantity defined in the same way for $L$ with $c=0$, and drop the subscript $L$ when it is clearly defined. The quantity $h_c^{\infty}(z,z')$ is a weak-KAM solution if we consider it as the function of $z$ or of $z'$. Let us consider the case that the $c$-minimal invariant measure has finitely many ergodic components. In this case, this function has some kind of continuity.
\[weakthm1\] Let $\{L_{\delta}\}$ be a sequence of Lagrangian, converging to $L$ in $C^2$-topology as $\delta\to 0$ when they are restricted on any bounded regions of $TM\times\mathbb{T}$. We assume that the minimal measure for $L$ consists of finitely many ergodic components $\mu^1,\mu^2$, $\cdots,\mu^m$ and the distance from $(x,\tau)\in\mathcal{M}^i$ to the Aubry set for $L_{\delta}$, $d((x,\tau), \mathcal{A}_{L_{\delta}})\to 0$ as $\delta\to 0$. Then $$\lim_{\delta\to 0}h_{L_\delta}^{\infty}((x,\tau),(x',\tau'))=h^{\infty}((x,\tau),(x',\tau')).$$
We only need to prove it on the time-1-section, e.g. for $\tau=\tau'=0$. So we omit the notation for the component $\tau$. For each $\epsilon>0$, there exists $k>0$ such that $|h^{\infty}(x,x') -h^{k}(x,x')|<\epsilon$. Let $\alpha$ and $\alpha_{\delta}$ denote the minimal average action of $L$ and $L_{\delta}$ respectively. Let $\gamma^k$: $[0,k]\to M$ be the curve such that $\gamma^k(0)=x$, $\gamma^k(k)=x'$ and $$[A_L(\gamma^k)]=\int_0^{k}L (d\gamma^{k}(t),t)dt+k\alpha=h^{k}(x,x').$$ For any $k'>k$, let $\zeta$: $[0,k']\to M$ be an absolutely continuous curve such that $\zeta(0)=x$, $\zeta(t-k'+k)=\gamma^k(t)$ for $t\in[k'-k,k']$ and $[A_{L_{\delta}}(\zeta|_{[0,k'-k]})]= \int_0^{k'-k}L_{\delta} (d\zeta(t),t)dt+(k'-k)\alpha_{\delta}=h_{L_{\delta}}^{k'-k}(x,x)$. Therefore, we have $$\begin{aligned}
[A_{L_{\delta}}(\zeta)]&\le h_{L_\delta}^{k'-k}(x,x)+h^{k}(x,x')+k|\alpha-\alpha_{\delta}|\\
& +\left|\int_0^k(L-L_{\delta})(d\gamma^k(t),t)dt\right|.\end{aligned}$$ Since $d(x, \mathcal{A}_{L_{\delta}}|_{t=0})\to 0$ as $\delta\to 0$ we see that $\liminf_{k'-k}h_{\delta}^{k'-k}(x,x) \to 0$ as $\delta\to 0$. Since $\alpha$ is continuous in the Lagrangian and $\epsilon$ is arbitrarily small we see that $$\limsup_{\delta\to 0}h_{L_\delta}^{\infty}(x,x')\le h^{\infty}(x,x').$$
So, in order to complete the proof, we only need to show $$\label{weakeq2}
\liminf_{\delta\to 0}h_{L_\delta}^{\infty}(x,x')\ge h^{\infty}(x,x').$$ Let $\gamma_{\delta}^{k_{\ell}}$: $[0,k_{\ell}]\to M$ be a curve such that $\gamma_{\delta}^{k_{\ell}}(0)=x$, $\gamma_{\delta}^{k_{\ell}}(k_{\ell})=x'$ and $$[A_{\delta}(\gamma_{\delta}^{k_{\ell}})]=h_{L_\delta}^{k_{\ell}}(x,x')\to h_{L_\delta}^{\infty}(x,x'),$$ where $k_{\ell}\to\infty$ is a sequence of integers. Let $O_{\epsilon}(S)$ denote the $\epsilon$-neighborhood of the set $S$. For small $\epsilon>0$, there exist some $j$ with $1\le j\le m$, an integer $k_j\in [0,k_{\ell}]$ and $x_j\in\mathcal {M}^j_{0}=\mathcal {M}^j|_{t=0}$ such that $\gamma_{\delta}^{k_{\ell}}(k_j)\in O_{\epsilon}(x_j)$ provided $k_{\ell}$ is sufficiently large.
Let us consider those ergodic components of minimal measure for $L$ of which the support is approached by $d\gamma^{k_{\ell}}_{\delta}$ as $\delta\to 0$: $\{d\gamma^{k_{\ell}}\}\cap O_{\epsilon} (\tilde{\mathcal{M}}^j)\neq\varnothing$. We number some $j$ as $j_1$ if some $x_1\in\mathcal{M}^{j_1}_{0}$ exists such that $\gamma^{k_{\ell}}_{\delta}(k_1)\in O_{\epsilon}(x_1)$ and for each $k<k_1$, $\gamma^{k_{\ell}}_{\delta}(k)$ does not fall into $\epsilon$-neighborhood of any $\mathcal{M}^{j}_{0}$. Let $k'_1\ge k_1$ be the integer such that $\gamma^{k_{\ell}}_{\delta}(k'_1)\in O_{\epsilon}(x_1)$ and $\gamma^{k_{\ell}}_{\delta}(k)\notin O_{\epsilon}(x_1)$ for all $k>k'_1$. We number some $j_2\ne j_1$ if some $k_2>k'_1$ and some $x_2\in\mathcal{M}^{j_2}_{0}$ exists such that $\gamma^{k_{\ell}}_{\delta}(k_2)\in O_{\epsilon}(x_2)$, let $k'_2\ge k_2$ be the integer such that $\gamma^{k_{\ell}}_{\delta}(k'_2)\in O_{\epsilon}(x_2)$ and $\gamma^{k_{\ell}}_{\delta}(k')\notin O_{\epsilon}(x_2)$ for all $k>k'_2$. Inductively, one obtains $x_i\in\mathcal{M}^{j_i}_{0}$ ($i=1,2\cdots m'\le m$) and $$0\le k_1\le k'_1\le\cdots\le k_{m'}\le k'_{m'}\le k_{\ell}.$$ Obviously, there exist small $\varepsilon=\varepsilon(\epsilon)>0$ and large integer $K=K(\epsilon)$ such that $$|k_j-k'_{j-1}|\le K, \qquad \forall\ k_{\ell}\to\infty$$ provided $|L_{\delta}-L|<\varepsilon$. Otherwise, there would exist also an ergodic component $\nu$ of the minimal invariant measure such that $\nu\neq\mu^j$ for all $0\le j\le m$, but it is absurd.
Given small $\epsilon>0$, let $k_{\ell}$ be the integer such that $|h_{L_\delta}^{k_{\ell}}(x,x') -h_{L_\delta}^{\infty}(x,x')|<\epsilon$. Let $\bar x_j=\gamma_{\delta}^{k_{\ell}}(k_j)$, $\tilde x_j= \gamma_{\delta}^{k_{\ell}}(k'_j)$, we choose an absolutely continuous curve $\zeta_j$: $[0,k_{\ell}^j]\to M$ such that $\zeta_j(0)=\bar x_j$, $\zeta_j(k_{\ell}^j)=\tilde x_j$ and $[A(\zeta_j)]=h^{k_{\ell}^j}(\bar x_j,\tilde x_j)$. As $\bar x_j, \tilde x_j\in O_{\epsilon}(x_j)$ we can choose sufficiently large $k_{\ell}^j$ such that $$|h^{k_{\ell}^j}(\bar x_j,\tilde x_j)|< C\epsilon,$$ where $C=C(L)$ is a constant depending on $L$ only. As $\|\bar x_j-\tilde x_j\|\le 2\epsilon$, for any positive integer $i$ we have $$h^{i}_{L_\delta}(\bar x_j,\tilde x_j)>-C\epsilon.$$ For any large integer $k'\in\mathbb{Z}$ with $k'\ge k$, we construct an absolutely continuous curve $\zeta$: $[0,k']\to M$ joining $x$ with $x'$ such that $$\zeta(t)=\begin{cases}
\gamma_{\delta}^{k_{\ell}}(t-\tau_{j-1}), &\text{\rm if}\ k'_{j-1}+\tau_{j-1}\le t\le k_j+\tau_{j-1},\\
\zeta_j(t-k_j-\tau_{j-1}), & \text{\rm if}\ k_j+\tau_{j-1}\le t\le k'_j+\tau_j
\end{cases}$$ where $\tau_j=\sum_{\jmath=1}^j(k_{\ell}^{\jmath}-k'_{\jmath}+k_{\jmath})$, $k'=k_{\ell}+\tau_{m'}$. The action of $L$ along this curve is easily estimated $$\begin{aligned}
h^{k'}(x,x')-h_{L_\delta}^k(x,x')&\le &[A(\zeta)]-h_{L_\delta}^k(x,x')\\
&\le & 2m'(C\epsilon +K|\alpha-\alpha_{\delta}|)\\
& &+\sum _{j=1}^{m'}\left|\int_{k'_{j-1}}^{k_j}(L-L_{\delta})(d\gamma_{\delta}^{\ell}(t),t)dt\right|.\end{aligned}$$ As $|k_j-k_{j-1}|\le K$, and $K$ is independent of $\delta$ when $\delta$ is sufficiently close to $0$, we see that the inequality (\[weakeq2\]) holds. This completes the proof.
Let $\{c_i\}$ be a sequence of cohomology classes such that $c_i\to c$. The $c$-minimal measure is assumed consisting of finitely many ergodic components $\mu_c^1$, $\mu_c^2$, $\cdots, \mu_c^m$, $(x,\tau)\in\mathcal{M}_{c}^j$ and $d((x,\tau),\mathcal{M}(c_i))\to 0$ for some $0\le j\le m$, as $c_i\to c$. Then $$\lim_{c_i\to c}h_{c_i}^{\infty}((x,\tau),(x',\tau'))=h_c^{\infty}((x,\tau),(x',\tau')).$$
From the proof one can see that the function $h_c^{\infty}$ is lower semi-continuous in $c$ if the $c$-minimal measure is assumed to have finitely many ergodic components: $$\liminf_{c'\to c}h_{c'}^{\infty}(z,z')\ge h_c^{\infty}(z,z').$$
Let us introduce the concept of [*elementary*]{} weak KAM solution if the $c$-minimal measure has finitely many ergodic components. One can choose finitely many non-negative functions $g_i$: $M\times\mathbb{T}\to\mathbb{R}$ such that its support has no intersection with a small neighborhood of $\mathcal{M}_c^i$ and the minimal measure for the Lagrangian $L_{c,i,\epsilon}=L_c+\epsilon g_i$ is uniquely supported on $\mathcal{M}_c^i$. By the theory of weak KAM ([@Fa2]), there is exactly one pair of weak KAM solutions denoted by $u_{c,i,\epsilon}^{\pm}$ and $$h^{\infty}_{L_{c,i,\epsilon}}(z,z')=u_{c,i,\epsilon}^{-}(z')-u_{c,i,\epsilon}^{+}(z).$$ Let $z\in\mathcal{M}_c^i$, in virtue of Theorem \[weakthm1\], one has $h^{\infty}_{L_{c,i,\epsilon}}(z,z')\to h^{\infty}_{c}(z,z')$ as $\epsilon\to 0$. Since $g_i=0$ in the neighborhood of $\mathcal{M}_c^i$, $u_{c,i,\epsilon}^{+}(z)$ remains unchanged as $\epsilon\to 0$. Thus, there is a Lipschitz function $u^{-}_{c,i}$ such that $u_{c,i,\epsilon}^{-} \to u^{-}_{c,i}$ as $\epsilon\to 0$. Clearly this $u^{-}_{c,i}$ is a weak-KAM solution for $L_c$. Similarly, we can see that $u_{c,i,\epsilon}^{+}\to u^{+}_{c,i}$ as $\epsilon\to 0$.
[(elementary weak-KAM solution)]{}. Assume that the minimal measure for $L_c$ consists of finitely many ergodic components $\mu_c^1$, $\mu_c^2,\cdots, \mu_c^m$. A weak KAM solution $u_{c,i}^{\pm}$ of $L_c$ is called [*elementary*]{} for $\mu_c^i$ if $u_{c,i}^{\pm}=\lim_{\epsilon\to 0}u_{c,i,\epsilon}^{\pm}$ where $u_{c,i,\epsilon}^{\pm}$ is the weak KAM solution of $L_{c,i,\epsilon}$, of which the minimal measure $\mu=\mu_c^i$ is uniquely ergodic and $L_{c,i,\epsilon}\to L_c$ as $\epsilon\to 0$.
It is not necessary that $u_{c,i}^{\pm}$ is a pair of conjugate weak KAM. Clearly, if $(x,t)\in\mathcal{M}_c^i$ $$\begin{aligned}
&u_{c,i}^{-}(x',t')=h_c^{\infty}((x,t),(x',t'))+u_{c,i}^{+}(x,t),\notag\\
&u_{c,i}^{+}(x',t')=u_{c,i}^{-}(x,t)-h_c^{\infty}((x',t'),(x,t)).\notag\end{aligned}$$
These elementary weak KAM solutions generate all weak KAM solutions in the following sense.
\[weakpro3\] Assume the minimal measure consists of $m$ ergodic components. For each weak KAM solution $u^{\pm}$, there exist $m'$ $(m'\le m)$ constants $d^{\pm}_1, \cdots,d^{\pm}_{m'}$ and $m'$ open domains $D^{\pm}_1,\cdots, D^{\pm}_{m'}$ such that they do not overlap each other, $M=\cup_{1\le i\le m'}\bar{D}^{\pm}_i$ and $$\label{weakeq4}
u^{\pm}|_{D^{\pm}_i}=u_i^{\pm}+d^{\pm}_i, \qquad \forall 1\le i\le m'.$$
It is deduced from the Lipschitz property of $u^-$ that it is differentiable almost every where. Let $x$ be a point where $u^-$ is differentiable, $du^-(x)$ determines a unique backward semi static orbit $d\gamma_c^i$: $(-\infty,0]\to M$ whose $\alpha$-limit set is in certain Aubry class $\tilde{\mathcal{A}}_c^i$. By definition we have $$u^-(x)-u^-(\gamma_c^i(-t))=\int_{-t}^0L_{c}(d\gamma_c^i(s),s)ds+\alpha(c)t.$$ Let $t_k\to\infty$ such that $\gamma_c^i(-t_k)\to x'\in\mathcal{A}_c^i$, it follows from Proposition \[weakpro2\] that $$\label{weakeq5}
u^-(x)=h_c^{\infty}(x',x)+u^+(x')=u_{c,i}^{-}(x)+d_i.$$ If $x^*\in M$ is another point where $du^-(x^*)$ determines a backward semi-static orbits whose $\alpha$-limit set is also contained in $\tilde{\mathcal{A}}_c^i$, we then obtain (\[weakeq5\]) for $u^-(x^*)$ with the same $d_i$. All these points constitute a set connected with $\mathcal{A}_c^i$. There are not more than $m$ connected sets such that (\[weakeq4\]) holds.
Let $c_i\to c$ be a sequence of cohomology and assume that the minimal measure consists of finitely many ergodic components for each $c_i$ and $c$. Let $\tilde{\mathcal{M}}^j_{c_i}$, $\tilde{\mathcal{M}}^j_c$ be the support for the ergodic minimal measure $\mu^j_{c_i}$ and $\mu^j_c$ respectively, let $u^-_{c_i}$ and $u^-_{c}$ be the corresponding elementary weak KAM solution. If $\mu^j_{c_i}\rightharpoonup\mu^j_c$ as $c_i\to c$, then $u^-_{c_i}\rightarrow u^-_{c}$ in $C^0$-topology.
It follows from the continuity of $h_c^{\infty}(x,x')$ in $c$ shown in Theorem \[weakthm1\] and the definition of the elementary weak KAM solution.
In terms of conjugate pair of weak KAM solution, one has a definition of Mañé set in [@Fa2]. For the purpose of this paper, we would like to use elementary weak KAM solution. Recall the definition of the barrier function in [@Ma2]: $$B_c^*(x)=\min_{\xi,\zeta\in\mathcal{M}_0(c)}\{h_c^{\infty}(\xi,x)-h_c^{\infty}(x,\zeta)+h_c^{\infty} (\xi,\zeta)\}.$$ If the minimal measure consists of finitely many ergodic components, we introduce barrier functions in terms of elementary weak KAM solutions: given $z=(x,\tau)\in M\times\mathbb{T}$, we set $$\label{weakeq6}
B_{c,i,j}(z)=u^-_{c,i}(z)-u^+_{c,j}(z).$$ which measures the minimum of the action along those curves passing through $z$ and joining $\mathcal{M}^i_c$ to $\mathcal{M}^j_c$. For autonomous systems, this barrier function is independent of time. Obviously, each $c$-semi static curve corresponds to a minimum of $u^-_{i,c}-u^+_{j,c}$ if its $\alpha$-limit set intersects $\tilde{\mathcal{M}}_c^i$ and its $\omega$-limit set intersects $\tilde{\mathcal{M}}_c^j$.
Minimal homoclinic orbits to Aubry Set
--------------------------------------
To extend the concept of elementary weak KAM solution to universal covering space, let us reveal some properties of minimal homoclinic orbit to Aubry set.
Given a curve $\gamma$: $\mathbb{R}\to M$, we call $d\gamma=(\gamma,\dot\gamma)$ a homoclinic orbit to some Aubry set $\tilde{\mathcal{A}}$ if it does not stay in the Aubry set, but its $\omega$-limit set as well as the $\alpha$-limit set is contained in the Aubry set: $$\alpha(d\gamma)\subseteq\tilde{\mathcal{A}}\ \ \ \ \ \text{\rm and}\
\ \ \ \ \omega(d\gamma)\subseteq\tilde{\mathcal{A}}.$$ Correspondingly, we call $\gamma$ homoclinic curve. The existence of homoclinic orbits to Aubry sets has been studied in a few papers, see [@Bo; @Be1; @Cui; @Zhe; @Zho2].
The existence of homoclinic orbits is closely related to the issue whether the $\check{\rm C}$ech homology group $H_1(M,\mathcal{A},\mathbb{R})$ is non-trivial ($H_1(M\times\mathbb{T},\mathcal{A},\mathbb{R})$ for time-periodically dependent Lagrangian). It is defined as the inverse limit $\lim_{\mathcal{A}\subset U}H_1(M,U,\mathbb{R})$, where $U$ is an open neighborhood of $\mathcal{A}$. There exists a small open neighborhood $U_0$ of $\mathcal {A}$ such that ${\rm rank}H_1(M,U,\mathbb{R})={\rm rank}H_1(M,\mathcal{A},\mathbb{R})$ provided $U\subseteq U_0$.
Let $\bar M$ be a covering of $M$ such that $\pi_1(\bar M)=\text{\rm ker}(\mathscr{H}: \pi_1(M)\to H_1(M,\mathbb{R}))$ where $\mathscr{H}$ denotes the Hurewicz homomorphism. The group of Deck transformation of this covering space is $$H=im(\mathscr{H}:\pi_1(M)\to H_1(M,\mathbb{R})).$$ Let $U$ be an open neighborhood of $\mathcal{A}$ such that ${\rm rank}H_1(M,U,\mathbb{Z})={\rm rank} H_1(M,\mathcal{A},\mathbb{R})$. Let $K=i_*H_1(U,\mathbb{Z}) \subset H$ and $G=H/K$, then $G$ is a free Abel group. To each orbit $(\gamma,\dot\gamma)$: $\mathbb{R}\to M$ homoclinic to $\tilde{\mathcal{A}}$, an element $[\gamma]\in G$ is associated.
If the group $G$ is non-trivial, there is a flat $\mathbb{F}$ of the $\alpha$-function containing the cohomology class. A set $\mathbb{F}\subset H^1(M,\mathbb{R})$ is called flat if the function $\alpha$ is affine when it is restricted on $\mathbb{F}$, not affine for any set properly contains $\mathbb{F}$. The dimension of this flat is not smaller than $r={\rm rank} H_1(M,\mathcal{A},\mathbb{Z})$ and the Aubry set is the same for all classes in the interior of $\mathbb{F}$ (see [@Ms]).
In this paper, we are interested in so-called [*minimal*]{} homoclinic orbits. Let $\check{M}$ be a covering manifold of $M$ such that $\pi_1(\check{M})=\pi_1(U)$. A curve $\gamma$: $\mathbb{R}\to M$ is called $\check{M}$ semi-static if the lift of $\gamma$ to $\check{M}$, $\check{\gamma}$: $\mathbb{R}\to\check{M}$ is semi-static. A homoclinic orbit $d\gamma$ is called [*minimal*]{} if the lift $\check{\gamma}$: $\mathbb{R} \to\check{M}$ is semi-static.
\[homothm1\] If there is only one Aubry class and $\text{\rm rank}\,H_1(M,\mathcal{A},\mathbb{R})=r>0$, then there are at least $r+1$ minimal homoclinic orbits. If $\mathcal{M}(c)\supsetneq \mathcal{M}(c')$ for $c\in\partial\mathbb{F}$ and $c'\in int\mathbb{F}$, then there are infinitely many $c$-minimal homoclinic orbits.
The existence of at least $r+1$ homoclinic orbits is proved in [@Be1], they are actually minimal. The infinity of minimal homoclinic orbits are proved in [@Zhe; @Zho2]. Let us briefly describe how to find these $r+1$ minimal homoclinic orbits. Given a point $x\in\mathcal{A}$ and $g\in G$, we denote by $\xi_k$: $[-k,k]\to M$ the minimizer of $$h_g^k(x)=\inf_{\stackrel {\xi_k(-k)=\xi_k(k)=x} {\scriptscriptstyle [\xi_k]=g}}\int_{-k}^kL(\xi_k(s),\dot\xi_k(s))ds+2k\alpha$$ Obviously, the set $\{\|\dot\xi_k(t)\|:\, t\in [-k,k]\}$ is uniformly bounded for $k\in\mathbb{Z}_+$. Because of positive definiteness of $L$, the set $\{\|\ddot{\xi}_k(t)\|:\, t\in [-k,k]\}$ is also uniformly bounded for each $k$. Let $$h_g^{\infty}(x)=\liminf_{k\to\infty}h_g^k(x),$$ there exists a subsequence of $k_j$ such that $h_g^{k_j}(x)\to h_g^{\infty}(x)$. The quantity $h_g^{\infty}$ keeps constant on each Aubry class. By diagonal extraction argument we can find a subsequence of $\xi_{k_j}$ which $C^1$-uniformly converges, on each compact interval, to a $C^1$-curve $\gamma$: $\mathbb{R}\to M$. In this sense, $\gamma$: $\mathbb{R}\to M$ is called an accumulation point of $\{\xi_{k_j}\}$. Each accumulation point is $\check M$ semi-static and there is at least one accumulation point $\gamma_1$ with non-zero homology $[\gamma_1]\neq 0$.
As the relative homology of the Aubry set is non-trivial, some $a>0$ exists such that $h_g^{\infty}\ge a$ holds for each class $g\in G$ and $h_g\to\infty$ as $|g|\to\infty$. Therefore, for each $g\in G$, there are finitely many accumulation points of $\{\xi_{k_j}\}$ with non-zero homology, denoted by $\gamma_1,\cdots,\gamma_i$. Clearly $\sum_{j=1}^i[\gamma_j]=g$. As $G$ is $r$-dimensional, at least $r+1$ geometrically different minimal homoclinic orbits exist.
Let us look at these homoclinic orbits from another point of view. For certain finite covering manifold, the lift of the Aubry set has several connected components (several Aubry classes). These Aubry classes are connected by semi-static orbits [@CP]. The projection of these semi-static orbits are nothing else but minimal homoclinic orbits. For a finite covering manifold $\tilde\pi$: $\tilde M\to M$, the fiber $\tilde\pi^{-1}x$ contains finitely many points. For a closed curve $\phi$: $[0,1]\to M$ such that $\phi(0)=\phi(1)=x$, there is a lift of $\bar\phi$ such that $\bar\phi(0)=\bar x_0\in \tilde\pi^{-1}x$. By the monodromy theorem, $\bar\phi(1)\in\tilde\pi^{-1}x$ is uniquely determined by its class $[\phi]\in\pi_1(M)$. Let $g_1,g_2,\cdots,g_{r}$ be the generators of $G$, $\phi_1,\phi_2,\cdots,\phi_r$ be closed path so that $[\phi_i]=g_i$, $\phi_i(0)=x$ for $i=1,2,\cdots,r$. If $\tilde M$ is chosen so that $\bar\phi_i(0)=\bar x_0$ and $\bar\phi_i(1)\neq \bar\phi_j(1)$, there will be at least $2r$ Aubry classes for this covering manifold. Among the semi-static orbits connecting different Aubry classes for the covering manifold, there are at least $r+1$ orbits whose projection is different from each other.
Let $G_{m}\subset G$ be defined such that $g\in G_m$ if and only if some minimal homoclinic orbit $d\gamma$ exists such that $[\gamma]=g$. We say that there are $k$-types of minimal homoclinic orbits if $G_m$ contains exactly $k$ elements.
\[homothm2\] If $M=\mathbb{T}^n$, $H_1(M,\mathcal{A},\mathbb{Z})\neq 0$ and $\mathcal{A}$ contains a set homeomorphic to $\mathbb{T}^{n-1}$, there exist exactly two types of minimal homolcinic orbits.
In this case $G=\mathbb{Z}$. By the condition, we can assume that each standard generator $e_i\in H_1(\mathbb{T}^n,\mathbb{Z})$ with $i>1$ can be represented by a closed curve in $\mathcal{A}$. Let $g=ke_1$ with $k>1$. If there is a minimal homoclinic orbits $(\gamma,\dot\gamma)$ such that $[\gamma]=g$, there must be some points $x=\gamma(t_0)\in\mathcal{A}$ but $(\gamma(t_0),\dot\gamma(t_0))\notin\tilde{\mathcal{A}}$.
As $x\in\mathcal{A}$, there is a unique vector $v$ such that $(x,v)\in\tilde{\mathcal{A}}$. Given any $\epsilon>0$, there is static curve $\xi$: $\mathbb{R}\to M$ and $s_0<s_1$ such that $\xi(s_0),\xi(s_1)$ are in $\epsilon$-neighborhood of $x$, $\|\dot\xi(s_0)-v\|<\epsilon$, $\|\dot\xi(s_1)-v\|<\epsilon$ and $[A(\xi)|_{[s_0,s_1]}]<\epsilon$.
Let $\tau_1^-=t_0-t^->0$, $\tau_2^+=t^+-t_0>0$, $\tau_1^+=s_0^+-s_0>0$ and $\tau_2^-=s_1-s_1^->0$ be suitably small numbers. We join $\gamma(t^-)$ to $\xi(s_0^+)$ by the curve $\zeta_1$: $[-\tau_1^-,\tau_1^+]\to M$ which minimizes the action $$[A(\zeta_1)|_{[-\tau_1^-,\tau_1^+]}]=\inf_{\stackrel{\zeta(-\tau_1^-)=\gamma(t^-)} {\scriptscriptstyle \zeta(\tau_1^+)= \xi(s_0^+)}}\int_{-\tau_1^-}^{\tau_1^+}L(\zeta(s),\dot\zeta(s))ds +(\tau_1^++\tau_1^-)\alpha,$$ and join $\xi(s_1^-)$ to $\gamma(t^+)$ by the curve $\zeta_2$: $[-\tau_2^-,\tau_2^+]\to M$ which minimizes the action $$[A(\zeta_2)|_{[-\tau_2^-,\tau_2^+]}]=\inf_{\stackrel{\zeta(-\tau_2^-)=\xi(s_1^-)} {\scriptscriptstyle \zeta(\tau_2^+)= \gamma(t^+)}}\int_{-\tau_2^-}^{\tau_2^+}L(\zeta(s),\dot\zeta(s))ds +(\tau_2^++\tau_2^-)\alpha.$$ We define a continuous curve $\gamma'$: $\mathbb{R}\to M$ by $$\gamma'(t)=\begin{cases}
\gamma(t), &t\in(-\infty, t^-],\\
\zeta_1(t-\Delta_1),&t-\Delta_1\in [-\tau_1^-,\tau_1^+],\\
\xi(t-\Delta_2),&t-\Delta_2\in [s_0^+,s_1^-],\\
\zeta_2(t-\Delta_3),&t-\Delta_3\in [-\tau_2^-,\tau_2^+],\\
\gamma(t-\Delta_4),& t-\Delta_4\in [t^+,\infty),
\end{cases}$$ where $\Delta_1=t^-+\tau_1^-$, $\Delta_2=t^-+\tau_1^-+\tau_1^+-s_0^+$, $\Delta_3=t^-+\tau_1^-+\tau_1^+ -s_0^++s_1^-+\tau_2^-$ and $\Delta_4=t^-+\tau_1^-+\tau_1^+ -s_0^++s_1^-+\tau_2^-+\tau_2^+-t^+$. By exploiting the [*curve shorten*]{} lemma in Riemannian geometry as did in [@Ma2] we find that $$[A(\gamma)|_{[t^-,t^+]}]+[A(\xi)|_{[s_0,s_0^+]\cup[s_1^-,s_1]}] >
[A(\zeta_1)|_{[-\tau_1^-,\tau_1^+]}]+[A(\zeta_2)|_{[-\tau_2^-,\tau_2^+]}]$$ if $\xi(s_0)=\xi(s_1)=x$ and $\dot\xi(s_0)=\dot\xi(s_1)=v\neq\dot\gamma(t_0)$. As $x\in\mathcal{A}$, $(\xi(s_0),\dot\xi(s_0))$, $(\xi(s_1),\dot\xi(s_1))$ can be arbitrarily close to $(x,v)$ by choosing suitable $s_0$ and $s_1$, this inequality still hold in our case. Note that the quantity $[A(\xi)|_{[s_0,s_1]}]$ can be arbitrarily close to zero, we see that $$[A(\gamma)|_{[t_{-1},t_1]}]>[A(\gamma')|_{[t_{-1},t_1+\Delta_4]}]$$ if $t_{-1}<t^-$ and $t_1>t^+$. As $[\gamma']=[\gamma]$, this property contradicts the fact that $\gamma$ is minimal. On the other hand, from Theorem \[homothm1\], we obtain the existence of 2 minimal homoclinic orbits. This completes the proof.
For each class $g\in G$, we define $$h_g^k(x,x)=\inf_{\stackrel {\xi(0)=\xi(k)=x}{\scriptscriptstyle \xi\in C^1,[\xi]=g}}\int_{0}^{k} L(\xi(s),\dot\xi(s))ds +k\alpha,$$ $$h_g^{\infty}(x,x)=\liminf_{k\to\infty}h_g^k(x,x).$$ It is easy to see that $h_g^{\infty}(x,x)\to\infty$ as $\|g\|\to\infty$. Indeed, it follows from the fact $H_1(\mathbb{T}^n,\mathcal{A},\mathbb{Z}) \neq 0$ that $h^{\infty}_g(x,x)>0$ for any $g\ne 0$. If $h_g^{\infty}(x,x)$ remains bounded as $\|g\|\to\infty$, there would be a minimal measure whose support is obviously not contained in $\tilde{\mathcal{A}}$, but it is absurd.
If the Aubry set contains only one class, as a function of $x$, $h_g^{\infty}(x,x)$ keeps constant on the Aubry set. So it makes sense let $h_g^{\infty}=h_g^{\infty}(x,x)$ for $x\in\mathcal{A}$. Obviously, one has $$h_{g_1+g_2}^{\infty}\le h_{g_1}^{\infty}+h_{g_2}^{\infty}.$$ and
\[homopro1\] If there is an infinite sequence $\{g_i\}\subset G$ such that $$h_{g_i+g_{i'}}^{\infty}< h_{g_i}^{\infty}+h_{g_{i'}}^{\infty},$$ then $G_m$ contains infinitely many elements.
The definition of $h_g^k(x,x)$ can be extended $h^k_g(x,x')$ for $x\ne x'$. Let us recall that the covering space $\check{\pi}$: $\check{M}\to M=\mathbb{T}^n$ is defined such that $\pi_1(\check{M})=\pi_1(U)$, where $U$ is a open neighborhood of $\mathcal {A}\subset\mathbb{T}^n$ so that $H_1(M,U,\mathbb{R})=H_1(M,\mathcal{A}, \mathbb{R})$. Let $D=\{\bar x: \bar x_i\in [0,1)\}\subset\mathbb{R}^n$ be the fundamental domain for $\mathbb{T}^n$ and use the same symbol to denote its projection to $\check M$ as well. For each closed path $\phi$: $[0,1]\to M$, there is a unique curve $\check{\phi}$ in the lift of $\phi$ such that $\check{\phi}(0)\in D$. Because of the monodromy theorem, $\check{\phi}(1)\in\check M$ is uniquely determined by the homological type $[\phi]\in H_1(\mathbb{T}^n,U,\mathbb{Z})$.
For a curve $\xi$: $[0,k]\to M$ with $\xi(0)=x$, $\xi(k)=x'$, we denote by $\check{\xi}$ the curve in the lift of $\xi$ such that $\check{\xi}(0)\in D$. We say $[\xi]=g$ if $[\phi]=g$ holds for any closed curve $\phi$ such that $\check\phi(0)\in D$ and $\check\phi(1)=\check\xi(k)$. Therefore, the following is well-defined: $$h_g^{k}(x,x')=\inf_{\stackrel{\stackrel{\stackrel {\xi(0)=x}{\scriptscriptstyle \xi(k)=x'}}{\scriptscriptstyle
[\xi]=g}}{\scriptscriptstyle \xi\in C^1}}\int_{0}^{k}L(\xi(s),\dot\xi(s))ds +k\alpha,$$ $$h_g^{\infty}(x,x')=\liminf_{k\to\infty}h_g^k(x,x').$$ Clearly, $h_g^{\infty}(x,x')\to\infty$ as $\|g\|\to\infty$. Indeed, let $\check x,\check x'\in D$ such that $\check{\pi}\check x=x, \check{\pi}\check x'=x'$, let $\check{\zeta}$: $[0,1]\to\bar M$ be a straight line such that $\check{\zeta}(0)=x'$, $\check{\zeta}(1)=x$ and denoted by $\zeta$ the projection of $\check{\zeta}$ down to $M$, we obviously have that $$h_g^{k+1}(x,x)\le h_g^k(x,x')+[A(\zeta)]$$ holds for each class $g$. As $[A(\zeta)]$ is a finite number, we verify the claim.
\[homopro2\] There exists positive number $a>0$ such that $h^{\infty}_g (x,x')\ge \|g\|a$ holds for each $(x,x')\in\mathbb{T}^n\times\mathbb{T}^n$ and for large $\|g\|$.
Obviously, there exists a positive number $a'>0$ such that $h^{T}_g (x,x)\ge a'$ holds for each $g\ne 0$ and each $T>0$. If the proposition does not hold, for any small $\epsilon_i>0$ there would exists $g_i$ and $T_i$ such that $$h^T_{g_i}(x,x)\le\epsilon_i\|g_i\|, \qquad \forall\, T\ge T_i.$$ Let $\gamma_i$: $[0,T_i]\to M$ be the minimizer of $h^{T_i}_{g_i} (x,x)$. Let $0=t_{i,0}<t_{i,1}<\cdots<t_{i,m_i}=T_i$ be a sequence so that $\gamma_i(t_{i,j})\in U$ and $H_1(\mathbb{T}^n,U,\mathbb{Z})\ni[\gamma_i|_{[t_{i,j},t_{i,j+1}]}]\ne 0$ and there does not exist $t'\in (t_{i,j},t_{i,j+1})$ such that $\gamma_i(t')\in U$, both $[\gamma_i|_{[t_{i,j},t']}]\ne 0$ and $[\gamma_i|_{[t',t_{i,j+1}]}]\ne 0$.
There are two possibilities for this sequence. Either some $t_{i,j}<t_{i,j+1}$ exists such that $t_{i,j+1}-t_{i,j}\to\infty$ as $i\to\infty$ or $t_{i,j+1}-t_{i,j}$ remains bounded for all $i,j$.
In the first case, let $\mu_i=d\gamma_i|_{[t_{i,j},t_{i,j+1}]}^*\nu_i$ where $\nu_i$ is a probability measure evenly distributed on the interval $[t_{i,j},t_{i,j+1}]$. By weak$^*$-compactness a probability measure $\mu$ exists such that $\mu_i\rightharpoonup\mu$. Clearly, $\mu$ is invariant for the Lagrange flow, $\int Ld\mu=0$ and the support of $\mu$ is not contained in the Aubry set. But it is absurd.
In the second case, one has $m_i\ge C\|g\|$. By choosing sufficiently small neighborhood $U$ of $\mathcal{A}$, the distance $d(\gamma_i(t_{i,j}),\mathcal{A})<\epsilon$ can be sufficiently small. As there is only one Aubry class, there is a closed curve $\zeta$ and a sequence of time $t'_j$ ($j=0,1,\cdots,m_i)$ such that $[A(\zeta)]<\epsilon$ and $d(\zeta(t'_j),\gamma_i(t_{i,m-j}))<\epsilon$. Let $\zeta'_j$ be the minimizer connecting $\gamma_i(t_{i,j})$ to $\gamma_j(t_{i,j-1})$, we have $|[A(\zeta'_j)]-[A(\zeta|_{[t'_{i,j-1}-t'_{i,j}]})]|\le C'\epsilon$, where the constant depends only on the Lagrangian. By construction, the curve $\gamma_i|_{[t_{i,j}-t_{i,j-1}]}\ast\zeta'_j$ is a closed curve with $[\gamma_i|_{[t_{i,j}-t_{i,j-1}]}\ast\zeta'_j]\ne 0$ and $[A(\gamma_i|_{[t_{i,j}-t_{i,j-1}]}\ast\zeta'_j)]>a'$. Therefore, $$\begin{aligned}
h^{T_i}_g(x,x)&\ge h^{T_i}_g(x,x)+[A(\zeta)]-\epsilon\\
&\ge\sum_{j} [A(\gamma_i|_{[t_{i,j}-t_{i,j-1}]}\ast\zeta'_j)]-(CC'\|g\|+1)\epsilon\\
&\ge C\|g\|a'-(CC'\|g\|+1)\epsilon.\end{aligned}$$ It contradicts the assumption. This proves the proposition in the case that $x=x'$.
For $x\ne x'$, we use a straight line connecting $x'$ to $x$. The action along this line is bounded. Therefore the proposition is also true for $x\ne x'$.
Globally elementary weak KAM solutions
--------------------------------------
For the configuration space $\mathbb{T}^n$, each weak KAM solution is 1-periodic in $x_i$ for $i=1,2,\cdots n$, where $(x_1,x_2,\cdots,x_n)=x$ denotes the configuration coordinate. If a finite covering of $\mathbb{T}^n$ is considered to be configuration space, weak KAM solution may not be 1-periodic for each coordinate.
We assume that the minimal measure contains finitely many ergodic components $\mu_c^1$, $\mu_c^2,\cdots, \mu_c^m$ for the cohomology class $c$. In this case, the elementary weak KAM solution for each $\mu_c^i$ is well-defined. The lift of $\mu_c^i$ to a finite covering $k\mathbb{T}^n$ may contain several ergodic components. For instance, if $\mathcal {M}\subset\{|x_1|\le\delta\}\times \mathbb{T}^{n-1}$, then there are two ergodic components in the lift of $\mathcal {M}$ for $2\mathbb{T}\times \mathbb{T}^{n-1}$. However, there are cases that the minimal measure is always uniquely ergodic for any finite covering manifold, for instance, if the measure is supported on a KAM torus.
Given $k=(k_1,k_2,\cdots,k_n)\in\mathbb{Z}^n$ with $k_i\ge 1$ for each $i=1,2,\cdots,n$, we define an equivalence relation $\sim_k$ in $\mathbb{R}^n$: we say $x\sim_k x'$ if $x_i-x'_i=2jk_i$ for some $j\in\mathbb{Z}$ ($i=1,2,\cdots n$). Clearly, $\pi_k$: $M_k =\mathbb{R}^n/\sim_k\to\mathbb{T}^n$ is a finite covering of $\mathbb{T}^n$. In the following, we shall also use the symbols: $\pi_{\infty,k}$: $\mathbb{R}^n\to M_k$ and $\pi_{\infty}$: $\mathbb{R}^n\to\mathbb{T}^n$ to denote the projection. For a bounded domain $\Omega\subset\mathbb{R}^n$, if the topology of $\pi_{\infty,k}\Omega\subset M_k$ is trivial, we use the same symbol to denote its projection $\Omega:=\pi_{\infty,k}\Omega$.
Let $\mathcal{M}_{\infty}$ and $\mathcal{M}_{k}$ be the lift of Mather set $\mathcal{M}$ to the universal covering space as well as to $M_k$ respectively. The connected components are denoted by $\mathcal{M}_{\infty}^i$ and $\mathcal{M}_{k}^i$ correspondingly. Obviously, the unit cube $D=[0,1)^n$ intersects finitely many connected components of $\mathcal{M}_{\infty}$, denoted by $\mathcal{M}_{\infty}^i$ with $i=0,1,\cdots,i_m$ ($i_m\ge m$).
Let $d_k=\min\{k_1,k_2,\cdots k_n\}$. Some $R_D>0$ exists such that for any $k\in\mathbb{Z}^n$ with $d_k\ge R_D$, $\pi_{\infty,k}\mathcal{M}_{\infty}^i\neq \pi_{\infty,k}\mathcal{M}_{\infty}^j$ holds for $0\le i,j\le i_m$ and $i\neq j$. In this case, we use the notation $\mathcal{M}_{k}^j=\pi_{\infty,k} \mathcal{M}_{\infty}^j$ for $0\le j\le i_m$. Let $u_{k,j}^{\pm}$ denote the elementary weak KAM for $\mathcal{M}_{k}^j$ with respect to the configuration manifold $M_k$.
\[globallem1\] For each bounded region $\Omega\subset\mathbb{R}^n$, there exists $R_{\Omega}>0$ such that for any $k, k'\in\mathbb{Z}^n$ with $d_k,d_{k'}\ge\max\{R_{\Omega},R_D\}$, $$u^{\pm}_{k,j}|_{\Omega}=u_{k',j}^{\pm}|_{\Omega}+\text{\rm constant}$$ holds for each $j=0,1,\cdots,i_m$.
We only need to study the case that the minimal measure is uniquely ergodic. If there are finitely many ergodic components, we obtain this result by perturbing the Lagrangian so that it is uniquely ergodic and applying Theorem \[weakthm1\].
Each weak KAM solution for $\mathbb{T}^n$ is a weak KAM solution for any $M_k$. If the lift of the minimal measure to any finite covering space is still uniquely ergodic, the elementary weak KAM solution remains the same.
Let us consider the case that there are more than one connected component in $\mathcal{M}_k$ with $d_k\ge R_D$. Remember $\mathcal {M}_{k}^j=\pi_{\infty,k}\mathcal{M}_{\infty}^j$ for $0\le j\le m$ where $\mathcal{M}_{\infty}^j$ intersects the fundamental domain $[0,1)^n$. Considered as a function defined in $\mathbb{R}^n$, the elementary weak KAM solution $u^-_{k,j}$ determined by $\mathcal {M}_{k}^j$ is $k_i$-periodic in the $i$-th coordinate. By the definition of elementary weak KAM solution, a sequence of functions $u^-_{k,j,\epsilon}$ exists such that $u^-_{k,j,\epsilon}\to u^-_{k,j}$ as $\epsilon\to 0$, where $u^-_{k,j,\epsilon}$ is the weak KAM solution for the Lagrangian $L_{k,\epsilon}: TM_k\to\mathbb{R}$. This Lagrangian satisfies the following conditions:
1, it is the same as $L$ when it is restricted on the tangent bundle of a neighborhood $U$ of $\mathcal{M}_{k}^j$, i.e. $L_{k,\epsilon}|_{TU}=L|_{TU}$;
2, the minimal measure is uniquely ergodic whenever $\epsilon\neq 0$, supported on $\mathcal{M}_{k}^j$;
3, $L_{k,\epsilon}\to L$ as $\epsilon\to 0$.
Starting from each $x\in M_k$, there exists at least one backward semi-static curve for $L_{k,\epsilon}$, $\gamma_{k,x, \epsilon}$: $(-\infty,0]$ with $\gamma_{k,x,\epsilon}(0)=x$. Clearly, $\pi\alpha(d\gamma_{k,x,\epsilon})\cap \mathcal{M}_{k}^j\ne\varnothing$. Let $t_i\to\infty$ be the sequence so that $\gamma_{k,x,\epsilon}(-t_i)\to x_0\in\mathcal{M}_{k}^j$, let $\alpha$ stand for the average action, then we have $$\label{globaleq1}
u^-_{k,\epsilon}(x)-u^-_{k,\epsilon}(x_0)=\lim_{t_i\to\infty}\int_{-t_i}^0L_{k,\epsilon} (d\gamma_{k,x,\epsilon}(s))ds+t_i\alpha.$$
Again, the lift of $\mathcal{M}_k^j$ to the universal covering space may contain many connected components, denoted by $\mathcal{M}_{\infty}^{j,\ell}$, among which only $\mathcal{M}_{\infty}^{j,0}$ intersects the fundamental domain $D$.
Let $D_k=\{\bar x: \bar x_i\in [-k_i,k_i)\}\subset\mathbb{R}^n$ so that $\pi_{\infty,k}$: $D_k\to M_k$ is an injection and $\pi_{\infty,k}D_k=M_k$. Let $\bar x\in D_k$ be the points such that $\pi_{\infty,k}\bar x=x$. Let $\bar\gamma_{k,\bar x,\epsilon}$ be the lift of $\gamma_{k,x,\epsilon}$ to the universal covering space so that $\bar\gamma_{k,\bar x,\epsilon}(0)=\bar x$. It is possible that $\pi\alpha(d\bar\gamma_{k,\bar x,\epsilon}) \cap\mathcal{M}_{\infty}^{j,0}=\varnothing$. The curve may approach to another connected component of $\mathcal{M}_{\infty}^{j,\ell}$. Let $\Omega_d=\{x:\max_i|x_i|\le d\}\subset\mathbb{R}^n$. Note that $L_{k,\epsilon}$ is a small perturbation of $L$. In virtue of Proposition \[homopro2\] we claim that $\bar\gamma_{k,\bar x,\epsilon}$ approaches to $\mathcal{M}_{\infty}^{j,0}$ provided $\bar x\in\Omega_d$, $\epsilon$ is suitably small and $d_k$ is sufficiently large. Let us assume the contrary, i.e. $\gamma_{k,\bar x,\epsilon}$ approaches to another connected component of $\mathcal{M}_{\infty}$. In this case, $\|[\pi_k\gamma_{k,\bar x,\epsilon}]\|$ would be sufficiently large provided $d_k$ is sufficiently large. By Proposition \[homopro2\] the action of $L$ along $\pi_k\gamma_{k,\bar x,\epsilon}$ $$\int L(d\pi_k\gamma_{k,\bar x,\epsilon}(t),t)dt\ge\|[\pi_k\gamma_{k,\bar x,\epsilon}]\|a$$ with certain $a>0$. As $L_{k,\epsilon}$ is a small perturbation of $L$, the action of $L_{k,\epsilon}$ along $\pi_k\gamma_{k,\bar x,\epsilon}$ would approach infinity as $d_k\to\infty$. The absurdity verifies the claim.
The set $\{\dot{\gamma}_{k,x,\epsilon}(0)\}$ is compact as $\epsilon\to 0$. For each accumulation point $v$, there is a subsequence of $\epsilon\to 0$ such that $\dot\gamma_{k,x,\epsilon}(0) \to v$. The initial value $(x,v)$ uniquely determines an orbit $(\gamma_{k,x},\dot\gamma_{k,x})$ of $L$. The curve $\gamma_{k,x}$: $(-\infty,0]\to M_k$ is a backward semi-static curve for $L$ which may not approach to $ \mathcal{M}_{k,0}$. When $\epsilon\to 0$, $\gamma_{k,x,\epsilon}$ may approach not only one but a family of semi-static curves for $L$ including the curves connecting different connected components of $\mathcal{M}_k$. More precisely, there might be several connected components $\mathcal{M}_{k}^{i_0}=\mathcal{M}_{k}^{j},\mathcal{M}_{k}^{i_1},\cdots,\mathcal{M}_{k}^{i_{\imath}}$ and semi-static curves $\gamma_{\ell,\ell+1}$ of $L$ for $M_k$ ($\ell=0,1,\cdots \imath-1$) such that $\pi\alpha(d\gamma_{\ell,\ell+1})\cap\mathcal{M}_{k}^{i_{\ell}}\ne\varnothing$, $\pi\omega(d\gamma_{\ell,\ell+1})\cap\mathcal{M}_{k}^{i_{\ell+1}}\ne\varnothing$, $\pi\alpha(d\gamma_{k,x})\cap\mathcal{M}_{k}^{i_{\imath}}\ne\varnothing$ and each $\gamma_{\ell,\ell+1}$ is approached by $\gamma_{k,x,\epsilon}$ as $\epsilon\to 0$. These curves have their natural projection down to $\mathbb{T}^n$, denoted by the same symbol.
We define the quantity $A_{i,j}$: $\mathcal{M}^i\times\mathcal{M}^j \to\mathbb{R}$ $$\label{globaleq2}
[A_{i,j}(x_i,x_j)]=\inf_{\stackrel{\stackrel{\gamma(0)=x_i}{\scriptscriptstyle \gamma(k)=x_j}}{\scriptscriptstyle k\in\mathbb{Z}_+}}\int^{k}_{0} L(d\gamma(s))ds+k\alpha.$$ By definition of weak KAM, for almost every point $x$, $(x,\partial _xu_{k,\epsilon}^-(x))$ uniquely determines a backward semi-static curve $\gamma_{k,x,\epsilon}$. Since this semi-static curve approaches to several curves: $\gamma_{k,x,\epsilon}\to\gamma_{1,2}\ast\cdots\ast \gamma_{\imath-1,\imath}\ast\gamma_{k,x}$ we obtain that $$\begin{aligned}
\label{globaleq3}
u^-_{k,0}(x)-u^-_{k,0}(x_0)=&\lim_{t_i\to\infty}\int_{-t_i}^0L(d\gamma_{x}(s))ds+t_i\alpha\\
&+\sum_{i=0}^{\imath-1}[A_{j_i,j_{i+1}}(x_i,x_{i+1})]\notag\end{aligned}$$ where $t_i\to\infty$ is a sequence such that $\gamma_{k,x}(-t_i)\to x_{\imath}\in \mathcal{M}_{k,i_{\imath}}$, $x_j\in\mathcal{M}_{k,i_j}$.
For each $x\in\Omega_d$, the backward semi-static curve $\gamma_{k,x,\epsilon}$ approaches to $\mathcal{M}_{k}^j$ provided $d_k$ is sufficiently large. For different $k,k'$ satisfying this condition, $\gamma_{k,x,\epsilon}$ and $\gamma_{k',x,\epsilon}$ may converge to different curves, $\gamma_{k,x,\epsilon}\to\gamma_{0,1}\ast\cdots\ast\gamma_{\imath-1,\imath}\ast\gamma_{k,x}$ and $\gamma_{k',x,\epsilon}\to\gamma'_{0,1}\ast\cdots\ast\gamma'_{\imath'-1,\imath'}\ast\gamma'_{k,x}$ as $\epsilon\to 0$. But the action of $L$ along $\gamma_{0,1}\ast\cdots\ast\gamma_{\imath-1,\imath}\ast\gamma_{k,x}$ is the same as along $\gamma'_{0,1}\ast\cdots\ast\gamma'_{\imath'-1,\imath'}\ast\gamma'_{k,x}$. Indeed, the action of $L_{k,\epsilon}$ along $\tilde\gamma_{k,x,\epsilon}$ is almost the same as the action of $L_{k',\epsilon}$ along $\tilde\gamma_{k',x,\epsilon}$ provided the perturbation is sufficiently small. Therefore, we obtain from the formula \[globaleq3\] that $$\bar u_{k,0}|_{\Omega}=\bar u_{k',0}|_{\Omega}+\text{\rm constant}$$ if both $d_k$ and $d_{k'}$ are sufficiently large.
The function $\bar u_i:\mathbb{R}^n\to\mathbb{R}$ is called globally elementary weak KAM solution for $\mathcal{M}_{\infty}^{j}$ if for each bounded domain $\Omega\subset\mathbb{R}^n$, there exists $R_{\Omega}>0$ such that for any $k\in\mathbb{Z}^n$ with $d_k\ge R_{\Omega}$, $$\bar u_{k,j}|_{\Omega}=\bar u_j|_{\Omega}+\text{\rm constant}$$ holds for each elementary weak KAM solution $\bar u_{k,j}$: $M_k\to\mathbb{R}$ for $\mathcal{M}_{k}^j$.
From Lemma \[globallem1\], we obtain the existence of a globally elementary weak KAM solution for each $\mathcal{M}_{\infty}^j$.
To investigate the properties of globally elementary weak KAM solution, let us consider a special case first, namely, the Mather set contains a connected component homeomorphic to $\mathbb{T}^{n-1}$. In this case, each $\mathcal{M}^i$ divided $\mathbb{R}^n$ into two parts, denoted by $R^-$ and $R^+$.
\[globalthm1\] If the Mather set contains a connected component contains a $\mathcal{M}_{\infty}^i$ homeomorphic to $\mathbb{T}^{n-1}$, then the globally elementary weak KAM solution $\bar u^{\pm}_i$: $\mathbb{R}^n\to\mathbb{R}$ has a decomposition $$\bar u^{\pm}_i=v^{\pm}_i+w^{\pm}_i,$$ where $v^{\pm}_i$ is periodic and $w^{\pm}_i$ is affine when they are restricted in the half space $R^+$ as well as in another half space $R^-$.
We only need to consider the case that the minimal measure is uniquely ergodic, as we did in the proof of Lemma \[globallem1\]. According to Theorem \[homothm2\], there are exactly two types of minimal homoclnic orbits to the Aubry set, we pick up two representative elements $\gamma_-$, $\gamma_+$: $\mathbb{R}\to\mathbb{T}^n$. Let $$h_{\pm}=\liminf_{t_i^{\pm}\to\infty}\int_{-t_i^-}^{t_i^+}L(d\gamma_{\pm}(t),t)dt+ (t_i^-+t_i^+)\alpha,$$ where $t_i^{\pm}$ is chosen such that $\gamma_{\pm}(t_i^+)\to 0$ and $\gamma_{\pm}(-t_i^-)\to 0$.
As the set $\mathcal{M}^i$ is co-dimension one, we are able to number all connected components by $\mathcal{M}^i_{\infty}$ ($i=\cdots -1,0,1,2,\cdots$) such that any path from $\mathcal{M}^{i-1}_{\infty}$ to $\mathcal{M}^{i+1}_{\infty}$ must pass through $\mathcal{M}^i_{\infty}$. Denote by $\Pi_i$ the strip bounded by $\mathcal{M}^i_{\infty}$ and $\mathcal{M}^{i+1}_{\infty}$. $\mathcal{M}^0_{\infty}$ separates $\mathbb{R}^n$ into two parts, denoted by $D^-$ and $D^+$ such that $\mathcal{M}_{\infty}^{-1}\subset D^-$ and $\mathcal{M}_{\infty}^1\subset D^+$.
Let $\bar\gamma_{\pm}$ denote a curve in the lift of $\gamma_{\pm}$ to $\mathbb{R}^n$ such that $\alpha(d\bar\gamma_{\pm})\subset\tilde{\mathcal {M}}_{\infty}^0$. Then, either $\omega(d\bar\gamma_{-})\subset\tilde{\mathcal{M}}_{\infty}^{-1}$, $\omega(d\bar\gamma_{+})\subset\tilde{\mathcal{M}}_{\infty}^1$, or $\omega(d\bar\gamma_{+})\subset\tilde{\mathcal{M}}_{\infty}^{-1}$, $\omega(d\bar\gamma_{-})\subset\tilde{\mathcal{M}}_{\infty}^1$. We only need to study one case, let’s say, the first case.
Given a bounded domain $\Omega\subset\mathbb{R}^n$. From the definition of globally elementary weak KAM solution, we see that $$\bar u^-_0|_{\Omega}=u^-_{k,0}|_{\Omega}$$ whenever $d_k$ is suitably large. Clearly, the function $\bar u_0$ is periodic when it is restricted $\Omega\cap\Pi_i$, i.e. $u^-_{k,0}(x)=u^-_{k,0}(x')$ if $x'-x\in\mathbb{Z}^n$ and $x,x'\in\Omega\cap\Pi_i$.
For each $x\in\Omega\cap\Pi_i$ with $i>0$, there exists at least one point $x_0\in\Omega\cap\Pi_0$ such that $x-x_0\in\mathbb{Z}^n$. By definition, we find that $u^-_{k,0}(x)=u^-_{k,0}(x_0)+ih_+$. Obviously, $u^-_{k,0}(x)=u^-_{k,0}(x_0)+(1+i)h_-$ if $x\in\Omega\cap\Pi_i$ with $i<0$.
Pick up a point $x_0\in\mathcal{M}_{\infty}^0$. For each non-zero integer vector $k\in\mathbb{Z}^n$, the point $x=x_0+k$ stays in certain $\mathcal{M}_{\infty}^i$. Along the ray $x=x_0+tk$ with $t>0$, we define $$v^-_0(x)= \begin{cases} u^-_0(x)-u^-_0(x_0)-tih_+,\hskip 0.5 true cm \text{if} \ i>0;\\
u^-_0(x)-u^-_0(x_0)-tih_-, \hskip 0.5 true cm\text{if} \ i>0,
\end{cases}$$ Clearly, $u^-_0-v^-_0$ is affine and $v^-_0$ is periodic when they are restricted in $D^-$ as well as in $D^+$.
Given an ergodic component of a minimal measure with higher co-dimensions, it is unclear what condition guarantees the decomposition of the globally elementary weak KAM solutions. It appears closely related to the problem whether there are infinitely many types of minimal homoclinic orbits to the Aubry class.
\[globalpro1\] Let $u^{\pm}_i$ be the globally elementary weak KAM solution for $\mathcal{M}^i$. Then, $u^{\pm}_i$ remains bounded on the whole ray $\{x_0+tg:t\in\mathbb{R}_+\}$ for each $g\in H_1(\mathcal{A}^i, \mathbb{Z})$, where $\mathcal{A}^i\supset\mathcal{M}^i$ is an Abury class; for $g\in H_1(M,\mathcal{A},\mathbb{Z})/K$, $u^{\pm}_i$ grows up linearly, or asymptotically linearly on the ray $\{x_0+tg:t\in\mathbb{R}_+\}$.
For arbitrarily large $t$, there exists $x\in\mathcal{M}_{\infty}^i$ such that $\text{\rm dist}(x_0+tg,x)\le 2$ and $\mathcal{M}_{\infty}^i\cap D\neq\varnothing$ where $D$ is the unit cube containing the origin. Let $x^*=\pi_{\infty}x$, $\bar\xi$ be a curve connecting $x^*$ to $x$, $\xi=\pi_{\infty}\bar\xi$, then $[\xi]\in H_1(\mathcal{A},\mathbb{Z})$. By definition, $$\inf_{[\xi]\in H_1(\mathcal{A},\mathbb{Z})}\inf_{\stackrel{\xi(0)=\xi(k)}{\scriptscriptstyle k\in\mathbb{Z}_+}}\int_0^k L(d\xi(t),t)dt=0$$ it proves the first conclusion.
For the second, one can see from Proposition \[homopro2\] that it grows up at least linearly. Given $g\in H_1(M,\mathcal{A}^i,\mathbb{Z})/K$ finitely many elements $g_0,g_1,\cdots,g_r\in H_1(\mathbb{T}^n,\mathbb{Z})$ exists such that for each $g=\sum_{i+0}^rj_ig_i$ with $j_i\in\mathbb{Z}_+$. Thus, $$h^{\infty}_g(x,x)\le\sum_{i=0}^r j_ih^{\infty}_{g_i}(x,x).$$ For each $\pi_{\infty}x\in\mathcal {M}^i$, $[x-x^*]=g$, we have $$u(x)-u(x^*)=h_g^{\infty}(\pi_{\infty}x,\pi_{\infty}x).$$ This completes the proof.
Dynamics around fixed point
===========================
Given a Tonelli Lagrangian $L$: $T\mathbb{T}^n\to\mathbb{R}$, let $c_0\in\arg\min\alpha$. Any minimal measure with zero-rotation vector must be $c_0$-minimal measure. In this section we study the dynamics around the Mather set for the class $c_0$. The motivation comes from following argument.
Let us consider the normal form of a nearly integrable Hamiltonian $$H(p,q)=h_0(p)+\epsilon P(p,q), \qquad (p,q)\in\mathbb{R}^{d}\times\mathbb{T}^{d}.$$ around a complete resonant point. Let $\omega(y)=\nabla h_0(y)$ denote the frequency vector of the unperturbed system. A frequency $\omega$ is called complete resonant of (minimal) period $T$ if $T\omega\in\mathbb{Z}^d$ and $t\omega\notin\mathbb{Z}^d$ for each $t\in (0,T)$. By finitely many steps of KAM iteration and one step of linear coordinate transformation on torus, one obtains a normal form of nearly integrable Hamiltonian (see Appendix A) $$\tilde H(\tilde x,\tilde y)=\tilde h(\tilde y)+\epsilon \tilde Z(x,\tilde y)+\epsilon \tilde R(\tilde x,\tilde y)$$ where $\tilde x=(x,x_d)$, $\tilde y=(y,y_d)$, $(x,y)\in\mathbb{T}^{d-1}\times\mathbb{R}^{d-1}$, $\tilde H$ is well-defined in $(\tilde x,\tilde y)\in\mathbb{T}^{d}\times B_d(\tilde y^*)$, $\partial\tilde h(\tilde y^*)=(0,\omega_{d})$ and $\epsilon\tilde R$ is a higher order term.
Since $\partial_{y_{d}}\tilde h(\tilde y^*)=\omega_{d}\ne 0$, there exists some function $Y(x,y,\tau)$ solving the equation $\tilde H(x,-\tau,y,Y(x,y,\tau))=E$ provided $E>\min\alpha_{\tilde H}$, which defines a time-periodic Hamiltonian system with $(d-1)$-degrees of freedom. Here $\tau=-x_{d}$ plays the role of time. One can write $$Y(x,y)=h(y)+\epsilon Z(x,y)+\epsilon R(x,y,\tau)$$ where $\epsilon R$ is a higher order term of $\epsilon$ and $\partial h(y^*)=0$, i.e. the complete resonance reduces to zero frequency. Omitting the higher order term, one obtains Hamiltonian with $d-1$ degrees of freedom $$\bar Y(x,y)=h(y)+\epsilon Z(x,y).$$ It determines a Lagrangian we shall study in this section.
Flat of the $\alpha$-function
-----------------------------
By definition, a subset is called a flat of certain $\alpha$-function if, restricted on this set, the $\alpha$-function is affine, and no longer affine on any set properly containing the flat. As $\alpha$-function is convex with super-linear growth, each flat is a convex and bounded set. Given an $n$-dimensional flat $\mathbb{F}$, a subset in $\partial \mathbb{F}$ is called an edge if it is contained in a $(n-1)$-dimensional hyperplane. Since each flat is convex, each edge is also convex.
\[flatthm1\] Given a class $c_0\in H^1(\mathbb{T}^n,\mathbb{R})$, if the minimal measure is uniquely ergodic, supported on a hyperbolic fixed point, then there exists an $n$-dimensional flat $\mathbb{F}_0\subset H^1(\mathbb{T}^n,\mathbb{R})$ such that this point supports a $c$-minimal measure for all $c\in\mathbb{F}_0$.
[**Remark**]{}: The condition of this theorem does not exclude topological non-triviality of the Aubry set. An example is the product of $n$ pendulums. The Aubry set covers the whole torus $\mathbb{T}^n$ if the Lagrangian $L$ is replaced by $L-\langle c,\dot x\rangle$ with $c$ being on the boundary of the flat.
By translation one can assume that the fixed point is at $(x,\dot x)=(0,0)$, by adding a closed 1-form and a constant to the Lagrangian, one can assume $c_0=0$ and $L(0,0)=0$.
To each closed curve $\xi$: $[-T,T]\to\mathbb{T}^n$ with $\xi(-T)=\xi(T)$ a first homology class $[\xi]=g\in H(\mathbb{T}^n,\mathbb{Z})$ is associated. We consider the quantity $$A(g)=\liminf_{T\to\infty}\inf_{\stackrel {\xi(-T)=\xi(T)}{\scriptscriptstyle [\xi]=g}} \int_{-T}^TL(d\xi(t))dt.$$ By the condition assumed on $L$, one has that $A(g)\ge0$ for any $g\ne 0$. There exist at least $n+1$ irreducible classes $g_i\in H(\mathbb{T}^n,\mathbb{Z})$ and $n+1$ minimal homoclinic orbits $d\gamma_i$ such that $A([g_i])=A(\gamma_i)$ [@Be1]. Clearly, $H_1(\mathbb{T}^n,\mathbb{Z})$ can be generated by the homology classes of all minimal homoclinic curves over $\mathbb{Z}_+$.
We abuse the notation $g$ to denote homology class $g\in H_1(\mathbb{T}^n,\mathbb{Z})$ or to denote a point $g\in\mathbb{Z}^n$. For each curve $\bar\gamma_T$: $[-T,T]\to\mathbb{R}^n$ with $\bar\gamma_T(-T)=0$ and $\bar\gamma_T(T)=g$, one has $$A(g)=\liminf_{T\to\infty}\inf_{\stackrel {\bar\xi(-T)=0}{\scriptscriptstyle \bar\xi(T)=g}} \int_{-T}^TL(d\bar\xi(t))dt.$$ Recall the definition of globally elementary weak-KAM and note that the point $x=0$ is the support of the minimal measure. Let $u_0^-$ ($u_0^+$) denote the backward (forward) globally elementary weak-KAM for $\mathcal{M}^0=\{x=0\}$, we have $$A(g)=u_0^-(g)-u_0^-(0),\qquad A(-g)=u_0^+(0)-u_0^+(g).\notag$$ By setting $u_0^-(0)=u_0^+(0)$, we claim $$\label{flateq1}
A(g)+A(-g)=u^-_0(g)-u^+_0(g)>0.$$
The quantity $A(g)$ is achieved may not by a curve connecting the origin to $g\in\mathbb{Z}^n$, but may by the conjunction of several curves $\bar\gamma_1\ast\bar\gamma_2\ast\cdots\ast\bar\gamma_m$. Let $\bar\gamma_i$: $\mathbb{R}\to\mathbb{R}^n$ denote a curve ($i=1,\cdots,m$), the conjunction implies that $\bar\gamma_i(-\infty)=\bar\gamma_{i-1}(\infty)$. Let $\gamma_i=\pi_{\infty}\bar\gamma_i$, where $\pi_{\infty}:\mathbb{R}^n\to\mathbb{T}^n$ denotes the standard projection. In this case, $\gamma_1,\cdots,\gamma_m$ are minimal homoclinic curves such that $g=\sum_{i=1}^m[\gamma_i]$. By the definition of elementary weak-KAM, each $\bar\gamma_i$ is a $(u_0^-,L)$-calibrated curve. Let $g_i=\sum_{j=1}^i[\gamma_j]$. Obviously, some large $t_0>0$ exists such that $(\bar\gamma_i(t),\dot{\bar\gamma}_i(t))$ stays in the local stable manifold of the point $(x,\dot x)=(g_i,0)$ whenever $t\ge t_0$. Therefore, some constant $C_i$ exists such that $$\label{flateq2}
u^-_0(\bar\gamma_i(t))=u^+_{g_i}(\bar\gamma_i(t))+C_i,\qquad \forall\ t\ge t_0,$$ where we use $u^-_{g_i}$ and $u^+_{g_i}$ to denote the globally elementary-KAM based on $x=g_i$. Clearly, $u^+_{g_i}$ and $u^-_{g_i}$ generate the local stable and unstable manifold around the point $(x,\dot x)=(g_i,0)$ respectively.
Because that $u^{\pm}_{g_i}$ is $L$-dominate function, for $x\in B_{\delta}(g_i)$ with suitably small $\delta>0$ we have (see Theorem 5.1.2 in [@Fa2]) $$\label{flateq2.1}
u^+_{g_i}(x)-u^{+}_{g_i}(g_i)\le u^{\pm}_{0}(x)-u^{\pm}_0(g_i)\le u^-_{g_i}(x)-u^{-}_{g_i}(g_i).$$ Remember that $u_0^-\ge u_0^+$. If $$A(g)+A(-g)=u_0^-(g)-u_0^+(g)=0,$$ substituting $u^+_{g_i}$ in (\[flateq2.1\]) by the expression in (\[flateq2\]) we see that some $t_0$ exists so that $$u_0^+(\bar\gamma_m(t))= u^-_0(\bar\gamma_m(t)), \qquad \forall\ t\ge t_0.$$ Here, $t_0$ is chosen so that $\bar\gamma_m(t))\in B_{\delta}(g_m)$ for $t\ge t_0$. Since $u_0^+$ is an $L$-dominate function and $\bar\gamma_i$ is a $(u_0^-,L)$-calibrated curve for each $1\le i\le m$, $$\begin{aligned}
u_0^+(\bar\gamma_i(t_0))-u_0^+(\bar\gamma_i(t_1))&\le\int_{t_1}^{t_0}L(d\bar\gamma_i(s))ds,\\
u_0^-(\bar\gamma_i(t_0))-u_0^-(\bar\gamma_i(t_1))&=\int_{t_1}^{t_0}L(d\bar\gamma_i(s))ds\end{aligned}$$ hold for any $t_1\le t_0$. This induces that $u_0^-(\bar\gamma_m(t))=u_0^+(\bar\gamma_m(t))$ holds for all $t\in\mathbb{R}$, and induces in turn the inequality for $i=m-1,m-2,\cdots$, and finally we have $$u_0^-(\bar\gamma_1(t))=u_0^+(\bar\gamma_1(t)), \qquad \forall\ t\in\mathbb{R}.$$ Since $u_0^-(x)\ge u_0^+(x)$ holds in a small neighborhood of $0$ $$u_0^-(\bar\gamma_1(t))= u_0^+(\bar\gamma_1(t)), \qquad \forall\ t\in\mathbb{R}.$$
On the other hand, as the fixed point $\{x=0\}$ is hyperbolic, some $\delta>0$ exists such that $$u^-_0(x)-u^+_0(x)>0, \qquad \forall\ x\in B_{\delta}(0)\backslash\{0\},\notag$$ if we set $u^-_0(0)=u^+_0(0)$. This contradiction proves the formula (\[flateq1\]).
Let $$\mathbb{G}_0=\{g\in H_1(\mathbb{T}^n,\mathbb{Z}):\exists\ \gamma: \mathbb{R}\to\mathbb{T}^n\ s.t. \ [\gamma]=g,\ A(\gamma)=0\}.$$ $\mathbb{G}_0$ is said to generate a rational direction $g\in\mathbb{Z}^n$ over $\mathbb{Z}_+$ if there exist $k,k_i\in\mathbb{Z}_+$ and $g_i\in \mathbb{G}_0$ such that $$kg=\sum k_ig_i.$$ It is an immediate consequence of the formula (\[flateq1\]) that once $\mathbb{G}_0$ generates a rational direction $g\in\mathbb{Z}^n$ over $\mathbb{Z}_+$, then it can not generate the direction $-g$ over $\mathbb{Z}_+$. Therefore, the set $$\text{\rm span}_{\mathbb{R}_+}\mathbb{G}_0=\{\Sigma a_ig_i:\ g_i\in\mathbb{G}_0,\ a_i\ge 0\}$$ is a cone properly restricted in half space. Thus, there exists an $n$-dimensional cone $\mathbb{C}_0$ such that $$\langle c,g\rangle >0,\qquad \forall\ c\in\mathbb{C}_0,\ g\in \text{\rm span}_{\mathbb{R}_+}\mathbb{G}_0.$$
Since the minimal measure for zero cohomology class is supported on the fixed point, $\tilde{\mathcal{N}}(0)$ is composed of those minimal homoclinic orbits along which the action equals zero. According to the upper semi-continuity of Mañé set in cohomology class, any minimal measure $\mu_c$ is supported by a set lying in a small neighborhood of these homoclinic orbits if $|c|$ is very small. Consequently. we have $\rho(\mu_c)\in\text{\rm span}_{\mathbb{R}_+}\mathbb{G}_0$, where $\rho(\mu_c)$ denotes the rotation vector of $\mu_c$.
Let us consider a cohomology class $c$ such that $-c\in\mathbb{C}_0$ and $|c|\ll 1$. We claim that the $c$-minimal measure is also supported on the fixed point. Indeed, if it is not true, we would have positive average action of $L$: $ A(\mu_c)>0$, since the minimal measure for zero class is assumed unique and supported on the fixed point. By the choice of $c$ one has that $\langle c,\rho(\mu_c)\rangle<0$. Thus, one obtains $$A_c(\mu_c)=A(\mu_c)-\langle c,\rho(\mu_c)\rangle>0=A_c(\mu),$$ it deduces absurdity. For this class $c$, the action of the Lagrangian $L_c=L-\langle c,\dot x\rangle$ along any minimal homoclinic curve $\gamma$ is positive, $$A(\gamma)-\langle c,[\gamma]\rangle >0,$$ namely, the Aubry set for this class is also a singleton. Consequently, $\mu_{c'}$ is also supported on this point if $c'$ is sufficiently close to $c$. This verifies the existence of $n$-dimensional flat.
[**Eigenvalues of the fixed point**]{}
Let us consider the eigenvalues of the fixed point by assuming the hyperbolicity, denoted by $\lambda_i$ $(i=1,2,\cdots, 2n)$. Under the hyperbolic assumption, half of these have positive real part, other half have negative real part. In general, these eigenvalues may have non-zero imaginary part. But in nearly integrable systems, all eigenvalues are real.
\[flatpro1\] If the Lagrangian is a small perturbation of integrable one $L=\ell(\dot x)+\epsilon P(x,\dot x)$ where $\ell$ is positive definite in $\dot x$, then for generic $P$ and for sufficiently small $\epsilon$, all eigenvalues at the fixed point are real and different.
Let $A=\partial^2_{\dot x\dot x}L$, $B=\partial^2_{x\dot x}P$, $C=\partial^2_{xx}P$ evaluated at the fixed point. As the minimal measure is supported on a fixed point, $C$ is positive definite. We consider the linearized equation and assume the solution with the form of $x=\xi \exp{\sqrt{\epsilon}\lambda t}$, then $$\label{flateq3}
\left |\lambda^2 A-\sqrt{\epsilon}\lambda(B-B^t)-C\right |_{n\times n}=0.$$ Let $A_0=\partial^2_{\dot x\dot x}\ell$, evaluated at the fixed point. For generic $C$, all solutions of the equation $$\label{flateq4}
\left |\lambda^2 A_0-C\right |_{n\times n}=0.$$ are real and different from each other: $\lambda=\pm\lambda_1,\pm\lambda_2,\cdots,\pm\lambda_n$, $\lambda_i\neq\lambda_j$ if $i\neq j$. Since (\[flateq3\]) is a small perturbation of (\[flateq4\]), all solutions of (\[flateq3\]) are different, and consequently, real. If there was a complex solution $\lambda=\sigma+i\omega$, $\pm\sigma\pm i\omega$ would be solution also, which is guaranteed by the Hamiltonian structure. It implies the existence of more than $2k$ solutions, but it is absurd.
[**The shape of the flat**]{}
Here, we are concerned about the flat $\mathbb{F}_0=\mathscr{L}_{\beta}(0)$. It is a $n$-dimensional flat if the $c$-minimal measure is supported on the hyperbolic fixed point for each $c\in\text{\rm int}\mathbb{F}_0$. By coordinate translation, we assume it is at the origin: $(\dot x,x)=(0,0)$. Correspondingly, in canonical coordinates the fixed point is also at the origin $(x,y)=(0,0)$. Let $(\xi_i^{\pm},\eta_i^{\pm})$ denote the eigenvector for $\pm\lambda_i$, where $\xi_i^{\pm}$ is for the $x$-coordinates, $\eta_i^{\pm}$ is for the $y$-coordinates. We assume
1, all eigenvalues are real number and different;
2, all minimal homoclinic curves approach to the fixed point in the direction $\xi_1^{\pm}$ as $t\to\mp\infty$.
The condition 1 is obviously generic. To see the genericity of the condition 2, let us remind reader that there is, generically, at most one minimal homoclinic curve for each homology class. By further perturbation, it approaches to the fixed point in the direction of $\xi_1^{\pm}$. Since there are countably many homology classes at most, the genericity is obtained.
For $\theta>0$ and $\xi\in\mathbb{R}^n\backslash\{0\}$, we define a cone $$C(\xi,\theta)=\{x\in\mathbb{R}^n:|\langle x,\xi\rangle|\ge\theta\|\xi\|\|x\|\},$$ and let $$C(\xi,\theta,d)=\{x\in C(\xi,\theta):\|x\|=d\}.$$
\[flatpro2\] Assume that $(x,y)=(0,0)\in\{H^{-1}(0)\}$ is a hyperbolic fixed point for $\Phi_H^t$, where all eigenvalues are real and different: $$Spec\{J\nabla H\}=\{\pm\lambda_1,\cdots,\pm\lambda_n; \ \ 0<\lambda_1<\cdots<\lambda_n\}.$$ Let $(\xi_i^{\pm},\eta_i^{\pm})$ denote the eigenvector for $\pm\lambda_i$, where $\xi_i^{\pm}$ is for the $x$-coordinates, $\eta_i^{\pm}$ is for the $y$-coordinates. Let $(x(t),y(t))\subset\{H^{-1}(0)\}$ be an orbit such that $x(t)$ passes through a ball $B_{\delta}(0)\subset\mathbb{R}^n$, $x(-T)\in\partial B_{\delta}(0)$, $x(T)\in\partial B_{\delta}(0)$ and $x(t)\in\text{\rm int}B_{\delta}(0)$ for all $t\in (-T,T)$. Then, for suitably small $\delta>0$ and $\theta=\frac 12$, there exist sufficiently large $T_0>0$ such that for $T\ge T_0$ one has $$(x(-T),x(T))\notin C(\xi_1^+,\theta,\delta)\times C(\xi_1^-,\theta,\delta).$$ For $T\to\infty$, one has $$x(-T)\in C(\xi_i^+,1-o(\delta),\delta) \ \ \ \text{\rm or}\ \ \ x(T)\in C(\xi_j^-,1-o(\delta),\delta).$$ for certain $i,j\ne1$.
By certain symplectic coordinate transformation, the Hamiltonian is assumed to have the normal form $$H(x,y)=\sum_{i=1}^n\frac 12\Big(y_i^2-\lambda_i^2x_i^2\Big)+P_3(x,y)$$ where $P_3=O(\|(x,y)\|^3)$ is a higher order term. By the method of variation of constants, we obtain the solution of the corresponding Hamilton equation $$\begin{aligned}
\label{flateq5}
x_i(t)=&e^{-\lambda_it}(b_{i}^{-}+F_i^-)+e^{\lambda_it}(b_{i}^{+}+F_i^+), \\
y_i(t)=&-\lambda_ie^{-\lambda_it}(b_{i}^{-}+F_i^-)+\lambda_ie^{\lambda_it}(b_{i}^{+}+F_i^+),\notag\end{aligned}$$ where $b_i^{\pm}$ are constants determined by boundary condition and $$\begin{aligned}
F_i^-=&\frac{1}{2\lambda_i} \int_0^te^{\lambda_is}(\lambda_i\partial_{y_i}P_3+\partial_{x_i}P_3)(x(s),y(s))ds, \\
F_i^+=&\frac{1}{2\lambda_i} \int_0^te^{-\lambda_is}(\lambda_i\partial_{y_i}P_3-\partial_{x_i}P_3) (x(s),y(s))ds.\end{aligned}$$ Substituting $(x,y)$ with the formula (\[flateq5\]) in the Hamiltonian we obtain a constraint for the constants $b_i^{\pm}$: $$\label{flateq6}
H(x(t),y(t))=-2\sum_{i=1}^n\lambda_i^2b_i^-b_i^++P_3((b^+_i+b^-_i),\lambda_i(b^+_i-b^-_i))$$ Let us estimate the size of the constants $c_i^{\pm}$ by the boundary conditions $x(T)=(x^+_1,x^+_2,\cdots,x^+_k)\in\partial B_{\delta}(0)$, $x(-T)=(x^-_1,x^-_2,\cdots,x^-_k) \in
\partial B_{\delta}(0)$ and assuming $$\label{flateq7}
\min\{|x_1^-|,|x_1^+|\}\ge \frac{\delta}2.$$ For $\theta=1/2$, $(x(-T),x(T))\in C(\xi_1^-,\theta,\delta)\times C(\xi_1^+,\theta,\delta)$ implies (\[flateq7\]) holds. Since the curve $x|_{[-T,T]}$ stays inside of the ball $B_{\delta}(0)$ and $T$ is sufficiently large, the orbit $(x,y)|_{[-T,T]}$ stays near the stable and unstable manifold of the fixed point. Note $P=O(\|(x,y)\|^3)$, we obtain from the theorem of Grobman-Hartman that $$\begin{aligned}
\label{flateq7.1}
x_i^-=&b_i^-e^{\lambda_iT}+b_i^+e^{-\lambda_iT}+o(\delta),\\
x_i^+=&b_i^-e^{-\lambda_iT}+b_i^+e^{\lambda_iT}+o(\delta).\notag\end{aligned}$$ For sufficiently large $T>0$, it deduces from the assumption (\[flateq7\]) that $$|b_1^{\pm}|\ge\frac {\delta}{3} e^{-\lambda_1T},$$ and $$|b_i^{\pm}|\le 2\delta e^{-\lambda_iT},\qquad \forall\ i=2,\cdots,k.$$ Since $\lambda_1<\lambda_i$ for each $i\ge 2$, $|b_i^{\pm}|\ll|b_1^{\pm}|$ if $T$ is sufficiently large. In this case, we obtain from (\[flateq6\]) that $$|H(x(t),y(t))|>|\lambda_1^2b_1^+b_1^-|>0.$$ It contradicts the assumption that $(x(t),y(t))\in\{H^{-1}(0)\}$. Let $T\to\infty$, one easily sees the last conclusion.
This proposition tells us following fact. In the energy level $\{H^{-1}(0)\}$ there does not exist such an orbit passing through $B_{\delta}(0)$ in the way that it enters into the ball in a direction close to $\xi_1^+$ and leaves in a direction close to $\xi_1^-$.
\[flatthm2\] Let $\mathbb{F}_0=\mathscr{L}_{\beta}(0)$ be an $n$-dimensional flat of the $\alpha$-function. Each minimal homoclinic curve $\gamma$ is assumed approaching to the fixed point in the direction of the eigenvectors corresponding to the smallest eigenvalue $$\lim_{t\to\pm\infty}\frac{\dot\gamma(t)}{\|\dot\gamma(t)\|}=\frac{\xi_1^{\mp}}{\|\xi_1^{\mp}\|},$$ all eigenvalues are assumed real and different. It is also assumed that, for each $c\in\mathbb{F}_0$ $($including the boundary$)$, the minimal measure is uniquely supported on the hyperbolic fixed point. Then, there exists a finite set $$H_{\mathbb{F}_0}=\{g_1,g_2,\cdots,g_m\}\subset H_1(\mathbb{T}^n,\mathbb{Z})$$ such that $[\gamma]\in H_{\mathbb{F}_0}$ if $\gamma$ is a minimal homoclinic curve. Consequently, the flat $\mathbb{F}_0$ is a polygon with finitely many edges, denoted by $\mathbb{E}_1,\cdots,\mathbb{E}_m$. Each edge $\mathbb{E}_i$ is associated with a homological class $g_i$ such that $\mathcal{A}(c)$ is composed by minimal homoclinic curves with homological type $g_i$ if $c$ is in the interior of $\mathbb{E}_i$ and $$\langle c-c',g_i\rangle =0, \qquad \forall\ c,c'\in\mathbb{E}_i.$$
As the minimal measure is uniquely ergodic and supported on a point for each class in $\mathbb{F}_0$, the Mañé set consists of homoclinic orbits and the point itself.
For each class $c\in\partial\mathbb{F}_0$, we claim that the Mañé set contains at least one homoclinic orbit. Otherwise, for each class $c'\notin\mathbb{F}_0$ very close to $c$, the homology of the Mañé set is trivial, the same as that for $c$. It is guaranteed by the upper semi-continuity of Mañé set in cohomology class. It follows that $\langle c,\rho(\mu_c)\rangle =\langle c',\rho(\mu_c')\rangle=0$ and $$-\alpha(c')=A(\mu_{c'})-\langle c',\rho(\mu_c')\rangle\ge A(\mu_c)=-\alpha(c).$$ However, as $c'\notin\mathbb{F}_0$, one has $\alpha(c')>\alpha(c)$. The contradiction verifies our claim.
Approached by minimal periodic curves, each minimal homoclinic curve $\gamma$ stays in certain Aubry set: $$\cup_{t\in\mathbb{R}}\gamma(t)\subset\mathcal{A}(c),\qquad \forall\ c\in\lim_{\delta\downarrow 0} \mathscr{L}_{\beta}(\delta[\gamma])\subset \mathbb{F}_0.$$ where the limit is in the sense of Hausdorff.
If $H_{\mathbb{F}_0}$ contains infinitely many elements, there would be infinitely many minimal homoclinic curves $\gamma_1,\gamma_2\cdots\gamma_k\cdots$ such that $[\gamma_i]\neq
[\gamma_j]$ provided $i\ne j$. Thus we have two possibilities.
1, a neighborhood $B_d(0)$ of the fixed point exists such that each minimal homoclinic curve $\gamma$ hits the sphere $\partial B_d(0)$ exactly twice, i.e. $\exists$ $t^-<t^+$ such that $\gamma(t)\in B_d(0)$ for all $t\in (-\infty,t^-]\cup [t^+,\infty)$ and $\gamma(t)\notin B_d(0)$ for all $t\in (t^-,t^+)$;
2, for any small $d>0$, there are infinitely many minimal homoclinic curves $\gamma_{i_1}, \gamma_{i_2}\cdots $ passing through the sphere $\partial B_d(0)$ in finite time, i.e. $\exists$ $t^-<t_0^-<t_0^+<t^+$ such that $\gamma(t)\in B_d(0)$ for all $t\in (-\infty,t^-]\cup[t^-_0,t_0^+]\cup [t^+,\infty)$ and $\gamma(t)\notin B_d(0)$ holds for some $t\in (t^-,t^-_0)$ as well as for some $t\in (t^+_0,t^+)$.
Let us study the first possibility. Denote by $t^-_i<t^+_i$ the time when the minimal homoclinic curve $\gamma_i$ hits the sphere $\partial B_d(0)$. For each $\gamma_i$, there is a segment $\gamma_i|_{(t^-_i,t^+_i)}$ staying outside of $B_d(0)$. Each $d\gamma_i|_{(t_i^-,t_i^+)}$ generates a probability measure $\mu_i$ on $T\mathbb{T}^n$ such that $$\int fd\mu_i=\frac 1{|t_i^+-t_i^-|}\int_{t_i^-}^{t_i^+}f(d\gamma_i(s))ds$$ holds for each continuous function $f$: $T\mathbb{T}^n\to\mathbb{R}$. As all these curves have different homology class, $\|[\gamma_i]\|\to\infty$ as $i\to\infty$. As the speed along these curves are uniformly bounded, we have $$|t_i^+-t_i^-|\to\infty, \qquad \text{\rm as }\ i\to\infty.$$ Let $c_i\in\partial\mathbb{F}_0$ be the class such that $\gamma_i\subset\mathcal{A}(c_i)$ and let $c^*\in\partial\mathbb{F}_0$ be an accumulation point of $\{c_i\}$, some invariant probability measure $\mu^*$ exists such that $\mu_i\rightharpoonup\mu^*$, it is $c^*$-minimal. Clearly, $\mu^*$ is not supported on the fixed point, it contradicts the assumption that the minimal measure is always uniquely ergodic for each $c\in\mathbb{F}_0$.
Let us study the second possibility. In this case, for suitably small $\delta>0$, there is an infinite sequence of homoclinic curves $\gamma_i$ and correspondingly the sequence of time $t^-_i<t^+_i$ such that $\gamma_i(t)\in B_{\delta}(0)$ for each $t\in [t^-_i,t^+_i]$, $\gamma_i(t_i^{\pm}\mp\epsilon) \notin B_{\delta}(0)$ and $|t_i^+-t_i^-|\to\infty$ as $i\to\infty$. Indeed, if $|t_i^+-t_i^-|$ remains bounded, one can choose $\delta'<\delta$ such that these curves hit the sphere $\partial B_{\delta'}(0)$ twice only. It is the first case again. By using Proposition \[flatpro2\], we find that some $\theta>0$ exists such that one of the inequalities in the following holds for each $i$ $$\label{flateq8}
\Big\|\frac{\dot\gamma_i(t_i^-)}{\|\dot\gamma_i(t_i^-)\|}-\frac{\xi_1^+}{\|\xi_1^+\|}\Big\|>\theta,\qquad
\Big\|\frac{\dot\gamma_i(t_i^+)}{\|\dot\gamma_i(t_i^+)\|}-\frac{\xi_1^-}{\|\xi_1^-\|}\Big\|>\theta$$ provided $i$ is sufficiently large. It implies that there exists some minimal homoclinic curve $\gamma$ as well as some eigenvector $\xi^-_{k_1}$ or $\xi^+_{k_2}$ with $k_1\ne 1$ and $k_2\ne 1$ such that at least one of the following holds $$\lim_{t\to-\infty}\frac{\dot\gamma(t)}{\|\dot\gamma(t)\|}=\frac{\xi_{k_2}^+}{\|\xi_{k_2}^+\|},\qquad
\lim_{t\to\infty}\frac{\dot\gamma(t)}{\|\dot\gamma(t)\|}=\frac{\xi_{k_1}^-}{\|\xi_{k_1}^-\|}.$$ This leads to a contradiction to the assumption, then verifies the finiteness of $H_{\mathbb{F}_0}$.
Let $\gamma,\gamma'$ be two minimal homoclinic curves contained in the Aubry set $\mathcal{A}(c)$, $\mathcal{A}(c')$ respectively. Let $\Gamma=\{\xi c+(1-\xi)c': \xi\in [0,1]\}$. If $\Gamma$ intersects the interior of $\mathbb{F}_0$, then $[\gamma]\neq [\gamma']$. Indeed, by definition we have $$\begin{aligned}
A(\gamma)-\langle c,[\gamma]\rangle&=0, \qquad A(\gamma')-\langle c,[\gamma']\rangle \ge 0;\\
A(\gamma')-\langle c',[\gamma']\rangle&=0, \qquad A(\gamma)-\langle c',[\gamma]\rangle \ge 0,\end{aligned}$$ it follows from $[\gamma]=[\gamma']$ that $A(\gamma)=A(\gamma')$. Consequently, $$\begin{aligned}
0&=\xi (A(\gamma)-\langle c,[\gamma]\rangle)+(1-\xi)(A(\gamma')-\langle c',[\gamma']\rangle) \\
&=A(\gamma)-\langle\xi c+(1-\xi)c',[\gamma]\rangle\\
&=A(\gamma')-\langle\xi c+(1-\xi)c',[\gamma']\rangle.\end{aligned}$$ It implies that both $\gamma$ and $\gamma'$ lie in the Aubry set for $\xi c+(1-\xi) c'$. On the other hand, the Aubry set for each class in the interior of $\mathbb{F}_0$ contains the fixed point only. The contradiction implies that $[\gamma]\neq [\gamma']$. Therefore, $\mathbb{F}_0$ is a polygon with exactly $m$ edges, each edge corresponds to one homology type of minimal homoclinic curve.
Let $\gamma$ be a minimal homoclinic curve lying in the Aubry set for $c\in\text{\rm int}\mathbb{E}_i$. Then, one has $A(\gamma)-\langle c,[\gamma]\rangle=0$. As the Aubry set remains the same for all classes in the interior of the edge, one has $A(\gamma)-\langle c',[\gamma]\rangle=0$ for each $c\in\text{\rm int}\mathbb{E}_i$. Consequently, one has $\langle c-c',[\gamma]\rangle =0$ for all $c,c'\in\text{\rm int}\mathbb{E}_i$. As it is $(n-1)$-dimensional, each edge $\mathbb{E}_i$ determines a unique homology class $g_i$ such that $[\gamma]=g_i$ if $\gamma$ is a minimal homoclinic curve lying in the Aubry set.
Modulus of continuity in terms of energy
----------------------------------------
Let $\gamma_0$ be a minimal homoclinic curve approaching to the fixed point in the direction of $\xi^{\pm}$ corresponding to the smallest eigenvalue $\pm\lambda_1$, let $\mathbb{E}_0$ be a edge of $\mathbb{F}_0$. In this subsection we assume that
1, for each $c\in\mathbb{F}_0$, the Mather set contains exactly one fixed point $(x,\dot x)=(0,0)$;
2, for each $c\in int\mathbb{E}_0$, the Aubry set consists of the fixed point and one minimal homoclinic curve $\gamma_0$: $\mathcal{A}(c)=\cup_{t\in\mathbb{R}}\gamma_0(t)\cup\{0\}$;
3, there exist a sequence of positive numbers $\nu_i\downarrow0$ and sequence of ergodic minimal measure $\mu_{i}$ such that $\rho(\mu_{i})=\nu_i[\gamma_0]$, and $\mathcal{A}(c)=\text{\rm supp}\mu_i$ for each $c\in\text{\rm int} \mathscr{L}_{\beta}(\nu_i[\gamma_0])$.
By definition, there exists an elementary weak KAM for each $\mu_{i}$ corresponding to the energy $E_i=\alpha(\mathscr{L}_{\beta}(\nu_i[\gamma_0]))$. The main purpose of this section is to study the modulus of continuity of some functions in terms of energy at $E=0$.
[**Dependence of the average speed on energy**]{}.
According to Birkhoff’s ergodic theorem, there is an orbit $d\zeta_i$: $\mathbb{R}\to\mathbb{T}^n$ of $\phi_L^t$ in the Mather set such that $$\frac 1{2T}A(\zeta_i|_{[-T,T]})\to A(\mu_i)\ \ \text{\rm and}\ \ \frac 1{2T}(\bar\zeta_i(T)-\bar\zeta_i(-T))\to\rho(\mu_i) \ \ \ \text{\rm as}\ T\to\infty,$$ where $\bar\zeta_i$ stands for a lift of $\zeta_i$ to the universal covering space. By the upper semi-continuity of Mañé set, the curve $\zeta_i$ passes through the ball $B_{\delta}(0)$ infinitely many times if $\nu_i$ is small. Denoted by $t_{i,k}^+$ and $t_{i,k}^-$ the time when $\zeta_{i}$ enters and leaves the ball respectively, i.e $\zeta_{i}(t)\in B_{\delta}(0)$ for each $t\in[t_{i,k}^+,t_{i,k}^-]$, $\zeta_{i}(t_{i,k}^{\pm}\mp\delta)\notin B_{\delta}(0)$. Clearly, $$|t_{i,k}^--t_{i,k}^+|\to\infty,\qquad \text{\rm as}\ E_i\to 0.$$ If $\zeta_i$ is a periodic curve, some $t_i>0$ exists so that $t_{i,k+1}^+=t_{i,k}^-+t_i$ holds for all $k\in\mathbb{Z}$.
Let $t^+,t^-\in\mathbb{R}$ such that the minimal homoclinic curve $\gamma_0$ enters $B_{\delta}(0)$ at $t=t^+$ and leaves $B_{\delta}(0)$ at $t=t^-$. By the upper semi-continuity of Mañé set one has $$(\zeta_{i}(t_{i,k}^{\pm}),\dot\zeta_{i}(t_{i,k}^{\pm}))\to(\gamma_0(t^{\pm}),\dot\gamma_0(t^{\pm})).$$ As each minimal homoclinic curve approaches to the fixed point in the direction $\xi_1^{\pm}$, $$\label{regularenergyeq1}
\Big\|\frac{\dot\zeta_{i}(t_{i,k}^{\pm})}{\|\dot\zeta_{i}(t_{i,k}^{\pm})\|}-\frac{\xi_1^{\mp}}{\|\xi_1^{\mp}\|} \Big\|<\frac 14$$ holds if ${\delta}>0$ is suitably small and $t_{i,k}^--t_{i,k}^+$ is suitably large.
Each segment $\zeta_{i}|_{[t_{i,k}^+,t_{i,k}^-]}$ solves the Hamilton equation. In the coordinates of normal form, it is given by Eq. (\[flateq7.1\]) and the integral constants $b_i^{\pm}$ satisfy the constraint (\[flateq6\]). The condition (\[regularenergyeq1\]) induces the following $$\frac 12 {\delta}e^{-\lambda_1|t_{i,k}^--t_{i,k}^+|/2}\le |b_1^{\pm}|\le 2{\delta} e^{-\lambda_1|t_{i,k}^--t_{i,k}^+|/2}.$$ For small ${\delta}>0$ and sufficiently large $|t_{i,k}^--t_{i,k}^+|$, we have $$|b_j^{\pm}|\le 2{\delta}e^{-\lambda_j|t_{i,k}^--t_{i,k}^+|/2},\qquad \forall\ j=2,\cdots,n,$$ $$|P_3((b^+_j+b^-_j),\lambda_j(b^+_j-b^-_j))|\le Ce^{-3\lambda_1|t_{i,k}^--t_{i,k}^+|/2}$$ where the constant $C$ depends only on the function $P_3$. So, for suitably small ${\delta}>0$ and sufficiently large $|t_{i,k}^--t_{i,k}^+|$, we obtain from (\[flateq6\]) that $$\begin{aligned}
E_i=&\Big|-2\sum_{j=1}^n\lambda_j^2b_i^+b_i^-+P_3((b^+_j+b^-_j),\lambda_j(b^+_j-b^-_j))\Big|\\
\ge&\frac 12\lambda_1^2{\delta}^2e^{-\lambda_1|t_{i,k}^--t_{i,k}^+|}-8\sum_{j=2}^n \lambda_j^2{\delta}^2e^{-\lambda_j|t_{i,k}^--t_{i,k}^+|}-Ce^{-3\lambda_1|t_{i,k}^--t_{i,k}^+|/2}\notag\\
\ge&\frac 14\lambda_1^2{\delta}^2e^{-\lambda_1|t_{i,k}^--t_{i,k}^+|}\end{aligned}$$ Under the same condition, $E_i$ is obviously upper bounded by $$\begin{aligned}
E_i=&\Big|-2\sum_{j=1}^n\lambda_j^2b_i^+b_i^-+P_3((b^+_j+b^-_j),\lambda_j(b^+_j-b^-_j))\Big|\\
\le&8\lambda_1^2{\delta}^2e^{-\lambda_1|t_{i,k}^--t_{i,k}^+|}+8\sum_{j=2}^n \lambda_j^2{\delta}^2e^{-\lambda_j|t_{i,k}^--t_{i,k}^+|}+Ce^{-3\lambda_1|t_{i,k}^--t_{i,k}^+|/2}\notag\\
\le&9\lambda_1^2{\delta}^2e^{-\lambda_1|t_{i,k}^--t_{i,k}^+|}.\end{aligned}$$ Therefore, we find the dependence of speed on the energy $$\label{regularenergyeq2}
|t_{i,k}^--t_{i,k}^+|=\frac 1{\lambda_1}|\ln E_i|-\frac 2{\lambda_1}|\ln \delta|+\tau_{i,k}$$ where $\tau_{i,k}$ is uniformly bounded for each $k\in\mathbb{Z}$: $$\frac 1{\lambda_1}(2\ln\lambda_1-2\ln 2)\le\tau_{i,k}\le \frac 1{\lambda_1}(2\ln\lambda_1+3\ln 3).$$ Obviously, $t_{i,k+1}^+-t_{i,k}^-\to t^+-t^-$ as $i\to\infty$, some $E_0>0$ exists such that $$\frac 12(t^+-t^-)\le t_{i,k+1}^+-t_{i,k}^-\le 2(t^+-t^-),\qquad\text{\rm if}\ E_i\le E_0.$$ Set $\tau_0=\frac 2{\lambda_1}|\ln 3\lambda_1|+2(t^+-t^-)$, one has $$\label{regularenergyeq3}
\Big|\frac 1\nu_i-\frac 1{\lambda_1}\Big|\ln\frac{E_i}{\delta^{2}}\Big|\Big|\le\tau_0.$$ Recall the meaning of $\nu_i$: $\nu_i[\gamma_0]$ is the rotation vector of the minimal measure$\mu_i$.
Around the two-dimensional flat
-------------------------------
In this section we restrict ourselves to the special case that the system has two degrees of freedom: $n=2$. The task of this section is to study the structure of the Mather sets as well as of the Mañé sets in a neighborhood of the resonant point. Under the coordinate transformation (\[normaleq10\]), it corresponds to a fixed point.
Since each Aubry set is a Lipschitz graph over the configuration manifold which is two dimensional here, each orbit in an Aubry set has to be [*parallel*]{} to any other orbit in the same set in the sense that these curves do not intersect each other. On the other hand, for autonomous system, $\beta(\lambda\omega)$, regarded as the function of $\lambda\in\mathbb{R}$, is differentiable at each $\lambda\neq 0$ (see [@Ms]). Thus, we have
\[flatpro3\] Assume that $L$ is an autonomous Tonelli Lagrangian defined on $\mathbb{T}^2$. For each non-zero rational vector, the Mather set consists of periodic orbits with the same rotation vector.
Each minimal measure with zero-rotation corresponds to the minimum of the $\alpha$-function. There are two [*nondegenerate*]{} cases for the set of minimal point $\mathbb{F}_0\subset H^1(\mathbb{T}^2,\mathbb{R})$. We call a case [*nondegenerate*]{} if it persists under small perturbation.
1, $\mathbb{F}_0$ is a two-dimensional flat. Typically, for each class in the interior of $\mathbb{F}_0$, the minimal measure is supported on a fixed point, or a shrinkable periodic orbit $(\gamma,
\dot\gamma)$, i.e. $[\gamma]=0$. The fixed point (periodic orbit) is of hyperbolic type.
2, $\mathbb{F}_0$ is one-dimensional. Typically, the minimal measure is supported on two periodic orbits $(\gamma_-, \dot\gamma_-)$ and $(\gamma_+, \dot\gamma_+)$ with the property: $$[\gamma_-]/\|[\gamma_-]\|=-[\gamma_+]/ \|[\gamma_+]\|.$$
The set $\mathbb{F}_0$ is a singleton only when the $\alpha$-function is differentiable at this point. Otherwise, the $\beta$ function can not have a two-dimensional flat. In this case, the Mather set contains two circles with different rotation direction, but it violates the Lipschitz property.
Let us study the first case and assume that $0\in\text{\rm int}\mathbb{F}_0$, the point $(x,\dot x)=(0,0)$ supports the minimal measure. Then, $\mathcal{A}(c)=\{0\}$ for all $c\in int\mathbb{F}_0$. The study is similar if it is supported on a shrinkable closed orbit.
Given the $\alpha$ as well as the $\beta$-function, let us recall the Fenchel-Legendre transformation $\mathscr{L}_{\beta}$: $H_1(M,\mathbb{R})\to H^1(M,\mathbb{R})$ is defined as $$\mathscr{L}_{\beta}(\omega)=\{c: \alpha(c)+\beta(\omega) =\langle
c,\omega\rangle\}.$$ Let $$\partial^*\mathbb{F}_0=\{c\in\partial\mathbb{F}_0: \ \mathcal{M}(c)\backslash\{x=0\}\neq\varnothing \},\qquad \Omega_{\mathbb{F}_0}=\mathscr{L}_{\beta}^{-1}(\partial^*\mathbb{F}_0),$$ it may be non-empty. Here is an example: $$L=\frac 12\dot x_1^2+\frac {\lambda^2}2\dot x_2^2+V(x)$$ where $|\lambda|\ne 1$, the potential satisfies the following conditions: $x=0$ is the minimal point of $V$ only; there exist two numbers $d>d'>0$ such that for any closed curve $\gamma$: $[0,1]\to\mathbb{T}^2$ passing through the origin with $[\gamma]\ne 0$ one has $$\int_0^1V(\gamma(s))ds\ge d;$$ $V=d'+(x_2-a)^2$ when it is restricted a neighborhood of circle $x_2=a$ with $a\ne 0$ mod 1. In this case, $\partial\mathbb{F}_0\cap\{c_2=0\}=\{c_1=\pm\sqrt{2d'}\}$. Indeed, $$L\pm c_1\dot x_1=\frac 12(\dot x_1\pm c_1)^2+\frac{\lambda^2}2\dot x_2^2+V(x)-\frac 12c_1^2,$$ the Mather set for $c=(\pm\sqrt{2d'},0)$ consists of the point $x=0$ and the periodic curve $x(t)=(x_{1,0}\mp\sqrt{2d'}t,a)$.
Clearly, the set $\partial^*\mathbb{F}_0$ is closed with respect to $\mathbb{F}_0$. If it is non-empty, the existence of infinitely many $\bar M$-minimal homoclinic orbits has been proved in [@Zhe; @Zho1]. These orbits are associated with different homological classes. If $\partial^*\mathbb{F}_0=\varnothing$, there are at least three minimal homoclinic orbits to the fixed point.
The existence of homoclinic orbit to some Aubry set is closely related to the existence of the flat of the $\alpha$-function.
\[flatlem1\] Given $c,c'\in\mathbb{F}$, let $c_{\lambda}=\lambda c+(1-\lambda)c'$. Then $$\tilde{\mathcal{A}}(c)\cap\tilde{\mathcal{A}}(c')=\tilde{\mathcal{A}}(c_{\lambda}), \qquad \forall\ \lambda\in (0,1).$$
Using argument in [@Ms], for any curve $\gamma$: $\mathbb{R}\to M$, we have $$[A_{c_{\lambda}}(\gamma|_{I})]=\lambda [A_c(\gamma|_{I})]+(1-\lambda)[A_{c'}(\gamma|_{I})],\qquad \forall\
I\subset\mathbb{R}.$$ As both $\lambda>0$ and $1-\lambda>0$, one has that $[A_c(\gamma)]=[A_{c'}(\gamma)]=0$ if $[A_{c_{\lambda}}(\gamma)]=0$.
\[flatlem2\] Let $\mathbb{F}_0$ be a 2-dimensional flat, the Mather set is a singleton for each class in the interior of $\mathbb{F}_0$, let $\mathbb{E}_i$ be an edge of $\mathbb{F}_0$, then $$\mathcal{A}(c')\supsetneq\mathcal{A}(c)$$ holds for $c'\in\partial\mathbb{F}_0$ $(\partial\mathbb{E}_i)$ and $c\in int \mathbb{F}$ $(int \mathbb{E}_i)$ respectively.
As the Mather set is a singleton for each $c\in\text{\rm int}\mathbb{F}_0$, each orbit in the Aubry set is either the fixed point itself, or a homoclinic orbit to the point with zero first homology. Indeed, let $[\gamma]$ denote its first homology of the homoclinic curve $\gamma$ in the Aubry set, then $$\int_{-\infty}^{\infty}L(d\gamma(t))dt-\langle c,[\gamma]\rangle=0$$ holds for each $c\in\text{\rm int}\mathbb{F}_0$. It follows that $\langle c-c',[\gamma]\rangle=0$ for $c,c'\in\text{\rm int}\mathbb{F}_0$. Since $\mathbb{F}_0$ shares the same dimension of the configuration space, $[\gamma]=0$. In fact, for classical mechanical system, the Aubry set consists of the fixed point only for $c\in\text{\rm int}\mathbb{F}_0$. If $c'\in\partial\mathbb{F}_0\backslash\partial^*\mathbb{F}_0$, as shown in the proof of Theorem \[flatthm2\], the Aubry set $\mathcal{A}(c')$ contains at least one minimal homoclinic curve with non-zero first homology. If $c'\in\partial^*\mathbb{F}_0$, the certain $c'$-minimal measure $\mu_{c'}$ exists with $\rho(\mu_{c'})\ne 0$. In both cases, $\mathcal{A}(c')\supsetneq\mathcal{A}(c)$ if $c\in\text{\rm int}\mathbb{F}_0$.
Let $\mathbb{E}_i$ be an edge. For $c\in\text{\rm int}\mathbb{E}_i$, the Aubry set contains one or more homoclinic curves, all of them share the same homology class, denoted by $g(\mathbb{E}_i)$ which is of course non-zero. If $\mathcal{M}(c)$ contains other curves, these curves also share the same rotation vector as $\langle c-c',g(\mathbb{E}_i)\rangle=0$ holds for $c,c'\in\text{\rm int}\mathbb{E}_i$.
Let $c'\in\partial\mathbb{E}_i$ and $c\in\text{\rm int}\mathbb{E}_i$, one chooses $c^*\in\partial\mathbb{F}_0 \backslash\mathbb{E}_i$ arbitrarily close to $c'$. As the straight line connecting $c$ to $c^*$ passes through the interior of $\mathcal{F}_0$, we obtain from Lemma \[flatlem1\] that $\mathcal{A}(c)\cap \mathcal{A}(c^*)=\mathcal{A}(c_0)$ with $c_0\in int\mathbb{E}_i$. For any curve $\zeta$ contained in $\mathcal{A}(c^*)\backslash\mathcal{A}(c_0)$, it follows from the formulation $$0=\int (L(d\zeta(t))-\langle c^*,\dot\zeta\rangle)dt=\int (L(d\zeta(t))-\langle c,\dot\zeta\rangle)dt+\langle c-c^*,[\zeta]\rangle$$ that $\langle c-c^*,[\zeta]\rangle\neq0$ holds. We claim $[\zeta]\neq g(\mathbb{E}_i)$. Let us assume the contrary and consider the case that $\zeta$ is a homoclinic curve and $\mathcal{A}(c)$ contains a homocilinic curve $\gamma$. In this case, by assuming that $\alpha(c)=0$ for $c\in\mathbb{F}_0$, we have $$\int_{-\infty}^{\infty} L(d\zeta)dt -\langle c^*,[\zeta]\rangle=0, \qquad \int_{-\infty}^{\infty} L(d\gamma)dt -\langle c,g(\mathbb{E}_i)\rangle=0.$$ Since the class $c^*$ is not on the straight line containing $\mathbb{E}_i$, we have $\langle c^*-c,g(\mathbb{E}_i)\rangle\ne0$. If $\langle c^*-c,g(\mathbb{E}_i)\rangle>0$ we would have $$\int_{-\infty}^{\infty} L(d\gamma)dt-\langle c^*,[\gamma]\rangle=\int_{-\infty}^{\infty} L(d\gamma)dt-\langle c,[\gamma]\rangle- \langle c^*-c,g(\mathbb{E}_i)\rangle<0$$ If $[\zeta]=g(\mathbb{E}_i)$ and $\langle c^*-c,g(\mathbb{E}_i)\rangle<0$ we would have $$\int_{-\infty}^{\infty} L(d\zeta)dt-\langle c,[\zeta]\rangle=\int_{-\infty}^{\infty} L(d\gamma)dt-\langle c^*,[\zeta]\rangle+ \langle c^*-c,g(\mathbb{E}_i)\rangle<0$$ Both cases are absurd as $\alpha(c)=\alpha(c^*)=0$. Because $[\zeta]\neq g(\mathbb{E}_i)$, some $x^*\in\mathcal{A}(c^*)$ remains far away from $\mathcal{A}(c)$. Let $c^*\to c'$, the accumulation point of these points does not fall into $\mathcal{A}(c)$, it implies $\mathcal{A}(c')\supsetneq\mathcal{A}(c)$. The proof is similar if $\xi$ as well as $\gamma$ is a curve lying in the Mather set.
Recall the definition of $G_m$ in the section 2: a first homology class $g\in G_m$ if and only if there exists a minimal homoclinic orbit $d\gamma$ such that $[\gamma]=g$. Let $G_{m,c}\subset G_m$ be defined such that $g\in G_{m,c}$ if and only if there exists a minimal homoclinic orbit $d\gamma$ in $\tilde{\mathcal{A}}(c)$ such that $[\gamma]=g$. We say that there are $k$-types of minimal homoclinic orbits in $\tilde{\mathcal{A}}(c)$ if $G_{m,c}$ contains exactly $k$ elements. For an edge we define $G_{m,\mathbb{E}_i}=G_{m,c}$ for each $c\in int \mathbb{E}_i$, from the proof of Lemma \[flatlem2\] one can see that it makes sense.
\[flatthm3\] Let $\mathbb{F}_0$ be a two dimensional flat, $\mathcal{M}(c_0)$ is a singleton for $c_0\in int\mathbb{F}_0$. Let $\mathbb{E}_i$ denote an edge of $\mathbb{F}_0$ (not a point), then
1, either $\mathbb{E}_i\cap\partial^*\mathbb{F}_0 =\varnothing$ or $\mathbb{E}_i\subset\partial^*\mathbb{F}_0$;
2, if $\mathbb{E}_i\cap\partial^*\mathbb{F}_0 =\varnothing$, then $G_{m,\mathbb{E}_i}$ contains exactly one element, if $\mathbb{E}_i\subset\partial^*\mathbb{F}_0$, all curves in $\mathcal{M}(\mathbb{E}_i)\backslash\{0\}$ have the same rotation vector;
3, if $c\in\partial\mathbb{E}_i$ and $c\notin\partial^*\mathbb{F}_0$ then $G_{m,c}$ contains exactly two elements;
4, if $\mathbb{E}_i, \mathbb{E}_j\subset\partial^*\mathbb{F}_0$, then either $\mathbb{E}_i$ and $\mathbb{E}_j$ are disjoint, or $\mathbb{E}_i=\mathbb{E}_j$;
5, if $\mathbb{E}_i\subset\partial^*\mathbb{F}_0$, $\mathcal{M}(c)=\mathcal{M}(c')$ holds for $c\in\partial \mathbb{E}_i$ and $c'\in int \mathbb{E}_i$.
For the conclusion 1, as $\mathcal{A}(c)=\mathcal{A}(c')$ if $c,c'\in\text{\rm int}\mathbb{E}_i$ [@Ms], we only need to consider $c\in\partial\mathbb{E}_i$. If it is not true, there would exist an invariant measure $\mu_c$, not supported on the singleton and minimizing the action $$\int Ld\mu_c-\langle \rho(\mu_c),c\rangle=-\alpha(c),$$ but not minimizing the $c'$-action for $c'\in\text{\rm int}\mathbb{E}_i$. As the configuration space is $\mathbb{T}^2$, the Lipschitz graph property of Aubry set will be violated if the rotation vector of the measure $\rho(\mu_c)$ is not parallel to $g\in G_{m,\mathbb{E}_i}$. So, $\langle\rho(\mu_c), c-c'\rangle=0$ holds for $c'\in\text{\rm int}\mathbb{E}_i$, thus $\mu_c$ also minimizes the action for $c'\in\text{\rm int}\mathbb{E}_i$. This leads to a contradiction. Since $\partial^*\mathbb{F}_0$ is closed, once $\text{\rm int}\mathbb{E}_i\subset\partial^*\mathbb{F}_0$, then whole edge is also contained in $\partial^*\mathbb{F}_0$.
The conclusion 2 follows from the fact that $\langle c-c',[\gamma]\rangle=0$ holds for any $c,c'\in\text{\rm int}\mathbb{E}_i$ and any $\gamma\in\mathcal{A}(c)$, the conclusion 3 follows from that $\mathcal{A}(c)\varsupsetneq\mathcal{A}(c')$ if $c'\in\text{\rm int}\mathbb{E}_i$.
If the conclusion 4 was not true, for the cohomology class in $\mathbb{E}_i\cap \mathbb{E}_j$ the Mather set would contain two closed circles with different homology, but it violates the Lipschitz graph property of Aubry set. With the same reason we have the conclusion 5.
By this theorem, each edge $\mathbb{E}_i\subset\partial^*\mathbb{F}_0$ aslo uniquely determines a class $g(\mathbb{E}_i)$ so that for each $c\in int \mathbb{E}_i$, the rotation vector of each $c$-minimal measure has the form $\nu g(\mathbb{E}_i)$ ($\nu>0$). For brevity, we also use the notation $\mathcal{M}(\mathbb{E}_i)= \mathcal{M}(c)$ for $c\in \mathbb{E}_i$.
![$\mathbb{E}_i\subset\partial^*\mathbb{F}_0$, $\mathcal{M}(\mathbb{E}_i)=\{0\}\cup
\{\gamma\}$. The blue curve is in $\mathcal{A}(c)$ for $c$ at one end point of $\mathbb{E}_i$, the red curve is in $\mathcal{A}(c')$ for $c'$ at another end point of $\mathbb{E}_i$.[]{data-label="Fig1"}](Ardiff1.eps){width="6.6cm" height="3.2cm"}
Given two homology classes $g,g'\in H_1(\mathbb{T}^2,\mathbb{Z})$, we call them adjacent if $g\in G_{m,\mathbb{E}}$, $g'\in G_{m,\mathbb{E}'}$, $\mathbb{E}\cap\partial^*\mathbb{F}_0=\varnothing$, $\mathbb{E}'\cap\partial^*\mathbb{F}_0=\varnothing$, $\mathbb{E}$ and $\mathbb{E}'$ are adjacent. The special topology of two-dimensional torus induces some restrictions on adjacent homologies.
\[flatlem3\] Let $\mathbb{E},\mathbb{E}'\subset\partial\mathbb{F}_0\backslash\partial^*\mathbb{F}_0$ be two adjacent edges and assume $c\in \mathbb{E}\cap \mathbb{E}'$. If $(m,n)=g\in G_{m,\mathbb{E}}$ and $(m',n')=g'\in G_{m,\mathbb{E}'}$, then one has that $m'n-mn'=\pm 1$.
The Aubry set $\tilde{\mathcal{A}}(c)$ contains homoclinic orbits with two two classes $(m,n)$ and $(m',n')$, both are irreducible. Guaranteed by the Lipschitz graph property, these curves intersect each other only at the fixed point. In the universal covering space $\mathbb{R}^2$, each curve in the lift of the homoclinic curves are determined by the equation $$mx_1+nx_2=k,\qquad m'x_1+n'x_2=k'.$$ The solution of the equations corresponds to the intersection point which are lattice points in $\mathbb{Z}^2$ for any $(k,k')\in\mathbb{Z}^2$. To guarantee this property, the necessary and sufficient condition is $mn'-m'n=\pm 1$.
For each indivisible homological class $0\neq g\in H_1(\mathbb{T}^2,\mathbb{Z})$, either $\mathscr{L}_{\beta}(\lambda g)\notin\partial\mathbb{F}_0$ for any $\nu>0$, or some $\lambda_0>0$ exists such that $\mathscr{L}_{\beta}(\lambda_0 g)\in\partial^*\mathbb{F}_0$.
In the first case, $\mathscr{L}_{\beta}(\lambda g)\to\partial\mathbb{F}_0\backslash\partial ^*\mathbb{F}_0$ as $\lambda\downarrow 0$, at least one periodic curve $\gamma_{\lambda}\subset\mathcal{M}(c)$ exists for $c\in\mathscr{L}_{\beta}(\lambda g)$ with $\lambda>0$. It is impossible that $d(c,\mathscr{L}_{\beta}(\lambda g))\to 0$ holds for $c\in\partial^*\mathbb{F}_0$, as in that case certain $c$-minimal measure $\mu_c$ would exist so that $\rho(\mu_c)$ is not parallel to $[\gamma]$. It will violate the Lipschitz property. Generically, $(\gamma_{\lambda},\dot\gamma_{\lambda})$ is hyperbolic and $\mathscr{L}_{\beta} (\lambda g)$ is an interval if $\lambda>0$. If $g\in G_{m,\mathbb{E}_i}$, then $\mathscr{L}_{\beta}(\lambda g)$ approaches to certain edge $\mathbb{E}_i$. If $g=k_ig_i+k_{i+1}g_{i+1}$ with indivisible $(k_i,k_{i+1})\in\mathbb{Z}^2_+$, $g_i\in G_{m,\mathbb{E}_i}$, $g_{i+1}\in G_{m,\mathbb{E}_{i+1}}$, $\mathbb{E}_i$ and $\mathbb{E}_{i+1}$ are two adjacent edges. As $\lambda\downarrow 0$, the interval will shrink to a vertex where $\mathbb{E}_i$ is joined to $\mathbb{E}_{i+1}$, and we have a sequence of closed orbits $\{d\gamma_{\lambda}\} =\{\cup_t(\gamma_{\lambda}(t),\dot\gamma_{\lambda}(t))\}$. Its Kuratowski upper limit set is obviously in the Aubry set for certain $c\in\partial\mathbb{F}_0\backslash\partial^*\mathbb{F}_0 $, thus, consists of minimal homoclinic orbits to the fixed point. As $c$ approaches to the vertex, the Mather set approaches to a set of figure eight: $$\mathcal{M}(c)\to \gamma_i\ast\gamma_{i+1},$$ where $\gamma_{\ell}\subset\mathcal{A}(E_{\ell})$ is a minimal homoclinic orbit such that $[\gamma_{\ell}]=g_{\ell}$ for ${\ell}=i,j$.
![For each $c$ on the red line in the channel, the Aubry set is a closed orbit in the cylinder, it approaches to a curve of figure eight.[]{data-label="fig2"}](Ardiff2.eps){width="7cm" height="3.0cm"}
To be more precise, let us study this phenomenon in the finite covering space $\bar M=\bar k_1\mathbb{T} \times\bar k_2\mathbb{T}$ where $\bar k_m=k_ig_{im}+k_{i+1}g_{i+1,m}$ for $m=1,2$ if we write $g_j=(g_{j1},g_{j2})$ for $j=i,i+1$. Let $\sigma$: $\{1,2,\cdots,k_i+k_{i+1}\}\to \{i,{i+1}\}$ be a permutation such that the cardinality $\#\sigma^{-1}(i)=k_i$ and $\#\sigma^{-1}({i+1})=k_{i+1}$. The lift of homoclinic curve $\gamma_i$ as well as $\gamma_{i+1}$ to $\bar M$ contains several curves, not closed. Pick up one curve $\bar\gamma_{\sigma(1)}$ in the lift of $\gamma_{\sigma(1)}$, it determines a unique curve $\bar\gamma_{\sigma(2)}$ such that the end point of $\bar\gamma_{\sigma(1)}$ is the starting point of $\bar\gamma_{\sigma(2)}$, and so on. See Figure \[fig3\].
![$[\gamma_1]=(1,0)$, $[\gamma_2]=(1,1)$, $k_1=1$, $k_2=2$. For each class $c$ on the red line, $\rho(\mu_c)=\lambda([\gamma_1]+2[\gamma_2])$. The solid blue line represent a periodic curve, the solid purple line represent the conjunction of the minimal homoclinic curves. The dashed lines represent the image of the Deck-transformation.[]{data-label="fig3"}](Ardiff3.eps){width="7.0cm" height="4.2cm"}
We are going to show below that there exists a unique permutation $\sigma$ such that one Aubry class in $\mathcal{A}(c,\bar M)$ $$\mathcal{A}_i(c,\bar M)\to \bar\gamma_{\sigma(1)}\ast
\bar\gamma_{\sigma(2)}\ast\cdots\ast\bar\gamma_{\sigma(k_i+k_{i+1})}$$ as $c$ approaches to the vertex along the path in the channel. As the minimal curve $\gamma_{\lambda}$ is periodic with the homological class $[\gamma_{\lambda}]=k_ig_i+k_{i+1}g_{i+1}$, the permutation $\sigma$: $\mathbb{Z}\to \{i,i+1\}$ is $(k_i+k_{i+1})$-periodic. Since $k_i$ is prime to $k_{i+1}$, we have $k_i=k_{i+1}=1$ if $k_i=k_{i+1}$.
\[flatlem4\] The permutation is uniquely determined by $k_i$ and $k_{i+1}$. If $k_{i}>k_{i+1}$, the following holds for $j=1,\cdots,k_i+k_{i+1}$ $$\begin{aligned}
&\sigma(j+j_0)=i, \hskip 1.0 true cm \text{\rm if}\ \ (a_j)\ne 0; \\
&\sigma(j+j_0)=i+1,\hskip 0.35 true cm \text{\rm if}\ \ (a_j)=0.\end{aligned}$$
By the assumption, there exists only one minimal homoclinic curve $\gamma_j$ such that $[\gamma_j]=g_j$ for $j=i,i+1$. Because of the lemma \[flatlem3\], we can assume that $g_i=(1,0)$ and $g_{i+1}=(0,1)$ by choosing suitable coordinates on $\mathbb{T}^2$. We choose two sections $I^-$ and $I^+$ in a small neighborhood of the origin such that, emanating from the origin, these homoclinic curves pass through $I^-$ and $I^+$ successively before they return back to the origin as $t\to\infty$. In the section $I^{\pm}$ we choose disjoint subsections $I^{\pm}_i$ and $I^{\pm}_{i+1}$ such that the curve $\gamma_j$ passes through $I^{\pm}_{j}$ for $j=i,i+1$.
Let $\gamma_{\lambda}$ be the minimal periodic curve with rotation vector $\lambda g$. For small $\lambda>0$, $\gamma_{\lambda}$ falls into a small neighborhood of these two homoclinic curves. So it has to pass either through $I^{\pm}_i$ or through $I^{\pm}_{i+1}$. Let $t_{\ell}^{\pm}$ be the time for $\gamma_{\lambda}$ passing through $I^{\pm}$ with $\cdots<t_{{\ell}-1}^-<t_{\ell}^+<t_{\ell}^-<t_{{\ell}+1}^+<\cdots$, and it does not tough these sections whenever $t\ne t_k^{\pm}$. By definition, the period of the curve equals $t_{k_1+k_2}^{\pm}-t_0^{\pm}$. If the curve intersects $I^+_i$ at $t^+_{\ell}$ and intersects $I^-_{i+1}$ at $t^-_{\ell}$, then the segment $\gamma_{\lambda}|_{[t_{{\ell}-1}^-,t_{\ell}^+]}$ keeps close to $\gamma_i$ and $\gamma_{\lambda}|_{[t_{\ell}^-,t_{\ell+1}^+]}$ keeps close to $\gamma_{i+1}$, so one has $\gamma_{\lambda}(t^-_{\ell-1})\in I_i^-$ and $\gamma_{\lambda}(t^+_{\ell+1})\in I^+_{i+1}$.
{width="7.8cm" height="3.5cm"} \[fig4\]
As the curve $\gamma_{\lambda}$ is minimal, it does not have self-intersection. Thus, once there exists $t_j^{\pm}$ such that $\gamma_{\lambda}(t^+_j)\in I^+_i$ and $\gamma_{\lambda}(t^-_j)\in I^-_i$, then there does not exist $t_{j'}^{\pm}$ such that $\gamma_{\lambda}(t^+_{j'})\in I^+_{i+1}$ and $\gamma_{\lambda}(t^-_{j'})\in I^-_{i+1}$. Therefore, there is a set $J\subset\{1,2,\cdots,k_i+k_{i+1}\}$ with cardinality $\#(J)=k_i-k_{i+1}$ such that for $j\in J$ one has $\gamma_{\lambda}(t^{\pm}_{j})\in I^{\pm}_{i}$, for $j\notin J$ one either has $\gamma_{\lambda}(t^{+}_{j})\in I^{+}_{i}$ and $\gamma_{\lambda}(t^{-}_{j})\in I^{-}_{i+1}$ or has $\gamma_{\lambda}(t^{+}_{j})\in I^{+}_{i+1}$ and $\gamma_{\lambda}(t^{-}_{j})\in I^{-}_{i}$.
By introducing coordinate transformation on $T$: $\mathbb{T}^2\to\mathbb{T}^2$ such that $T_*g=g$ $\forall\,g\in H_1(\mathbb{T}^2,\mathbb{Z})$, let us think the curve $T\gamma_{\lambda}$ as a straight line projected down to the unit square, a fundamental domain of $\mathbb{T}^2$. Starting from a point $z^h_0=(x_0,0)$, the line successively reaches to the points $z^h_1=(x_1,0),\cdots,z^h_m=(x_m,0),\cdots,z^h_{k_i}=z^h_{0}$ where $x_m=(x_0+mk_{i+1}/k_i\mod 1,0)$ with small $x_0>0$. To connect the point $(x_{m-1},0)$ to the point $(x_m,1)$, the curve $T\gamma_{\lambda}$ does not touch the vertical boundary lines if $$\Big[(m-1)\frac{k_{i+1}}{k_i}\Big]=\Big[m\frac{k_{i+1}}{k_i}\Big],$$ where $[a]$ denote the largest integer not bigger than the number $a$, and it has to pass through the vertical lines at some point $z^v_m=(0\mod 1,y_m)$ if $$\Big[(m-1)\frac{k_{i+1}}{k_i}\Big]+1=\Big[m\frac{k_{i+1}}{k_i}\Big].$$ We define an order $\prec$ for these $k_i+k_{i+1}$ points such that $z^h_j\prec z^h_k$ iff $j<k$ and $z^h_j\prec z^v_{j+1}\prec z^h_{j+1}$ iff $[jk_i/k_{j+1}]+1=[(j+1)k_i/k_{j+1}]$.
Returning back to the original coordinates, the curve $\gamma_{\lambda}$ falls into a neighborhood of the curves $\gamma_{i}$ and $\gamma_{i+1}$, intersects the horizontal line $\Gamma_h=T^{-1}\{(x_1,x_2):x_1=\frac12\mod 1\}$ at $T^{-1}z_j^h$ and intersects the vertical line $\Gamma_v=T^{-1}\{(x_1,x_2):x_2=\frac12\mod 1\}$ at $T^{-1}z_j^v$, $[\Gamma_h]=g_{i+1}$ and $[\Gamma_v]=g_i$. Naturally, the map $T$ induces the order among these points: $T^{-1}z^{h,v}_j\prec T^{-1}z^{h,v}_{\ell}$ if and only if $z^{h,v}_j\prec z^{h,v}_{\ell}$. If the curve passes the point $T^{-1}z^h_j$ at $t\in(t^-_j,t^+_{j+1})$, the segment $\gamma_{\lambda}|_{[t^-_j,t^+_{j+1}]}$ falls into a neighborhood of $\gamma_i$, otherwise, it falls into a neighborhood of $\gamma_{i+1}$. In this way, we obtained a unique permutation $\sigma$ up to a translation.
{width="7cm" height="3.5cm"} \[fig5\]
In the second case, some $\lambda_0>0$ exists such that $\mathscr{L}_{\beta}(\lambda g)\in\partial^* \mathbb{F}_0$. It is typical that certain edge $\mathbb{E}_i$ exists such that $\mathscr{L}_{\beta}(\lambda g)=\mathbb{E}_i\subset\partial^*\mathbb{F}_0$, the $c$-minimal measure is supported on a hyperbolic periodic orbit for each $c\in\mathbb{E}_i$. Thus, there is a channel $\mathbb{C}$ endding at $\mathbb{E}_i$ such that the family of periodic orbits $\cup_{c\in\mathbb{C}}\tilde{\mathcal{M}}(c)$ constitutes a normally hyperbolic cylinder.
![For each $c\in\mathbb{C}$, the Aubry set is a closed orbit in the cylinder (in purple), the closed circle in blue is in the Aubry set for $c\in\mathbb{F}_0\cap\mathbb{C}$, the closed orbit in green is not global minimum.[]{data-label="fig6"}](Ardiff6.eps){width="7cm" height="3.5cm"}
In the following, we shall study the dynamics on certain energy level of truncated normal form $H(x,y,y_3)=E$. Since one obtains the condition $\partial_{y_3}H\neq 0$ from the normal form, some function $y_3=Y(x,y)$ solves the equation $H=E$. Treat $Y(x,y)$ as the new Hamiltonian and let $\tau=-x_3$ be the time, one obtains a system with two degrees of freedom, which is equivalent to the dynamics on the energy level $\{H^{-1}(E)\}$.
\[flatthm5\] For the Hamiltonian $H(x,y,x_n,y_n)$ we assume that $\partial_{y_n}H\neq 0$ on $\{H^{-1}(E)\}\cap\{y_n\in [y_n^-,y_n^+]\}$. Let $y_n=Y(x,y,\tau)$ be the solution of $H=E$ $(\tau=-x_n)$. Let $\alpha_H$ and $\alpha_G$ be the $\alpha$-function for $L_H$ and $L_G$ respectively, where $$L_H(x,x_n,\dot x,\dot x_n)=\max_{y,y_n}\langle (\dot x,\dot x_n),(y,y_n)\rangle -H(x,y,x_n,y_n),$$ $$L_Y(x,y,\tau)=\max_{y}\langle \dot x,y\rangle -Y(x,y,\tau),$$ Then for $\alpha_Y(c)\in [y_n^-,y_n^+]$ we have $(c,\alpha_Y(c))\in \alpha_H^{-1}(E)$.
Let $\tilde c=(c,\alpha_Y(c))$, $\tilde\gamma=(\gamma,\gamma_n)$, $\tilde x=(x,x_n)$ and $\tilde y=(y,y_n)$. Let $\gamma$ be $c$-minimal curve for the Lagrange flow $\phi_{L_F}^t$, $\tilde\gamma$ is then $\tilde c$-minimal curve for the Lagrange flow $\phi_{L_H}^t$ if $\gamma_n=x_n$ and $\tilde\gamma$ is re-parameterized $\tau\to t$. If $x=x(\tau)$ is a solution of $\phi_{L_F}^t$, one obtains $y=y(\tau)$ from the Hamiltonian equations. Since $H(\tilde x(t),\tilde y(t))\equiv E$, we find $$\begin{aligned}
[A_Y(\gamma)]&=\int\Big(\Big\langle\frac{dx}{d\tau},y-c\Big\rangle-y_n+\alpha_Y(c)\Big)d\tau\\
&=\int(\langle\dot{\tilde x},\tilde y-\tilde c\rangle -H+E)dt\\
&=[A_H(\tilde\gamma)].\end{aligned}$$ This completes the proof.
Let $\pi_3: \mathbb{R}^3\to\mathbb{R}^{n-1}$ be the projection $\pi_3\tilde x=x$. By this theorem, $\pi_3^{-1}:H^1(\mathbb{T}^{2},\mathbb{R})\to \alpha_H(E)$ is a homeomorphism for $c\in\mathbb{F}_0+d$, the $d$-neighborhood of the flat $\mathbb{F}_0$. Thus, what we obtained in this subsection have their counterpart in the energy level set $\{H^{-1}(E)\}$ where the class $\tilde c\in\pi_3^{-1}(\mathbb{F}_0+d)\cap\alpha^{-1}_H(E)$.
Normally hyperbolic invariant cylinder
======================================
In this section, normally hyperbolic invariant cylinder is proved to exist in certain neighborhood of double resonant point. It uses the normal form which is obtained in the appendix where several steps of KAM iteration and one step of linear coordinate transformation were carried out. All these coordinate transformations are symplectic. Since Aubry set and Mañé set are symplectic invariants [@Be2], it is good enough to study these objects by considering the normal form.
Homogenized Hamiltonian
-----------------------
The normal form of the Hamiltonian takes the form $$\label{homogenizedeq1}
H=\tilde h(\tilde y)+\epsilon\tilde Z(x,\tilde y)+\epsilon\tilde R(\tilde x,\tilde y),$$ where $\tilde x=(x,x_3)=(x_1,x_2,x_3)$ and $\tilde y=(y,y_3)=(y_1,y_2,y_3)$. Since $h$ is positive definite, a unique curve exists along which $\partial_{y}h=(0,0)$. This curve passes through the energy level $\{h^{-1}(\tilde E)\}$ transversally at a unique point $\tilde y_0$. As $\tilde E>\min\alpha$, one has $\partial_{y_3}h(\tilde y_0)=\omega_3\ne 0$. By coordinate translation, we assume $\tilde y_0=0$.
As the dynamics we are going to study is restricted on an energy level $\{H^{-1}(\tilde E)\}$, it can be reduced to a system with two and half degrees of freedom. Since $\omega_3\ne 0$, the equation $H(\tilde x,\tilde y)=\tilde E$ uniquely determines a smooth function $y_3=y_3(x,y,x_3)$ in certain neighborhood of $y_3=0$. Treating $-\omega_3 y_3=Y$ as a new Hamiltonian and $\omega_3^{-1}x_3=\tau$ as a new time variable, we obtain a time-periodic system with two degrees of freedom. Correspondingly, we have the normal form $$\label{homogenizedeq2}
Y(x,y,\tau)=h(y)+\epsilon Z(x,y)+\epsilon R(x,y,\tau)$$ where $h(0)=0$, $\partial_yh(0)=0$ and $\epsilon R$ is as small as $\epsilon\tilde R$.
Each $\text{\bf k}\in\mathbb{Z}^2$ determines a resonant curve $\Gamma_{\text{\bf k}}=\{y\in\mathbb{R}^2:\langle\text{\bf k},\partial_yh(y)\rangle=0\}$. Normally hyperbolic cylinder is searched when $y$ varies along this curve. Recall that the normal form remains valid in the domain $\{\|y\|\le O(\epsilon^{\kappa})\}$ ($\frac 16<\kappa\le\frac 13$). The $\sqrt{\epsilon}$-neighborhood of the curve is covered by as many as $O(\epsilon^{-\kappa+\frac 12})$ small balls with radius $O(\sqrt{\epsilon})$. Given a ball centered at $y=y_j\in\Gamma_{\text{\bf k}}$, by rescaling variables $y-y_j=\sqrt{\epsilon}p$, $s=\sqrt{\epsilon}\tau$, one obtains an equivalent Hamiltonian equation $$\begin{aligned}
\frac{dx}{ds}&=\frac {\omega_j}{\sqrt{\epsilon}}+A_jp+\Big(\frac{\partial h(\sqrt{\epsilon}p)} {\epsilon\partial p}-\frac {\omega_j}{\sqrt{\epsilon}}-A_jp\Big) +\frac{\partial Z}{\partial p}
+\frac{\partial R}{\partial p},\\
\frac{dp}{ds}&=-\frac{\partial Z}{\partial x}-\frac{\partial R}{\partial x}\end{aligned}$$ which corresponds to the Hamiltonian $$G_{\epsilon}=\frac 1{\sqrt{\epsilon}}\langle\omega_j,p\rangle+\frac 12\langle A_jp,p\rangle+V_j(x)+ Z_{\epsilon}(x,\sqrt{\epsilon}p)+R_{\epsilon}(x,\sqrt{\epsilon}p,s/\sqrt{\epsilon})$$ where $\omega_j=\partial h(y_j)$, $A_j=\partial^2 h(y_j)$, $V_j(x)=Z(x,y_j)$ and $$Z_{\epsilon}=\frac{1}{\epsilon}h(y_j+\sqrt{\epsilon}p)-\frac 1{\sqrt{\epsilon}}\langle\omega_j,p\rangle-\langle A_jp,p\rangle+ Z(x,\sqrt{\epsilon}p+y_j)-Z(x,y_j).$$ According to Appendix A, one has $Z_{\epsilon}=O(\sqrt{\epsilon})$ and $\|R_{\epsilon}\|_{C^2}=O(\epsilon^{3\sigma-2\rho})$, where the $C^2$-norm is with respect to $(x,p)$ only. To guarantee the covering property shown in the appendix, one choose $\sigma=\frac 17$, so we have $3\sigma-2\rho=\frac 1{21}>0$. Obviously, one has
\[homogenizationpro\] Each orbit $(x(s),p(s))$ of the Hamiltonian flow $\Phi^s_{G_{\epsilon}}$ uniquely determines an orbit $(x(\tau),y(\tau))=(x(s/\sqrt{\epsilon}),y_j+\sqrt{\epsilon}p(s/\sqrt{\epsilon}))$ of $\Phi^{\tau}_{G}$. If $G_{\epsilon}(x(s),p(s))=E_{\epsilon}$ and $G(x(\tau),y(\tau))=E$, then $E=\epsilon E_{\epsilon}$.
The Hamiltonian $G_{\epsilon}$ is a small perturbation of the homogenized Hamiltonian $$\bar G=\frac 1{\sqrt{\epsilon}}\langle\omega_j,p\rangle+\frac 12\langle A_jp,p\rangle+V_j(x),$$ which determines the Lagrangian $$\bar L=\frac 12\Big\langle A^{-1}_j\Big(\dot x-\frac{\omega_j}{\sqrt{\epsilon}}\Big),\dot x-\frac{\omega_j}{\sqrt{\epsilon}}\Big\rangle-V_j(x).$$ Let us first consider the case when $y_j=0$, it follows that $\omega_j=0$. The variable $y$ is restricted in the domain $\|y\|\le K\sqrt{\epsilon}$, where the higher order term is bounded by $\|R_{\epsilon}\|_{C^2}=O(\epsilon^{5\sigma-\frac 16})$ (see (\[normaleq1\])). Correspondingly, let $A=A_j$ and $V=V_j$ for $y_j=0$: $$\bar G=\frac 12\langle Ap,p\rangle+V(x), \qquad \bar L=\frac 12\langle A^{-1}\dot x,\dot x\rangle-V(x).$$ For this Hamiltonian system, the maximal point of $V$ determines a stationary solution which corresponds to a minimal measure of $\bar L$, where the matrix $$\left (\begin{matrix}0 & A\\
-\partial^2_xV & 0
\end{matrix}\right )$$ has 4 real eigenvalue $\pm\lambda_1,\pm\lambda_2$. By translation of coordinates, it is generic that
([**H1**]{}): [*$V$ attains its maximum at $x=0$ only, the Hessian matrix of $V$ at $x=0$ is negative definite. All eigenvalues are different: $-\lambda_2<-\lambda_1<0<\lambda_1<\lambda_2$*]{}.
Such a hypothesis leads to certain hyperbolicity of minimal homoclinic orbits. Let us consider the case: for $c\in\partial\mathbb{F}_0$ the Aubry set $\mathcal{A}(c)=\cup_{t\in\mathbb{R}}\zeta(t)$, where $\zeta$: $\mathbb{R}\to M$ is a minimal homoclinic curve. By the assumption [**H1**]{}, the fixed point $z=(x,y)=0$ has its locally stable manifold $W^+$ as well as the locally unstable manifold $W^-$. They intersect each other transversally at the origin. As each homoclinic orbit entirely stays in the stable as well as in the unstable manifolds, along such orbit their intersection can not be transversal in the standard definition, but transversal module the curve: $$T_xW^-\oplus T_xW^+=T_xH^{-1}(E)$$ holds for $x$ is on minimal homoclinic curves. Without danger of confusion, we also call the intersection transversal.
If we denote by $\Lambda^+_i=(\Lambda_{xi},\Lambda_{yi})$ the eigenvector corresponding to the eigenvalue $\lambda_i$, where $\Lambda_{xi}$ and $\Lambda_{yi}$ are for the $x$- and $y$-coordinate respectively, then the eigenvector for $-\lambda_i$ will be $\Lambda^-_i=(\Lambda_{xi},-\Lambda_{yi})$. it is also a generic condition that
([**H2**]{}): [*with each $g\in H_1(\mathbb{T}^2,\mathbb{Z})$, there is at most one minimal orbit associated, the stable manifold intersects the unstable manifold transversally along each minimal homoclinic orbit. Each minimal homoclinic orbit approaches to the fixed point along the direction $\Lambda_1$: $\dot\gamma(t)/\|\dot\gamma(t)\| \to\Lambda_{x1}$ as $t\to\pm\infty$.*]{}
Recall the set $\partial^*\mathbb{F}_0$. If $c\in\partial^*\mathbb{F}_0\subset\partial\mathbb{F}_0$, the $c$-minimal measure consists of two or more ergodic components. Because of Theorem \[flatthm3\] and the countability of homology classes of all homoclinic curves, we have another generic condition
([**H3**]{}): [*For each $c\in\partial^*\mathbb{F}_0$, the Aubry set does not contain minimal curve homoclinic to the origin $($fixed point$)$.*]{}
Cylinder for truncated Hamiltonian: near double resonance
---------------------------------------------------------
Let us start with a Hamiltonian with two and half degrees of freedom: $$\label{cylindereq3}
G_{\epsilon}=\frac 12\langle Ap,p\rangle+V(x)+Z_{\epsilon}(x,p)+R_{\epsilon}(x,\sqrt{\epsilon}p,s/\sqrt{\epsilon})$$ where $(x,\sqrt{\epsilon}p,s/\sqrt{\epsilon})\in\mathbb{T}^2\times\mathbb{R}^2\times\mathbb{T}$, $Z_{\epsilon}=O(\sqrt{\epsilon})$, $\|R_{\epsilon}\|_{C^2}=O(\epsilon^{5\sigma-\frac 16})$ (see Theorem \[normalthm1\]) where the $C^2$-norm is with respect to $(x,p)$. Recall the homogenized Hamiltonian as well as the homogenized Lagrangian $$\bar G(x,p)=\frac 12\langle Ap,p\rangle+V(x),\qquad \bar L(x,\dot x)=\frac 12\langle A^{-1}\dot x,\dot x\rangle-V(x)$$ where $\dot x=\frac {dx}{ds}$.
By the assumption ([**H1**]{}), the fixed point $(x,\dot x)=0=\tilde{\mathcal{M}}(c)$ each $c\in\mathbb{F}_0$ which is a 2-dimensional flat. As classified before, for each $c\in\partial\mathbb{F}_0\backslash \partial^*\mathbb{F}_0$ the Aubry set consists of minimal homoclinic orbits plus the fixed point.
Given an irreducible class $g\in H_1(\mathbb{T}^2,\mathbb{Z})$, it is not necessary that some minimal homoclinic curve $\gamma$ exists such that $[\gamma]=g$. Let us consider the $c$-minimal measure for $c\in\mathscr{L}_{\beta}(\nu g)$ with $\nu>0$. As we shall see later, it is generic that the minimal measure is supported on at most two periodic orbits, denoted by $d\gamma_{\nu}$, both are hyperbolic, namely, it has the stable and unstable manifold. These periodic orbits constitute a two-dimensional cylinder. However, it appears not reasonable to assume the normal hyperbolicity for $\Phi_{\bar G}^s$ in usual sense as the speed along the orbit may undergo large variation, especially, when it is very close to some homoclinic orbit. Therefore, one can not see the separation of the spectrum of $D\Phi_{\bar G}^s$ in normal and in tangent direction. As the first step, let us study the case when $c\to\partial\mathbb{F}_0$.
There are two cases alternatively as $\nu$ decreases: $\mathscr{L}_{\beta}(\nu g)\to\partial \mathbb{F}_0$ as $\nu$ decrease to zero, or $\exists$ $\nu_0>0$ such that $\mathscr{L}_{\beta} (\nu_0g)\in\partial\mathbb{F}_0$.
In the first case, the cohomology class approaches to some edge $\mathbb{E}_i\subset\partial \mathbb{F}_0\backslash\partial^* \mathbb{F}_0$ or to some vertex where two adjacent edges $\mathbb{E}_i,\mathbb{E}_{i+1}\subset\partial\mathbb{F}_0\backslash \partial^*\mathbb{F}_0$ joint together. Under the hypothesis ([**H2**]{}), for class $c$ in the interior of the edge, the Aubry set $\mathcal{A}(c)$ contains exactly one minimal homoclinic curve and the point of the origin. Let $\gamma_j$ be the minimal homoclinic curve related to $\mathbb{E}_j$ for $j=i,i+1$, it determines the minimal homoclinic orbit $(x_i(s),p_i(s))\subset\bar G^{-1}(0)$. Denote by $g_j=[\gamma_j]$ the homology class. If $g=g_i$, $\gamma_{\nu}\to\gamma_i$ as $\nu$ decreases to zero. If there exist two positive integers $k_i,k_{i+1}$ such that $g=k_ig_i+k_{i+1}g_{i+1}$, the curve $x_{\nu}$ approaches to the set $\cup_{t\in\mathbb{R}}x_i(t)\cup x_{i+1}(t)$ (figure eight) as $\nu\to 0$, folding $k_i$ and $k_{i+1}$-times along $x_i$ and $x_{i+1}$ respectively.
The periodic curve $\gamma_{\nu}$ determines a periodic orbit $(\gamma_{\nu},y_{\nu})$ in the phase space. It stays in certain energy level set $H^{-1}(E)$. For $g=k_ig_i+k_{i+1}g_{i+1}$ and suitably small $E>0$, by the study in Section 3.2 (Eq.(\[regularenergyeq2\])), the period $T$ is related to the energy by the formula $$T=T(E,g)=\tau_{E,g}-\frac 1{\lambda_1}(k_i+k_{i+1})\ln E$$ where $\tau_{E,g}\to k_i\tau_{E,g_i}+k_{i+1}\tau_{E,g_{i+1}}$ as $E\to 0$, both $\tau_{E,g_i}$ and $\tau_{E,g_{i+1}}$ is bounded.
To study the dynamics around the minimal homoclinic orbits $z_{\ell}=(x_{\ell},p_{\ell})$ ($\ell=i,i+1$), we use a new canonical coordinates $(x,p)$ such that, restricted in a small neighborhood of $z=0$, one has the form $$\bar G=\frac 12(p_1^2-\lambda_1^2x_1^2)+\frac 12(p_2^2-\lambda_2^2x_2^2)+P_3(x)$$ where $P_3(x)=O(\|x\|^3)$. Without losing generality, we assume $x_{\ell,1}(s)\downarrow 0$ as $s\to -\infty$, $x_{\ell,1}(s)\uparrow 0$ as $s\to \infty$ and $\dot x_{\ell}(s)/ \|\dot x_{\ell}(s)\|\to(1,0)$ as $s\to\pm\infty$. Here the notation is taken as granted: $x_{\ell}=(x_{\ell,1},x_{\ell,2})$. We choose 2-dimensional disk lying in $\bar G^{-1}(E)$ $$\Sigma^{\mp}_{E,\delta}=\{(x,p)\in\mathbb{R}^4:\|(x,p)\|\le d,\bar G(x,p)=E, x_1=\pm\delta\}.$$ Because of the special form of $\bar G$, one has $$\Sigma^{\mp}_{0,\delta}=\{x_1=\pm\delta, p_1^2+p_2^2-\lambda_2^2x_2^2 =\lambda_1^2\delta^2 -2P_3(\pm\delta,x_2),\|(x,p)\|\le d\}.$$ Let $W^-$ ($W^+$) denote the unstable (stable) manifold of the fixed point which entirely stays in the energy level set $\bar G^{-1}(0)$. If $P_3=0$, the tangent vector of $W^-\cap\Sigma^-_{0,\delta}$ has the form $(0,\pm 1,0,\pm\lambda_2)$. So, the tangent vector of $W^-\cap\Sigma^-_{0,\delta}$ takes the form $$\label{tangentvector}
v_{\delta}^-=(v_{x_1},v_{x_2},v_{p_1},v_{p_2})=(0,\pm 1,p_{1,\delta},\pm\lambda_2+p_{2,\delta})\in T_{z^-_{\delta}}(W^-\cap\Sigma^-_{0,\delta})$$ where both $p_{1,\delta}$ and $p_{2,\delta}$ are small.
Denote by $T^{\pm}_{\delta,\ell}$ the time when the homoclinic orbit $z_\ell(s)$ passes through $\Sigma^{\pm}_{0,\delta}$. As $\partial_{y_1}\bar G>0$ holds at the point $z_{\ell}\cap\{x_1=\pm\delta\}$, both homoclinic orbits $z_i(s)$ and $z_{i+1}(s)$ approach in the same direction to the fixed point, the section $\Sigma^+_{0,\delta}$ as well as $\Sigma^-_{0,\delta}$ intersects these two homoclinic orbits transversally. Let $z^{\pm}_{\delta,\ell}$ denote the intersection point of $z_\ell(s)$ with $\Sigma^{\pm}_{0,\delta}$. In a small neighborhood of that point $B_{\varepsilon}(z^-_{\delta,\ell})$, one obtains a map $\Psi_{0,\delta}$: $\Sigma^-_{0,\delta}\cap B_{\varepsilon}(z^-_{\delta,\ell})\to \Sigma^+_{0,\delta}$ in following way, starting from a point $z$ in this neighborhood, there is a unique orbit which moves along $z_{\ell}(s)$ and comes to a point $\Psi_{0,\delta}(z)\in \Sigma^+_{0,\delta}$ after a time approximately equal to $T_{\delta,\ell}^+-T_{\delta,\ell}^-$.
Let us fix small $D>0$. There exists $C_0>1$ (depending on $D$) such that $$C_0^{-1}\le \|D\Psi_{0,D}(z^-_{D,\ell})|_{T(W^-\cap\Sigma^-_{0,D})}\|, \|D\Psi_{0,D}^{-1}(z^+_{D,\ell})|_{T(W^+\cap\Sigma^+_{0,D})}\|\le C_0$$ holds for both $\ell=i$ and $\ell=i+1$. Clearly, one has $C_0\to\infty$ as $D\to 0$.
As the homoclinic curves approach to the origin in the direction of $(1,0)$ in $x$-space, for small $\delta\ll D$, there exists a constant $\mu_1>0$ such that $\mu_1\downarrow 0$ as $D\to 0$ and $$\frac {1}{\lambda_1+\mu_1}\ln\Big(\frac{D}{\delta}\Big)\le T^{-}_{D,\ell}-T^{-}_{\delta,\ell}, T^+_{\delta, \ell}-T^+_{D,\ell}\le \frac {1}{\lambda_1-\mu_1}\ln\Big(\frac{D}{\delta}\Big).$$ The Hamiltonian flow $\Phi^t_{\bar G}$ defines a map $\Psi^-_{0,\delta,D}$: $\Sigma^-_{0,\delta}\to\Sigma^-_{0,D}$ and a map $\Psi^+_{0,\delta,D}$: $\Sigma^+_{0,D}\to\Sigma^+_{0,\delta}$: emanating from a point in $\Sigma^-_{0,\delta}$ ($\Sigma^+_{0,D}$) there exists a unique orbit which arrives $\Sigma^-_{0,D}$ ($\Sigma^+_{0,\delta}$) after a time bounded by the last formula.
Restricted in the ball $B_D$, let us consider the variational equation of the flow $\Psi_{\bar G}^s$ along the homoclinic orbit $z_j(s)$. It follows from the normal form of the homogenized Hamiltonian $\bar G$ that the tangent vector $(\Delta x,\Delta p)=(\Delta x_1,\Delta x_2,\Delta p_1,\Delta p_2)$ satisfies the variational equation $$\label{cylindereq4}
\Delta\dot x_i=\Delta p_i, \qquad \Delta\dot p_i=\lambda_i^2\Delta x_i-\Psi_{1i}(x_{\ell}(s))\Delta x_1-\Psi_{2i}(x_{\ell}(s)) \Delta x_2, \ \ \ i=1,2$$ where $\Psi_{ij}=\partial_{x_i}\partial_{x_j}P_3$. Clearly, $|\Psi_{ij}(x_{\ell}(s))|\le C_1\|x_{\ell}(s)\|$ with $C_1>0$ if $\|x_{\ell}(s)\|$ is small. Since the homoclinic orbit approaches to the fixed point in the direction of $(\dot x,\dot p)=(1,0,\lambda_1^2,0)$, one has $$De^{-(\lambda_1+\mu_1)(s-T^+_{D,\ell})}\le\|x(s)|_{[T^+_{D,\ell},\infty)}\|\le De^{-(\lambda_1-\mu_1)(s-T^+_{D,\ell})}.$$ For the initial value $\Delta z(T^+_{D,\ell})=(\Delta x(T^+_{D,\ell}),\Delta p(T^+_{D,\ell}))$ satisfying the condition $$|\langle\Delta z(T^+_{D,\ell}),v^-_{\delta}\rangle|\ge 2/3\|\Delta z(T^+_{D,\ell})\|\|v^-_{\delta}\|$$ $(v^-_{\delta}=(0,\pm 1,p_{1,\delta},\pm\lambda_2+p_{2,\delta}))$ one obtains from the hyperbolicity that $$C_2^{-1}\|\Delta z(T^+_{D,\ell})\|e^{(\lambda_2-\mu_1)(T^+_{\delta,\ell}-T^+_{D,\ell})}\le\|\Delta z(T^+_{\delta,\ell})\|\le C_2\|\Delta z(T^+_{D,\ell})\|e^{(\lambda_2+\mu_1)(T^+_{\delta,\ell}-T^+_{D,\ell})}$$ holds for some constant $C_2>1$ depending on $\lambda_i$ as well as on $P$. Therefore, for each vector $v\in T_{z_D^+}\Sigma^+_{0,D}$ which is nearly parallel to $T_{z_D^+}(W^-\cap\Sigma^+_{0,D})$: $|\langle v,v'\rangle|\ge\frac 23\|v\|\|v'\|$ holds for $v'\in T_{z_D^+}(W^-\cap\Sigma^+_{0,D})$ we obtain from the last two formulae that $$C_2^{-1}\Big(\frac D{\delta}\Big)^{\frac{\lambda_2}{\lambda_1}-\mu_2}\le
\lim_{\|v\|\to 0}\frac{\|D\Psi^+_{0,\delta,D}(z^+_{D,\ell})v\|}{\|v\|}\le C_2\Big(\frac D{\delta}\Big)^{\frac{\lambda_2}{\lambda_1}+\mu_2}.$$ Similarly, one has $$C_3^{-1}\Big(\frac D{\delta}\Big)^{\frac{\lambda_2}{\lambda_1}-\mu_2}\le \|D\Psi^-_{0,\delta,D}(z^-_{\delta,\ell})|_{T_{z^-_{\delta}}(W^-\cap\Sigma^-_{0,\delta})}\|\le C_3\Big(\frac D{\delta}\Big)^{\frac{\lambda_2}{\lambda_1}+\mu_2},$$ where $C_3>1$ also depends on $\lambda_i$ as well as on $P$, $\mu_2>0$ and $\mu_2\to 0$ as $D\to 0$.
By the construction, the 2-dimensional disk $\Sigma^{-}_{0,\delta}$ intersects the unstable manifold $W^-$ along a curve. Let $\Gamma^{-}_{\delta,\ell}\subset W^{-}\cap\Sigma^{-}_{0,\delta}$ be a very short segment of the curve, passing through the point $z^{-}_{\delta,\ell}$. Pick up a point $z^*_\ell$ on the homoclinic orbit $z_\ell$ far away from the fixed point and take a 2-dimensional disk $\Sigma^*_\ell\subset\bar G^{-1}(0)$ containing the point $z^*_\ell$ and transversal to the flow $\Phi^s_{\bar G}$ in the sense that $T_{z^*_\ell}\bar G^{-1}(0)=\text{\rm span}(T_{z^*_\ell} \Sigma_\ell,J\nabla\bar G(z^*_\ell))$. The Hamiltonian flow $\Phi^s_{\bar G}$ sends each point of $\Gamma^{-}_{\delta,\ell}$ to this disk provided it is close to $z^-_\ell$. In this way, one obtains a map $\Psi^{-,*}_{\delta,\ell}$: $\Sigma^-_{0,\delta}\to\Sigma^*_\ell$. Let $\Gamma^{-,*}_{\delta,\ell}= \Psi^{-,*} _{\delta,\ell} \Gamma^{-}_{\delta,\ell}$. According to the assumption ([**H2**]{}), one has $T_{z^*_{\ell}}\bar G^{-1}(0)=\text{\rm span}(T_{z^*_{\ell}}W^+,T_{z^*_{\delta,\ell}}W^-)$. Thus, one also has $T_{z^*_{\ell}}\bar G^{-1}(0)=\text{\rm span}(T_{z^*_{\ell}}W^+,T_{z^*_{\ell}} \Gamma^{-,*}_{\ell})$. It follows from the $\lambda$-lemma that $\Psi_{0,\delta}(\Gamma^-_{\delta,\ell})$ keeps $C^1$-close to $W^-\cap\Sigma^+_{0,\delta}$ at the point $z^+_{\delta,\ell}$ and $\Psi^{-1}_{0,\delta}(\Gamma^+_{\delta,\ell})$ keeps $C^1$-close to $W^+\cap\Sigma^-_{0,\delta}$ at the point $z^-_{\delta,\ell}$ provided $\delta>0$ is sufficiently small. As $\Psi_{0,\delta}=\Psi^-_{0,\delta,D}\circ\Psi_{0,D}\circ\Psi^+_{0,\delta,D}$, one obtains $$C_4^{-1}\left(\frac D{\delta}\right)^{2(\frac {\lambda_2}{\lambda_1}-\mu_2)}\le \|D\Psi_{0,\delta}(z^-_{\delta})|_{T_{z^-_{\delta}}(W^-\cap\Sigma^-_{0,\delta})}\|
\le C_4\left(\frac D{\delta}\right)^{2(\frac {\lambda_2}{\lambda_1}+\mu_2)},$$ and $$C_4^{-1}\left(\frac D{\delta}\right)^{2(\frac {\lambda_2}{\lambda_1}-\mu_2)}\le \|D\Psi_{0,\delta}^{-1} (z^+_{\delta}) |_{T_{z^+_{\delta}}(W^+\cap\Sigma^+_{0,\delta})}\|
\le C_4\left(\frac D{\delta}\right) ^{2(\frac {\lambda_2}{\lambda_1}+\mu_2)},$$ where $C_4=C_0C_2C_3>1$. See the figure below.
![[]{data-label="fig7"}](Ardiff7.eps){width="6.5cm" height="6.7cm"}
Recall the definition, $\Sigma^{\pm}_{E,\delta}$ is a two-dimensional disk lying in the energy level set $\bar G^{-1}(E)$. For $E>0$ sufficiently small, $\Sigma^{\pm}_{E,\delta}$ is $C^{r-1}$-close to $\Sigma^{\pm}_{0,\delta}$ respectively. Let $z_{E}(s)=(x_{E}(s),p_{E}(s))$ be the minimal periodic orbit staying in the energy level set $\bar G^{-1}(E)$, it approaches to the homoclinic orbit as $E$ decreases to zero. Thus, for sufficiently small $E>0$, it passes through the section $\Sigma^{-}_{E,\delta}$ as well as $\Sigma^{+}_{E,\delta}$ $k_1+k_2$ times for one period. We number these points as $z^{\pm}_{E,k}$ ($k=1,2,\cdots k_1+k_2$) by the role that emanating from a point $z^-_{E,k}$, the orbit reaches to the point $z^+_{E,k+1}$ after time $\Delta t^-_{E,k}$, then to the point $z^-_{E,k+1}$ and so on. Note that $\Delta t^-_{E,k}$ remains bounded uniformly for any $E>0$. Restricted on small neighborhoods of these points, denoted by $B_d(\bar z^{\pm}_{E,k})$, the flow $\Phi_{\bar G}^t$ defines a local diffeomorphism $\Psi_{E,\delta}$: $\Sigma^{-}_{E,\delta}\supset B_d(\bar z^{-}_{E,k})\to \Sigma^{+}_{E,\delta}$. Because of the smooth dependence of ODE solutions on initial data, a small $\varepsilon>0$ exists such that, for the vector $v^{\pm}$ $\varepsilon$-parallel to $T_{z^{\pm}_{\delta}}(W^{\pm}\cap\Sigma^{\pm}_{0,\delta})$ in the sense that $|\langle v^{\pm},v^{\pm}_0\rangle|\ge (1-\varepsilon) \|v^{\pm}\|\|v^{\pm}_0\|$ holds for some $v^{\pm}_0\in T_{z^{\pm}_{\delta}}(W^{\pm}\cap\Sigma^{\pm}_{0,\delta})$, we obtain from the hyperbolicity of $\Psi_{0,\delta}$ (see the formulae above Figure \[fig7\]) that $$C_5^{-1}\left(\frac D{\delta}\right)^{2(\frac {\lambda_2}{\lambda_1}-\mu_3)}\le \frac {\|D\Psi_{E,\delta}(z^-_{E,k})v^-\|}{\|v^-\|}
\le C_5\left(\frac D{\delta}\right)^{2(\frac {\lambda_2}{\lambda_1}+\mu_3)},$$ and $$C_5^{-1}\left(\frac D{\delta}\right)^{2(\frac {\lambda_2}{\lambda_1}-\mu_3)}\le \frac {\|D\Psi_{E,\delta}^{-1}(z^+_{E,k})v^+\|}{\|v^+\|}
\le C_5\left(\frac D{\delta}\right) ^{2(\frac {\lambda_2}{\lambda_1}+\mu_3)}$$ where $C_5\ge C_4>1$, $0<\mu_3\to 0$ as $D\to 0$. If the vector $v^-$ is chosen $\varepsilon$-parallel to $T_{z^{-}_{\delta}}(W^{-}\cap\Sigma^{-}_{0,\delta})$ then the vector $D\Psi_{E,\delta}(z^-_{E,k})v^-$ is $\varepsilon$-parallel to $T_{z^{+}_{\delta}}(W^{+}\cap\Sigma^{+}_{0,\delta})$.
For $E>0$, the Hamiltonian flow $\Phi_{\bar G}^t$ defines local diffeomorphism $\Psi^+_{E,\delta,\delta}$: $\Sigma^{+}_{E,\delta}\supset B_d(\bar z^{+}_{E,k})\to\Sigma^{-}_{E,\delta}$. To make $\Psi^+_{E,\delta,\delta}(B_d(\bar z^{+}_{E,k}))\subset\Sigma^{-}_{E,\delta}$ one has $d\to 0$ as $E\to 0$. According to the study in Section 3.2 (cf. formula (\[regularenergyeq2\])), starting from $\Sigma^{+}_{E,\delta}$, the periodic orbit comes to $\Sigma^{-}_{E,\delta}$ after a time approximately equal to $$T=\frac 1{\lambda_1}\Big|\ln\Big(\frac{\delta^2}{E}\Big)\Big|+\tau_{\delta}$$ in which $\tau_{\delta}$ is uniformly bounded as $\delta\to 0$. Given a vector $v$, we use $v_i$ denote the $(x_i,p_i)$-component. For a vector $v^+$ $\varepsilon$-parallel to $T_{z^+_{0,\delta}}(W^-\cap\Sigma^+_{0,\delta})$, there is $C>0$ such that $\|v^+_2\|\ge C\|v^+_1\|$. From Eq.(\[cylindereq3\]) one obtains $$\begin{aligned}
\label{cylindereq5}
\|v^+_2\|e^{(\lambda_2-\mu)T}\le&\|D\Psi^+_{E,\delta,\delta}(z^+_{E,k})v^+_2\|\le\|v^+_2\|e^{(\lambda_2+ \mu)T},\\
\|v^+_1\|e^{(\lambda_1-\mu)T}\le&\|D\Psi^+_{E,\delta,\delta}(z^+_{E,k})v^+_1\|\le\|v^+_1\|e^{(\lambda_1+ \mu)T}\notag\end{aligned}$$ where $0<\mu\to 0$ as $\delta\to0$. It follows that the vector $D\Psi^+_{E,\delta,\delta}(z^+_{E,k})v^+$ is $\varepsilon$-parallel to $T_{z^-_{0,\delta}}(W^-\cap\Sigma^-_{\delta})$ and $$C_6^{-1}\Big(\frac {\delta^2}E\Big)^{\frac{\lambda_2}{\lambda_1}-\mu_4}
\le\frac{\|D\Psi^+_{E,\delta,\delta}(z^+_{E,k})v^+\|}{\|v^+\|}
\le C_6\Big(\frac {\delta^2}E\Big)^{\frac{\lambda_2}{\lambda_1}+\mu_4}$$ where $C_6>1$ and $\mu_4\downarrow 0$ as $\delta\downarrow 0$. Similarly, for a vector $v^-$ $\varepsilon$-parallel to $T_{z^-_{0,\delta}}(W^+\cap\Sigma^-_{0,\delta})$, one sees that the vector $D{\Psi^+_{E,\delta,\delta}(z^-_{E,j})}^{-1}v^-$ is $\varepsilon$-parallel to $T_{z^-_{0,\delta}}(W^-\cap\Sigma^-_{0,\delta})$ and $$C_6^{-1}\Big(\frac {\delta^2}E\Big)^{\frac{\lambda_2}{\lambda_1}-\mu_4}
\le\frac{\|D{\Psi^+_{E,\delta,\delta}}^{-1}(z^-_{E,k})v^-\|}{\|v^-\|}
\le C_6\Big(\frac {\delta^2}E\Big)^{\frac{\lambda_2}{\lambda_1}+\mu_4}.$$
The composition of the two maps constitutes a Poinćare map $\Phi_{E,\delta}=\Psi^+_{E,\delta,\delta}\circ\Psi_{E,\delta}$, it maps a small neighborhood of the point $z^-_{E,k}$ in $\Sigma^{-}_{E,\delta}$ to a small neighborhood of the point $z^-_{E,k+1}$ in $\Sigma^{-}_{E,\delta}$. For a vector $v^-$ $\varepsilon$-parallel to $T_{z^-_{0,\delta}}(W^-\cap\Sigma^-_{0,\delta})$ the vector $D\Phi_{E,\delta}(z^-_{E,k})v^-$ is still $\varepsilon$-parallel to $T_{z^-_{0,\delta}}(W^-\cap\Sigma^-_{0,\delta})$ $$\label{cylindereq6}
\Lambda^{-1}\left(\frac {D^2}{E}\right)^{\frac{\lambda_2}{\lambda_1}-\mu_5}
\le\frac {\|D\Phi_{E,\delta}(z^-_{E,k})v^-\|}{\|v^-\|}
\le \Lambda\left(\frac{D^2}{E}\right)^{\frac {\lambda_2}{\lambda_1}+\mu_5},$$ and for a vector $v^+$ $\varepsilon$-parallel to $T_{z^-_{0,\delta}}(W^+\cap\Sigma^-_{0,\delta})$ the vector $D\Phi_{E,\delta}^{-1}(z^-_{E,k})v^+$ is still $\varepsilon$-parallel to $T_{z^-_{0,\delta}}(W^+\cap\Sigma^-_{0,\delta})$ $$\label{cylindereq7}
\Lambda^{-1}\left(\frac {D^2}{E}\right)^{\frac{\lambda_2}{\lambda_1}-\mu_5}
\le\frac{\|D\Phi_{E,\delta}^{-1}(z^-_{E,k})v^+\|}{\|v^+\|}
\le \Lambda\left(\frac {D^2}{E}\right)^{\frac{\lambda_2}{\lambda_1}+\mu_5}$$ holds for each $k$, where $\Lambda\ge C_5C_6>1$, $0<\mu_5\to 0$ as $D\to 0$. Therefore, each point $z^-_{E,k}$ is a hyperbolic fixed point for the map $\Phi^{k_i+k_{i+1}}_{E,\delta}$, $\{z^-_{E,k}:k=1,\cdots,k_i+k_{i+1}\}$ is a hyperbolic orbit of $\Phi_{E,\delta}$. By Lemma \[flatlem4\], these points are uniquely ordered, $k_i+k_{i+1}$ is the minimal period. As these points approach to the fixed point as $E\downarrow0$, the hyperbolicity guarantees the uniqueness. It also guarantees the smooth continuation of periodic orbits. Therefore, we have
\[cylinderlem1\] Assume the conditions and and let $g\in H_1(\mathbb{T}^2,\mathbb{Z})$ be a class. If $\mathscr{L}_{\beta}(\nu g)\to\partial \mathbb{F}_0$ as $\nu\downarrow 0$, then there exists $E'>0$ such that for each $c\in\mathscr{L}_{\beta}(\nu g)$ with $\alpha(c)=E\in (0,E']$ the Mather set $\tilde{\mathcal{A}}(c)$ consists of exactly one periodic orbit.
Let $\Sigma_E\subset\bar G^{-1}(E)$ be a two-dimensional disk transversally intersecting the orbit at $x_1=\delta$ such that $T_z\bar G^{-1}(E)=\text{\rm span}(T_z\Sigma_E,J\nabla\bar G(z))$ for $z\in\Sigma_E$ and let $\Phi_{E}$: $\Sigma_E\to \Sigma_E$ be the return map naturally determined by the flow $\Phi_{\bar G}^t$, there exists some $\lambda>1,C>0$ independent of $E\le E'$ such that $$\|D\Phi_{E}(z_{E,0})v^-\|\ge C E^{-\lambda}\|v^-\|,\qquad \forall\ v^-\in T_{z_{E,0}}W^-_E;$$ $$\|D\Phi_{E}(z_{E,0})v^+\|\le C^{-1}E^{\lambda}\|v^-\|,\qquad \forall\ v^+\in T_{z_{E,0}}W^+_E,$$ where $z_{E,0}$ is the point where the periodic orbit intersects $\Sigma_{E}$, $W^{\pm}_E$ denotes the stable $($unstable$)$ manifold of the periodic orbit. Therefore, the periodic orbits for $E\in (0,E']$ constitute a smooth cylinder.
In the second case, some $\nu_0>0$ exists such that $\mathscr{L}_{\beta}(\nu_0g)\in \partial^*\mathbb{F}_0$. It is typical that $\mathscr{L}_{\beta}(\nu_0g)$ is an edge of $\mathbb{E}_i\subset\partial^*\mathbb{F}_0$, where each Mather set consists of a hyperbolic periodic orbit $z_0(s)\subset\bar G^{-1}(0)$ and the fixed point. The uniqueness of minimal periodic orbit for $\nu$ close to $\nu_0$ follows from the implicit function theorem.
Given $g\in H_1(\mathbb{T}^2,\mathbb{Z})$, the set $\cup_{\nu>0}\mathscr{L}_{\beta}(\nu g)$ is a channel in $H^1(\mathbb{T}^2,\mathbb{Z})$. There is a path $\Gamma_g\subset\cup_{\nu>0} \mathscr{L}_{\beta}(\nu g)$ reaching to the boundary $\partial\mathbb{F}_0$, along which the $\alpha$-function monotonely decreases to the minimum as the cohomology class approaches to $\partial\mathbb{F}_0$. With the argument as above and Theorem \[AppenHyperTh1\] shown in the appendix, we find the following condition also generic
([**H4**]{}): [*Given a class $g\in H_1(\mathbb{T}^2,\mathbb{Z})$ and an energy $E^*>0$, there are at most finitely many $E_i\in (0,E^*]$ such that for $c\in\Gamma_g$ with $\alpha(c)=E_i$ the Mather set consists of two periodic orbits, for all other $c\in\Gamma_g$ with $\alpha(c)\ne E_i$, the Mather set consists of exactly one periodic orbit. All these periodic orbits are hyperbolic.*]{}
We call these $\{E_i\}$ bifurcation points. Let $E_1>0$ be the smallest one. Each energy $E<E_1$ uniquely determines a hyperbolic periodic orbit $\{x_E(s),p_E(s):s\in\mathbb{R}\}$. Thus, we introduce a notation $$\Pi_{E_0,E_1,g}=\{(x_{E}(s),p_{E}(s)):[x_{E}]=g,E\in [E_0,E_1],s\in\mathbb{R}\},$$ with small $E_0>0$. It is a two-dimensional cylinder composed of a family of periodic orbits, may approach to a curve of figure-of-eight as $E_0$ decreases to zero. Obviously, the cylinder is invariant for the Hamiltonian flow $\Phi_{\bar G}^t$ of $\bar G$. Let $T(E)$ denote the period of the periodic orbit in $\bar G^{-1}(E)$, one has $$\int_{\Pi_{E_0,E_1,g}}\omega=\int_{E_0}^{E_1}\int_{0}^{T(E)}dE\wedge dt>0.$$
The cylinder may be slant and crumpled, not standard. To see how the symplectic area is related to the usual area of the cylinder, let us study the dependence of the fixed point $(x_2(E),p_2(E))$ of the Poincar' e return map $\Phi_{E,\delta}$ on $E$. By definition, the fixed point is a solution of the equation $$\label{cylindereq8}
\Phi_{E,\delta}(x_2(E),p_2(E))-(x_2(E),p_2(E))=0.$$ Emanating from a point $(\delta,p_1,x_2,p_2)\in \bar G^{-1}(E)$ the orbit reach a point $z\in\{x_1=-\delta\}$ after a time $\tau(E,\delta)$ which remains bounded as $E\downarrow 0$. Let $z'\in\{x_1=-\delta\}$ be the point corresponding to $(\delta,p'_1,x_2,p_2)\in \bar G^{-1}(E')$, obtained in the same way. The difference of the $(x_2,y_2)$-coordinate of these two points is bounded by $d_0|p_1-p'_1|$ where $d_0$ depends on $\delta$. Let $(\Delta x, \Delta y)$ be the solution of the variational equation (\[cylindereq3\]) along the periodic solution $(x_{E}(s),p_{E}(s))$ passing through a neighborhood of the origin, let $s_0<s_1$ be the time such that the first coordinate $x_{E,1}(s_0)=-\delta$ and $x_{E,1}(s_1)=\delta$, the quantity $s_1-s_0$ is bounded by (\[regularenergyeq2\]). In virtue of the formula (\[cylindereq5\]), it yields $$\|(\Delta x, \Delta y)(s_1)\|\le C_7E^{-\frac{\lambda_2}{\lambda_1}-\mu_6}\|(\Delta x, \Delta y)(s_0)\|$$ where $0<\mu_6\to 0$ as $\delta\to 0$. Therefore, we find that $$\Big\|\frac{\partial\Phi_E}{\partial p_1}\Big\|\le C_8E^{-\frac{\lambda_2}{\lambda_1}-\mu_6}.$$ As the quantity $\|\frac{\partial\Phi_E}{\partial (x_2,p_2)}\|$ is bounded by (\[cylindereq6\]), the quantity for the inverse of $\Phi_E$ is bounded by (\[cylindereq7\]), we obtain from the equation (\[cylindereq8\]) that $$\label{cylindereq9}
\Big\|\frac{\partial x_2}{\partial p_1}\Big\|,\Big\|\frac{\partial p_2}{\partial p_1}\Big\|\le C_9E^{-2\mu_6}.$$ It yields a relation between the symplectic area $\omega$ and the usual area $S$ of the cylinder $$\label{cylindereq10}
|\omega|\ge C_{10}E^{2\mu_6}|S|.$$
\[cylinderthm1\] We assume the conditions . For each $E_0\in (0,E_1]$, the cylinder $\Pi_{E_0,E_1,g}$ is normally hyperbolic for the map $\Phi_{\bar G}^{\Delta t_{E}}$, where $\Delta t_{E_0}=2\lambda_1^{-1}|\ln E_0|$.
For any $0<E_0<E_1$, the cylinder $\Pi_{E_0,E_1,g}$ is a 2-dimensional symplectic sub-manifold, invariant for the Hamiltonian flow $\Phi^s_{\bar G}$. However, it is not clear whether this cylinder is normally hyperbolic for the time-1-map $\Phi_{\bar G}=\Phi^s_{\bar G}|_{s=1}$, as it is possible that $$\begin{aligned}
m(D\Phi_{\bar G}|_{T\Pi_{E_0,E_1,g}})=&\inf\{|D\Phi_{\bar G}v|:v\in T\Pi_{E_0,E_1,g}, |v|=1\}<1,\\
&\|D\Phi_{\bar G}|_{T\Pi_{E_0,E_1,g}}\|>1,\end{aligned}$$ and we do not have the estimate on the norm of $D\Phi_{\bar G}$ acting on the normal bundle.
As the first step of the proof, let us search for the normal hyperbolicity of $\Phi^s_{\bar G}$ with large $s$ for the cylinder $\Pi_{E_0,E',g}$, where $E'$ denotes the largest value such that the formulae (\[cylindereq6\]) and (\[cylindereq7\]) hold for each $E\le E'$. From these formulae, one sees that the smaller the energy reaches, the stronger hyperbolicity the map $\Phi_{E,\delta}$ obtains. The strong hyperbolicity is obtained by passing through small neighborhood of the fixed point. However, on the other hand, the smaller the energy decreases, the longer the return time becomes.
Let $\Delta t_{E,k}$ denote the time interval such that, starting from $z^-_{E,k}$, the periodic orbit comes to $z^-_{E,k+1}$ after time $\Delta t_{E,k}$. In virtue of the study in Section 3, Eq. (\[regularenergyeq2\]), $$\Delta t_{E,k}\approx\tau_{E,g_{\ell}}-\lambda_1^{-1}\ln E, \qquad \ell=i, \ \text{\rm or}\ i+1,$$ where $\tau_{E,g_{\ell}}$ is uniformly bounded and we take $\tau_{E,g_{\ell}}$ if the segment of the periodic orbit keeps close to the homoclinic orbit $\gamma_{\ell}$. Note $\Delta t_{E}=\frac 2{\lambda_1}|\ln E|$ is much larger than $\max_k\Delta t_{E,k}$. Thus, starting from any point $z$ on the minimal periodic orbit $z_{E}(s)$, $\Phi_{\bar G}^s(z)$ passes through the neighborhood of the fixed point after time $\Delta t_{E}$. It implies that the map $\Phi_{\bar G}^s|_{s=T_{E}}$ obtains strong hyperbolicity on normal bundle.
To measure how the map $D\Phi^s_{\bar G}$ acts on the tangent bundle, let us study how the map $\Phi^s_{\bar G}$ elongates or shortens small arc of the periodic orbit. As the orbit passes through the neighborhood of the origin $O_{\delta}(0)$ in a time approximately equal to $-\lambda_1^{-1}\ln \delta^{-2}E$ the variation of the length of short arc is between $O(E_0^{1+\mu_7})$ and $O(E_0^{-1-\mu_7})$, where $\mu_7>0$ is small. Because of the relation between the symplectic area $\omega$ and the usual area $S$ of the cylinder, given by the formula (\[cylindereq10\]), the variation of $\|D\Phi^s_{\bar G}\|$, restricted on the tangent bundle of the cylinder, is between $O(E_0^{1+\mu_7+2\mu_6})$ and $O(E_0^{-1-\mu_7-2\mu_6})$. Because of periodicity, it is independent of $s$.
Thus, the normally hyperbolic property becomes clear: the tangent bundle of $M$ over $\Pi_{E_0,E_1,g}$ admits $D\Phi_{\bar G}^s|_{s=T_{E}}$-invariant splitting $$T_zM=T_zN^+\oplus T_z\Pi_{E_0,E_1,g}\oplus T_zN^-$$ and some $\Lambda_1\ge 1$, $\Lambda_2\ge 1$ and small $\nu>0$ exist such that $$\label{cylindereq11}
\Lambda_1^{-1}E_0^{1+\nu}<\frac {\|D\Phi^s_{\bar G}(z)v\|}{\|v\|}<\Lambda_1E_0^{-1-\nu},\qquad \forall\ v\in T_z\Pi_{E_0,E_1,g},$$ $$\frac{\|D\Phi_{\bar G}^{s}(z)v\|}{\|v\|}\le \Lambda_2 E_0^{\frac{\lambda_2}{\lambda_1}-\nu},\qquad \forall\ v\in T_zN^+,$$ $$\frac{\|D\Phi_{\bar G}^{s}(z)v\|}{\|v\|}\ge\Lambda_2^{-1} E_0^{-\frac{\lambda_2}{\lambda_1}+\nu}, \qquad \forall\ v\in T_zN^-,$$ hold for $s\ge\Delta t_{E}$ (cf. (\[cylindereq6\]) and (\[cylindereq7\])). Note that $\lambda_2/\lambda_1-\nu>1+\nu$ provided $\nu>0$ is suitably small. The formula (\[cylindereq11\]) satisfies the definition of normal hyperbolicity.
For each $E\in[E' ,E_1]$, let $z_{E}(s)$ be the minimal periodic orbit, $\Sigma_{E}\subset\bar G^{-1}(E)$ be a 2-dimensional disk intersecting $z_{E}(s)$ transversally at the point $z_{E,0}$, $\Phi_{E}$: $\Sigma_{E}\to\Sigma_{E}$ be the Poincaré return map. By the generic property , $z_{E,0}$ is the hyperbolic fixed point of $\Sigma_{E}$ and $\Lambda_{2}>1$ exists such that $$\|D\Phi_{E}(z_{E,0})v^-\|\ge\Lambda_{2}\|v^-\|, \qquad \forall\ v^-\in T_{z_{E,0}}(W^-_{E}\cap\Sigma_{E}),$$ $$\|D\Phi_{E}(z_{E,0})v^+\|\le\Lambda_{2}^{-1}\|v^+\|, \qquad \forall\ v^+\in T_{z_{E,0}}(W^+_{E}\cap\Sigma_{E}).$$ As the cylinder is foliated into periodic orbits and $\Phi_{\bar G}$ preserves the symplectic form, some $\Lambda_{1}\ge 1$ exists such that $$\Lambda_{1}^{-1}\|v\|\le\|D\Phi_{\bar G}^{s}(z)v\|\le\Lambda_{1}\|v\|, \qquad \forall\ v^-\in T_{z}\Pi_{E',E_1,g}$$ holds for any $s>0$. Choosing $m\in\mathbb{N}$ such that $\Lambda_{2}^m\ge 2\Lambda_{1}$, one obtains the normal hyperbolicity for $\Phi_{\bar G}^{s}(z)$ with $s\ge mT(E')$, where $T(E')$ is the period of the periodic solution in $\bar G^{-1}(E')$, $z\in\Pi_{E',E_1,g}\cap \bar G^{-1}(E)$ with $E\in [E',E_1]$. By choosing suitably small $E_0>0$, we have $\Delta t_{E_0}\ge mT(E')$.
Persistence of cylinder: near double resonance
----------------------------------------------
To apply the theorem of normally hyperbolic manifold [@HPS] to the Hamiltonian $G_{\epsilon}$ of (\[cylindereq3\]), we note that the Hamiltonian $\bar G_{\epsilon}=\bar G+Z_{\epsilon}$ is autonomous with two degrees of freedom. As $Z_{\epsilon}=O(\sqrt{\epsilon})$, and because of the non-degeneracy assumption for $V$ (([**H1**]{},[**H2**]{})), we see that for each suitably small $\epsilon>0$, the map $\Phi_{\bar G_{\epsilon}}$ admits invariant cylinder also, denoted by $\Pi_{E_0,E_1,g}$ still, with the normally hyperbolic properties (see Formulae (\[cylindereq11\])), independent of the size of $\epsilon$.
Let us consider the persistence of $\Pi_{E'_0,E_1,g}$ with $E'_0=\epsilon^{2d}$ with $d>0$. As the perturbation depends on time $s$, we use $\Phi^{s,s_0}_{G_{\epsilon}}$ to denote the map from the time $s_0$-section to the time $s$-section, omit the symbol $s_0$ if $s_0=0$. Since these hyperbolic properties are posed for the map $\Phi^{s,s_0}_{\bar G_{\epsilon}}$ with large $s-s_0$, one has to measure how large the quantity $\|\Phi_{\bar G_{\epsilon}}^{s,s_0}-\Phi_{G_{\epsilon}}^{s,s_0}\|$ will be. As $\bar G_{\epsilon}$ is autonomous, $\Phi_{\bar G_{\epsilon}}^{s,s_0}=\Phi_{\bar G_{\epsilon}}^{s-s_0}$.
\[cylinderlem2\] Let the equation $\dot z=F_{\epsilon}(z,t)$ be a small perturbation of $\dot z=F_0(z,t)$, let $\Phi_{\epsilon}^t$ and $\Phi_0^t$ denote the flow determined by these two equations respectively. Then $$\|\Phi_{\epsilon}^t-\Phi_{0}^t\|_{C^1}\le \frac BA(1-e^{-At})e^{2At}$$ where $A=\max_{t,\lambda=\epsilon,0}\|F_{\lambda}(\cdot,t)\|_{C^2}$ and $B=\max_t\|(F_{\epsilon}-F_0)(\cdot,t)\|_{C^1}$.
Let $z_{\lambda}(t)$ denote the solution of the equations $\dot z=F_{\lambda}(z,t)$ for $\lambda =\epsilon,0$ respectively, and $z_{\epsilon}(0)=z(0)$. Let $\Delta z(t)=z_{\epsilon}(t)-z(t)$, then $\Delta z(0)=0$ and $$\Delta\dot z=\partial_zF_{\epsilon}((\nu z+(1-\nu)z_{\epsilon})(t),t)\Delta z+(F_{\epsilon}-F_0)(z(t),t)$$ where $\nu=\nu(t)\in [0,1]$. Therefore, one has $$\|\Delta\dot z\|\le\max\|\partial_zF_{\epsilon}\|\|\Delta z\|+\max\|F_{\epsilon}-F_0\|.$$ Let $\Delta z=y-\frac BA$, we have $\dot y\le Ay$. It follows from Gronwell’s inequality that $$\|\Delta z(t)\|\le\frac BA(e^{At}-1).$$
Along the orbit $z_{\lambda}(t)$, the differential of the flow $\Phi^t_{\lambda}$ obviously satisfies the equation $$\frac d{dt}D\Phi^t_{\lambda}=\partial_zF_{\lambda}(z_{\lambda}(t),t)D\Phi^t_{\lambda},\qquad \lambda=\epsilon, 0.$$ Therefore, for each tangent vector $v$ attached to $z_{\lambda}(0)$ one has $$\|D\Phi^t_{\lambda}v\|\le\|v\|e^{At}.$$
To study the differential of $\Phi_{\epsilon}^t-\Phi_{0}^t$, let us consider the equation of secondary variation. Let $\delta z_{\lambda}$ be the solution of the variational equation $\delta\dot z_{\lambda}=\partial_zF_{\lambda}(z_{\lambda}(t),t)\delta z_{\lambda}$ for $\lambda=\epsilon,0$ respectively, where $z_{\lambda}(t)$ solves the equation $\dot z_{\lambda}= F_{\lambda}(z_{\lambda},t)$ and $z_{\epsilon}(0)=z(0)$. To measure the size $\Delta\delta z=\delta z_{\epsilon}-\delta z$ with the condition $z_{\epsilon}(0)=z(0)$, we make use of the relations such as $v=\delta z_{\epsilon}(0)=\delta z(0)$, $\|\delta z(t)\|\le\|v\|e^{At}$ and find that $$\begin{aligned}
\Big\|\frac {d(\Delta\delta z)}{dt}\Big\|\le &\max\|\partial _zF_{\epsilon}\|\|\Delta\delta z\| +\max\|\partial^2_zF\|\|\Delta z(t)\|\|\delta z(t)\| \\
&+\max\|\partial_z(F_{\epsilon}-F)\|\|\delta z(t)\| \\
\le &A\Delta\delta z+B\|v\|e^{2At}.\end{aligned}$$ Let $\Delta\delta z=y+\frac BA\|v\|e^{2At}$, we have $\dot y\le Ay$. Using Gronwell’s inequality again, one obtains an upper bound of the variation of the differential $$\|\Delta\delta z(t)\|\le\frac BA\|v\|(1-e^{-At})e^{2At}.$$ Note that $v$ represents initial tangent vector, it completes the proof.
Let us applying this lemma to the Hamiltonian $G_{\epsilon}$. Treating $R_{\epsilon}$ as the function of $(x,p)$ we find that there exist some constants $C_{11},C_{12}>0$ independent of $\epsilon$ such that $$\max_s\|J\nabla\bar G_{\epsilon}-J\nabla G_{\epsilon}\|_{C^1}\le\max\Big|\frac{\partial^2 R'_{\epsilon}} {\partial x\partial p}\Big|\le C_{11}\epsilon^{5\sigma-\frac 16}$$ as $\|\epsilon R_{\epsilon}(\cdot,s)\|_2=O(\epsilon^{\frac 56+5\sigma})$ (see Theorem \[normalthm1\]). Since the function $\bar G_{\epsilon}$ comes from $h+\epsilon Z$ which is $C^r$-smooth ($r\ge 8$), one has $\max_s\|J\nabla\bar G_{\epsilon}\|_{C^2}=\max_s\|\bar G_{\epsilon}\|_{C^3}<C_{12}$. For $s-s_0=\frac 2{\lambda_1}|\ln\epsilon^{2d}|$ one obtains from Lemma \[cylinderlem2\] that $$\|\Phi^{s,s_0}_{\bar G_{\epsilon}}-\Phi^{s,s_0}_{G_{\epsilon}}\|_{C^1}\le \frac{C_{11}}{C_{12}}\epsilon^{5\sigma-\frac 16-\frac {8C_{12}d}{\lambda_1}}$$ If the condition $0<d<\frac{\lambda_1}{8C_{12}}(5\sigma-\frac 16)$ is satisfied, then $\|\Phi^{s,s_0}_{\bar G_{\epsilon}}-\Phi^{s,s_0}_{G_{\epsilon}}\|_{C^1}\to 0$ as $\epsilon\to 0$. It allows one to apply the theorem of normally hyperbolic manifold to obtain the existence of invariant cylinder for the flow $\Phi^{s,s_0}_{G_{\epsilon}}$ in the extended phase space $\mathbb{T}^2\times\mathbb{R}^2\times\sqrt{\epsilon}\mathbb{T}$, which is a small deformation of $\Pi_{E'_0,E_1,g}\times\sqrt{\epsilon}\mathbb{T}$.
Be aware of the fact that $\Pi_{E'_0,E_1,g}$ is a cylinder with boundary, normally hyperbolic and invariant for $\Phi^s_{\bar G_{\epsilon}}$, where $s=\frac 2{\lambda_1}|\ln \epsilon^{2d}|$, we do not expect that the whole cylinder survives small perturbation, it may lose some part close to the boundary. To measure to what range the cylinder survives, we see that the variation of the energy along each orbit of $\Phi^{s,s_0}_{ G_{\epsilon}}$ is bounded by $$\label{cylindereq12}
\Big |\frac d{ds}G_{\epsilon}(z(s),s)\Big |=|\partial _sG_{\epsilon}(z,s)|=\frac 1{\sqrt{\epsilon}}\Big|\frac{\partial R_{\epsilon}}{\partial\tau}\Big|\le C_{13}\epsilon^{5\sigma-\frac 16}$$ here, the estimate $\|\epsilon R_{\epsilon}\|_{1}\le O(\epsilon^{\frac 43+5\sigma})$ is used (see Theorem \[normalthm1\]). Assume that the number $d$ satisfies the condition $$\label{cylindereq13}
d<\min\Big\{\frac 12\Big(5\sigma-\frac 16\Big),\frac{\lambda_1}{8C_{12}}\Big(5\sigma-\frac 16\Big),\frac 12\Big\}$$ one sees that, starting from the energy level $G_{\epsilon}^{-1}(\epsilon^{d})$, after a time of $s-s_0=\frac {2}{\lambda_1}|\ln\epsilon^{2d}|$, the orbit of $\Phi^{s,s_0}_{G_{\epsilon}}$ can not reach the energy level $G_{\epsilon}^{-1}(\epsilon^{2d})$ if $\epsilon$ is suitably small so that $2\epsilon^d(1+2C_{13}\lambda^{-1}|\ln\epsilon^{2d}|)<1$. Indeed, under such condition one has $$\begin{aligned}
\label{cylindereq14}
G_{\epsilon}(z(s),s)&\ge G_{\epsilon}(z(s_0),s_0)-\int_{s_0}^s\Big |\frac d{dt}G_{\epsilon}(z(t),t)\Big|dt\\
&\ge\epsilon^d-\frac {2C_{13}}{\lambda_1}\epsilon^{2d}|\ln\epsilon^{2d}|>\frac12\epsilon^d+\epsilon^{2d}.\notag\end{aligned}$$
To use the theorem of normally hyperbolic invariant manifold, let us introduce a modified Hamiltonian. Let $u$: $\mathbb{R}\to\mathbb{R}_+$ be a smooth function so that $u=0$ for $t\le 1$ and $u=1$ for $t\ge 2$, $$G'_{\epsilon}=\bar G_{\epsilon}+u\Big(\frac 2{\epsilon^{d}}\Big(\bar G_{\epsilon}-\epsilon^{2d}\Big)+1\Big)R_{\epsilon}$$ it coincides with $G_{\epsilon}$ for $(x,p)\in\bar G_{\epsilon}^{-1}(E)$ with $E\ge\frac 12 \epsilon^d+ \epsilon^{2d}$ and coincides with $\bar G_{\epsilon}$ for $(x,p)\in\bar G_{\epsilon}^{-1}(E)$ with $E\le\epsilon^{2d}$. For small $d$ satisfying the condition (\[cylindereq13\]) and small $\epsilon$, the cylinder $\Pi_{E'_0,E_1,g}$ survives the perturbation $\Phi^{s,s_0}_{G'_{\epsilon}}\to\Phi^{s,s_0}_{\bar G_{\epsilon}}$ and the bottom remains invariant for $\Phi^{s,s_0}_{G'_{\epsilon}}$.
By the definition, one has $G_{\epsilon}=G'_{\epsilon}$ for $(x,p)\in\bar G_{\epsilon}^{-1}(E)$ with $E\ge\frac 12\epsilon^d+\epsilon^{2d}$ and it follows from (\[cylindereq14\]) that $G_{\epsilon}(\Phi^{s,s_0}_{G'_{\epsilon}}(x,p),s)\ge\frac12\epsilon^d+\epsilon^{2d}$ provided $G_{\epsilon}(x,p,s_0)\ge\epsilon^d$ and $s\in [0,\frac 2{\lambda_1}|\ln\epsilon^{2d}|]$. So, for $E_0=\epsilon^d$ the invariant cylinder $\Pi_{E_0,E_1,g}\times\sqrt{\epsilon}\mathbb{T}$ persists under the perturbation $\Phi^{s,s_0}_{\bar G_{\epsilon}}\to\Phi^{s,s_0}_{G_{\epsilon}}$, denoted by $\tilde\Pi_{E_0,E_1,g}$. A point $(x,p,s)\in\tilde\Pi_{E_0,E_1,g}$ implies $G_{\epsilon}(x,p,s)\in[E_0,E_1]$. The invariance is in the sense that, emanating from any point in $\tilde\Pi_{E_0,E_1,g}$, the orbit has to pass through the bottom of the cylinder if it is going to leave the cylinder.
[**Location of Aubry set in the cylinder**]{}
As the working space here is phase space, we say that an Aubry set $\tilde{\mathcal{A}}(c)$ is located in the cylinder $\tilde\Pi_{E_0,E_1,g}$ if for each $c$-static curve $\gamma$, the orbit in the phase space $(x(s)=\gamma(s), p(s)=\partial_{\dot x}L_{G_{\epsilon}}(\gamma(s), \dot\gamma(s),s),s)\in\tilde\Pi_{E_0,E_1,g}$.
Recall the Hamiltonian $G_{\epsilon}$ defined in (\[cylindereq3\]) and note the Hamiltonian $\bar G_{\epsilon}=G_{\epsilon}-R_{\epsilon}$ is autonomous. Let $\alpha_{G_{\epsilon}}$ and $\alpha_{\bar G_{\epsilon}}$ denote the $\alpha$-function for the Lagrangians determined by $G_{\epsilon}$ and $\bar G_{\epsilon}$ respectively.
Since $\Pi_{0,E_1,g}$ is a hyperbolic cylinder, invariant for the Hamiltonian flow $\Phi^t_{\bar G_{\epsilon}}$, the channel $\mathbb{W}_g=\cup_{\nu\in (0,\nu_1]}\mathscr{L}_{\beta}(\nu g)$ has a foliation into a family of segments of line (one-dimensional flat), where $\nu_{1}>0$ is chosen so that $\alpha_{\bar G_{\epsilon}}(c)\le E_1$ for each $c\in\mathscr{L}_{\beta}(\nu g)$ if $\nu\le\nu_1$. We claim that the $\alpha$-function $\alpha_{\bar G_{\epsilon}}$ is smooth in $\mathbb{W}_g$. Indeed, restricted on each of these flats the function $\alpha_{\bar G_{\epsilon}}$ keeps constant, while restricted on a line $\Gamma_g$ orthogonal to these flats, the function is smooth because $\bar G_{\epsilon}$ can be treated as a Hamiltonian with one degree of freedom when it is restricted on the cylinder. If $g=k_ig_i+k_{i+1}g_{i+1}$ we consider the Hamiltonian in the finite covering space $\bar M=\bar k_1\mathbb{T} \times\bar k_2\mathbb{T}$ where $\bar k_m=k_ig_{im}+k_{i+1}g_{i+1,m}$ for $m=1,2$ if we write $g_j=(g_{j1},g_{j2})$ for $j=i,i+1$. In that space there are $k_i+k_{i+1}$ fixed points. Because of Hartman’s theorem for two-dimensional system, it is $C^1$-conjugate to a linear equation $\dot x=\lambda_1y$, $\dot y=\lambda_1x$ around each fixed point. Therefore, for small $E>0$ some $C^1$-function $\tau_g(E)$ exists such that the period of the frequency $\nu g$ is $T_{\nu g}=\lambda_1^{-1}(k_i+k_{i+1})(-\ln E+\tau_g(E))$ (see (\[regularenergyeq2\])). Since $\partial\alpha_{\bar G_{\epsilon}}=\nu g$ $$\label{cylindereq15}
\frac{\lambda_1}{-\ln E+\tau_g(E)}=\|\partial\alpha_{\bar G_{\epsilon}}\|,\qquad \forall\ c\in \Gamma_g,$$ As $\alpha_{\bar G_{\epsilon}}(c)=E$ remains constant when it is restricted on each of its flats which are orthogonal to the line $\Gamma_g$, we find that for $0<\alpha_{\bar G_{\epsilon}}\ll 1$ $$\label{cylindereq16}
\langle\partial^2\alpha_{\bar G_{\epsilon}}v,v\rangle=\frac{\lambda_1^2(1-\alpha_{\bar G_{\epsilon}} \tau'_g(\alpha_{\bar G_{\epsilon}}))} {\alpha_{\bar G_{\epsilon}}(-\ln\alpha_{\bar G_{\epsilon}}+ \tau_g(\alpha_{\bar G_{\epsilon}}))^3}>0,$$ where $v$ is the direction of $\Gamma_g$ and $\|v\|=1$. It follows that $$\label{cylindereq17}
\alpha_{\bar G_{\epsilon}}(c)-\alpha_{\bar G_{\epsilon}}(c_{\omega}) \ge\langle\omega,c-c_{\omega}\rangle+\frac12\langle\partial^2\alpha_{\bar G_{\epsilon}}(c)v,v\rangle|c-c_{\omega}|^2$$ holds for $\omega=\nu g$, $c_{\omega}\in\mathscr{L}_{\beta}(\omega)\cap\Gamma_g$ $c\in\Gamma_g$, $\alpha_{\bar G_{\epsilon}}(c)>\alpha_{\bar G_{\epsilon}}(c_{\omega})$.
Let $c^*$ be the class so that $\alpha_{G_{\epsilon}}(c)=\alpha_{\bar G_{\epsilon}}(c^*)$. To measure the difference of $c^*-c$, we find $\frac 12|\langle c-c^*,\omega(c^*)\rangle|\le|\alpha_{\bar G_{\epsilon}}(c^*)-\alpha_{\bar G_{\epsilon}}(c)|=|\alpha_{G_{\epsilon}}(c)-\alpha_{\bar G_{\epsilon}}(c)|$. As the $\alpha$-function undergoes small variation: $|\alpha_L(c)-\alpha_{L'}(c)|\le\varepsilon$ for small perturbation $L'\to L$ with $\|L'-L\|_{C^1}\le\varepsilon$ [@Ch], we find $|\alpha_{G_{\epsilon}}(c)-\alpha_{\bar G_{\epsilon}}(c)|\le \epsilon^{2d}$ when $c$ is restricted on the path $\Gamma_g$. Therefore, we obtain that $$\label{cylindereq18}
|\langle c^*-c,\omega(c^*)\rangle|\le 2\epsilon^{2d}.$$
In the Aubry set for $c\in\Gamma_g\cap\alpha^{-1}_{G_{\epsilon}}(E)$ with $E\ge 2\epsilon^{d}$, any orbit does not hit the energy level set $G_{\epsilon}^{-1}(E)$ with $E\le\epsilon^d$.
If the lemma does not hold, there would exist an orbit $(x(s),p(s))$ in the Aubry set for $c\in\Gamma_g\cap\alpha^{-1}_{G_{\epsilon}}(2\epsilon^{d})$, which hits the energy level $G_{\epsilon}^{-1}(\epsilon^d)$ at the time $s=s_0\mod\sqrt{\epsilon}$, i.e. $G_{\epsilon}(x(s_0),y(s_0),s_0)=\epsilon^d$. Since the orbit entirely stays in the invariant cylinder and the perturbation is of order $\epsilon^{2d}$, it returns back to the neighborhood of $(x(s_0),y(s_0))$ after a time $S=\lambda_1^{-1}(k_i+k_{i+1})\ln\epsilon^d+\tau_{\epsilon}$ where $\tau_{\epsilon}$ remains bounded as $\epsilon\to 0$. To see how close it could be, we obtain from (\[cylindereq12\]) that $$\begin{aligned}
\label{cylindereq19}
&|G_{\epsilon}(x(S+s_0),p(S+s_0),S+s_0)-G_{\epsilon}(x(s_0),p(s_0),s_0)|\\
&\le \int_{s_0}^{S+s_0}\Big |\frac d{ds}G'_{\epsilon}(z(s),s)\Big|ds\le C_{15}\epsilon^{2d}|\ln\epsilon^{d}|. \notag\end{aligned}$$ As $\bar G_{\epsilon}^{-1}(E)\cap\Pi_{0,E_1,g}$ is an invariant circle for $\Phi^t_{\bar G_{\epsilon}}$, the perturbed cylinder is $O(\epsilon^{2d})$-close to the original one [@BLZ] and the cylinder may be crumpled but at most up to the order $O(E^{-2\mu_6})$ (cf. (\[cylindereq9\])), some large $k\in\mathbb{Z}$ exists such that $S=k\sqrt{\epsilon}$ and $$\|(x(S+s_0),p(S+s_0))-(x(s_0),p(s_0))\|\le C_{14}\epsilon^{2d(1-\mu_6)}|\ln\epsilon^{d}|.$$ Since the curve $x(s)$ is assumed $c$-static, it follows that $$\label{cylindereq20}
\Big|\int_{s_0}^{S+s_0}(L_{G_{\epsilon}}(x(s),\dot x(s),s)-\langle c,\dot x(s)\rangle+\alpha_{G_{\epsilon}}(c))ds \Big|\le C_{16}\epsilon^{2d(1-\mu_6)}|\ln\epsilon^{d}|.$$
As the cylinder $\Pi_{E_0,E_1,g}\times\sqrt{\epsilon}\mathbb{T}$ is $\epsilon^{2d}$-close to $\tilde\Pi_{E_0,E_1,g}$, there is a $c'$-minimal orbit $(x'(s),y'(s))$ of the Hamiltonian flow $\Phi^s_{\bar G_{\epsilon}}$ on $\Pi_{E_0,E_1,g}$ such that $\alpha_{\bar G_{\epsilon}}(c')=\epsilon^d$ and $\|(x'(s_0),y'(s_0))-(x(s_0),y(s_0))\|\le O(\epsilon^{2d(1-\mu_6)})$. Let $\Gamma_x=\bigcup_{s=s_0}^{s_0+S}(x(s),y(s))$ and $\Gamma_{x'}=\bigcup_{s=s_0}^{s_0+S'}(x'(s),y'(s))$ where $S'$ is the period of $x'(s)$, we have an estimate on the Hausdorff distance $d_H(\Gamma_x,\Gamma_{x'})\le O(\epsilon^{2d(1-\mu_6)}|\ln\epsilon^d|)$. Consequently, we have $$\int_{\Gamma_x}\langle y,dx\rangle-\int_{\Gamma_{x'}}\langle y,dx\rangle=O(\epsilon^{2d(1-\mu_6)}|\ln\epsilon^d|).$$ As $\bar G_{\epsilon}(x'(s),y'(s))\equiv\alpha_{G_{\epsilon}}(c')$ we have $$\begin{aligned}
0&=\int(L_{\bar G_{\epsilon}}(x'(t),\dot x'(t))-\langle c',\dot x'(t)\rangle+\alpha_{\bar G_{\epsilon}}(c'))dt \notag\\
&=\int \langle y'(s)-c',\dot x'(s)\rangle ds\notag\end{aligned}$$ Let $\bar x(s)$ be the lift of $x(s)$ to the universal covering space, it follows that $$\begin{aligned}
&\int_{s_0}^{S+s_0}\langle y(s)-c,\dot x(s)\rangle ds\notag\\
=&\int_{s_0}^{S+s_0}\langle y(s)-c',\dot x(s)\rangle ds-\int_{s_0}^{S'+s_0}\langle y'(s)-c',\dot x'(s)\rangle ds \notag\\
&-\langle c-c',\bar x(S+s_0)-\bar x(s_0)\rangle\notag\\
=&\int_{\Gamma_x}\langle y,dx\rangle-\int_{\Gamma_{x'}}\langle y',dx'\rangle+O(\epsilon^{2d(1-\mu_6)}|\ln\epsilon^d|)\notag\\
&-\langle c-c',\bar x(S+s_0)-\bar x(s_0)\rangle\notag\\
=&-\langle c-c',\bar x(S+s_0)-\bar x(s_0)\rangle+O(\epsilon^{2d(1-\mu_6)}|\ln\epsilon^{2d}|)\notag\end{aligned}$$ Since it follows from (\[cylindereq19\]) that $$\alpha_{G_{\epsilon}}(c)-G_{\epsilon}(x(s),y(s),s)\ge\alpha_{G_{\epsilon}}(c)-\alpha_{\bar G_{\epsilon}}(c')-O(\epsilon^{2d}|\ln\epsilon^{d}|)$$ holds for all $s\in[s_0,S+s_0]$, we find $$\begin{aligned}
\label{cylindereq21}
&\int_{s_0}^{S+s_0}(L_{G_{\epsilon}}(x(s),\dot x(s),s)-\langle c,\dot x(s)\rangle+\alpha_{G_{\epsilon}} (c))ds\\
=&\int_{s_0}^{S+s_0}\Big(\langle y(s)-c,\dot x(s)\rangle+(\alpha_{G_{\epsilon}}(c)- G_{\epsilon}(x(s),y(s),s))\Big)ds\notag\\
\ge&\, (\alpha_{G_{\epsilon}}(c)-\alpha_{\bar G_{\epsilon}}(c'))S-\langle c-c',\bar x(S+s_0)-\bar x(s_0)\rangle-O(\epsilon^{2d}|\ln\epsilon^{2d}|)\notag\\
\ge&\, C_{17}\epsilon^d.\notag\end{aligned}$$ To verify the second inequality, let $c^*$ be the class such that $\alpha_{\bar G_{\epsilon}}(c^*)=\alpha_{G_{\epsilon}}(c)$, then we obtain from the formula (\[cylindereq18\]) that $$|c^*-c|\le 3\lambda_1^{-1}\epsilon^{2d}|\ln\epsilon^d|.$$ For small $\epsilon$ such that $\epsilon^d\ge 4\epsilon^{2d}$ we find from (\[cylindereq15\]) that $$|c'-c^*|\ge \frac 1{\|\partial\alpha_{\bar G_{\epsilon}}\|}\Big(\alpha_{\bar G_{\epsilon}}(c')-\alpha_{\bar G_{\epsilon}}(c^*)\Big)\ge C_{18}\epsilon^{d}|\ln\epsilon^{d}|$$ holds for $c^*,c'\in\Gamma_g$ and $\alpha_{\bar G_{\epsilon}}(c')>\alpha_{\bar G_{\epsilon}}(c^*)$. Therefore, one obtains from (\[cylindereq16\]) and (\[cylindereq17\]) that $$\alpha_{\bar G_{\epsilon}}(c')-\alpha_{\bar G_{\epsilon}}(c^*)-\langle c'-c^*,\omega\rangle\ge C_{19}\frac{\epsilon^{d}}{|\ln\epsilon^{d}|},$$ from which one obtains the second inequality of (\[cylindereq21\]) from the first one. As $\mu_6$ is very small, the formula (\[cylindereq21\]) contradicts (\[cylindereq20\]). It completes the proof.
For $d>0$ satisfying the condition (\[cylindereq13\]), going back to the original coordinates ($E\to\epsilon E$, $y=\sqrt{\epsilon}p$ and $s=\sqrt{\epsilon}\tau$), we obtain
\[cylinderthm2\] For an irreducible class $g\in H_1(\mathbb{T}^2,\mathbb{Z})$, there exists a 3-dimensional cylinder $\tilde \Pi_{E_0,E_1,g}\subset\mathbb{R}^2 \times\mathbb{T}^3$ associated with a channel $\mathbb{W}_g\subset H^1(\mathbb{T}^2,\mathbb{R})$ such that
1, the cylinder $\tilde\Pi_{E_0,E_1,g}$ is a small deformation of the cylinder $$\{(x_{E}(\tau+\tau^*),y_{E}(\tau+\tau^*),\tau^*): [x_{E}]=g,(\tau,\tau^*)\in\mathbb{T}^2,E\in[E_0,E_1]\},$$ where $E_0=\epsilon^{\frac 1d}$, $x_{E}$ is the minimal periodic curve for $L_{\bar Y}-\langle c,\dot x\rangle+E$, the function $\bar Y$ solves the equation $(\tilde h+\epsilon\tilde Z)(x,y,\bar Y(x,y))=\tilde E$, $c\in\mathbb{W}_g$ and $y_E=\partial_{\dot x}L_{\bar Y}(x_E,\dot x_E)$;
2, the cylinder is invariant for the Hamiltonian flow $\Phi^{\tau,\tau_0}_{Y}$: $\forall$ $(x,y,\tau_0)\in \tilde\Pi_{E_0,E_1,g}$, if $\Phi^{\tau,\tau_0}_{Y}(x,y)\notin\tilde\Pi_{\epsilon,g}$ holds for $\tau>\tau_0$ then $\exists$ $\tau'\in(\tau_0,\tau)$ such that $\Phi^{\tau',\tau_0}_{Y}(x,y)$ is on the boundary of $\tilde\Pi_{E_0,E_1,g}$, where $Y$ solves the equation $H(x,y,-\tau,Y(x,y,\tau))=\tilde E$;
3, $\tilde\Pi_{E_0,E_1,g}$ is normally hyperbolic for $\Phi^{\tau,\tau_0}_{Y}$ with $\tau-\tau_0=\frac{2d}{\lambda_1}\frac {\ln\epsilon}{\sqrt{\epsilon}}$;
4, this channel reaches to a small neighborhood of the flat $\mathbb{F}_0$ in the sense $$\min_{c\in\mathbb{W}_g}\alpha_Y(c)-\min_{c\in H^1(\mathbb{T}^2,\mathbb{R})}\alpha_Y(c)=2\epsilon^{1+d}.$$ For each $c\in\mathbb{W}_g$ with $\alpha(c)\ge 2\epsilon^{1+d}$, the Aubry set entirely stays in the cylinder.
Let us consider the autonomous Hamiltonian $H$ with the form of (\[homogenizedeq1\]). Let $\tilde E>\min\alpha_H$, $Y(x,y,\tau)$ be the function solves the equation $H(x,y,x_3,Y(x,y,-x_3))=\tilde E$. Then $Y$ has the form of (\[homogenizedeq2\]). The energy $E$ of $G$ corresponds to the coordinate $y_3$ for the autonomous Hamiltonian $H$. Applying Theorem \[flatthm5\] one obtains
\[cylinderthm3\] Assume $\tilde E>\min\alpha_H$. Given an irreducible $g\in H_1(\mathbb{T}^2,\mathbb{Z})$, there is a $3$-dimensional cylinder $\tilde\Pi_{E_0,E_1,g}\subset H^{-1}(\tilde E)$ associated with a channel $\tilde{\mathbb{W}}\subset \alpha^{-1}(\tilde E)$ such that
1, the cylinder $\tilde\Pi_{E_0,E_1,g}$ is a small deformation of the cylinder $$\{(x_{E}(x_3),y_{E}(x_3),x^*_3,E): [x_{E}]=g,(x_3,x^*_3)\in\mathbb{T}^2,E\in[E_0,E_1]\};$$
2, the cylinder $\tilde\Pi_{E_0,E_1,g}$ is invariant for the Hamiltonian flow $\Phi^t_{H}$: for each $(\tilde x,\tilde y)\in \tilde\Pi_{E_0,E_1,g}$, if $\Phi^t_{H}(\tilde x,\tilde y)\notin\tilde\Pi_{\epsilon,g}$ holds for certain $t>0$ then $\exists$ $t'\in(0,t)$ such that $\Phi^{t'}_{H}(\tilde x,\tilde y)$ is on the boundary of $\tilde\Pi_{E_0,E_1,g}$;
3, $\tilde\Pi_{E_0,E_1,g}$ is normally hyperbolic for $\Phi^t_H$ with $t=\frac{2d}{\lambda_1}\frac {\ln\epsilon}{\sqrt{\epsilon}}$;
4, for each $\tilde c=(c,c_3)\in\tilde{\mathbb{W}}$ with $c_3\ge 2\epsilon^{1+d}$, the Aubry set is contained in that cylinder $\tilde{\mathcal{A}}(c)\subset\tilde\Pi_{E_0,E_1,g}$.
By [**H4**]{}, it is generic that there are finitely many bifurcation points, denoted by $c_1,c_2,\cdots,c_m\in\Gamma_g$ corresponding to the energy $E_1,E_2,\cdots E_m$. For each point $c\neq c_i$, the $c$-minimal measure for the Hamiltonian $\bar G_{\epsilon}$ is uniquely supported on a hyperbolic periodic orbit. For the class $c_i$, the Mather set is composed of two hyperbolic periodic orbits. The periodic orbits for $c\in (c_{i},c_{i+1})$ constitute a piece of cylinder $\Pi_i$. As these periodic orbits are hyperbolic, the cylinder $\Pi_i$ can be extended a little bit consisting of periodic orbits which are hyperbolic also, but do not support minimal measure. They are local minimal. These cylinders are clearly invariant and normally hyperbolic for the Hamiltonian flow $\Phi_{\bar G}^t$ with suitably large $t$. Applying the theorem of normally hyperbolic manifold, one can see that there exists some $\tilde\Pi_{i,\epsilon}$ which is invariant for the Hamiltonian flow determined by $G$, and keeps close to $\Pi_{i}\times\sqrt{\epsilon}\mathbb{T}$. For each $c\in\Gamma_g$, the Mather set for $G$ stays in the cylinder. Except for finitely many $c_{i,\epsilon}$ very close to $c_{i}$, the time-$\sqrt{\epsilon}$-section of the Mather set for other $c\in\Gamma_{g}$ is an invariant circle, or periodic points or Aubry-Mather set in the cylinder, for bifurcation point $c=c_{i,\epsilon}$, the Mather set consists of two parts, one stays in $\tilde\Pi_{i,\epsilon}$ another one stays in $\tilde\Pi_{i+1,\epsilon}$.
Transition from double to single resonance
------------------------------------------
Some normally hyperbolic invariant cylinder has been shown to reach $O(\epsilon^{\frac12+d})$-neighborhood of the double resonant point. This cylinder extends to the place a bit far away from the double resonant point. To see how to transit from double resonance to single resonance, let us homogenize the Hamiltonian in a region $\|y-y_j\|\le O(\sqrt{\epsilon})$ and choose different $y_j$. Recall that the normal form remains valid in the domain $\{\|y\|\le O(\epsilon^{\kappa})\}$ ($\frac 16<\kappa\le\frac 13$). The $\sqrt{\epsilon}$-neighborhood of the curve is covered by as many as $O(\epsilon^{\kappa-\frac 12})$ small balls with radius $O(\sqrt{\epsilon})$. Such approach is based on the following:
For nearly integrable Lagrangian $L(x,\dot x,t)=\ell(\dot x)+\epsilon\ell_1(x,\dot x,t)$, each orbit in Mather set $(\gamma,\dot\gamma)$ $$\|\dot\gamma(t)-\dot\gamma(0)\|\le O(\sqrt{\epsilon}),\qquad \forall\ t\in\mathbb{R}.$$
The result is proved in [@BK] for time-1-map. It is also true for the Hamiltonian flow i.e. $\|y(t)-y(0)\|\le O(\sqrt{\epsilon})$. It makes sense for us to homogenize the Hamiltonian in the range $\|y-y_j\|\le K\sqrt{\epsilon}$ with suitably large $K>0$.
Using new variables $y-y_j=\sqrt{\epsilon}p$ and $s=\sqrt{\epsilon}\tau$, the homogenized Hamiltonian equation turns out to be the following form $$\frac{dx}{ds}=\frac{\omega}{\sqrt{\epsilon}}+Ap,\qquad
\frac{dp}{ds}=-\frac{\partial V}{\partial x}(x,y_j),$$ where $A=\partial^2h(y_j)$, the frequency $\omega=\partial h(y_j)$ satisfies a resonant condition. The corresponding Lagrangian reads $$L(\dot x,x)=\frac 12\Big\langle A^{-1}\Big(\dot x-\frac{\omega}{\sqrt{\epsilon}}\Big),\Big(\dot x-\frac{\omega} {\sqrt{\epsilon}}\Big)\Big\rangle-V(x)$$
With the potential $V(x)$ on the torus one associates its time average $[V]$ along the orbits of the linear flow defined by $\omega$: $x\to x+\omega t$ $$[V](x)=\frac 1T\int_0^TV(x+\omega t)dt,$$ where $T$ is the period of the frequency $\omega$. The function $[V]$ is then defined on a circle. Let $x_0$ be the maximal point of $[V]$, it corresponds to a circle on $\mathbb{T}^2$. The averaged Hamiltonian is also associated with a Lagrangian $$[L](\dot x,x)=\frac 12\Big\langle A^{-1}\Big(\dot x-\frac{\omega}{\sqrt{\epsilon}}\Big),\Big(\dot x-\frac{\omega} {\sqrt{\epsilon}}\Big)\Big\rangle-[V](x).$$ Let $T_{\omega,\epsilon}$ be the period of the frequency $\omega/\sqrt{\epsilon}$, $\xi_{\omega,\epsilon}$: $[0,T_{\omega,\epsilon}]\to\mathbb{T}^2$ be the minimizer of the action $$\inf_{[\xi]=g_{\omega}}\int_0^{T_{\omega,\epsilon}}[L](d\xi(s))ds,$$ then it is a curve of maximal points of $[V]$ with constant speed $\dot\xi_{\omega,\epsilon}= \omega/\sqrt{\epsilon}$. Consider $[V]$ as a function defined on $\mathbb{T}^2/\xi_{\omega,\epsilon}$ and denote by $[V]''$ the second derivative for the variable of $\mathbb{T}^2/\xi_{\omega,\epsilon}$, where we use $\xi_{\omega,\epsilon}$ to denote the circle $\cup_{t\in[0,T_{\omega,\epsilon}]} \xi_{\omega,\epsilon}(t)$.
Let $\xi_{\omega,\epsilon}+\delta$ denote a translation of $\xi_{\omega,\epsilon}$ such that $d(\xi_{\omega,\epsilon}+\delta,\xi_{\omega,\epsilon})=\delta$ and let $\gamma_{\omega,\epsilon}$: $[0,T_{\omega,\epsilon}]\to\mathbb{T}^2$ be the minimizer of the action $$\inf_{[\xi]=g_{\omega}}\int_0^{T_{\omega,\epsilon}}L(d\xi(s))ds,$$ then we have
Assume $-[V]$ is non-degenerate at its minimal point $-[V]''>\Lambda$, and assume some $\lambda>0$ exists so that $T_{\omega,\epsilon}=\epsilon^{\lambda}$. Then there exist some constants $D,D'>0$ such that the minimizer $\gamma_{\omega,\epsilon}$ entirely stays in $D\epsilon^{\lambda}$-neighborhood of the circle $\xi_{\omega,\epsilon}+\delta$, i.e. $d(\gamma(s), \xi_{\omega,\epsilon}+\delta)< D\epsilon^{\lambda}$ holds for each $s\in [0,\epsilon^{\lambda}]$ and $|\delta|\le D'\epsilon^{\lambda/2}$.
Since the minimizer solves the Lagrange equation, its second derivative remains bounded. Thus, as the first step, we claim that the minimizer stays entirely in $D\epsilon^{\lambda}$-neighborhood of $\xi_{\omega,\epsilon}+\delta$, a translation of the circle $\xi_{\omega,\epsilon}$. If not, the oscillation of $\gamma_{\omega,\epsilon}$ in the direction perpendicular to $\omega$ would not be smaller than $2D\epsilon^{\lambda}$. As the average speed is $O(\epsilon^{-\lambda})$, there would be a point on the minimizer where $\|\dot{\gamma}_{\omega,\epsilon} -\omega/\sqrt{\epsilon}\|>D$. Since the potential is bounded $|V|<C_{20}$, one obtains that $A_L(\gamma_{\omega,\epsilon})\ge (C_{21}D^2-C_{20})\epsilon^{\lambda}>C_{20}\epsilon^{\lambda}$ if we choose $D^2>2C_{21}^{-1}C_{20}$. On the other hand, the action along the curve $\gamma(t)=x_0+\omega/\sqrt{\epsilon}t$ would be not bigger then $C_{20}\epsilon^{\lambda}$. The contradiction implies our claim.
Let us compare the action of $L$ along the curve $\gamma_{\omega,\epsilon}$ with that along the curve $\xi_{\omega,\epsilon}$. If $|\delta|>D'\epsilon^{\lambda/2}$, by what we have proved, some $x_1\in\mathbb{T}^2/\xi_{\omega,\epsilon}$ exists such that $$|\gamma_{\omega,\epsilon}(t)-(x_1+\omega/\sqrt{\epsilon}t)|\le D\epsilon^{\lambda}, \qquad |(x_1-x_0)/\xi_{\omega,\epsilon}|\ge D'\epsilon^{\lambda/2}.$$ It follows that $$\begin{aligned}
A(\gamma_{\omega,\epsilon})-A(\xi_{\omega,\epsilon})=&\frac 12\int_0^{\epsilon^{\lambda}} \Big\langle A^{-1}\Big(\dot\gamma_{\omega,\epsilon}(t)-\frac{\omega}{\sqrt{\epsilon}}\Big), \Big(\dot\gamma_{\omega,\epsilon}(t)-\frac{\omega}{\sqrt{\epsilon}}\Big)\Big\rangle dt\\
&-\int_0^{\epsilon^{\lambda}}\Big(V(\gamma_{\omega,\epsilon}(t))-V\Big(x_0+\frac{\omega} {\sqrt{\epsilon}} t\Big) \Big)dt\\
>&-\int_0^{\epsilon^{\lambda}}\Big(V(\gamma_{\omega,\epsilon}(t))-V\Big(x_1+\frac{\omega} {\sqrt{\epsilon}} t\Big) \Big)dt\\
&+(-[V](x_1)+[V](x_0))\epsilon^{\lambda}\\
>&\Big(\frac 12{C_{22}}D'^2-|\max\partial V|\Big)\epsilon^{2\lambda}\end{aligned}$$ it contradicts the minimality of the curve $\gamma_{\omega,\epsilon}$ if $D'>0$ is chosen suitably large.
Recall the picture of minimal periodic orbit close to double resonance, one can see from this proposition how the shape of the periodic orbit changes when it moves away from double resonance to single resonance.
Annulus of incomplete intersection
==================================
Let us also start with the Hamiltonian $G_{\epsilon}$ defined by Formula (\[cylindereq3\]), it has two and half degrees of freedom. Given any two homology class $g,g'\in H_1(\mathbb{T}^2,\mathbb{Z})$, The theorem \[cylinderthm2\] confirms the existence of two wedge-shaped regions $\mathbb{W}$ and $\mathbb{W}'$ which reach to the boundary of the annulus $$\mathbb{A}_0=\Big\{c\in H^1(M,\mathbb{R}):0<\alpha_{G_{\epsilon}}(c)-\min\alpha_{G_{\epsilon}}<D\epsilon^{1+d}
\Big\},$$ For each class in $\mathbb{W}$ and $\mathbb{W}'$, the Aubry set lies in the normally hyperbolic cylinder and, by the result for [*a priori*]{} unstable systems, can be connected to other Aubry set lying in the cylinder under certain generic conditions. However, it seems unclear whether these two wedges can reach to the flat $\mathbb{F}_0$. Thus, a notable difficulty rises as these cylinders are separated by an annulus $\mathbb{A}_0$ around the flat $\mathbb{F}_0$, it is the problem of crossing double resonance.
It is the goal of this section to find an annulus $\mathbb{A}\supsetneq\mathbb{A}_0$ where those two wedge-shaped regions are plugged into and for each class in that annulus, the stable set of the Aubry set “intersects" the unstable set non-trivially, possibly incomplete. In other words, for each class in this region, the Mañé set does not cover the whole configuration space.
The Mañé set for $c\in\partial^* \mathbb{F}_0$
----------------------------------------------
As the first step, let us consider the Hamiltonian $\bar G$ and study all cases when the Mañé set covers the whole configuration space.
For each $c\in\partial^* \mathbb{F}_0$, except for the minimal measure $\mu$ supported on the fixed point $(x,\dot x)=0$, some minimal measure exists with non-zero rotation vector. In the covering space $\bar\pi$: $\mathbb{R}^2\to\mathbb{T}^2$, a disk $B_{\delta}(0)$ is contained in a strip bounded by two $c$-static curves $\xi_c$ and $\xi'_c$, both curves are in the Mather set: $\bar\pi\xi_c, \bar\pi\xi'_c\subset\mathcal{M}(c)$ no other $c$-static curve in the Mather set touches the interior of this strip. Let $U^{\pm}_c$ and $U'^{\pm}_c$ denote the elementary weak KAM solution determined by $\xi_c$ and $\xi'_c$ respectively, we investigate what happens when $U^-_c-U'^+_c=0$ holds in this strip. As the configuration space is two dimensional, for each $x$ in this region, $(x,y)=(x,\partial U^-_c(x))=(x,\partial U'^+_c(x))$ uniquely determines a $c$-semi static curve which lies entirely in this strip. The $c$-semi static curves considered here are all determined by $U_c^-=U_c'^+$. It is possible that some curve approaches to the origin as $t\to\infty(-\infty)$, in this case, because of [**H3**]{}, it approaches to the curve $\xi_c$ ($\xi_c'$) as $t\to -\infty$($\infty$).
Supported on the fixed point, the measure $\mu$ is minimal for all $c\in\mathbb{F}_0$. Thus, there always exists some semi-static curve $\gamma_c^{\pm}$ connecting the fixed point to the support of other $c$-minimal measure $\mu_c$ $$\lim_{t\to\pm\infty}\gamma_c^{\pm}(t)=0, \ \ \text{\rm and}\ \ \lim_{t\to\mp\infty}\gamma_c^{\pm}(t)\to\pi_x\text{\rm supp}\mu_c.$$ As all eigenvalues are assumed different, generically, all minimal homoclinic curves approach to the fixed point in the direction $\Lambda_{1,x}$, associated to the smallest eigenvalue: $$\lim_{t\to\pm\infty}\frac {\dot\gamma_{c_i}^{\pm}(t)}{\|\dot\gamma_{c_i}^{\pm}(t)\|}
=\pm\Lambda_{1,x}.$$ But this does not exclude the possibility that some $c$-semi static curves approach to the point in the direction of $\Lambda_{2,x}$. It provides us a criterion to classify the cases when the Mañé set covers the whole configuration manifold.
[**Case 1**]{}: no $c$-semi static curve approaches the origin in the direction of $\Lambda_{1,x}$. In this case, as $|\lambda_1|<|\lambda_2|$, there exist exactly two semi-static curves $\gamma^{\pm}_c$ such that $\gamma^{\pm}_c(t)\to 0$ as $t\to\pm\infty$. They approach the origin in the direction $\Lambda_{2,x}$ and $$\lim_{t\to\infty}\frac {\dot\gamma_{c}^{+}(t)}{\|\dot\gamma_{c}^{+}(t)\|}
=\lim_{t\to-\infty}\frac {\dot\gamma_{c}^{-}(t)}{\|\dot\gamma_{c}^{-}(t)\|}.$$
Other cases are classified under the condition that there exist some $c$-static curves approaching the origin in the direction of $\Lambda_{1,x}$. Since the curves $\xi_c$ as well as $\xi'_{c}$ is disjoint with the origin, some number $\delta>0$ exists such that these two curves do not hit the ball $B_{\delta}(0)$. The number $\delta$ seems depending on $c$. Let $\gamma_c^+$ be a semi-static curve approaching the origin as $t\to\infty$, it intersects the circle $\partial B_{\delta}(0)$ at some point. Let $I^{\pm}\subset\partial B_{\delta}(0)$ be such a set that passing through each point $x\in I^{\pm}$ a $c$-semi static curve approaches to the origin, as $t\to\pm\infty$, in the direction of $\Lambda_{1,x}$. By assumption, the set $I^+$ is not empty. Obviously, $I^+$ does not occupy the whole circle and can be made closed by adding at most two points, through which some semi-static curves approach the origin in the direction of $\Lambda_{2,x}$.
Passing through a point $x\in \partial B_{\delta}(0)$ very close to $I^+$, there is a unique $c$-semi static curve, determined by $U^-_c=U_c'^+$. Because of Proposition \[flatpro2\], the curve $\gamma_c$ will get very close to the origin and leave in a direction far away from $\Lambda_{1,x}$. Let $I^+_i$ be a connected component of $I^+$, it may be a point or an interval. If it is a point, let $x_i,x'_i\in\partial B_{\delta}(0)$ be two sequences of points such that they approach $I^+_i$ from different sides. Let $\gamma_i$ ($\gamma'_i$) be the semi static curve passing through $x^+_i$ ($x'^+_i$), it shall intersect the circle $\partial B_{\delta}(0)$ at a point $x^-_i$ ($x'^-$) respectively. Some $x^-,x'^-\in\partial B_{\delta}(0)$ exist so that $x^-_i\to x^-$, $x'^-_i\to x'^-$ as $i\to\infty$.
If $x^-=x'^-$, it determines a $c$-semi static curve approaches the origin as $t\to-\infty$. Because of Proposition \[flatpro2\], it approaches in the direction of $\Lambda_{2,x}$. This leads to
[**Case 2**]{}: there exists exactly one $c$-semi static curve approaching origin in the direction of $\Lambda_{2,x}$.
If $x^-\ne x'^-$, let $I^-_i$ denote the arc bounded by these two points, not containing $I^+_i$. One can see from the proof of Proposition \[flatpro2\] that the angle of this arc is not smaller than $\pi/2$. Passing from each point in the interior of the arc, the $c$-semi static curve approaches to the origin as $t\to -\infty$ and these curves constitute a sector. Since the fixed point is hyperbolic, it has its stable and unstable manifolds $W_0^{\pm}$. Thus, some some generating function $U^{\pm}$ and $r>0$ exist such that $W_0^{\pm}|_{B_{r}(0)}=\text{\rm graph}dU^{\pm} |_{B_{r}(0)}$. As the orbits determined by the curves entirely lie in the unstable manifold of the fixed point, one has $U_c^-=U'^+_c=U^-$ in the sector. Therefore, the size of the sector-shaped region is independent of the size of $\delta$. This leads to
[**Case 3**]{}: in the disk $B_{r}(0)$ there is a sector-shaped region with the field angle not smaller than $\pi/2$. In this sector, one has $U_c^-=U'^+_c=U^-$.
Let $\gamma^+_c$ ($\gamma^-_c$) be $c$-semi static curve passing through a point in $I_i^+$ ($I^-_i$) respectively, then they approach the origin in opposite direction as $t\to\pm\infty$ respectively, i.e. $\lim_{t\to\infty}\dot\gamma_c^+(t)\|\dot\gamma_c^+(t)\|^{-1}=\lim_{t\to-\infty}\dot\gamma_c^-(t) \|\dot\gamma_c^-(t)\|^{-1}$. To verify this claim, let us assume the contrary. Thus, these two curves cut the ball $B_{\delta}(0)$ into two parts, one is sharp wedge-shaped, denoted by $W$.
{width="5.0cm" height="2.5cm"} \[fig8\]
We choose a $c$-semi static curve lying in $W$ and keeping very close to the curves $\gamma_c^-$ and $\gamma_c^+$. In canonical coordinates such that $\bar G=\frac 12(p_1^2-\lambda_1x_1^2)+\frac 12(p_2^2-\lambda_2x_2^2)+O(\|(x,p)\|^3)$, the set $W$ has a vertex at the origin. As both $\gamma_c^{+}$ and $\gamma_c^{-}$ approach the origin in the direction of $\Lambda_{1,x}$, there exists $\delta_1\le\delta$ such that $|x_2|\le |x_1|^3$ if $(x_1,x_2)\in W$ and $|x_1|\le\delta_1$. Since the fixed point is hyperbolic, it has local stable and unstable manifold, determined by the generating functions $U^+$ and $U^-$ respectively. Restricted in $W$, these functions satisfy the condition $$U^-(x)-U^-(0)\ge\frac{\lambda_1^2}{3}\|x\|^2,\qquad U^+(0)-U^+(x)\ge\frac{\lambda_1^2}{3}\|x\|^2, \ \ \ \forall\ \|x\|\le\delta.$$ Pick up two points $x$ and $x'$ very close to $\gamma_c^{\pm}$ respectively, through which some $c$-semi static curve $\gamma_c$ passes, namely, some $t'>t$ exist such that $\gamma_c(t)=x$ and $\gamma_c(t')=x'$. Note the orbit determined by $\gamma_c^+$ ($\gamma_c^+$) lies in the stable (unstable) manifold, by definition ones has $$A[\gamma_c|_{[t,t']}]\ge\frac 34\Big(U_c^-(x')-U_c^+(x)\Big)\ge\frac {\lambda_1^2}4(\|x'\|^2+\|x\|^2).$$ If we choose $x$ sharing the same first coordinate with $x'$ and connect them with a straight line $\zeta$: $[0,|x_2-x'_2|]\to\mathbb{T}^2$, then $|\dot\zeta|\le O(1)$ and the action along this curve one has $A[\zeta]\le O(\|x\|^3)$. It contradicts the minimality of $\gamma_c$, thus the claim is proved.
We claim that $I^+$ has only one connected component. Otherwise, there would be two connected component $I^+_k$ and $I^+_i$. By the definition, passing through a point $x\in I^+_j$ ($x'\in I^+_k$) there is a $c$-semi static curve $\gamma_x$ ($\gamma_{x'}$) which approaches the curve $\xi_c$ as $t\to\infty$ and approaches the origin as $t\to\infty$. These two curves divided the strip into two parts $S=S_1\cup S_2$, where $S_1$ is such a strip that passing through any point $x^*\in S_1$, the $c$-semi static curve will approach the origin as $t\to\infty$. However, there exists a point $x^*\in S_1\cap(\partial B_{\delta}\backslash I^+)$, it implies that passing thorough $x^*$, the $c$-semi-static curve will approach the curve $\xi'_c$, namely, it would intersect either $\gamma_x$ or $\gamma_{x'}$. It is absurd. Thus, we obtain the left picture in Figure \[fig9\].
![[]{data-label="fig9"}](Ardiff9.eps){width="9.7cm" height="3.7cm"}
By similar argument applying to the set $I^-$, we have either the case 2 again or
[**Case 4**]{}: in the disk $B_{r}(0)$ there is a sector-shaped region with the field angle not smaller than $\pi/2$, where $U_c^-=U'^+_c=U^+$, see the right picture in Figure \[fig9\].
We claim that all of these cases do not occur for generic potential $V$. The first two cases takes place at most for four invariant measures, as there are only four curves which approaches the origin in the direction of $\Lambda_{2,x}$. Each of these curves approaches at most one Mather set. These Mather sets correspond to at most four edges of $\mathbb{F}_0$. Let $V_{\delta}-V$ be a non-negative function such that its support does not touch these four curves as well as the support of the minimal measure. By perturbing the potential $V\to V_{\delta}$, one can see that the Mather set remains unchanged, but the Mañé set does not cover $\mathbb{T}^2$ for each of these four cohomology classes.
Both case 3 and 4 take place also for at most four Mather sets, as each sector-shaped region has the field angle not smaller than $\pi/2$, and the orbit determined by $(x,y)=(x,\partial_xU^{\pm})$ approaches one Mather set only. Let us destruct it one by one. If some sector-shaped region $S^+\subset B_r(0)$ exists where $U_c^-=U_c'^+=U^+$, $\pi_x\text{\rm supp}\mu_c\cap S^+=\varnothing$. We divide it into three sub-sectors $S^+=S^+_1\cup S^+_2\cup S^+_3$, each of which is composed of $c$-semi static curves approaching the origin as $t\to\infty$ and $S^+_1$ is disjoint with $S^+_3$. We introduce another potential $V_{\delta}$ such that the function $V_{\delta}-V$ is non-negative, $\text{\rm supp}(V_{\delta}-V)\subset S^+_2\backslash B_{r_1}$ ($r_1<r$).
For the perturbed Lagrangian determined by $\bar G_{\delta}=\frac 12\langle Ap,p\rangle+V_{\delta}(x)$, the minimal measure for the class $c$ is the same as that for unperturbed Hamiltonian. Let $U^-_{c,\delta}$, $U'^+_{c,\delta}$ be the elementary weak KAM solutions of the perturbed Hamiltonian, associated to the minimal measure $\mu_c$ and $\mu'_c$ respectively, one has $$\arg\min(U^-_{c,\delta}-U'^+_{c,\delta})\cap \text{\rm supp}(V_{\delta}-V)=\varnothing,\qquad
\arg\min(U^-_{c,\delta}-U'^+_{c,\delta})\supset S^+_1\cup S^+_3.$$
Under such perturbation, there might be another cohomology class $c'$ such that $U_{c'}^--U'^+_{c'}=0$ holds on the whole torus and a sector $S^-$ exists where $U_{c'}^-=U'^+_{c'}=U^-$. Note $\pi_x\text{\rm supp}\mu_{c'}\cap S^-=\varnothing$, we split it into three sub-sectors $S^-=S^-_1\cup S^-_2\cup S^-_3$, each of which is composed by $c'$-semi static curves approaching to the origin as $t\to -\infty$ and $S^-_1$ is disjoint with $S^-_3$. We introduce again a perturbed potential $V_{\delta}$ such that the function $V'_{\delta}-V$ is non-negative, $\text{\rm supp}(V'_{\delta}-V)\subset S^-_2\backslash B_{r_1}$ ($r_1<r$).
For the new perturbed Lagrangian, the minimal measure for the class $c'$ is the same as that for unperturbed one. Let $U^-_{c',\delta}$, $U'^+_{c',\delta}$ be the elementary weak KAM solutions of the perturbed Hamiltonian, determined by $\xi_{c'}$ and $\xi'_{c'}$ respectively, one also has $$\arg\min(U^-_{c',\delta}-U'^+_{c',\delta})\cap \text{\rm supp}(V_{\delta}-V)=\varnothing,\qquad
\arg\min(U^-_{c',\delta}-U'^+_{c',\delta})\supset S^-_1\cup S^-_3.$$
For suitably small $r>0$, the Hamiltonian flow determined by $\bar G$ is well approximated by its linearized flow when they are restricted in the ball $B_r(0)$. For the linearized flow, if $(x(t),y(t))$ is an orbit in the unstable manifold, $(x(-t),-y(-t))$ is an orbit in the stable manifold. Therefore, some sectors $\check S^{\pm}_k$ ($k=1,3$) exist so that $\check S^{\pm}_k\subset S^{\pm}_k$, $\check S^{-}_k\cap S_2^+=\varnothing$ and $\check S^{+}_k\cap S_2^-=\varnothing$ hold for $k=1,3$, each $\check S^{+}_k$ consists of $c$-semi static curves which approach the origin as $t\to\infty$ with $$\arg\min(U^-_{c,\delta}-U'^+_{c,\delta})\supset\check S^{+}_1\cup\check S^{+}_3,\ \ \ \arg\min(U^-_{c,\delta}-U'^+_{c,\delta})\cap \text{\rm supp}(V_{\delta}-V)=\varnothing,$$ each $\check S^{-}_k$ consists of $c'$-semi static curves which approach the origin as $t\to -\infty$ and $$\arg\min(U^-_{c',\delta}-U'^+_{c',\delta})\supset\check S^{-}_1\cup\check S^{-}_3,\ \ \ \arg\min(U^-_{c',\delta}-U'^+_{c',\delta})\cap \text{\rm supp}(V_{\delta}-V)=\varnothing.$$ Since there are at most two sectors corresponding to unstable manifold and two sectors corresponding to stable manifold, there are at most four pairs of static curves $(\xi_{c_i},\xi'_{c_i})$ ($i=1,2,3,4$) for which the case 3 and 4 takes place.
These four pairs of static curves $(\xi_{c_i},\xi'_{c_i})$ ($i=1,2,3,4$) corresponds to four edges (points) contained in $\partial^*\mathbb{F}_0$. By construction, the Mañé set does not cover the whole torus $\mathbb{T}^2$ for each cohomology class one these four edges. For any other class $c\in\partial^*\mathbb{F}_0$, the Mañé set can not cover the whole torus also. Otherwise, there would be a sector $S^{\pm}$ of $B_r(0)$ where $U_c^-=U_c'^+=U^{\pm}$, but it is absurd since some $\check S^{\pm}_i\subset S^{\pm}$ where $U_{c_i}^-=U_{c_i}'^+=U^{\pm}$ holds for some $i\in (1,2,3,4)$. For each $x\in\check S^{\pm}_i$, $(x,v=\partial_y\bar G(x,\partial U^{\pm}(x))$ determines an orbit of the Lagrangian flow which approaches both to the support of $\mu_{c_i}$ and to the support of $\mu_{c}$ as $t\to\pm\infty$, it is impossible. Therefore, we have
\[beltlem1\] It is an open and dense condition for the potential $V$ that for all class $c\in\partial^*\mathbb{F}_0$, the Mañé set does not cover the torus: $\mathcal{N}(c)\subsetneq\mathbb{T}^2$.
The Mañé set for $c\in\partial\mathbb{F}_0\backslash\partial^* \mathbb{F}_0$
----------------------------------------------------------------------------
This set contains at most countably many vertexes. Indeed, if both $\partial\mathbb{F}_0\backslash\partial^*\mathbb{F}_0$ and $\partial^*\mathbb{F}_0$ are non-empty, there do exist countably many vertexes (cf. Theorem \[flatthm3\]).
Let $E_i\subset\partial\mathbb{F}_0\backslash\partial^*\mathbb{F}_0$ be an edge joined to other two edges at the vertex $c_i$, $c_{i+1}$ respectively. By Theorem \[flatthm3\], the Aubry set for $c_j$ consists of two minimal homoclinic curves $\gamma_{j-1}$ and $\gamma_j$. Denote by $g_j\in\mathbb{Z}^2$ the homology class of $\gamma_j$, then the matrix $(g_{j-1},g_j)$ is uni-module. By introducing suitable coordinates on $\mathbb{T}^2$, we can assume $g_i=(1,0)$. In this coordinate system, $g_{i-1}=(k,1)$ and $g_{i+1}=(k',-1)$.
![[]{data-label="fig10"}](Ardiff10.eps){width="7.0cm" height="2.5cm"}
In this figure, each unit square represents a fundamental domain of $\mathbb{T}^2$ in the universal covering space, the horizontal line represents the lift of the homoclinic curve $\gamma_i$, which stays in the Aubry set for each $c\in E_i$. The blue dashed lines represent the lift of the $\gamma_{i-1}$ which stays in the Abury set for the class at one end-point of $E_i$. The purple dashed lines represents the lift of the $\gamma_{i+1}$ which stays in the Abury set for the class at another end-point of $E_i$.
Let us consider weak KAM solution $U_{i,\pm}^{\pm}$ in the strip bounded by the lines $L_+$ and $L_-$. According to Lemma \[cylinderlem1\], for $E-\min\alpha>0$ very small, there exists an interval $I_E\subset H^1(\mathbb{T}^2,\mathbb{R})$ such that for each $c\in I_E$, the Mather set consists of only one closed curve $\gamma_E$ such that $[\gamma_E]=g_i$. In the universal covering space, let $\bar\gamma_E$ be a component of the lift of $\gamma_E$ which approaches $L_+$ as $E\downarrow\min\alpha$. Let $U^{\pm}_{i,E}$ be the elementary weak KAM solution determined by $\bar\gamma_E$. As $E\downarrow\min\alpha$, $U^{\pm}_{i,E}\to U^{\pm}_{i,+}$ in $C^0$-topology. The function $U^{\pm}_{i,-}$ is obtained in the same way. The function $U_{i,\pm}^{-}$ determines backward semi-static curves approaching to the line $L_{\pm}$ as the time approaches to minus infinity, $U_{i,\pm}^{+}$ determines forward semi-static curves approaching to the line $L_{\pm}$ as the time approaches to positive infinity respectively.
If we remove the coercive condition on these weak KAM solutions that $(x,\partial U^{\pm}(x))$ determines a backward (forward) semi-static curve approaching $L_+$ ($L_-$), then the weak KAM depends on $c\in E_i$ (no longer elementary). As $A_c(\gamma_{i\pm1})>0$ for each $c\in\text{\rm int}E_i$, starting from a point close to the line $L_+$ ($L_-$), the backward (forward) semi-static curve will approach to $L_+$ ($L_-$).
Let $c_{\lambda}=\lambda c_i+(1-\lambda)c_{i+1}$. For each $\lambda\in (0,1)$, by Proposition \[weakpro3\], the strip is divided into two connected parts $D_{\lambda}^+$ and $D_{\lambda}^-$ such that $U_{c_\lambda}^{+}|_{D_{\lambda}^+}=U^{+}_{i,+}$ and $U_{c_\lambda}^{+}|_{D_{\lambda}^-}=U^{+}_{i,-}$. Let $\gamma_{x,{\pm}}^+$ be the forward semi-static curve determined by $U_{i,\pm}^+$, starting from the point $x$. If $x\in D_{\lambda}^+$, the curve $\gamma^+_{x,+}$ is calibrated for $U^+_{i,+}$ which induces $$\begin{aligned}
\label{belteq-1}
A_L(\gamma_{x,{+}}^+)-\langle \gamma_{x,{+}}^+(\infty)-x,c_{\lambda}\rangle&= U^+_{c_{\lambda}}(\gamma^+_{x,+}(\infty))-U^+_{c_{\lambda}}(x)\\
&<A_L(\gamma_{x,{-}}^+)-\langle \gamma_{x,{-}}^+(\infty)-x,c_{\lambda}\rangle,\notag\end{aligned}$$ where both $\gamma^+_{x,+}(\infty)$ and $\gamma^+_{x,-}(\infty)$ exist. Note $\pi_{\infty}\gamma^+_{x,+}(\infty)= \pi_{\infty}\gamma^+_{x,-}(\infty)$ where $\pi_{\infty}:\mathbb{R}^2\to\mathbb{T}^2$ is the standard projection. One can see from Figure \[fig10\] that $$\langle \gamma_{x,{+}}^+(\infty)-\gamma_{x,{-}}^+(\infty),c_{i}-c_{i+1}\rangle>0.$$ It follows that the inequality (\[belteq-1\]) also holds for $\lambda'>\lambda$, which implies that $x\in D_{\lambda'}^+$ also. Clearly, $D_{\lambda}^+$ expands and $D_{\lambda}^-$ shrinks as $\lambda$ increases. As the limit, we see that $D_{1}^-$ and $D_{0}^+$ occupies the whole strip. Therefore, we have:
\[beltpro1\] Assume $E_i\subset\partial\mathbb{F}_0\backslash\partial^*\mathbb{F}_0$ be an edge joined to other two edges at the vertex $c_i$, $c_{i+1}$ respectively. Let $U^{\pm}_j$ be the globally elementary weak KAM for $c_{\lambda}=c_j$ with $j=i,i+1$. Then, for each $c=\lambda c_i+(1-\lambda)c_{i+1}\in E_i$, the weak KAM solution is completely determined by $U^{\pm}_i$ and $U^{\pm}_{i+1}$ in the following sense: there is a partition $\mathbb{T}^2=D_{i,\lambda}^{\pm}\cup D_{i+1,\lambda}^{\pm}$ such that $D^{\pm}_{j,\lambda}$ is connected, $$U_{c_{\lambda}}^{\pm}|_{D_{j,\lambda}^{\pm}}=U_j^{\pm}|_{D_{j,\lambda}^{\pm}},$$ where $D_{i,\lambda}^{\pm}\subset D_{i,\lambda'}^{\pm}$ and $D_{i+1,\lambda}^{\pm}\supset D_{i+1,\lambda'}^{\pm}$ if $\lambda<\lambda'$, and $D_{i+1,0}^{\pm}=D_{i,1}^{\pm}=\mathbb{T}^2$.
In virtue of this proposition, we see that, for all $c\in\partial\mathbb{F}_0\backslash\partial^*\mathbb{F}_0 $, the set of barrier functions $\{U_c-U_c^+\}$ are determined by countably many weak KAM solutions. For each edge $E_i$, there is an open-dense set in $C^{r}$ space such that $\arg\min(U_c-U_c^+)\subsetneq \mathbb{T}^2$ holds for each $c\in E_i$ if the potential $V$ takes value in this set. Therefore, in virtue of the lemma \[beltlem1\], we have
\[beltthm1\] Let $\bar L=\frac 12\langle A^{-1}\dot x,\dot x\rangle-V(x)$. A residual set $\mathfrak{V}\subset C^r(\mathbb{T}^2,\mathbb{R})$ exists such that for each $V\in\mathfrak{V}$ and for each $c\in\partial\mathbb{F}_0$, the Mañé set does not cover the whole configuration space: $\arg\min(U_c^--U_c^+)\subsetneq \mathbb{T}^2$.
Because of this theorem, we have generic hypothesis
([**H5**]{}): The potential is chosen so that the Mañé set does not cover the whole torus $\mathcal{N}(c)\subsetneq \mathbb{T}^2$ for each $c\in\partial\mathbb{F}_0$.
Thickness of the annulus
------------------------
The Hamiltonian under consideration is given by the formula (\[homogenizedeq1\]) which is autonomous. To start with, let us assume assume $\min\alpha_G=0$ and study elementary weak KAM solutions of the Hamilton-Jacobi equation $$\label{belteq0}
\partial_{\tau}u+G(x,\tau,\partial_xu+c)=\epsilon\Delta, \qquad c\in\partial\mathbb{F}_0$$ where $G$ solves the equation $H(x,x_3,y,G)=\tilde E$ and $\tau=-x_3$. Expanding the Hamiltonian into Taylor series of $\epsilon$, replacing $u$ by $\sqrt{\epsilon}u$ and rescaling $\tau$ by $s=\sqrt{\epsilon}\tau$ we obtain the equation $$\label{belteq1}
\frac{\partial u}{\partial s}+\frac 12\Big\langle A\Big(\frac{\partial u}{\partial x}+c\Big),\frac{\partial u}{\partial x}+c\Big\rangle +V(x)+O(\sqrt{\epsilon})=\Delta$$ where $A$ is the Hessian matrix of $h$ in $y$ at $y=0$, $V(x)=Z(x, 0)$ (cf. formula (\[cylindereq3\])). In this equation, only higher order term depends on the time $s$. As the first step to study the weak KAM of this Hamilton-Jacobi equation, we omit the higher order term and let $\Delta=0$. Since the potential $V$ is independent of the time $s$, all elementary weak KAM solutions discussed in the last two subsections solve the equation $$\label{belteq2}
\frac 12\Big\langle A\frac{\partial u}{\partial x}+c, \frac{\partial u}{\partial x} +c\Big\rangle +V(x)=0,$$ and for each $V\in\mathfrak{V}\subset C^r(\mathbb{T}^2,\mathbb{R})$ and each $c\in\partial\mathbb{F}_0$, the weak KAM solutions of this equation define a Mañé set which does not cover the whole torus: $\arg\min(U_c^--U'^+_c)\subsetneq\mathbb{T}^2$. By the upper semi-continuity of the set of semi-static curves, some small $\Delta_0>0$ exists such that for each positive $\Delta\le\Delta_0$ and each $c\in\alpha^{-1}_{\bar G}(\Delta)$ (We use $\alpha_{\bar G}$ to denote the $\alpha$-function determined by the Hamiltonian $\bar G$) the Mañé set does not cover the torus. It implies that $\arg\min(U_c^--U'^+_c) \subsetneq\mathbb{T}^2$ holds if $c\in\alpha_{\bar G}^{-1}(\Delta)$, both $U^-_c$ and $U'^+_c$ are the weak KAM solutions of the equation \[belteq2\] with $\Delta\le\Delta_0$.
For each average action $\Delta>\min\alpha_{\bar G}$, the dynamics on the energy level $\bar G^{-1}(\Delta)$ is similar to twist and area-preserving maps. First of all, the rotation vector of each minimal measure is not zero. Thus, any minimal measure is not supported on fixed points. Secondly, for each class $c\in\alpha_{\bar G}^{-1}(\Delta)$, all $c$-minimal measures share the same rotation rotation direction, otherwise, the Lipschitz graph property will be violated. If the rotation direction is rational, each ergoidc minimal measure is supported on a periodic orbit.
From these properties one derives the following: for each $c\in\alpha^{-1}_{\bar G}(\Delta)$, there exists a circle $\Gamma_c\subset\mathbb{T}^2$ such that each semi-static curve passes through it transversally and $\arg\min(U_c^--U_c^+)\cap\Gamma_c\subsetneq\Gamma_c$. Since the set $\arg\min(U_c^--U_c^+)$ is closed, there exist finitely many intervals $I_{c,i}\subset\Gamma_c$ disjoint to each other such that $(\arg\min(U_c^--U_c^+)\cap\Gamma_c)\subset\cup I_{c,i}$.
As these functions are independent of $s=\sqrt{\epsilon}\tau$, all of these functions can be thought as the weak KAM solutions of the homogenized Hamilton-Jacobi equation $$\frac{\partial u}{\partial s}+\frac 12\Big\langle A\Big(\frac{\partial u}{\partial x}+c\Big),\frac{\partial u}
{\partial x}+c\Big\rangle +V(x)=\Delta$$ if they are thought as the function of the variable $(x,s)$. Here, the cohomology class takes value on the circle: $c\in\alpha^{-1}_{\bar G}(\Delta)$. It follows that, for each class $c\in\alpha^{-1}_{\bar G}(\Delta)$, there exists non-degenerate embedded two-torus $\Gamma_c\times\mathbb{T}\subset\mathbb{T}^3$ and finitely many intervals $I_{c,i}\subset\Gamma_c$ disjoint to each other such that each $c$-semi-static curve passes through the two-torus transversally and $\arg\min(U_c^--U_c^+)\cap(\Gamma_c\times\mathbb{T})\subset\cup I_{c,i}\times\mathbb{T}$. Here the circle $\mathbb{T}$ is for the time $s=\sqrt{\epsilon}\tau$.
Let us return back to the Hamilton-Jacobi equation (\[belteq1\]). Recall the normal form of the Hamiltonian, we see that in the remainder $O(\sqrt{\epsilon})$, one contribution is from $R(x,\sqrt{\epsilon}p,\sqrt{\epsilon}^{-1}s)$ (see the formula (\[cylindereq3\])), other contributions are independent of $s$. Again, by the upper semi-continuity of the set of semi-static curves, we have
\[beltthm2\] Under the hypotheses $($[**H1$\sim$H3, H5**]{}$)$, some positive numbers $\Delta_0>0$ and $\epsilon_0>0$ exist, depending on the potential $V$, such that for each $\Delta<\Delta_0$, each $\epsilon\in (0,\epsilon_0)$ and each $c\in\alpha^{-1}(\Delta)$, all semi-static curves pass through transversally the two-torus $\Gamma_c\times\{s\in\mathbb{T}\}$, $$\label{belteq3}
\arg\min(U_c^--U_c^+)\cap(\Gamma_c\times\{s=\text{\rm const}.\})\subset\bigcup I_{c,i}$$ where $\Gamma_c$ is a circle located in a 2-torus $\{s=\rm constant\}\subset\mathbb{T}^3$, $I_{c,i}\subset\Gamma_c$ are closed intervals, disjoint to each other and independent of the time $s$.
Here, the semi-static curves are in the sense of extended configuration space, i.e. if $\gamma:\mathbb{R}\to M$ is a curve, we also call its graph a curve $\tilde{\gamma}(s)=(\gamma(s),s)\in M\times\mathbb{T}$.
Let us go back to the original scale. By Theorem \[beltthm2\], there exist a annulus-shaped region $$\mathbb{A}=\{c:0<\alpha_G(c)-\min\alpha_G\le\epsilon\Delta_0\}$$ such that the condition (\[belteq3\]) holds for each $c\in\mathbb{A}$. Recall this time-periodic system is deduced from the autonomous system restricted on certain energy level $H^{-1}(\tilde E)$. In virtue of Theorem \[flatthm5\], the counterpart of $\mathbb{A}$ in $H^1(\mathbb{T}^3,\mathbb{R})$ is $$\tilde{\mathbb{A}}=\{\tilde c=(c,c_3)\in\alpha^{-1}_H(\tilde E):0<c_3\le\epsilon\Delta_0\},$$ where we notice that the sphere $\alpha^{-1}_H(\tilde E)$ is located in the upper half space of $\mathbb{R}^3$ and touches the plane $\{c_3=0\}$ where $c\in\mathbb{F}_0$.
In the original coordinates $(\tilde x,\tilde y)=(x,x_3,y,y_3)$, Theorem \[beltthm2\] states such a fact: for each $\tilde c\in\tilde{\mathbb{A}}$, all $\tilde c$-semi static curves pass through the 2-torus $\Gamma_c\times\{x_3\in\mathbb{T}\}$ transversally and all intersection points are restricted in the strips $\cup I_{c,i}\times\{x_3\in\mathbb{T}\}$. However, the condition (\[belteq3\]) dost not guarantee complete “intersection" of the stable set with unstable set, in the sense that the set $\arg\min(U_c^--U_c^+)\cap\{\tau=0\}$ contains some disconnected points. Therefore, we call $\tilde{\mathbb{A}}$ the annulus of incomplete intersection. In this case, we do not expect to construct orbits connecting each Aubry set to any other Aubry set nearby, possible incompleteness may block some direction. However, once non-trivial intersection exists, it opens way to connect some Aubry set nearby. We shall show it in the subsection 7.2.
As $\epsilon\Delta_0\gg 2\epsilon^{1+d}$ provided $\epsilon>0$ is sufficiently small and $d>0$, we obtain the following:
[**Overlap Property**]{}: [*Given any two irreducible $g,g'\in H_1(\mathbb{T}^2,\mathbb{Z})$, there exists a positive number $\epsilon_0= \epsilon_0(V,g,g')>0$ such that the wedge-shaped regions intersects the annulus-shaped region: $\mathbb{W}_g\cap\mathbb{A} \ne\varnothing$ and $\mathbb{W}_{g'}\cap\mathbb{A}\ne\varnothing$*]{} provided $0<\epsilon\le\epsilon_0$.
Local connecting orbits
=======================
To construct orbits connecting some Aubry set to another one nearby, we introduce two types of modified Tonelli Lagrangian, namely, the time-step and the space-step Lagrangian. They satisfy the conditions of positive definiteness, super-linear growth and completeness. The time-step Lagrangian $L:TM\times\mathbb{R}\to\mathbb{R}$ is not periodic in $t$ on the whole $\mathbb{R}$, instead, it is periodic when it is restricted either on $(-\infty,-\delta)$ or on $(\delta,\infty)$, i.e. $L(\cdot,t)=L(\cdot,t+1)$ if $t,t+1\in (-\infty, -\delta)$ or $t,t+1\in (\delta,\infty)$. The second type of Lagrangian is defined on some covering space. Let $\pi :\bar M=\mathbb{R}\times\mathbb{T}^{n-1}\to M$. The space-step Lagrangian $L:T\bar M\times\mathbb{T}\to\mathbb{R}$ is not periodic in one component of spaces coordinates $x_1$. It is periodic in $x_1$ when it is restricted either on $(-\infty,-\delta)$ or on $(\delta,\infty)$, i.e. $L(x_1,\cdot)=L(x_1+1,\cdot)$ if $x_1,x_1+1\in (-\infty, -\delta)$ or $x_1,x_1+1\in (\delta,\infty)$.
The existence of local connecting orbits is established based on some upper semi-continuity of minimal curves for the modified Lagrangian.
Upper semi-continuity of minimal curves
---------------------------------------
[**Time-step Lagrangian**]{}: Let us consider time-step Lagrangian first. A curve $\gamma:\mathbb{R}\to M$ is called minimal if $$\int_{\tau}^{\tau'}L(\gamma(t),\dot\gamma(t),t)dt\le\int_{\tau}^{\tau'}L(\zeta(t),\dot\zeta(t),t)dt$$ holds for any $\tau<\tau'$ and for any absolutely continuous curve $\zeta:[\tau,\tau']\to M$ with $\zeta(\tau)=\gamma(\tau)$ and $\zeta(\tau')=\gamma(\tau')$. Let $\mathscr{G}(L)$ denote the set of minimal curves for $L$. Let $\tilde{\mathcal{G}}(L)=\bigcup_{\gamma\in\mathscr{G}(L)}(\gamma(t),
\dot\gamma(t),t)$, $\mathcal{G}(L)=\pi\tilde{\mathcal{G}}(L)$ where $\pi:TM\times\mathbb{R}\to M\times\mathbb{R}$ is the standard projection.
\[semicontinuitythm1\] The set-valued map $L\to\mathscr{G}(L)$ is upper semi-continuous. Consequently, the map $L\to\tilde{\mathcal{G}}(L)$ is also upper semi-continuous.
Let $K$ be the diameter of the closed manifold $M$, namely, $$K=\max_{x,x'\in M}\ell(x,x')$$ where $\ell(x,x')$ denotes the length of the shortest geodesic connecting $x$ with $x'$. Let $$K_1=\sup_{\stackrel{(x,t)\in M\times\mathbb{R}}{\scriptscriptstyle \|v\|\le K}}L(x,v,t).$$ As $L$ is assumed periodic for $t\le 0$ as well as for $t\ge 1$, $K_1$ is finite.
Let $\gamma$ be a shortest geodesic connecting the point $x$ to the point $x'$. Given time interval $[\tau,\tau']$ with $\tau'-\tau\ge 1$, we re-parameterize the geodesic $\gamma(s)$ by $\gamma'(\ell(x,x')(t-\tau)/(\tau'-\tau))$, then $\gamma'$: $[\tau,\tau']\to M$ is $C^1$-curve such that $\gamma'(\tau)=x$, $\gamma'(\tau')=x'$. Clearly, the action along this curve is not bigger than $K_1(\tau'-\tau)$. Obviously, there is an upper bound uniformly for all minimizing action of $L'$ if it is close to $L$ on $\{\|v\|\le K\}$, still denoted by $$h_{L'}((x,\tau),(x',\tau'))\le K_1(\tau'-\tau).$$ If the Lagrangian has super-linear growth, some positive numbers $C,D>0$ exist such that $L'(x,\dot x,t)\ge C\|\dot x\|-D$ for all $(x,\dot x,t)\in TM\times\mathbb{R}$ and for all $L'$ close to $L$. Therefore, if $\gamma$ is a minimizer, one obtains $$\label{semicontinuityeq1}
\frac{\text{\rm dist}(\gamma(\tau),\gamma(\tau'))}{\tau'-\tau}\le\frac 1{\tau'-\tau}\int_{\tau}^{\tau'}\|d\gamma\|\le\frac{K_1+D}C.$$ As (\[semicontinuityeq1\]) holds for any $\tau'-\tau\ge 1$, it implies that there must be some $t_i\in [\tau+i,\tau+i+1]$ for each $i\in\mathbb{Z}$ such that $\|\dot\gamma(t_i)\|\le C^{-1}(K_1+D)$. As it holds for any $x,x'\in M$, therefore, some positive number $K_2>0$ exists such that $$\phi^s\Big(\Big\{x,v,t_i:\|v\|\le\frac {K_1+D}C\Big\}\Big)\subset\Big\{x,v, t_i+s:\|v\|\le K_2\Big\}$$ holds for all $s\in [0,2]$ and for all relevant $i$. It implies that $\|\dot\gamma(t)\|\le K_2$ holds for all $t\in[\tau,\tau']$.
Let $L_i\in C^r(TM\times\mathbb{R},\mathbb{R})$ be a sequence converging to $L$ in the following sense: there exists some $U_k\supset \{x,v,t:\| v\|\le K_2\}$, as well as a sequence of $\epsilon _i\to 0$ as $i\to\infty$ such that $\|L-L_i\|_{C^2(U_k,\mathbb{R})}\le\epsilon _i$. Let $\gamma _i$: $[\tau,\tau']\to M$ be the minimizer of $L_i$ with $\tau'- \tau\ge 1$, we then have $\|\dot\gamma_i(t)\|\le K_2$ for all $t\in [\tau,\tau']$. The set $\{\gamma_i\}$ is compact in the $C^1([\tau,\tau'],M)$-topology. Indeed, since $\partial ^2L/\partial\dot x^2$ is positive definite one can write the Lagrange equations in the form of $\ddot x=f(x,\dot x,t)$, which implies $\gamma _i$ is bounded in $C^2$-topology.
Let $\gamma $: $[\tau,\tau']\to M$ be one of the accumulation points of this set. Clearly, $\gamma$: $[\tau,\tau']\to M$ is the minimizer of $L$. Let $I_i=[\tau_i,\tau'_i]$ and let $\tau_i\to-\infty$ and $\tau'_i\to\infty$, we obtain a sequence of minimizers of $L_i$, $\gamma_i$: $I_i\to M$. By diagonal extraction argument some subsequence of $\gamma_i$ which converges $C^1$-uniformly on each compact set to a $C^1$-curve $\gamma$: $\mathbb{R}\to M$. Obviously, it is a minimal curve of $L$. This proves the upper semi-continuity.
In application, the set $\mathscr{G}(L)$ seems too big for the construction of connecting orbits. For time-periodic Lagrangian, Mañé set can be a proper subset of $\tilde{\mathcal{G}}(L)$, $\tilde{\mathcal{N}}(L)\subsetneq\tilde{\mathcal{G}}(c)$. It is closely related to the problem whether the Lax-Oleinik semi-group converges or not (cf. [@FM]). For time-step Lagrangian, pseudo connecting curve is introduced to play roles similar to what semi-static curve does.
Each time-step Lagrangian $L$ uniquely determines two time-periodic Lagrangian $L^+$ and $L^-$ such that $L^+|_{(\delta,\infty)}=L|_{(\delta,\infty)}$ and $L^-|_{(-\infty,-\delta)}=L|_{(-\infty,-\delta)}$. Let $-\alpha^{\pm}$ denote the minimal average action of $L^{\pm}$. For $m_0,m_1\in M$ and $T_0,T_1>0$, we define $$h_{L}^{T_0,T_1}(m_0,m_1)=\inf_{\stackrel{\gamma(-T_0)=m_0}{\scriptscriptstyle\gamma(T_1)=m_1}} \int_{-T_0}^{T_1}L(d\gamma(t),t)dt+T_0\alpha^-+T_1\alpha^+.\notag$$ Clearly the limit infimum is bounded $$|h_{L}^{\infty}(m_0,m_1)|=|\liminf_{T_0,T_1\to\infty}h_{L}^{T_0,T_1}(m_0,m_1)|<\infty.$$ Let $\{T_0^i\}_{i\in\mathbb{Z}_+}$ and $\{T_1^i\}_{i\in\mathbb{Z}_+}$ be the sequence of positive integers such that $T_j^i\to\infty$ ($j=0,1$) as $i\to\infty$ and the following limit exists $$\lim_{i\to\infty}h_{L}^{T_0^i,T_1^i}(m_0,m_1)=h_{L}^{\infty}(m_0,m_1).$$ Let $\gamma_i(t,m_0,m_1)$: $[-T^i_0,T^i_1]\to M$ be a minimizer connecting $m_0$ and $m_1$ $$h^{T_0^i,T_1^i}_{L}(m_0,m_1)=\int_{-T_0^i}^{T_1^i}L(d\gamma_i(t),t)dt +T^i_0\alpha^-+T^i_1\alpha^+.$$ From the proof of Theorem \[semicontinuitythm1\] one can see that for any compact interval $[a,b]$ there is some $I\in\mathbb{Z}_+$ such that the set $\{\gamma_i\}_{i\ge I}$ is pre-compact in $C^1([a,b],M)$.
\[semicontinuitylem1\] Let $\gamma$: $\mathbb{R}\to M$ be an accumulation point of $\{\gamma_i\}$. Then for $s,\tau\ge\delta$ $$\begin{aligned}
\label{semicontinuityeq2}
A_{L}(\gamma|[-s,\tau])=&\inf_{\stackrel{s_1-s\in\mathbb{Z}, \tau_1-\tau\in\mathbb{Z}} {\stackrel{s_1,\tau_1\ge\delta}{\stackrel{\gamma^*(-s_1)=\gamma(-s)}{\scriptscriptstyle \gamma^*(\tau_1) =\gamma(\tau)}}}}\int_{-s_1}^{\tau_1}L(d\gamma^*(t),t)dt \\
&+(s_1-s)\alpha^-+(\tau_1-\tau)\alpha^+.\notag\end{aligned}$$
: To prove the lemma let us suppose the contrary. Thus there would exist $\Delta>0$, $s_1,\tau_1\ge\delta$, $s_1-s\in\mathbb{Z}$, $\tau_1-\tau\in\mathbb{Z}$ and a curve $\gamma^*$: $[s_1,\tau_1]\to M$ with $\gamma^*(-s_1)=\gamma(-s)$, $\gamma^*(\tau)=\gamma(\tau_1)$ such that $$A_{L}(\gamma|[-s,\tau])\ge\int_{-s_1}^{\tau_1}L(d\gamma^*(t),t)dt+(s_1-s)\alpha^-+(\tau_1-\tau)\alpha^+ +\Delta.$$ Let $\epsilon=\frac 13\Delta$. By the definition of limit infimum there exist $T^{i_0}_0>s$ and $T^{i_0}_1>\tau$ such that $$h_{L}^{T_0,T_1}(m_0,m_1)>h_{L}^{\infty}(m_0,m_1)-\epsilon, \qquad \forall \ \ T_0\ge T_0^{i_0}, \ T_1\ge T_1^{i_0},$$ and there exist subsequences $T_j^{i_k}$ ($j=0,1$; $k=0,1,2,\cdots$) such that $T_0^{i_k}-T_0^{i_0}\ge |s-s_1|$, $T_1^{i_k}-T_1^{i_0}\ge |\tau-\tau_1|$ and $$|h_{L}^{T_0^{i_k},T_1^{i_k}}(m_0,m_1)-h_{L}^{\infty} (m_0,m_1)|<\epsilon$$ holds for each $k>0$. Let $\gamma_{i_k}$ be the minimizer of $h_{L}^{T_0^{i_k},T_1^{i_k}}(m_0,m_1)$. By taking a subsequence further one can assume $\gamma_{i_k}\to\gamma$. In this case, for sufficiently large $k$, we are able to construct a curve $\gamma_{i_k}^*$: $[s_1,\tau_1]\to M$ which has the same endpoints as $\gamma_{i_k}$: $\gamma_{i_k}^*(-s_1)=\gamma_{i_k}(-s)$, $\gamma_{i_k}^*(\tau_1) =\gamma_{i_k}(\tau)$ and satisfies the following $$A_{L}(\gamma_{i_k}|[-s,\tau])\ge\int_{-s_1}^{\tau_1}L(d\gamma_{i_k}^*(t),t)dt +(s_1-s)\alpha^-+(\tau_1-\tau)\alpha^++\frac 23\Delta.$$ Extending $\gamma_{i_k}^*$ from $[s_1,\tau_1]$ to the $[-T_0^{i_k}-(s_1-s), T_1^{i_k}+(\tau_1-\tau)]$ by $$\gamma_{i_k}^*=\begin{cases}\gamma_{i_k}(t+s_1-s),\hskip 0.95 true cm t\le -s_1,\\
\gamma_{i_k}^*(t),\hskip 2.35 true cm -s_1\le t\le\tau_1,\\
\gamma_{i_k}(t-\tau_1+\tau),\hskip 0.9 true cm t\ge\tau_1,
\end{cases}$$ and defining $T'_0=T_0^{i_k}+(s_1-s)$, $T'_1=T_1^{i_k}+(\tau_1-\tau)$ we find that $$\begin{aligned}
h_{L}^{T'_0,T'_1}(m_0,m_1)\le & A_{L}(\gamma_{i_k}^*|[-T'_0,T'_1])+T'_0\alpha^-+T'_1\alpha^+\\
\le &A_{L}(\gamma_{i_k}|[-T_0^{i_k},T_1^{i_k}])+T^{i_k}_0\alpha^-+T_1^{i_k}\alpha^+-\frac 23\Delta\\
\le &h_{L}^{\infty}(m_0,m_1)-\epsilon.\end{aligned}$$ But this contradicts the definition of the limit infimum as $T'_0\ge T_0$ and $T'_1\ge T_1$.
We define so-called pseudo connecting curve set $$\mathscr{C}(L)=\{\gamma\in\mathscr{G}(L):\ (\ref{semicontinuityeq2})\ \text{\rm hold}\ \}.$$ In application, we usually choose time-step Lagrangian so that the Aubry set of $L^-$ is different from that of $L^+$. Clearly, for $\gamma\in\mathscr{C}(L)$, the orbit $(\gamma(t),\dot\gamma(t))$ approaches the Aubry set $\tilde{\mathcal{A}}(L^-)$ as $t\to -\infty$ and approaches $\tilde{\mathcal{A}}(L^+)$ as $t\to \infty$. That is why we call it pseudo connecting curve. Let $$\tilde{\mathcal{C}}(L)=\bigcup_{\gamma\in\mathscr{C}(L)}(\gamma(t),\dot\gamma(t),t),\qquad \mathcal{C}(L)=\bigcup_{\gamma\in\mathscr{C}(L)}(\gamma(t),t).$$ Clearly, if $L$ is periodic in $t$, then $\tilde{\mathcal{C}}(L)=\tilde{\mathcal{N}}(L)$ and $\mathcal{C}(L)=\mathcal{N}(L)$.
\[semicontinuitythm2\] The map $L\to\mathscr{C}(L)$ is upper semi-continuous. As the special case, the map $c\to\tilde{\mathcal{N}}(c)$ as well as the map $c\to\mathcal{N}(c)$ is upper semi-continuous.
: Let $L_i\to L$ be a sequence of time-step Lagrangian, let $\gamma_i\in\mathscr{C}(L_i)$ and let $\gamma$ be an accumulation point of the set $\{\gamma_i\in\mathscr{C}(L_i)\}_{i\in\mathbb{Z}^+}$. We claim that $\gamma\in\mathscr{C}(L)$. If $\gamma\notin\mathscr{C}(L)$, there would be two point $\gamma(s)$,$\gamma(\tau)\in M$ connected by another curve $\gamma^*$: $[-s-n_1,\tau+n_2]\to M$ and $\Delta>0$ such that $$A_{L}(\gamma^*)<A_{L}(\gamma|[-s,\tau])-n_1\alpha^--n_2\alpha^++\Delta$$ where $s,s+n_1\ge\delta$, $\tau,\tau+n_2\ge\delta$. Since $\gamma$ is an accumulation point of $\gamma_i$, for any small $\epsilon>0$, there would be sufficiently large $i$ such that $\|\gamma-\gamma_i\|_{C^1[s,t]}<\epsilon$, and above inequality also holds for $A_{L_i}(\gamma_i|_{[s,\tau]})$. It follows that $\gamma_i\notin\mathscr{C}(L_i)$, contradicting to the assumption.
[**Space-step Lagrangian**]{}: Let $M=\mathbb{T}^n$ and $\pi :\bar M=\mathbb{R}\times\mathbb{T}^{n-1}\to M$, where $\mathbb{R}$ is for the coordinate $x_1$. The space-step Lagrangian $L$ is introduced to handle the problem of incomplete intersection. A space-step Lagrangian also uniquely determines two Lagrangian $L^-$ and $L^+$: $TM$ such that $L^-(x_1,\cdot)|_{(-\infty,-\delta)}=L(x_1,\cdot)|_{(-\infty,-\delta)}$ and $L^+(x_1,\cdot)|_{(\delta,\infty)}=L(x_1,\cdot)|_{(\delta,\infty)}$ if we treat $L^{\pm}$ as its natural lift to $T\bar M$. Let $\mu^{\pm}$ denote minimal measure of $L^{\pm}$ with $0$-cohomology class, $\omega(\mu^{\pm})=(\omega_1(\mu^{\pm}),\cdots,\omega_n(\mu^{\pm}))$ denote the rotation vector. We assume some conditions on the Lagrangian:
1, $\omega_1(\mu^{\pm})>0$ for each ergodic minimal measure $\mu^{\pm}$;
2, $\min\beta_{L^-}=\min\beta_{L^+}$, without losing of generality, it equals zero;
3, $|L^--L^+|\le\frac 12\min_{\omega_1=0}\{ \beta_{L^-}(\omega'),\beta_{L^+}(\omega')\}$.
It is shown in [@Lx] that some coordinates exists such that the first condition holds provided $\alpha(0)>\min\alpha$. As the minimal average action of $L^{\pm}$ is achieved on $\text{\rm supp}\mu^{\pm}$ with $\omega_1(\mu^{\pm})\neq 0$, one can see that $\min_{\omega_1(\nu)=0}\int L^{\pm}d\nu>\min\int L^{\pm}d\nu$, so the third condition makes sense. To introduce minimal curve for space-Lagrangian, we define $$h_{L}^{T}(\bar m_0,\bar m_1)=\inf_{\stackrel{\bar\gamma(-T)=\bar m_0} {\scriptscriptstyle \bar\gamma(T)=\bar m_1}}\int_{-T}^{T}L(\bar\gamma(t),\dot{\bar\gamma}(t))dt, \qquad \forall\ \bar m_0,\bar m_1\in\bar M.$$
\[semicontinuitylem2\] If the rotation vector of each ergodic minimal measure has positive first component $\omega_1(\mu^{\pm})>0$, $\bar m_0\neq\bar m_1$, then $$\lim_{T\to 0}h_{L}^{T}(\bar m_0,\bar m_1)=\infty \ \ \ \
and\ \ \ \ \lim_{T\to\infty}h_{L}^{T}(\bar m_0,\bar m_1)=\infty.$$
Let $\bar\gamma^{T}_{L}$: $[-T,T]\to\bar M$ be the minimizer of $h_{L}^{T}(\bar m_0,\bar m_1)$. Let $m_0=\pi \bar m_0$, $m_1=\pi \bar m_1$, $\zeta$: $[0,1]\to M$ be a smooth curve connecting $m_1$ to $m_0$, $\dot\zeta(0)=\dot{\bar\gamma}^{T}_{L}(T)$ and $\dot\zeta(1)=\dot{\bar\gamma}^{T}_{L}(-T)$. The action of $L^+$ along $\zeta$ is clearly bounded, thus for any $\epsilon>0$, one has $A_{L^+}(\zeta)\le 2T\epsilon$ provided $T$ is sufficiently large. The curve $\xi=\zeta\ast\pi \bar\gamma^{T}_{L}$ determines a holonomic probability measure $\nu^{T}_{L}\in\mathfrak{H}$ such that $$\int fd\nu^{T}_{L}=\frac 1{2T+1}\int_{-T}^{T+1}f(\xi(t),\dot\xi(t))dt\qquad \forall\ f\in C(TM,\mathbb{R}).$$ Since $|\bar\gamma_L^T(T)-\bar\gamma_L^T(-T)|$ is bounded for any $T>0$, one has $\omega_1(\nu^{T}_{L})\to 0$ as $T\to\infty$. By using the third condition, we obtain $$\begin{aligned}
\frac 1{2T}h_{L}^{T}(\bar m_0,\bar m_1)=&\frac{2T+1}{2T}\int L^+d\nu^{T}_{L}-\frac 1{2T}
\int_0^1 L^+(\zeta(t),\dot\zeta(t))dt \\
&+\frac 1{2T}\int_{-T}^{T}(L-L^+)(\bar\gamma^{T}_{L}(t), \dot{\bar\gamma}^{T}_{L}(t))dt\\
\ge&\int L^+d\nu^{T}_{L}-\frac 12\min_{\omega_1=0}\beta_{L^+}(\omega)-\epsilon>0.\end{aligned}$$ It implies that $\lim_{T\to\infty}h_{L}^{T}(\bar m_0,\bar m_1)=\infty$. The case for $T\to 0$ is a consequence of the super-linear growth of $L$ in $\dot x$.
As an intermediate step in introducing pseudo-connecting curve, we define a set of minimal curve $\mathscr{G}(L)$.
A curve $\bar\gamma:\mathbb{R}\to\bar M$ is in $\mathscr{G}(L)$ if $$A_L(\bar\gamma|_{[-T,T]})=\inf_{T'\in\mathbb{R}_+} h_{L}^{T'}(\bar\gamma(-T),\bar\gamma(T)).$$
We claim that $\mathscr{G}(L)\neq\varnothing$. Indeed, denote by $\bar\gamma_{L}(\cdot,\bar m_0,\bar m_1):[-T,T]\to M$ the minimizer such that $\bar\gamma_{L}(-T)=\bar m_0$, $\bar\gamma_{L}(T)=\bar m_1$ and $$A(\bar\gamma_{L})=\int_{-T}^{T}L(\bar\gamma_{L}(t),\dot{\bar\gamma}_{L}(t))dt
=\inf_{T'\in\mathbb{R}_+} h_{L}^{T'}(\bar m,\bar m').$$ Because of Lemma \[semicontinuitylem2\], this infimum is attained for finite $T>0$ if $\bar m_0$ and $\bar m_1$ are two different points in $\bar M$. The super-linear growth of $L$ in $\dot x$ guarantees that $T\to\infty$ as $-\bar m_{01},\bar m_{11}\to\infty$, where $\bar m_{i1}$ denotes the first coordinate of $\bar m_i$. Given an interval $[-T,T]$, for sufficiently large $-\bar m_{01},\bar m_{11}$, the set $\{\bar\gamma_{L}(\cdot,\bar m_0,\bar m_1)|_{[-T,T]}\}$ is pre-compact in $C^1([-T,T],\bar M)$. Let $T\to\infty$. By diagonal extraction argument, there is a subsequence of $\{\bar\gamma_{L}(\cdot,\bar m_0,\bar m_1)\}$ which converges $C^1$-uniformly on any compact set to a $C^1$-curve $\bar\gamma$: $\mathbb{R}\to\bar M$. Obviously, $\bar\gamma\in\mathscr{G}(L)$, and
\[semicontinuitypro1\] Some number $K>0$ exists so that $|h_{L}^{T}(\bar\gamma(-T), \bar\gamma(T))|\le K$ holds for any curve $\bar\gamma\in \mathscr{G}(L)$ and any $T>0$.
By the assumption, one has $\alpha_{L^{\pm}}(0)=\min\beta_{L^{\pm}}=0$. So, some $K'>0$ exists such that $|A(\gamma|_I)|\le K'$ holds for any interval $I\subset\mathbb{R}_+(\mathbb{R}_-)$ provided it is a forward (backward) semi-static curves for $L^{+}$ ($L^-$). Also, some $K''>0$ exists such that $$-K''\le\max_{\bar x,\bar x'\in \{x\in\bar M:|x_1|\le 1\}}\inf_{T\ge 0}h^T_L(\bar x,\bar x')\le K''.$$ We claim that $K\le 2K'+K''$.
If there exists some $\bar\gamma\in\mathscr{G}(L)$ and some $T>0$ such that $h_{L}^{T}(\bar\gamma(-T), \bar\gamma(T))>2K'+K''$, we join $\bar\gamma(-T)$ to $\bar\gamma(T)$ by another curve $\xi=\bar\gamma_-\ast\zeta\ast\bar\gamma_+$ where $\bar\gamma_-$ is a lift of backward semi-static curve $\gamma_-$ for $L_-$ such that $\bar\gamma(-T)=\bar\gamma_-(0)$, denote by $\bar x_-$ the intersection point of this curve with the section $\{\bar x\in\bar M:\bar x_1=-1\}$, $\bar\gamma_+$ is a lift of forward semi-static curve $\gamma_+$ for $L_+$ such that $\bar\gamma(T)=\bar\gamma_+(0)$, denote by $\bar x_+$ the intersection point of this curve with the section $\{\bar x\in\bar M:\bar x_1=1\}$, $\zeta$ is a minimal curve of $L$ that connects the point $\bar x_-$ to $\bar x_+$. Obviously, one has $A_L(\xi)\le 2K'+K''<h_{L}^{T}(\bar\gamma(-T), \bar\gamma(T))$, but it contradicts the definition of $\mathscr{G}(L)$.
Each $k\in\mathbb{Z}$ defines a Deck transformation ${\bf k}:\bar M\to\bar M$: ${\bf k}x=(x_1+k,x_2,\cdots,x_n)$. Let $\bar M^-_{\delta}=\{x\in\bar M:x_1<-\delta\}$, $\bar M^+_{\delta}=\{x\in\bar M:x_1>\delta\}$.
\[semicontinuitydef2\] A curve $\bar\gamma\in\mathscr{G}(L)$ is called pseudo connecting curve if the following holds $$A_L(\bar\gamma|_{[-T,T]})=\inf_{\stackrel{\stackrel{T'\in\mathbb{R}_+} {\scriptscriptstyle {\bf k}^-\bar\gamma(-T)\in \bar M^-_{\delta}}}{\scriptscriptstyle {\bf k}^+ \bar\gamma(T)\in \bar M^+_{\delta}}} h_{L}^{T'}({\bf k}^- \bar\gamma(-T),{\bf k}^+\bar\gamma (T))$$ for each $\bar\gamma(T)\in \bar M^-_{\delta}$ and $\bar\gamma(T)\in\bar M^+_{\delta}$. Denote by $\mathscr{C}(L)$ the set of pseudo connecting curves.
\[semicontinuitylem3\] The set $\mathscr{C}(L)$ is non-empty.
Let us start with a curve $\bar\gamma\in\mathscr{G}(L)$. Given $\Delta>0$, if some interval $[t^-_i,t^+_i]$ exists such that ${\bf k}^-_i \bar\gamma(t^-_i)$ can be connected to ${\bf k}^+_i\bar\gamma(t_i^+)$ by another curve $\zeta_i$ with smaller action $$A_L(\gamma|_{[t_i^-,t_i^+]})-A_L(\zeta_i)\ge\Delta>0,$$ then one obtain a curve $\bar\gamma_i={\bf k}^-_i \bar\gamma|_{(-\infty,t^-_i]}\ast\zeta\ast{\bf k}^-_i \bar\gamma|_{[t^+_i,\infty)}$ by one step of such surgery.
Given any $\Delta>0$, we claim that there are finitely many intervals $[t_i^-,t_i^+]$ with $t_i^+\le t_{i+1}^-$ such that ${\bf k}^-_i \bar\gamma(t^-_i)$ can be connected to ${\bf k}^+_i\bar\gamma(t_i^+)$ by another curve $\zeta_i$ with the action $\Delta$ smaller than the original one. Let us assume the contrary. Then, for any positive integer $m$, some large $T>0$ exists such that $[-T,T]\supset \cup_{i=1}^m [t_i^-,t_i^+]$. We can choose arbitrarily many of such intervals such that either $t_1^->\delta$ or $t^+_m<-\delta$. In the first case, let $\bar x^-=\bar\gamma(-T)$ and $\bar x^+=\Pi_{\ell=1}^m{\bf k}^-_{\ell}{\bf k}^+_{\ell}\bar\gamma(T)$. By assumption, these two points can be connected by a curve $\zeta$ along which the action $A_L(\zeta)\le K-m\Delta$ as it follows from Proposition \[semicontinuitypro1\] that $A_L(\bar\gamma|_{[-T,T]})\le K$. Since $m$ can be arbitrarily large, it implies the existence of a curve along which the action of $L$ approaches to minus infinity, it also contradicts Proposition \[semicontinuitypro1\].
Given a curve $\bar\gamma\in\mathscr{G}(L)$ and any small $\epsilon_i>0$, by finitely many steps of such surgery, we obtain a curve $\bar\gamma_i:\mathbb{R}\to\bar M$ with following properties:
1, for each small $\epsilon_i>0$, some large $T_i$ exists such that $\bar\gamma(-T_i)\in\bar M^-_{\delta}$, $\bar\gamma(T)\in\bar M^+_{\delta}$ and $$A_L(\bar\gamma_i|_{[-T_i,T_i]})\le\inf_{\stackrel{\stackrel{T'\in\mathbb{R}_+} {\scriptscriptstyle {\bf k}^-\bar\gamma(-T)\in \bar M^-_{\delta}}}{\scriptscriptstyle {\bf k}^+ \bar\gamma(T)\in \bar M^+_{\delta}}}h_{L}^{T'}({\bf k}^-\bar\gamma_i(-T),{\bf k}^+\bar\gamma_i(T)) +\epsilon_i.$$
2, $\bar\gamma_i$ is smooth everywhere except for two points which fall beyond the region $\{x\in\bar M: |x_1|\le\Theta_i\}$, and $\Theta_i\to\infty$ as $\epsilon_i\to 0$.
Let $T'_i>0$ such that $\bar\gamma_{i1}(\pm T'_i)=\pm\Theta_i$. Because of Lemma \[semicontinuitylem2\], we see that $T'_i\to\infty$ as $\Theta_i\to\infty$. In virtue of the argument before, for any large $T$ $\exists$ $i_0>0$ such that the set $\{\bar\gamma_i|_{[-T,T]}:i\ge i_0\}$ is pre-compact in $C^1([-T,T],\bar M)$. Let $T\to\infty$, by diagonal extraction argument, there is a subsequence of $\{\bar\gamma_i\}$ which converges $C^1$-uniformly on each compact set to a $C^1$-curve $\bar\gamma$: $\mathbb{R}\to\bar M$. Obviously, $\bar\gamma\in\mathscr{C}(L)$.
\[semicontinuitythm3\] The map $L\to\mathscr{C}(L)$ is upper semi-continuous.
Let $\bar\gamma_i\in\mathscr{C}(L_i)$, $L_i\to L$. If $\{\bar\gamma_i\}$ converges $C^1$-uniformly on each compact set to a $C^1$-curve $\bar\gamma$, it is obvious that $\bar\gamma\in\mathscr{C}(L)$.
It is an immediate consequence of Definition \[semicontinuitydef2\] that
\[semicontinuitypro\] If the space-step Lagrangian $L$ is periodic in $x_1$, then a curve $\bar\gamma\in\mathscr{C}(L)$ if and only if its projection $\gamma=\pi \bar\gamma$: $\mathbb{R}\to M$ is semi-static.
Similar to the definition for time-step Lagrangian, we define $$\tilde{\mathcal{C}}(L)=\bigcup_{\bar\gamma\in\mathscr{C}(L)}(\bar\gamma(t),\dot{\bar\gamma}(t)), \qquad \mathcal{C}(L)=\bigcup_{\bar\gamma\in\mathscr{C}(L)}\bar\gamma(t).$$ If $L$ is periodic in $x_1$, then $\pi \tilde{\mathcal{C}}(L)=\tilde{\mathcal{N}}(L)$ and $\pi \mathcal{C}(L)=\mathcal{N}(L)$.
Local connecting orbits of type-$c$
-----------------------------------
An orbit $d\gamma$ (A curve $\gamma$) is said connecting one Aubry set $\tilde{\mathcal{A}}(c)$ to another one $\tilde{\mathcal{A}}(c')$ if the $\alpha$-limit set of the orbit $d\gamma$ is contained in $\tilde{\mathcal{A}}(c)$ and the $\omega$-limit set is contained in $\tilde{\mathcal{A}}(c')$. It is called local connecting orbit if these two classes are close to each other. It is called global when the two classes are far away from each other. In this subsection, we show how to construct local connecting orbits of type-$c$ by using so-called $c$-equivalence. This type of connecting orbits are found in the annulus of incomplete intersection and plays key role in establishing transition chain crossing strong double resonance.
For this purpose, we use the new version of $c$-equivalence introduced in [@LC]. The concept of $c$-equivalence was introduced in [@Ma2] for the first time, but it does not apply in interesting problems of autonomous system. The new version is defined not on the whole $M$, but on a non-degenerate embedded $(n-1)$-dimensional torus. We call $\Sigma_c$ non-degenerately embedded ($n-1$)-dimensional torus by assuming a smooth injection $\varphi$: $\mathbb{T}^{n-1}\to\mathbb{T}^n$ such that $\Sigma_c$ is the image of $\varphi$, and the induced map $\varphi_*$: $H_1(\mathbb{T}^{n-1}, \mathbb{Z})\to H_1(\mathbb{T}^{n},\mathbb{Z})$ is an injection.
Let $\mathfrak{C}\subset H^1(\mathbb{T}^n,\mathbb{R})$ be a connected set where we are going to define $c$-equivalence. For each class $c\in \mathfrak{C} $, we assume that there exists a non-degenerate embedded $(n-1)$-dimensional torus $\Sigma_c\subset\mathbb{T}^n$ such that each $c$-semi static curve $\gamma$ transversally intersects $\Sigma_c$. Let $$\mathbb{V}_{c}=\bigcap_U\{i_{U*}H_1(U,\mathbb{R}): U\, \text{\rm is a neighborhood of}\, \mathcal {N}(c) \cap\Sigma_c\},$$ here $i_U$: $U\to M$ denotes inclusion map. $\mathbb{V}_{c}^{\bot}$ is defined to be the annihilator of $\mathbb{V}_{c}$, i.e. if $c'\in H^1(\mathbb{T}^n,\mathbb{R})$, then $c'\in \mathbb{V}_{c}^{\bot}$ if and only if $\langle c',h \rangle =0$ for all $h\in \mathbb{V}_c$. Clearly, $$\mathbb{V}_{c}^{\bot}=\bigcup_U\{\text{\rm ker}\, i_{U}^*: U\, \text{\rm is a neighborhood of}\, \mathcal {N}(c) \cap\Sigma_c\}.$$ Note that there exists a neighborhood $U$ of $\mathcal {N}(c)\cap\Sigma_c$ such that $\mathbb{V}_c=i_{U*}H_1(U,\mathbb{R})$ and $\mathbb{V}_{c}^{\bot}=\text{\rm ker}i^*_U$ (see [@Ma2]).
We say that $c,c'\in H^1(M,\mathbb{R})$ are $c$-equivalent if there exists a continuous curve $\Gamma$: $[0,1]\to \mathfrak{C}$ such that $\Gamma(0)=c$, $\Gamma(1)=c'$, $\alpha(\Gamma(s))$ keeps constant for all $s\in [0,1]$, and for each $s_0\in [0,1]$ there exists $\delta>0$ such that $\Gamma(s)-\Gamma(s_0)\in \mathbb{V}_{{\Gamma}(s_0)}^{\bot}$ whenever $s\in [0,1]$ and $|s-s_0|<\delta$.
Let $\{e_i\}_{1\le i\le n-1}$ be the standard basis of $H_1(\mathbb{T}^{n-1}, \mathbb{Z})$, one obtains $n$-dimensional vectors $\{g_{i+1}=\varphi_*(e_i)\in H_1(\mathbb{T}^{n}, \mathbb{Z})\}_{1\le i\le n-1}$. Because $\varphi$ is injection, there is a vector $g_1\in\mathbb{Z}^n$ such that the $n\times n$ matrix $G=(g_1,g_2,\cdots, g_n)$ is uni-module, i.e. $\text{\rm det}G=\pm 1$. In new coordinates system $x\to G^{-1}x$, the Lagrangian $\tilde L(\dot x,x)=L(G\dot x,Gx)$ is also $2\pi$-periodic in $x$. In new coordinates, let $\bar M=\mathbb{R}\times\mathbb{T}^{n-1}= \{x_1\in\mathbb{R},(x_2,\cdots,x_n)\in \mathbb{T}^{n-1}\}$ be a covering space of $\mathbb{T}^n$, $\pi:\bar M\to M=\mathbb{T}^n$. The lift of $\Sigma_c$, $\pi^{-1}(\Sigma_c)$ has infinitely many compact components $\{\Sigma_c^i\}_{i\in\mathbb{Z}}$. If $\varphi$ is linear, $\Sigma_c^i=\{x_1=2i\pi\}$. For the section $\Sigma=\{x_1=0\mod 1\}$ we have $\pi^{-1}(\Sigma)=\cup_{k\in\mathbb{Z}}\{x_1=k\}$ while for $\Sigma=\{x_1=x_2\}$, the lift $\pi^{-1}(\Sigma)$ consists of only one connected component.
\[typecthm1\] Assume the cohomology class $c^*$ is $c$-equivalent to the class $c'$ through the path $\Gamma$: $[0,1]\to H^1(\mathbb{T}^n,\mathbb{R})$. For each $s\in [0,1]$, the following are assumed:
1, there exists a coordinate systems $G_s^{-1}x$ where the first component of rotation vector is positive, $\omega_1(\mu_{\Gamma(s)})>0$ for each ergodic $\Gamma(s)$-minimal measure $\mu_{\Gamma(s)}$;
2, for the covering space $\bar M_s=\mathbb{R}\times\mathbb{T}^{n-1}$ in the coordinate system the lift of non-degenerately embedded codimension-one torus $\Sigma_{\Gamma(s)}$ has infinitely many connected and compact components, each of which is also a codimension-one torus.
Then there exist some classes $c^*=c_0, c_1,\cdots,c_k=c'$ on this path, closed 1-forms $\eta_i$ and $\bar\mu_i$ on $M$ with $[\eta_i]=c_i$ and $[\bar\mu_i]=c_{i+1}-c_i$, and smooth functions $\varrho_i$ on $\bar M$ for $i=0,1,\cdots,k-1$, such that the pseudo connecting curve set $\mathscr{C}(L_i)$ for the space-step Lagrangian $$L_i=L-\eta_i-\varrho\bar\mu_i$$ possesses the properties:
, each curve $\bar\gamma\in\mathscr{C}(L_i)$ determines an orbit $(\gamma,\dot{\gamma})$ of $\phi_L^t$;
, the orbit $(\gamma,\dot{\gamma})$ connects $\tilde{\mathcal{A}}(c_{i})$ to $\tilde{\mathcal{A}}(c_{i+1})$, i.e., the $\alpha$-limit set $\alpha(d\gamma)\subseteq\tilde{\mathcal{A}} (c_{i})$ and $\omega$-limit set $\omega(d\gamma)\subseteq\tilde{\mathcal{A}}(c_{i+1})$.
By the definition of $c$-equivalence, there exists a path $\Gamma$: $[0,1]\to H^1(M,\mathbb{R})$ with $\Gamma(0)=c^*$, $\Gamma(1)=c'$ such that for each $c=\Gamma(s)$ ($s\in [0,1]$) on the path, there exists $\epsilon>0$ such that $\Gamma(s')-c\in \mathbb{V}_{{\Gamma}(s)}^{\bot}$ whenever $s'\in [0,1]$ and $|s-s'|<\epsilon$. Thus, there exist a non-degenerately embedded ($n-1$)-dimensional torus $\Sigma_c$, a closed form $\bar\mu_{c}$ and a neighborhood $U$ of $\mathcal {N}(c)\cap \Sigma_{c}$ such that $[\bar\mu_{c}]=\Gamma(s')-c$ and $\text{\rm supp}\bar\mu_{c}\cap
U=\varnothing$.
In the new coordinates $x\to G^{-1}_{c}x$ on the torus as above, the codimension one hypersurface $\Sigma_c^0$ separates $\bar M$ into two parts, the upper part $\bar M^+$ and the lower part $\bar
M^-$. $\bar M^{\pm}$ extends to where the first coordinate $x_1\to\pm\infty$. Let $\Sigma_c^0+\delta$ denotes the $\delta$-neighborhood of $\Sigma_c^0$ in $\bar M$, we introduce a smooth function $\varrho\in C^r(\bar M,[0,1])$ such that $\varrho=0$ if $x\in\bar M^- \backslash (\Sigma_c^0+\delta)$, $\varrho=1$ if $x\in\bar M^+\backslash(\Sigma_c^0+\delta)$. Let $\eta$ and $\bar\mu$ are closed 1-forms on $M$ such that $[\eta]=c$ and $[\eta+\bar\mu]=c'$. These forms have natural lift on $\bar M$, with the same notation.
A sufficiently small $\delta>0$ can be chosen so that $$(\Sigma_{c}^0+\delta)\cap(\mathcal {C}(L+\eta)+2\delta)\subset U,$$ It follows from the upper semi-continuity of $\mathcal {C}(L)$ w.r.t. $L$, we find $$\label{typeceq1}
(\Sigma_{c}^0+\delta)\cap(\mathcal{C}(L+\eta+\varrho\bar\mu)+\delta)\subset U,$$ if $\varrho\bar\mu$ is $C^0$-sufficiently small. As $\bar\mu$ is carefully chosen so that its support is disjoint from $U$, each curve $\bar\gamma\in\mathscr{C}(L+\eta+\varrho\bar\mu)$ is clearly a solution of the Euler-Lagrange equation determined by $L$, the term $\varrho\bar\mu$ has no contribution to the equation along $\bar\gamma$. In other words, each curve in $\mathscr{C}(L+\eta+\varrho\bar\mu)$ generates an orbit $d\gamma$ of $\phi_L^t$: $\mathbb{R}\to TM$.
The definition of $\mathscr{C}$ tells us that for each curve $\bar\gamma\in\mathscr{C}$, $\gamma|_{(-\infty,t_0]}$ is backward $\Gamma(s)$-semi static once $\bar\gamma|_{(-\infty,t_0]}$ falls entirely into $\bar M^-\backslash (\Sigma_c^0+\delta)$, $\gamma|_{[t_1,\infty)}$ is forward $\Gamma(s')$-semi static once $\bar\gamma|_{[t_1,\infty)}$ falls entirely into $\bar M^+\backslash (\Sigma_c^0+\delta)$. Therefore, $(\gamma(t),\dot\gamma(t))\to\tilde{\mathcal {A}}(\Gamma(s))$ as $t\to -\infty$ and $(\gamma(t),\dot\gamma(t))\to\tilde{\mathcal {A}}(\Gamma(s'))$ as $t\to\infty$.
Because of the compactness of $[0,1]$, there are finitely many numbers $s_0,\cdots,s_k\in [0,1]$ such that above argument applies if $s$ and $s'$ are replaced respectively by $s_i$ and $s_{i+1}$. Set $c_i=\Gamma(s_i)$.
\[typeccor1\] Let $c_i$, $\eta_i$, $\bar\mu_i$ and $\varrho_i$ be evaluated as in Theorem \[typecthm1\]. Let $U_i$ be an open neighborhood of $\mathcal {N}(c_i)\cap \Sigma_{c_i}^0$ such that $U_i\cap\text{\rm supp}\bar\mu_i=\varnothing$. Then, there exist large $K_i>0$, $T_i>0$ and small $\delta>0$ such that for each $\bar m,\bar m'\in\bar M$, with $-K_i\le\bar m_1\le -K_i+2\pi$, $K_i-2\pi \le\bar m'_1\le K_i$, the quantity $h_{\eta_i,\mu_i}^{T}(\bar m,\bar m')$ reaches its minimum at some $T<T_i$ and the corresponding minimizer $\bar\gamma_{i}(t,\bar m,\bar m')$ satisfies the condition $$\label{typeceq2}
\text{\rm Image}(\bar\gamma_{i})\cap(\Sigma_{c_i}^0+\delta)\subset U_i.$$
There is some flexibility to choose the coordinate system and the non-degenerately embedded codimension one torus. Let $\pi_s$: $\bar M_s\to M=\mathbb{T}^n$ be a covering space such that $\bar M_s=\mathbb{R} \times\mathbb{T}^{n-1}$ in the coordinate system $G_s^{-1}x$.
\[typecdef1\] For $s\in [0,1]$, the non-degenerately embedded codimension one torus $\Sigma_s$ is called admissible for the coordinate system $G_s^{-1}x$ if the lift of $\Sigma_s$ to the covering space $\bar M_s$ consists of infinitely many connected and compact components, the first component of the rotation vector is positive $\omega_1(\mu_{\Gamma(s)})$ for each ergodic $\Gamma(s)$-minimal measure.
Let us describe how the equivalence relation is established between two classes near strong double resonance. Let $\Gamma\subset\mathbb{A}\subset\alpha^{-1}(E)$ be a curve skirting around the flat $\mathbb{F}_0$, along which the $\alpha$-function keeps constant and the third coordinate $c_3$ keeps constant as well. For each $c\in\Gamma$, there exists certain coordinate system and finitely many intervals $I_{c,i}$ for $x_2$-coordinates such that each $c$-semi static curve passes through the section $\Sigma_c=\{x_1=0\}$ and $$\mathcal{N}(c)\cap\Sigma_c\subset\{(x_1,x_2,x_3):x_1=0,x_2\in\cup I_{c,i},x_3\in\mathbb{T}\}.$$ Clearly, some open set $U\supset\mathcal{N}(c)\cap\Sigma_c$ such that $V_c=i_{U*}H_1(U,\mathbb{R})=\text{\rm span}\{(0,0,1)\}$, from which one obtains that $V_c^{\perp}=\text{\rm span}\{(1,0,0),(0,1,0)\}$. For each class $c'\in\Gamma$ very close to $c$, one has $c'-c=(\Delta c_1,\Delta c_2,0)\in V_c^{\perp}$, thus, there exists a closed 1-form $\bar\mu$ such that $[\bar\mu]=c'-c$ and $$\text{\rm supp}\bar\mu\cap\mathcal{N}(c)\cap\Sigma_c=\varnothing.$$ Therefore, all classes along the curve $\Gamma$ are equivalent in this case.
Local connecting orbits of type-$h$
-----------------------------------
Another type of local connecting orbits look like heteroclinic orbits. Therefore, we call them local connecting orbits of type-$h$.
It is used to handle a typical case when an Aubry set falls in a neighborhood $N$ of some lower dimensional torus such that $H_1(M,N,\mathbb{Z})\neq 0$. Equivalence relation seems not exist among those classes if the Aubry sets is located in $N$. However, each of these Aubry sets has homoclinic orbit, it may lead to the existence of heteroclinic orbits. Towards this goal, let us work in suitable finite covering manifold $\check{\pi}$: $\check{M}\to M$. In this covering space, these homoclinic orbits turn out to be semi-static orbits. We assume that the Aubry set $\mathcal{A}(c,\check{M})$ consists of finitely many classes $\mathcal{A} (c)=\mathcal{A}_1\cup\cdots\cup\mathcal{A}_k$ $(k>1)$, $\check{M}$ is chosen so that the lift of $N$, $\check{N}=N_1\cup\cdots\cup N_k$ with $k>1$, $\check{\pi}N_i=N$ and $\text{\rm dist}(N_i,N_j)>0$ provided $i\neq j$. In the following, we denote by $N_i$ the open neighborhood such that each $N_i$ contains one Aubry class $N_i\supset\mathcal{A}_i$.
If an Aubry set contains finitely many static classes only, denoted by $\tilde{\mathcal{A}}_i$ ($i=1,2,\cdots,k$), then these classes are transitive in the following sense: by rearranging the subscripts, there exist $k$ semi-static curves $\gamma_{i,i+1}$ $(\text{\rm mod}\ k)$ such that $\omega(d\gamma_{i,i+1})\subseteq\tilde{\mathcal{A}}_{i+1}$ and $\alpha(d\gamma_{i,i+1})\subseteq
\tilde{\mathcal{A}}_i$ [@CP]. It does not exclude the case that some semi-static curve $\gamma_{i,j}$ exists such that $j\neq i+1$ $(\text{\rm mod}\ k)$, $\alpha(d\gamma_{i,j})\subseteq \tilde{\mathcal{A}}_{i}$ and $\omega(d\gamma_{i,j})\subseteq\tilde{\mathcal{A}}_j$. We say that $\tilde{\mathcal{A}}_i$ is connected to $\tilde{\mathcal{A}}_j$ through $\tilde{\mathcal{A}}_{i'}$ with $i'=i+1,i+2,\cdots,j-1$ if there exist semi-static curves $\gamma_{i',i'+1}$ such that $\omega(d\gamma_{i',i'+1})\subseteq \tilde{\mathcal{A}}_{i'+1}$ and $\alpha(d\gamma_{i',i'+1})\subseteq \tilde{\mathcal{A}}_{i'}$.
The Aubry set $\mathcal{A}_i$ is said to be directly connected to the Aubry set $\mathcal{A}_j$ if a semi-static curve $\gamma$: $\mathbb{R}\to M$ exists such that $\omega(d\gamma)\subseteq \tilde{\mathcal{A}}_{j}$ and $\alpha(d\gamma)\subseteq \tilde{\mathcal{A}}_{i}$. That $\mathcal{A}_i$ is directly connected to $\mathcal{A}_j$ does not imply that $\mathcal{A}_j$ is directly connected to $\mathcal{A}_i$.
Pick up two points $x_i\in\mathcal{A}_i$, $x_j\in\mathcal{A}_j$, we consider the quantity $$h_c^{T}(x_i,x_j)=\inf_{\stackrel{\gamma(-T) =x_i}{\scriptscriptstyle \gamma(T)=x_j}} \int_{-T}^{T}
L_{c}(d\gamma(t))dt +2T\alpha(c).$$ By standard notation, $$h_c^{\infty}(x_i,x_j)=\liminf_{T\to\infty}h_c^{T}(x_i,x_j).$$ Let $\gamma^T$: $[-T,T]\to M$ be the minimal curve realizing the quantity $h_c^{T}(x_i,x_j)$. Let $[t_{i,T},t_{j,T}]$ be the sub-interval of $[-T,T]$ such that $\gamma^{T}(t)\notin N_i\cup N_j$ for $t\in(t_{i,T},t_{j,T})$ but $\gamma^{T}(t_{i,T})\in\bar N_i$ and $\gamma^{T}(t_{j,T})\in\bar N_j$. In the case that $\mathcal{A}_i$ is directly connected only to $\mathcal{A}_j$, $t_{j,T}-t_{i,T}$ is upper bounded uniformly for $T>0$. Some sequence of time $t_{T}$ and a positive number $\Delta>0$ such that $[t_{T}-\Delta, t_{T}+\Delta]\subset (t_{i,T},t_{j,T})$ for sufficiently large $T$. The set of curves $\{\gamma^{T}(t-t_{T}) |_{[-\Delta,\Delta]}\}$ is compact in $C^1$-topology. Let $\gamma|_{[-\Delta, \Delta]}$ be the accumulation point which can be uniquely extended to whole line $\gamma$: $\mathbb{R}\to M$. Clearly, $\alpha(d\gamma)\subset\tilde{\mathcal{A}}_i$ and $\omega(d\gamma)\subset \tilde{\mathcal{A}}_j$. If $\mathcal{A}_i$ is directly connected also to other $\mathcal{A}_k$, one can also obtain such a sequence of curves by introducing small perturbation so that $\mathcal{A}_i$ is directly connected only to $\mathcal{A}_j$ and the support of the perturbation does not touch the semi-static curves connecting $\mathcal{A}_i$ to $\mathcal{A}_j$.
Given a semi-static curve one can choose an $(n-1)$-dimensional disk $\Sigma$ intersecting the curve transversally. This disk also intersects semi-static curves nearby. A semi-static curve is said [*disconnected*]{} to other semi-static curves if the intersection point is disconnected to the intersection points of all other semi-static curves.
\[typehthm1\] [(Connecting Lemma)]{} Assume that the Aubry set contains finitely many classes $\mathcal{A}(c)=\mathcal{A}_1\cup\cdots \cup\mathcal{A}_k$, there exist open domains $N_1\cdots N_k$ such that $\mathcal{A}_i\subset N_i$ for each $1\le i\le k$ and $\text{\rm dist}(N_i,N_j)>0$ provided $i\neq j$. If each semi-static curves connecting different Aubry sets is disconnected to all other semi-static curve, then there exists some orbit $d\gamma'$ of $\phi_L^t$ connecting $\tilde{\mathcal{A}}(c)$ to $\tilde{\mathcal {A}}(c')$ provided $\alpha(c)=\alpha(c')$, the class $c'$ is close to the class $c$, $\mathcal{A}(c')\subset \cup_{i=1}^k N_i$ and two sets $N_i$, $N_j$ exist such that $\mathcal{A}(c')\cap N_i\neq\varnothing$ and $\mathcal{A}(c')\cap N_j\neq\varnothing$.
In autonomous case, $\tilde{\mathcal{A}}(c)$ can be connected to $\tilde{\mathcal{A}}(c')$ only if $\alpha(c)=\alpha(c')$. If both $c$ and $c'$ are the minimal points of the $\alpha$-function, then $\tilde{\mathcal {A}}(c)\cap\tilde{ \mathcal {A}}(c')\neq\varnothing$ (see [@Ms]), it is trivial to connect an Aubry set to itself. Thus we only need to work on the energy level set $H^{-1}(E)$ with $E>\min\alpha$, the minimum of the $\alpha$-function. In this case, we obtain from [@Lx] that
\[typehpro1\] Let $L:\mathbb{T}^n\to\mathbb{R}$ be an autonomous Lagrangian of Tonelli type, the class $c$ not be the minimal point of the $\alpha$-function, and $\Omega_c$ be the flat of the $\beta$-function such that $$\omega\in\Omega_c\ \ \Rightarrow\ \ \alpha(c)+\beta(\omega)=\langle c,\omega\rangle.$$ Then, there exists a coordinate system such that each rotation vector in this flat has positive first component $\omega_1>0$.
The existence of such connecting orbits is derived from the upper-semi continuity of pseudo-connecting orbit set (see Definition \[semicontinuitydef2\]). For the definition of this set in autonomous case, we need to work in certain covering space $\pi:\bar M=\mathbb{R}\times\mathbb{T}^{n-1}$ where $\omega_1(\mu_c)>0$ holds for each ergodic minimal measure $\mu_c$. By Proposition \[typehpro1\], it is possible if we choose suitable coordinate system. Let $\bar\gamma$ denote the lift of the curve $\gamma:\mathbb{R}\to M$, $\bar\gamma_1$ denote the first coordinate.
Let $\Sigma_0=\{x:x_1=0\}$ be a codimension one hyperplane separating $\bar M$ into two parts, the upper part $\bar M^+$ connected to $\{x_1=\infty\}$ and the lower part $\bar M^-$ connected to $\{x_1=-\infty\}$. Let $\Sigma_0+\delta$ denote the $\delta$-neighborhood of $\Sigma_0$ in $\bar M$, we introduce a smooth function $\rho\in C^r(\bar M,[0,1])$ such that $\rho=0$ if $x\in\bar M^- \backslash (\Sigma_0+\delta)$, $\rho=1$ if $x\in\bar M^+\backslash (\Sigma_0+\delta)$. Let $\eta$ and $\bar\mu$ be closed 1-forms on $M$ such that $[\eta]=c$ and $[\eta+\bar\mu]=c'$. They have natural lift on $\bar M$. Let $\mu=\rho\bar\mu$. We carefully choose smooth function $\psi=\psi(x,\dot x)$ such that $\psi=0$ as $x_1\in (-\infty,-1)\cup (1,\infty)$ (the construction will be demonstrated later) and let $$L_{\eta,\mu,\psi}=L-\eta-\mu-\psi.$$ Let $\bar m,\bar m'$ be two points in $\bar M$, we define $$h_{\eta,\mu,\psi}^{T}(\bar m,\bar m')=\inf_{\stackrel{\gamma(-T)=\bar m}{\scriptscriptstyle \gamma(T)=\bar m'}}\int_{-T}^T(L_{\eta,\mu,\psi}(d\gamma(t))+\alpha(c))dt.$$ For small $\mu$ and $\psi$, the Lagrangian $L_{\eta,\mu,\psi}$ satisfies the conditions required for space-step Lagrangian. In the following we shall use the notation $\mathscr{C}_{\eta,\mu,\psi}= \mathscr{C}(L_{\eta,\mu,\psi})$ to denote the relevant set of the pseudo-connecting curves.
Let us recall a graph property. Given two Aubry classes $\tilde{\mathcal{A}}_i$ and $\tilde{\mathcal{A}}_j$, let $\tilde{\mathcal{N}}_{ij}$ be the set of all semi-static orbits whose $\alpha$-limit set is in $\tilde{\mathcal{A}}_i$ and the $\omega$-limit set is in $\tilde{\mathcal{A}}_j$. Let $\mathcal{N}_{ij}=\pi_x\tilde{\mathcal{N}}_{ij}$, where $\pi_x$ denotes the standard projection $TM\to M$. Then, the inverse of $\pi_x$, restricted on $\mathcal{N}_{ij}$, is Lipschitz. The proof is the same as that for the graph property of Aubry set.
If $\tilde{\mathcal{A}}_i$ is directly connected to $\tilde{\mathcal{A}}_j$, there exists a semi-static orbit $d\zeta_{ij}$ connecting $\tilde{\mathcal{A}}_i$ to $\tilde{\mathcal{A}}_{j}$. Pick up a curve $\bar\zeta_{ij}$ in the lift of $\zeta_{ij}$ to $\bar M$ such that its intersection point $x_0$ with the section $\{x:x_1=0\}$ is not close to ${\cup\bar N_i}$, the lift of ${\cup N_i}$ to $\bar M$. Denote by $v_0=\dot{\zeta}_{ij}$ the velocity of $\zeta_{ij}$ at $\pi x_0$, obviously, $v_0\neq 0$. Let $\varrho'$ be a smooth function in $s$ such that $\varrho'=0$ for $s\le 0$, $\varrho'=1$ for $s>\delta$ and $\dot\varrho'>0$ for $s\in (0,\delta)$, where $\delta>0$ is suitably small. Let $\varrho_{ij}(x)=\varrho'(\langle x-x_0,v_0\rangle)$, then $\langle\partial\varrho_{ij}(x),v \rangle=\langle v_0,v\rangle\dot\varrho'(s)$ where $s=\langle x-x_0,v_0\rangle$.
We choose an $(n-1)$-dimensional plane $\Sigma_{ij,s}=\{x:\langle x-x_0,v_0\rangle=s\}$. Since the set of semi-static curves is totally disconnected, we can choose, for each $s\in [0,\delta]$, two suitably small $(n-1)$-dimensional topological disks $D'_{ij,s},D_{ij,s}$ located in $\Sigma_{ij,s}$ and small $\delta_1>0$ such that $D'_{ij,s} \cap(\cup N_j)=\varnothing$, $D'_{ij,s}\supset D_{ij,s}+\delta_1$, certain semi-static curve $\zeta_{ij}$ passes through the disk $D_{ij,s}$ and no semi-static curve in $\mathcal{N}_{ij}$ passes through $D'_{ij,s}\backslash D_{ij,s}$. These disks can be chosen so that the Hausdorff distance $d_H(D_{ij,s},D_{ij,s'})\to 0$ and $d_H(D'_{ij,s},D'_{ij,s'})\to 0$ as $s'\to s$. Let $D'_{ij}=\cup_{s\in [0,\delta]}D'_{ij,s}$, $D_{ij}=\cup_{s\in [0,\delta]}D_{ij,s}$. We choose a smooth non-negative function $w_{ij}$: $\bar M\to\mathbb{R}$ such that $\text{\rm supp}w_{ij}\cap\{x:0\le\langle x-x_0,v_0\rangle \le\delta\}=D'_{ij}$ and $w_{ij}\equiv\lambda$ if $x\in D_{ij}$.
For different $(i,j)\neq(i',j')$, it is possible that $D_{ij}\cap\mathcal{N}_{i',j'}\neq\varnothing$. But it does not make trouble, as $\tilde{\mathcal{N}}_{ij}\cap\tilde{\mathcal{N}}_{i'j'}=\varnothing$. Let $S_{ij}$ be the graph of a Lipschitz map $x\to\dot x$ containing $\tilde{\mathcal{N}}_{ij}$. Therefore, we can choose a smooth function $\upsilon_{ij}$: $TM\to [0,1]$ such that $\upsilon_{ij}\equiv 1$ when $(x,\dot x)\in (S_{ij}+\delta_2)\cap TD'_{ij}$ and $\upsilon_{ij}\equiv 0$ when $(x,\dot x)\notin S_{ij}+\delta_3$, where $\delta_3>\delta_2>0$ are small numbers. As there are finitely many Aubry classes, we have $\text{\rm supp}\upsilon_{ij}\cap \text{\rm supp}\upsilon_{i'j'}=\varnothing$ if $(i,j)\neq(i',j')$.
Let us consider what curves contained in the set $\mathscr{C}_{\eta,0,\psi}$ by assuming $$\psi=\sum\upsilon_{ij}w_{ij}\langle\partial\varrho_{ij},\dot x\rangle.$$ Since the term $\langle\partial\varrho_{ij},\dot x\rangle=0$ for $\{\langle x-x_0,v_0\rangle\le 0\}\cup \{ \langle x-x_0,v_0\rangle\ge\delta\}$, we do not care about how $w_{ij}$ is defined on those $(n-1)$-dimensional plane $\Sigma_{ij,s}$ with $s\notin [0,\delta]$. The set $\{\langle x-x_0,v_0\rangle=s\}\subset\bar M$ may extend to infinity, but it does not make trouble since the support of $w_{ij}$ is contained in $D_{ij,s}$ when $\langle x-x_0,v_0\rangle=s$.
Let $\psi_0$ be the function defined on $TM$ such that $\pi\psi|_{\{x_1\in [-\pi,\pi]\}} =\psi_0$. By the construction of $\psi$ we see that $\psi_0$ is well-defined and smooth. The Aubry set for the Lagrangian $L-{\eta}-\psi_0$ is the same as for $L-{\eta}$. As there is no semi-static curve of $L-{\eta}$ touches the tube $D'_{ij}\backslash D_{ij}$, each semi-static curve of $L-\eta$ also solves the Euler-Lagrange equation determined by $L-{\eta}-\psi_0$. Because of the upper semi-continuity of $L\to\mathcal{N}(L)$, each semi-static curve for $L-\eta-\psi_0$ stays in a small neighborhood of $\mathcal{N}(L)$. Since $\psi<0$ if $x\in\cup D_{ij}$ and $\psi=0$ if $x\notin \cup D'_{ij}$, a curve is still semi-static for $L-{\eta}-\psi_0$ if it is semi-static for $L-{\eta}$ and passes through the solid cylinder $D_{ij}$. It is based on following argument. Since the 1-form $w_{ij}\langle\partial \varrho_{ij}, dx\rangle$ is closed in $D_{ij}$, $\langle\partial\varrho_{ij},\dot x\rangle=\dot\varrho'(s)\langle\dot x,v_0\rangle=0$ on each $\Sigma_{ij,s}$ with $s\notin [0,\delta]$, this term has no contribution to the Euler-Lagrange equation along this semi-static curve, i.e. this curve solves the Euler-Lagrange equation determined by $L-\eta-\psi_0$ also. Any other semi-static curve for $L-\eta$ is no longer minimal for $L-\eta-\psi_0$ if it connects $\mathcal{A}_i$ to $\mathcal{A}_{j}$ but does not pass through $D_{ij}$, for there exists some semi-static curve $\zeta_{ij}$ of $L-\eta$ passing through $D_{ij}$, along which the action is smaller than the action along $\gamma_{ij}$.
Let us go back to the covering space $\bar M$. For small $\psi$, realized by choosing small $w_{ij}$, each curve $\bar\gamma\in\mathscr{C}_{\eta,0,\psi}$ stays in a small neighborhood of certain curve belong to the lift of the semi-static curve. It is due to the upper semi-continuity of $L\to\mathscr{C}(L)$. By the discussion above, a curve does not belong to $\mathscr{C}_{\eta,0,\psi}$ if its projection does not belong to the Aubry set for $L-\eta$, or dose not pass through $D_{ij}$ although it is semi-static and connects $\mathcal{A}^i$ to $\mathcal{A}^j$.
Let $\bar\zeta_{ij}$ be a curve in $\mathscr{C}_{\eta,0,\psi}$ passing through $D_{ij}$. Its projection $\pi\bar\zeta$ connects $\mathcal{A}^i$ to $\mathcal{A}^j$. Let $k^*\bar\zeta_{ij}=\bar\zeta_{ij} +(k,0,\cdots,0)$ with $k\in\mathbb{Z}$ denote its shift. Each of these curves solves the Euler-Lagrange equation determined by $L_{\eta,0,\psi}$. However, except for $\bar\zeta_{ij}$, any other curve $k^*\bar\zeta_{ij}$ with $k\neq 0$ does not belong to $\mathscr{C}_{\eta,0,\psi}$ because they do not pass through $D_{ij}$, the action along these curves is bigger than the action along $\bar\zeta_{ij}$. It can be easily seen from the definition \[semicontinuitydef2\]: minimal property persists under translation.
In the cylinder $\bar M$ we choose two sections $\Sigma^+$ and $\Sigma^-$ such that:
1, both are the deformation of the section $\{x:x_1=\text{\rm constant}\}$, they divide $\bar M$ into three parts, $\bar M^+$, $\bar M^-$ and $\bar M_0$. $\bar M^+$ is homeomorphic $(0,\infty)\times\mathbb{T}^{n-1}$, $\bar M^-$ is homeomorphic $(-\infty,0)\times\mathbb{T}^{n-1}$ and $\bar M_0$ is homeomorphic to $(0,1)\times\mathbb{T}^{n-1}$. Let $\Sigma_{\pm}$ denote the boundary of $\bar M^{\pm}$ shared with $\bar M_0$;
2, there exists $\delta_4>0$ such that $\cup D'_{ij}+\delta_4\subset\bar M_0$;
3, for each $\bar\zeta_{ij}\in\mathscr{C}_{\eta,0,\psi}$, both $\text{\rm Image}\bar\zeta\cap\bar M^+$ and $\text{\rm Image}\bar\zeta\cap\bar M^-$ are connected, i.e. if one moves into $\bar M_{\pm}$ along the curve as $t\to\pm\infty$ then it stays in $\bar M_{\pm}$ forever.
Let $U^+_{ij}$ be the tube connecting $D_{ij}$ to $\bar M^+$, $U^+_{ij}\cap D_{ij}=D_{ij,\delta}$, each $\bar\zeta_{ij}\in\mathscr{C}_{\eta,0,\psi}$ passes through $U^+_{ij}$, does not touch the boundary of $U^+_{ij}$ before it moves forward into $\bar M^+$. Similarly, we define the tube $U^-_{ij}$ connecting $D_{ij}$ to $\bar M^-$ such that $U^-_{ij}\cap D_{ij}=D_{ji,0}$, each of those curve passes through $U^-_{ij}$, does not touch the boundary of $U^-_{ij}$ before it is going to retreat back into $\bar M^-$.
Since there are finitely many Aubry classes only, by choosing suitably small $D'_{ij}$ we can assume $\text{\rm dist}(D'_{ij},D'_{i'j'})>0$ if $(i,j)\neq (i',j')$. A closed 1-form $\bar\mu$ clearly exists such that $[\bar\mu]=c'-c$ and $\text{\rm supp}\bar\mu \cap(\cup D_{ij})=\varnothing$. Let $\rho':\mathbb{R}\to [0,1]$ be a smooth function such that $\rho'=0$ for $s\le 0$, $\rho'=1$ for $s>\delta$ and let $U'_{ij}$ be an open set containing the closure of $U^+_{ij} \cup D'_{ij}\cup U^-_{ij}$ and $\text{\rm dist}(U'_{ij},U'_{i'j'})>0$ if $(i,j)\neq (i',j')$. We define a smooth function $\rho$: $\bar M\to [0,1]$ such that $\rho(x)=\rho'(\langle x-x_0,v_0\rangle)$ if $x\in D_{ij}$ where $x_0=\bar\zeta_{ij} (t_0)$ and $v_0=\dot{\bar\zeta}_{ij}(t_0)$, $\rho=1$ if $x\in \bar M^+\cup (\cup U^+_{ij})$ and $\rho(x)=0$ if $x\in \bar M^-\cup(\cup U^-_{ij})$. By the construction of $\bar M^{\pm}$, $U^{\pm}_{ij}$ and $D_{i,j}$, we see the existence of such function.
Let us now study the Lagrangian $L_{\eta,\mu,\psi}$ with $\mu=\rho\bar\mu$. By condition, $\mathcal{A}(c')\cap N_i\neq\varnothing$, $\mathcal{A}(c')\cap N_j\neq\varnothing$ and $i\neq j$. Thus, there exist $x_i\in\mathcal{M}(c)\cap N_i$ and $x_j\in\mathcal{M}(c')\cap N_j$. Let $\bar x_i$ and $\bar x_j$ be two points in $\bar M$ such that $\pi\bar x_i=x_i$ and $\pi\bar x_j=x_j$ and let $\bar x_{ik}=\bar x_i-ke_1$ and $\bar x_{jk}=\bar x_j+ke_1$ where $e_1=(1,0,\cdots,0)$. Let $\bar\gamma_k$: $[-T,T] \to\bar M$ be the minimizer of $$\inf_{T'>0}h_{\eta,\mu,\psi}^{T'}(\bar x_{ik},\bar x_{jk})=
\int_{-T}^{T} L_{\eta,\mu,\psi}(d\bar\gamma_k(t))dt+2T\alpha(c),$$ and let $k\to\infty$, we obtain a sequence of $\{\bar\gamma_k\}$. Let $\bar\gamma$: $\mathbb{R}\to \bar M$ be the accumulation point of the sequence. Due to the upper semi-continuity of $\mathscr
{C}_{\eta,\mu,\psi}$ with respect to $(\eta,\mu,\psi)$, the curve $\bar\gamma$ must pass through $\cup D_{ij}$ if $|c'-c|$ is suitably small. Thus, along the curve $\bar\gamma$ the term $\rho\bar\mu$ does not contribute the Lagrange equation, namely, the curve determines an orbit of $\phi_L^t$. Since this curve is in the set $\mathscr{C}_{\eta,\mu,\psi}$, therefore, it connects $\tilde{\mathcal{A}}(c)$ to $\tilde{\mathcal{A}}(c')$. This completes the proof.
Locally minimal property
------------------------
The orbit $d\gamma$ obtained in Theorem \[typehthm1\] is locally minimal in the sense we define in the following. It is crucial for the variational construction of global connecting orbits. The set of local minimal curve will not be empty if the Aubry set $\mathcal{A}(c)$ has some totally disconnected minimal homoclinic orbit, the 1-form $\mu$ as well as the function $\psi$ is carefully chosen for the modified Lagrangian.
Here is the definition for autonomous systems:
\[localdef2\] Let $N_1,\cdots,N_k\subset M$ be open domains such that $\text{\rm dist}(N_i,N_j)>0$ $(k>1)$. We assume that $\mathcal{A}(c), \mathcal{A}(c')\subset\cup N_i$, $[\eta]=c$, $[\eta+ \bar\mu]=c'$, $\alpha(c)=\alpha(c')$ and the first component of both $c$- and $c'$-minimal measures is positive $\omega_1(\mu_c)>0$, $\omega_1(\mu_{c'})>0$. Let $\pi:\bar
M=\mathbb{R}\times\mathbb{T}^{n-1}\to M$ be the covering space, denote by $\bar\gamma$ the lift of a curve $\gamma:\mathbb{R}\to M$. Then, $d\gamma$: $TM\to\mathbb{R}$ is called local minimal orbit of type-$h$ that connects $\tilde{\mathcal{A}}(c)$ to $\tilde{\mathcal{A}}(c')$ if
1, $d\gamma$ is an orbit of $\phi_L^t$, $\alpha(d\gamma)\subset\tilde{\mathcal{A}}(c)$ and $\omega(d\gamma)\subset\tilde{\mathcal{A}}(c')$. There exist $1\le i\neq j\le k$ such that $\alpha(d\gamma)\subset TN_i$ and $\omega(d\gamma)\subset TN_j$;
2, there exist two $(n-1)$dimensional disks $V_i^-$, $V_j^+\subset\bar M$ and positive numbers $T,d>0$ such that $\pi V_i^-\subset N_i\backslash\mathcal{A}(c)$, $\pi V_j^+\subset N_j\backslash\mathcal{A}(c')$, $\gamma$ transversally passes $\pi V_i^-$ and $\pi V_j^+$ at the time $-T$ and $T$ respectively, and $$\begin{aligned}
\label{localmineq1}
&h_c^{\infty}(x^-,\pi\bar m_0)+h_{\eta,\mu,\psi}^{T'}(\bar m_0,\bar m_1)+h_{c'}^{\infty}(\pi\bar m_1,x^+) \\
&-\lim_{\stackrel{t^-_i\to\infty}{\scriptscriptstyle t^+_i\to\infty}} \int_{-t^-_i}^{t^+_i}L_{\eta,\mu,\psi} (d\gamma(t))dt-(t_i^-+t_i^+)\alpha(c)>0\notag\end{aligned}$$ holds for each $(\bar m_0,\bar m_1,T')\in\partial(V_i^-\times V_j^+\times [T-d,T+d])$, $x^-\in N_i\cap\pi_x(\alpha(d\gamma))$ and $x^+\in N_j\cap\pi_x(\omega(d\gamma))$. Where $t^-_i\to\infty$ and $t^+_i\to\infty$ are the sequences such that $\gamma(-t^-_i)\to x^-$ and $\gamma(t^+_i)\to x^+$.
In this definition, the term $h_c^{\infty}(x^-,\pi\bar m_0)+ h_{\eta,\mu,\psi}^{T'}(\bar m_0,\bar m_1)+h_{c'}^{\infty} (\pi\bar m_1,x^+)$ measures the smallest action of $L_{\eta,\mu,\psi}$ along those curves which join $m_0$ to $m_1$ with time $2T'$ such that $x^-$ is an accumulation point of these curves as $t\to-\infty$ and $x^+$ is an accumulation point of the curves as $t\to\infty$.
[**Remark**]{}. In the space of curves, a neighborhood of the curve $\gamma$ consists of those curves that start from $V^-$ and reach $V^+$ within a time between $2(T-d)$ and $2(T+d)$. Different time scale determine orbits in different energy levels, that is why we consider the time scale $T'\in [T-d,T+d]$ as variable while we search for the local minimum.
[**Remark**]{}. This definition applies also to the case that there exists only one Aubry class staying in the small neighborhood of lower-dimensional torus. In that case, we can consider a suitable finite covering of the configuration manifold. In the finite covering configuration space, there are more than one Aubry class.
The following is the version for time-periodic systems
\[localdef3\] Let $N_1,\cdots,N_k\subset M$ $(k>1)$ be open domains with$\text{\rm dist}(N_i,N_j)>0$. We assume that $\mathcal{A}_0(c), \mathcal{A}_0(c')\subset\cup N_i$, $[\eta]=c$, $[\eta+ \bar\mu]=c'$. Then, $d\gamma$: $TM\to\mathbb{R}$ is called local minimal orbit of type-$h$ that connects $\tilde{\mathcal{A}}(c)$ to $\tilde{\mathcal{A}}(c')$ if
1, $d\gamma$ is an orbit of $\phi_L^t$, the $\alpha$-limit and the $\omega$-limit sets of $d\gamma$ are contained in $\tilde{\mathcal{A}}(c)$ and $\tilde{\mathcal{A}}(c')$ respectively, $\alpha(d\gamma)|_{t=0}\subset TN_i$ and $\omega(d\gamma)|_{t=0}\subset TN_j$ with $i\neq j$;
2, there exist two open balls $V_i^-$, $V_j^+$ and two positive integers $t^-,t^+$ such that $\bar V_j^-\subset N_i\backslash\mathcal {A}_0(c)$, $\bar V_j^+\subset N_j\backslash\mathcal {A}_0(c')$, $\gamma(-k^-)\in V_i^-$, $\gamma(k^+)\in V_j^+$ and $$\begin{aligned}
\label{localeq1}
&h_c^{\infty}(x^-,m_0)+h_{\eta,\mu,\psi}^{k^-,k^+}(m_0,m_1)+h_{c'}^{\infty}(m_1,x^+)\notag\\
&-\liminf_{\stackrel {k^-_i\to\infty}{\scriptscriptstyle k_i^+\to\infty}}\int_{-k^-_i}^{k^+_i}
L_{\eta,\mu,\psi}(d\gamma(t),t)dt-k^-_i\alpha(c)-k^+_i\alpha(c')>0\notag\end{aligned}$$ holds $\forall$ $(m_0,m_1)\in\partial(V_i^-\times V_j^+)$, $x^-\in N_i\cap\pi_x(\alpha(d\gamma))_{t=0}$, $x^+\in N_j\cap\pi_x(\omega(d\gamma))|_{t=0}$, where $k^-_i, k^+_i\in\mathbb{Z}^+$ are the sequences such that $\gamma(-k^-_i)\to x^-$ and $\gamma(k^+_i)\to x^+$.
The set of curves starting from $V^-$ and reaching $V^+$ with time $k^-+k^+$ constitutes a neighborhood of the curve $\gamma$ in the space of curves. Once a curve $\tilde\gamma$ touches the boundary of this neighborhood, the action of $L_{\eta,\mu,\psi}$ along $\tilde\gamma$ will be larger than the action along $\gamma$. As $V^-$, $V^+$ and therefore $d>0$ can be chosen arbitrarily small, it is reasonable to call it locally minimal.
Variational construction of global connecting orbits
====================================================
In this section we show how to construct global connecting orbits by variational method, provided a generalized transition chain exists. In the next section, the main result (Theorem \[mainthm\]) is proved by showing the genericity of such transition chain.
Generalized Transition chain
----------------------------
The concept of transition chain was proposed by Arnold in his celebrated paper [@Ar1] where it is formulated in geometric language. The generalized transition chain formulated in our previous work [@CY1; @CY2] is in variational version which need less information about the geometric structure.
\[chaindef1\] [(Autonomous Case)]{} Let $c$, $c'$ be two cohomolgy classes in $H^1(M,\mathbb{R})$. We say that $c$ is joined with $c'$ by a generalized transition chain if a continuous curve $\Gamma$: $[0,1]\to H^1(M,\mathbb{R})$ exists such that $\Gamma(0)=c$, $\Gamma(1)=c'$, $\alpha(\Gamma(s))\equiv E>\min\alpha$ and for each $s\in [0,1]$ at least one of the following cases takes place:
$\text{\rm (H1)}$, the Aubry set is composed of finitely many classes only. There exist certain finite covering: $\check{\pi}:\check{M}\to M$, two open domains $N_1,N_2\subset\check M$ with $d(N_1,N_2)>0$, an $(n-1)$ dimensional disk $D_{s}$ and small numbers $\delta_s,\delta'_s>0$ such that
[i]{}, the Aubry set $\mathcal{A}(\Gamma(s))\cap N_1\neq\varnothing$, $\mathcal{A}(\Gamma(s))\cap N_2\neq\varnothing$ and $\mathcal{A}(\Gamma(s'))\cap (N_1\cup N_2)\neq\varnothing$ for each $|s'-s|<\delta_s$,
[ii]{}, $\check\pi\mathcal{N} (\Gamma(s),\check M)|_{D_{s}}\backslash (\mathcal{A}(\Gamma(s))+\delta'_s)$ is non-empty and totally disconnected;
$\text{\rm (H2)}$, For each $s'\in (s-\delta_s,s+\delta_s)$, $\Gamma(s')$ is equivalent to $\Gamma(s)$. Some section $\Sigma_s$ and some neighborhood $U$ of $\mathcal{N}(\Gamma(s))\cap \Sigma_{s}$ exist such that $\Gamma(s')-\Gamma(s)\in\text {\rm ker}\,i^*_U$. Each class $\Gamma(s')$ is associated with an admissible section $\Sigma_{s'}$ and an admissible coordinate system $G_{s'}^{-1}x$.
[**Remark**]{}. Because of upper semi-continuity of Mañé set, it is possible that there exist some classes for which both cases take place.
In the case (H1), if the Aubry set contains only one Aubry class, one can take some finite covering $\check\pi:\check M\to M$ non trivial if $H_1(M,\mathcal{A},\mathbb{Z})\neq 0$. A typical case is that $\mathcal{A}(\Gamma(s))$ is contained in a small neighborhood of lower dimensional torus. One takes suitable finite covering space so that $\mathcal{A}(\Gamma(s),\check M)$ contains exactly two connected components. If $\mathcal{A}(\Gamma(s))$ contains more than one class, we choose $\check M=M$.
The existence of generalized transition chain implies that there exists a sequence of local connecting orbits. More precisely, there exists a sequence of locally minimal curve $\gamma_i$, a sequence of numbers $s_i$ $(s=0,1,\cdots,m)$ such that $\alpha(d\gamma_i)\subset \mathcal{A}(\Gamma(s_i))$ and $\omega(d\gamma_i))\subset\mathcal {A}(\Gamma(s_{i+1}))$. Global connecting orbits are constructed shadowing these local connecting orbits.
One can also define generalized transition chain for time-periodic systems.
[(Time-periodic Case)]{} Let $c$, $c'$ be two classes in $H^1(M,\mathbb{R})$. We say that $c$ is joined with $c'$ by a generalized transition chain if a continuous curve $\Gamma$: $[0,1]\to H^1(M,\mathbb{R})$ exists such that $\Gamma(0)=c$, $\Gamma(1)=c'$ and for each $s\in [0,1]$ at least one of the following cases takes place:
$\text{\rm (H1)}$, the Aubry set is composed of finitely many classes only. There exist certain finite covering: $\check{\pi}:\check{M}\to M$, two open domains $N_1, N_2$ with $d(N_1,N_2)>0$ and small number $\delta_s,\delta'_s>0$ such that
[i]{}, the Aubry set $\mathcal{A}_0(\Gamma(s))\cap N_1\neq\varnothing$, $\mathcal{A}_0(\Gamma(s))\cap N_2\neq\varnothing$ and $\mathcal{A}_0(\Gamma(s'))\cap(N_1\cup N_2)\neq\varnothing$ for each $|s'-s|<\delta_s$,
[ii]{}, $\check\pi\mathcal{N}_0 (\Gamma(s),\check M)\backslash(\mathcal{A}_0(\Gamma(s))+\delta'_s)$ is non-empty and totally disconnected;
$\text{\rm (H2)}$, For each $s'\in (s-\delta_s,s+\delta_s)$, $\Gamma(s')$ is equivalent to $\Gamma(s)$, namely, there exists a neighborhood of $\mathcal{N}_0(\Gamma(s))$, denoted by $U$, such that $\Gamma(s')-\Gamma(s)\in\text {\rm ker}\,i^*_U$.
Variational construction
------------------------
Given $x\in M$ and $c\in H^1(M,\mathbb{R})$, there exists at least a forward (backward) $c$-semi static curve $\gamma^+_c$: $[0,\infty)\to M$ ($\gamma^-_c$: $(-\infty,0]\to M$) such that $\gamma_c ^{\pm}(0)=x$. It determines certain velocity $v_{x,c}^{\pm}=\dot\gamma_c^{\pm}(0)$, for almost all points, the velocity is uniquely determined. Before proving the main theorem of this subsection, let us formulate and prove a proposition.
\[constructionpro1\] Given an Aubry set, the Aubry distance from any Aubry class $\mathcal{A}^i$ to all other Aubry classes is assumed have positive lower bound, namely, some $d>0$ exists such that $d_c(\mathcal{A}^i,\mathcal{A}^j)\ge d>0$ for all $j\neq i$. Let $$N_i=\{m\in M: h^{\infty}(m,x)+h^{\infty}(x,m)\le\frac d6,\ \forall\ x\in\mathcal{A}^i\},$$ then for all $m_0,m_1\in N_i$ and for any $x\in\mathcal{A}^i$ one has $$\label{constructioneq1}
h^{\infty}(m_0,x)+h^{\infty}(x,m_1)=h^{\infty}(m_0,m_1);$$ for any $m_0,m_1\in N_i$ and any $x\in\mathcal{A}\backslash\mathcal{A}^i$ one has $$\label{constructioneq2}
h^{\infty}(m_0,x)+h^{\infty}(x,m_1)\ge h^{\infty}(m_0,m_1)+\frac d2.$$
: For each pair of points $(m_0,m_1)\in M\times M$, we claim that there exists some Aubry class $\mathcal{A}^j$ such that $$h^{\infty}(m_0,m_1)=h^{\infty}(m_0,x)+h^{\infty}(x,m_1)$$ holds for each $x\in \mathcal{A}^j$. Indeed, let $k_i\to\infty$ be a subsequence of integers such that $$\lim_{i\to\infty}h^{k_i}(m_0,m_1)=h^{\infty}(m_0,m_1),$$ let $\gamma^{k_i}$: $[-k_i,k_i]\to M$ be the minimizer for $h^{k_i}(m_0,m_1)$. There exists at least one point $x\in\mathcal{A}$ which is the accumulation point of $\{\gamma^{k_i}(t_i)\}_{i\in\mathbb{Z}}$. Otherwise, the quantity $h^{k_i}(m_0,m_1)\to\infty$ as $k_i\to\infty$.
Given $m\in N_i$, we claim that (\[constructioneq1\]) and (\[constructioneq2\]) hold if $m_0=m_1=m$. Let $k_{\ell}\to\infty$ be a sequence such that $\lim_{k_{\ell}\to\infty}h^{k_{\ell}}(m,m)
=h^{\infty}(m,m)$ and let $\gamma^{k_{\ell}}_m(t)$: $[-k_{\ell}, k_{\ell}]\to M$ be the minimizer of $h^{k_{\ell}}(m,m)$. There is a positive number $d_1>0$ such that the ordinary distance $d(\gamma^{k_{\ell}}_m(t), \mathcal{A}^j)\ge d_1>0$ for any $t\in [-k_{\ell},k_{\ell}]$ and $j\neq i$. Otherwise along the curve $\gamma_m^{k_{\ell}}(t)$ there exists a point getting closer and closer to a point $x_j\in\mathcal{A}^j$. Consequently, one would obtain from the property that $d_c(\mathcal{A}^i,\mathcal{A}^j)\ge d>0$ for each $j\neq i$ that $$\begin{aligned}
h^{\infty}(m,m)=& h^{\infty}(m,x_j)+h^{\infty}(x_j,m)\\
\ge &h^{\infty}(x_i,x_j)-h^{\infty}(x_i,m)+h^{\infty}(x_j,x_i)-h^{\infty}(m,x_i) \\
\ge &\frac 56d\end{aligned}$$ where $x_i\in\mathcal{A}^i$. On the other hand, we have $$h^{\infty}(m,m)\le h^{\infty}(m,x_i)+h^{\infty}(x_i,m)\le\frac 16d.$$ It is a contradiction. Therefore, some $x_i\in\mathcal{A}^i$ and $t_{\ell}\in [0,k_{\ell}]$ exist such that $t_{\ell}\to\infty$ as $k_{\ell}\to\infty$ and $\gamma^{k_{\ell}}_m(t_{\ell})\to x_i$. This proves (\[constructioneq1\]) in case $m_1=m_2$.
For different points $m_0,m_1\in N_i$ and $x\in\mathcal{A}^j$ with $j\neq i$, let $\zeta^{k}_s(t,m_0,x)$: $[-k,k]\to M$ be the curve which minimizes the quantity $h^{k}(m_0,x)$, let $k_{j}$ be the subsequence of $k$ such that $\lim_{k_{j}\to\infty}h^{k_{j}}(m_0,x) =h^{\infty}(m_0,x)$. In autonomous case, it converges as $k\to\infty$. Similarly, we let $\zeta^{k}_u(t,x,m_1)$: $[-k,k]\to M$ be the curve which minimizes the quantity $h^{k}(x,m_1)$, let $k'_j$ be the sequence of $k$ such that $\lim_{k'_{j}\to\infty}h^{k'_{j}}(x,m_1) =h^{\infty}(x,m_1)$. Let $\ell=0,1$, $\gamma_{\ell}^{k}$: $[-k,k]\to M$ be the minimizer of $h^{k}(m_{\ell},m_{\ell})$ and let $k_{\ell}$ be the subsequence of $k$ such that $h^{k_{\ell}}(m_{\ell},m_{\ell})\to h^{\infty}(m_{\ell},m_{\ell})$. By the proof we just finished, there exists $x_{\ell}\in\mathcal{A}^i$ and integer $t_{\ell}^i\in [-k_{\ell},k_{\ell}]$ such that $\gamma_{\ell}^{k_{\ell}} (t^i_{\ell})\to x_{\ell}$ and $t^i_{\ell}\to\infty$ as $k_{\ell}\to\infty$. Let $\xi^k_{01}$: $[-k,k]\to M$ be the minimizer of $h^{k}(x_0,x_1)$, $k_{01}^i$ be the subsequence of $k$ such that $h^{k^i_{01}}(x_0,x_1)\to h^{\infty} (x_0,x_1)$, let $\xi^i_{10}$: $[0,k]\to M$ be the minimizer of $h^{k}(x_1,x_0)$, $k_{10}^i$ be the subsequence of $k$ such that $h^{k^i_{1}}(x_1,x_0)\to h^{\infty} (x_1,x_0)$. Given arbitrarily small $\delta>0$, we have sufficiently large $k_j$, $k'_j$, $k_0^i$, $k_1^i$, $k^i_{01}$ and $k^i_{10}$ such that $$\begin{aligned}
&|h^{\infty}(m_0,x)-h^{k_j}(m_0,x)|<\delta,\\
&|h^{\infty}(x,m_1)-h^{k'_j}(x,m_1)|<\delta,\\
&|h^{\infty}(m_{\ell},m_{\ell})-h^{k^i_{\ell}}(m_{\ell},m_{\ell})|
<\delta,\qquad \ell=0,1 \\
&|h^{\infty}(x_0,x_1)-h^{k^i_{01}}(x_0,x_1)|<\delta,\\
&|h^{\infty}(x_1,x_0)-h^{k^i_{10}}(x_1,x_0)|<\delta.\end{aligned}$$ Since $x_0,x_1\in \mathcal{A}^i$, we have $d_c(x_1,x_0)=0$. Consequently, $$\begin{aligned}
\label{constructioneq3}
&h^{t_0^i}(m_0,x_0)+h^{k^i_{01}}(x_0,x_1)+h^{k_1^i-t_1^i}(x_1,m_1)\\
+&h^{t_1^i}(m_1,x_1)+h^{k^i_{10}}(x_1,x_0)+h^{k_0^i-t_0^i}(x_0,m_0)\notag\\
\le&\frac 13d+6\delta.\notag\end{aligned}$$ Since $x$ is in Aubry class $\mathcal{A}^j$, while $x_0,x_1\in\mathcal{A}^i$, one has $$\begin{aligned}
\label{constructioneq4}
&h^{k'_j}(x,m_1)+h^{t_1^i}(m_1,x_1)+h^{k^i_{10}}(x_1,x_0)\\
&+h^{k_0^i-t_0^i}(x_0,m_0)+h^{k_j}(m_0,x)\notag\\
\ge&d-5\delta.\notag\end{aligned}$$ Because $\delta$ can be arbitrarily small, by subtracting (\[constructioneq3\]) from (\[constructioneq4\]) we obtain $$\begin{aligned}
&h^{\infty}(m_0,x)+h^{\infty}(x,m_1)-\frac 23d\\
\ge &h^{\infty}(m_0,x_0)+h^{\infty}(x_0,x_1)+h^{\infty}(x_1,m_1)\\
\ge &h^{\infty}(m_0,m_1)\end{aligned}$$ it verifies (\[constructioneq2\]). Since (\[constructioneq2\]) holds for each $x\in\mathcal{A}^j$ with $j\neq i$ and for any $m_0,m_1\in N_i$, (\[constructioneq1\]) hold for each $x\in\mathcal{A}^i$ and for any $m_0,m_1\in N_i$. This completes the proof of the proposition.
\[constructionthm1\] If $c$ is connected to $c'$ by a generalized transition chain, then
1, there exists an orbit of the Lagrange flow $\phi_L^t$, $d\gamma$: $\mathbb{R}\to TM$ which connects the Aubry set $\tilde{\mathcal{A}}(c)$ to $\tilde{\mathcal{A}}(c')$, namely, $\alpha(d\gamma)\subseteq\tilde{\mathcal{A}}(c)$ and $\omega(d\gamma)\subseteq\tilde{\mathcal{A}}(c')$;
2, given $x,x'\in M$ and arbitrarily small $\delta>0$, there exists an orbit $(\gamma,\dot\gamma)$ of $\phi_L^t$ passing through $\delta$-neighborhood of the points $(x,v_{x,c}^{+})$ and $(x',v_{x,c'}^{-})$ successively, namely, $t<t'$ such that $(\gamma(t),\dot\gamma(t))\in B_{\delta}(x,v_{x,c}^{+})$ and $(\gamma(t'),\dot\gamma(t'))\in B_{\delta}(x',v_{x,c'}^{-})$.
We only need to study the autonomous case. Time periodic case can be treated in the same way. Therefore, one has that $\alpha(\Gamma(s))\equiv E> \min\alpha$. By adding suitable constant on the Lagrangian, we assume $E=0$ to simplify notation.
First of all, as a generalized transition chain $\Gamma(s)$ is assumed, the Aubry set $\tilde{\mathcal{A}} (\Gamma(s))$ can be connected to some $\tilde{\mathcal{A}} (\Gamma(s'))$ by locally minimal orbits of either type-$h$, or type-$c$ if $s'$ close to $s$. So, there is a sequence $0=s_0<s_1<\cdots<s_k=1$ such that for each $0\le j<k$, $\tilde{\mathcal{A}} (\Gamma(s_j))$ is connected to $\tilde{\mathcal{A}}(\Gamma(s_{j+1}))$ by some local minimal orbits. The global connecting orbits are constructed shadowing such a sequence of orbits.
Let $c_j=\Gamma(s_j)$. We divide the set $\{0,1,\cdots,k\}$ into $m$ parts $$\{0,1,\cdots
k\}=\{0,1,\cdots,i_1\}\cup\{i_1+1,\cdots,i_2\}\cup\cdots\cup\{i_{m-1}+1,\cdots, i_m=k\}.$$ The rule to make such a partition is that for all $i=i_{j},i_{j}+1,\cdots, i_{j+1}-1$, $\tilde{\mathcal{A}} (c_i)$ is connected to $\tilde{\mathcal{A}}(c_{i+1})$ by a local minimal orbit of the same type. More precisely, let $\Lambda_c$ and $\Lambda_h$ be the subset of $\{i_1,i_2,\cdots, i_m\}$, $\Lambda_c\cup\Lambda_h= \{i_1,i_2,\cdots,i_m\}$, $\Lambda_c\cap\Lambda_h=\varnothing$. If $i_j\in\Lambda_{\imath}$, then for all $i=i_j,i_j+1,\cdots,i_{j+1}-1$, $\tilde{\mathcal{A}} (c_i)$ is connected to $\tilde{\mathcal{A}}(c_{i+1})$ by a local minimal orbit of type-$\imath$ ($\imath$ =$c$, or $h$).
Since the map $c\to\tilde{\mathcal{N}}(c,M)$ is upper semi-continuous, once the Mañé set $\tilde{\mathcal{N}}(\Gamma(s))$ is in the case (H1) (or H2), then for $s'$ sufficiently close to $s$, the set $\tilde{\mathcal{N}}(\Gamma(s))$ is also in the case (H1) (or H2). Thus, for each $i_j\in\Lambda_h$, by choosing $c_{i_j-1}$ and $c_{i_{j+1}}$ sufficiently close to $c_{i_j}$ and $c_{i_{j+1}-1}$ respectively, we can assume that both $c_{i_j-1}$ and $c_{i_{j+1}}$ satisfy the condition (H1) also.
With the class $c_i$ we associate an admissible coordinate system $x\to G_i^{-1}x$ and let $G_i^{-1}=[g_{i,1}^{-1}, g_{i,2}^{-1},\cdots, g_{i,n}^{-1}]^t$ denote the inverse of $G_i$. Because we consider the problem on $H^{-1}(E)$ with $E>\min\alpha$, we can choose $G_i$ for each $i\in\Lambda_c$ (see Proposition \[typehpro1\]) such that, in the new coordinate system, the first component of $\omega(\mu_{c_i})$ is positive for each ergodic $c_i$-minimal measure. In virtue of the upper semi-continuity of Mañé set on cohomology class, one can assume that $$\langle g_{j,1}^{-1},\omega(\mu_{c_i})\rangle >0, \qquad \forall \ j=i-1,i$$ holds for each ergodic component $\mu_{c_i}$ as $c_{i-1}$ is chosen suitably close to $c_i$. It means that the $\omega_1(\mu_{c_i})>0$ holds in the coordinates not only determined by $G_i$, but also determined by $G_{i-1}$ as well as by $G_{i+1}$. Therefore, $\exists$ $x_{i,1}>0$ such that $$\begin{aligned}
\label{constructioneq5}
&\langle g_{i,1}^{-1}, \Delta\tilde\gamma_i\rangle\ge 2\pi,\ \ \ \text{\rm whenever}\ \ \langle g_{i-1,1}^{-1}, \Delta\tilde\gamma_i\rangle\ge x_{i,1}, \notag\\
&\langle g_{i-1,1}^{-1}, \Delta\tilde\gamma_i\rangle\ge 2\pi,\ \ \ \text{\rm whenever}\ \ \langle g_{i,1}^{-1}, \Delta\tilde\gamma_i\rangle\ge x_{i,1}\end{aligned}$$ holds for each $c_i$-semi-static curve $\gamma_i$, where $\tilde\gamma_i$ denotes a curve in the lift of $\gamma_i$ to universal covering space and $\Delta\tilde\gamma_i=\tilde\gamma_i(t')-\tilde\gamma_i(t)$ with $t'>t$.
As the second step, let us describe the minimal properties of local connecting orbits of type-$h$ as well as of type-$c$.
[**The case of type-$h$**]{}. For each integer $i\in\bigcup_{i_j\in\Lambda_h}\{i_j,i_j+1,\cdots,i_{j+1}-1\}$, the condition (H1) holds for generalized transition chain. Namely, in certain finite covering space $\check{\pi}:\check{M}\to M$, the Aubry set for $i$ and $i+1$ consists of more than one but finitely many classes $\mathcal{A}(c_{\ell}, \check{M})=\cup \mathcal{A}_{\ell}^j$ for $\ell=i,i+1$. By the assumption of (H1), some open domains $N^-_{i},N^+_{i+1}\subset\check M$ exist such that $d(N^-_i,N^+_{i+1})>0$, $\mathcal{A}(c_i,\check{M})\cap N^-_{i}\neq\varnothing$, $\mathcal{A}(c_i,\check{M})\cap N^+_{i+1}\neq\varnothing$, $\mathcal{A}(c_{i+1},\check{M})\cap N^+_{i+1}\neq \varnothing$ and $\mathcal {N}(c_i,\check M)\backslash (\mathcal{A}(c_i)+\delta'_i)\neq\varnothing$ is totally disconnected, with small $\delta'_i>0$.
The new coordinate system $x\to G_i^{-1}x$ of $\mathbb{T}^n$ is introduced such that (\[constructioneq5\]) holds. If writing $\check M=\{(x_1,x_2,\cdots,x_n):x_i\in\mathbb{T}\}$, we introduce the covering space $\bar M=\{(x_1,x_2,\cdots,x_n):x_1\in\mathbb{R}, x_i\in\mathbb{T}\ \text{\rm for}\ i\ge 2\}$. We shall work with this covering space $\pi:\bar M\to\check M$. For closed 1-forms $\eta_i$ and $\bar\mu_i$ on $\check{M}$ we use the same symbol to denote their natural lift to $\bar M$.
Recall the proof of Theorem \[typehthm1\]. Some decomposition of $\bar M$ exists such that $\bar M=\bar M_i^+\cup\bar M_{i,0}\cup\bar M_i^-$ such that $\bar M_i^+$ is diffeomorphic to $[0,\infty)\times \mathbb{T}^{n-1}$, $\bar M_i^-$ is diffeomorphic to $(-\infty,0]\times\mathbb{T}^{n-1}$ and $\bar M_{i,0}$ is diffeomorphic to $(0,1)\times \mathbb{T}^{n-1}$. Some open and connected disks $U^+_i,U^-_i,D_i,D'_i\subset \bar M_{i,0}$ and $\delta_i>0$ exist such that $D_i+\delta_i \subset D'_i$, $(\pi D'_i+\delta_i)\cap N^-_i=\varnothing$ and $(\pi D'_i+\delta_i)\cap N^+_{i+1}=\varnothing$, the intersection of any two of these sets is empty and the closure of $\bar M_i^+\cup U^+_i\cup D_i\cup U^-_i\cup\bar M_i^-$, denoted by $\bar M^c_i$, is connected, see Figure \[fig11\].
![[]{data-label="fig11"}](Ardiff11.eps){width="10cm" height="2.2cm"}
As it was studied in the subsection of 6.3 (local connecting orbit of type-$h$), some function $w_i$, $\rho_i$: $\bar M\to [0,1]$, some closed 1-form $\eta_i$, $\bar\mu_i$, $\varrho_i$ and some small constant $\delta_i>0$ exist such that $[\eta_i]=c_i$, $[\bar\mu_i]=c_{i+1}-c_i$, $\text{\rm supp}\bar\mu_i\cap\bar D_i=\varnothing$, $\text{\rm supp}w_i\subset D'_i$, $w_i|_{D_i}=\text{\rm constant}$, $\rho_i(x)=1$ if $x\in\bar M_i^+\cup U_i^+$ and $\rho_i(x)=0$ if $x\in U_i^-\cup\bar M_i^- $.
Let $\mu_i=\rho_i\bar\mu_i$, $\psi_i=w_i\varrho_i$ we introduce a space-step Lagrangian for the coordinate system $G_i^{-1}x$ $$L_{\eta_i,\mu_i,\psi_i}=L-\eta_i-\mu_i-\psi_i.$$ In virtue of Theorem \[typehthm1\], some curve $\bar\zeta_i\in\mathscr{C}_{\eta_i,\mu_i,\psi_i}$ (pseudo connecting orbit set) such that its projection down to $\check{M}$, $\zeta_i=\pi\bar\zeta_i$, determines an orbit $d\zeta_i=(\zeta_i,\dot\zeta_i)$ connecting certain Aubry class $\tilde{\mathcal{A}}^j_i$ to another Aubry class $\tilde{\mathcal{A}}^{j'}_{i+1}$. Here, the subscript $i$ indicates the Aubry set is for the cohomology class $c_i$, the superscript $j$ indicates which Aubry class it belongs to. Such curve stays entirely in the interior of $\bar M_i^c$. Therefore, along such curve both $\mu_i$ and $\psi_i$ do not contribute to the Euler-Lagrange equation, i.e. $d\zeta_i$ is an orbit of $\phi_L^t$.
As pointed out in Definition \[localdef2\], such local connecting orbit of type-$h$ is endowed with certain kind of local minimality. There exist two $(n-1)$-dimensional disks $V_i^-$ and $V_{i+1}^+$ with $\pi V^-_i \subset N^-_i\backslash\mathcal{A}^j_i$, $\pi V_{i+1}^+\subset N^+_{i+1}\backslash\mathcal{A}^{j'}_{i+1}$, large number $T_i^+>0$, suitably small $d_i>0$ and quite small $\epsilon_i^*>0$ such that $\bar\zeta_i(-T_i^+)\in V_i^-$, $\bar\zeta_i(T_i^+)\in V_{i+1}^+$ and $$\begin{aligned}
\label{constructioneq6}
\min&\Big\{h_{c_i}^{\infty}(x^-,m_0)+h_{\eta_i,\mu_i,\psi_i}^{T}(\bar m_0,\bar m_1)+h_{c_{i+1}}^{\infty} (m_1,x^+):\\
&(\bar m_0,\bar m_1,T)\in\partial (V_i^-\times V_{i+1}^+\times[T_i^+-d_i,T_i^++d_i])\Big\}\notag\\
\ge\min&\Big\{h_{c_i}^{\infty}(x^-,m_0)+h_{\eta_i,\mu_i,\psi_i}^{T}(\bar m_0,\bar m_1) +h_{c_{i+1}}^{\infty}(m_1,x^+):\notag\\
&(\bar m_0,\bar m_1,T)\in V_i^-\times V_{i+1}^+\times[T_i^+-d_i,T_i^++d_i] \Big\} +5\epsilon_i^*,\notag\end{aligned}$$ where $x^-\in\pi_x\alpha(d\zeta_i)\subseteq\mathcal{A}^j_i$, $x^+\in\omega(d\zeta_i)\subseteq\pi_x \mathcal{A}^{j'}_{i+1}$. The disks $V^-_i$ and $V^+_{i+1}$ are chosen so that $\bar\zeta_i$ intersects them transversally, for each $(\bar m_0,\bar m_1,T')$ $\in V_i^-\times V_{i+1}^+\times[T_i^+-d_i,T_i^++d_i]$, the minimizer of $h_{\eta_i,\mu_i,\psi_i} ^{T}(\bar m_0,\bar m_1)$, $\bar\gamma_i(t,\bar m_0,$ $\bar m_1,T)$ has the property $$\label{constructioneq7}
\bar\gamma_i(t)\in \bar M_i^c\qquad\forall \ t\in [-T,T].$$
Let $\zeta_{i-1}$ be a locally minimal curve such that the orbit $d\zeta_{i-1}$ connecting $\tilde{\mathcal{A}}_{i-1}$ to $\tilde{\mathcal{A}}_{i}$. Denote by $\tilde{\mathcal{A}}^{j'}_i$ the Aubry class which contains the $\omega$-limit set of $d\zeta_{i-1}$. It is possible that $\mathcal{A}^{j'}_i$ is different from $\mathcal{A}^j_i$ which contains the $\alpha$-limit set of $d\zeta_i$. Remember that we have assumed that each Aubry set consists of finitely many classes, let’s say, $k_i$ classes. By the result in [@CP], the subscript of Aubry classes can be rearranged such that some $c_i$-semi-static curve $\gamma_{i,j}$ exists such that $\alpha(d\gamma_{i,j})\subset\tilde{\mathcal{A}}^j_i$ and $\omega(d\gamma_{i,j}) \subset\tilde{\mathcal{A}}^{j+1}_i$ for $j=1,\cdots,k_i$ $(\text{\rm mod}\ k_i)$. So some positive integer $k\le k_i$ exists such that $j_i-j'_i=k\le k_i$ $(\text{\rm mod}\ k_i)$. Let $d_i=\min d_c(\mathcal{A}^j_i,\mathcal{A}^{j'}_i)$.
We choose some $(n-1)$-dimensional small disks $V^{\pm}_{i,j}$ with $j=j'_i,\cdots,j_i$ such that $V^+_{i,j'_i}=V^+_i$, $V^-_{i,j_i}=V^-_i$, $V_{i,j}^{\pm}$ is located within $N_{i,j}$, a small neighborhood of $\mathcal{A}^j_i$ such that $d_c(m,x)\le\frac{d_i}6$ holds for each $m\in N_{i,j}$ and each $x\in\mathcal{A}^j_i$ (the definition of $N_{i,j}$ is the same as $N_i$ in Proposition \[constructionpro1\]), $\gamma_{i,j}$ intersects $V^-_{i,j}$ as well as $V^+_{i,j+1}$ transversally. These curves also have locally minimal property similar to the form of (\[constructioneq6\]): $$\begin{aligned}
\label{constructioneq8}
\min&\Big\{h_{c_i}^{\infty}(x^-,m_0)+h_{c_i}^{T}(m_0,m_1)+h_{c_{i}}^{\infty}(m_1,x^+):\notag\\
&(\bar m_0,\bar m_1,T)\in\partial (V_{i,j}^-\times V_{i,j+1}^+\times[T^+_{i,j}-\tau_i,T^+_{i,j}+\tau_i]) \Big\}\\
\ge\min&\Big\{h_{c_i}^{\infty}(x^-,m_0)+h_{c_i}^{T}(m_0,m_1)+h_{c_{i}}^{\infty}(m_1,x^+):\notag\\
&(\bar m_0,\bar m_1,T)\in V_{i,j}^-\times V_{i,j+1}^+\times[T^+_{i,j}-\tau_i,T^+_{i,j}+\tau_i] \Big\}
+5\epsilon_i^*,\notag\end{aligned}$$ where $x^-\in\mathcal{A}^j_i$, $x^+\in\mathcal{A}^{j+1}_i$, $T^+_{i,j}$ is the time such that $\gamma_{i,j}(2T^+_{i,j})\in V_{i,j+1}^+$ if $\gamma_{i,j}(0)\in V^-_{i,j}$. As semi-static curves are totally disconnected, $V^-_{i,j}$ and $V^+_{i,j+1}$ are chosen so that any curve in $\mathcal{N}(c_i)$ does not touch the boundary of $V^-_{i,j}$ and of $V^+_{i,j+1}$.
Note that $h_c^{\infty}=\lim_{T\to\infty}h_c^T$ in the autonomous case [@Fa1], we find from Proposition \[constructionpro1\] that, for any $\epsilon^*_i>0$, there exists $T^-_{i,j}=T^-_{i,j}(\epsilon^*)>0$ such that $$|h_{c_i}^T(m,m')-h^\infty_{c_i}(m,x)-h^\infty_{c_i}(x,m')|\le\epsilon^*_i\notag$$ holds for each $T\ge T^-_{i,j}$, each $m,m'\in N_{i,j}$ and $x\in\mathcal{A}^j_i$.
Let $t^+_i=t^+_{i,j'}<t^-_{i,j'}<t^+_{i,j'+1}\cdots<t^+_{i,j}<t^-_{i,j}=t^-_i$, $2\Delta t^+_{i,j}=t^+_{i,j+1}-t^-_{i,j}$, $2\Delta t^-_{i,j}=t^-_{i,j}-t^+_{i,j}$. A curve $\gamma$: $[t^+_i,t^-_i]\to\check{M}$ is called [*admissible*]{} for $V^{\pm}_{i,j}$ if $$\label{constructioneq9}
\gamma(t^{\pm}_{i,j})=x^{\pm}_{i,j}\in V^{\pm}_{i,j}, \qquad \forall\ j'_i\le j\le
j_i,$$ where $\Delta t^+_{i,j}$ and $\Delta t^-_{i,j}$ are chosen to satisfy the condition $$\label{constructioneq10}
T^+_{i,j}-\tau_i\le\Delta t^+_{i,j}\le T^+_{i,j}+\tau_i$$ where $\tau_i$ is chosen in the inequality (\[constructioneq8\]), the condition $$\label{constructioneq11}
T^-_{i,j}+T^+_{i,j}+\tau_{i}\le \Delta t_{i,j}^-+\Delta t_{i,j}^+\le T^-_{i,j}+T^+_{i,j}+\tau^*_i$$ which is set so that the minimal curve does not touch the boundary of $V^-_{i,j}$ provided it passes through $V^+_{i,j}$ at $t=t^+_{i,j}$ and through $V^+_{i,j+1}$ at $t=t^+_{i,j+1}$ and the condition $$\label{constructioneq12}
T^-_{i,j}+T^+_{i,j-1}+\tau_{i}\le \Delta t_{i,j}^-+\Delta t_{i,j-1}^+ \le T^-_{i,j}+T^+_{i,j-1}+\tau^*_i.$$ which is set so that the minimal curve does not touch the boundary of $V^+_{i,j}$ provided it passes through $V^-_{i,j-1}$ at $t=t^-_{i,j-1}$ and through $V^-_{i,j}$ at $t=t^-_{i,j}$. These conditions define non-empty set for $(\Delta t^+_{i,j},\Delta t^-_{i,j})$ if we choose suitably large $\tau^*_i>0$.
We consider the minimum of the following action among all admissible curves: $$h_{c_i}^{t^+_i,t^-_i}(x^+_i,x^-_i)=\inf_{\stackrel{\scriptscriptstyle \gamma(t^+_i)=x^+_i\in V^+_{i,j'_i} } {\scriptscriptstyle \gamma(t^-_i)=x^-_i\in V^-_{i,j_i}}} \int_{t^+_i}^{t^-_i}(L-\eta_i)(d\gamma)dt.$$ Let $\gamma(t,t^{\pm}_i,x^{\pm}_i)$: $[t^+_i,t^-_i]\to\check{M}$ be the minimizer of the action. If $t^-_{i,j}-t^+_{i,j}$ is sufficiently large, the minimizer is smooth at each $t^-_{i,j'_i}<t^+_{i,j'_i+1}<\cdots<t^+_{i,j_i}$. First of all, we claim that $(x_{i,j}^-,x^+_{i,j+1}, \Delta t^+_{i,j})\in\text{\rm int}(V^-_{i,j}\times V^+_{i,j+1}\times [T^+_{i,j}-\tau_i, T^+_{i,j}+\tau_i])$ holds for each $j'_i\le j<j_i$. If it does not hold for certain $j'_i\le j<j_i$, one obtains from (\[constructioneq8\]) that $$\begin{aligned}
&h_{c_{i}}^{\Delta t_{i}^-}(x^+_{i,j},x^-_{i,j})+h^{\Delta t_i^+}_{c_i}(x^-_{i,j},x^+_{i,j+1})+h^{\Delta t_{i+1}^-}_{c_{i}}(x^+_{i,j+1}, x^-_{i,j+1})\notag \\
\ge &h_{c_{i}}^{\infty}(\xi,x^-_{i,j})+ h^{\Delta t_i^+}_{c_i}(x^-_{i,j},x^+_{i,j+1})+h^{\infty}_{c_{i}} (x^+_{i,j+1},\zeta)\notag \\
&+h_{c_{i}}^{\infty}(x^+_{i,j},\xi) +h^{\infty}_{c_{i+1}}(\zeta,x^-_{i,j+1})-2\epsilon_i^*\notag \\
\ge &h_{c_{i}}^{\infty}(\xi,\hat x^-_{i,j})+ h^{\Delta t_i^+}_{c_i}(\hat x^-_{i,j},\hat x^+_{i,j+1})
+h^{\infty}_{c_{i}}(\hat x^+_{i,j+1},\zeta)\notag \\
&+h_{c_{i}}^{\infty}(x^+_{i,j},\xi) +h^{\infty}_{c_{i+1}}(\zeta,x^-_{i,j+1})+3\epsilon_i^* \notag\\
\ge &h_{c_{i}}^{\Delta t_{i}^-}(x^+_{i,j},\hat x^-_{i,j})+h^{\Delta t_i^+}_{c_i}(\hat x^-_{i,j},\hat x^+_{i,j+1}) +h^{\Delta t_{i+1}^-}_{c_{i}} (\hat x^+_{i,j+1}, x^-_{i,j+1})+\epsilon_i^*\notag\end{aligned}$$ where $\xi\in\mathcal{A}^j_i$, $\zeta\in\mathcal{A}^{j+1}_i$, $\hat x_{i}^-$ as well as $\hat x_{i+1}^+$ is the intersection point of a semi-static curve $\gamma_{j,j+1}$ with $V^-_{i,j}$ and with $V^+_{i,j+1}$ respectively. The orbit $d\gamma_{j,j+1}$ connects $\mathcal{A}^j_i$ to $\mathcal{A}^{j+1}_i$. This contradicts the minimality of $\gamma$. The smoothness follows from the property that $(x_{i,j}^-,x^+_{i,j+1},\Delta t^+_{i,j})$ is in the interior of the domain. If the minimizer $\gamma(t,t^{\pm}_i,x^{\pm}_i)$ is not smooth at $x^-_{i,j}$, we join the points $\gamma(t^-_{i,j}-\delta,t^{\pm}_i,x^{\pm}_i)$ and $\gamma(t^-_{i,j}+\delta,t^{\pm}_i,x^{\pm}_i)$ by the minimizer of $$h_{c_i}^{\delta}(\gamma(t^-_{i,j}-\delta,t^{\pm}_i,x^{\pm}_i),\gamma(t^-_{i,j}+\delta,t^{\pm}_i,x^{\pm}_i)).$$ As the minimizer approaches $x^-_{i,j}$ from both sides of $V^-_{i,j}$ as $t$ approaches $t^-_{i,j}$ from opposite direction [@BCV], this minimizer also passes through $V^-_{i,j}$. Thus, one obtains a curve $\gamma'$ by replacing the segment of $\gamma(t,t^{\pm}_i,x^{\pm}_i)|_{t^-_{i,j}-\delta,t^-_{i,j} +\delta}$ with this minimizer. Let $t'$ be the time of this curve passing through $V^-_{i,j}$, clearly, $t^+_{i,j}-t'\in(T^+_{i,j}-\tau_i, T^+_{i,j}+\tau_i)$, the action along this curve is clearly smaller than the original one. But this is absurd.
[**The case of type-$c$**]{}. For each integer $i\in\bigcup_{i_j\in\Lambda_c}\{i_j,i_j+1,\cdots,i_{j+1}-1\}$, there exist an admissible section $\Sigma_{c_i}$, a neighborhood $U_i$ of $\mathcal{N}(c_i)\cap \Sigma_{c_i}$, two closed 1-forms $\eta_i$ and $\bar\mu_i$ on $M$ with $[\eta_i]=c_i$, $[\bar\mu_i]=c_{i+1}-c_i$ and $\text{\rm supp}\bar\mu_i\cap U_i=\varnothing$. Correspondingly, an admissible coordinate system $q=G_i^{-1}x$ on $M$ is chosen such that in the new coordinates, one has a covering space $\pi_i$: $\bar M_i=\mathbb{R}\times\mathbb{T}^{n-1}\to M$, the set $\pi^{-1}_i\Sigma_{c_i}$ consists of infinitely many compact components $\Sigma_{c_i}^j= \{q=(q_1+j,q_2,\cdots,q_n):q\in\Sigma_{c_i}^0\}$ $(j\in\mathbb{Z})$. $\bar M_i$ is separated by $\Sigma_{c_i}^0$ into upper part $\bar M^+_i$ and lower part $\bar M^-_i$. A smooth function $\rho_i$: $\bar M\to [0,1]$ is constructed such that $\rho_i=0$ for $q\in\bar M^-_i \backslash(\Sigma_{c_i}^0 +\delta_i)$ and $\rho_i=1$ for $q\in\bar M^+_i\backslash (\Sigma_{c_i}^0+\delta_i)$. The number $\delta_i>0$ is chosen so small such that $(\Sigma_{c_i}^0 +\delta_i)\cap(\mathcal{N}([\eta_i+\mu_i])+\delta_i) \subset U_i$, (cf. Formula (\[typeceq1\])). Let $\mu_i=\rho_i\bar\mu_i$.
To make notation simpler, for each integer $i\in\bigcup_{i_j\in\Lambda_c} \{i_j,i_j+1,\cdots, i_{j+1}-1\}$, we let $\psi_i=0$, $V_i^{-}=\{q_1=-K_i\}$ and $V_{i+1}^{+}=\{q_1=K_i\}$ in the coordinate system $q=G_i^{-1}x$ (see the corollary \[typeccor1\] for the definition of $K_i$). Again, let $$L_{\eta_i,\mu_i,\psi_i}=L-\eta_i-\mu_i-\psi_i.$$ Since the class $c_i$ is equivalent to the class $c_{i+1}$ and they are close to each other, one sees from Theorem \[typecthm1\] that each curve $\bar\gamma \in\mathscr{C}_{\eta_i,\mu_i,\psi_i}$ determines a locally minimal orbit type-$c$ $d\gamma$ which is an orbit of $\phi_L^t$ and connects $\tilde{\mathcal{A}}(c_i)$ to $\tilde{\mathcal {A}}(c_{i+1})$.
Let $\bar m\in V_i^-$, $\bar m'\in V^+_{i+1}$ and let $\bar\gamma_i(t,\bar m,\bar m'):[-T,T]\to\bar M$ be the minimizer of $$h_{\eta_i,\mu_i,\psi_i}^T(\bar m,\bar m')=\inf_{T'>0}h_{\eta_i,\mu_i,\psi_i}^{T'}(\bar m,\bar m').$$ According to Lemma \[semicontinuitylem2\] and Corollary \[typeccor1\], $\exists$ $K_i>0$, $T_i^+=T_i^+(K_i)>0$, there exists $T< T_i^+$ such that $h_{\eta_i,\mu_i,\psi_i}^T(\bar m,\bar m')= \inf_{T'>0}h_{\eta_i,\mu_i,\psi_i}^{T'}(\bar m,\bar m')$ provided the first coordinate of $\bar m$ as well as of $\bar m'$ satisfies the condition that $\bar m_1\le -K_i$ and $\bar m'_1\ge K_i$. The minimizer $\bar\gamma_i(t,\bar m,\bar m'):[-T,T]\to\bar M$ satisfies $$\label{constructioneq13}
\bar\gamma_i(t,\bar m,\bar m')\in U_i, \qquad \text{\rm whenever}\ \ \bar\gamma_{i}(t,\bar m,\bar m')\in \Sigma_{c_i}^0+\delta_i.$$
By the definition, the disks $V^{+}_i$ and $V^-_i$ are codimension one torus in different coordinate systems, their relative position needs to be fixed in the universal covering space. For this purpose, we define the following covering spaces: $$\mathbb{R}^n\xrightarrow{\bar\pi_i}\bar M_i\xrightarrow{\pi_i}\check{M}_i,$$ where $\check M_i=\{(q_1,\cdots,q_n): q_i\ \text{\rm mod}\ 2i_j\pi \}$ and $\bar M_i=\mathbb{R}\times\{(q_2,\cdots,q_n): q_i\ \text{\rm mod}\ 2i_j\pi \}$ in the coordinate system $q=G_i^{-1}x$. For simplicity of notation and without danger of confusion, we use the same symbol for a fundamental domain of $V^{\pm}_i$ in $\mathbb{R}^n$, i.e. restricted on $V^{\pm}_i$ the projection is a homeomorphism, $\bar\pi_iV^{-}_i=V^{-}_i$ and $\bar\pi_iV^-_{i+1}=V^-_{i+1}$. Both $V^-_i$ and $V^+_{i+1}$ are some translation of unit $(n-1)$-dimensional disk $\{q_1=0,q_i\in [0,2\pi)\,\text{\rm for}\, i=2,\cdots,n\}$. Clearly, $\bar\pi^{-1}_iV^-_i$ is parallel to $\bar\pi^{-1}_iV^+_{i+1}$ and there exist $(n-1)$ irreducible integer vectors $(v^i_2,v^i_3,\cdots v^i_n)$ tangent to $V^{-}_i$ ($V^+_{i+1}$) such that $$\bar\pi^{-1}_iV^-_i=\bigcup_{k_{\ell}\in\mathbb{Z},\ell=2,\cdots n}V^-_i+k_{\ell}v^i_{\ell},\qquad
\bar\pi^{-1}_iV^+_{i+1}=\bigcup_{k_{\ell}\in\mathbb{Z},\ell=2,\cdots n}V^+_{i+1}+k_{\ell}v^i_{\ell}.$$
Given a fundamental domain $V^+_i$ and a $c_i$-semi static curve $\gamma_i$, there exists a curve in the lift $\gamma_i$ to the universal covering space, denoted by $\tilde\gamma_i$, which intersects $V^+_i\subset\mathbb{R}^n$. Clearly, one can choose a fundamental domain $V^-_i\subset\mathbb{R}^n$ (up to a translation) so that it intersects the curve $\tilde\gamma_i$ and $\bar\pi_{i-1}V^-_i$ is “above" the $V^+_i\subset\bar M_{i-1}$ in the following sense: some suitably large $K'_i>0$ exists such that $\min\{q^-_1-q^+_1:q^-\in\bar\pi_{i-1}V^-_i,q^+\in V^+_i\}=K'_i$. Note that $V^-_i$ is usually not parallel to $V^+_i$. We say that the two fundamental domains $V^+_i$ and $V^-_i$ are $(c_i,K'_i)$-related if they satisfy this condition.
Let $V^+_i$ and $V^-_i$ be $(c_i,K'_i)$-related fundamental domains. Given positive integers $k^+_i,k^-_i$, we define $${\bf k}^{\pm}_iV^{\pm}_i=\bigcup_{|k_{\ell}|\le k^{\pm}_i,\ell=2,\cdots n}V^{\pm}_i+k_{\ell}v^i_{\ell},$$ then $\bar\pi_{i-1}{\bf k}^{+}_iV^{+}_i=V^+_i\subset\bar M_{i-1}$. Let $\tilde x^+_i\in{\bf k}^{+}_iV^{+}_i$, $\tilde x^-_i\in{\bf k}^{-}_iV^{-}_i$ one defines the minimal action of $L_{c_i}$ connecting these two points $$h_{c_i}(\tilde x^+_i, \tilde x^-_i)=\inf_{T>0}\inf_{\stackrel{\scriptscriptstyle\tilde\zeta(-T) = \tilde x^+_i}{\scriptscriptstyle \tilde\zeta(T)=\tilde x^-_{i}}} \int_{-T}^TL_{c_i}(d\tilde\zeta(s))ds.$$ Let $$h_{c_i}({\bf k}^{+}_iV^{+}_i, {\bf k}^{-}_iV^{-}_i)=\min_{\stackrel{\scriptscriptstyle\tilde x^+_i\in {\bf k}^{+}_iV^{+}_i}{\scriptscriptstyle \tilde x^-_i\in {\bf k}^{-}_iV^{-}_i}} h_{c_i}(\tilde x^+_i, \tilde x^-_i).$$ Clearly, for fixed $k^+_i$, some positive number $\epsilon_i>0$ and suitably large integer $k^-_i$ exist such that ${\bf k}^{+}_iV^{+}_i$ does not touch ${\bf k}^{-}_iV^{-}_i$, $$\label{constructioneq14}
h_{c_i}(\tilde x^+_i, \tilde x^-_i)>h_{c_i}({\bf k}^{+}_iV^{+}_i, {\bf k}^{-}_iV^{-}_i)+\epsilon'_i, \qquad \text{\rm if}\ d(\tilde x^-_i,\partial{\bf k}^{-}_iV^{-}_i)\le 1.$$ To understand this property let us consider those curves in the lift of $c_i$-semi static curves which pass through $V_i^+$. There exists $k_i>0$ such that the intersection points of these curves with $\bar\pi_i^{-1}V^-_i$ locate in the disk $\cup_{|k_{\ell}|\le k_i}V^-_i+k_{\ell}v^i_{\ell}$. The “rotation vector" of this segment of the orbit can not be too far away from $\rho(\mu_i)$.
One can also define related fundamental domains $V^-_i$ and $V^+_{i+1}$. For a fundamental domain $V^-_i$ and a curve $\gamma\in\mathscr{C}_{\eta_i,\mu_i}$, we pick up a curve in the lift of this curve to the universal covering space, denoted by $\tilde\gamma$, which intersects the section $V^-_i$. Some fundamental domain $V^+_{i+1}$ exists where this curve intersects. Recall the projection of the two fundamental domains takes the form $V^+_{i+1}=\{q_1=K_i\}$ and $V^-_{i}=\{q_1=-K_i\}$ in the configuration space $\bar M_i$. We say that the two fundamental domains $V^-_i$ and $V^+_{i+1}$ are $(\eta_i,\mu_i,K_i)$-related.
As the Lagrangian $L_{\eta_i,\mu_i}$ is well-defined in the universal covering space, let us consider its action in the universal covering space: $$h_{\eta_i,\mu_i}(\tilde x^-_i,k^*\tilde x^+_{i+1})=\inf_{T>0}\inf_{\stackrel{\scriptscriptstyle\tilde\zeta(-T) = \tilde x^-_i}{\scriptscriptstyle \tilde\zeta(T)=k^*\tilde x^+_{i+1}}} \int_{-T}^TL_{\eta_i,\mu_i}(d\tilde\zeta(s))ds,$$ where $k^*\tilde x^+_{i+1}=\tilde x^+_{i+1}+\sum_{\ell=2,\cdots n}k_{\ell}v^i_{\ell}$ stands for a translation of $\tilde x^+_{i+1}$ and $k=(k_2,\cdots k_n)$. Obviously, one has $$\inf_{k\in\mathbb{Z}^{n-1}}h_{\eta_i,\mu_i}(\tilde x^-_i,k^*\tilde x^+_{i+1})=h_{\eta_i,\mu_i}(\tilde x^-_i,\tilde x^+_{i+1})= \inf_{T>0}h_{\eta_i,\mu_i}^T(q^-_i,q^+_{i+1})$$ where the term $\inf_{T>0}h_{\eta_i,\mu_i}^T(q^-_i,q^+_{i+1})$ was defined before by considering the action in the configuration space $\bar M_i$. As above, one defines $$h_{\eta_i,\mu_i}({\bf k}^{-}_iV^{-}_i,{\bf k}^+_{i+1}V^+_{i+1})=\min_{\stackrel{\scriptscriptstyle\tilde x^-_i\in {\bf k}^{-}_iV^{-}_i}{\scriptscriptstyle \tilde x^+_{i+1}\in {\bf k}^{+}_{i+1}V^{+}_{i+1}}} h_{\eta_i,\mu_i}(\tilde x^-_i, \tilde x^+_{i+1}).$$ Again, for fixed $k^-_i$, some positive number $\epsilon_i>0$ and suitably large $k^+_{i+1}$ exist such that ${\bf k}^{-}_iV^{-}_i$ does not touch ${\bf k}^{+}_{i+1}V^{+}_{i+1}$ and $$\label{constructioneq15}
h_{\eta_i,\mu_i}(\tilde x^-_i, \tilde x^+_{i+1})>h_{\eta_i,\mu_i}({\bf k}^{-}_iV^{-}_i, {\bf k}^{+}_{i+1}V^{+}_{i+1})+\epsilon_i, \qquad \text{\rm if}\ d(\tilde x^+_{i+1},\partial{\bf k}^{+}_iV^{+}_{i+1})\le 1.$$
Let $\tilde V^{\pm}_i={\bf k}^{\pm}_iV^{\pm}_i$. By induction, these sections $\tilde V^{\pm}_i$ are well defined such that $V^+_i$ and $V^-_i$ are $(c_i,K'_i)$-related, $V^-_i$ and $V^+_{i+1}$ are $(\eta_i,\mu_i,K_i)$-related, the formulae (\[constructioneq14\]) and (\[constructioneq15\]) are satisfied.
As the third step of the construction, let us clarify what conditions the candidates of minimal curve are required to satisfy.
Let $\gamma$: $[-K,K']\to M$ be an absolutely continuous curve joining $m$ to $m'$, i.e. $\gamma(-K)=m$ and $\gamma(K')=m'$. We split the interval $[-K,K']$ into $2i_m+1$ subintervals $$[-K,K']=[t_0^+,t^-_0]\cup [t^-_0,t^+_1]\cup\cdots\cup [t^+_{i_m},t_{i_m}^-],$$ where $t_0^+=-K$, $t_{i_m}^-=K'$. Correspondingly, we divide the curve into $2i_m+1$ segments $\gamma_i^-=\gamma|_{[t_{i}^+,t_i^-]}$, $\gamma_i^+=\gamma|_{[t_{i}^-,t_{i+1}^+]}$ for $i=0,1,2,\cdots, i_m-1$, and $\gamma_{i_m}^-=\gamma|_{[t_{i_m}^+,t_{i_m}^-]}$.
We fix a curve $\tilde\gamma$ in the lift of $\gamma$ to the universal covering space $\mathbb{R}^n$ by choosing $\bar\pi G_0^{-1}\tilde\gamma (t_0^-)\in V_0^-$. Correspondingly, each $\gamma_i^{\pm}$ has its lift $\tilde\gamma_i^{\pm}$ to $\mathbb{R}^n$.
The curve $\gamma$ is required to satisfy the conditions:
1, for each $i=0,1,2,\cdots i_m-1$, there is some $k_i\in\mathbb{Z}$ such that $$\begin{aligned}
\label{constructioneq16}
&\bar\pi_iG_i^{-1}\tilde\gamma_i^+(t_i^-)-(2k_i\pi,0,\cdots,0)\in V_i^-,\notag \\
&\bar\pi_iG_i^{-1}\tilde\gamma_i^+(t_{i+1}^+)-(2k_i\pi,0,\cdots,0)\in V_{i+1}^+;\end{aligned}$$
2, for $i\in\bigcup_{i_j\in\Lambda_c} \{i_j+1,\cdots,i_{j+1}-1\}$, $\tilde\gamma(t^{\pm}_i)\in\tilde V^{\pm}_i$. Let $\Delta t_i^+=\frac 12(t_{i+1}^+-t_{i}^-)$ and $\Delta t_i^-=\frac 12(t_i^--t_i^+)$. To formulate the conditions for $\Delta t_i^{\pm}$, let us consider the quantity $$h_{c_i}^{\Delta t}(\tilde x_i^+,\tilde x_i^-)=\inf_{\stackrel{\scriptscriptstyle\tilde\xi(-\Delta t)=\tilde x_i^+\in \tilde V^+_{i}} {\scriptscriptstyle \tilde\xi(\Delta t)=\tilde x_i^-\in \tilde V_i^-}}\int_{-\Delta t}^{\Delta t}(L-\eta_{i})(d\tilde\xi(t))dt.$$ One obtains from the proof of Lemma \[semicontinuitylem2\] that $h_{c_i}^{\Delta t}(\tilde x_i^+, \tilde x_i^-)\to\infty$ as $\Delta t\to 0$ or $\to\infty$. Thus, if $T_i^-=T_i^-(\tilde x_i^+, \tilde x_i^-)$ is defined as the quantity such that $h_{c_i}^{T_i^-}(\tilde x_i^+, \tilde x_i^-)=\min_{\Delta t} h_{c_i}^{\Delta t}(\tilde x_i^+, \tilde x_i^-)$, then we find $0<T_i^-(\tilde x_i^+, \tilde x_i^-)<\infty$. Since both $V_{i,-}^+$ and $V_{i,+}^-$ are compact, there exist $0<\hat T_i^-<\breve{T}_i^-<\infty$ such that $\hat T_i^-<T_i^-(\tilde x_i^+, \tilde x_i^-)<\breve{T}_i^-$ holds for each $\tilde x_i^+\in\tilde V_{i,-}^+$ and $\tilde x_i^-\in\tilde V_{i,+}^-$. Let $$\label{constructioneq17}
\Delta T_i^-=[\hat T_i^-,\breve{T}_i^-].$$
The range of $\Delta t_i^{\pm}$ is somehow implicitly defined. Let $$\label{constructioneq18}
\Delta T_i^+=[T_i^+-d_i,T_i^++d_i],\qquad \forall i\in\bigcup_{i_j\in\Lambda_h}\{i_j,i_j+1, \cdots,i_{j+1}-1\}$$ $$\Delta T_i^+=(0,T_i^+], \qquad \Delta T_i^-=[\hat T_i^-,\breve{T}_i^-],\qquad \text{\rm for other}\ i\le i_m.$$ See (\[constructioneq6\]), (\[constructioneq13\]) for the definition of $T_i^+$ and (\[constructioneq17\]) for the definition of $\Delta T_i^-$ respectively.
The conditions for $\Delta t_i^{\pm}$ are the following:
1, $\Delta t^+_i\in\Delta T_i^+$ for all $0\le i<i_m$;
2, $\Delta t_i^-\in\Delta T_i^-$ for $i\in\bigcup_{i_j\in\Lambda_c}\{i_j+1, i_j+2,\cdots,i_{j+1}-1\}$;
3, for $i\in\bigcup_{i_j\in\Lambda_h}\{i_j, i_j+1,\cdots,i_{j+1}-1\}$, as it is assumed that the Aubry set $\mathcal{A}(c_i)$ contains finitely many classes, an orbit connects $\mathcal{A}(c_{i-1})$ to $\mathcal{A}(c_i)$ by approaching the Aubry class $\mathcal{A}^{j'}_i$ as $t\to\infty$, another orbit connects $\mathcal{A}(c_i)$ to $\mathcal{A}(c_{i+1})$ by approaching the Aubry class $\mathcal{A}^j_i$ as the time retreat back to $-\infty$. For the time interval $[t_i^+,t_i^-]$, one has the partition $$[t_i^+,t_i^-]= [t^+_i,t^-_{i,j'}]\cup[t^-_{i,j'},t^+_{i,j'+1}]\cup\cdots\cup[t^+_{i,j}, t^-_i],$$ and has restrictions for these quantities, formulae (\[constructioneq10\]) ,(\[constructioneq11\]), (\[constructioneq12\]) and $$\label{constructioneq19}
T^-_{i,j}+T_{i}^++d_{i}\le \Delta t_{i,j}^-+\Delta t_{i}^+\le T^-_{i,j}+T_{i}^++d^*_{i},$$ $$\label{constructioneq20}
T^-_{i+1,j'}+T_{i}^++d_{i}\le \Delta t_{i+1,j'}^-+\Delta t_{i}^+\le T^-_{i+1,j'}+T_{i}^++d^*_{i};$$ with suitably large $d^*_i>0$.
4, for $i=i_j$ with $i_j\in\Lambda_h$, by definition, $c_i$ is equivalent to $c_{i-1}$, one has $$\label{constructioneq21}
\hat T_i^-+T_i^+\le\Delta t^+_i+\Delta t^-_i\le \breve{T}_i^-+T_i^++d^*_i;$$
5, for $i=i_j$ with $i_j\in\Lambda_c$, by choosing $c_{i-1}$ suitable close to $c_i$ one can also assume that $c_i$ is equivalent to $c_{i-1}$. Thus, one has $$\label{constructioneq22}
\hat T_i^-+T_{i-1}^+\le\Delta t^+_{i-1}+\Delta t^-_i\le \breve{T}_i^-+T_{i-1}^++d^*_{i-1}.$$
As the system is autonomous, by choosing sufficiently large $T_i^-$, these conditions defines non-empty set for $(\Delta t_i^+,\Delta t_i^-)$.
Finally, let us introduce a modified Lagrangian and verify the smoothness of the minimizer of the action. Recall $\mu_i$ and $\psi_i$ are defined on $\mathbb{R}\times\mathbb{T}^{n-1}$ in the coordinate system $q=G_i^{-1}x$, $G_i^*(\mu_i+\psi_i)
(d\tilde\gamma)=(\mu_i+\psi_i)(\bar\pi G_i^{-1}d\tilde\gamma)$ is well defined. We introduce a modified Lagrangian $$L_{\eta_i,\mu_i,\psi_i}\to L-\eta_i-(k_iG_i)^*(\mu_i+\psi_i)$$ where $k_i^*$ is a translation of $q_1$: $(k_i)^*\phi(q,\dot q)=\phi(q_1-2\pi k_i,\hat q,\dot q)$ on $T\bar M_i$ and the integer $k_i$ is chosen so that (\[constructioneq16\]) holds.
Let $\tilde\pi$: $\mathbb{R}^n\to M$ be the universal covering space. For a curve $\tilde\gamma$: $[-K,K']\to\mathbb{R}^n$, let $\gamma=\tilde\pi\tilde\gamma$: $[-K,K']\to M$. Let $\vec{t}=(t_0^-,t_1^{\pm},\cdots, t_{i_m-1}^{\pm},t_{i_m}^+)$, $\vec{x}=(\tilde x_0^-,\tilde x_1^{\pm},\cdots, \tilde x_{i_m-1}^{\pm},\tilde x_{i_m}^+)$, we consider the minimal action $$\begin{aligned}
\label{constructioneq23}
h_{L}^{K,K'}(m,m',\vec{x},\vec{t})&=\inf\sum_{i=0}^{i_m}\int^{t_{i}^-}_{t_i^+}(L-\eta_i) (d\tilde\gamma_i^-(t))dt \notag\\
&+\sum_{i=0}^{i_m-1}\int^{t_{i+1}^+}_{t_{i}^-}(L-\eta_i-(k_iG_i)^*(\mu_i+\psi_i))(d\tilde\gamma_i^+(t))dt\end{aligned}$$ where the infimum is taken over all absolutely continuous curves $\tilde\gamma$: $[-K,K']\to\mathbb{R}^n$ with the boundary conditions $\tilde\gamma_i^+(t_i^-)=\tilde x_i^-$, $\tilde\gamma_i^+(t_{i+1}^+)=\tilde x_{i+1}^+$ for $i=0,1,\cdots, i_{m}-1$, $\gamma(-K)=m$, $\gamma(K')=m'$ and satisfying the condition (\[constructioneq16\]). Moreover, restricted on $[t^+_i,t^-_i]$, $\gamma$ is admissible for the condition (\[constructioneq9\]).
As the system is autonomous, the quantity $h_{L}^{K,K'}(m,m',\vec{z},\vec{t})$ remains constant if $(\vec{t},K,K')$ is subject to a translation. Thus, it is a function of $K'-t_{i_m}^+$, $t_0^-+K$ and $\Delta\vec{t}=\{\Delta t^+_0,\Delta t^{\pm}_1,\cdots,\Delta t^{\pm}_{i_m-1}\}$. Denote by $\Delta\vec{T}$ the domain where $\Delta\vec{t}$ takes its value. Let $\vec{V}=(\tilde V_0^-,\tilde V_1^{\pm},\cdots, \tilde V_{i_m-1}^{\pm}, \tilde V_{i_m}^+)$, where all entries have been well defined in the previous proof.
Denote by $\gamma(t;K,K',m,m',\vec{x},\Delta\vec{t})$ the curve along which the quantity of (\[construction 24\]) is realized, it obviously depends on the value $K,K',m,m',\vec{x},\Delta\vec{t}$ and it may not be smooth at $\vec{t}$. Let $\vec{x}$ and $\Delta\vec{t}$ range over the set $\vec{V}$ and $\Delta\vec{T}$ respectively, one obtains a minimizer. The purpose of the following steps is to show that the minimizer is a solution of the Euler-Lagrange equation determined by $L$.
Let $h_{L}^{K,K'}(m,m')$ be the minimum of $h_{L}^{K,K'}(m,m',\vec{z},\vec{t})$ over $\vec{V}$ in $\vec{x}$ and over $\Delta\vec{T}$ in $\Delta\vec{t}$ respectively: $$h_{L}^{K,K'}(m,m')=\min_{\Delta\vec{t}\in\Delta\vec{T},\vec{x}\in\vec{V}} h_{L}^{K,K'} (m,m',\vec{x},\vec{t}),$$ denote the minimal curve by $\gamma(t;K,K',m,m')$, we claim that $d\gamma(t;K,K',m,m')$ is a solution of the Euler-Lagrange equation of $L$ if $K$ and $K'$ are sufficiently large. To verify this claim, we need to show that
1, $d\gamma_i^+=d\gamma|_{\Delta t_i^+}$ solves the Euler-Lagrange equation determined by $L$. Restricted on $\Delta t_i^-$, it obviously solves the Euler-Lagrange equation.
2, $\gamma(t;K,K',m,m')$ has no corner at $\tilde x_i^-$ and $\tilde x_i^+$ for each $i=0,1,\cdots i_m-1$, i.e. it is smooth for the whole $t\in [-K,K']$. For each $i\in\Lambda_h$, as $\gamma_i^-=\gamma|_{\Delta t_i^-}$ is the minimizer for the curves admissible for the condition (\[constructioneq9\]), it is smooth at each $t^+_{i,j'+1}<\cdots<t^+_{i,j}$.
Indeed, if $i\in\bigcup_{i_j\in\Lambda_h}\{i_j,i_j+1,\cdots,i_{j+1}-1\}$, we obtained from (\[constructioneq7\]) that $$\bar\gamma_i^+(t)\in U_i\qquad \text{\rm when }\ \bar\gamma^+_{i,1}(t)-2k_i\pi\in\Sigma^0_{c_i}+\delta_i,$$ where $\bar\gamma_i^+=\bar\pi_iG_i\tilde\gamma_i^+$. Since the support of $\bar\mu_i$ has no intersection with $U_i$ and $\psi_i$ is closed in $U_i$, while $\mu_i$ is closed and $\psi_i=0$ in the region $\{\bar\gamma^+_{i,1}(t)-2k_i\pi\not\in [-\Delta_i,\Delta_i]\}$, the term $\mu_i$ and $\psi_i$ have no contribution to the Euler-Lagrange equation along $\bar\gamma_i^+$. For other $i$, the conclusion is obtained from (\[constructioneq13\]) by similar argument. This proves the first conclusion.
Recall the disks $\tilde V^-_i$ and $\tilde V^+_{i+1}$ are defined in the covering space $\mathbb{R}^n$. We claim that $\tilde\gamma$ does not touch the boundary of $\tilde V_i^-\times \tilde V_{i+1}^+\times [T_i^+-d_i,T_i^++d_i]$ for $i\in\bigcup_{i_j\in\Lambda_h}\{i_j, i_j+1,\cdots,i_{j+1}-1\}$. Let us assume the contrary, i.e. $(\tilde x_{i}^-,\tilde x_{i+1}^+, \Delta t_{i}^+)\in\partial (\tilde V_i^-\times \tilde V_{i+1}^+\times [T_i^+-d_i,T_i^++d_i])$ holds for some $i\in\bigcup_{i_j\in\Lambda_h}\{i_j, i_j+1,\cdots,i_{j+1}-1\}$. Let $\hat x_i^-=\bar\pi_i G_i^{-1}\tilde x_i^--(2k_i\pi,0,\cdots,0)$ and $\hat x_{i+1}^+=\bar\pi_i G_i^{-1}\tilde x_{i+1}^+-(2k_i\pi,0,\cdots,0)$. By the condition (\[constructioneq16\]) we see that $\hat x_i^-\in V_i^-$ and $\hat x_{i+1}^+\in V^+_{i+1}$. Then, in $x\to G_i^{-1}x$-coordinates, we obtain from (\[constructioneq6\]) and (\[constructioneq18\]) that $$\begin{aligned}
&h_{c_{i}}^{\Delta t_{i}^-}(\pi_i\hat x_{i}^+,\pi_i\hat x_{i}^-)+h^{\Delta t_i^+} _{\eta_i,\mu_i,\psi_i}(\hat x_{i}^-,\hat x_{i+1}^+)+h^{\Delta t_{i+1}^-}_{c_{i+1}}(\pi_i\hat x_{i+1}^+,\pi_i\hat x_{i+1}^-)\\
\ge &h_{c_{i}}^{\infty}(\xi,\pi_i\hat x_{i}^-)+ h^{\Delta t_i^+}_{\eta_i,\mu_i,\psi_i} (\hat x_{i}^-,\hat x_{i+1}^+)
+h^{\infty}_{c_{i+1}}(\pi_i\hat x_{i+1}^+,\zeta)+h_{c_{i}}^{\infty}(\pi_i\hat x_{i}^+,\xi)\\
&+h^{\infty}_{c_{i+1}}(\zeta,\pi_i\hat x_{i+1}^-)-2\epsilon_i^*\\
\ge &h_{c_{i}}^{\infty}(\xi,\pi_i\bar x_{i}^-)+h^{T_i^+}_{\eta_i,\mu_i,\psi_i}(\bar x_{i}^-,\bar x_{i+1}^+) +h^{\infty}_{c_{i+1}}(\pi_i\bar x_{i+1}^+,\zeta)+h_{c_{i}}^{\infty}(\pi_i\hat x_{i}^+,\xi)\\
&+h^{\infty}_{c_{i+1}}(\zeta,\pi_i\hat x_{i+1}^-)+3\epsilon_i^* \\
\ge &h_{c_{i}}^{\Delta t_{i}^-}(\pi_i\hat x_{i}^+,\pi_i\bar x_{i}^-)+ h^{T_i^+}_{\eta_i,\mu_i,\psi_i} (\bar x_{i}^-,\bar x_{i+1}^+)
+h^{\Delta t_{i+1}^-}_{c_{i+1}}(\pi_i\bar x_{i+1}^+,\pi_i\hat x_{i+1}^-)+\epsilon_i^*\end{aligned}$$ where $\bar x_{i}^-$ and $\bar x_{i+1}^+$ are the intersection points of a curve in $\mathscr{C}_{\eta_i,\mu_i,\psi_i}$ with $V_{i}^-$ and with $V_{i+1}^+$ respectively, $\xi\in\mathcal{M}(c_{i-1})$ and $\zeta\in\mathcal{M}(c_i)$. This contradicts the minimality of $\gamma$, thus it verifies our claim.
To see that the curve $\tilde\gamma$ is smooth at $x_{i}^-$, let us assume the contrary again. Let $x'=\tilde\gamma(t_{i}^--\delta)$ and $x^*=\tilde\gamma(t_{i}^-+\delta)$, here $\delta$ is chosen so small that $\Delta t_{i}^+\pm\delta\in [T_i^+-d_i,T_i^++d_i]$. This is possible since $(\hat x_{i}^-,\hat x_{i+1}^+,\Delta t_{i}^+)\not\in\partial (V_i^-\times V_{i+1}^+\times [T_i^+-d_i,T_i^++d_i])$ implies that $T_i^+-d_i<\Delta t_{i}^+<T_i^++d_i$. We join these two points by a minimizer $\xi:[-\delta,\delta]\to M$ with $\xi(-\delta)=x'$ and $\xi(\delta)=x^*$ $$[A_{c_{i}}(\xi|_{[-\delta,\delta]})]=\inf_{\stackrel{\zeta(-\delta)=x'}{\scriptscriptstyle \zeta(\delta)=x^*}} \int_{-\delta}^{\delta}(L-\eta_{i})(d\xi(s))ds.$$ If $\xi$ passes through $V^-_{i}$, we obtain a curve $\gamma'$ by replacing the segment of the minimizer $\gamma|_{[t_{i}^--\delta,t_{i}^-+\delta]}$ with $\xi:[-\delta,\delta]\to M$. Let $t'^-_i$ be the time for $\gamma'$ passing through $V^-_i$, then $\frac 12(t_{i+1}^+-t'^-_i)\in\Delta T_i^+$ and $\frac 12(t'^-_{i}-t^+_i)\in\Delta T_i^-$. Thus, we obtain an absolutely continuous curve which is admissible for each required condition (see (\[constructioneq18\]) and (\[constructioneq20\])). Along this curve we obtain smaller action $h_{L}^{K,K'}(m,m')$, but this is absurd. So, we only need to show that $\xi$ passes through $V^-_{i}$. Indeed, as $V_i^-$ is chosen small and transversal to the local connecting curve in $\mathscr{C}_{\eta_i,\mu_i,\psi_i}$, $\gamma(t)$ approaches $V_i^-$ from different sides as $t\downarrow t_i^-$ and $t\uparrow t_i^-$ respectively. Otherwise, the minimality of $\gamma$ would be violated. One refers to [@BCV] for the details. The smoothness at $x_{i+1}^+$ can be proved similarly.
The smoothness of $\gamma$ at $t=t_i^{\pm}$ for $i\in\bigcup_{i_j\in\Lambda_c}\{i_j, i_j+1,\cdots,i_{j+1}-1\}$ is obvious. Because of the formulae (\[constructioneq14\]) and (\[constructioneq15\]), $\tilde\gamma$ does not touch the boundary of $\tilde V^{\pm}_i$ at the time of $t^{\pm}_i$ respectively. Indeed, $\tilde\gamma$ approaches $\tilde V_i^{\pm}$ from different sides as $t\downarrow t_i^{\pm}$ and $t\uparrow t_i^{\pm}$ respectively. If $\tilde\gamma$ has a corner at $t=t^{\pm}_i$, let $\zeta$: $[t^{\pm}_i-\delta, t^{\pm}_i+\delta]\to\mathbb{R}^n$ be the minimizer of the action $$A(\zeta|_{[t^{\pm}_i-\delta, t^{\pm}_i+\delta]})=
\inf_{\stackrel{\scriptscriptstyle\xi(t^{\pm}_i-\delta) = \tilde\gamma(t^{\pm}_i-\delta)}{\scriptscriptstyle \xi(t^{\pm}_i+\delta) = \tilde\gamma(t^{\pm}_i+\delta)}} \int_{t^{\pm}_i-\delta}^{t^{\pm}_i+\delta}L_{c_i}(d\xi(s))ds,$$ then the curve $\zeta$ passes through the disk $\tilde V^{\pm}_i$. Replacing $\tilde\gamma|_{[t^{\pm}_i-\delta, t^{\pm}_i+\delta]}$ by this minimizer $\zeta$ one obtains a curve with smaller action. The contradiction verifies the smoothness.
If $c_{i-1}$ is connected to $c_i$ by a type-$h$ orbit and $i\in\bigcup_{i_j\in\Lambda_c}\{i_j, i_j+1,\cdots,i_{j+1}-1\}$, then $V^+_i$ is a small disk. By the same argument as above, one obtains the smoothness of $\tilde\gamma$ at $t=t^-_i$ and the smoothness at $t=t^+_i$ from the arguments for type-$h$.
As the system is autonomous, the following limit exists $$h_{L}^{\infty}(m,m')=\lim_{K,K'\to\infty}h_{L}^{K,K'}(m,m').$$ We pick out a sequence of $\gamma(t;K,K',m,m')$ for large $K$ and $K'$. Obviously, the set $\{\gamma (t;K,K',m,m')\}$ has at least one accumulation point $\gamma_{\infty}$: $\mathbb{R}\to M$ with the property $\alpha(d\gamma_{\infty})\subseteq\tilde{\mathcal{A}}(c)$ and $\omega(d\gamma_{\infty}) \subseteq\tilde{\mathcal{A}}(c')$. As we have shown, it is an orbit of $\phi_{L}^t$. This proves the first conclusion of the theorem.
For any two points $x,x'\in M$, the sequence $\{\gamma(t;K,K',x,x')|_{[0,K]}\}$ approaches to a forward $c$-semi static curve as $K\to\infty$, which starts from the point $x$, and $\{\gamma(t;K,K',x,x')|_{[t^-_{i_m},K']}\}$ approaches to a backward $c'$-semi static curve as $K'\to\infty$, which approach to the point $x'$. Therefore, for sufficiently large $K,K'$, the initial value $(\gamma,\dot\gamma)|_{t=0}$ falls into any prescribed $\delta$-neighborhood of the points $(x,v_{x,c}^{+})$ and the orbit reaches the $\delta$-neighborhood of $(x',v_{x,c'}^{-})$ at the time $t=K'$. This completes the proof.
The proof for time-periodic system is similar, and a bit easier from technical point of view, since one can treat the time variable $t$ as the first angle variable and take $\{t=0\}$ the section for all classes. One does not need to introduce various coordinate systems $\{G_i^{-1}\}$ for different cohomology class. We omit the details here.
Proof of the main theorem
=========================
Once one obtains the existence of a generalized transition chain in the system (\[introeq1\]), Theorem \[mainthm\] is proved by applying Theorem \[constructionthm1\]. Therefore, the main purpose of this section is to show the genericity of such transition chains.
Candidate of transition chain
-----------------------------
Let us consider the Hamiltonian (\[introeq1\]). Any integer vector $\tilde k\in\mathbb{Z}^3$ determines a plane $\Sigma_{\tilde k}=\{\tilde\omega\in\mathbb{R}^3:\langle\tilde\omega,\tilde k\rangle=0\}$, which passes through the origin. Let $\Omega_E=\{\tilde\omega=\nabla h(\tilde y): \tilde y\in h^{-1}(E)\}\subset\mathbb{R}^3$, it is diffeomorphic to a $2$-sphere with the origin inside if $E>\min h$, because the Hamiltonian $h$ is assumed convex. Thus, the set $\Sigma_{\tilde k}\cap\Omega_E$ is a closed curve, denoted by $\Gamma_{\tilde\omega,\tilde k}$. Given any positive number $\delta>0$, some positive integer $K_{\delta}$ exists such that $\cup_{\|\tilde k\|\le K_{\delta}}\Gamma_{\tilde\omega,\tilde k}$ constitutes a $\delta$-grid on $\Omega_E$ in the sense that the $M_h^{-1}\delta$-neighborhood of $\cup_{\|\tilde k\|\le K_{\delta}}\Gamma_{\tilde\omega,\tilde k}$ cover the whole sphere $\Omega_E$, where $M_h=\max_{\tilde y\in h^{-1}(E)}\|\partial^2 h(\tilde y)\|$. Therefore, there exists a resonant path $$\Gamma_{\tilde\omega}=\Gamma_{\tilde\omega,0}\ast\Gamma_{\tilde\omega,1}\ast\cdots\ast \Gamma_{\tilde\omega,m}.$$ such that each rotation vector falls into its $M_h^{-1}\delta$-neighborhood, where $\Gamma_{\tilde\omega,\ell}$ represents a resonant path determined by one resonant relation (one integer vector). It is possible that $\Gamma_{\tilde\omega,\ell}$ and $\Gamma_{\tilde\omega,\ell'}$ are determined by the same resonant relation $\tilde k_{\ell}=\tilde k_{\ell'}$.
![The resonant path in the surface of $h^{-1}(E)$.[]{data-label="fig12"}](Ardiff12.eps){width="4cm" height="4cm"}
Obviously, the map $\partial h$ is a global diffeomorphism, which maps each closed curve $\Gamma_{\tilde\omega,\ell}$ onto the $2$-sphere $h^{-1}(E)$, $\Gamma_{\ell}=\partial h^{-1}\Gamma_{\tilde\omega,\ell}$. The $\delta$-grid on $\Omega_E$ induces a $\delta$-grid on $h^{-1}(E)$: the $\delta$-neighborhood of $\cup_{\|\tilde k\|\le K_{\delta}}\Gamma_{\tilde k}$ covers the whole sphere. Under the inverse of the frequency map $\tilde\omega\to\tilde y=(\nabla h)^{-1}(\tilde\omega)$ we obtain a path $$\Gamma=\Gamma_{0}\ast\Gamma_{1}\ast\cdots\ast\Gamma_{m}$$ in action variable space, where $\Gamma_{\ell}=(\nabla h)^{-1}\Gamma_{\tilde\omega,\ell}$.
Let $\ell(\dot{\tilde x})=\max_{\tilde y}(\langle\dot{\tilde x},\tilde y\rangle-h(\tilde y))$ be the Lagrangian determined by the Hamiltonian $h$, $\phi_{\ell}^t$ be the Lagrange flow. As the system is integrable, the action variable $\tilde y$ keeps constant along each orbit of $\phi_{\ell}^t$ which obviously lies in the support of certain $c$-minimal measure with $\tilde c=\tilde y$. In this sense, one obtains a path $\Gamma_c\subset H^1(\mathbb{T}^3, \mathbb{R})$ and $\Gamma_c=\Gamma$ if we identify $H^1(\mathbb{T}^3,\mathbb{R})=\mathbb{R}^3$.
By the study of normal form, finitely many points $\tilde y_0,\tilde y_1,\cdots \tilde y_N\in\Gamma$ exist such that each $\tilde\omega_i=\nabla h(\tilde y_i)$ is rational frequency vector with period $T_i\le K_0\epsilon^{-\varrho}$ and $$\bigcup_{0\le i\le N}\{\tilde y:\|\tilde y-\tilde y_i\|\le\mu T_i^{-1}\epsilon^{\sigma}\}\supset \Gamma+\frac {\mu}2 \epsilon^{\frac 13},$$ where $\varrho=(1-3\sigma)/3$, $\sigma<1/6$, see (\[normaleq9\]). Obviously, $N$ depends on $\epsilon$, the size of perturbation. Under five steps of KAM iteration, we obtain the normal form $$H_i(\tilde x,\tilde y)=\tilde h(\tilde y)+\epsilon\tilde Z_{\epsilon,i}(\tilde x,\tilde y)+\epsilon\tilde R_{\epsilon,i}(\tilde x,\tilde y),$$ which is valid in the domain $\{\tilde y:\|\tilde y-\tilde y_i\|\le\mu T_i^{-1} \epsilon^{\sigma}\}\times\mathbb{T}^3$. In which $\|\tilde R_{\epsilon,i}\|_{C^2}= O(\epsilon^{\frac 1{21}})$ and $\tilde Z_{\epsilon,i}$ is resonant with respect to $\omega_i$ $$\tilde Z_{\epsilon,i}(\tilde x,\tilde y)=\sum_{\langle\tilde k,\omega_i\rangle=0}\tilde Z_{\epsilon,i,\tilde k}(\tilde y)e^{i\langle\tilde k, \tilde x \rangle},$$ where the summation is made over all those $\tilde k$ spanned by $(\tilde k_i,\tilde k'_i)$: $\tilde k=j_1\tilde k_i+j_2\tilde k'_i$ in which $\|\tilde k_i\|,\|\tilde k'_i\|\le K_{\delta}$, $(\tilde k_i,\tilde k_i)$ is irreducible and the integer vector $\tilde k_i$ is used to determine the resonant path $\Gamma_i$, i.e. $\langle\nabla\tilde h(\tilde y),\tilde k_i\rangle=0$ holds for each $\tilde y\in\Gamma_i$.
For these two integer vectors $(\tilde k_i,\tilde k_i)$ there is another $\tilde k^*_i\in\mathbb{Z}^3$ such that the matrix $I_i=(\tilde k_i,\tilde k'_i,\tilde k^*_i)$ is uni-module. There are infinitely many $\tilde k^*_i$ satisfying the condition, we choose one so that its norm is as small as it can be. The coordinate transformation: $$\label{chaineq1}
\tilde q=I_i^t\tilde x,\qquad \tilde p=I_i^{-1}\tilde y.$$ is obviously symplectic and $H_i(\tilde p,\tilde q)=H(I_i^{-t}\tilde x,I_i\tilde y)$ is also a function of $\tilde q$ defined in $\mathbb{T}^n$. Let $\tilde y$ be the point where $\nabla h_i(\tilde y)=\tilde\omega$, then the gradient of $h_i(\tilde p)=h(I_i\tilde y)$ satisfies $$\tilde\omega_i=\nabla h_i(p)=(0,0,\omega_{i3}),$$ and $\tilde Z_{\epsilon,i}(\tilde p,\tilde q)=\tilde Z_{\epsilon,i}(I_i\tilde y,I_i^{-t}\tilde x)$ is independent of $q_3$, namely, $\tilde Z_{\epsilon,i}=\tilde Z_{\epsilon,i}(p,p_3,q)$ if we write $p=(p_1,p_2)$ and $q=(q_1,q_2)$.
Let us still use $(\tilde x,\tilde y)$ to denote the new coordinate system. Therefore, around a strong resonance point $\tilde\omega_i$ the normal form takes the form $$\label{chaineq2}
H_i(\tilde x,\tilde y)=h_i(\tilde y)+\tilde Z_{\epsilon,i}(x,y,y_3)+\tilde R_{\epsilon,i}(\tilde x,\tilde y)$$ in the new coordinate system (\[chaineq1\]). This form remains valid in $\mathbb{T}^3\times\{\|I_i(\tilde y-\tilde y_i)\|<T_i^{-1}\epsilon^{\sigma}\}$ and at $\tilde y=\tilde y_i$ one has $\nabla\tilde h_i=(0,0,\omega_3)$ with $\omega_3\neq 0$.
For our purpose, it is not necessary consider the Hamiltonian $H_i$ on the whole disk $\{\tilde y:\|\tilde y-\tilde y_i\|\le\mu T_i^{-1}\epsilon^{\sigma}\}$. Instead, we choose finitely many $\tilde y_{ij}\in\Gamma_i$ with $\tilde y_{i0}=\tilde y_i$ such that $$\cup_j\{\|\tilde y-\tilde y_{ij}\|<2K\sqrt{\epsilon}\}\supseteq\Gamma_i+ K\sqrt{\epsilon},$$ and $$\text{\rm dist}(\tilde y_{ij'},\tilde y_{ij})\ge K\sqrt{\epsilon}\ \ \ \ \ \forall \ j'\neq j.$$ where $K>0$ is a suitably large number. The results obtained in Section 4 and 5 can be applied to the Hamiltonian when it is restricted on each domain $\mathbb{T}^3\times\{\|y-y_{ij}\|<K\sqrt{\epsilon}\}$, especially on the domain $\mathbb{T}^3\times\{\|y-y_{i0}\|<K\sqrt{\epsilon}\}$.
Let $Y_{i}(x,y,\tau)$ be the solution of the equation $H_i(x,-\tau,y,Y_{i})=E$, where $H_i$ is given by (\[chaineq2\]). It can be written in the form of $$Y_{i}=h_{i}(y)+\epsilon Z_{i}(x,y)+\epsilon R_{i}(x,y,\tau).$$ The truncated form of $Y_{i}$ $$Y_{i,T}=h_{i}(y)+\epsilon Z_{i}(x,y)$$ is determined by $H_{i,T}=\tilde h(y)+\tilde Z_{\epsilon,i}(x,\tilde y)$, the truncated form of $H_i$. Denote $\tilde y_{ij}=(y_{ij},y_{ij,3})$ and let $y-y_{ij}=\sqrt{\epsilon}p$, $s=\sqrt{\epsilon}\tau$, we obtain from $Y_i$ the Hamiltonian $$G_{ij,\epsilon}=\frac 1{\sqrt{\epsilon}}\langle\omega_{ij},p\rangle+\frac 12\langle A_{ij}p,p\rangle+V_{ij}(x)+ Z_{ij,\epsilon}(x,\sqrt{\epsilon}p)+R_{ij,\epsilon}(x,\sqrt{\epsilon}p, s/\sqrt{\epsilon}),$$ for $j=0$ we have $$G_{i0,\epsilon}=G_{i,\epsilon}=\frac 12\langle A_{i}p,p\rangle+V_{i}(x)+ Z_{i,\epsilon}(x,\sqrt{\epsilon}p) +R_{i,\epsilon}(x,\sqrt{\epsilon}p,s/\sqrt{\epsilon}),$$ where $\omega_{ij}=\partial h_{i}(y_{ij})$, $A_{ij}=\partial^2 h_{i}(y_{ij})$, $A_{i}=\partial^2 h_{i}(y_i)$, $V_{ij}(x)=Z_{j}(x,y_{ij})$, $V_{i}(x)=Z_{j}(x,y_{i})$, $\|R_{ij,\epsilon}\|_{C^2},\|R_{i,\epsilon}\|_{C^2} =O(\epsilon^{\frac 1{21}})$ and $\|Z_{ij,\epsilon}\|_{C^2},\|Z_{i,\epsilon}\|_{C^2}=O(\sqrt{\epsilon})$ where the $C^2$-norm is with respect to $(x,p)$ only. By the choice of $y_{ij}$ and the convexity of $h$ we can see that $$\|\omega_{ij}\|\ge m_hK\sqrt{\epsilon}$$ where $m_h$ is the lower bound of the eigenvalues of $\partial^2h$. Therefore, the $\alpha$-function $\alpha_{G_{ij,\epsilon}}$ for the Lagrangian determined by $G_{ij,\epsilon}$ does not reach its minimum when the action variable is restricted on the disk $\|p\|\le K$. As the frequency $\mathbb{R}^2\ni\omega_{ij}\neq 0$ satisfies certain resonant condition, the existence of normally hyperbolic cylinder is guaranteed by Theorem \[AppenHyperTh1\] (see Appendix B) for generic $V_{ij}$. Therefore, all functions $G_{ij,\epsilon}$ ($j\neq 0$) are treated as [*a priori*]{} unstable Hamiltonian and the Hamiltonian $G_{i,\epsilon}$ is considered as the problem of double resonance.
Recall the Fenchel-Legendre transformation $\mathscr{L}_{\beta}$: $H_1(M,\mathbb{R})\to H^1(M,\mathbb{R})$, determined by the $\beta$-function. Let $\beta_h$, $\beta_{H_{i,T}}$ and $\beta_{H_{i}}$ be the $\beta$-function for $h$, $H_{i,T}$ and $H_{i}$ respectively. Obviously, $\mathscr{L}_{\beta_h}(\Gamma_{\omega,i})$ is still a curve. As it was studied in Subsection 4.3, $\mathscr{L}_{\beta_{H_{i}^T}}(\Gamma_{\omega,i})$ is composed of a flat $\mathbb{F}_0$ joined with two channels. See Figure \[fig13\] below. These channels are joined to the flat either at a point or along an edge. The former case was thought difficult to handle.
![The transition chain under $\pi_3:\alpha^{-1}(E)\to\mathbb{R}^2$, represented by the thick solid red curve. Along the segment from $B$ to $C$, $c_3$ keeps constant. The purple dashed curve represents the curve $\mathscr{L}_{\beta_h}$.[]{data-label="fig13"}](Ardiff13.eps){width="9.5cm" height="4cm"}
Transition chain of incomplete intersection
-------------------------------------------
Let $\alpha_{H_i}, \alpha_{H_{i,T}}$ be $\alpha$-function determined by the Hamiltonian $H_i,H_{i,T}$ respectively. As it has been studied in the subsection 5.3, the double resonance corresponds to a flat $\mathbb{F}_0\subset\alpha_{H_{i}}^{-1}(E)$, around which there exists a annulus of incomplete intersection $$\tilde{\mathbb{A}}_T=\{(c_1,c_2,c_3)\in\alpha^{-1}_{H_{i,T}}(E):0<c_3\le\epsilon\Delta_0\}.$$ The following has been proved generic in Theorem \[beltthm2\]. For each $\tilde c\in \tilde{\mathbb{A}}_T$, the Mañé set does not cover the whole $3$-torus. Thus, some $d_i>0$ exists such that for each $\tilde c\in \tilde{\mathbb{A}}_T$ the set $$N_{\tilde c,d_i}=\{x\in\mathbb{T}^3:U^-_{\tilde c}(x)-U'^+_{\tilde c}(x)<d_i\epsilon\}$$ does not cover the whole $3$-torus, where $U^-_{\tilde c}$ and $U'^+_{\tilde c}$ are the elementary weak KAM solutions. Such results are obtained under the hypothesis ([**H1$\sim$4**]{}) proposed in the section 5.
As the truncated system is independent of $x_3$, the Mañé set for $H_{i,T}$ is independent of $x_3$ $$\mathcal{N}_{H_{i,T}}(\tilde c)|_{\Sigma_s}=\mathcal{N}_{H_{i,T}}(\tilde c)|_{\Sigma_{s'}}, \qquad N_{\tilde c,d_i}|_{\Sigma_s}=N_{\tilde c,d_i}|_{\Sigma_{s'}}$$ where $\Sigma_s$ is a co-dimension one section on which $x_3=s$. Let $\pi_3$: $\mathbb{R}^3\to\mathbb{R}^2$ be the standard projection: $\pi_3(x_1,x_2,x_3)=(x_1,x_2)$, let $\alpha_{Y_{i,T}}$ and $\alpha_{Y_{i}}$ be the Lagrangian determined by $Y_{i,T}$ and $Y_i$ respectively, $\mathcal{N}_{Y_{i,T}}$ and $\mathcal{N}_{Y_i}$ denote the Mañé set for the Lagrangian determined by $Y_{i,T}$ and $Y_i$ respectively. As $Y_{i,T}(x,y)$ solves the equation $H_{i,T}(x,y,Y_{i,T})=\alpha_{H_i}(c)$, one has $\pi_3\mathcal{N}_{H_{i,T}}(\tilde c)= \mathcal{N}_{Y_{i,T}}(c)$. If $\mathcal{N}_{H_{i,T}}(\tilde c)$ does not cover the $3$-torus, $\mathcal{N}_{Y_{i,T}}(c)$ does not cover the $2$-torus. Because of $\alpha_{Y_{i,T}}(c)>\min\alpha_{Y_{i,T}}$, each $c$-minimal measure possesses non-zero rotation vector. Therefore, there exists some circle $\Sigma^1_c$ non-degenerately embedded into the $2$-torus such that each $c$-minimal curve passes through $\Sigma^1_c$ transversally and $\mathcal{N}_{Y_{i,T}}(c)|_{\Sigma^1_c}$ is topologically trivial, i.e. some open intervals $I_{\ell}\subset\Sigma^1_c$ exist such that $$\bigcup I_{\ell}\supset\mathcal{N}_{Y_{i,T}}(c)|_{\Sigma^1_c},\qquad I_{\ell}\cap I_{\ell'}=\varnothing,\ \ \forall\ \ell\neq\ell'.$$ One can suitably choose $I_{\ell}$ so that $$\label{chaineq3}
\bigcup I_{\ell}\times\{x_3\in\mathbb{R}:\mod 2\pi\}\supset N_{\tilde c,d_i}.$$
As $\|H_i- H_{i,T}\|_{C^2}\le O(\epsilon^{1+\frac 1{21}})$, some $\epsilon_i>0$ exists such that the Mañé set for the Hamiltonian $H_i$ $$\label{chaineq4}
\mathcal{N}_{H_i}(\tilde c)\subset N_{\tilde c,d_i}, \qquad \forall \ \epsilon<\epsilon_i.$$
Let $\Gamma_{Y_i}=\tilde{\mathbb{A}}\cap\{c_3=Y_i\}$ where $$\tilde{\mathbb{A}}=\{(c_1,c_2,c_3)\in\alpha^{-1}_{H_i}(E):0<c_3\le\epsilon\Delta_0\}.$$ It is a closed curve. By the preliminary works as above, some $c$-equivalence along the curve is established. Indeed, for each $\tilde c\in\Gamma_{Y_i}$, let $$\Sigma_{\tilde c}=\Sigma_c^1\times\{x_3\in\mathbb{R}\mod 2\pi\}.$$ By the construction, each $\tilde c$-semi static curve passes through the section $\Sigma_{c}$ transversally. Recall $$V_{\tilde c}=\bigcap_U\{i_{U*}H_1(U,\mathbb{R}): U\, \text{\rm is a neighborhood of}\, \mathcal {N}(\tilde c)\cap\Sigma_{\tilde c}\},$$ one sees that $\tilde c'-\tilde c\in V^{\perp}_{\tilde c}$ provided $\tilde c'$ is close to $\tilde c$, $c'_3=c_3$ and $\alpha_{H_i}(\tilde c')=\alpha_{H_i}(\tilde c)$, i.e. $c'\in\Gamma_{Y_i}$. In this case, some open set $U\supset\mathcal{N}_{H_i}(c)\cap\Sigma_{\tilde c}$ such that $V_{\tilde c}= i_{U*}H_1(U,\mathbb{R})=\text{\rm span}\{(0,0,1)\}$, from which one obtains that $V_{\tilde c}^{\perp} =\text{\rm span}\{(1,0,0),(0,1,0)\}$. For each class $\tilde c'\in\Gamma_{Y_i}$ close to $\tilde c$, one has $\tilde c'-\tilde c=(\Delta c_1,\Delta c_2,0)\in V_{\tilde c}^{\perp}$, thus, there exists a closed 1-form $\bar\mu$ such that $[\bar\mu]=c'-c$ and $$\text{\rm supp}\bar\mu\cap\mathcal{N}_{H_i}(\tilde c)\cap\Sigma_{\tilde c}=\varnothing.$$ Thus, any two classes along the curve $\Gamma_{Y_i}$ is equivalent. Therefore, a transition chain for incomplete intersection is established, see Figure \[fig13\], the thick solid red curve from the point $B$ to the point $C$.
Transition chain for complete intersection
------------------------------------------
By the study in the subsection 4.2 (see Theorem \[cylinderthm2\]), there are two wedge-shaped channels $\tilde{\mathbb{W}}_g =\cup_{\lambda\ge\lambda_0>0}\mathscr{L}_{\beta}(\lambda g)$ and $\tilde{\mathbb{W}}_{g'} =\cup_{\lambda\ge\lambda'_0>0} \mathscr{L}_{\beta}(\lambda g')$ which extend into the annulus $\tilde{\mathbb{A}}$. Corresponding to these two channels there exist two normally hyperbolic cylinder $\tilde\Pi_{E_0,E_1,g}$ and $\tilde\Pi_{E'_0,E'_1,g'}$ respectively, which are three-dimensional and invariant for the Hamiltonian flow: for each $\tilde c=(c_1,c_2,c_3)\in \tilde{\mathbb{W}}_g$, the Mañé set $\tilde{\mathcal{N}}_{H_i}(\tilde c)\subset\tilde\Pi_{E_0,E_1,g}$ if $c_3\ge 2\epsilon^{1+d}$.
We are now in the situation that there is a normally hyperbolic cylinder $\tilde\Pi$ homeomorphic to $I\times\mathbb{T}^2$, the Aubry set is located on this cylinder for each cohomology class under consideration. If the Aubry set is a two-dimensional torus, it has its own stable and unstable manifold. It implies that the forward (backward) weak KAM solution is differentiable when it is restricted in a neighborhood of this 2-torus. Because weak KAM is a viscosity solution, any $C^1$ viscosity solution for Tonelli Hamiltonian must be $C^{1,1}$ [@CS; @FS; @Ri]. Therefore, in a small neighborhood of the Aubry set, the stable and unstable manifold are Lipschitz graphs. As the cylinder is smooth, the Aubry set is also a Lipschitz graph over two-torus.
Let $\Sigma\subset H^{-1}(E)$ be a four-dimensional section intersecting each orbit in the Aubry sets transversally. The set $\Pi=\Sigma\cap\tilde\Pi$ is a two-dimensional cylinder. In a neighborhood of $\Pi$ the Hamiltonian flow defines a return map on the section $\Sigma$. Restricted on the cylinder $\Pi$, each Aubry set is either periodic orbit, or Aubry-Mather set or invariant circle. Each circle is a Lipschitz curve. A piece of the cylinder $\Pi$, bounded by two invariant circles, is invariant for the return map which preserves some “area" element. Let $\psi$: $\Pi_0=[0,1]\times\mathbb{T}\to\Pi$ be the map, it pulls back the standard closed 2-form $\omega=dx\wedge dy$ to a 2-form on $\Pi$. Since the second de Rham cohomology of a cylinder is trivial, by Moser’s theorem on the isotopy of symplectic forms, there exists a diffeomorphism $\psi_1$ which transforms this form to the standard 2-form, namely $$(\psi\circ\psi_1)^*\omega=d\theta\wedge dI.$$ Since the return map $\Phi_H$ preserves the form $\omega$, one has $$((\psi\circ\psi_1)^{-1}\circ\Phi_H\circ(\psi\circ\psi_1))^*d\theta\wedge dI=d\theta\wedge dI.$$ Let us consider those Aubry sets which are invariant two-torus, denoted by $\Upsilon_c$. We use the same notation for their intersection with $\Pi$, which are circles. Fix one circle $\Upsilon_{c_0}$, other circles are parameterized by the “area" $\sigma$. Given any other circle $\Upsilon_c$, we obtain the algebraic area $\sigma$ of the region bounded by these two circles. If each circle is regarded as the graph of a function, then there is a regularity result [@CY1] $$\|\Upsilon_{c(\sigma)}-\Upsilon_{c(\sigma')}\|_{C^0}\le C_1\sqrt{|\sigma-\sigma'|}.$$ Because the cylinder is normally hyperbolic, there is an segment of a line $I_{\sigma}\subset \alpha^{-1}(E)$ such that all cohomology classes located in this segment share the same Aubry set, an invariant 2-torus, so we have a map $\sigma\to I_{\sigma}$.
For a small segment of cylinder, some neighborhood $N\subset\mathbb{T}^3$ of a two-torus exists so that all Aubry sets on this cylinder fall into this neighborhood: $\mathcal{A}(c)\subset N$. In a suitably coordinate system we take a finite covering space $\check{M}$ so that the lift of $N$ consists of two connected components $N_l$ and $N_r$. The Mañé set satisfies the condition $$\mathcal{N}(c,\check M)\backslash(N_{l}\cup N_{r})\neq\varnothing.$$ To construct transition chain in this situation, one need to show it consists of totally disconnected semi-static curves when the Aubry set is a two-torus.
Let us consider the covering space $\pi_1: \bar M=\mathbb{R}\times\mathbb{T}^2$ such that the lift of $N$ contains infinitely many connected components, each of which is still a neighborhood of two-torus. We consider two adjacent components $N_l$ and $N_r$ in the lift of $N$, i.e. $\pi_{1}N_l=\pi_{1}N_r=N$ and no other component in the lift is located between them. The subscript $r$ means “right" and $l$ means “left". Correspondingly, denote by $\Upsilon_{l,\sigma}$ and $\Upsilon_{r,\sigma}$ the connected component in the lift of $\Upsilon_{\sigma}$ respectively, $\Upsilon_{l,\sigma}\subset N_l$ and $\Upsilon_{r,\sigma}\subset N_r$. The barrier function takes the form $$u^-_{l,\sigma}-u^+_{r,\sigma}\ \ \ \ \text{\rm or}\ \ \ \ u^-_{r,\sigma}-u^+_{l,\sigma}$$ where $u^{\pm}_{l,\sigma}$ and $u^{\pm}_{r,\sigma}$ are the elementary weak KAM solution determined by $\Upsilon_{l,\sigma}$ and $\Upsilon_{r,\sigma}$ respectively. The elementary weak-KAM solution $u^{\pm}_{l,\sigma}$ is uniquely determined by $I_{\sigma}$, all classes in $I_{\sigma}$ share the same elementary weak-KAM solution. It is why we use the subscript $\sigma$. A point $\pi_1x\in\mathcal{N}(c)$ if and only if $$x\in\arg\min(u^-_{l,\sigma}-u^+_{r,\sigma}), \ \ \ \text{\rm or} \ \ \ x\in\arg\min(u^-_{r,\sigma}-u^+_{l,\sigma}).$$ Let $M_0$ be a segment of $\mathbb{R}\times\mathbb{T}^2$ bounded by $\Upsilon_{l,c}$ and $\Upsilon_{r,c}$. The problem turns out to be the version: whether does the set $\arg\min(u^{-}_{l,\sigma}-u^{+}_{r,\sigma})|_{M_0\backslash(N_l\cup N_r)}$ consist of totally disconnected semi-static curves?
We only need to follow the argument in [@CY1; @CY2; @LC] if we are satisfied with the generic property in the category of Lagrangian, where the perturbations are functions also defined on $TM$: $L(x,\dot x)\to L(x,\dot x)-L_{\delta}(x,\dot x)$. In this paper, we are also going to prove the generic property in the sense of Mañé, i.e. the perturbations are imposed on the potential $L(x,\dot x)\to L(x,\dot x)-V(x)$.
Let us construct the potential perturbations. Choose a 2-dimensional disk $D$ which transversally intersects the backward semi-static curves $\gamma_{x,\sigma_0}^-:(-\infty,0]\to\bar M$ with $\gamma_{x,\sigma_0}^-(0)=x\in D$. These curves approach $\Upsilon_{l,\sigma_0}$ as $t\to -\infty$. In suitable coordinate system we can assume that $D$ is located in the section $$D+d_1=\{(x_1,x_2,x_3):x_1=x_{10},|x_2-x_{20}|\le d+d_1, |x_3-x_{30}|\le d+d_1\}$$ where $(x_{10},x_{20},x_{30})=x_0$. Let $D=(D+d_1)|_{d_1=0}$. We write the curve $\gamma^-_{x_0,\sigma_0}$ in the coordinate form $$\gamma^-_{x_0,\sigma_0}(t)=(x_{10}(t),x_{20}(t),x_{30}(t))$$ where $x_{10}$ is monotonely increases for $t\in [-T,0]$. Since continuous function can be approximated by smooth function, for any small $\delta>0$, a tubular neighborhood of the semi-static curve $\gamma_{x_0,\sigma_0}^-|_{[-T,0]}$ admits smooth foliation of curves $\zeta_{x}$: $(x,t)\in (D+d_1)\times[-T,0]\to\mathbb{T}^3$ such that each semi-static curve $\gamma^-_{x,\sigma_0}|_{[-T,0]}$ remains $\delta$-close to $\zeta_{x}$ in the sense that $d(\zeta_x(t),\gamma_{x,\sigma_0}^-(t))<\delta$ for all $t\in[-T,0]$. The tubular neighborhood is defined by the form $$\text{\uj C}=\cup_{-T\le t\le 0}\{\zeta_{x}(t):x\in D+d_1\}.$$
Let $\rho$: $(D+d_1)\times\mathbb{R}\to\mathbb{R}$ be a smooth function such that $\rho(x,t)=\rho(x',t)$, $\rho(x,t)=0$ if $t\notin[-T+t_0,-t_0]$ with small $t_0>0$ and $\rho(x,t)>0$ if $x\in (-T+t_0,-t_0)$. As $\zeta_x$ is a smooth foliation of the tubular domain, it can be thought as a differeomorphism $\Psi$: $(D+d_1)\times [-T,0]\to\text{\uj C}$, namely, for $x'\in\text{\uj C}$ there exists unique $(x,t)\in (D+d_1)\times [-T,0]$ such that $\Psi(x,t)=\zeta_x(t)=x'$. With a smooth function $V$: $D+d_1\to\mathbb{R}$ we obtain a smooth function $\bar V$ defined on $\text{\uj C}$ $$\label{completeeq5}
\bar V(x')=\rho(\Psi^{-1}(x'))V(\zeta_{x}(0)),$$ Since unique $(x,t)\in (D+d_1)\times [-T,0]$ is determined by certain $x'\in\text{\uj C}$, some constant $C_2>0$ exists such that $$\label{completeeq6}
\int_{-T+t_0}^{-t_0}\bar V(\zeta_{x}(t))dt=C_2V(x), \qquad \forall x\in D+d_1.$$
We construct the potential perturbation in the form of (\[completeeq5\]) where $V$ ranges over the function space spanned by $$\begin{aligned}
\mathfrak{V}_{2}=&\varepsilon\Big(\sum_{\ell=1,2}a_{\ell}\cos2\ell\pi(x_2-x_{20}) +b_{\ell}\sin2\ell\pi(x_2-x_{20})\Big),\\
\mathfrak{V}_{3}=&\varepsilon\Big(\sum_{\ell=1,2}c_{\ell}\cos2\ell\pi(x_3-x_{30}) +d_{\ell}\sin2\ell\pi(x_3-x_{30})\Big),\end{aligned}$$ where each parameter of $(a_{\ell},b_{\ell},c_{\ell},d_{\ell})$ ranges over an unit interval $[1,2]$. If we construct a grid for the parameters $(a_{\ell},b_{\ell},c_{\ell},d_{\ell})$ by splitting the domain equally into a family of cubes and setting the size length by $$\Delta a_{\ell}=\Delta b_{\ell}=\Delta c_{\ell}=\Delta d_{\ell}=\varepsilon,$$ the grid consists of as many as $[\varepsilon^{-8}]$ cubes.
Let us choose a neighborhood $\mathbb{I}_{\sigma_0}$ of the point $\sigma_0$ which satisfies the conditions:
1, for each $(x,\sigma)$ with $x\in D$ and $\sigma\in\mathbb{I}_{\sigma_0}$, there is a unique backward semi-static curve $\gamma^-_{x,\sigma}$ such that $\gamma^-_{x,\sigma}(0)=x$ and $\gamma^-_{x,\sigma}(t)\to\Upsilon_{l,\sigma}$ as $t\to -\infty$. It is guaranteed by the existence of unstable manifold and if $D$ is chosen close to $\Upsilon_{l,\sigma_0}$. By the definition, $\gamma_{x,\sigma}^-(t)\in\text{\uj C}$ for $t\in [-T,0]$ and $x\in D$, so each $\sigma\in\mathbb{I}_{\sigma_0}$ defines a linear operator $$\label{completeeq7}
\mathscr{K}_{\sigma}\bar V=\int_{-T}^0\bar V(\gamma_{x,\sigma}^-(t))dt;$$
2, as each curve $\gamma_{x,\sigma_0}^-(t)$ stays in $\delta$-neighborhood of the fiber $\zeta_{x}$ for $t\in [-T,0]$ with small $\delta>0$, by choosing suitably small neighborhood $\mathbb{I}_{\sigma_0}$ (depending on the size of $D$) some constant $C_3>0$ exists such that $$\begin{aligned}
\label{completeeq8}
\text{\rm Osc}_{x\in D}(\mathscr{K}_{\sigma}\bar V-\mathscr{K}_{\sigma}\bar V')&=\max_{x,x'\in D}|\mathscr{K}_{\sigma}\bar V(x)-\mathscr{K}_{\sigma}\bar V'(x')|\notag\\
&>2^{-1}C_2\text{\rm Osc}_{x\in D}(V-V')\\
&>C_3\varepsilon\Delta\notag\end{aligned}$$ with $\Delta=\max\{|a_{\ell}-a'_{\ell}|,|b_{\ell}-b'_{\ell}|,|c_{\ell}-c'_{\ell}|, |d_{\ell}-d'_{\ell}|\}$. Indeed, as $V$ is a linear combination of the functions $\{\sin\ell x_j,\cos\ell x_j:\ell =1,2,j=2,3\}$, there exists some number $d=d(D)>0$ depending on the size of $D$ only such that the Hausdorff distance $$d_H(V_{D}^{-1}(\min_DV+\frac 14\Delta),V_{D}^{-1}(\max_DV-\frac 14\Delta))\ge d(D)$$ where $V_{D}^{-1}(\min_DV+\frac 14\Delta)=\{x\in D:V(x)\le\min _DV+\frac 14|(\max_DV-\min_DV)\}$ and $ V_{D}^{-1}(\max_DV-\frac 14\Delta)=\{x\in D:V(x)\ge\max _DV-\frac 14|(\max_DV-\min_DV)\}$. By requiring $\sigma$ suitably close to $\sigma_0$ and using the notation $\pi_x(x,t)=x$, we have $$\pi_x\Psi^{-1}\gamma_{x,\sigma}(t)\in V_{D}^{-1}(\min_DV+\frac 14\Delta)\qquad \text{\rm if}\ V(x)=\min_D V;$$ and $$\pi_x\Psi^{-1}\gamma_{x,\sigma}(t)\in V_{D}^{-1}(\max_DV-\frac 14\Delta)\qquad \text{\rm if}\ V(x)=\max_D V.$$ Therefore, one obtains (\[completeeq8\]) from (\[completeeq5\]), (\[completeeq6\]) and (\[completeeq7\]);
3, for each $\sigma\in \mathbb{I}_{\sigma_0}$ and each $x\in D$, the forward semi-static curve $\gamma_{x,\sigma}^+$, determined by $u^+_{r,\sigma}$ with $\gamma_{x,\sigma}(0)=x\in D$, does not touch the support of $\rho\subset\text{\uj C}$ and approaches $\Upsilon_{r,\sigma}$ as $t$ increases to infinity.
For the perturbed system $L(\dot x,x)-\bar V(x)$, we use $u^+_{r,\sigma,\bar V}$ and $u^-_{l,\sigma,\bar V}$ to denote the weak KAM solution. By the construction of perturbation, the invariant cylinder remains unchanged. Restricted on the disk $D$, the forward weak-KAM solution $u^+_{r,\sigma,\bar V}$ is also unchanged $(u^+_{r,\sigma,V}-u^+_{r,\sigma})|_{x\in D}=0$, but the backward weak KAM solution undergoes small perturbation $u^-_{l,\sigma,\bar V}\neq u^-_{l,\sigma}$. To see how it is related to the potential, let us recall the following relations $$u^-_{l,\sigma}(\gamma_{x,\sigma}(0))-u^-_{l,\sigma}(\gamma_{x,\sigma}(-t))= \int_{-t}^0(L-\eta_{c})(d\gamma_{x,\sigma}(t))dt+Et$$ if $\gamma_{x,\sigma}$ is a semi-static curve determined by $u^-_{l,\sigma}$ with $\gamma_{x,\sigma}(0)=x$ and $c\in I_{\sigma}$. We also have $$u^-_{l,\sigma,\bar V}(\gamma_{x,\sigma}(0))-u^-_{l,\sigma,\bar V}(\gamma_{x,\sigma}(-t))\le \int_{-t}^0(L-\bar V-\eta_{c})(d\gamma_{x,\sigma}(t))dt+Et.$$ Clearly, for suitably large $t$ the backward weak-KAM solution $\gamma_{x,\sigma}(-t)$ shall retreat into a small neighborhood of $\Upsilon_{l,\sigma}$ where the weak KAM solution $u^-_{l,\sigma}$ also remains unchanged. Therefore we deduce from the last two formulae that $$u^-_{l,\sigma,\bar V}(x)-u^-_{l,\sigma}(x)\ge\int^0_{-T}\bar V(\gamma_{x,\sigma}(t))dt.$$ In a similar way, we find $$u^-_{l,\sigma,\bar V}(x)-u^-_{l,\sigma}(x)\le\int^0_{-T}\bar V(\gamma_{x,\sigma,\bar V}(t))dt$$ where $\gamma_{x,\sigma,\bar V}$ stands for the backward semi-static curve determined by the elementary weak-KAM solution $u^-_{l,\sigma,\bar V}$ with $\gamma_{x,\sigma,\bar V}(0)=x$. As $x$ is located in the region where the weak KAM solution is differentiable, we have $|\gamma_{x,\sigma,\bar V}(t)-\gamma_{x,\sigma}(t)| \to 0$ as $\bar V\to 0$, guaranteed by the upper-semi continuity of semi-static curves. Therefore, it follows that for $x\in D$ $$\begin{aligned}
\label{completeeq9}
u^-_{l,\sigma,V}(x)-u^-_{l,\sigma,V'}(x)=&\int_{-T}^0(\bar V-\bar V')(\gamma_{x,\sigma,\bar V}^-(t))dt+o(\|\bar V-\bar V'\|),\\
=&(\mathscr{K}_{\sigma}+ \mathscr{R}_{\sigma})(\bar V-\bar V')\notag\end{aligned}$$ where the linear operator $\mathscr{K}_{\sigma}$ is defined in (\[completeeq7\]) and $\mathscr{R}_{\sigma}(\bar V-\bar V')=o(\|V-V'\|)$.
Next, let us consider all backward weak-KAM solutions for $\sigma\in\mathbb{I}_{\sigma}$. Each parameter $\sigma\in\mathbb{I}_{\sigma}$ determines an interval $I_{c(\sigma)}$ for cohomology class. We restricted ourselves on a curve of first cohomology classes contained in the set $\cup I_{c(\sigma)}$ and intersecting each $I_{c(\sigma)}$ transversally. In this sense, we think the class defined on the interval $\mathbb{I}_c\ni c$ and the map $\sigma\to c(\sigma)$ is continuous. As $h^{\infty}_{c(\sigma)}(x,x')=u^-_{l,\sigma}(x')-u^-_{l,\sigma}(x)$ if $x\in\Upsilon_{l,\sigma}$ and $h^{\infty}_{c(\sigma)}(x,x')=u^+_{r,\sigma}(x')-u^+_{r,\sigma}(x)$ if $x'\in\Upsilon_{r,\sigma}$, we obtain from Lemma 6.4 in [@CY2] $$\begin{aligned}
\label{completeeq10}
&|u^-_{l,\sigma}(x)-u^-_{l,\sigma'}(x)|\le
C_4(\sqrt{|\sigma-\sigma'|}+|c(\sigma)-c(\sigma')|),\\
&|u^+_{r,\sigma}(x)-u^+_{r,\sigma'}(x)|\le
C_4(\sqrt{|\sigma-\sigma'|}+|c(\sigma)-c(\sigma')|).\notag\end{aligned}$$
We split the interval $\mathbb{I}_{\sigma}$ equally into $K_{\sigma}[\varepsilon^{-2}]$ parts and split the interval $\mathbb{I}_c$ equally into $K_c[\varepsilon^{-1}]$, where $$K_{\sigma}=\Big[L_{\sigma}\Big(\frac{12C_4}{C_3}\Big)^2\Big],\qquad
K_c=\Big[L_c\frac{12C_4}{C_3}\Big],$$ $L_{\sigma}$ and $L_c$ are the length of $\mathbb{I}_{\sigma}$ and of $\mathbb{I}_c$ respectively. The grid over $\mathbb{I}_c\times\mathbb{I}_{\sigma}$ consists of as many as $K_{\sigma}K_c[\varepsilon^{-3}]$ cuboids in which $K_{\sigma},K_c$ are independent of $\varepsilon$. We pick up all cuboids which contain the points $(c,\sigma(c))$ and denote them by $\text{\uj c}_j$ with $j\in\mathbb{J}$, then the cardinality of the set $\mathbb{J}$ is not bigger than $K_{\sigma}K_c[\varepsilon^{-3}]$.
According to the definition, a point $(c_j,\sigma(c_j))\in\text{\uj c}_j$ corresponds to a barrier function $u^-_{l,\sigma_j}-u^+_{r,\sigma_j}$. Let us assume that some parameters $(a_{\ell,j},b_{\ell,j})$ exist such that $$\text{\rm Osc}_{x\in D}\min_{x_3}\Big(u^-_{l,\sigma_j}-u^+_{r,\sigma_j}-(\mathscr{K}_{\sigma_j}+ \mathscr{R}_{\sigma_j})\bar V_j\Big)=0$$ where $\bar V_j=\rho\Psi^{-1} V_j$ is defined as in (\[completeeq5\]) with $V_j\in\mathfrak{V}_2$ determined by the parameters. We consider another perturbation determined by the parameters $(a'_{\ell},b'_{\ell})$ $$V'=\varepsilon\Big(\sum_{\ell=1,2}a'_{\ell}\cos2\ell \pi(x_2-x_{20})+b'_{\ell}\sin2\ell\pi(x_2-x_{20})\Big)$$ and set $\bar V'=\rho\Psi^{-1} V'$. By using the formula (\[completeeq9\]) we write the identity $$\begin{aligned}
u^-_{l,\sigma,\bar V'}-u^+_{r,\sigma,\bar V'}&=(u^-_{l,\sigma,\bar V'}-u^-_{l,\sigma_j,\bar V'})- (u^+_{r,\sigma,\bar V'}- u^+_{r,\sigma_j,\bar V'})\\
&+(u^-_{l,\sigma_j}-u^+_{r,\sigma_j})-(\mathscr{K}_{\sigma_j}+\mathscr{R}_{\sigma_j})\bar V_j\\
&+(\mathscr{K}_{\sigma_j}+\mathscr{R}_{\sigma_j})(\bar V_j-\bar V').\end{aligned}$$ For any point $(c,\sigma(c))\in\text{\uj c}_j$, in virtue of the formulae in (\[completeeq10\]) the first term on the right-hand-side of the identity is not bigger than $C_3\varepsilon^2/3$. For small $\|\bar V_j-\bar V'\|$ we have $\|(\mathscr{K}_{\sigma_j}+\mathscr{R}_{\sigma_j})(\bar V_j-\bar V')\|<\frac 13\|\mathscr{K}_{\sigma_j}(\bar V_j-\bar V')\|$. As both $V'$ and $V_j$ are independent of $x_3$, if the parameters $(a'_{\ell},b'_{\ell})$ satisfy $$\max\{|a_{\ell,j}-a'_{\ell}|,|b_{\ell,j}-b'_{\ell}|\}\ge\varepsilon$$ we find from above identities and the estimate (\[completeeq8\]) that $$\label{completeeq11}
\text{\rm Osc}_{x\in D}\min_{x_3}\Big(u^-_{l,\sigma}-u^+_{u,\sigma}-(\mathscr{K}_{\sigma}+ \mathscr{R}_{\sigma})\bar V'\Big)\ge \frac 13C_3\varepsilon^2>0.$$ It implies that, for each small rectangle $\text{\uj c}_j$ we only need to cancel out at most $2^4$ $\varepsilon$-cubes from the grid for $\{\Delta a_{\ell},\Delta b_{\ell}:\ell=1,2\}$ so that the formula (\[completeeq11\]) holds for the all other cubes. Let $j$ ranges over the set $\mathbb{J}$, we obtain a set $\text{\uj S}^c_2\subset\{a_{\ell}\in [1,2],b_{\ell}\in[1,2]:\ell=1,2\}$ with Lebesgue measure $$\text{\rm meas}\text{\uj S}^c_2\ge 1-2^4K_{\sigma}K_c\varepsilon,$$ such that the formula (\[completeeq11\]) holds for each $(a'_{\ell},b'_{\ell})\in\text{\uj S}^c_2$ and for each $\sigma\in\mathbb{I}_{\sigma_0}$.
By taking $V'\in\mathfrak{V}_3$, in the same way we can see that some set $\text{\uj S}^c_3\subset\{c_{\ell}\in [1,2],d_{\ell}\in[1,2]:\ell=1,2\}$ with Lebesgue measure $$\text{\rm meas}\text{\uj S}^c_3\ge 1-2^4K_{\sigma}K_c\varepsilon,$$ such that the formula $$\label{completeeq12}
\text{\rm Osc}_{x\in D}\min_{x_2}\Big(u^-_{l,\sigma}-u^+_{u,\sigma}-(\mathscr{K}_{\sigma}+ \mathscr{R}_{\sigma})\bar V'\Big)>0$$ for each $(c'_{\ell},c'_{\ell})\in\text{\uj S}^c_3$ and each $\sigma\in\mathbb{I}_{\sigma_0}$.
Therefore, for each $(a_{\ell},b_{\ell},c_{\ell},d_{\ell},)\in\text{\uj S}^c_2\times\text{\uj S}^c_3$, the formulae (\[completeeq11\]) and (\[completeeq12\]) implies that for all $\sigma\in\mathbb{I}_{\sigma_0}$ the diameter of each connected component of the set $$\arg\min(u^-_{l,\sigma,\bar V}-u^+_{r,\sigma,\bar V})|_D$$ is smaller than $D$. As $\varepsilon>0$ can be arbitrarily small, for each disk $D$, an open-dense set $\mathfrak{V}_D$ exists such that this disconnect property holds for the system $L-\bar V$ with $\bar V\in\mathfrak{V}_D$. Since $\sigma$ is restricted on a closed set in the line which can be covered by finitely many $\mathbb{I}_{\sigma_i}$, this property is also open-sense for all $\sigma$ under our consideration.
Each section $D$ admits a hierachy of partition of small disks $D=\cup_jD_{kj}$ such that the size $D_{jk}$ approaches zero as $k\to\infty$, the intersection $\cap_k\mathfrak{V}_{D_{kj}}$ is a residual set. Therefore, we have proved
\[chainthm1\] It is an open-dense condition for $H$ such that the set $$\arg\min(u^-_{l,\sigma}-u^+_{r,\sigma})\backslash((\Upsilon_{l,\sigma}\cup\Upsilon_{r,\sigma})+\delta)$$ consists of totally disconnected semi-static curves.
Criterion for strong and weak resonance
---------------------------------------
Given a perturbation $\epsilon P(\tilde x,\tilde y)$, it is natural to ask, along the resonant path $\Gamma$, how many many double resonant points need to be treated as strong resonance. Along a segment of resonant path $\Gamma_{\tilde\omega,\ell}$ the resonance condition $$\langle\tilde k,\tilde \omega\rangle=0$$ is always satisfied and at each double resonant point some other $\tilde k'\in\mathbb{Z}^3$ exists such that $\tilde k'$ is linearly independent of $\tilde k$ and $$\langle\tilde k',\tilde \omega\rangle=0.$$ Recall the process of KAM iteration, the main part of the resonant term is obtained by averaging the perturbation over a circle determined by these two resonant relations. It takes the form $$Z=Z_{\tilde k}(\langle \tilde k,\tilde x\rangle,\tilde y)+Z_{\tilde k,\tilde k'}(\langle \tilde k,\tilde x\rangle,\langle \tilde k',\tilde x\rangle,\tilde y)$$ where $$Z_{\tilde k}=\sum_{j\in\mathbb{Z}\backslash\{0\}}P_{j\tilde k}(\tilde y)e^{j\langle \tilde k,\tilde x\rangle i}, \qquad
Z_{\tilde k,\tilde k'}=\sum_{(j,l)\in\mathbb{Z}^2, l\neq 0}P_{jk+lk'}(\tilde y)e^{(j\langle \tilde k,\tilde x\rangle+l\langle\tilde k_i,\tilde x\rangle)i}.$$ Since $P$ is $C^r$-function, the coefficient $P_{j\tilde k+l\tilde k'}$ is bounded by $$|P_{j\tilde k+l\tilde k'}|\le 8\pi^3\|P\|_{C^r}\|j\tilde k+l\tilde k'\|^{-r},$$ which deduces the estimation $$\label{criterioneq1}
\|Z_{\tilde k,\tilde k'}\|_2\le d\|P\|_{C^r}\|\tilde k'\|^{-r+2}$$ where $d=d(\tilde k)$ depends on $\tilde k$. The function $Z_{\tilde k}$ is periodic in $\tilde q=\langle \tilde k,\tilde x\rangle$. In virtue of the theorem \[appenBthm1\] (see Appendix B), the following hypotheses is obviously open and dense:
([**H1.1**]{}): For each $\tilde y\in\Gamma_{\ell}$, $Z_{\tilde k}$ is non-degenerate at its maximal point, i.e. $\partial^2_{qq}Z_{\tilde k}(\tilde q)>0$ holds provided $\tilde q$ is a maximal point.
Given some $Z_{\tilde k}$ satisfying the hypothesis ([**H1.1**]{}), certain $\lambda>0$ exists such that for each $\tilde y\in\Gamma_{\ell}$, $\partial^2_{qq}Z_{\tilde k}\ge\lambda$ holds at the maximal point. Assume at $\tilde y'\in\Gamma_{\ell}$, the second resonant condition $\langle\tilde k',\tilde \omega(\tilde y')\rangle=0$ is also satisfied. One thus obtains the normal form (\[chaineq2\]), by performing the coordinate transformation (\[chaineq1\]). The homogenized form of the truncated Hamiltonian takes the form $$G=\langle Ay,y\rangle+V_{\tilde k}(x_2)+V_{\tilde k,\tilde k'}(x).$$ The Hamiltonian flow determined by $\langle Ay,y\rangle+V_{\tilde k}(x_2)$ admits a normally hyperbolic invariant cylinder $\Pi_{\tilde k,\tilde k'}^0=\{y=0,x_2=x^*_2\}\times\mathbb{T}$ if $x^*_2$ is a non-degenerate maximal point of $V_{\tilde k}$. Applying the theorem of normally hyperbolic manifold, one obtains from the estimate (\[criterioneq1\]) that some positive number $d_1=d_1(\lambda)>0$ exists such that $\Phi_{\bar G}^t$ also admits a normally hyperbolic and invariant cylinder $\Pi_{\tilde k,\tilde k'}$ close to $\Pi_{\tilde k,\tilde k'}^0$ provided $$\label{criterioneq2}
\|\tilde k'\|^{r-2}\ge\frac d{d_1}\|P\|_{C^r}.$$ It is a criterion to see whether the double resonance is thought as weak resonance and can be treated in the way for [*a priori*]{} unstable system. There are only finitely many $\tilde k'\in\mathbb{Z}^3$ not satisfying this condition, thus are treated as strong double resonance.
Therefore, once a perturbation $P$ is chosen so that ([**H1.1**]{}) is satisfied, there are finitely many double resonant frequencies which need to be treated as strong double resonance. The number is independent of the size of $\epsilon$. At strong double resonance, it is also open and dense condition that
([**H1.2**]{}): at each strong double resonance point, the maximal point of $Z_{\tilde k}+Z_{\tilde k,\tilde k'}$ is non-degenerate, two eigenvalues of the Hessian matrix are positive and different $\lambda_{k,j}>0$ for $j=1,2$. Indeed, there exists $\nu>0$ such that $\lambda_{\tilde k,2}\ge\nu\|\tilde k\|^{r-2}$ and $\lambda_{\tilde k,1}\ge\nu\|\tilde k\|^{r-2}\|\tilde k'\|^{r-2}$.
Proof of the main theorem
-------------------------
Given $y'_0,y'_1,\cdots,y'_k$ we have chosen a resonant path $\Gamma_{\omega}$ so that $\mathscr{L}_{\beta_h}(\Gamma_{\omega})$ passes through each $\delta$-neighborhood of these points. Let $\epsilon P$ satisfy all hypothesis above. In order to make things convenient for readers, we formulate them here again:
([**H1**]{}) [*for each strong double resonance, the potential $V_i$ attains its maximum at one point only, the Hessian matrix of $V_i$ at that point is negative definite. All eigenvalues are different: $-\lambda_2<-\lambda_1<0<\lambda_1<\lambda_2$*]{}. (see [**H1**]{} in Subsection 5.1, [**H1.1**]{} and [**H1.2**]{} in Subsection 8.4);
([**H2**]{}) [*for the Hamiltonian flow $\Phi_{Y_{i,T}}^t$, the stable and unstable manifold of the fixed point intersect transversally along each minimal homoclinic orbit. Each minimal homoclinic orbit approaches to the fixed point along the direction $\Lambda_1$: $\dot\gamma(t)/\|\dot\gamma(t)\| \to\Lambda_{x1}$ as $t\to\pm\infty$.*]{} (see [**H2**]{} in Subsection 4.1. The function $Y_{i,T}$ solves the equation $H_{i,T}(\tilde x,y,Y_{i,T})=E$, the transversality is in the sense that, at the intersection points, the tangent space of the stable and unstable manifold span the tangent space of the energy level set.)
([**H3**]{}): [*For each $c\in\partial^*\mathbb{F}_{0,i}$, the Aubry set does not contain minimal curve homoclinic to the origin $($fixed point$)$.*]{} (see [**H2**]{} in Subsection 4.1, each strong double resonance is related to a flat $\mathbb{F}_{0,i}$ corresponding to the Hamiltonian $Y_{i,T}$.)
([**H4**]{}): [*For each $g\in H_1(\mathbb{T}^2,\mathbb{Z})$, there are finitely many $\theta_i\in\mathbb{R}$ such that, for each rotation vector $\theta_i g$, the Mather set consists of two periodic orbits, for other rotation vector $\theta g$, the Mather set consists of one periodic orbit only. All these periodic orbits are hyperbolic.*]{} (see [**H4**]{} in Subsection 4.2, also formulated for the Hamiltonian $Y_{i,T}$.)
([**H5**]{}): [*For each $c\in\partial^*\mathbb{F}_{0,i}$ there is a disk disjoint either with the support of $\mu_c$ or of $\mu$, restricted on which, the set $\arg\min(U_c^--U'^+_c)$ is non-empty. The size of the disk is independent of $c$.*]{} (see [**H5**]{} in Subsection 4.2, also formulated for the Hamiltonian $Y_{i,T}$.)
Along the resonant path $\Gamma_{\omega}$, the strong double resonance points are denoted by $\omega_0,\omega_1,\cdots, \omega_m$, where the number $m$ depends on $P$. Each flat $\mathscr{L}_{\beta_H}(\omega_i)$ is surrounded by a annulus $\tilde{\mathbb{A}}_i\subset\alpha^{-1}_H(E)$. For each segment of $\Gamma_{\omega}$ connecting $\omega_i$ to $\omega_{i+1}$, denoted by $\Gamma_{\omega,i}$, $\mathscr{L}_{\beta_H}(\Gamma_{\omega,i})$ constitutes a channel connecting $\tilde{\mathbb{A}}_i$ to $\tilde{\mathbb{A}}_{i+1}$.
Split the unit interval into $2m+1$ segments $$[0,1]=[0=s_{0,i},s_{0,c}]\cup[s_{0,c},s_{1,i}]\cup\cdots\cup [s_{m-1,i},s_{m,c}] \cup[s_{m,c},s_{m,i}=1]$$ and let $\Gamma_{j,c}$: $[s_{j,i},s_{j,c}]\to\alpha^{-1}_H(E)$, $\Gamma_{j,i}$: $[s_{j,c},s_{j+1,i}]\to\alpha^{-1}_H(E)$ denote the paths such that $\Gamma_{j,c}(s_{j,c}) =\Gamma_{j,i}(s_{j,c})$, $\Gamma_{j,i}$ falls into the annulus $\tilde{\mathbb{A}}_j$ along which the component $c_3$ keeps constant in the local coordinate system and $\Gamma_{j,c}$ falls into the channel $\mathscr{L}_{\beta_H}(\Gamma_{\omega,j})$ connecting $\tilde{\mathbb{A}}_i$ to $\tilde{\mathbb{A}}_{i+1}$. Let $\Gamma_{0,c}(0)\in\mathscr{L}_{\beta_H}(\nabla h(y'_0))$ and $\Gamma_{m,c}(1)\in\mathscr{L}_{\beta_H}(\nabla h(y'_k))$. The subscript “$c$" is used to indicate complete intersection and the subscript “$i$" denotes the incomplete intersection. We choose the conjunction of these curves as candidate of generalized transition chain $$\label{proofeq1}
\Gamma=\Gamma_{0,c}\ast\Gamma_{0,i}\ast\cdots\ast\Gamma_{m-1,i}\ast\Gamma_{m,c}.$$ Indeed, restricted on the segment $\Gamma_{j,i}$ ($j=0,\cdots, m-1$) it has been proved satisfying the condition (H2) in the Definition \[chaindef1\] ($c$-equivalence) by using the hypothesis ([**H1$\sim$5**]{}). To guarantee the condition (H1) in the Definition \[chaindef1\] when it is restricted on each segment $\Gamma_{j,c}$ ($j=0,\cdots, m$), one need to impose some condition which has been proved to be generic in Subsection 8.3, Theorem \[chainthm1\]:
([**H6**]{}): [*if the Aubry set covers certain $2$-torus in $\mathbb{T}^3$ for $c\in\Gamma_{j,c}$, then certain finite covering manifold $\check M$ and certain two-dimensional section $\Sigma_c$ exist such that $$\mathcal{N}(c,\check M)|_{\Sigma_c}\backslash(\mathcal{A}(c,\check M)+\delta)|_{\Sigma_c}\neq\varnothing.$$ is totally disconnected.*]{}
Under these hypothesis, namely ([**H1$\sim$6**]{}), the path $\Gamma$ defined in (\[proofeq1\]) is a transition chain. Choose suitably many $c_i\in\Gamma$ ($i=0,1,\cdots,i_m)$ such that
1, each $\tilde{\mathcal{A}}(c_i)$ is connected to $\tilde{\mathcal{A}}(c_{i+1})$ by local minimal orbit either of type-$c$ or of type-$h$;
2, among these classes, some classes $c_{i_j}$ $(j=0,1,\cdots,k)$ exist such that $c_{i_j}$ is very close to $y'_j$ (the prescribed action variables in Theorem \[mainthm\]) if one thinks both $c_{i_j}$ and $y_j$ as points in $\mathbb{R}^3$.
Recall the proof of Theorem \[constructionthm1\]. Let $\gamma$: $[-K,K']\to\mathbb{T}^3$ be the minimizer of the action (see (\[constructioneq23\])) satisfying the boundary conditions $\gamma(-K)=x_0$ and $\gamma(K')=x_k$. Dividing the time interval $[-K,K']$ into $2i_m+1$ parts $$[-K,K']=[t_0^+,t^-_0]\cup [t^-_0,t^+_1]\cup\cdots\cup [t^+_{i_m},t_{i_m}^-],$$ imposing some constraints on $\gamma$ at $t=t_i^{\pm}$ and conditions on sufficiently large $t_{i+1}^+-t_i^-$ and $t^-_i-t^+_i$, one then proves that $\gamma$ is a solution of the Lagrange equation determined by $H$. The curve $\gamma$ determines an orbit of the Hamiltonian flow $\Phi_H^t$: $$x(t)=\gamma(t),\qquad y(t)=\frac {\partial L}{\partial \dot x}(\gamma(t),\dot\gamma(t)).$$
For each $x\in M$, the set $V^-(c,x,L)\subset T_xM$ is defined as follows: a vector $v\in V^-(c,x,L)$ if and only if a backward $c$-semi static curve $\gamma^{-}$ for the Lagrangian $L$ exists such that $v=\dot\gamma^{-}(0)$. The set $V^+(c,x,L)$ is defined for forward semi-static curve similarly. Clearly, one has
The set-valued map $L\to V^{\pm}_{c,x,L}$ is upper-semi continuous.
To see that this orbit visits the ball $B_{\delta}(x_0,y_0),B_{\delta}(x_k,y_k)\subset\mathbb{T}^3 \times\mathbb{R}^3$ and the balls $B_{\delta}(y_i)\subset \mathbb{R}^3$ $(i=1,\cdots,k-1)$, we use this proposition. As $h$ is integrable, any backward (forward) $c$-semi static curve is $c$-static for all $t\in\mathbb{R}$. Along any $c$-minimal curve it holds that the action variable $y=c$. Since the perturbation $h\to h+\epsilon P$ is small, $\dot\gamma(-K)$ is close to $V^+_{x,c_0,L}$ if $t_0^--t_0^+$ is sufficiently large. It follows that $\|y(-K)-y_0\|$ is very small provided $\epsilon$ is sufficiently small. In the same way, one can see that $\|y(K')-y_k\|$ is also very small. Since Mañé set is upper semi continuous with respect to Lagrangian, at the time $t_i=(t_i^-+t_i^+)/2$, $(\gamma(t_i),\dot\gamma(t_i))$ is very close to $\tilde{\mathcal{A}}(c_i)$, $\|y(t_i)-y_i\|$ is very small. This proves that the Hamiltonian flow $\Phi_H^t$ admits an orbit that visits these balls in turn.
To complete the proof of Theorem \[mainthm\], we only need to show the generic property. Towards this goal, let us observe a fact: under the rescaling $y\to\sqrt{\lambda}y$, $t\to\sqrt{\lambda}^{-1}t$, the Hamiltonian equation determined by $\frac12\langle Ay,y\rangle+\lambda V(x)$ is the same as it for the function $\frac12\langle Ay,y\rangle+V(x)$. Therefore, some open-dense set $\mathfrak{O}\subset\mathfrak{S}_1\subset C^r$ exists, some $\epsilon_P>0$ is associated to each $P\in\mathfrak{O}$ such that the Hamiltonian flow $\Phi_H^t$ satisfies the conditions [**H1$\sim$5**]{} provided $\epsilon\le\epsilon_P$, because the number of strong double resonant points is independent of the size of $\epsilon$. From the proof of Theorem \[chainthm1\] in the subsection 8.3, the condition [**H6**]{} is required for the intersection of countably many open-dense set contained in $\mathfrak{B}_{\epsilon_0}$: $\cap_i\mathfrak{O}_i$. Clearly, there exists a residual set $\mathfrak{R}_{\epsilon_0}\subset \mathfrak{S}_{\epsilon_0}$, for each $P\in\mathfrak{R}_{\epsilon_0}$ there exists a set $R_P$ residual in $[0,\epsilon_0]$ such that $\{\lambda P:P\in\mathfrak{R}_{\epsilon_0},\lambda\in R_P\}\subset\cap_i\mathfrak{O}_i$. Take the intersection of these sets, we obtain the cusp-residual property. [ $\;\;\; \Box$]{}
Normal form
===========
In this appendix, we study the normal form of nearly integrable Hamiltonian, from which one obtains some information about the relevant Mather sets, Aubry sets as well as Mañé sets. Here, the system is assumed to have arbitrary $n$-degrees of freedom $$H(x,y,t)=h(y)+P_{\epsilon}(x,y,t),\qquad (x,y,t)\in\mathbb{T}^n\times \mathbb{R}^n\times\mathbb{T}.$$ The perturbation can be autonomous as well as time-1-periodic. As $t$ can be treated as the $(n+1)$-th angle coordinate, we replace $n$ by $n+1$ when we consider time-1-periodic perturbation $P_{\epsilon}(x,y,t)$. Thus, we consider autonomous Hamiltonian only.
KAM iteration at strong resonance
---------------------------------
Let $\omega(y)=\nabla h(y)$ denote the frequency vector of the unperturbed system. For autonomous case, a frequency $\omega$ is called rational of (minimal) period $T$ if $T\omega\in\mathbb{Z}^n$ and $t\omega\notin\mathbb{Z}^n$ for each $t\in (0,T)$.
Let the frequency $\omega$ be rational of period $T$. With a function $g(x,y)$ on the torus one associates its time average $[g]$ along the orbits of the linear flow defined by $\omega$: $x\to x+\omega t$ $$[g](x,y)=\frac 1T\int_0^Tg(x+\omega t,y)dt.$$ We say that $g$ is [*resonant*]{} (with respect to $\omega$) if $g=[g]$, it implies that $g$ is constant along the orbits of the linear flow $(x,y)\to (x+\omega t,y)$.
Let $B_R\subset\mathbb{R}^n$ be the ball of radius $R$ around the origin, then there are positive numbers $M=M(R)\ge m=m(R)>0$ such that $$m\|v\|^2\le\langle\nabla^2h(y)v,v\rangle\le M\|v\|^2, \qquad \forall\ y\in B_R,\ v\in\mathbb{R}^n.$$ Let $\sigma$ and $\varrho$ denote positive number such that $$\sigma<\frac 13, \qquad K=K(\epsilon)=K_0\epsilon^{-\varrho},\qquad \varrho=\frac 13(1-3\sigma),$$ the value of $\sigma$ will be specified later to satisfy certain covering property.
Denoted by $\{\omega_{\lambda}:\lambda\in\Lambda_{K,R}\}\subset B_{MR}$ the set of frequencies which are rational of period $T$ with $T\le K$. Clearly, $\Lambda_K$ is a finite index set. Let $y_{\lambda}=\nabla^{-1} h(\omega_{\lambda})$.
Let $i=(i_1,i_2,\cdots,i_n)\in \mathbb{Z}^n_+$, namely, $i_j$ is non-negative $\forall$ $j\in\{1,2,\cdots,n\}$. Let $|i|=\sum_{j=1}^ni_j$, $Y_i(y)= \prod_{j=1}^ny_{j}^{i_{j}}$. Let $\|\cdot\|_{j,D}$ denote the $C^j$-norm on the domain $D$, we omit the notation $D$ when it is clearly implied.
\[normalthm1\] For a nearly integrable Hamiltonian $H(x,y)=h(y)+P_{\epsilon}(x,y)$ we assume that both $h$ and $P_{\epsilon}$ are $C^r$-smooth with $r\ge 8$, and $\|P_{\epsilon}\|_{r,B_R\times\mathbb{T}^{n+1}}\le \epsilon$. Some small $\epsilon_0=\epsilon_0(M,m,n,r)>0$ exists such that for each $\epsilon\le\epsilon_0$ and each rational frequency $\omega_{\lambda} =\omega(y_{\lambda})$ with a period $T\le K(\epsilon)$, a canonical transformation $\mathscr{F}_{\lambda}$ is well defined on $$D_{y_{\lambda},\epsilon}=\{(x,y)\in\mathbb{T}^n\times\mathbb{R}^n: \|y-y_{\lambda}\|\le T^{-1}\epsilon^{\sigma}\}$$ which reduce the Hamiltonian into the normal form $$\label{normaleq1}
H\circ\mathscr{F}_{\lambda}(x,y)=h(y)+Z(x,y)+R(x,y)$$ where $Z$ is resonant with respect to $\omega_{\lambda}$ with $\|Z\|_r\le 2\epsilon$, $R=R_h+R_r$ is a higher order term when it is restricted in $D_{y_{\lambda},\epsilon}:$ $$R_h=\sum_{|i|=5}Y_i(y-y_{\lambda})R_{h,i}(x,y),$$ $$\begin{aligned}
&\|R_{h}\|_{2,D_{y_{\lambda},\epsilon}}\le D\epsilon^{\frac 13+5\sigma}, \\
&\|R_r\|_{2,D_{y_{\lambda},\epsilon}}\le D\epsilon^{\frac 43+2\sigma}\end{aligned}$$ where the constant $D>0$ is independent of $\epsilon$. Restricted in the region $\{\|y-y_{\lambda}\|=O(\sqrt{\epsilon})\}$, we have a sharper estimate $$\|R_{h}\|_1\le D\epsilon^{\frac 43+5\sigma},\qquad \|R_{h}\|_2\le D\epsilon^{\frac 56+5\sigma}.$$
The canonical transformation $\mathscr{F}_{\lambda}$ is the composition of five steps of coordinate transformations $\mathscr{F}_{\lambda}=\mathscr{F}_4\circ\cdots\circ\mathscr{F}_1 \circ\mathscr{F}_0$. Each step of transformation $\mathscr{F}_j$ is defined as the time-1-map of $\phi_{W_j}^t$, the Hamiltonian flow determined by the generating function $W_j$. For the first step of coordinate transformation $\mathscr{F}_0$, we set $$W_0(x,y)=-\frac 1T\int_0^TP(x+\omega t,y)tdt$$ which solves the equation $$\Big\langle\omega, \frac{\partial W_0}{\partial x}\Big\rangle=-P+[P].$$ Let $Z=[P]$, it follows that $$H_1=h(y)+Z(x,y)+R_{1,1}(x,y)+R_{1,2}(x,y)$$ where $$\begin{aligned}
R_{1,1}&=\Big\langle\frac{\partial h}{\partial y}-\omega,\frac{\partial W_0}{\partial x}\Big\rangle,\\
R_{1,2}&=\int_0^1(1-t)\{\{H,W_0\},W_0\}\circ\phi_{W_0}^tdt.\end{aligned}$$ Obviously, $[R_{1,1}]=0$ and one has the form $R_{1,1}=\sum_{|i|=1}Y_i(y-y_{\lambda})R_{1,1,i}(x,y)$. If we write $H_j=H_{j,1}+R_{j,2}$ where $$H_{j,1}=h(y)+Z(x,y)+R_{j,1}(x,y),$$ and for the coordinate transformation $\mathscr{F}_j$ ($j=1,\cdots,4$), we set by induction $$W_j(x,y)=-\frac 1T\int_0^TR_{j,1}(x+\omega s,y)sds$$ then the Hamiltonian $H_{j+1}$ takes the form $$H_{j+1}=h(y)+Z(x,y)+R_{j+1,1}(x,y)+R_{j+1,2}(x,y)$$ where $$\begin{aligned}
R_{j+1,1}=&\Big\langle\frac{\partial h}{\partial y}-\omega,\frac{\partial W_j}{\partial x}\Big\rangle,\\
R_{j+1,2}=&\int_0^1(1-t)\{\{H_{j,1},W_j\},W_j\}\circ\phi_{W_j}^tdt\\
&+R_{j,2}\circ\mathscr{F}_j.\end{aligned}$$ Also, $[R_{j+1,1}]=0$ and we can write $R_{j+1,1}=\sum_{|i|=j+1}Y_i(y-y_{\lambda})R_{j+1,1,i}(x,y)$.
By the construction, we see that $H_{j,1}$ is $C^{r-j}$-smooth and $R_{j,2}$ is $C^{r-j-1}$-smooth. Some constants $D_{j}>0$, independent of $\epsilon$, exists such that for $(x,y)\in D_{y_{\lambda},\epsilon}:$ $$\begin{aligned}
\label{estimate}
&\|W_0\|_1\le D_0K_0\epsilon^{1-\varrho},\notag\\
&\|W_j\|_{1}\le D_{j}K_0^{j+1}\epsilon^{1+(j-1)\sigma-2\varrho},\ \ \ (j\ge 1), \notag\\
&\|R_{j+1,1,i}\|_{2}\le D_{j}K_0^{j+1}\epsilon^{1-(j+1)\varrho},\\
&\|R_{j+1,2}\|_{2}\le D_{j}K_0^{2(j+1)}\epsilon^{2-2\varrho},\notag\end{aligned}$$ where we have used the relations that $T\le K_0\epsilon^{-\varrho}$, $\varrho=\frac 13(1-3\sigma)$ and $\sigma<\frac13$. If we write the canonical transformation $\mathscr{F}_j$ in the form $$\mathscr{F}_j:(x,y)\Rightarrow (x+U_j(x,y),y+V_j(x,y))$$ then $$(U_j,V_j)=\int_0^1\Big(\frac{\partial W_j}{\partial y},-\frac{\partial W_j}{\partial x}\Big)\circ\phi^t_{W_j}dt$$ It maps $\mathbb{T}^n\times B_{\delta_{j+1}}\to \mathbb{T}^n\times B_{\delta_j}$ for $\epsilon\le\epsilon_0$ if we set $$\delta_j=\Big(2-\frac{j+1}5\Big)\frac {\epsilon^{\sigma}}T,\qquad \epsilon_0\le\max_{j\le 5}\frac{1}{(5D_jK_0^{j+2})^{\frac 1{(j+1)\sigma}}}.$$ The estimate on $R_r=R_{5,1}$ and $R_h=R_{5,2}$ follows from the formulae (\[estimate\]).
If the system is real analytical, the higher order term can be reduced to the order $O(\exp(-\frac 1{\epsilon^\sigma}))$ (see [@Lo]), with which one obtains the Nekhoroshev’s estimate.
Covering property
-----------------
Recall that the set of frequencies $\{\omega_{\lambda}:\lambda\in\Lambda_{K,R}\}\subset B_{MR}$ each of which is rational of period $T$ with $T\le K$, and the domains $\{D_{y_{\lambda},\epsilon}:\ \lambda\in\Lambda_{K,R}\}$ where the iteration of KAM is carried (see Theorem \[normalthm1\] for definition).
The following covering property holds $$\label{normaleq2}
\bigcup_{\lambda\in\Lambda_{K,R}}\mathscr{F}_{\lambda}^{-1}D_{y_{\lambda},\epsilon}\supset \mathbb{T}^{n}\times B_R \hskip 0.5 true cm \text{\rm provided}\hskip 0.4 true cm \sigma<\frac 1{3n+3}.$$
To show the covering property, we use Dirichlet’s approximation theorem. For real $x$ one has $$x=[x]+\{x\},$$ where $[x]\in\mathbb{Z}$ the integer part, and $\{x\}\in(0,1)$. We use notation $$\|x\|_{\mathbb{Z}}=\inf\{\{x\},1-\{x\}\}=\text{\rm dist}(x,\mathbb{Z}).$$ If $x=(x_1,x_2,\cdots,x_n)\in\mathbb{R}^n$ one sets $$\|x\|_{\mathbb{Z}}=\sup_{i=1,2,\cdots,n}\|x_i\|_{\mathbb{Z}}.$$
\[normalpro1\] [(Dirichlet, see for examples, [@Cas; @Sch])]{} Let $\omega\in\mathbb{R}^n$ and $K$ a real number with $K>1$. There exists an integer $k$, $1\le k<K$, such that $$\|k\omega\|_{\mathbb{Z}}\le K^{-\frac 1n}.$$
For any $\omega\in\mathbb{R}^n$, by applying Dirichlet’s theorem, we find some rational vector $\omega^*$ existing such that $K'\omega^*\in \mathbb{Z}^n$ with $K'\le K$ and $$\text{\rm dist}(K'\omega,K'\omega^*)\le\sqrt{n}K^{-\frac 1n},$$ here, $\omega^*$ is a rational vector of period $T$. Since $h$ is assumed strictly convex, there exist two points $y,y^*\in B_R$ such that $\nabla h(y)=\omega$, $\nabla h(y^*)=\omega^*$ and $$\text{\rm dist}(y,y^*)\le\frac 1m\text{\rm dist}(\omega,\omega^*).$$ The condition $\|y-y^*\|\le T^{-1}\epsilon^{\sigma}$ is guaranteed if we choose $$\label{normaleq8}
K^{\frac 1n}=\frac {\sqrt{n}}{m}\epsilon^{-\sigma}.$$ As $T\le K$ is required, the following should be satisfied: $$K\le K_0\epsilon^{-\frac 13(1-3\sigma)},$$ that is, referring to (\[normaleq8\]), $$\epsilon^{\frac{1-3\sigma}3-n\sigma}\le \Big(\frac {m}{\sqrt{n}}\Big)^nK_0,$$ which determines a threshold for $\epsilon$ provided: $$\sigma<\frac 1{3n+3}.$$ As we choose $\sigma$ satisfying this condition, the covering property (\[normaleq2\]) is proved.
For the purpose of this paper, the covering property (\[normaleq2\]) for the whole space is not necessary, instead, it is good enough to cover a neighborhood of a resonant path. Denote by ${\bf k}=(k_1,\cdots,k_{n-1})$ a $n\times(n-1)$ matrix, where $k_1,\cdots,k_{n-1}$ are integer vectors. We consider the $n-1$ resonance line $$\Gamma_{\bf k}=\{y\in\mathbb{R}^n:\langle k_i,\partial h(y)\rangle=0\ \forall\ i=1,\cdots n-1\}.$$ If the covering property (\[normaleq2\]) in Theorem \[normalthm1\] is replaced by covering a neighborhood of the line $$\label{normaleq9}
\bigcup_{\lambda\in\Lambda_{K,R}}D_{y_{\lambda},\epsilon}\supset\mathbb{T}^{n}\times\{\|y-y_0\|<\mu
K^{-1}\epsilon^{\sigma}:y_0\in\Gamma_{\bf k}\cap B_R\}$$ then it works if $$\sigma<\frac 16.$$ Indeed, as all frequencies are on a $(n-1)$-resonance line, by using Dirichlet approximation theorem (Proposition \[normalpro1\]) for $n=1$ we obtain a threshold $\sigma< 1/6$.
Recall that the term $Z$ in (\[normaleq1\]) is resonant with respect to $\omega$, some rational frequency of period $T\le K$, namely, it has the form $$Z(x,y)=\sum_{\langle k,\omega\rangle=0}Z_{k}(y)e^{i\langle k,x\rangle}.$$ Note that $T\omega$ is an indivisible integer vector, i.e. $\mu T\omega\notin \mathbb{Z}^n$ for any $\mu\in (0,1)$. There are $n-1$ integer vectors $I_2,I_3,\cdots,I_n$ such that the matrix $(I_2,I_3,\cdots,I_n)$ is indivisible, $\text{\rm rank}(I_2,I_3,\cdots,I_n)=n-1$ and $\langle I_i,\omega\rangle =0$ holds for each $i\in\{2,3,\cdots,n\}$. Clearly, there is another integer vector $I_1$ such that the matrix $I=(I_1,I_2,\cdots,I_n)$ is uni-module. Each integer vector $k\in\mathbb{Z}^n$ with $\langle k,\omega\rangle=0$ uniquely determines an integer vector $\bar k\in\mathbb{Z}^{n-1}$ such that $k=\sum_{j=1}^{n-1} \bar k_jI_{j+1}$.
We introduce a coordinate transformation: $(x,y)\to (p,q)$ such that $$\label{normaleq10}
\tilde q=I^tx,\qquad \tilde p=I^{-1}y.$$ This coordinate transformation is symplectic, $H(I^{-t}\tilde q,I\tilde p)$ is also a function defined on $\mathbb{T}^n$ with respect to $\tilde q$. Let $y$ be the point where $\nabla h(y)=\omega$, then the gradient of $\tilde h(\tilde p)=h(I\tilde p)$ satisfies $$\tilde\omega=\nabla \tilde h(\tilde p)=(0,\cdots,0,\tilde\omega_n),$$ and $Z(I\tilde p,I^{-t}\tilde q)$ is independent of $q_n$, thus we can write $\tilde Z(\tilde p,\tilde q) =\tilde Z(p,q,p_n)$ if we use the natation $\tilde p=(p,p_n)$ and $\tilde q=(q,q_n)$.
Next, let us consider the time-1-periodical non-autonomous case. Assume $T(\omega,1)\in \mathbb{Z}^{n+1}$ is an indivisible integer vector. As $Z$ is resonant with respect to $\omega$, we have $$Z(x,y,t)=\sum_{\langle k,\omega\rangle+l=0}Z_{k,l}(y)e^{i\langle k,x\rangle+lt}.$$ Thus, there are $n$ integer vectors $I_1,\cdots,I_n,J\in\mathbb{Z}^n$ such that $\langle I_i, \omega \rangle+J_i=0$ for each $i\in\{1,\cdots,n\}$. For each $(k,l) \in\mathbb{Z}^{n+1}$ with $\langle k, \omega\rangle +l=0$, there is uniquely determined $(\bar k_1,\cdots,\bar k_n)\in\mathbb{Z}^n$ such that $$(k,l)=\sum_{i=1}^n\bar k_i(I_i,J_i).$$ By choosing suitable $I_i$, we can make $I=(I_1,I_2,\cdots,I_n)$ be uni-module. Introduce the coordinate transformation (\[normaleq10\]), let $y$ be the point where $\nabla h(y)=\omega$, then the gradient of $\bar h(p)=h(Ip)$ satisfies $$\bar\omega=\nabla \bar h(p)=-J.$$ Note that each $(k,l)$ with $\langle k,\omega\rangle+l=0$ uniquely determines $\bar k\in \mathbb{Z}^n$ such that $(k,l)=\bar k(I^t,J)$. As we have $$Z(I^{-t}q,Ip,t)=\sum_{\bar k\in\mathbb{Z}^n}Z_{k,l}(Ip)e^{i\langle\bar k, q+Jt\rangle},$$ in the new coordinates the resonant term $\bar Z=\bar Z(p,q+Jt)$. Let $q'=q+Jt, p'=p$ and let $h'(p')=\bar h(p')+\langle J,p\rangle$, we find the Hamiltonian equation of $h'(p')+\bar Z(p',q')$ is the same as the Hamiltonian equation of $\bar h (p)+\bar Z(p,q+Jt)$.
Hyperbolicity of minimal periodic orbits
========================================
[*by*]{} [Chong-Qing Cheng and Min Zhou]{}
In the section 4, we made the hypothesis ([**H4**]{}) on the hyperbolicity of minimal periodic orbits in Hamiltonian systems with two degrees of freedom. In the subsetion 8.4, we need the hypothesis ([**H1.1**]{}). In $C^r$-topology with $r\ge 4$ these hypotheses are shown generic in [@CZ]. For the sake of completeness and convenience of reader, we present the proof in this appendix.
Non-degeneracy of global minimum
--------------------------------
\[appenBthm1\] Let $F_{\lambda}$: $\mathbb{T}\to\mathbb{R}$ be a family of $C^{r}$-functions depending on the parameter $\lambda\in [\lambda_0,\lambda_1]$ $(r\ge 4)$. If $F_{\lambda}$ is Lipschitz continuous in $\lambda$, then there exists an open-dense set $\mathfrak{O}\subset C^{r}(\mathbb{T},\mathbb{R})$ such that for each $V\in\mathfrak{O}$ and each $\lambda\in[\lambda_0,\lambda_1]$, each global minimum of $F_{\lambda}-V$ is non-degenerate, namely, the second derivative is positive at each global minimizer.
Since the openness is obvious, we only need to show the density. For this goal, we introduce a set of perturbations with four parameters: $$\mathfrak{V}=\Big\{V=\sum_{i=1}^2(A_i\cos ix+B_i\sin ix):\ (A_1,B_1,A_2,B_2)\in\mathbb{I}^4\Big\},$$ where $\mathbb{I}^4=[1,2]\times[1,2]\times[1,2]\times[1,2]$. Let $M=12^{-1}\sup_{x,\lambda}|\partial^4_xF_{\lambda}|$, we are going to show that, for any small numbers $\epsilon,d>0$ there exists $(A_1,B_1,A_2,B_2)\in I^4$ such that $$\label{appenBeq1}
(F_{\lambda}-\epsilon V)(x)-\min_x(F_{\lambda}-\epsilon V)\ge M|x-x^*|^4, \qquad \forall\ |x-x^*|\le d$$ holds for each $\lambda\in[\lambda_0,\lambda_1]$ whenever the point $x^*$ is a global minimizer of $F_{\lambda}-\epsilon V$. It implies the second derivative is positive. Indeed, if it equals zero, the third derivative will be zero also. Consequently, the above formula does not hold.
By choosing sufficiently large integer $k$, $\epsilon=\sqrt{\pi/k}$ can be arbitrarily small. Let $x_i=2i\pi/k$, $I_i=[x_i-d,x_i+d]$ and $d=\pi/k$, then $\epsilon=\sqrt{d}$ and $$\bigcup_{i=0}^{k-1}I_i=\mathbb{T}.$$ Restricted on each interval $I_i$, each function $V\in\mathfrak{V}$ is approximated by Taylor series (module constant) $$V_i(x)=a_i(x-x_i)+b_i(x-x_i)^2+c_i(x-x_i)^3+O(|x-x_i|^4).$$ Given two points $(a_i,b_i,c_i)$ and $(a'_i,b'_i,c'_i)$, we have two functions $V_i$ and $V'_i$ in the form of Taylor series. Let $\Delta V=V_i-V'_i$, $\Delta a=a_i-a'_i$, $\Delta b=b_i-b'_i$ and $\Delta c=c_i-c'_i$, we have $\Delta V(x_i)=0$ and $$\begin{aligned}
&\Delta V(x_i+d)+\Delta V(x_i-d)=2\Delta bd^2+O(d^4),\\
&\Delta V(x_i+d)-\Delta V(x_i-d)=2(\Delta a+\Delta cd^2)d+O(d^4),\\
&\Delta V\Big(x_i\pm\frac 12d\Big)=\Big(\pm\frac 12\Delta a+\frac 14\Delta bd\pm\frac 18\Delta cd^2\Big)d+O(d^4).\end{aligned}$$ Using the notation $$\text{\rm Osc}_{I}V=\sup\{V(x)-V(x'):x,x'\in I\},$$ it follows from the identities above that $$\label{appenBeq2}
\text{\rm Osc}_{I_i}(V'_i-V_i)\ge\max\Big\{\frac 14|\Delta a|d,|\Delta b|d^2,\frac 12|\Delta c|d^3\Big\}+O(d^4).$$ We construct a grid for the parameters $(a_i,b_i,c_i)$ by splitting the domain for $(a_i,b_i,c_i)$ equally into a family of cuboids and setting the size length by $$\Delta a_i=8Md^{5/2},\ \ \Delta b_i=2Md^{3/2},\ \ \Delta c_i=4Md^{1/2}.$$ These cuboids are denoted by $\text{\uj c}_{ij}$ with $j\in\mathbb{J}_i=\{1,2,\cdots\}$, the cardinality of the set of the subscripts is bounded by $$\#(\mathbb{J}_i)=K[d^{-9/2}],$$ where the integer $K$ is independent of $d$. Let $(a_{ij},b_{ij},c_{ij})$ denote the center of each cuboid and let $$V_{ij}(x)=a_{ij}(x-x_i)+b_{ij}(x-x_i)^2+c_{ij}(x-x_i)^3+O(|x-x_i|^4).$$ Define $$\ell_{j,j'}=\max\Big\{\frac{|a_{ij}-a_{ij'}|}{8Md^{5/2}},\frac{|b_{ij}-b_{ij'}|}{2Md^{3/2}}, \frac{|c_{ij}-c_{ij'}|}{4Md^{1/2}}\Big\},$$ we find from the formula (\[appenBeq2\]) that following holds for suitably small $d>0$ $$\label{appenBeq4}
\text{\rm Osc}_{I_i}(\epsilon V_{ij}-\epsilon V_{ij'})\ge 2\ell_{j,j'}Md^{4}.$$
Let us define a subset $\mathbb{J}'_i\subset\mathbb{J}_i$ in the following way. A subscript $j\in\mathbb{J}'_i$ if and only the set $$\Lambda_j=\{\lambda\in[\lambda_0,\lambda_1]: \text{\rm Osc}_{I_i}(F_{\lambda}-\epsilon V_{ij})< 2Md^4\}\neq\varnothing.$$ is non-empty. By definition, we have $$\text{\rm Osc}_{I_i}(F_{\lambda}-\epsilon V_{ij})\ge 2Md^{4}\qquad \forall \ \lambda\in[\lambda_0,\lambda_1]\ \text{\rm and}\ j\in\mathbb{J}_i\backslash\mathbb{J}'_i.$$ Using the Lipschitz property $\lambda\to F_{\lambda}$, we claim an estimate on the cardinality of this subset $$\#(\mathbb{J}'_i)\le 27K_d[d^{-2-\frac 12}], \ \ \ \text{\rm where}\ \ \ \frac{\log K_d}{|\log d|}\to 0 \ \ \text{\rm as}\ \ \ d\to 0.$$ In order to prove it, let us replace $F_{\lambda}(x)$ by $F_{\lambda}(x)-F_{\lambda}(x_i)=\int_{x_i}^x\partial_xF_{\lambda}(x)dx$, which is still Lipschitz in $\lambda$, and denote the set of functions by $$\mathfrak{F}=\{F_{\lambda}:\lambda\in [\lambda_0,\lambda_1]\}.$$ It follows from the Lipschitz property that the box dimension of the set $\mathfrak{F}$ equals one in $C^0$-topology. Let $\text{\uj C}_{D\epsilon d}(0)$ denote a cube in $C^0$-function space, centered at the origin with the size equal to $D\epsilon d$, where $D>0$ depends on the upper bound of $\{|a_{ij}|, |b_{ij}|, |c_{ij}|\}$. The set $\Lambda_j$ is non-empty only if $F_{\lambda}\in\text{\uj C}_{D\epsilon d}(0)$ holds for $\lambda\in\Lambda_j$. Since the box dimension of the set $\mathfrak{F}$ equals one, we see that the set $$\mathfrak{F}\cap\text{\uj C}_{D\epsilon d}(0),$$ can be covered by as many as $K_d[\epsilon d^{-3}]$ cubes with the size of $2Md^{4}$, where the number $K_d$ satisfies the condition that $\log K_d/|\log d|\to 0$ as $d\to 0$.
Let us keep in mind that, by the definition, each $j\in\mathbb{J}'_i$ corresponds to a non-empty set $\Lambda_j\subset [\lambda_0,\lambda_1]$. If the cardinality $\#(\mathbb{J}'_i)>28K_d[\epsilon d^{-3}]$, by Pigeonhole principle, there would be at least 28 different subscripts $j_m\in \mathbb{J}'_i$ such that certain $\lambda_{j_m}\in\Lambda_{j_m}$, and the 28 functions $\{F_{\lambda_{j_m}}:m=1,2,\cdots,28\}$ fall into one small cube with the size of $2Md^{4}$. On the other hand, since the parameter space is three dimensional, in these 28 different subscripts, there must be $m\neq m'$ such that $$\ell_{j_m,j_{m'}}=\max_{1\le \ell,\ell'\le 28} \Big\{\frac{|a_{ij_{\ell}}-a_{ij_{\ell'}}|}{8Md^{5/2}},\frac{|b_{ij_{\ell}}-b_{ij_{\ell'}}|}{2Md^{3/2}}, \frac{|c_{ij_{\ell}}-c_{ij_{\ell'}}|}{4Md^{1/2}}\Big\}\ge 4,$$ it follows from (\[appenBeq4\]) that $$\text{\rm Osc}_{I_i}(\epsilon V_{ij_m}-\epsilon V_{ij_{m'}})\ge 8Md^{4}.$$ On the other hand, as both $F_{\lambda_{j_m}}$ and $F_{\lambda_{j_{m'}}}$ fall into the same cube where $$\text{\rm Osc}_{I_i}|F_{\lambda_{j}}-\epsilon V_{ij}|<2Md^4, \qquad \text{\rm for}\ j=j_m,j_{m'}$$ we find that $$\text{\rm Osc}_{I_i}(\epsilon V_{ij_m}-\epsilon V_{ij_{m'}})\le 6Md^4.$$ This contradiction proves the claim that $\#(\mathbb{J}'_i)\le 27 K_d[\epsilon d^{-3}]$.
By the definition of the cube, we have that if $(a_i,b_i,c_i)\in\text{\uj c}_{ij}$ then $$\text{\rm Osc}_{I_i}(\epsilon V_{ij}-\epsilon V_{i})\le Md^{4}.$$ It follows from the definition for $\mathbb{J}'$ that for $(a_i,b_i,c_i)\in\text{\uj c}_{ij}$ with $j\in\mathbb{J}_i\backslash \mathbb{J}'_i$ $$\label{appenBeq5}
\text{\rm Osc}_{I_i}(F_{\lambda}-\epsilon V_{i})\ge Md^4, \ \ \ \ \forall\ \lambda\in[\lambda_0,\lambda_1].$$
The gird for $(a_i,b_i,c_i)$ induces a grid for the parameters $(A_1,B_1,A_2,B_2)$ determined by the equation $$\label{appenBeq6}
\left[\begin{matrix}a_i\\ b_i\\ c_i
\end{matrix}\right]=
\left[\begin{matrix}
-\sin x_i & \cos x_i & -2\sin 2x_i & 2\cos 2x_i \\
-\cos x_i & -\sin x_i & -4\cos 2x_i & -4\sin 2x_i \\
\sin x_i & -\cos x_i & 8\sin 2x_i & -8\cos 2x_i
\end{matrix}\right]
\left[\begin{matrix} A_1\\ B_1\\ A_2\\ B_2
\end{matrix}\right]$$ the coefficient matrix is non-singular for each $x_i\in\mathbb{T}$. Indeed, let ${\bf M}_1$ be the $3\times 3$ matrix formed by first three columns and let ${\bf M}_2$ be the matrix formed by the first, second and the fourth column, we have $$\text{\rm det}({\bf M}_1)(x_i)=6\sin 2x_i, \qquad \text{\rm det}({\bf M}_2)(x_i)=-6\cos 2x_i.$$ Note that $\inf_{x_i}\{|{\rm det}{\bf M}_1(x_i)|,|{\rm det}{\bf M}_2(x_i)|\}=3\sqrt{2}$, the grid for $(a_i,b_i,c_i)$ induces a grid for $(A_1,B_1,A_2,B_2)$. It contains as many as $K[d^{-9/2}]$ 4-dimensional strips, denoted by $\text{\uj s}_{ij}$ with $j\in\mathbb{J}_i$. Each $\text{\uj s}_{ij}$ is mapped onto $\text{\uj c}_{ij}$ by the equation (\[appenBeq6\]). For each $(A_1,B_1,A_2,B_2)\in\text{\uj s}_{ij}$ with $j\in\mathbb{J}_i\backslash\mathbb{J}'_i$, the inequality (\[appenBeq5\]) holds for any $\lambda\in [\lambda_0,\lambda_1]$.
Let us consider all intervals $I_i$ with $i=0,1,\cdots, k-1$. Different $I_i$ induces different gird for the parameters $(A_1,B_1,A_2,B_2)$. In general, $\text{\uj s}_{ij}$ is not parallel to $\text{\uj s}_{i'j'}$ if $i\ne i'$. For each $I_i$, one can define two set of subscripts $\mathbb{J}_i\supset\mathbb{J}'_i$ in the way as above. Thus, one obtains the cardinality of the disjoint union set $$\#(\vee_{i=0}^{k-1}\mathbb{J}'_i)\le 27K_d\pi[\epsilon d^{-4}]=27K_d\pi[d^{-7/2}]\ll K[d^{-9/2}].$$ Since $\log K_d/|\log d|\to 0$ as $d\to 0$, we obtain a Lebesgue measure estimate $$\text{\rm meas}\Big(\bigcup_{\stackrel {j\in\mathbb{J}'_i}{\scriptscriptstyle 0\le i\le k-1}}\text{\uj s}_{ij}\Big)\le\frac{27K_d\pi}{K}d\to 0\qquad \text{\rm as}\ d\to 0.$$ Let $\text{\uj S}^c=I^4\backslash\cup_{j\in\mathbb{J}'_i,\, 0\le i\le k-1}\text{\uj s}_{ij}$, we obtain the Lebesgue measure estimate $$\text{\rm meas}(\text{\uj S}^c)\ge 1-\frac{27K_d\pi}{K}d\to 1,\qquad \text{\rm as}\ \ d\to 0.$$ By the definition $\mathbb{J}_i$ and $\mathbb{J}'_i$, one can see that for any $(A_1,B_1,A_2,B_2)\in\text{\uj S}^c$ and any $\lambda\in [\lambda_0,\lambda_1]$ the variation of $F_{\lambda}-\epsilon V$ is bounded from below $$\text{\rm Osc}_{I_i}(F_{\lambda}-\epsilon V)\ge Md^4, \qquad \forall\ 0\le i<k.$$ It implies that the inequality (\[appenBeq1\]) holds. This completes the proof.
Hyperbolicity of minimal periodic orbits
----------------------------------------
Let $L\in C^r(T\mathbb{T}^2,\mathbb{R})$ be a Tonelli Lagrangian with two degrees of freedom ($r\ge 4$), let $H$ be the Hamiltonian determined by $L$. Because of topological property of two torus, each ergodic minimal measure is supported on closed orbits if the rotation vector satisfies certain resonant condition. It is a natural question whether these periodic orbits are hyperbolic. Once it is true, one then obtains normally hyperbolic cylinder composed by these periodic orbits.
Given a Lagrangian $L$ and a rotation direction $g\in H_1(\mathbb{T}^2, \mathbb{Z})$, by Fenchel-Legendre transformation, we obtain a channel in $H^1(\mathbb{T}^2,\mathbb{R})$ $$\mathbb{C}_g=\bigcup_{\lambda>0}\mathscr{L}_{\beta_L}(\lambda g)\subset H^1(\mathbb{T}^2,\mathbb{R}).$$ Typically, it is foliated into segments of line (flat of the $\alpha$-function), along which the $\alpha$-function keeps constant, all cohomology classes share the same Mather set. Thus, it makes sense to write $\tilde{\mathcal{M}}(c)=\tilde{\mathcal{M}}(E,g)$ with $E=\alpha(c)$ and $c\in\mathbb{C}_g$.
\[AppenHyperTh1\] Given a class $g\in H_1(\mathbb{T}^2, \mathbb{Z})$ and a closed interval $[E_a,E_d]\subset \mathbb{R}_+$ with $E_a>\min\alpha$, there exists an open-dense set $\mathfrak{O}\subset C^{r}(\mathbb{T}^2,\mathbb{R})$ with $r\ge 4$ such that for each $P\in\mathfrak{O}$, each $E\in[E_a,E_d]$, the Mather set $\tilde{\mathcal{M}}(E,g)$ for $L+P$ consists of hyperbolic periodic orbits. Indeed, except for finitely many $E_j\in[E_a,E_d]$ where the Mather set consists of two hyperbolic periodic orbits, for all other $E\in[E_a,E_d]$ it consists of exactly one hyperbolic periodic orbit.
This theorem will be proved by showing the non-degeneracy of the minimal point of certain action function. Toward this goal, let us split the interval into suitably many subintervals $[E_a,E_d]=\cup_{i=0}^{k}[E_i-\delta_{E_i},E_i+\delta_{E_i}]$ with suitably small $\delta_{E_i}>0$. Once the open-dense property holds for each small subinterval, then it hold for the whole interval.
Let us explain how the interval $[E_a,E_d]$ is split. In the channel, one can choose a path along which the $\alpha$-function monotonely increases. Restricted on this path, we obtain a family of Lagrangians with one parameter. By using the method of [@BC], we can see that it is typical that the minimal measure is supported at most on two periodic orbits for each class on this path. Thus, the Mather set $\tilde{\mathcal{M}}(E,g)$ consists of at most two periodic orbits for each $E\in[E_a,E_d]$.
Without of losing generality, we assume $g=(0,1)$, all of these minimal curves are associated with the homological class. Restricted on the neighborhood $\mathbb{S}_{\gamma_{E_i}}\subset \mathbb{T}^2$ of a minimal curve $\gamma_{E_i}\in\mathcal{M}(E_i,g)$ for certain energy $E_i$, we introduce a configuration coordinate transformation $x=X(u)$ such that along the curve $\gamma_{E_i}$ one has $u_1=\text{\rm constant}$. In the new coordinates, the Lagrangian reads $$L'(\dot u,u)=L(DX(u)\dot u,X(u))$$ which is obviously positive definite in $\dot u$. As $\gamma_E(t)$ is a solution of the Euler-Lagrange equation determined by $L$, the curve $X^{-1}(\gamma_E)(t)$ solves the equation determined by $L'$ and is minimal for the action of $L'$. As there are at most two minimal curves for each energy, the neighborhood of these two curves can be chosen not to overlap each other. Therefore, one can extend the coordinate transformation to the whole torus.
Let $H'$ be the Hamiltonian determined by $L'$ through the Legendre transformation. the minimal curve determines a periodic solution for the Hamiltonian equation. By construction, $\partial_{v_2}H'>0$ holds along the periodic solution which entirely stays in the energy level set $H'^{-1}(E)$. We choose suitably small $\delta_{E_i}>0$ such that for $E\in[E_i-\delta_{E_i},E_i+\delta_{E_i}]$ each minimal periodic curve in $\mathcal{M}(E,g)$ falls into the strip $\mathbb{S}_{\gamma_{E_i}}$ and $\partial_{v_2}H'>0$ holds along each minimal periodic orbit.
For brevity of notation, we still use $x$ to denote the configuration coordinates, use $L$ and $H$ to denote the Lagrangian and Hamiltonian, for which the condition $\partial_{y_2}H>0$ holds along minimal periodic orbits for $E\in[E_i-\delta_{E_i},E_i+\delta_{E_i}]$. Under such conditions, the Lagrangian as well as the Hamiltonian can be reduced to a time-periodic system with one degree of freedom when it is restricted on energy level set. The new Hamiltonian $\bar H(x_1,y_1,\tau,E)$ solves the equation $H(x_1,y_1,x_2,\bar H)=E$ with $\tau=-x_2$, from which one obtains a new Lagrangian $\bar L=\dot x_1y_1-\bar H(x_1,y_1,\tau,E)$ where $y_1=y_1(x_1,\dot x_1,\tau)$ solves the equation $\dot x_1=\partial_{y_1}\bar H(x_1,y_1,\tau)$. In the following we omit the subscript “1", i.e. let $(x,y,\dot x)=(x_1,y_1,\dot x_1)$ if no danger of confusion occurs.
We introduce a function of Lagrange action $F(\cdot,E)$: $\mathbb{T}\to\mathbb{R}$: $$F(x,E)=\inf_{\gamma(0)=\gamma(2\pi)=x} \int_{0}^{2\pi}\bar L(d\gamma(\tau),\tau,E)d\tau.$$ A periodic curve $\gamma$ is called the minimizer of $F$ if the Lagrange action along this curve reaches the quantity $F(\gamma(0),E)$. As there might be two or more minimizers if $x$ is not a minimal point, the function $F$ may not be smooth in global. However, we claim that it is smooth in certain neighborhood of minimal point.
To verify our claim, we let $T_i=\frac {2\pi i}m$ and define the function of action $F_i(x,x',E)$ $$F_i(x,x',E)=\inf_{\stackrel {\gamma(T_i)=x}{\scriptscriptstyle \gamma(T_{i+1})=x'}} \int_{T_i}^{T_{i+1}}\bar L(d\gamma(\tau),\tau)d\tau.$$ There will be two or more minimizers of $F_i(x,x',E)$ if the point $x$ is in the “cut locus" of the point $x'$. However, the minimizer is unique if $x$ is suitably close to $x'$, denoted by $\gamma_i(\cdot,x,x',E)$. In this case, it uniquely determines a speed $v=v(x,x')$ such that $\dot\gamma_i(T_i,x,x',E)=v(x,x')$. Let $\vec{x}= (x_0,x_1,\cdots,x_{m})$ denote a periodic configuration ($x_0=x_m$), we introduce a function of action $$\text{\bf F}(\vec{x},E)=\sum_{i=0}^{m-1}F_i(x_i,x_{i+1},E).$$ As $T_{i+1}-T_i$ is suitably small and the Lagrangian is positive definite in the speed, the boundary condition $\gamma(T_j)=x_j$ for $j=i,i+1$ uniquely determines the speed $v_j=\dot\gamma(T_j)$ for $j=i, i+1$. Indeed, the function $F_i$ generates an area-preserving twist map from the time-$T_i$-section to the time-$T_{i+1}$-section $\Phi_i$: $(x_i,y_i)\to(x_{i+1},y_{i+1})$ $$y_{i+1}=\partial_{x_{i+1}}F_i(x_{i+1},x_i), \qquad y_{i}=-\partial_{x_{i}}F_i(x_{i+1},x_i).$$ where $y_i=\partial_{\dot x}L(x_i,v_i,T_i)$. As the Lagrangian is positive definite in $\dot x$, it implies that the initial condition $(x_i,v_i)$ smoothly depends on the boundary condition $(x_i,x_{i+1})$ in this case. Because of the smooth dependance of solution of ordinary differential equation on initial condition, the function is smooth. Obviously, each minimal point of $\text{\bf F}(\cdot,E)$ uniquely determines a $c$-minimal measure for $c\in\alpha^{-1}(E)\cap \mathbb{C}_g$, supported on a periodic orbit $(\gamma_E,\dot\gamma_E)$ with $[\gamma_E]=g$. Let $x_i=\gamma_E(T_i)$, it satisfies the discrete Euler-Lagrange equation $$\frac{\partial F_i}{\partial x'}(x_{i-1},x_i,E)+\frac{\partial F_{i+1}}{\partial x}(x_{i},x_{i+1},E)=0.$$
We shall show later that the periodic orbit is hyperbolic if and only if the minimal configuration is non-degenerate, namely, the Jacobi matrix is positive definite: $$\text{\bf J}=\left[\begin{matrix}
A_0 & B_0 & 0& \cdots & B_{m-1}\\
B_0 & A_1 & B_1 & \cdots & 0 \\
0 & B_1 & A_2 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & B_{m-2}\\
B_{m-1} & 0 & 0 & B_{m-2} & A_{m-1}
\end{matrix}\right]$$ where $$A_i=\frac{\partial^2F_{i-1}}{\partial x'^2}(x_{i-1},x_i)+\frac{\partial^2F_{i}}{\partial x^2}(x_i,x_{i+1}),\ \ B_i=\frac{\partial^2F_{i}}{\partial x\partial x'}(x_{i},x_{i+1})$$ and $x_{-1}=x_{m-1}$.
Let $\vec{x}=(x_0,x_1,\cdots,x_m=x_0)$ be a minimal configuration of the function $F(\vec{x},E)$, where the Jacobi matrix is non-negative and the smallest eigenvalue is simple. Indeed, as the Lagrangian is positive definite, the generating function $F_i(x,x',E)$ determines an area-preserving and twist map $\Phi_i$, we have $B_i<0$. Consequently, by using a theorem in [@vM], we find that the smallest eigenvalue is simple. Let $\lambda_i$ denote the $i$-th eigenvalue of the matrix, at the minimal configuration one has $$0\le\lambda_0<\lambda_1\le\lambda_2<\cdots\le\lambda_{m-1}.$$ Let $\xi_i=(\xi_{i,0},\xi_{i,1},\cdots,\xi_{i,m-1})$ be the eigenvector for $\lambda_i$. By choosing $\xi_{0,0}=1$ we have $\xi_{0,i}>0$ for $1\le i<m$ (see Lemma 3.4 in [@An]). At the minimal configuration, we find the following matrix is positive definite: $$\text{\bf J}_{m-1}=\left[\begin{matrix}
A_1 & B_1 & \cdots & 0 \\
B_1 & A_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & B_{m-2}\\
0 & 0 & B_{m-2} & A_{m-1}
\end{matrix}\right].$$ If not, there will be a vector $\hat v=(v_1,\cdots,v_{m-1})\in\mathbb{R}^{m-1}\backslash\{0\}$ such that $\hat v^t\text{\bf J}_m\hat v=0$. It follows that $v^t\text{\bf J}v=0$ if we set $v=(0,\hat v)\in\mathbb{R}^m$. As the matrix $\text{\bf J}$ is non-negative, it implies that $v=\mu\xi_0$, but it contradicts the fact that all entries of $\xi_{0}$ have the same sign, either positive or negative.
In a suitably small neighborhood $\vec{U}=U_0\times U_1\times\cdots\times U_{m-1}$ of the minimal configuration, let us consider the equations $$\label{variationeq1}
\frac{\partial\text{\bf F}}{\partial x_i}(x_0,x_1,\cdots,x_{m-1},E)=0,\qquad \forall\ i=1,2,\cdots,m-1.$$ Since the matrix $\{\frac{\partial^2\text{\bf F}}{\partial x_i\partial x_j}\}_{i,j=1}^{m-1}=\text{\bf J}_{m-1}$ is positive definite at the minimal point, by the implicit function theorem, this equation has a unique smooth solution $x_i=X_i(x_0,E)$ when $x_0\in U_0$. Let $\gamma$: $[0,2\pi]\to\mathbb{R}$ be a minimizer of $F(x_0)$ with $\gamma(0)=\gamma(2\pi)=x_0$, we obtain a configuration $x_i=\gamma(2i\pi/m)$. Clearly, $\partial_{x_i}\text{\bf F}=0$ holds at this configuration for each $i\ge 1$. It implies the uniqueness of the minimizer of $F(x_0,E)$ for $x_0\in U_0$. The minimal point of $F$ uniquely determines a minimal configuration of $\text{\bf F}$, therefore, the function $F$ is smooth in certain neighborhood of its minimal point.
[**Non-degeneracy of minimizers**]{}
In a neighborhood of the minimal point, let us study what change the function of action undergoes when the Lagrangian is under a perturbation of potential $L\to L+P$, where $P$: $\mathbb{T}^2\to\mathbb{R}$ is a potential. Let $\bar L'$ denote the reduced Lagrangian of $L+P$ and let $G=-(\partial_{y_2}H)^{-1}$, one has $$\bar L'=\bar L+GP+O(\|P\|^2).$$ We denote the minimal curve of $F(x,E)$ by $\gamma(t,x,E)$ such that $\gamma(0,x,E)=x$. Let $\gamma'(t,x,E)$ and $F'(x,E)$ be the quantities defined for $\bar L'$ as the quantities $\gamma(t,x,E)$ and $F(x,E)$ defined for $\bar L$. By the definition of minimizer, we have $$\begin{aligned}
F'(x,E)-F(x,E)&=\int_0^{2\pi}\bar L'(d\gamma'(\tau))d\tau -\int_0^{2\pi}\bar L(d\gamma(\tau))d\tau\\
&\ge \int_0^{2\pi}G(d\gamma'(\tau),\tau)P(\gamma'(\tau))d\tau+o(\|\gamma'-\gamma\|,\|P\|),\end{aligned}$$ and $$\begin{aligned}
F'(x,E)-F(x,E)&=\int_0^{2\pi}\bar L'(d\gamma'(\tau))d\tau-\int_0^{2\pi}\bar L(d\gamma(\tau))d\tau\\
&\le \int_0^{2\pi}G(d\gamma(\tau),\tau)P(\gamma(\tau))d\tau+o(\|\gamma'-\gamma\|,\|P\|).\end{aligned}$$ Since the distance between these two curves $\gamma$ and $\gamma'$ is due to the perturbation $P$, we finally obtain $$\label{nondegenerate1}
F'(x,E)=F(x,E)+\mathscr{K}_EP(x)+\mathscr{R}_EP(x)$$ where $\mathscr{R}_EP=o(\|P\|)$ and $$\mathscr{K}_EP(x)=\int_0^{2\pi}G(d\gamma(\tau,x,E),\tau)P(\gamma(\tau,x,E))d\tau.$$ The field of smooth curves $\{\gamma(\cdot,x,E)\}$ defines an operator $P\to\mathscr{K}_EP$, which maps functions defined on $\mathbb{T}^2$ into the function space defined on $\mathbb{T}$. Obviously, both $\mathscr{K}_EP$ and $\mathscr{R}_EP$ are smooth in $x\in U_0$ and in $E$.
Unless the point $x$ is a minimizer of $F(\cdot,E)$, the curve $\gamma(\cdot,x,E)$ may have corner at $\tau=0\mod 2\pi$.
\[hyperlem1\] There exist constants $\varepsilon,\theta>0$ such that if $F(x,E)-\min F(\cdot,E)<\varepsilon$ and if $\gamma:$ $[0,2\pi]\to\mathbb{R}$ is a minimizer of $F(x,E)$, then $$\label{hyperbolicorbit2}
\|\dot\gamma(0)-\dot\gamma(2\pi)\|<\theta\sqrt{F(x,E)-\min F(\cdot,E)}.$$
Let us consider the derivative of $F(\cdot,E)$. As the Lagrangian is positive definite, some positive constants $m_L>0$ exist such that $$\frac{\partial ^2\bar L}{\partial\dot x^2}\ge m_L, \qquad \forall\ (x,\dot x)\in T\mathbb{T}^2.$$ Since $\gamma(0,x,E)=\gamma(2\pi,x,E)=x$, one has $\partial_{x}\gamma(0)=\partial_{x}\gamma(2\pi)=1$ and $$\begin{aligned}
\Big|\frac{\partial F}{\partial x}\Big|&=\Big|\int_{0}^{2\pi}\Big (\frac{\partial\bar L}{\partial\dot x} (d\gamma(\tau),\tau)\frac{\partial\dot\gamma}{\partial x}+\frac{\partial\bar L}{\partial x} (d\gamma(\tau),\tau)\frac{\partial\gamma}{\partial x}\Big )d\tau\Big|\\
&=\Big|\frac{\partial\bar L}{\partial\dot x}(\dot\gamma(0),\gamma(0),0)-\frac{\partial\bar L}{\partial\dot x}(\dot\gamma(2\pi),\gamma(2\pi),2\pi)\Big|\\
&\ge m_L|\dot\gamma(0)-\dot\gamma(2\pi)|,\end{aligned}$$ where the second equality follows from that $\gamma$ solves the Euler-Lagrange equation. If $\frac{\partial F}{\partial x}>0$ and if the lemma does not hold, by choosing $x'-x=-\sqrt{\Delta}$ ($\Delta=F(x,E)-\min F(\cdot,E)$) we obtain from the Taylor series up to second order that $$\begin{aligned}
F(x',E)-\min F(\cdot,E)&=F(x',E)-F(x,E)+F(x,E)-\min F(\cdot,E)\\
&\le -\partial_{x}F(x,E)\sqrt{\Delta}+\frac M2\Delta+\Delta<0\end{aligned}$$ if $\theta>\frac 1{m_L}(1+\frac M2)$, where $M=\max\partial^2_{x}F$. But it is absurd. The case $\frac{\partial F}{\partial x}<0$ can be proved by choosing $x'-x=\sqrt{\Delta}$. This completes the proof.
Let $x\in(x^*-\delta_{x^*},x^*+\delta_{x^*})$, where $x^*$ is the minimal point of $F(\cdot,E_0)$. As it was shown above, $\gamma(\frac{2i\pi}m,x,E_0)$ smoothly depends on $x$, we have a smooth foliation of curves in a neighborhood of the curve $\gamma(\cdot,x^*,E_0)$. The corner at $\gamma(0,x,E_0)$, i.e. $\dot\gamma(2\pi,x,E_0)-\dot\gamma(0,x,E_0)$ approaches to zero as $F(x,E_0)\downarrow\min F(\cdot,E_0)$. For each $x$, if there is a corner at $\gamma(0,x,E_0)=\gamma(2\pi,x,E_0)$, we construct a curve $\gamma_{x}$ that smoothly connects the point $\gamma(2\pi-\delta,x,E_0)$ to the point $\gamma(\delta,x,E_0)$ with $\gamma_x(0)=x$, where $\delta>0$ is suitably small. Replacing the segment $\gamma(\cdot,x,E_0)|_{[0,\delta]\cup[2\pi-\delta,2\pi]}$ by this curve, we obtain a smooth curve $\gamma_{x}$ such that $\gamma_x(t)=\gamma(t,x,E_0)|_{[\delta,2\pi-\delta]}$ and $\gamma_x(0)=x$. Indeed, as the curve $\gamma(t,x,E_0)$ is $C^3$-smooth in $x$ and $\gamma(t,x^*,E_0)$ is also $C^3$-smooth in $t$, for small number $\varepsilon$ some $\mu_0>0$ exists such that the quantities $$\Big|\frac{d^k\gamma}{dt^k}(t,x,E_0)-\frac{d^k\gamma}{dt^k}(t,x^*,E_0))\Big|< \varepsilon,\qquad \forall\ x\in[x^*-\delta_{x^*},x^*+\delta_{x^*}], \ k=0,1,2,3.$$ Let the curve $\zeta_x(\cdot)$: $[-\delta,\delta]\to\mathbb{R}$ be an interpolation polynomial of degree eight such that $$\frac{d^k\zeta_x}{dt^k}(t)=\frac{d^k\gamma}{dt^k}(t,x,E_0)-\frac{d^k\gamma}{dt^k}(t,x^*,E_0) \qquad \forall\ t=\pm\delta,$$ and $\zeta_x(0)=\gamma(0,x,E_0)-\gamma(0,x^*,E_0)$, then the coefficients of the polynomial are smooth in $x$. Let $\gamma_{x}(t)=\gamma(t,x^*,E_0)+\zeta_x(t)$, we see that the foliation of the curves $\gamma_{x}$ is smooth in $x$ and as a function of $t$, $\gamma_{x}-\gamma(\cdot,x,E_0)$ is small in $C^3$-topology.
For each point $(\tau,x)\in\mathbb{S}$, there is a curve $\gamma_{x_0}$ such that $x=\gamma_{x_0}(\tau)$. It uniquely determines a speed $v=v(x,\tau)=\dot\gamma_{x_0}(\tau)$. By the construction, $v(x,\tau)$ is $C^3$-smooth in $(x,\tau)$. As $-G^{-1}=\partial_{y_2}H>0$ when it is restricted to a neighborhood of the minimal curves, both $v$ and $G$ can be approximated by $C^r$-function $v_s$ and $G_s$ in $C^3$-topology respectively i.e. $\|v-v_s\|_{C^3}<\varepsilon$ and $\|G-G_s\|_{C^3}<\varepsilon$ hold for small $\varepsilon>0$. Given a $C^r$-function $\bar P$: $\mathbb{T}\to\mathbb{R}$ we obtain a $C^r$-function $P=\mathscr{T}_{E_0}\bar P$: $\mathbb{T}^2\to\mathbb{R}$ defined by $$\label{nondegenerate2}
P(x,\tau)=\mathscr{T}_{E_0}\bar P(x_0)=G^{-1}_s(v_s(x,\tau),x,\tau)\bar P(x_0),$$ if $x=\gamma_{x_0}(\tau)$. By the definition, we have $$\label{nondegenerate3}
\mathscr{K}_E\mathscr{T}_{E_0}\bar P(x)= \int_0^{2\pi}\frac{G(d\gamma(\tau,x,E),\tau)}{G_s(v_s(\gamma (\tau,x,E),\tau),\gamma (\tau,x,E),\tau)}\bar P(x+\Delta\gamma (\tau,x,E))d\tau$$ where $\Delta\gamma(\tau,x,E)$ is defined as follows: passing through the point $\gamma (\tau,x,E)$ there is a unique $x'$ such that $\gamma_{x'}(\tau)=\gamma(\tau,x,E)$. We set $\Delta\gamma(\tau,x,E)=x'-x$.
We introduce a set of perturbations with four parameters: $$\bar{\mathfrak{P}}=\Big\{\sum_{\ell=1}^2(A_{\ell}\cos \ell x+B_{\ell}\sin \ell x):\ (A_1,B_1,A_2,B_2)\in\mathbb{I}^4\Big\},$$ where $\mathbb{I}^4=[1,2]\times[1,2]\times[1,2]\times[1,2]$. By applying the formula (\[nondegenerate3\]) to the function $\cos\ell x$ and $\sin\ell x$ we find that $$\begin{aligned}
\label{nondegenerate4}
\mathscr{K}_E\mathscr{T}_{E_0}\cos\ell x&=u_{\ell}(x,E)\cos\ell x-v_{\ell}(x,E)\sin\ell x,\\
\mathscr{K}_E\mathscr{T}_{E_0}\sin\ell x&=u_{\ell}(x,E)\sin\ell x+v_{\ell}(x,E)\cos\ell x,\notag\end{aligned}$$ where $$\begin{aligned}
u_{\ell}(x,E)&=\int_0^{2\pi}\frac{G(d\gamma(\tau,x,E),\tau)}{G_s(v_s(\gamma (\tau,x,E),\tau),\gamma (\tau,x,E),\tau)}\cos \ell\Delta\gamma(\tau,x,E)d\tau,\\
v_{\ell}(x,E)&=\int_0^{2\pi}\frac{G(d\gamma(\tau,x,E),\tau)}{G_s(v_s(\gamma (\tau,x,E),\tau),\gamma (\tau,x,E),\tau)}\sin \ell\Delta\gamma(\tau,x,E)d\tau.\end{aligned}$$ Let us study the dependence of the terms $u_{\ell}(x,E)$ and $v_{\ell}(x,E)$ on the point $x$. We claim that there exists constant $\theta_1>0$ as well as small numbers $\delta_{E_0}>0$ and $\delta_{x^*}>0$ such that for each $E\in (E_0-\delta_{E_0},E_0+\delta_{E_0})$, each $x\in(x^*-\delta_{x^*},x^*+\delta_{x^*})$, $j=0,1,2,3$ and $\ell=1,2$, we have $$|u_{\ell}(x,E)|\ge 1-\theta_1\delta, \qquad |v_{\ell}(x,E)|\le\theta_1\delta,$$ $$\label{nondegenerate5}
\max_{j=1,2,3}\Big\{\Big|\frac {\partial^ju_{\ell}}{\partial x^j}(x,E)\Big|,\Big|\frac {\partial^j v_{\ell}}{\partial x^j}(x,E)\Big|\Big\} \le\theta_1\delta.$$
By the construction of the curves $\gamma_x$, for $\tau\in\mathbb{T}\backslash (-\delta,\delta)$ and for $x\in(x^*-\delta_{x^*},x^*+\delta_{x^*})$ we have $$\Delta\gamma(\tau,x,E_0)=0 \ \ \text{\rm and}\ \ \frac{G(d\gamma(\tau,x,E_0),\tau)}{G(v(\gamma (\tau,x,E_0),\tau),\gamma (\tau,x,E_0),\tau)}=1$$ and $\partial_{x}^j\Delta\gamma (\tau,x,E_0)$ is small for $\tau\in(-\delta,\delta)$ and for $j=0,1,2,3$. Integrating the them over the set with Lebesgue measure $2\delta$, we find that some small $\theta_1>0$ exists such that the formulae in (\[nondegenerate5\]) hold for $E=E_0$ with $\theta_1$ being replaced by $\theta_1/4$ if $G_s$ and $v_s$ in the formula (\[nondegenerate3\]) are replaced by $G$ and $v$ respectively. As both $v$ and $G$ are approximated by $v_s$ and $G_s$ in $C^3$-topology, by choosing $\varepsilon>0$ suitably small, all formulae in (\[nondegenerate5\]) hold for $E=E_0$ with $\theta_1$ being replaced by $\theta_1/2$.
For other energy $E$, let us recall the solution $x_i=X_i(x_0,E)$ of Eq. (\[variationeq1\]) is smooth. As the map $\Phi_i$: $(x_i,y_i)\to(x_{i+1},y_{i+1})$ is area-preserving and twist, it uniquely determines the initial speed $v_0=v_0(x_0,E)$, namely, the initial speed smoothly depends on the initial position and such dependence is also smooth in the parameter $E$. As solution of ODE smoothly depends on its initial conditions, the minimal curve $\gamma(\cdot,x,E)$ of $F(x,E)$ smoothly depends on the parameters $x$ and $E$. Thus, the formulae in (\[nondegenerate5\]) hold if the numbers $\delta_{E_0}>0$ and $\delta_{x^*}>0$ are suitably small.
\[theo2\] There exists an open-dense set $\mathfrak{O}\subset C^r(M,\mathbb{R})$ with $r\ge4$ such that for each $P\in\mathfrak{O}$ and each $E\in [E_0-\delta_{E_0},E_0+\delta_{E_0}]$, all minimizers of $F(\cdot,E)$, determined by $L+P$, are non-degenerate.
To show the non-degeneracy of the global minimum of $F(\cdot,E)$ located at the point $x$, we only need to verify that $$\label{nondegenerate6}
F(x+\Delta x,E)-F(x,E)\ge M|\Delta x|^4$$ holds for small $|\Delta x|$, where $M=12^{-1}\max\partial^4_{x}F$. Assume $I$ is an interval, we define $\text{\rm Osc}_{I}F=\max_{x,x'}|F(x)-F(x')|$. To show the non-degeneracy, it is sufficient to verify that $$\text{\rm Osc}_{I}F(\cdot,E)\ge M|I|^4$$ if the minimal point $x\in I$, where $|I|$ denotes the length of the interval.
The openness is obvious of $\mathfrak{P}$. To show the density, we are concerned only about the configurations where $F$ takes the value close to the minimum and consider small perturbations from the following set where the parameters $(A_1,B_1,A_2,B_2)$ range over the cube $\mathbb{I}^4=[1,2]\times[1,2]\times[1,2]\times[1,2]$ $$\mathfrak{V}_E=\Big\{(\mathscr{K}_E+\mathscr{R}_E)\mathscr{T}_{E_0}\sum_{\ell=1}^2\epsilon (A_{\ell}\cos\ell x +B_{\ell}\sin\ell x):(A_1,B_1,A_2,B_2)\in\mathbb{I}^4\Big\}$$ where each element is a function of $(x,E)$, see the formulae (\[nondegenerate4\]). Recall that both operators $\mathscr{K}_E$ and $\mathscr{T}_{E_0}$ are linear and $\|\mathscr{R}_E(\epsilon P)\|=o(\epsilon)$, see the formula (\[nondegenerate1\]) and the formula (\[nondegenerate2\]).
We choose sufficiently large integer $K$ so that $\epsilon=\sqrt[4]{\pi/K}$ can be arbitrarily small. Let $x_k=\frac{2k\pi}K$, $I_k=[x_k-d,x_k+d]$ and $d=\pi/K$, then $\bigcup_{k=0}^{K-1}I_k=\mathbb{T}$. Restricted on each interval $I_k$, each $C^4$-function $V\in\mathfrak{V}_E$ is approximated by the Taylor series (module constant) $$V_k(x)=\epsilon\Big(a_k(x-x_{k})+b_k(x-x_{k})^2+c_k(x-x_{k})^3+O(|x-x_{k}|^4)\Big).$$
Given two points $(a_k,b_k,c_k)$ and $(a'_k,b'_k,c'_k)$, we have two functions $V_k$ and $V'_k$ in the form of Taylor series. Let $\Delta V=V'_k-V_k$, $\Delta a=a'_k-a_k$, $\Delta b=b'_k-b_k$ and $\Delta c=c'_k-c_k$, we have $\Delta V(x_k)=0$ and $$\begin{aligned}
&\Delta V(x_k+d)+\Delta V(x_k-d)=2\epsilon\Delta bd^2+O(\epsilon d^4),\\
&\Delta V(x_k+d)-\Delta V(x_k-d)=2\epsilon(\Delta a+\Delta cd^2)d+O(\epsilon d^4),\\
&\Delta V\Big(x_k\pm\frac 12d\Big)=\epsilon\Big(\pm\frac 12\Delta a+\frac 14\Delta bd\pm\frac 18\Delta cd^2\Big)d+O(\epsilon d^4).\end{aligned}$$ It follows that $$\label{nondegenerate7}
\text{\rm Osc}_{I_k}(V'_k-V_k)\ge\epsilon\max\Big\{\frac 13|\Delta a|d,|\Delta b|d^2,\frac 12|\Delta c|d^3\Big\}.$$
We construct a grid for the parameters $(a_k,b_k,c_k)$ by splitting the domain for them equally into a family of cuboids and setting the size length by $$\Delta a_k=9Md^{\frac{11}4},\ \ \Delta b_k=3Md^{\frac{7}4},\ \ \Delta c_k=6Md^{\frac 34}.$$ These cuboids are denoted by $\text{\uj c}_{kj}$ with $j\in\mathbb{J}_k=\{1,2,\cdots\}$, the cardinality of the set of the subscripts is up to the order $$\#(\mathbb{J}_k)=N[d^{-\frac{21}4}],$$ where the integer $0<N\in\mathbb{N}$ is independent of $d$. If $\text{\rm Osc}_{I_k}F(\cdot,E)\le Md^4$, we obtain from the formula (\[nondegenerate7\]) that $$\text{\rm Osc}_{I_k}(F(x,E)+V(x))\ge 2Md^4$$ if $V(x)=\epsilon(a(x-x_k)+ b(x-x_k)^2+c(x-x_k)^3+O(|x-x_k|^4))$ with $$\max\Big\{\frac 13|a|d^{-\frac{11}4},|b|d^{-\frac 74}, \frac 12|c|d^{-\frac 34}\Big\}\ge 3M.$$
The coefficients $(a_k,b_k,c_k)$ depend on the parameters $(A_1,B_1,A_2,B_2)$, the energy $E$ and the position $x_k$. The gird for $(a_k,b_k,c_k)$ induces a grid for the parameters $(A_1,B_1,A_2,B_2)$, determined by the equation $$\label{nondegenerate8}
\left[\begin{matrix} a_k\\ b_k\\ c_k
\end{matrix}\right]=({\bf C_1}{\bf U}+{\bf C_2})
\left[\begin{matrix} A_1\\ B_1\\ A_2\\ B_2
\end{matrix}\right]\Big(1+T_{\epsilon,E,x_{k}}(A_1,B_1,A_2,B_2)\Big)$$ where the map $T_{\epsilon,E,x_{k}}$: $\mathbb{R}^4\to\mathbb{R}^3$ is as small as of order $O(\epsilon)$, $${\bf C_1}=\left[\begin{matrix}
-\sin x_{k} & \cos x_{k} & -2\sin 2x_{k} & 2\cos 2x_{k}\\
-\cos x_{k} & -\sin x_{k} & -4\cos 2x_{k} & -4\sin 2x_{k} \\
\sin x_{k} & -\cos x_{k} & 8\sin 2x_{k} & -8\cos 2x_{k}
\end{matrix}\right],$$ $${\bf U}=\text{\rm diag}\left\{
\left[\begin{matrix}
u_1(x_{k}) & v_1(x_{k})\\
-v_1(x_{k}) & u_1(x_{k})
\end{matrix}\right],
\left[\begin{matrix}
u_2(x_{k}) & v_2(x_{k})\\
-v_2(x_{k}) & u_2(x_{k})
\end{matrix}\right]\right\},$$ each entry of ${\bf C_2}$ is a linear function of $\partial^{j}_{x}u_{\ell}\cos \ell x_{k}$, $\partial^{j}_{x}v_{\ell}\cos\ell x_{k}$, $\partial^{j}_{x}u_{\ell}\sin\ell x_{k}$ and $\partial^{j}_{x}v_{\ell}\sin\ell x_{k}$ with $j=1,2,3$, $\ell=1,2$. Both matrices ${\bf U}$ and ${\bf C}_2$ depend on the energy $E$, ${\bf U}$ is close to the identity matrix. Let ${\bf M_1}$ be the matrix composed by the first three columns of ${\bf C_1}{\bf U}+{\bf C_2}$, ${\bf M_2}$ be the matrix composed by the first, the second and the fourth column of ${\bf C_1}{\bf U}+{\bf C_2}$. As we are only concerned about those positions where $F$ takes value close to the minimum and about the energy $E$ close to $E_0$, in virtue of (\[nondegenerate5\]) we obtain $$\begin{aligned}
\text{\rm det}({\bf M}_1)(x_k)&=6\sin 2x_{k}(1-O(\theta_1\delta)), \\
\text{\rm det}({\bf M}_2)(x_k)&=-6\cos 2x_{k}(1-O(\theta_1\delta)).\end{aligned}$$ Since $\inf_{x_{k}}\{|{\rm det}{\bf M}_1(x_k)|,|{\rm det}{\bf M}_2(x_k)|\}= 3\sqrt{2}(1-O(\theta_1\delta))$, the grid for $(a_k,b_k,c_k)$ induces a grid for $(A_1,B_1,A_2,B_2)$ which contains as many as $N_1[d^{-\frac{21}4}]$ 4-dimensional strips ($N_1>0$ is independent of $d$). Note that the induced partition for the parameters $(A_1,B_1,A_2,B_2)$ depends on the energy $E$.
Given an energy $E\in[E_0-\delta_{E_0},E_0+\delta_{E_0}]$, if there exist Taylor coefficients $(a_k,b_k,c_k)$ which determines a perturbation $V$ such that $$\text{\rm Osc}_{I_k}(F(\cdot,E)+V)\le Md^{4}$$ then for $(a'_k,b'_k,c'_k)$ which determines a perturbation $\Delta V'$ and satisfies the condition $$\max\Big\{\frac{|a_k-a'_k|}{9Md^{\frac{11}4}},\frac{|b_k-b'_k|}{3Md^{\frac 74}}, \frac{|c_k-c'_k|}{6Md^{\frac 34}}\Big\}\ge 1$$ one obtains from the formula (\[nondegenerate7\]) that $$\label{nondegenerate9}
\text{\rm Osc}_{I_k}(F(\cdot,E)+V')\ge 2Md^{4}.$$
Under the map defined by the formula (\[nondegenerate8\]), the inverse image of a cuboid $\text{\uj c}_{k}$ with the size $18Md^{\frac{11}4}\times6Md^{\frac{7}4}\times12Md^{\frac{3}4}$ is a strip in the parameter space of $(A_1,B_1,A_2,B_2)$, denoted by $\text{\uj S}_{k}(E)$, with the Lebesgue measure as small as $N_1^{-1}d^{\frac{21}4}$. If the cuboid $\text{\uj c}_{k}$ is centered at $(a_k,b_k,c_k)$, then for $(a'_k,b'_k,c'_k)\notin\text{\uj c}_{k}$ the inequality (\[nondegenerate9\]) holds.
Splitting the interval $[E_0-\delta_{E_0},E_0+\delta_{E_0}]$ equally into small sub-intervals $I_{E,j}$ with the size $|I_{E,j}|=M_1^{-1}d^4$, we obtain as many as $[M_1d^{-4}]$ small intervals. Since the function $F$ is Lipschitz in $E$, suitably large positive number $M_1$ can be chosen so that $$\max_{x\in I_k}|F(x,E)-F(x,E')|<\frac 12Md^4, \qquad \forall\ E,E'\in I_{E,j}.$$ Therefore, for $V\in\mathfrak{V}_E$ with $(\Delta A_1,\Delta B_1,\Delta A_2,\Delta B_2)\notin\text{\uj S}_{k}(E)$, one has $$\label{nondegenerate10}
\text{\rm Osc}_{I_k}(F(\cdot,E)+\Delta V')\ge Md^{4}.$$ Pick up one energy $E_{j}$ in each small interval $I_{E,j}$, there are $[M_1d^{-4}]$ strips $\text{\uj S}_{k}(E_j)$. Finally, by considering all small intervals $I_k$ with $k=0,1,\cdots K-1$, we find $$\text{\rm meas}\Big(\bigcup_{k,j}\text{\uj S}_{k}(E_j)\Big)\le M_1N_1^{-1}\sqrt[4]{d}.$$ Let $\text{\uj S}^c=\mathbb{I}^4\backslash\cup_{j,k}\text{\uj S}_{kj}(E_j)$, we obtain the Lebesgue measure estimate $$\text{\rm meas}(\text{\uj S}^c)\ge 1-M_1N_1^{-1}\sqrt[4]{d}\to 1,\qquad \text{\rm as}\ \ d\to 0.$$ Obviously, for each $(A_1,B_1,A_2,B_2)\in\text{\uj S}^c$, each $E\in [E_0-\delta_{E_0},E_0+\delta_{E_0}]$ and each $k=1,2,\cdots,K$ the formula (\[nondegenerate10\]) holds. This proves that it is open-dense that all minimal points of $F(\cdot,E)$ are non-degenerate when the energy ranges over the interval $[E_0-\delta_{E_0},E_0+\delta_{E_0}]$.
[**Hyperbolicity**]{}
Let $x^*$ be a minimal point of the function $F(\cdot,E)$ and let the curve $\gamma(\cdot,x^*,E)$: $\mathbb{T}\to\mathbb{R}$ be the minimal curve of $F(x^*,E)$ which is smooth and determines a periodic orbit $(\tau,\gamma(\tau),\frac d{d\tau}\gamma(\tau))$ of the Lagrange flow $\phi^{\tau}_{\bar L}$. Back to the autonomous system, it determines a periodic orbit $(\gamma_1(t),\dot\gamma_1(t),\gamma_2(t),\dot\gamma_2(t))$ of the Lagrange flow $\phi_L^t$, where $\gamma_2(t)=-\tau$, $\gamma_1(t)=\gamma(\gamma_2(t))$.
\[theo3\] If $x^*$ is a non-degenerate minimal point of the function $F(\cdot,E)$, then the periodic orbit $\gamma(\cdot,x^*,E)$ is hyperbolic.
If a periodic orbit is hyperbolic, it has its stable and unstable manifold in the phase space. Consequently, any orbit staying on the stable (unstable) manifold approaches to the periodic orbit exponentially fast as the time approaches to positive (negative) infinity.
In a neighborhood of the minimal periodic curve $\gamma$, each point $x$ on the section $\{\tau=0\}$ determines at least one forward (backward) semi-static curve $\gamma^+_x$: $\mathbb{R}_+\to\mathbb{T}$ ($\gamma^-_x$: $\mathbb{R}_-\to\mathbb{T}$) such that $\gamma^{\pm}_x(0)=x$. These curves determine forward (backward) semi-static orbits $d\gamma^{\pm}_x$ of which the $\omega$-set ($\alpha$-set) is the periodic orbit $d\gamma$. In the configuration space $(x,\tau)\in\mathbb{T}^2$, these two curves intersect with the section $\{\tau=0\}$ infinitely many times at the points $\gamma^+_x(2k\pi)$ and $\gamma^-_x(-2k\pi)$. These points are denoted by $x_i$, they are well ordered $\cdots\prec x_{i+1}\prec x_i\cdots\prec x_0$. It is possible that $\gamma^+_x(2k\pi)=\gamma^-_x(-2k'\pi)$. In this case, we count the point twice. For each point $x_i$, there is a curve joining $(x_i,0)$ to $(x_i,2\pi)$ which is composed by some segments of $\gamma^+_x$ as well as of $\gamma^-_x$. For instance, in the following figure, by starting from the point $(x_2,0)$ and following a segment of $\gamma^-_x$ to the point $A$, then following a segment of $\gamma^+_x$ to the point $B$ and finally following a segment of $\gamma^-_x$ to the point $(x_2,2\pi)$, we obtain a circle. Clearly, the Lagrange action along this circle is not smaller than the quantity $F(x_2)$.
{width="5.2cm" height="5.0cm"} \[\]
Let us consider the whole sequence $\{x_i\}$, we obtain infinitely many circles in that way. Therefore, the sum of the quantities $F(x_i)|_{i=0}^{\infty}$ is obviously not bigger than the total action along all of these circles $$\label{hypereq1}
\sum_{i=0}^{\infty}F(x_i)\le \lim_{k\to\infty}\Big\{\int_{0}^{2k\pi}L(d\gamma^+(\tau),\tau)d\tau+\int_{-2k\pi}^0L(d\gamma^-(\tau), \tau)d\tau\Big\}.$$ The right hand side is nothing else but the barrier function valued at $x_0$.
As the periodic orbit supports the minimal measure, both $\gamma^+_x(2k\pi)$ and $\gamma^-_x(-2k\pi)$ approach the point $x^*$ where the periodic curve intersects the section $\{\tau=0\}$ as $k\to\infty$. If the periodic orbit is not hyperbolic, the sequence of $\{x_i\}$ approach $x$ slower than exponentially, i.e., for any small $\lambda>0$ there exists $\delta>0$ such that $$\begin{aligned}
|\gamma^+_x(2(k+1)\pi)-x^*)|&\ge (1-\lambda)|\gamma^+_x(2k\pi)-x^*)|,\\
|\gamma^-_x(-2(k+1)\pi)-x^*)|&\ge (1-\lambda)|\gamma^-_x(-2k\pi)-x^*)|,\end{aligned}$$ if $|\gamma^{\pm}_x(0)-x^*|\le\delta$. It follows that $|x_{i+1}-x^*|\ge(1-\lambda)|x_{i}-x^*|$. As the periodic curve is assumed non-degenerate minimizer, some $\lambda_0>0$ exists such that $$\label{hypereq2}
\sum_{i=0}^{\infty}(F(x_i)-F(x^*))\ge\lambda_0\sum_{i=0}^{\infty}(x_i-x^*)^2\ge \lambda_0\frac{(x_0-x^*)^2}{1-(1-\lambda)^2}.$$ By subtracting $\min F$ from the Lagrangian $L$ we obtain that $F(x^*)=0$ and $$\text{\rm right-hand-side of (\ref{hypereq1})}=u^-(x,0)-u^+(x,0)$$ where $u^{\pm}$ represents the backward (forward) weak-KAM solution. Since $u^-$ is semi-concave and $u^+$ is semi-convex, $u^--u^+$ is semi-concave. Since $(x^*,0)$ is a minimal point where $u^-(x^*,0)-u^+(x^*,0)=0$, there exists some number $C_L>0$ such that (cf. [@Fa2]) $$u^-(x_0,0)-u^+(x_0,0)\le C_L(x_0-x^*)^2.$$ Comparing this with the inequality (\[hypereq2\]), we obtain from (\[hypereq1\]) a contradiction $$\lambda_0\frac{(x_0-x^*)^2}{1-(1-\lambda)^2}\le C_L(x_0-x^*)^2$$ if $\lambda>0$ is suitably small. This proves the hyperbolicity of the periodic orbit.
We are now ready to prove the main result.
[*Proof of Theorem \[AppenHyperTh1\]*]{}. According to Theorem \[theo2\] and \[theo3\], for each $E_i\in[E_a,E_d]$, a neighborhood $[E_i-\delta_{E_i},E_i+\delta_{E_i}]$ of $E_i$ and an open-dense set $\mathfrak{O}(E_i)\subset C^r(\mathbb{T}^2,\mathbb{R})$ exist such that for each $P\in\mathfrak{O}(E_i)$ and each $E\in[E_i-\delta_{E_i},E_i+\delta_{E_i}]$ each minimal orbit of $\phi_{L+P}^t$ with homological class $g$ is hyperbolic. As each $\delta_{E_i}$ is positive, there exists finitely many $E_i$ such that $[E_a,E_d]\subset\cup_i[E_i-\delta_{E_i},E_i+\delta_{E_i}]$. We take $P\in\cap\mathfrak{O}(E_i)$, the hyperbolicity for $L+P$ holds for all $E\in[E_a,E_d]$.
Once a minimal point is non-degenerate for certain $E$, by the theorem of implicit function it has natural continuation to a neighborhood of $E$. Namely, there exists a curve of minimal points passing through this point, it either reaches to the boundary of $[E_a,E_d]$, or extends to some point $E'$ where the critical point is degenerate. Since each global minimal point is non-degenerate, the critical point becomes local minimum when it enters into certain neighborhood of $E'$. As each non-degenerate minimal point is isolated to other minimal points for the same energy $E$, there are finitely many such curves, denoted by $\Gamma_i$.
For a curve $\Gamma_i$: $I_i=(E_i,E'_i)\to\mathbb{T}$, the definition domain $I_i$ contains finitely many closed sub-intervals $I_{i,j}$ such that $F(\Gamma_i(E),E)=\min_xF(\cdot,E)$ for all $E\in I_{i,j}$. By definition, $I_{i,j}\cap I_{i,j'}=\varnothing$ for $j\ne j'$. Let $\Gamma_{i,j}=\Gamma_i|_{I_{i,j}}$, we have finitely many curves $\{\Gamma_{i,j}\}$ such that $F(\cdot, E)$ reaches global minimum at the point $x$ if and only if $x=\Gamma_{i,j}(E)$ for certain subscript $(i,j)$.
For each $E\in\partial I_{i,j}$, by the definition of $I_{i,j}$, some other subscript $(i',j')$ exists such that $F(\cdot,E)$ reaches the global minimum at the points $\Gamma_{i,j}(E)$ and $\Gamma_{i',j'}(E)$. It is obviously an open-dense property that $$\frac {dF(\Gamma_{i,j}(E),E)}{dE}\ne \frac {dF(\Gamma_{i',j'})(E),E)}{dE}.$$ Thus, it is also open-dense that $[E_a,E_d]=\cup I_{i,j}$ and $[E_a,E_d]\backslash\cup\text{\rm int}I_{i,j}$ contains finitely points. This completes the whole proof.[ $\;\;\; \Box$]{}
[**Acknowledgement**]{} This work is supported by NNSF of China (Grant 11171146, Grant 10531050), National Basic Research Program of China (973, 2007CB814800), Basic Research Program of Jiangsu Province (BK2008013) and a program PAPD of Jiangsu Province, China.
I would like to thank my colleagues J. Cheng, W. Cheng, X. Cui, J. Yan and M. Zhou for helpful discussions. I also thank Marc Chaperon, Alain Chenciner and Hakan Eliasson for inviting me to give talks on this result at their seminars while I was visiting University of Paris 7 for its hospitality. The main ideas of the proof were presented on the conferences at Edinburgh and at IAS, Princeton in October of 2011.
[WW]{}
Arnol’d V. I., [*Instability of dynamical systems with several degrees of freedom,*]{} (Russian, English) Sov. Math., Dokl., [**5**]{}(1964), 581-585; translation from Dokl. Akad. Nauk SSSR, [**156**]{}(1964), 9-12.
Arnol’d, V.I., [*Small denominators and problems of stability of motion in classical and celestial mechanics,*]{} Russ. Math. Survey [**18**]{} (1963) 85-192.
Arnol’d V.I. Kozlov V.V. and Neishtadt A.I., [*Mathematical Aspects of Classical and Celestial Mechanics*]{}, [Dynamical Systems III, Encyclopaedia of Mathematical Sciences,]{} [**3**]{} Springer-Verlag Berlin Heidelberg, (1988).
Angenent S., [*The periodic orbits of an area preserving twist map*]{}, Commun. Math. Phys. [**115**]{} (1988) 353-374.
Bates P., Lu. K. and Zeng C., [*Persistence of overflowing manifolds for semiflow*]{}, Commun. Pure Appl. Math. [**52**]{} (1999) 983-1046.
Bernard P., [*Homoclinic orbits to invariant sets of quasi-integrable exact maps*]{}, Ergod. Theory Dynam. Syst. [**20**]{} (2000) 1583-1601.
Bernard P., [*Symplectic aspects of Mather theory*]{}, Duke Math. J. [**136**]{} (2007) 401-420.
Bernard P., [*The dynamics of pseudographs in convex Hamiltonian systems*]{}, Journal Amer. Math. Soc. [**21**]{} (2008) 615-669.
Bernard P., [*Large normally hyperbolic cylinders in a priori stable Hamiltonian systems*]{}, Annales H. Poincare [*11*]{} (2010) 929-942.
Bernard P. & Contreras G., [*A generic property of families of Lagrangian systems*]{}, Annals of Math. [**167**]{} (2008) 1099-1108.
Bernard P., Kaloshin V. and Zhang K., [*Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinder*]{} arXiv: 1112.2773v1 (2011).
Berstein D. and Katok A., [*Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians,*]{} Invent. Math., [**88**]{} (1987) 225-241.
Bessi U., [*An approach to Arnold’s diffusion through the calculus of variations,*]{} Nonlinear Anal., [**26(6)**]{}(1996) 1115-1135.
Bessi U., Chierchia L. and Valdinoci E., [*Upper bounds on Arnold diffusion times via Mather theory,*]{} J. Math. Pures Appl., [**80(1)**]{}(2001) 105-129.
Bolotin S. [*Homoclinic orbits in invariant tori of Hamiltonian systems,*]{} Dynamical systems in classical mechanics, 21–90, Amer. Math. Soc. Transl. Ser. 2, [**168**]{}, Amer. Math. Soc., Providence, RI, 1995.
Cassels J.W.S., [*An introduction to Diophantine approximation,*]{} Cambridge Tracts in Mathematics and Mathematical Physics 45, Cambridge Univ. Press, New York 1957.
Cannarsa P. & Sinestrari C., [*Semiconcave functions, Hamilton-Jacobi equations and Optimal Control*]{}, Progress in Nonlinear Differential Equations and Their Appications [**58**]{} (2004) Birkhäuser.
Cheng C.-Q., [*Non-existence of KAM torus*]{}, Acta Math. Sinica, [**27**]{} (2011) 397-404.
Cheng C.-Q. & Yan J., [*Existence of diffusion orbits in a priori unstable Hamiltonian systems,*]{} J. Differential Geometry, [**67**]{} (2004) 457-517.
Cheng C.-Q. & Yan J., [*Arnold diffusion in Hamiltonian Systems: a priori Unstable Case,*]{} J. Differential Geometry, [**82**]{} (2009) 229-277.
Cheng C.-Q. & Zhou M., [*Hyperbolicity of minimal periodic orbits*]{}, Preprint (2012).
Contreras G., Delgado J. & Iturriaga R., [*Lagrangian flows: the dynamics of globally minimizing orbits II*]{}, Bol. Soc. Bras. Mat. [**28**]{} (1997) 155-196.
Contreras G. & Paternain G. P., [*Connecting orbits between static classes for generic Lagrangian systems,*]{} Topology, [**41**]{} (2002) 645-666.
Cui X. et al, [*Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems*]{}, J. Diff. Eqns. [**214**]{} (2005) 176-188.
Delshams A., de la Llave R. and Seara T. M., [*Geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristic and rigorous verification of a model*]{}, Memoirs Amer. Math. Soc. [**179**]{}(844), (2006).
Delshames A. and Huguet G., [*Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems*]{}, Nonlineairty [**22**]{} (2009) 1997-2077.
Delshames A. and Huguet G., A geometric mechanism of diffusion: rigorous verification in a priori unstable Hamiltonian systems, J. Diff. Eqns. [**250**]{} (2011) 2601-2623.
E W., [*Aubry-Mather theory and periodic solutions of the forced Burgers equation*]{}, Commun Pure Appl. Math. [**52**]{} (1999) 811-828.
Fathi A., [*Théorème KAM faible et théorie de Mather sue les systèmes*]{}, C. R. Acad. Sci. Paris Sér. I Math. [**324**]{} (1997) 1043-1046.
Fathi A., [*Weak KAM Theorem in Lagrangian Dynamics*]{}, Cambridge Studies in Adavnced Mathematics, Cambridge University Press, (2009).
Fathi A. and Mather J., [*Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case,*]{} Bull. Soc. Math. France, [**128(3)**]{}(2000) 473-483.
Fathi A. and Siconolfi A., [*Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation*]{}, Invent. Math. [**155**]{} (2004) 363-388.
Fontich E. and Martin P., [*Arnold diffusion in perturbations of analytic integrable Hamiltonian systems*]{}, Discrete Contin. Dyn. Syst. [*7*]{} (2001) 61-84.
Gidea M. and de la Llave R., [*Topological methods in the instability problem of Hamiltonian systems*]{}, Discrete and Continuous Dynamical Systems [*14*]{} (2006) 294-328.
Gidea M. and Robinson C., [*Shadowing orbits for transitive chains of invariant tori alternating with Birkhoff zone of instability*]{} Nonlinearity [**20**]{} (2007) 1115-1143.
Gidea M. and Robinson C., [*Obstruction argument for transition chains of tori interspersed with gaps*]{}, Discrete and Continuous Dynamical Systems series S, [**2**]{} (2009) 393-416.
Hirsch M. Pugh C. and Shub M., [*Invariant Manifolds*]{}, Lecture Notes Math. [**583**]{} (1977) Springer-Verlag.
Kaloshin V. and Levi M., [*An example of Arnold diffusion for near-integrable Hamiltonians*]{}, Bulletin AMS, [**45**]{} (2008) 409-427.
Kaloshin V. and Levi M., [*Geometry of Arnold diffusion*]{}, SIAM Review [**50**]{} (2008) 702-720.
Li X. [*On c-equivalence*]{}, Science in China, series A, [**52**]{} (2009) 2389-2396.
Li X. & Cheng C-Q., [*Connecting orbits of autonomous Lagrangian systems*]{}, Nonlinearity [**23**]{} (2009) 119-141.
Lochak P., [*Canonical perturbation theory via simultaneous approximation*]{}, Russian Math. Surveys [**47**]{} (1992) 57-133.
Marco J.-P., [*Generic hyperbolic propertires of classical systems on the torus $\mathbb{T}^2$*]{}, preprint (2013).
Mather J., [*Action minimizing invariant measures for positive definite Lagrangian systems,*]{} Math. Z., [**207(2)**]{}(1991) 169-207.
Mather J., [*Variational construction of connecting orbits,*]{} Ann. Inst. Fourier (Grenoble), [**43(5)**]{}(1993) 1349-1386.
Mather J., [*Variational construction of trajectories for time-periodic Lagrangian systems on the two torus*]{}, unpublished manuscript.
Mather J., [*Arnold diffusion, I: Announcement of results,*]{} J. Mathematical Sciences, [**124**]{}(5) (2004) 5275-5289.(Russian translation in Sovrem. Mat. Fundam. Napravl, [**2**]{} (2003) 116-130).
Mather J., [*Examples of Aubry Sets*]{}, Ergodic Theory and Dynamical Systems, [**24**]{} (2004) 1667-1723.
Mather J., [*Near double resonance*]{}, talks at conferences on dynamical systems at Nice (2009), Toronto, Nanjing, Oberwolfach and Eidinburgh (2011).
Mañé R., [*Generic properties and problems of minimizing measures of Lagrangian systems*]{}, Nonlinearity [**9**]{} (1996) 273-310.
Massart D., [*On Aubry sets and Mather’s action functional*]{}, Israel J. Math. [**134**]{} (2003) 157-171.
Osuna O., [*Vertices of Mather’s beta function*]{}, Ergod. Theory Dynam. Syst. [**25**]{} (2005) 949-955.
Rifford L., [*On viscosity solutions of certain Hamilton-Jacobi equations: regularity results and generalized Sard’s theorem*]{}, Commun. Partial Diff. Eqns. [**33**]{} (2008) 517-559.
Schmidt W.M., [*Diophantine approximation,*]{} Lecture Notes in Math. 785 (1980).
Treschev D.V., [*Evolution of slow variables in a priori unstable Hamiltonian systems*]{}, Nonlinearity, [**17**]{} (2004) 1803-1841.
van Moerbeke P., [*The spectrum of Jacobi matrices*]{}, Invent. Math., [**37**]{} (1976) 45-81.
Xia Z., [*Arnold diffusion: a variational construction*]{}, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, Extra Vol. II, 867-877.
Zhang K., [*Speed of Arnold diffusion for analytical Hamiltonian systems*]{}, Invent. Math. [**186**]{} (2009) 255-290.
Zheng Y. & Cheng C.-Q., [*Homoclinic orbits of positive definite Lagrangian systems*]{}, J. Diff. Eqns. [**229**]{} (2006) 297-316.
Zhou M., [*Hölder regularity of barrier functions in a priori unstable case*]{}, Math. Res. Lett., [**18**]{} (2011) 77-94.
Zhou M., [*Infinity of minimal homoclinic orbits*]{}, Nonlinearity, [**24**]{} (2011) 931-939.
|
---
author:
- |
[^1]\
LHEP JINR\
E-mail:
- |
E. Ardashev, V. Golovkin, S. Golovnya, S. Gorokhov, A. Kholodenko, A. Kiryakov, I. Lobanov, M. Polkovnikov, V. Ronzhin, V. Ryadovikov, Yu. Tsyupa, A. Vorobiev\
IHEP, Protvino, Russia, 142281
- |
V. Avdeichikov, V. Balandin, V. Dunin, O. Gavrishchuk, A. Isupov, N. Kuzmin, V. Nikitin, Yu. Petukhov, S. Reznikov, V. Rogov, I. Rufanov, N. Zhidkov, L. Zolin\
Joint Institute for Nuclear Research, Dubna, Russia, 141980
- |
G. Bogdanova, V. Popov, V. Volkov,\
Lomonosov Moscow State University Scobeltsyn Institute of Nuclear Physics, Russia, 110000
- |
A. Kutov\
DM Komi SC UrD RAS, Syktyvkar, Russia 167982
- |
A. Kazakov\
Smolensk State University, Russia, 214000
- |
G. Pokatashkin, R. Salyanko\
Gomel State University,Belarus, 246000
title: Soft photon registration at Nuclotron
---
Introduction
============
The experiments with relativistic heavy ions point to the manifestation of a quark-gluon matter. The understanding of the nature of the deconfinement (transition of hadrons to quarks and gluons) is important for formulation of the nuclear substance equation of state. Physicists consider that at their interactions with proton and deuteron beams are formed cold nuclear matter. These researches permit to compare properties of hot quark-gluon matter formed in collisions of heavy ions and cold nuclear matter producing in $pp$ or $p(d)$A interactions [@CNM]. The SVD Collaboration carries out studies of $pp$, $p$A and AA interactions. Experiments with 50 GeV-proton beams are fulfilled at U-70 in IHEP, Protvino [@Therm]. The SVD-2 Collaboration also works at Nuclotron (JINR) with 3.5 GeV/nucleon nuclear beams.
The SVD Collaboration has carried out studies in the unique region of high multiplicity where the collective behaviour of secondary particles is observed. One of the most important results obtained in these studies is the rapid growth of the scaled variance, $\omega = D/<N_0>$, with the increasing of total pion multiplicity [@Fluct]. Here $D$ is the variance of a number of neutral mesons at the fixed total multiplicity, $<N_0>$ – their mean multiplicity. The growth of the experimental value $\omega $ as compared to the Monte Carlo predictions has been confirmed at the level of 7 standard deviations. This result is one of the evidences of Bose-Einstein condensate (BEC) formation [@Gor].
The theoretical description of this phenomenon has been developed by Begun and Gorenstein [@Gor] at the specific conditions of the SVD-2 experiment. S. Barshay predicts [@Bars] that the pion condensation may be accompanied by an increased yield of soft photons (SP). The anomalous SP have being studied experimentally during more than 30 years [@Chlia; @WA83; @HELIOS; @Perep1]. There are some theoretical models worked out for explanation of SP yield [@Van; @Wong; @GDM]. Unfortunately, an incompleteness of data does not permit disclosing of the physical essence of this phenomenon completely.
To understand the picture of SP formation more comprehensive and in particular to test a connection between their excess yield and the BEC formation, SP electromagnetic calorimeter (SPEC) has been manufactured and tested by SVD Collaboration at U-70 [@SPEC]. This calorimeter is a stand-alone device and it differs from many similar ones by its extremely low threshold of gamma-quantum registration – of order of 2 MeV. The SPEC technique permits execution of the unique research program of $pp$, $p$A and AA interactions with registration of SP.
The report is organised in the following way. The previous SP observations are reviewed in section 2. In section 3 we give the description and technical characteristics of electromagnetic calorimeter manufactured by SVD Collaboration. The first spectra of SP obtained with the deuterium and lithium beams on a carbon target at Nuclotron are also presented in this section.
Review of experimental data on the soft photon yield
====================================================
Experimental and theoretical studies of direct photon production in hadron and nuclear collisions essentially expand our knowledge of multi-particle production mechanisms. These photons are useful probes to investigate nuclear matter at all stages of the interaction. SP play a particular role in these studies. Until now we do not have total explanation for the experimentally observed excess of SP yield. These photons have low transverse momenta $p_{T}$ < 0.1 GeV/c and Feynman variable |x| < 0.01. In this domain their yield exceeds the theoretical estimates by 3 $\div $ 8 times.
This anomalous phenomenon has been discovered at the end of 1970s with the Big Europe Bubble Chamber at the SPS accelerator, in CERN, in the experiment with a 70 GeV/c $K^+$-meson and antiproton beams [@Chlia]. The SP yield had exceeded the theoretical predictions by 4.5 $\pm $ 0.9. The following electronic experiments such as [@WA83; @HELIOS; @Perep1] have confirmed an anomalous behaviour of SP.
WA83 Collaboration studied the direct SP yield at OMEGA spectrometer in $\pi ^-$ + $p$ interactions at hydrogen target with 280 GeV/$c$ $\pi^-$-mesons. Excess yield of SP turned out to be equal to 7.9 $\pm $ 1.4 [@WA83]. Last experimental study of SP had been carried out at the LEP accelerator with DELPHI setup in CERN [@Perep1]. The processes investigated were: $e^+ + e^-\to Z^0 \to $ jet + $\gamma $ and $e^+$ + $e^- \to \mu ^+ + \mu ^- $. In processes with formation of jets the DELPHI Collaboration had revealed SP excess yield over of Monte Carlo estimations at the level 4.0 $\pm $ 0.3 $\pm $ 1.0 times. For the first time the SP yield at maximum number of neutral pions (7-8) had amounted to about 17-fold exceeding [@Perep1] in comparison with bremsstrahlung of charged particles. On the contrary, in the lepton processes without formation of hadron jets the yield of SP turned out to agree well with theoretical predictions.
Existent theoretical models try to explain anomalous yield of SP. The SVD Collaboration has developed a gluon dominance model [@GDM] explained an excessive SP yield by the production of soft gluons in quark-gluon system. These gluons do not have enough energy to fragment into hadrons, so they are scattered on the valency quarks of secondary particles and form SP [@GDM]. This model gives two(three)fold exceeding of common accepted of strong interaction area in accordance with estimations of the region of their emission.
Design, manufacture and testing of SPEC. SP spectra
===================================================
The SPEC has been manufactured on the base of the BGO scintillators (bismuthortogermanate) [@SPEC]. The BGO crystals have a small radiation length X$_0$ = 1.12 cm, that permits to reduce considerably the volume of the device. At manufacturing of such calorimeter the problems of uniform distributions of activator in the crystal volume do not appear. While many of inorganic scintillators have the long-term of emission. BGO crystals shows relative small afterglow.
SPEC scheme is shown in Fig. 2, a left panel. It is a square matrix composed of 49 (3$\times $3$\times $18 cm$^3$) counters [@SPEC]. The front side is covered by the high-reflective film VM2000. The PMT 9106SB are used (ET Enterprises). They have 8 dynodes and high quantum efficiency in the green part of spectrum. The photocathode diameter is equal to 25 mm. The tube has the permalloy magnetic protection. PMT is glued to the crystal by the optic EPO-TEK 301 glue.
The plastic veto-detector of charged particles (23$\times $23$\times $1 cm$^3$) is placed before the crystals. Behind it an assembly of 4 plastics of a pre-shower (18$\times $4.5$\times $1 cm$^3$) is shown. Lead 2mm-convertor is put between the front-veto and plastics. In Fig. 1, right panel, the target and counters of a trigger system are shown. A trigger is produced at the signal from any 2 of 4 pre-shower counters. In front of the target, there are two large veto-counters to forbid a response from the beam halo. Time-stamp is given by the 4.5$\times $4.5$\times $0.1 cm$^3$ beam counter (not shown), also upstream from the target.
{width="40.00000%"} {width="40.00000%"}
{width="51.50000%"} {width="47.00000%"}
{width="50.00000%"} {width="50.00000%"}
SPEC is set at an angle of 16$^\circ $, the front plane of crystals is away from a target at the distance 203 cm. The digitization of plastic scintillators is realised with a CAMAC ADCs (Lecroy 2249A) and TDCs (LeCroy 2228A), the digitization of analog signals of calorimeter - by ADC CC-008.
We used CAMAC and a LE-88K crate-controller with input for a trigger signal. The crate-controller has been connected to PC with PCI-QBUS interface. Data acquisition software has been developed in MIDAS framework (http://midas.psi.ch). Time of flight between the beam counter and the pre-shower for neutral particles (no signal in the front-veto) gives time resolutions 632 ps for d+C and 532 ps for Li+C (Fig.2, left panel) interactions. In Fig.2, right panel spectrum of $\gamma $ quanta deposited in pre-shower plastic with time selection for neutral particles is presented for Li+C interactions. A solid line shows Compton peak at 1 MIP energy and more intensive peak of gamma quanta conversion at 2 MIP. In this Fig. with dotted line this structure is almost unnoticeable.
Selection criterions of events were as the following: 1) energy in the front veto-counter smaller than 0.3 MIPs; 2) energy in pre-shower 0.5 < $E$ < 4 MIPs; 3) time of flight - 1200 < t - t$_\gamma $ < 600 ps; 4) more than 2 MeV is registered in one of BGO crystals; 5) location of shower in BGO crystal must overlay throughout vertical with the triggered pre-shower counter; 6) energy deposition in the outer BGO layer should be no more than 1/3 of a total to prevent significant leakages.
In 2014 two experimental runs have been carried out at Nuclotron in LHEP JINR with 3.5 A GeV deuterium and lithium beams. SPEC was installed at the location of NIS-GIBS setup. After data processing we have obtained SP spectra of energy release in deuterium-carbon (Fig. 3, left panel) and lithium-carbon (Fig. 3, right panel) interactions. In the region of energy below than 50 MeV, a noticeable excess over Monte-Carlo simulation (uRQMD+Geant-3.21) has been observed. It agrees with other SP experiments.
[99]{} A. Andronic et al. e-Print: arXiv:1506.03981 \[nucl-ex\]. V. V. Avdeichikov et al., Proposal “*Termalization*” (in Russian), JINR-P1-2004-190 (2005). E. S. Kokoulina (On behalf of the SVD-2 Collaboration). Progr. Theor. Phys., [**193**]{} (2012) 306; V. N. Ryadovikov (On behalf of the SVD-2 Collaboration). Phys. Atom. Nucl., [**75**]{} (2012) 989. V. V. Begun and M. I. Gorenstein. Phys. Lett. **B**653 (2007) 190; V. V. Begun and M. I. Gorenstein. Phys. Rev. **C**77 (2008) 064903. S. Barshay. Phys. Lett. **B**227 (1989) 279. P. V. Chliapnikov et al., Phys. Let. **B**141 (1984) 276. S. Banerjee et al. SOPHIE/WA83. **B** 305 (1993) 182. J. Schukraft. HELIOS Collaboration. Nucl.Phys. **A** 498 (1989) 79. J. Abdallah et al. DELPHI Collaboration. Eur. Phys. J. **C** 47 (2006) 273. L. Van Hove . Ann.of Phys. (NY), **192** (1989) 66; P. Lichard and L. Van Hove. Phys.Lett. **B** 245 (1990) 605. Wong Cheuk-Yin. Phys.Rev. Lett. **C** 81(2010) 064903. E. Kokoulina. Acta Phys. Polon. **B**35 (2004) 295; M. K. Volkov, E. Kokoulina, and E. A. Kuraev. Pepan Letters. N **5** (2004) 16. E. .N. Ardashev et al. Instr. Exp. Tech. **58** (2015) 18.
[^1]: On behalf of the SVD-2 Collaboration.
|
---
abstract: 'The transformer [@vaswani2017attention] has been shown to outperform recurrent neural network-based sequence-to-sequence models in various word-level NLP tasks. The model offers other benefits as well: It trains faster and has fewer parameters. Yet for character-level transduction tasks, e.g. morphological inflection generation and historical text normalization, few shows success on outperforming recurrent models with the transformer. In an empirical study, we uncover that, in contrast to recurrent sequence-to-sequence models, the batch size plays a crucial role in the performance of the transformer on character-level tasks, and we show that with a large enough batch size, the transformer does indeed outperform recurrent models. We also introduce a simple technique to handle feature-guided character-level transduction that further improves performance. With these insights, we achieve state-of-the-art performance on morphological inflection and historical text normalization. We also show that the transformer outperforms a strong baseline on two other character-level transduction tasks: grapheme-to-phoneme conversion and transliteration. Code is available at <https://github.com/shijie-wu/neural-transducer>.'
author:
- |
Shijie Wu$^{{\normalfont \text{\textipa{Z}}}}$ **Ryan Cotterell**$^{{\text{\normalfont \textipa{Q}}},{\normalfont \text{\textipa{6}}}}$ **Mans Hulden**$^{{\textrm{\normalfont \textipa{X}}}}$\
$^{{\normalfont \text{\textipa{Z}}}}$Johns Hopkins University $^{{\normalfont \text{\textipa{6}}}}$University of Cambridge\
$^{{\text{\normalfont \textipa{Q}}}}$ETH Z[ü]{}rich $^{\textrm{\normalfont \textipa{X}}}$University of Colorado Boulder\
`shijie.wu@jhu.edu` `ryan.cotterell@inf.ethz.ch` `mans.hulden@colorado.edu`
bibliography:
- 'acl2020.bib'
- 'anthology.bib'
title: 'Applying the Transformer to Character-level Transduction'
---
Introduction
============
The transformer [@vaswani2017attention] has become a popular architecture for sequence-to-sequence transduction in NLP. It has achieved state-of-the-art performance on a range of common word-level transduction tasks: neural machine translation [@barrault-etal-2019-findings], question answering [@devlin-etal-2019-bert] and abstractive summarization [@dong2019unified]. In addition, the transformer forms the backbone of the widely-used BERT [@devlin-etal-2019-bert]. Yet for character-level transduction tasks like morphological inflection, the dominant model has remained a recurrent neural network-based sequence-to-sequence model with attention [@cotterell-etal-2018-conll]. This is not for lack of effort—but rather, it is the case that the transformer has consistently underperformed in experiments on average [@tang-etal-2018-self]. [^1] As anecdotal evidence of this, we note that in 2019, the most recent addition of the SIGMORPHON shared task on cross-lingual transfer for morphological inflection, no participating system was based on the transformer [@mccarthy-etal-2019-sigmorphon].=-1
![Development set accuracy for 5 languages on morphological inflection with different batch sizes. We evince our two primary contributions: (1) we set the **new state of the art** morphological inflection using the transformer and (2) we demonstrate the transformer’s **dependence on the batch size**.[]{data-label="fig:batch-size"}](content/batch-size.pdf){width="1\columnwidth"}
{width="2\columnwidth"}
Character-level transduction tasks often have fewer data than their word-level counterparts: In contrast to machine translation, where millions of training samples are available, the 2018 SIGMORPHON shared task [@cotterell-etal-2018-conll] high-resource setting only provides $\approx$ 10k training examples per language. It is also not obvious that non-recurrent architectures such as the transformer should provide an advantage at many character-level tasks: For instance, and suggest that transformers (and convolutional models in general) should help remember long-range dependencies better. In the case of morphology, none of these considerations seem relevant: inflecting a word (a) requires little capacity to model long-distance dependencies and is largely monotonic transduction; (b) it involves no semantic disambiguation, the tokens in question being letters; (c) it is not a task for which parallelization during training appears to help, since training time has never been an issue in morphology tasks.[^2]=-1
In this work, we provide state-of-the-art art numbers for morphological inflection and historical text normalization, a novel result in the literature. We also show the transformer outperforms a strong recurrent baseline on two other character-level tasks: grapheme-to-phoneme (g2p) conversion and transliteration. We find that a single hyperparameter, batch size, is largely responsible for the previous poor results. Despite having fewer parameters, the transformer outperforms the recurrent sequence-to-sequence baselines on all four tasks. We conduct a short error analysis on the task of morphological inflection to round out the paper.
The Transformer for Characters
==============================
#### The Transformer.
The transformer, originally described by , is a self-attention-based encoder-decoder model. The encoder has $N$ layers, consisting of a multi-head self-attention layer and a two-layer feed-forward layer with ReLU activation, both equipped with a skip connection. The decoder has a similar structure as the encoder except that, in each decoder layer between the self-attention layer and feed-forward layer, a multi-head attention layer attends to the output of the encoder. Layer normalization [@ba2016layer] is applied to the output of each skip connection. Sinusoidal positional embeddings are used to incorporate positional information without the need for recurrence or convolution. Here, we describe two modifications we make to the transformer for character-level tasks.=-1
#### A Smaller Transformer.
As the dataset sizes in character-level transduction tasks are significantly smaller than in machine translation, we employ a smaller transformer with $N=4$ encoder-decoder layers. We use 4 self-attention heads. The embedding size is $d_{\textit{model}} = 256$ and the hidden size of the feed-forward layer is $d_{\textit{FF}} = 1024$. In the preliminary experiments, we found that using layer normalization before self-attention and the feed-forward layer performed slightly better than the original model. It is also the default setting of a popular implementation of the transformer [@vaswani2018tensor2tensor]. The transformer alone has around 7.37M parameters, excluding character embeddings and the linear mapping before the softmax layer. We decode the model left to right in a greedy fashion.
#### Feature Invariance.
Some character-level transduction is guided by features. For example, in the case of morphological reinflection, the task requires a set of morphological attributes that control what form a citation form is inflected into (see \[fig:position\] for an example). The order of the features is irrelevant. In a recurrent neural network, features are input in some predefined order as special characters and pre- or postpended to the input character sequence representing the citation form. The same is true for a vanilla transformer model, as shown on the left-hand side of \[fig:position\]. This leads to different relative distances between a character and a set of features.[^3] To avoid such an inconsistency, we propose a simple remedy: We set the positional encoding of features to 0 and only start counting the positions for characters. Additionally, we add a special token to indicate whether a symbol is a word character or a feature. The right-hand side of \[fig:position\] evinces how we have the same relative distance between characters and features.=-1
Empirical Findings
==================
#### Tasks.
We consider four character-level transduction tasks: morphological inflection, grapheme-to-phoneme conversion, transliteration, and historical text normalization. For morphological inflection, we use the 2017 SIGMORPHON shared task data [@cotterell-etal-2017-conll] with 52 languages. The performance is evaluated by accuracy (ACC) and edit distance (Dist). For the g2p task, we use the unstressed CMUDict [@CMUDict] and NETtalk [@Sejnowski1987ParallelNT] resources. We use the splits from . We evaluate under word error rate (WER) and phoneme error rate (PER). For transliteration, we use the NEWS 2015 shared task data [@zhang-etal-2015-whitepaper].[^4] For historical text normalization, we follow and use datasets for Spanish [@sanchez2013open], Icelandic and Swedish [@pettersson2013smt], Slovene [@scherrer2013modernizing; @scherrer2016modernising; @ljubevsic2016normalising], Hungarian and German [@pettersson2016spelling].[^5] We evaluate using accuracy (ACC) and character error rate of incorrect prediction (CER$_i$).
![Distribution of incorrectly inflected forms in the test set of the inflection task over all 52 languages grouped by desired output word length.[]{data-label="fig:errorlengths"}](content/errorlengths.pdf){width="1\columnwidth"}
#### Optimization.
We use Adam [@kingma2014adam] with a learning rate of $0.001$ and an inverse square root learning rate scheduler [@vaswani2017attention] with 4k steps during the warm-up. We train the model for 20k gradient updates and save and evaluate the model every 400 gradient updates. We select the best model out of 50 checkpoints based on development set accuracy. The number of gradient updates and checkpoints are roughly the same as , the single model state of the art on the 2017 SIGMORPHON dataset. We use their model as a baseline model. For all experiments, we use a single predefined random seed.
A Controlled Hyperparameter Study {#sec:ablation}
---------------------------------
To demonstrate the importance of hyperparameter tuning for the transformer on character-level tasks, we perform a small controlled hyperparameter study. This is important since researchers had previously failed to achieve high-performing results with the transformer on character-level tasks. Here, we look at morphological inflection on the five languages in the 2017 SIGMORPHON dataset where submitted systems performed the worst: Latin, Faroese, French, Hungarian, and Norwegian (Nynorsk). We set the dropout to 0.3, $\beta_2$ of Adam to 0.999 (the default value), and do not use label smoothing. We do not tune any other hyperparameter except the following three hyperparameters.
#### The Importance of Batch Size.
While recurrent models like @wu-cotterell-2019-exact use a batch size of 20, halving the learning rate when stuck and employing early stopping, we find that a less aggressive learning rate scheduler, allowing the model to train longer, outperforms these hyperparameters. \[fig:batch-size\] shows that the *single most important hyperparameter* when training is the batch size. The transformer performance increases steadily as the batch size is increased, similarly to what observe for machine translation. The transformer only outperforms the recurrent baseline when the batch size is above 128. Note that the model of @wu-cotterell-2019-exact has 8.66M parameters, 17% more than the transformer model. To get an apples-to-apples comparison, we apply the same learning rate scheduler to @wu-cotterell-2019-exact; this does not yield similar improvements and underperforms with respect to the traditional learning rate scheduler. Our feature invariant transformer also outperforms the vanilla transformer model. We set the batch size to 400 for our main experiments. Note the batch size of 400 is especially large (4% of training data) consider the training size is only 10k.
#### Other Hyperparameters.
apply label smoothing [@szegedy2016rethinking] of 0.1 to the transformer model and show that it hurts perplexity, but improves BLEU scores for machine translation. Instead of the default 0.999 $\beta_2$ for Adam, use 0.98 and we find that both choices benefit character-level transduction tasks as well (see \[table:optim\]).
New State-of-the-Art Results
----------------------------
We train our feature invariant transformer on the four character-level tasks, exhibiting state-of-the-art results on morphological inflection and historical text normalization.
#### Morphological Inflection.
As shown in \[table:morph-inflect\], the feature invariant transformer produces state-of-the-art results on the 2017 SIGMORPHON shared tasks, improving upon ensemble-based systems by 0.27 points. We observe that as the dataset decreases in size, a model with a larger dropout value performs slightly better. A brief tally of phenomena that are difficult to learn for many machine learning models, categorized along typical linguistic dimensions (such as word-internal sound changes, vowel harmony, circumfixation, ablaut, and umlaut phenomena) fail to reveal any consistent pattern of advantage to the transformer model. In fact, errors seem to be randomly distributed with an overall advantage of the transformer model. Curiously, errors grouped along the dimension of word length reveal that as word forms grow longer, the transformer advantage shrinks (\[fig:errorlengths\]).
#### Historical Text Normalization.
\[table:hist-norm\] shows that the transformer model with dropout of 0.1, like with morphological inflection, improves upon the previous state of the art, although the model with a dropout of 0.3 yields a slightly better CER$_i$.
#### G2P and Transliteration.
\[table:g2p-trans\] shows that the transformer outperforms previously published strong recurrent models on two tasks despite having fewer parameters. A dropout rate of 0.3 yields significantly better performance on the transliteration task while a dropout rate of 0.1 is stronger on the g2p task. This shows that transformers can and do outperform recurrent transducers on common character-level tasks when properly tuned.
Related Work
============
Character-level transduction is largely dominated by attention-based LSTM sequence-to-sequence [@luong-etal-2015-effective] models [@cotterell-etal-2018-conll]. Character-level transduction tasks usually involve input-output pairs that share large substrings and alignments between these are often monotonic. Models that address the task tend to focus on exploiting such structural bias. Instead of learning the alignments, use external monotonic alignments from the SIGMORPHON 2016 shared task baseline . use this approach to win the CoNLL-SIGMORPHON 2017 shared task on morphological inflection [@cotterell-etal-2017-conll]. shows that explicitly modeling alignment (hard attention) between source and target characters outperforms soft attention. further show that enforcing monotonicity in a hard attention model improves performance.
Conclusion
==========
Using a large batch size and feature invariant input allows the transformer to achieve strong performance on character-level tasks. However, it is unclear what linguistic errors the transformer makes compared to recurrent models on these tasks. Future work should analyze the errors in detail as does for recurrent models. While @wu-cotterell-2019-exact shows that the monotonicity bias benefits character-level tasks, it is not evident how to enforce monotonicity on multi-headed self-attention. Future work should consider how to best incorporate monotonicity into the model., either by enforcing strictly [@wu-cotterell-2019-exact] or by pretraining the model to copy [@anastasopoulos-neubig-2019-pushing].
[^1]: This claim is also based on the authors’ personal communication with other researchers in morphology in the corridors of conferences and through email.
[^2]: Many successful CoNLL–SIGMORPHON shared task participants report training their models on laptop CPUs.
[^3]: While the features could be encoded with a binary vector followed by MLP, it introduces a representation bottleneck for encoding features.
[^4]: We do not have access to the test set.
[^5]: We do not include English due to licensing issues.
|
---
abstract: 'The stock market of China experienced an abrupt crash in 2015 and evaporated over one third of the market value. Given its associations with fear and fine-resolutions in frequency, the illiquidity of stocks may offer a promising perspective of understanding and even signaling the market crash. In this study, by connecting stocks that mutually explain illiquidity fluctuations, a illiquidity network is established to model the market. It is found that as compared to non-crash days, the market is more densely connected on crash days due to heavier but more homogeneous illiquidity dependencies that facilitate abrupt collapses. Critical socks in the illiquidity network, in particular the ones in sector of finance are targeted for inspection because of their crucial roles in taking over and passing on the losing of illiquidity. The cascading failures of stocks in market crash is profiled as disseminating from small degrees to high degrees that usually locate in the core of the illiquidity network and then back to the periphery. And by counting the days with random failures in previous five days, an early single is implemented to successfully warn more than half crash days, especially those consecutive ones at early phase. Our results would help market practitioners like regulators detect and prevent risk of crash in advance.'
address:
- 'School of Economics and Management, Beihang University, Beijing, China'
- 'Beijing Advanced Innovation Center for Big Data and Brain Computing, Beijing, China'
author:
- Xiaoling Tan
- Jichang Zhao
---
illiquidity ,complex network ,market crash ,cascading failures ,warning signals
Introduction {#sec:intro}
============
The stock market occupies the most profound role in the financial systems of modern economies like China. An abrupt stock market crash, like the one of 2015 that evaporated around 15 trillion yuan in wealth, therefore could be a cartographic shock to the economics and bring about huge losses to the whole society. In fact, how to understand the market crash and implement early warnings has been an important issue and trending topic not only in finance but also interdisciplinary fields after the crisis. While it is conventionally thought that market crash might be a typical black-swan event, which is hardly predicted due to sophisticated factors beyond and unexpected entanglements with external systems. Nevertheless, the associations between investor behaviors, like expectations, emotions and imitations and the market performance, especially their power in return predictions [@ZhouTales; @luemotion], imply that trading behaviors may provide a new but promising perspective of probing and warning the market crash. In particular, details of every trading decision in high-frequency records further offer a manner of big-data proxy to investigate the collective behavior of investors, either before, during or after the market crash.
Liquidity, referring to the spread between bid price and ask price, inherently reflects expectations of investors towards the future performance of stocks in their elementary trading decisions. And illiquidity, which inversely originates from the pessimism of investors, would thus increase the crash risk since it dissolves the effective price information and disseminates panic across the market. Given the significant impact from investor emotions, especially the negative ones [@ChiuInvestor; @FloriCommunities], illiquidity can also be contagious, e.g., scared investors on stocks of illiquidity incline to sell out other stocks on hands to keep their own liquidity and reluctantly result in more stocks of illiquidity. Hence, in order to model the market crash from a system view, it would be natural to connect stocks of similar illiquidity fluctuations and build a network to represent the market. In the accordingly established illiquidity network, links among stocks stands for the possibilities of cascading crash across the market, suggesting a new angle of profiling the market crash dynamics. Though it is indeed not a new idea to transform a market into a network, linking stocks in terms of illiquidity is rarely visited. More importantly, different from previous networking models of mutual fund sharing [@LuHerding] or price co-movements, illiquidity can be captured dynamically in a fine-resolution, i.e., in the most minimum decision granularity of bid and ask. It means that in terms of elementary decisions in trading and their contagions, the illiquidity network provides a very micro-perspective of the market crash.
Although there are lots of literatures on stock market crash, results on crash forecasts are still inadequate and more efforts are desperate. Unlike many emerging financial markets, however, the China stock market is unique since it is dominated by individual investors [@ShenzhenNews]. Contrary to their institutional counterparts, individual investors are more emotional and susceptible, meaning they are more likely to be scared, spread panic and overly react to external disturbs. They even imitate trading strategies and help forge the herding in market. These characteristics might undermine the challenges that make crash hard to predict and suggest the possibility of detecting the crash of China market at early days. In terms of illiquidity, the trading behaviors in extreme market situations can be finely examined from the micro perspective, helping identify the sources of market volatility and extreme stock price movements. In addition, the anomaly in the evolution of illiquidity networks can also be probed from the differences between crash days to non-crash days, which paves the way to develop the warning signals of market crash.
Inspired by above motivations, this study aims to profile, explain and warn the China market crash through the illiquidity network. The illiquidity of stocks is defined and derived from 2.3 billion trades in 2015, from which profound associations between illiquidity and negative emotions of investors like fear are disclosed. The illiquidity dependency between stocks, measured by the mutual information, can surprisingly distinguish crash days from those non-crash ones. And it is also inspiring that the market is more connected and homogeneous due to heavier and lower-deviated illiquidity dependencies on crash days. While in the illiquidity network, influential stocks in crash are found to be the ones with large capital values or belonging to the sector of finance. The dynamics of the crash is also profiled in the illiquidity network as cascading failures of losing illiquidity from stocks of smaller degrees to the ones of higher degrees that usually locate in the core and then out to the fringe. More importantly, an early signal, which simply counts the days without systemic failures in a window of previous five days is presented to accurately warn more than half crash days in 2015. Our results decently demonstrate the power of illiquidity network in understanding market crash of China and would help practitioners in particularly the regulators inspect risky stocks and prevent possible crash in advance.
The rest of the paper is organized as follows. Section 2 reviews literatures. Section 3 introduces our datasets and the methodology of measuring illiquidity. Section 4 presents the results from illiquidity networks. Section 5 concludes the paper with a brief summary and suggestions for future research.
Literature review {#sec:lr}
=================
Due to the late development of China’s stock market and the obvious gap with developed foreign markets, there have been some unique features of the Chinese stock market discussed among the academic scholars and practitioners. On the one hand, Yao et al. indicated that Chinese investors exhibit different levels of herding behavior [@YaoInvestor]. On the other hand, Xing and Yang found that the increased correlation among the stocks could ignite market crash [@XingHow]. Further, Tian et al. found that institutional investors (primarily pension funds) provide stabilizing effect during extreme market-down days [@TianWho], unlike Dennis and Strickland who revealed that institutional investors magnify extreme market movements by buying (selling) more on return-up (return-down) days in the U.S. markets [@DennisWho]. Although there are many related studies in either China market or foreign ones, no detailed explanations and early warning signals of stock market crash have been given to prevent risks. In the meantime, the dominant occupation of individual investors in China market also implies the possible abnormality in trading behaviors that can be sensed and detected as warnings before the crash.
In fact, previous efforts have already suggested that the stock market crash is closely related to illiquidity. Amihud et al. presented evidence linking the decline in stock prices to increased illiquidity using the method of bid-ask spread during the market crash [@Amihud1990Liquidity]. As return is more comparable to price, related research on associations between return and illiquidity has increased rapidly. Amihud and Bekaert et al. stated that there is a positive correlation between stock returns and illiquidity in terms of the daily ratio of absolute stock return to its dollar volume and the proportion of zero return days, respectively [@Amihud2002Illiquidity; @BekaertLiquidity]. Furthermore, Nagel indicated that the main reason of the evaporation of liquidity during crash is the increasing expected returns of liquidity [@NagelEvaporating]. Even more inspiring, measuring illiquidity, e.g., through bid-ask spread, is deeply rooted in the minimum decision granularity of daily trading and thus can be inherently derived from highly frequent trading records of investors. And also, illiquidity contains future economic information which can be employed for stock market forecasting [@StollInferring; @ChenMicro]. Therefore, it is feasible to explore stock market crash from the perspective of illiquidity, but existing examinations still lack explanations, cascading dynamics, and warning signals of the crash.
Illiquidity may also be influenced by both internal and external factors including stock attributes, policies and industry, which should be considered in understanding the market crash. Stoll et al. suggested that stock attributes such as market value, volume and volatility can significantly reshape the stock illiquidity [@StollInferring; @ChenMicro; @ChordiaMarket]. On the other hand, An et al. found that macro economic factors such as media independence, policy uncertainty, default risk and funding conditions have a remarkable impact on illiquidity [@AnTheImpact; @ChungUncertainty; @BrogaardStock; @BrunnermeierMarket]. These evidences imply that stocks can be well profiled in terms of illiquidity and more importantly, external shocks to the market can also be absorbed and thus sensed through illiquidity. In addition, the illiquidity of individual stocks co-varies with each other [@ChordiaCommonality; @HasbrouckCommon; @HubermanSystematic; @AcharyaAsset; @DengForeign], suggesting in essence that illiquidity can be contagious across the market.
Modeling market as a network of stocks to examine the crash is a new and promising approach in recent efforts. Stocks can be connected due to price correlations or common investors [@LuHerding]. By removing failed stocks, e.g., reaching the down-limit and transactions being suspended, the market crash can then be reflected through the falling apart of the network. The topology evolution before and after the 2008 financial crisis of South African, Korean and China’s stock markets were investigated [@MajapaTopology; @NobiEffects; @YangAnalysis], respectively, in which the minimum spanning trees (MST) are carefully examined. Li and Pi proposed a complex network based method to understand the effects of the 2008 global financial crisis on global main stock index [@LiAnalysis]. besides, Bosma et al. use network centrality to identify the position of the financial industry in the network, which can be a significant predictors of bailouts [@BosmaToo]. In particular, the turbulence in 2015–2016 were probed by transforming China stock market into a complex network, showing that there exist influential stocks and sectors within the market crash [@LuHerding; @khoojineNetwork]. Nevertheless, connecting stocks because of illiquidity associations is rarely considered in constructing the market network. The absence of establishing illiquidity networks in existing studies on market crash will spark up new perspectives in this paper.
To sum up, although extensive efforts have been devoted on the association between stock illiquidity and market crash, few insights are available on illiquidity networks based on high-frequency transaction data. Given the closeness between stock illiquidity and both internal and external factors of the market, probing the crash from the perspective of illiquidity networks could offer more insightful observations and explanations. Moreover, the dominance of individual investors in China stock market also indicates that the trading abnormality, which can be grasped by illiquidity and its contagion in a fine resolution could produce novel signals to warn risks before the crash. From a interdisciplinary view, a big-data proxy based on tremendous trading records before, during and after the 2015 crash of China’s stock market will be employed to measure illiquidity, establish networks, examine crash dynamics and detect warning signals.
Dataset and methods {#sec:data}
===================
Dataset {#subsec:data}
-------
The data sample employed in this study consists of stocks selected from the Shenzhen Stock Exchange and the Shanghai Stock Exchange in 2015, i.e., more than 2500 stocks and a total of 244 trading days. In particular, transaction records of the minimum trading decision granularity include ask price, ask volume, bid price, and bid volume for every second of every stock. The dataset is provided by the Wind Information (Wind Info), a leading integrated service provider of financial data in China.
![Review of key events of the market crash in 2015.[]{data-label="fig:crash_event"}](figure_2015.pdf){width="120mm"}
Then, for identifying the stock market crash, the crash days are defined as days whose number of stocks being sell-off to the down limit (the allowed maximum one-day drop of a stock, i.e., ten percent of its closing price last day) is more than 800 . Specifically, as seen in Figure \[fig:crash\_event\], in 2015, there are 17 trading days on which the stock market was crashed, including June 19th, June 26th, June 29th, July 1st, July 2nd, July 3rd, July 6th, July 7th, July 8th, July 15th, July 27th, Aug.18th, Aug.24th, Aug.25th, Sept.1st, Sept.14th, Oct.21st. And other days before or after these crash ones will be defined as non-crash days and consist the counterparts for further comparison.
Measuring illiquidity {#subsec_mi}
---------------------
The transaction data is full of noise due to the too much frequent occurrences of quote. In order to filter out noise and smooth the data, a fixed time window of one minute is selected to average the spread. Note that as compared to previous study, one minute is short enough to reflect the investment behavior of investors at the smallest decision granularity. Besides, it is necessary to convert the length of data sequence into 237 minutes for every stock in a day for the reason that the Shenzhen Stock Exchange adopts collective bid for the last three minutes. With respect to the illiquidity, various methods have been presented to calculate it for different occasions and purposes. The methods on low-frequency data work great when high-frequency data is not available [@Amihud2002Illiquidity; @LesmondA; @RollA; @CorwinA; @ChengLiquidity], but it is still undeniable that approaches based on high-frequency data perform better since richer information and higher accuracy [@ChungUncertainty; @GoyenkoDo]. Here the illiquidity is expected to sense the minimum decisions in trading behavior, hence the bid-ask spread based on high-frequency records, which is always considered to be the best method, is selected to measure illiquidity [@EasleyChapter; @KyleContinuous]. Moreover, it is known that the size of the transaction has a great impact on illiquidity, we further update the measure by adding the quoted amount as the weight of the spread. The illiquidity can be noted as $$\label{eq:ill}
{I_t} = \frac{{\frac{1}{{10}}(\sum\limits_{i = 1}^{10} {{A_{it}}{V_{it}} - \sum\limits_{j = 1}^{10} {{B_{jt}}{V_{jt}}} } )}}{{{P_{mid,t}}}} \cdot 10000,$$ where $A_{it}$ is the ask price of investor $i$ at time t, $V_{it}$ is the ask volume of investor $i$ at time $t$, $B_{jt}$ is the bid price of investor $j$ at time $t$, $V_{jt}$ is the ask volume of investor $j$ at time $t$, $P_{mid,t}$ is the mean of ask price and bid price at time $t$. It can be learned from the definition that the lower the weighted spread, the lower the transaction coast and the lower the illiquidity.
![Illiquidity with stock index. $I_t$ is the illiquidity we measured and market index represents the CSI 300 Index. The correlation between illiquidity and market index is -0.64 with $p$-value 0.00. The red dots indicate the crash days.[]{data-label="fig:iindex"}](figure_1.pdf){width="120mm"}
The potential capability of the illiquidity in understanding the market crash can be simply illustrated in Figure \[fig:iindex\], in which the market index is negatively associated with the fluctuation of illiquidity we measured. In fact, China’s stock market had experienced a period of ups and downs in the year of 2015, in which period more than ten days of crash erupted in succession. Specifically, the illiquidity continued a decreasing trend before June and at this stage investors easily completed transactions due to lowering cost and the market index kept soaring. In contrast, the illiquidity demonstrated an abrupt increase in June and August, in which months the crash densely occurred and resulted high transaction cost, inactive investors and falling market index. These observations confirm the previously disclosed association between illiquidity and crash in China’s stock market and inspire the following investigations from the novel perspective of illiquidity network.
Results {#sec:results}
=======
Illiquidity and crash {#subsec:illcrash}
---------------------
![Trading behaviors in crash and non-crash days. (a) shows the ask and bid volume in market crash day of June 26th, the stocks are randomly selected from the sample. The first sub-graph shows the stock that not losing liquidity in crash day, and the ohter two show stocks that losing liquidity in crash day. When one of the ask or the bid does not exist, or neither of them exists, the stock loses liquidity. (b) shows the quotation of buyers and sellers in crash days, in which frequency is defined as how often each action occurred every minute of the crash day, no ask means no buyers quote and no bid means no buyers quote. (c) shows the quotation of buyers and sellers in non-crash days.[]{data-label="fig:tf_crash"}](figure_2.pdf){width="120mm"}
It is supposed that trading behaviors, especially the elementary actions like ask and bid of high frequency, would be essentially influenced by shocks like market crash. As can be seen in Figure \[fig:tf\_crash\](a), when stocks approached down limit on crash days, the volume of bid experienced an abrupt decline and then vanished, contrarily the ask volume soared, implying that many investors were forced to sell off shares owing to panic selling and risk prevention. However, approaching down limit might also happen on non-crash days. To further testify the impact from market crash on trading behaviors, we randomly select ten crash days and non-crash days to compose two different groups and compare the occurrence occupations of no-ask, no-bid, and no-quote when stocks experienced down limit. It is unexpected that crash days can be surprisingly distinguished from non-crash days. Specifically, as can be seen in Figure \[fig:tf\_crash\](b), no quotations, which would result in liquidity losing, mainly comes from no-bid on crash days instead of no-ask on non-crash days. This disguising impact from market crash to trading behaviors further suggest that in terms of illiquidity, whose calculations are based on both ask and bid, would inherently sense the footprints of market crash from the novel angle of trading decisions.
![Max duration of illiqudity due to no bid. (a) shows the correlation between the max ask volume and the max duration of illiqudity, which indicates the longest duration of losing liquidity. Note that there may be several periods of losing liquidity per stock in a day. The stock is randomly selected from the sample, and other stocks have similar relationships. (b) shows the correlation between investors’ fear and the illiquidity, whose value is 0.44 with $p$-value 0.00.[]{data-label="fig:max_duration"}](figure_3.pdf){width="120mm"}
Zero volume of bid but soaring amount of ask suggests that on crash days investors are anxious and their anxiety are accumulating. As can be seen in Figure \[fig:max\_duration\](a), it is interesting that the maximum volume of ask in fact logarithmically grows with the duration of losing illiquidity, i.e., no-bid. At this stage, investors can be easily affected by others, especially the spread of pessimism. This logarithmic-like relationship also indicates that the longer the no-bid lasts, the more anxious the investors are and the soar of ask eventually slows down. The saturation of ask volume can be explained that investors will become less panic when more information is obtained. From this perspective, trading actions like ask can be directly connected to investor emotions and intuitively, illiquidity that based on spread of ask and bid should be coupled with emotions, especially the negative ones.
In order to empirically verify the possible associations between illiquidity and investor emotions, the correlations between illiquidity and investor emotions sensed in social media are examined. Specifically, daily emotions towards China’s stock market delivered by investors in social media are split into fear, sadness, disgust, joy and anger [@ZhouTales]. The averaged sequence of illiquidity of the market is accordingly aggregated into a daily sequence and its significant associations with fear can be found in Figure \[fig:max\_duration\][b]{}. The cointegration regression also proves the accuracy since the coefficient of determination is greater than 0.7. The found positive correlation implies that the illiquidity can well reflect, even in a better resolution, the fear in China’s stock market, in which individual investors dominate. In fact, it is difficult for individual investors to be completely rational, they usually like to follow suit blindly and catch up and sell down, causing disorder fluctuations and then spread negative emotions like fear across market. Individual investors may follow and imitate institutional investors for believing that institutional investors possessing more capacity to collect and process information owing to the professional knowledge. The key, however, is that many institutional investors are not rational as assumed, and they are also susceptible to external shocks when dealing with information and making decisions. Besides, even for financial professionals, fear, a potential mechanism underlying risk aversion, might make investors divest more stocks [@fear_aer]. Then fear from those institutional investors might be magnified by following individual investors and reignite much stronger disturb that would lead to a market crash. Hence from the perspective of negative emotions and their contagions, the illiquidity can be contagious among stocks, suggesting that establishing a network by connecting stocks due to mutual illiquidity dependency could offer a new proxy of emotion contagion to finely probe the dynamics of crash.
Illiquidity networks and crash {#subsec:inc}
------------------------------
In the stock market of China, the actual interactions coupled within stocks are extraordinarily important because of susceptible investors. While most existing models forge links between stocks mainly based on the similarity of time-series, e.g., of price and measures of Pearson and Partial correlations are extensively employed [@MantegnaHierarchical; @XuTopological; @WangCorrelation; @KenettPartial]. However, the relationship between stocks is too complicated and should not be too much simplified to neglect trading behaviors, investor emotions and their possible contagions. Taking the limitations of linear correlations into account, here we use mutual information to measure the nonlinear dependency between illiquidity of stock pairs. In fact ,the power in reflecting nonlinear dependency of mutual information in networking market have been previously demonstrated and emphasized [@khoojineNetwork; @BarbiNonlinear; @MenezesOn].
![The normalized mutual information (NMI) of illiquidity. (a) shows the distributions of NMI of illiquidity on both crash(June 26th, June 29th) and non-crash(June 24th, June 25th) days. It is clear that the globally averaged NMI is getting larger while the standard deviation is getting smaller when the stock market is approaching a turmoil. (b) shows the mean and standard deviation of NMI of illiquidity with all the transaction days over the year of 2015.[]{data-label="fig:nmi"}](figure_4.pdf){width="120mm"}
By calculating the normalized mutual information (NMI) of illiquidity series in minute between all pairs of stocks, we first try to profile the distributions of illiquidity dependency of the market on both crash and non-crash days. As can be found in randomly selected samples in Figure \[fig:nmi\](a), the globally averaged NMI is getting larger while the standard deviation (e.g., the broadness of the distribution) is getting smaller when the stock market is approaching a turmoil. Drawing a mean-standard deviation graph with all the transaction days over the year of 2015 for ease of observation, see as Figure \[fig:nmi\](b), it is clear that the average mutual information will increase and the standard deviation will decrease while in the crash days, indicating that the illiquidity network will become more closely connected and more homogeneously coupled when the market is in a bad situation. Because of pessimism, investors become cautious and unwilling to participate in the transaction, which abruptly increases and spreads illiquidity across the market and results in a crash. Besides, we also find that the market crash demonstrates a lasting effect because the days after the crash show the same characteristics as the day in the crash. However, regarding to the days before the crash, as seen in Figure \[fig:nmi\](b), they overlap with those of non-crash and hardly demonstrate any distinct features, suggesting that from the global and static view there is no warning signal can be detected. It inspires us to investigate the illiquidity network from more in-depth and dynamic perspectives further.
In building an illiquidity network, links are weighted as NMI between their ends’ illiquidity, while not all links are necessarily kept and those with less weights, which might relatively represent random dependency among stocks instead of plausible paths for illiquidity contagion, would be removed. Specifically, the size of the giant connected component (GCC) is taken into account for locating the critical threshold of link weight [@KenettDominating; @LuHerding], i.e., the value beyond which the size of GCC starts to decline rapidly will be set as the threshold for each trading day (see Appendix Figure A1(a)). And links with weights below the threshold will then be omitted since their removals trivially influence the connectivity of the market structure. The ratio of GCC in illiquidity networks fluctuates and significantly increase on crash days, suggesting consistently that the market will be more connected and coupled in crashes (see Appendix Figure A1(b)). High illiquidity dependency could facilitate spread of illiquidity across the market and low deviation of illiquidity dependency would further lead to an abrupt collapse of the network. The positive associations between GCC ratios of illiquidity networks and market crash indicate that the refined structures by thresholds of link weights can be proper models of networking market.
![The degree-weighted occurrence proportions on different sections and capital styles in illiquidity networks.(a) shows the proportions on different sections. Note that the proportions in other sections are very similar except for the financial. Therefore, only a few representative industries are selected to simplify the picture. (b) shows the proportions on different capital styles of stock values, the large-cap-value is the most critical group in market crash. As for growth stocks and balanced stocks, the results are the same, they are not shown here in the figure.[]{data-label="fig:opro"}](figure_5.pdf){width="120mm"}
The illiquidity network of stock market evolves in forms of adding new links or removing existing connections. It is found that the China’s stock market evolves in a high frequency, especially on crash days and only 10% links kept on average for consecutive two trading days (see Appendix Figure A2). Highly varying structures suggest that to target critical stocks that function profoundly in crash can help inspect market risk. In terms of grouping stocks into different sections or capital styles (see Appendix Table A1), a degree-weighted proportion, denoted as $R_{ij}$, is defined to identify key group $i$ of stocks on trading day $j$. Specifically, $$\label{eq:Rij}
{R_{ij}} = \frac{n_{ij}/n_j}{N_{ij}/N_j},$$ where $n_{ij}$ is the occurrence of stocks belonging to group (sector or style) $i$ and it is summed over all links in the network of $j$ day, $n_j$ is the occurrence of all stocks and it is summed over all links in the network of $j$ day, $N_{ij}$ is the number of group $i$ in the network of $j$ day, $N_j$ is the number of unique stocks in the network of $j$ day. Accordingly, the group of stocks with higher $R_{ij}$ will occupy more links in the market, meaning heavier dependency on other stocks’ illiquidity and greater odds of taking over or passing on crash risk. It is unexpected that, as can be seen Figure \[fig:opro\](a), the sector of finance constantly occupies the highest proportion in China’s stock market, especially on crash days. As for the capital style, the style of large capitalization, i.e., the large-cap-value is the most critical group in market crash (see Figure \[fig:opro\](b)). Both observations suggest that stocks in finance, especially those of large capital values, should be targets of inspection for market regulators.
![The significance of failing before peak. The red dots indicate crash days, and the blue dots indicate non-crash days. $S_{ij}=R_{ji}^{bp}- R_{ji}^{bpr}$, so $S_{ij}$ may be positive or negative. If $S_{ij}$ is positive, which means stocks within $i$ tend to fail before peaks. It is obvious that the sector of finance failed most before peaks on crash days. In contrast, the sectors like manufacturing and information technology perform similarly both on crash and non-crash days. Note that $S_{ij}$ can not be calculated for all stocks since some of them might not appear in the illiquidity network due to good liquidity, especially on non-crash days.[]{data-label="fig:peaks"}](figure_6.pdf){width="120mm"}
The falling-apart of China’s market in crash was consisted by waves of stocks completely losing illiquidity, i.e., declining to the down limit [@lusmall]. These failure waves produced peaks in number of newly failed stocks (see Appendix Figure A3). Assuming that each wave of failure can be identified by a peak, then stocks that failed before the peak could be seeding failures that lead to the corresponding wave of losing illiquidity. Then sectors with more stocks failed before peaks might be causes of the following collapse and thus could be targets for early inspection and even sources of warning signals. A new ratio, denoted as $R_{ij}^{bp}$ is thus defined to target critical sectors, which can be calculated as $$R_{ij}^{bp}=\frac{n_{ij}^{bp}/N_j^{bp}}{N_{ij}/N_j},$$ where $n_{ij}^{bp}$ is the number of stocks failed before peaks in group $i$ on day $j$, $N_j^{bp}$ is the number of stocks failed before peaks on day $j$. To testify the significance of failing before peaks, the timings of fail for all the stocks of one trading day are also randomly shuffled to get a random value of $R_{ij}^{bp}$, which is denoted as $R_{ij}^{bpr}$ for comparison to test significance. Then for group $i$, its significance of being seeds that probably lead to a wave of failures on day $j$ can be defined simply as $S_{ij}=R_{ij}^{bp}- R_{ij}^{bpr}$. Intuitively, $S_{ij}$ will be much greater than 0 if stocks within $i$ tend to fail before peaks. Consistent with our above observation, the sector of finance, as can be seen in Figure \[fig:peaks\], failed most before peaks, especially on crash days. In the contrary, the significance of sectors like manufacturing and information technology just fluctuates around zero with trivial deviations. It again suggests that stocks of finance in China’s market might be sinks or even triggers that produce illiquidity and spread it across the market. In terms of inspecting these stocks of finance, market practitioners, in particular the regulators, could sense warnings from their abnormal variations on illiquidity.
![The correlation between stock degree and the timing distance of losing illiquidity to the peak. (a) shows the degrees of stocks that decline to the down limit before and after the peak. How to find and determine the peak of stocks decline to the down limit is illustrated in Appendix Figure A3. (b) shows that greater the absolute distance, smaller the degrees of stocks (y-axis is logarithmic). The correlation between the maximum degree and the absolute distance is -0.66 with $p$-value 0.00.[]{data-label="fig:tdistance"}](figure_7.pdf){width="120mm"}
The illiquidity network can also track the dynamics of market crash. Considering peaks of newly failed stocks can be interfaces to split failure cascades, the timing distance between the timing of losing illiquidity to the peak timing inherently measures at which stage the stock join the crash cascade. Specifically, for negative distances, smaller ones stand for the early collapse, while for positive distances, greater ones represent the later failures in the crash (see Figure \[fig:tdistance\](a)). We then examine the function between stock degree and the absolute value of time distance, as can be seen in Figure \[fig:tdistance\](b), it is found that the degree, in particular the maximum degree in each bin, is negatively correlated with the distance. This negative association indicates that stocks fails nearly the peak timing are those with high degrees, while these fail at the early state or at the ending of the crash possess small degrees. That’s to say, the crash ignites from stocks of small degrees, then spread to stocks of high degrees which usually locate at the core of the network and finally cascades to the periphery. Though market crash in essence originates from failure of these crucial nodes in the core, those with small degrees collapsed at the early stage might be the real triggers. Consisting with the previous study [@lusmall], this finding discloses the unexpected role of small-degree stocks in market crash and inspire regulators pay more attention on those conventionally might be overlooked, especially the ones in finance sector.
Illiquidity networks and a warning signal {#subsec:wsignal}
-----------------------------------------
Above illustrations solidly suggest the associations between illiquidity network and market crash. Assuming market crash being systemic failure rather than random error, stocks failed together in a short interval, e.g., ten minutes, should be inherently entangled with each other due the contagion of losing illiquidity and therefore connected in our built illiquidity network. Then the non-randomness of failures within a short interval $i$ can be defined as $w_i=\frac{e_{n_f}}{n_f(n_f-1)/2}$, where $n_f$ stocks got to the down limit in $i$ simultaneously, $e_{n_f}$ is the number of links among them that captured in the illiquidity network built on the corresponding day and $n_f(n_f-1)/2$ is the maximum number of possible links among them. In line with this, higher $w_i$ represents more likelihoods of systemic failures instead of random errors, i.e., signs of crash. And $wd_j=\textless w_i\textgreater$ from all intervals of trading day $j$ can be accordingly measured to value the daily non-randomness. As can be seen in Appendix Figure A4(a), most values of the daily non-randomness are zero and greater fluctuations significantly occur as approaching crash days, which implies a warning signal could be accordingly forged.
![The warning signal of $N_{w_d=0}=0.$ Grey bars stands for non-crash days, red bars mark crash days that can be warned in advance of one day, while those can not be warned in advance are colored to blue.[]{data-label="fig:signal"}](figure_12.pdf){width="120mm"}
Given the fluctuations of daily non-randomness (see Appendix Figure A4(a)), a sliding window of $t$ days, meaning historical information of previous $t$ days is supposed to be helpful, is set to smooth the daily sequence and then we simply count the occurrences of $w_d=0$ within the window, which is denoted as $N_{w_d=0}$ to construct a warning signal. Specifically, smaller $N_{w_d=0}$ suggests more systemic failures and greater odds of leading to market crash. As can be seen in Appendix Figure S4(b), as $t=5$, an abrupt decline of $N_{w_d=0}$ can be detected one day earlier than more than half of 2015 crash days in China’s stock market, in particular for those consecutive ones occurred at the early phase. It indicates that if $N_{w_d=0}=0$ in the previous five days, a warning signal should be sent out because there would be a market crash in the next day, as seen in Figure \[fig:signal\]. Note that $t=5$ is the optimal setting as we vary $t$ from 1 to 15 days. It is interesting that time windows with length shorter than five days result in insensitive $N_{w_d=0}$, while those of longer than five days result in disappearance of signals in advance. The one day ahead of the crash is vital because it indicates that the presented early warning signal can help prevent systematic risk of the market in reality. It should also be noted that not all the crashes of 2015 can be effectively and correctly warned (see Figure \[fig:signal\] and Appendix Figure A4(b)) and those on which the signal failed might be caused by shocks that similar to random ones on non-crash days. Besides, former crash might essentially re-structured the stock market and make the later crash hard to predict.
Conclusions {#sec:con}
===========
Financial systems like stock markets are vital components of modern economics and function profound roles in economic growth. The market crash, however, occurs occasionally and brings about huge shocks to the entire social-economic system and even lead to a global recession. For example, the crash of 2015 in China’s stock market erupted unexpectedly and abruptly evaporated over one third of market value. How to understand and warn the crash has been an open and trending problem in not only finance but also interdiscipline. In fact, from the view of system science, the stock market can be modeled as a complex network and the crash can thus be cascading failures of stocks that decline to the down limit. Nevertheless, in previous study, trading behaviors, in particular the emotions of investors are rarely considered in networking the market, which in essence motivates the present study.
Given the dominance of individual investors in China’s stock market, it is assumed that abnormal decisions and negative emotions could help profile and even warn the market crash. Illiquidity, which is defined as weighted spread between ask price and bid price, can capture trading decisions of ask and bid in a fine resolution of minute and is significantly associated with fear of investors, suggesting a novel perspective of modeling the market and crash. By connecting stocks with illiquidity mutually associated, it is found that the market is more densely and homogeneously coupled due to great mean and low deviation of illiquidity dependencies, which can explain the abrupt collapse of the market in the crash. Stocks are not randomly connected and the ones with large capital value or from the sector of finance are targeted as most influential parts in the market crash. What is even more interesting is that the negative correlation between maximum degrees and distances to the peak of down limit suggests the pattern of periphery-core-periphery propagations in crash. And by simply counting the days without systemic failures in previous five days, an early signal is also derived from the illiquidity network to warn in advance more than half crash days in 2015. Our results could help market practitioners like regulators inspect risky stocks like the ones from finance sector or with small degrees and sense the early warning signal to prevent the crash. Our approach can also be easily adjusted and extended to stock markets of other countries.
While we must admit that not all the crash can be warned accurately by the proposed early signal ($N_{w_d=0}=0$). Those crashes that our signal failed to warn imply that the causes beyond crash can be sophisticated and some of them might be truly caused by random shocks. How to group crashes into categories of can be warned or not will be an interesting direction in the future work. In the meantime, the possible entanglement between different crashes also deserves more efforts.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by National Natural Science Foundation of China (Grant No. 71871006). The authors also appreciate valuable comments and constructive suggestions from Dr. Shan Lu.
[10]{} url \#1[`#1`]{}urlprefixhref \#1\#2[\#2]{} \#1[\#1]{}
Z. Zhou, K. Xu, J. Zhao, Tales of emotion and stock in [China]{}: volatility, causality and prediction, World Wide Web 21 (4) (2018) 1093–1116. [](http://dx.doi.org/10.1007/s11280-017-0495-4).
H. Wang, S. Lu, J. Zhao, Aggregating multiple types of complex data in stock market prediction: A model-independent framework, Knowledge-Based Systems 164 (2019) 193 – 204. [](http://dx.doi.org/https://doi.org/10.1016/j.knosys.2018.10.035).
J. Chiu, H. Chung, K.-Y. Ho, C.-C. Wu, Investor sentiment and evaporating liquidity during the financial crisis, International Review of Economics & Finance 55 (2018) 21–36. [](http://dx.doi.org/10.1016/j.iref.2018.01.006).
A. Flori, F. Pammolli, S. V. Buldyrev, L. Regis, H. E. Stanley, Communities and regularities in the behavior of investment fund managers, Proceedings of the National Academy of Sciences of the United States of America 116 (14) (2019) 6569–6574. [](http://dx.doi.org/10.1073/pnas.1802976116).
S. Lu, J. Zhao, H. Wang, R. Ruoen, Herding boosts too-connected-to-fail risk in the stock market of [China]{}, Physica A: Statistical Mechanics and its Applications 505 (2018) 945–964. [](http://dx.doi.org/10.1016/j.physa.2018.04.020).
Investor structure and characteristics of behavioral changes in the shenzhen market: Evidence from data of recent years, News, Shenzhen Stock Exchange, <http://www.szse.cn/main/en/QFII/SZSENews/39749848.shtml> (2013).
J. Yao, C. Ma, W. P. He, Investor herding behaviour of [Chinese]{} stock market, International Review of Economics & Finance 29 (2014) 12–29. [](http://dx.doi.org/10.1016/j.iref.2013.03.002).
K. Xing, X. Yang, How to detect crashes before they burst: Evidence from [Chinese]{} stock market, Physica A: Statistical Mechanics and its Applications 528 (2019) 121392. [](http://dx.doi.org/10.1016/j.physa.2019.121392).
S. Tian, E. Wu, Q. Wu, Who exacerbates the extreme swings in the [Chinese]{} stock market?, International Review of Financial Analysis 55 (2018) 50–59. [](http://dx.doi.org/10.1016/j.irfa.2017.10.00).
P. J. Dennis, D. Strickland, Who blinks in volatile markets, individuals or institutions?, The Journal of Finance 57 (5) (2002) 1923–1949. [](http://dx.doi.org/10.1111/0022-1082.00484).
Y. Amihud, H. Mendelson, R. A. Wood, Liquidity and the 1987 stock market crash, Journal of Portfolio Management 16 (3) (1990) 65–69. [](http://dx.doi.org/10.3905/jpm.1990.409268).
Y. Amihud, J. Noh, Illiquidity and stock returns: cross-section and time-series effects, Journal of Financial Markets 5 (1) (2002) 31–56. [](http://dx.doi.org/10.1016/S1386-4181(01)00024-6).
G. Bekaert, C. R. Harvey, C. Lundblad, Liquidity and expected returns: Lessons from emerging markets, Review of Financial Studies 20 (6) (2007) 1783–1831. [](http://dx.doi.org/10.1093/rfs/hhm030).
S. Nagel, Evaporating liquidity, The Review of Financial Studies 25 (7) (2012) 2005–2039. [](http://dx.doi.org/10.3386/w17653).
H. R. Stoll, Inferring the components of the bid‐ask spread: Theory and empirical tests, Journal of Finance 44 (1) (1989) 115–134. [](http://dx.doi.org/10.1111/j.1540-6261.1989.tb02407.x).
Y. Chen, G. W. Eaton, B. S. Paye, Micro(structure) before macro? [The]{} predictive power of aggregate illiquidity for economic activity and stock returns, Journal of Financial Economics 130 (1) (2018) 48–73. [](http://dx.doi.org/10.1016/j.jfineco.2018.05.011).
T. Chordia, Market liquidity and trading activity, Journal of Finance 56 (2) (2001) 501–530. [](http://dx.doi.org/10.1111/j.1540-6261.2001.tb00794.x).
Z. An, W. Gao, D. Li, F. Zhu, The impact of firm-level illiquidity on crash risk and the role of media independence: International evidence, International Review of Finance 18 (4) (2017) 547–593. [](http://dx.doi.org/10.1111/irfi.12162).
K. H. Chung, C. Chuwonganant, Uncertainty, market structure, and liquidity, Journal of Financial Economics 113 (3) (2014) 476–499. [](http://dx.doi.org/10.1016/j.jfineco.2014.05.008).
J. Brogaard, L. Dan, X. Ying, Stock liquidity and default risk, Journal of Financial Economics 124 (3) (2017) 486–502. [](http://dx.doi.org/10.1016/j.jfineco.2017.03.003).
M. Brunnermeier, L. Pedersen, Market liquidity and funding liquidity, Review of Financial Studies 22 (6) (2009) 2201–2238. [](http://dx.doi.org/10.3386/w12939).
T. Chordia, R. Roll, A. Subrahmanyam, Commonality in liquidity, Journal of Financial Economics 56 (1) (2000) 3–28. [](http://dx.doi.org/10.1016/S0304-405X(99)00057-4).
J. Hasbrouck, D. J. Seppi, Common factors in prices, order flows and liquidity, Journal of Financial Economics 59 (3) (2001) 383–411. [](http://dx.doi.org/10.1016/S0304-405X(00)00091-X).
G. Huberman, D. Halka, Systematic liquidity, Journal of Financial Research 24 (2) (2001) 161–178. [](http://dx.doi.org/10.1111/j.1475-6803.2001.tb00763.x).
V. V. Acharya, L. H. Pedersen, Asset pricing with liquidity risk, Journal of Financial Economics 77 (2) (2005) 375–410. [](http://dx.doi.org/10.1016/j.jfineco.2004.06.007).
B. Deng, Z. Li, Y. Li, Foreign institutional ownership and liquidity commonality around the world, Journal of Corporate Finance 51 (2018) 20–49. [](http://dx.doi.org/10.1016/j.jcorpfin.2018.04.005).
M. Majapa, S. J. Gossel, Topology of the [South African]{} stock market network across the 2008 financial crisis, Physica A Statistical Mechanics & Its Applications 445 (2016) 35–47. [](http://dx.doi.org/10.1016/j.physa.2015.10.108).
A. Nobi, S. E. Maeng, J. W. Lee, Effects of global financial crisis on network structure in a local stock market, Physica A: Statistical Mechanics and its Applications 407 (2014) 135–143. [](http://dx.doi.org/10.1016/j.physa.2014.03.083).
R. Yang, X. Li, T. Zhang, Analysis of linkage effects among industry sectors in [China]{}’s stock market before and after the financial crisis, Physica A Statistical Mechanics & Its Applications 411 (2014) 12–20. [](http://dx.doi.org/10.1016/j.physa.2014.05.072).
L. Bentian, P. Dechang, Analysis of global stock index data during crisis period via complex network approach, PloS one 13 (7) (2018) 0200600. [](http://dx.doi.org/10.1371/journal.pone.0200600).
J. Bosma, M. Koetter, M. Wedow, Too connected to fail? [Inferring]{} network ties from price co-movements, Journal of Business and Economic Statistics 37 (1) (2017) 67–80. [](http://dx.doi.org/10.1080/07350015.2016.1272459).
A. S. khoojine, D. Han, Network analysis of the [Chinese]{} stock market during the turbulence of 2015–2016 using log-returns, Physica A: Statistical Mechanics and its Applications 523 (2019) 1091–1109. [](http://dx.doi.org/10.1016/j.physa.2019.04.128).
C. Trzcinka, D. Lesmond, J. Ogden, A new estimate of transaction costs, Review of Financial Studies 12 (5) (1999) 1113–1141. [](http://dx.doi.org/10.1093/rfs/12.5.1113).
R. Roll, A simple implicit measure of the effective bid-ask spread in an efficient market, Journal of Finance 39 (4) (1984) 1127–39. [](http://dx.doi.org/10.1111/j.1540-6261.1984.tb03897.x).
S. A. Corwin, P. Schultz, A simple way to estimate bid‐ask spreads from daily high and low prices, Journal of Finance 67 (2) (2011) 719–760. [](http://dx.doi.org/10.2307/41419709).
P. Cheng, Z. Lin, Y. Liu, Liquidity risk of private assets: Evidence from real estate markets, Financial Review 48 (4) (2013) 671–696. [](http://dx.doi.org/10.1111/fire.12020).
R. Goyenko, C. Holden, C. Trzcinka, Do liquidity measures measure liquidity?, Journal of Financial Economics 92 (2) (2009) 153–181. [](http://dx.doi.org/10.1016/j.jfineco.2008.06.002).
D. Easley, M. O’Hara, Chapter 12 market microstructure, Handbooks in Operations Research and Management Science 9 (1995) 357–383. [](http://dx.doi.org/10.1016/S0927-0507(05)80056-8).
A. S. Kyle, Continuous auctions and insider trading, Econometrica: Journal of the Econometric Society 53 (6) (1985) 1315–1335. [](http://dx.doi.org/10.2307/1913210).
A. Cohn, J. Engelmann, E. Fehr, M. A. Maréchal, Evidence for countercyclical risk aversion: An experiment with financial professionals, American Economic Review 105 (2) (2015) 860–85. [](http://dx.doi.org/10.1257/aer.20131314).
R. Mantegna, Hierarchical structure in financial markets, Quantitative Finance Papers 11 (1) (1998) 193–197. [](http://dx.doi.org/10.1007/s100510050929).
R. Xu, W. K. Wong, G. Chen, S. Huang, Topological characteristics of the [Hong Kong]{} stock market: A test-based p-threshold approach to understanding network complexity, Scientific Reports 7 (2017) 41379. [](http://dx.doi.org/10.1038/srep41379).
G. Wang, C. Xie, H. Stanley, Correlation structure and evolution of world stock markets: Evidence from pearson and partial correlation-based networks, Computational Economics 51 (2016) 1–29. [](http://dx.doi.org/10.1007/s10614-016-9627-7).
D. Y. Kenett, X. Huang, I. Vodenska, S. Havlin, H. E. Stanley, Partial correlation analysis: applications for financial markets, Quantitative Finance 15 (4) (2014) 569–578. [](http://dx.doi.org/10.1080/14697688.2014.946660).
A. Q. Barbi, G. A. Prataviera, Nonlinear dependencies on [Brazilian]{} equity network from mutual information minimum spanning trees, Physica A: Statistical Mechanics and its Applications 523 (2019) 876–885. [](http://dx.doi.org/10.1016/j.physa.2019.04.147).
R. Menezes, A. Dionisio, H. Hassani, On the globalization of stock markets: An application of vector error correction model, mutual information and singular spectrum analysis to the [G7]{} countries, The Quarterly Review of Economics & Finance 52 (4) (2012) 369–384. [](http://dx.doi.org/10.1016/j.qref.2012.10.002).
D. Y. Kenett, M. Tumminello, A. Madi, G. Gur-Gershgoren, M. R. N., B.-J. Eshel, Dominating clasp of the financial sector revealed by partial correlation analysis of the stock market, PloS one 5 (12). [](http://dx.doi.org/10.1371/journal.pone.0015032).
S. Lu, J. Zhao, H. Wang, The emergence of critical stocks in market crash, Frontiers in Physics 8 (2020) 49. [](http://dx.doi.org/10.3389/fphy.2020.00049).
Appendix {#appendix .unnumbered}
========
-- -- -- -- --
-- -- -- -- --
: Sector and cap style of stocks
{width="120mm"}
{width="120mm"}
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|
---
abstract: 'By combining the linear theory and numerical simulations, we study the response of a radially bounded axisymmetric plasma to relativistic charged particle beams in a wide range of plasma densities. We present analytical expressions for the magnetic field generated in the dense plasma, prove vanishing of the wakefield potential beyond the trajectory of the outermost plasma electron, and follow the wakefield potential change as the plasma density decreases. At high plasma densities, wavefronts of electron density and radial electric field are distorted because of beam charge and current neutralization, while wavefronts of wakefield potential and longitudinal electric field are not. At plasma densities lower than or of the order of beam density, multiple electron flows develop in and outside the plasma, resulting in nonzero wakefield potential around the plasma column.'
author:
- 'A.A.Gorn'
- 'P.V.Tuev'
- 'A.V.Petrenko'
- 'A.P.Sosedkin'
- 'K.V.Lotov'
title: Response of narrow cylindrical plasmas to dense charged particle beams
---
Introduction
============
Plasma response to relativistic particle beams is a classical problem of plasma physics, actively studied since early 1970s.[@PF13-182; @PF13-1831; @JETP34-93; @TP16-1989; @PP15-429] The advent of particle beam-driven plasma wakefield acceleration (PWFA)[@PRL54-693; @Sci.Am.260; @PFB5-2363; @IEEE-PS24-252] renewed the interest in this problem.[@PAcc20-171; @PAcc22-81; @PF30-252; @PoP3-2753; @PRE69-046405; @PRST-AB7-061302; @PRL96-165002; @PoP13-056709] PWFA is now pursued as a prospective path to future high-energy accelerators.[@NIMA-410-388; @NIMA-410-532; @PRST-AB5-011001; @PoP13-055503; @UFN55-965; @PPCF56-084013; @IPAC14-3791; @EPJC76-463; @RAST9-63; @RAST9-85] Development of this concept gave impetus to in-depth studies of various special cases, one of which is the response of radially-bounded plasmas to ultra-relativistic particle beams.
Studies of radially-bounded plasmas at setups having a direct relationship to PWFA also started in the 1970s.[@JETP39-661; @FP2-49] The problem was solved in the linear approximation for uniform plasmas and beams of densities much lower than the plasma density. Later studies focus on effects of radial plasma non-uniformity[@PRE60-6210; @PoP23-013109], long-term evolution of nonlinear plasma waves[@PRL112-194801], and beam instabilities[@PoP21-056703].
{width="\textwidth"}
Recently, the experiment AWAKE[@PPCF56-084013] at CERN have generated interest in interaction of dense proton beams with low-density plasmas. In AWAKE, three overlapping beams (laser, proton, and electron) propagate through the 10 meter long gas cell filled with the rubidium vapor (Fig.\[fig1-AWAKE\]).[@NIMA-829-76; @PPCF60-014046] The short laser pulse creates the uniform plasma column with a sharp boundary.[@NIMA-740-197] The proton beam self-modulates[@PRL104-255003; @PoP22-103110] and drives a high-amplitude plasma wave that is witnessed by the electron beam.[@PoP21-123116; @NIMA-829-3] Since the laser pulse cannot penetrate foils, there are orifices between the gas cell and high-vacuum upstream and downstream beam lines.[@NIMA-829-3] The rubidium vapor leaks through the orifices and condenses on cold walls of expansion volumes attached to both ends of the gas cell. The loss of vapor is refilled by two rubidium sources near both orifices so that the vapor flows only near the ends of the cell.[@JPDAP-51-025203] The laser pulse ionizes the vapor and creates a radially uniform plasma of an approximately constant radius and the density that gradually reduces away from the orifice. The wakefields excited in the low-density plasma before the orifice by the particle beams are rather weak to disturb the high-energy proton beam, but sufficient for changing trajectories of lower-energy electrons and modifying electron trapping conditions.[@PoP21-123116; @NIMA-829-3]
In this paper, we consider wakefields driven in radially-bounded low-density plasmas by beams of both charge signs. We follow the plasma response from the linear to strongly nonlinear interaction regime as the plasma density reduces. Beam instabilities possible in plasmas at long interaction times are beyond the scope of this paper. In Sec.\[s2\], we introduce the model and discuss possible regimes of beam-plasma interaction. In Sec.\[s3\] and Sec.\[s4\] we respectively study linear and nonlinear regimes of the plasma response. In Sec.\[s5\] we discuss how the wakefield in the low-density part of the plasma column modifies electron trapping conditions in the AWAKE experiment. Then in Sec.\[s6\] we summarize the main findings.
![Geometry of the problem.[]{data-label="fig2-setup"}](fig2-setup.pdf){width="\columnwidth"}
Problem definition and interaction regimes {#s2}
==========================================
We consider axisymmetric beams and use cylindrical coordinates $(r, \phi, z)$ and the co-moving coordinate $\xi=z-ct$, where $c$ is the speed of light. The plasma column has the radius $R$ and constant density $n_0$ (Fig.\[fig2-setup\]). The plasma is collisionless, and the plasma ions are immobile. The beam density $n_b (r, \xi)$ does not evolve in the co-moving frame. The latter approximation is valid if the beam is ultra-relativistic, and the time scale of beam evolution is much longer than the beam duration. We fix the spotlight on the wakefield potential $\Phi$ that characterizes both focusing and accelerating properties of the plasma wave: the force components acting on an axially moving ultra-relativistic elementary charge $e>0$ are $$\label{e0a}
e (E_r - B_\phi) = -e \frac{\partial \Phi}{\partial r}, \qquad e E_z = -e \frac{\partial \Phi}{\partial z},$$ where $\vec{E}$ and $\vec{B}$ are electric and magnetic fields. We also focus on the density of plasma electrons $n_e$, as it gives a general idea of plasma response, and discuss other wakefield features as necessary.
Parameter, notation Value
---------------------------------- -----------------------------------
Maximum plasma density, $n_{e0}$ $7\times 10^{14}\,\text{cm}^{-3}$
Plasma radius, $R$, 1.4mm
Maximum beam density, $n_{b0}$ $4\times 10^{12}\,\text{cm}^{-3}$
Beam total length, $L$ 30cm
Beam radius, $\sigma_r$ 0.2mm
: Baseline parameter set for simulations.[]{data-label="t1"}
Some of the considered interaction regimes are intractable analytically. To get insight into their properties, we make numerical simulations with two-dimensional axisymmetric fully kinetic quasistatic code LCODE[@PRST-AB6-061301; @NIMA-829-350]. Since our study is motivated by AWAKE experiment, we take baseline AWAKE parameters[@PoP21-123116] as the reference case (Table \[t1\]) and vary the plasma density only. This will limit the variety of interaction regimes to those of known practical importance. Note that in our case the peak beam current is much smaller than $mc^3/e \approx 17$kA, where $m$ is the electron mass, and there is the hierarchy of scales $$\label{e2}
L \gg R \gg \sigma_r.$$ For a larger beam current and different ratio $\sigma_r/R$, the interaction regimes could be different.
The particular beam shape is $$\label{e1}
n_b (r, \xi) = \begin{cases}
n_{b0} e^{-r^2/2 \sigma_r^2} \bigl(1 + \cos(\pi\xi/L)\bigr)/2, & -L<\xi<0, \\
0, & \text{otherwise}.
\end{cases}$$ It is rather convenient for basic studies because it has both slowly varying (long tail) and rapidly changing (hard leading edge) parts, so the study can inform of the plasma response on beams of different timescales. While our focus is on positively charged beams, we also consider electron beams wherever comparison of the two cases is helpful.
Equality Effect Plasma density $n_0/n_{b0}$
--------------------------------- --------------------- ------------------------------------ --------------
$k_p R = 1$ plasma boundary $1.5 \times 10^{13}\text{cm}^{-3}$ 3.6
$n_{b0} = n_0$ plasma nonlinearity $4\times 10^{12}\,\text{cm}^{-3}$ 1.0
$n_0 R^2 = 2 n_{b0} \sigma_r^2$ plasma self-fields $4\times 10^{10}\,\text{cm}^{-3}$ 0.01
: Boundaries between the interaction regimes and effects responsible for this boundaries.[]{data-label="t2"}
For the selected relation of scales, we can distinguish four regimes of plasma response (Table \[t2\]). The first regime corresponds to high plasma densities, where the plasma radius $R$ is much larger than the plasma skin-depth $k_p^{-1} = c/\omega_p$, where $\omega_p = \sqrt{4 \pi n_0 e^2/m}$ is the plasma frequency. In this regime, there is no difference between unbounded and radially bounded plasmas.
In the second regime, effects of the plasma boundary are important ($k_p R \lesssim 1$), but still $n_0 \gg n_{b0}$, and nonlinear effects are weak. The first two regimes allow for a unified analytical description (Sec.\[s2\]).
In the third regime, $n_0 \lesssim n_{b0}$, and the plasma response is strongly nonlinear (Sec.\[s3\]). Still, the plasma column contains enough plasma electrons to neutralize the beam charge and current.
The fourth regime corresponds to very low plasma densities. In this regime (also described in Sec.\[s3\]), the beam linear charge exceeds that of the plasma column, $n_b \sigma_r^2 > n_0 R^2$, and plasma fields has a negligible effect on the motion of plasma electrons.
The transition between the regimes is smooth and the equalities presented in Table \[t2\] show the transition borders only approximately.
Linear plasma response {#s3}
======================
The expressions for wakefields induced in the radially-bounded uniform plasma by a low-density particle beam have an easy-to-use form[@PoP21-056703; @notebook] if the beam density is separable (as in our case), $$\label{e4a}
n_b(r,\xi) = n_{b0} f(r) g(\xi).$$ Then the potential is also separable $$\begin{aligned}
\label{e6a} \Phi (r,\xi) &= \begin{cases}
\ds q\frac{mc^2 n_{b0}}{e n_0} F(r) G(\xi), & r<R, \\
0, & r>R,
\end{cases} \\
\label{e7b} G(\xi) &= k_p \int_\xi^\infty d\xi' \sin \bigl( k_p (\xi'-\xi) \bigr) g(\xi'), \\
\nonumber F(r) &= k_p^2 \int_0^R \left[\frac{K_0(k_p R)}{I_0(k_p R)}I_0(k_p r_>) - K_0(k_p r_>)\right] \\
\label{e7a} &\times I_0(k_p r_<) f(r') r'dr',\end{aligned}$$ where $$\label{e7c}
r_< = \min (r, r'), \qquad r_> = \max (r, r'),$$ $q=\pm 1$ is the beam charge sign, and $I_0$ and $K_0$ are modified Bessel functions. Note the same longitudinal periodicity of the potential at all radial positions and no boundary effect on the oscillation frequency in the near-boundary regions.
![Longitudinal functions $g(\xi)$ (beam shape) and $G(\xi)$ (wakefield potential) for beams with (a) sharp leading edge and (b) localized short-scale fragment.[]{data-label="fig-G"}](fig3-G.pdf){width="236bp"}
Properties of the longitudinal function are best seen after integrating by parts $$\begin{gathered}
\label{e8}
G(\xi) = g(\xi) + \sin (k_p \xi) \int_\xi^\infty \sin (k_p \xi') \frac{d g (\xi')}{d \xi'} \, d \xi' \\
+ \cos (k_p \xi) \int_\xi^\infty \cos (k_p \xi') \frac{d g (\xi')}{d \xi'} \, d \xi'.\end{gathered}$$ The first term follows the beam density profile and can be slowly-varying. The second and third terms oscillate with the plasma frequency, and their total amplitude is proportional to the Fourier component of the derivative $d g/d \xi$ at the this frequency. If the beam has a sharp leading edge \[Fig.\[fig-G\](a)\], then the amplitude of the oscillating component always equals $g(\xi)$ at the edge location. In the general case, a localized short-scale fragment can initiate oscillations of an arbitrary amplitude \[Fig.\[fig-G\](b)\].
![Radial dependences of the wakefield potential term $F(r)$, radial force $E_r - B_\phi$, and fields $E_r$ and $B_\phi$ for $k_p R = 1$, $G(\xi) = 0.5$, $g(\xi) = 1$. Dashed lines show the corresponding dependences for the unbounded plasma. The dotted line is the vacuum magnetic field of the beam.[]{data-label="fig-sum"}](fig4-sum.pdf){width="228bp"}
The oscillating part of the wakefield appears due to Langmuir waves, which are potential and produce no magnetic field. Accordingly, the expression for the magnetic field $B_\phi$ contains no oscillations at the plasma frequency $$\begin{gathered}
\label{e9} B_\phi (r,\xi) = -qE_{b0} k_p g(\xi)\int_0^R dr' r' \frac{d f(r')}{dr'} \\
\times \begin{cases}
\left[\alpha I_1(k_p r_>) + K_1(k_p r_>)\right] I_1(k_p r_<), & r<R, \\
I_1(k_p r') \left(\alpha I_1(k_p R) + K_1(k_p R)\right) R / r, & r>R,
\end{cases}\end{gathered}$$ where $$\alpha = \frac{K_0(k_p R)}{I_0(k_p R)},$$ and $$\label{e12a}
E_{b0} = \frac{mc\omega_p n_{b0}}{e n_0}$$ is a convenient field unit for our problem. We obtained the formula similarly to the infinite plasma case[@PP15-429], but with two additional interface conditions for continuity of $B_\phi$ and $\partial B_\phi / \partial r$ at $r=R$. The expression relates to the ultra-relativistic beam case and, therefore, differs from that of Ref. , which corresponds to moderately relativistic beams. Unlike the wakefield potential, the magnetic field does not vanish outside the plasma. Consequently, the radial electric field in the outer region equals the magnetic field for any beam shape and radius (Fig.\[fig-sum\]). Thus, the plasma fields can be conceived as composed of two parts. One part is the plasma wave excited by longitudinal beam non-uniformities. Its frequency equals the plasma frequency, and its field is purely electric and does not extend outside the plasma column. The other part is incompletely neutralized electric and magnetic self-fields of the beam, which may have different radial dependence inside the plasma, but are identical outside. Both parts have the same radial dependence of the wakefield potential. The surface wave[@JETP39-661; @PoP10-4563] is not excited in our case, as its phase velocity is smaller than the beam (light) velocity.
![ Illustration to the derivation of field equality outside the plasma.[]{data-label="fig-elflux"}](fig5-elflux.pdf){width="\columnwidth"}
The equality of $E_r$ and $B_\phi$ outside the plasma comes from the electron flux conservation in the co-moving frame (Fig.\[fig-elflux\]) that necessarily takes place in the context of the quasistatic approximation. The number of electrons passing through black circles in Fig.\[fig-elflux\] is the same and equals $$\label{e10}
\int_0^{r_e} n_e (c - v_z) \, 2 \pi r \, dr = \int_0^R n_0 c \, 2 \pi r \, dr,$$ where $v_z (r,\xi)$ and $n_e(r,\xi)$ are longitudinal velocity and density of plasma electrons, $n_0$ is the unperturbed electron density equal to the ion density $n_i$, and $r_e$ is the radius of the outermost electron. Since the beam current $j_{bz} = e n_b c$, from Maxwell and Poisson equations we have $$\label{e11}
\frac{\partial}{\partial r} r (E_r - B_\phi) = 4 \pi e r \left( n_i - n_e + n_e \frac{v_z}{c} \right),$$ which, after integrating and using , gives $E_r (r) = B_\phi (r)$, if $r > r_e$ and $r > R$. This is also valid for the nonlinear plasma response and proves the equality of $E_r$ and $B_\phi$ beyond the trajectory of the outermost electron.
Wakefield strength scales differently with the decrease of the plasma density in bounded and unbounded plasmas. The amplitude of the longitudinal function does not depend on the plasma density for our beam, so the difference comes from the radial function $F(r)$. In the unbounded plasma, the low-density limit corresponds to $k_p \sigma_r \ll 1$, for which[@PoP12-063101] $$\label{e12}
F(0) \approx k_p^2 \sigma_r^2 [0.05797 - \ln (k_p \sigma_r)],$$ and the potential amplitude grows in absolute value, as the density decreases (Fig.\[fig-evolphi\]): $$\label{e13}
\Phi(0) \propto 0.05797 - \ln (k_p \sigma_r).$$ Consequently, the lower the plasma density, the larger emittance the beam must have to stay in equilibrium with the wakefield in the unbounded plasma[@PoP24-023119]. For the same reason, particles side-injected into the wakefield[@NIMA-829-3; @JPP78-455] in a low-density plasma gain a larger transverse momentum than in a high-density plasma. The longitudinal electric field $E_z$, however, is smaller at low densities, as decrease of the derivative $\partial / \partial z \approx k_p \propto \sqrt{n_0}$ in prevails over the slow logarithmic growth .
![ Plasma density dependence of the on-axis potential calculated in the linear approximation for bounded and unbounded plasmas (solid lines) and simulated for the bounded plasma (dots). Vertical lines show the boundaries between the interaction regimes from Table \[t2\].[]{data-label="fig-evolphi"}](fig6-evolphi.pdf){width="\columnwidth"}
In the bounded plasma, the first term in square brackets in Eq. dominates at $k_p R \ll 1$, and the scaling at low plasma densities is $$\label{e14}
F(0) \approx -\frac{k_p^2 \sigma_r^2}{2} \left(\ln{\frac{R^2}{2 \sigma_r^2}} + \Gamma(0, R^2/(2 \sigma_r^2)) + \gamma \right),$$ where $$\label{e14a}
\Gamma (0, \beta) = \int_\beta^\infty t^{-1} e^{-t} dt, \qquad \gamma \approx 0.577215,$$ so the potential well depth tends to a constant (Fig.\[fig-evolphi\]). At even lower densities, for which the linear theory is not applicable and simulations are needed, the potential well gradually disappears (Fig.\[fig-evolphi\]). The linear theory thus gives a correct value of the potential well depth up to the onset of nonlinear effects at $n_0 \sim n_{b0}$.
Nonlinear plasma response {#s4}
=========================
It is commonly believed that the linear theory of plasma response to the beam is fully applicable if the plasma density is much higher than the beam density, $$\label{e15}
n_0 \gg n_b.$$ Evaluation of all neglected nonlinear terms[@PP15-429; @PoP3-2753] formally gives a stronger limitation $$\label{e16}\
n_0 \gg n_b k_p^2 L^2,$$ but weakly restricts the validity of the linear theory results, as account of the nonlinear terms does not considerably change the plasma response.
![ Maps of the radial force $E_r -B_\phi$ and wakefield potential $\Phi$ in wide (a), (b) and zoomed in (c), (d) areas and their radial slices (e) at $\xi = -12.15\,\text{cm}$ (black dashed line) for the plasma density $n_0 \approx 3.6 n_{b0}$ (at which $k_p R = 1$).[]{data-label="fig-drift"}](fig7-drift.pdf){width="\columnwidth"}
However, if the beam has two different scales (as in our case), the applicability condition for the linear theory is much stronger, $$\label{e17}
n_0 \gg n_b k_p L,$$ where $L$ is the larger scale. The limitation comes from changing the local plasma frequency due to beam charge neutralization and from the drift of plasma electrons, neutralizing the beam current. Plasma electron density perturbation $\delta n$ and longitudinal velocity $v_z$ are $$\label{e17a}
\delta n = qn_{b0} f(r)G(\xi), \qquad v_z = qc\frac{n_{b0}}{n_0}F(r)G(\xi).$$ These quantities, averaged over the plasma wave period, determine the plasma frequency shift. For smooth drivers \[$|dg(\xi)/d \xi| \ll k_p$\], the averaging takes a simple form $$\label{e17c}
\langle G(\xi) \rangle = g(\xi).$$ Therefore, the average density perturbation copies the shape of the driver beam while the speed of plasma electrons copies the shape of the wakefield potential.
If the ratio $n_b/n_0$ is small, the plasma frequency changes by $\delta \omega_p \sim q \omega_p n_b/(2n_0)$ in the beam area. Plasma electrons move with the average velocity $v_z \sim c n_b / n_0$ (in the case of local neutralization for $k_p \sigma_r \gg 1$) or less (if the plasma current flows in a wider area for $k_p \sigma_r \lesssim 1$). The electron motion causes the Doppler shift of the oscillation frequency by about $q\omega_p n_b/n_0$ or less. Two effects add together and result in deformation of wavefronts, which accumulates towards the beam tail \[Fig.\[fig-drift\](a)\]. The sense of curvature depends on the beam charge sign. The limitation comes from the requirement of a small accumulated phase shift. The analogue to this is a limitation on the distance $|\xi|$ from the beam part that generates the plasma wave: $$\label{e18}
|\xi| \ll k_p^{-1} n_0 / n_{b0}.$$ At these distances, the linear theory gives correct fields, velocities, and electron densities (orange dashed lines in Fig.\[fig-drift\]).
Surprisingly, the distortion of wavefronts does not result in distortion of the wakefield potential pattern \[Fig.\[fig-drift\](b)\]. The potential $\Phi$ and its longitudinal derivative $E_z$ oscillate exactly with the plasma frequency, while patterns of the radial force $E_r - B_\phi$ and plasma electron density $n_e$ are distorted and have a period, which is shorter or longer than the plasma period, depending on the driver charge sign. This unusual feature appears due to low-amplitude, short-scale radial rippling of the potential \[Fig.\[fig-drift\](c)\] and violation of separability .
![ Maps of the plasma electron density $n_e$ for smooth electron (a) and proton (b) beams of the shape . The initial plasma density is $n_0= 0.1 n_{b0}.$[]{data-label="fig-channel"}](fig8-channel.pdf){width="211bp"}
At densities $n_0 \lesssim n_{b0}$, the plasma response is strongly nonlinear. We started studying it from the case of long smooth beams of both charge signs with the density distribution $$\label{e19}
n_b (r, \xi) = \begin{cases}
n_{b0} e^{-r^2/2 \sigma_r^2} \bigl(1 - \cos(2\pi\xi/L)\bigr)/2, & -L < \xi < 0, \\
0, & \text{otherwise}.
\end{cases}$$ For electron beams, the main difference from the unbounded plasma case is that electrons initially located in outer layers leave the plasma, as the driver current increases, and carries away the excessive negative charge \[Fig.\[fig-channel\](a)\]. When the driver current later decreases, these electrons cannot quickly return, so the plasma acquires a positive charge and generates a radial electric field. The column of plasma electrons shrinks in radius to keep the charge balance inside, so the total positive charge of the plasma is that of bare ions in the outer layer. Small plasma-frequency oscillations of the radial electric field, which are always present in this system, cause some boundary electrons to gain a large inward momentum and form a multiple flow inside the plasma. Formation of the ion channel (or bubble) near the axis does not differ from the unbounded plasma case.[@PRE69-046405] For the proton driver, no electrons escape the plasma \[Fig.\[fig-channel\](b)\], and the excessive positive charge of the beam transfers to an “ion tube” in the outer region of the plasma column.
![ Maps of the plasma electron density $n_e$ for electron (a) and proton (b) beams of the shape . Map of the focusing force $E_r-B_\phi$ for the proton beam case (c). The initial plasma density is $n_0= 0.5 n_{b0}.$[]{data-label="fig-outer"}](fig9-outer.pdf){width="212bp"}
If the beam efficiently generates both low-frequency and plasma-frequency perturbations, the number of electrons escaping the plasma is even higher. For the electron beam, nonlinear oscillations of the plasma electron density near the axis cause oscillations of the electron boundary \[Fig.\[fig-outer\](a)\]. During each oscillation period, groups of electrons separate from the boundary and either leave the plasma column, or propagate towards the axis forming a multiple flow. The escaping electrons appear also as a result of wave breaking in the near-axis region. For the proton beam, the escaping electrons appear from the wave breaking only \[Fig.\[fig-outer\](b)\].
The electrons that escape the plasma column carry a non-zero wakefield potential to the region of their reach and, therefore, make this region defocusing for proton beams and focusing for electron beams \[Fig.\[fig-outer\](c)\]. The property of no radial force exerted on ultra-relativistic beams outside the narrow plasma thus disappears at low plasma densities.
![Simulated plasma density dependence of the wakefield period number N where the wave first breaks (points). The solid line helps to see the linear scaling.[]{data-label="fig-wbreak"}](fig10-wbreak.pdf){width="\columnwidth"}
Appearance of escaping electrons is directly related to distortion of wavefronts discussed earlier. As the phase difference of electron oscillations at different radii reaches some critical value, electron trajectories cross: inner electrons become outer and vice versa. The escaping electrons are those initially located at smaller radii. After the trajectories cross, these electrons experience the radial expelling force from the increased negative charge inside and, therefore, escape the plasma rather than continue oscillating around some radial position. The lower the plasma density, the stronger the distortion of wavefronts, the sooner the wavebreaking occurs (Fig.\[fig-wbreak\]). Note also that the oscillating component of the radial force almost disappears at radii of wavebreaking \[Fig.\[fig-outer\](c)\], only the slowly varying component remains.
![ Maps of the plasma electron density $n_e$ for electron (a) and proton (b) beams and initial plasma density $n_0= 0.14 n_{b0}$.[]{data-label="fig-waves"}](fig11-waves.pdf){width="212bp"}
For plasma densities satisfying conditions $n_{b0} \gg n_0 \gg n_{b0} \sigma_r^2/R^2$, two different oscillation scales are visible at density maps (Fig.\[fig-waves\]). In parallel with plasma-frequency oscillations, there appears radial oscillations of electrons ejected out of the plasma. The time scale of the radial oscillations depends on the linear charge of the plasma column and on the beam current. Two oscillation types do not continuously evolve into another, thus forming a chaos-like plasma response, if both are present and strong.
![ Maps of the plasma electron density $n_e$ for electron (a) and proton (b) beams and initial plasma density $n_0= 0.05 n_{b0}$.[]{data-label="fig-chaos"}](fig12-chaos.pdf){width="212bp"}
At very low plasma densities, the behavior of plasma electrons and the wakefields are determined by the beam fields (Fig.\[fig-chaos\]). For electron beams, all plasma electrons are ejected out of the plasma and return back well after the beam transit. For proton drivers, the plasma electrons oscillate around the beam. The interaction regime changes to this “low density” mode well before the linear charge of plasma electrons equals the beam linear density, as is illustrated in Fig.\[fig-chaos\].
Problem of electron injection {#s5}
=============================
Quantitative measures of the plasma response depend on particular beam and plasma parameters. For this reason, we have discussed mostly qualitative features in the previous section. Here we quantitatively study the effect of smooth density transition at the beginning of the plasma section on propagation of the witness electron beam in the AWAKE experiment. We take the longitudinal plasma density profile[@NIMA-829-3; @JPDAP-51-025203] $$\label{e20}
n_0 = \frac{n_{e0}}{2} \left( 1 - \frac{\delta z / D}{\sqrt{(\delta z / D)^2+0.25}} \right),$$ where $\delta z$ is the distance to the orifice that separates the plasma cell and expansion volume at $z=0$, and $D=1$cm is the orifice diameter.
![ The focusing force $E_r - B_\phi$ (a) and the “integral” radial force $F_\text{r,int} (r)$ (b) for the proton beam and plasma of the density $n_0= n_{b0}$. []{data-label="fig-outline"}](fig13-outline.pdf){width="\columnwidth"}
The optimum injection parameters[@NIMA-829-3] were obtained from simulations of electron beam propagation through the whole plasma section, including the density transition areas. In this Section, we present a less precise, but more intuitive explanation why the optimum is like that. For this, we reduce full maps of the radial forces available for each plasma density value \[Fig.\[fig-outline\](a)\] to simple radial dependencies $F_\text{r,int} (r)$ showing the integral effect of the radial force \[Fig.\[fig-outline\](b)\]. The “integral” radial force $F_\text{r,int} (r)$ is the half-sum of the maximum and minimum forces at the given radius. If the wakefield oscillates as a function of $\xi$, as is typical for $r < R$, then the ‘integral’’ force presents the average slow-varying force component. If the force has a definite sign, as for $r > R$, then the ‘integral’’ force is half of the maximum force at this radius.
![ The “integral” radial force $F_\text{r,int} (r)$ in the density transition area near the orifice. The arrow shows the optimum electron trajectory for the best injection efficiency (from Ref.). The green shading shows the location of the constant density area, in which electrons can be trapped by the stationary-phase wakefield.[]{data-label="fig-map"}](fig14-map.pdf){width="\columnwidth"}
The map of the “integral” radial force (Fig.\[fig-map\]) shows that the plasma column always defocuses axially propagating electrons. The typical defocusing force is several MeV/m, which is sufficient to deflect a 16MeV electron beam (the AWAKE design energy) by 1.4mm (plasma radius) in several centimeters of propagation. Thus, no collinearly injected electrons can cross blue areas of Fig.\[fig-map\] and get trapped by the driver wakefield in areas of the constant density inside the plasma cell.
Summary {#s6}
=======
We combined the linear analytical theory and numerical simulations to study the response of a radially bounded plasma to highly relativistic charged particle beams in a wide range of plasma densities.
We discovered that the wakefield potential vanishes outside the plasma column. This result is valid for any plasma density, as it is a direct consequence of the axial symmetry and charge conservation. For a strongly nonlinear plasma response, some electrons may leave the plasma and carry away a nonzero potential. If so, the potential vanishes beyond the outermost electron trajectory.
The wakefield potential is strongest in absolute value at plasma densities close to the peak beam density ($n_0 \sim n_b$). At higher plasma densities, the wakefield is tractable analytically and is weaker for the radially bounded case, as compared with the infinite plasma, if the plasma skin depth is longer than or of the order of the plasma radius. At lower plasma densities, analytical expressions overestimate the wakefield amplitude that falls to zero as the density decreases.
For long low-density beams ($n_b \ll n_0$), the nonlinearity of the plasma response manifests itself as wavefront distortion caused by compensation of beam charge and current in the plasma. This imposes strong limitations or on the applicability of the linear theory. Patterns of the wakefield potential and the longitudinal electric field, however, are not distorted and keep the unperturbed plasma period.
Even at low plasma densities ($n_0 \lesssim n_b$), the plasma maintains average quasi-neutrality in most of its volume. In the case of electron beams, this is achieved by pushing a certain number of plasma electrons out of the plasma column. A positively charged beam pulls all electrons from the near-boundary region and leaves a “tube” of bare ions there. Plasma oscillations initiated by the beam, if any, produce electron jets that form multiple flows inside and outside the plasma column. The jets originate either from the plasma boundary or from wavebreaking regions near the axis, and their origin is locked to certain oscillation phases. Timescales of jet dynamics differ from the period of plasma oscillations, so the multiple flow in the presence of strong jets looks chaos-like. The plasma response at very low plasma densities is expectedly fully determined by the beam fields.
The wakefield created by the beam in the plasma is, in average, focusing for this beam and for witness particles of alike charge and is defocusing for particles of the opposite charge sign. If some electrons leave the plasma column in the case of nonlinear response, then the wakefield carried by these electrons is always focusing for electron beams. This clarifies the choice of oblique injection as the baseline scenario of AWAKE experiment.[@NIMA-829-3]
This work is supported by The Russian Science Foundation, grant No. 14-50-00080. The computer simulations are made at Siberian Supercomputer Center SB RAS and Computing Center of Novosibirsk State University.
[88]{} J.L.Cox, W.H.Bennett, Phys. Fluids **13**, 182 (1970). D.A.Hammer, N.Rostoker, Phys. Fluids **13**, 1831 (1970). A.A. Rukhadze, V.G. Rukhlin, Sov. Phys. JETP **34**, 93 (1972). S.E.Rosinskii, A.A.Rukhadze, Sov. Phys. Tech. Phys. **16**, 1989 (1972). G. Küppers, A. Salat, H. K. Wimmel, Plasma Physics **15**, 429-439 (1973). P.Chen, J.M.Dawson, R.W.Huff, and T.Katsouleas, Phys. Rev. Lett. **54**, 693 (1985); Phys. Rev. Lett. **55**, 1537 (1985). J.Dawson, Sci. Am. **260**(3), 54 (1989). J.S.Wurtele, Phys. Fluids B **5** 2363 (1993). E.Esarey, P.Sprangle, J.Krall, and A.Ting, IEEE Trans. Plasma Sci. **24**, 252 (1996). P.Chen, Part. Accel. **20**, 171 (1987). T.Katsouleas, S.Wilks, P.Chen, J.M.Dawson, and J.J.Su, Part.Accel. **22**, 81 (1987). R.Keinigs and M.E.Jones, Phys. Fluids **30**, 252 (1987). K.V.Lotov, Phys. Plasmas **3**, 2753 (1996). K.V.Lotov, Phys. Rev. E **69**, 046405 (2004). J.B.Rosenzweig, N.Barov, M.C.Thompson, and R.B.Yoder, Phys. Rev. ST Accel. Beams **7**, 061302 (2004). W.Lu, C.Huang, M.Zhou, W.B.Mori, and T.Katsouleas, Phys. Rev. Lett. **96**, 165002 (2006). W.Lu, C.Huang, M.Zhou, M.Tzoufras, F.S.Tsung, W.B.Mori, T.Katsouleas, Phys. Plasmas **13**, 056709 (2006). A.M.Kudryavtsev, K.V.Lotov, A.N.Skrinsky, Plasma wake-field acceleration of high energies: Physics and perspectives. Nuclear Instr. Meth. A **410**, 388 (1998). J.Rosenzweig, N.Barov, A.Murokh, E.Colby, P Colestock, Nuclear Instr. Meth. A **410**, 532 (1998). S.Lee, T.Katsouleas, P.Muggli, W.B.Mori, C.Joshi, R.Hemker, E.S.Dodd, C.E.Clayton, K.A.Marsh, B.Blue, et al., Phys. Rev. ST Accel. Beams **5**, 011001 (2002). T.Katsouleas, Phys. Plasmas **13**, 055503 (2006). V.D.Shiltsev, Physics – Uspekhi **55**, 965 (2012). R.Assmann, R.Bingham, T.Bohl, C.Bracco, B.Buttenschon, A.Butterworth, A.Caldwell, S.Chattopadhyay, S.Cipiccia, E.Feldbaumer, et al., Plasma Phys. Control. Fusion **56**, 084013 (2014). J.P.Delahaye, E.Adli, S.J.Gessner, M.J.Hogan, T.O.Raubenheimer, W.An, C.Joshi, and W.Mori, “A beam driven plasma-wakefield linear collider from Higgs factory to multi-TeV,” in Proceedings of the IPAC2014 (Dresden, Germany), pp. 3791-3793. A.Caldwell and M.Wing, Eur. Phys. J. C **76**, 463 (2016). M.J.Hogan, Reviews of Accelerator Science and Technology 9, 63 (2016). E.Adli and P.Muggli, Reviews of Accelerator Science and Technology 9, 85 (2016). S.E. Rosinskii, E.V. Rostomyan, A.A. Rukhadze, and V.G. Rukhlin, Sov. Phys. JETP **39**, 661 (1975). S.E. Rosinskii, E.V. Rostomyan, and V.G. Rukhlin, Fizika Plasmy **2**, 49 (1976) (in Russian). A.G.Khachatryan, Phys. Rev. E **60**, 6210 (1999). Y. Golian, M. Aslaninejad, and D. Dorranian, Phys. Plasmas **23**, 013109 (2016). K.V.Lotov, A.P.Sosedkin, A.V.Petrenko, Phys. Rev. Lett. **112**, 194801 (2014). Y. Fang, J. Vieira, L. D. Amorim, W. Mori, and P. Muggli, Phys. Plasmas **21**, 056703 (2014). <https://anaconda.org/Gorn/easy-linear-field-theory> E. Gschwendtner, E. Adli, L. Amorim, R. Apsimon, R. Assmann, A.-M. Bachmann, F. Batsch, J. Bauche, V.K. Berglyd Olsen, M. Bernardini, et al., Nuclear Instr. Methods A **829**, 76 (2016). P.Muggli, E.Adli, R.Apsimon, F.Asmus, R.Baartman, A-M.Bachmann, M.Barros Marin, F.Batsch, J.Bauche, V.K.Berglyd Olsen, et al., Plasma Phys. Control. Fusion 60, 014046 (2018). E.Oz, P.Muggli, Nucl. Instr. Meth. A **740**, 197 (2014). N.Kumar, A.Pukhov, and K.Lotov, Phys. Rev. Lett. **104**, 255003 (2010). K. V. Lotov, Phys. Plasmas **22**, 103110 (2015). K.V.Lotov, A.P.Sosedkin, A.V.Petrenko, L.D.Amorim, J.Vieira, R.A.Fonseca, L.O.Silva, E.Gschwendtner, and P.Muggli, Phys. Plasmas **21**, 123116 (2014). A. Caldwell, E. Adli, L. Amorim, R. Apsimon, T. Argyropoulos, R. Assmann, A.-M. Bachmann, F. Batsch, J. Bauche, V.K. Berglyd Olsen, et al., Nuclear Instr. Methods A **829**, 3 (2016). G Plyushchev et al, J. Phys. D: Appl. Phys. **51** 025203 (2018) K.V.Lotov, Phys. Rev. ST - Accel. Beams **6**, 061301 (2003). A.P.Sosedkin, K.V.Lotov, Nuclear Instr. Methods A **829**, 350 (2016). L.M. Gorbunov, P. Mora, R.R. Ramazashvili, Phys. Plasmas **10**, 4563 (2003). W. Lu, C. Huang, M. M. Zhou, W. B. Mori, and T. Katsouleas, Phys. Plasmas **12**, 063101 (2005). K.V. Lotov, Phys. Plasmas **24**, 023119 (2017). K.V.Lotov, J. Plasma Phys. **78**, 455 (2012).
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abstract: 'In this paper, we propose a new inexact version of the projected subgradient method to solve nondifferentiable constrained convex optimization problems. The method combine $\epsilon$-subgradient method with a procedure to obtain a feasible inexact projection onto the constraint set. Asymptotic convergence results and iteration-complexity bounds for the sequence generated by the method employing the well known exogenous stepsizes, Polyak’s stepsizes, and dynamic stepsizes are stablished.'
author:
- 'A. A. Aguiar [^1]'
- 'O. P. Ferreira'
- 'L. F. Prudente'
bibliography:
- 'SubgradInexP.bib'
title: Subgradient method with feasible inexact projections for constrained convex optimization problems
---
[**Keywords:**]{} Subgradient method, feasible inexact projection, constrained convex optimization.
[**AMS subject classification:**]{} 49J52, 49M15, 65H10, 90C30.
Introduction
============
The Subgradient method is one of the most interesting iterative method for solving nondifferentiable convex optimization problems, which has its origin and development in the 60’s, see [@Ermolev1966; @Shor1985]. Since then, the subgradient method has attracted the attention of the scientific community working on optimization. One of the factors that explains this interest is its simplicity and ease of implementation. In particular, allowing a low cost of storage and ready exploitation of separability and sparsity. For these reasons, several variants of this method have emerged and properties of it have been discovered throughout the years, resulting in a wide literature on the subject; see, for exemple [@AlberIusemSolodov1998; @Yunier2013; @Bertsekas1999; @GoffinKiwiel1999; @KiwielBook1985; @NedicBertsekas2010] and the references therein.
The aim of this paper is to present an inexact version of the projected subgradient method, which consists in using an inexact projection instead of the exact one, for minimizing a convex function $f: \mathbb{R}^n \to \mathbb{R}$ onto a closed and convex subset $C$ of $\mathbb{R}^n$. The proposed method, that we call [*Subgradient-InexP method*]{}, generates a sequence $\{x_k\}$ where each iteration consists of two stages. The first stage performs a step from the current iterate $x_k$ in the opposite direction of a $\epsilon$-subgradient of $f$ at $ x_k $ and the second inexactly projects the resulting vector onto the feasible set $C$. From the theoretical point of view, considering methods that use inexact projections are particularly interesting for the following reasons. Even when the projection onto a convex set is an easy problem, iterative methods provide only approximated solutions with small errors, due to round-off errors in floating-point arithmetics. Therefore, the study of inexact methods gives theoretical support for real computational implementations of exact schemes. On the other hand, in general, one drawback of methods that use exact projections is having to solve a quadratic problem at each stage, which may substantially increasing the cost per iteration if the number of unknowns is large. In fact, it may not be justified to compute exact projection when the current iterate $x_k$ is far from the solution of the problem in consideration. Moreover, a procedure for computing a feasible inexact projection may present a low computation cost per iteration in comparison with one that computes the exact projection. Thus, it seems reasonable to consider versions of projected subgradient method that compute the projection only approximately. In order to present formally and analyze the Subgradient-InexP method, we use the concept of feasible inexact projection with relative error, which was appeared in [@VillaSalzBaldassarre2013] (see also[@OrizonFabianaGilson2018]). It is worth noting that the concept of feasible inexact projection also accepts an exact projection when it is easy to obtain. For instance, the exact projections onto a box or a second order cone is very easy to obtain; see, respectively, [@NocedalWright2006 p. 520] and [@FukushimaTseng2002 Proposition 3.3]. A feasible inexact projection onto a polyhedral closed convex set can be obtained using quadratic programming methods that generate feasible iterates, such as feasible active set methods and interior point methods; see, for example, [@NicholasPhilippe2002; @NocedalWright2006; @Robert1996]. It is worth mentioning that, if the exact projection is used, then Subgradient-InexP method becomes the projected subgradient method considered in [@AlberIusemSolodov1998]. Several methods similar to the projected subgradient method have been studied in different papers, see [@GoffinKiwiel1999; @Mainge2008]. However, as far as we know, none of them use the concept of feasible inexact projection.
The main tool used in our analysis of Subgradient-InexP method is a version of the inequality obtained in [@CorreaLemarecha1993 Lemma 1.1]; see also a variant of it in [@nedic_bertsekas2001 Lemma 2.1]. By using this inequality, we establish asymptotic convergence results and iteration-complexity bounds for the sequence generated by our method employing the well known exogenous stepsizes, Polyak’s stepsizes, and dynamic stepsizes. We point out that these stepsizes have been discussed extensively in the related literature, including [@AlberIusemSolodov1998; @GoffinKiwiel1999; @nedic_bertsekas2001rate; @nedic_bertsekas2001; @NedicBertsekas2010; @xmwang2018], where many of our results were inspired. Let us describe the results in the present and their relationship with the literature on the subject. With respect to the exogenous stepsize we establish convergence results without any compactness assumption, existence of a solution, and the iteration-complexity bound, which are similar to the well known bound presented in [@AlberIusemSolodov1998; @nedic_bertsekas2001]. In particular, for $C = \mathbb{R}^n$, the convergence results merge into the ones presented in [@CorreaLemarecha1993] and the iteration-complexity bound into [@Nesterov2004 Theorem 3.2.2]. The asymptotic convergence result and the iteration-complexity bound obtained using Polyak’s stepsizes are similar to the corespondent ones in [@nedic_bertsekas2001; @Nesterov2014; @Polyak1969] and [@Nesterov2014], respectively. Regarding to the dynamic stepsize, we establish global convergence in objective values as address, for example, in [@GoffinKiwiel1999; @nedic_bertsekas2001]. In [@nedic_bertsekas2001rate Proposition 2.15], the authors presented the rate of convergence for another variant of subgradient method, known as incremental subgradient algorithms. This study allowed us to estimate an iteraction-complexity bound for the dynamic stepsize.
The organization of the paper is as follows. In Section \[sec:int.1\], we present some notation and basic results used in our presentation. In Section \[Sec:SubInexProj\] we describe the Subgradient-InexP method with different choices for the stepsize. The main results of the present paper, including the converge theorems and iteration-complexity, are presented in Section \[Sec:aca\]. Some numerical experiments are provided in Section \[Sec:NumExp\]. We conclude the paper with some remarks in Section \[Sec:Conclusions\].
Notation and definitions {#sec:int.1}
========================
In this section, we present some notations, definitions, and results used throughout the paper. We are interested in $$\label{eq:OptP}
\min \{ f(x) :~ x\in C\},$$ where $C$ is a closed and convex subset of $\mathbb{R}^n$, $f:\mathbb{R}^n \to \mathbb{R}$ is a convex function. We denote by $$\label{eq:ValueOpt}
f^*:= \inf_{x\in C} f(x),$$ its infimal value (possibly $-\infty$) and by $\Omega^*$ its solution set (possibly $\Omega^*= \varnothing$). The next concept will be useful in the analysis of the sequence generated by the subgradient method to solve .
\[def:QuasiFejer\] A sequence $\{y_k\}\subset \mathbb{R}^n$ is said to be quasi-Fejér convergent to a nonempty set $W\subset \mathbb{R}^n$ if, for every $w\in W$, there exists a sequence $\{\delta_k\}\subset\mathbb{R}$ such that $\delta_k\geq 0$, $\sum_{k=1}^{\infty}\delta_k<+\infty$, and $$\|y_{k+1}-w\|^2\leq \|y_k-w\|^2+\delta_k, \qquad \forall~k=0, 1, \ldots.$$ When, $\delta_k= 0$, for all $k=0, 1, \ldots.$, $\{y_k\}$ is called Fejér convergent to a set $W$.
The main property of the quasi-Fejér convergent sequence is stated in the next result, and its proof can be found in [@burachik1995full].
\[teo.qf\] Let $\{y_k\}$ be a sequence in $\mathbb{R}^n$. If $\{y_k\}$ is quasi-Fejér convergent to a nomempty set $W\subset \mathbb{R}^n$, then $\{y_k\}$ is bounded. If furthermore, a cluster point $y$ of $\{y_k\}$ belongs to $W$, then $\lim_{k\rightarrow\infty}y_k=y$.
To describe the method for solving the problem we need to define, for each $\epsilon \geq 0$, the $\epsilon$-subdifferential $\partial_{\epsilon} f(x)$ of a convex function $f$ at $x\in {\mathbb R}^n$, $$\label{eq:e-subdif}
\partial_{\epsilon} f(x):=\{ s\in {\mathbb R}^n:~f(y)\geq f(x)+\langle s, y-x\rangle -\epsilon, ~\forall y\in {\mathbb R}^n\}.$$ We end this section by presenting important properties of the set $\epsilon$-subdifferential of a convex function, which proofs follow by combining [@Bertsekas2003 Proposition 4.3.1(a)] and [@UrrutyLemarechal1993_II Proposition 4.1.1, Proposition 4.1.2].
\[pr:CompE-subdif\] Let $f:\mathbb{R}^n \to \mathbb{R}$ be a convex function and $\epsilon \geq 0$. The set $\partial_{\epsilon} f(x)$ is nonempty, convex, and compact. Moreover, if $B\subset \mathbb{R}^n$ is a bounded set, then there exists a real number $L>0$ such that $\|s\|<L$, for all $s\in \cup_{x\in B} \partial_{\epsilon} f(x)$. In addition, if $\{\epsilon_k\}$ is a bounded sequence of nonnegative real numbers, the sequence $\{x_k\}$ converges to $x \in \mathbb{R}^n$, and $s_k \in \partial_{\epsilon_k} f(x_k)$ for all $k$, then the sequence $\{s_k\}$ is bounded.
Subgradient-InexP method {#Sec:SubInexProj}
========================
Next, we present the subgradient method with a feasible inexact projections, which will be called [*Subgradient-InexP method*]{}. We begin by presenting the concept of relative feasible inexact projection, which is a variation of those presented in [@OrizonFabianaGilson2018; @VillaSalzBaldassarre2013].
\[def:InexactProj\] Let $C\subset {\mathbb R}^n$ be a closed convex set and $\varphi_{\gamma, \theta, \lambda}: {\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}^n \to {\mathbb R}_{+}$ be a relative error tolerance function such that $$\label{eq:fphi}
\varphi_{\gamma, \theta, \lambda}(u, v, w)\leq \gamma \|v-u\|^2 + \theta \|w-v\|^2 + \lambda \|w-u\|^2, \qquad \forall~ u, v, w \in \mathbb{R}^n,$$ where $ \gamma, \theta, \lambda \geq 0$ are given forcing parameters. The [*feasible inexact projection mapping*]{} relative to $u \in C$ with relative error tolerance function $\varphi_{\gamma, \theta, \lambda}$, denoted by ${\cal P}_C(\varphi_{\gamma, \theta, \lambda},u, \cdot): {\mathbb R}^n \rightrightarrows C$ is the set-valued mapping defined as follows $$\label{eq:ProjI}
{\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v) := \left\{w\in C:~\left\langle v-w, z-w \right\rangle \leq \varphi_{\gamma, \theta, \lambda}(u, v, w), \quad \forall~ z \in C \right\}.$$ Each point $w\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ is called a [*feasible inexact projection of $v$ onto $C$ relative to $u$ and with relative error tolerance function $\varphi_{\gamma, \theta, \lambda}$*]{}.
In the following, we present some remarks about the definition of the feasible inexact projection mapping onto the convex set $C$.
\[rem: welldef\] Let $C\subset {\mathbb R}^n$, $u\in C$ and $\varphi_{\gamma, \theta, \lambda}$ be as in Definition \[def:InexactProj\]. Therefore, for all $v\in {\mathbb R}^n$, it follows from that ${\cal P}_C(0, u, v)$ is the exact projection of $u$ onto $C$; see [@Bertsekas1999 Proposition 2.1.3, p. 201]. Moreover, ${\cal P}_C(0, u, v) \in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ concluding that ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)\neq \varnothing$, for all $u\in C$ and $v\in {\mathbb R}^n$. Consequently, the set-valued mapping ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, \cdot) $ is well-defined.
Next lemma is a variation of [@Reiner_Orizon_Leandro2019 Lemma 6]. It will play an important role in the remainder of this paper.
\[pr:cond\] Let $v \in {\mathbb R}^n$, $u \in C$, $\gamma, \theta, \lambda \geq 0$ and $w\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$. Then, there holds $$\|w-x\|^2 \leq \|v-x\|^2 + \frac{2\gamma+2\lambda}{1-2\lambda}\|v-u\|^2, \qquad \forall ~x \in C,$$ for all $ \lambda, \theta \in [0, 1/2)$.
Let $x \in C$. First note that $\|w-x\|^2 = \|v-x\|^2 - \|w-v\|^2 + 2 \langle v-w, x-w \rangle$. Since $w \in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ and $0\leq \theta < 1/2$, combining the last equality with and we obtain $$\begin{aligned}
\|w-x\|^2 &\leq \|v-x\|^2 - (1-2\theta)\|v-w\|^2 + 2\gamma \|v-u\|^2 + 2\lambda \|w-u\|^2\notag\\
&\leq \|v-x\|^2 + 2\gamma \|v-u\|^2 + 2\lambda \|w-u\|^2 \label{eq:fg}.
\end{aligned}$$ On the other hand, we also have $$\begin{aligned}
\|w-u\|^2 &= \|v-u\|^2 + \|w-v\|^2 + 2 \langle v-w,u-v \rangle \\
&= \|v-u\|^2 + \|w-v\|^2 + 2 \langle v-w,u-w \rangle - 2 \|w-v\|^2 \\
&= \|v-u\|^2 - \|w-v\|^2 + 2 \langle v-w,u-w \rangle.
\end{aligned}$$ Thus, due to $w\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ and $u \in C$, using , , $0\leq \theta < 1/2$ and $0 \leq \lambda <1/2$, we have $$\|w-u\|^2 \leq \frac{1+2\gamma}{1-2\lambda}\|v-u\|^2 - \frac{1-2\theta}{1-2\lambda} \|w-v\|^2\leq \frac{1+2\gamma}{1-2\lambda}\|v-u\|^2.$$ Therefore, combining the last inequality with , we obtain the desired inequality.
The conceptual subgradient method with feasible inexact projections for solving the Problem is formally defined as follows:\
\[H\]
Step 0.
: Let $\{\epsilon_k\}$, $\{\theta_k\}$, and $\{\lambda_k\}$ be sequences of nonnegative real numbers. Let $x_0\in C$ and set $k=0$.
Step 1.
: If $0\in \partial f(x_k)$, then [**stop**]{}. Otherwise, choose a non-null element $s_k \in \partial_{\epsilon_k} f(x_k)$, compute a stepsize $t_k>0$, (to be specified later), and take the next iterate as any point such that $$x_{k+1} \in {\cal P}_C\left(\varphi_{ \gamma_k, \theta_k, \lambda_k}, x_k, x_k-t_ks_k\right).$$
Step 2.
: Set $k\gets k+1$, and go to **Step 1**.
Let us describe the main features of the subgradient-InexP method. Firstly, we check if the current iterate $x_k$ is a solution of Problem . If $x_k$ is not a a solution, then we choose a non-null element $s_k \in \partial_{\epsilon_k} f(x_k)$, compute a stepsize $t_k>0$, and take the next iterate $x_{k+1}\in C$ as any feasible inexact projection of $x_k-t_ks_k$ onto $C$ relative to $x_{k}$ with error tolerance given by $\varphi_{ \gamma_k, \theta_k, \lambda_k}(x_k, x_k-t_ks_k,x_{k+1}) $, i.e., $x_{k+1} \in {\cal P}_C\left(\varphi_{ \gamma_k, \theta_k, \lambda_k}, x_k, x_k-t_ks_k\right)$. We remark that if $\varphi_{ \gamma_k, \theta_k, \lambda_k}\equiv~0$, then ${\cal P}_C\left(0, x_k, x_k-t_ks_k\right)$ is the exact projection of $x_k-t_ks_k$ onto $C$, and Algorithm \[Alg:INP\] amounts to the projected subgradient method studied in [@AlberIusemSolodov1998]. Among the several possible choices that appeared in literature on subject, see for example [@Bertsekas1999; @nedic_bertsekas2001; @Shor1985], we studied three well known strategies, beginning with exogenous stepsize.
\[Exogenous.Step\] Let $ \mu\geq 0$. Take exogenous sequences $\{\alpha_k\}$ and $\{\epsilon_k\}$ of nonnegative real numbers satisfying the following conditions: the sequence $\{\epsilon_k\}$ is nonincreasing and $$\label{eq:ExogSeq}
\sum_{k=0}^{\infty}\alpha_k=+\infty, \qquad \qquad \sum_{k=0}^{\infty}\alpha_k^2<+\infty, \qquad \qquad \epsilon_k\leq \mu \alpha_ k, \qquad \qquad ~k=0, 1, \ldots.$$ Given $s_k \in \partial_{\epsilon_k} f(x_k)$, define the stepsize $t_k $ as the following nonnegative real number $$\label{eq:StepSize1}
t_k:=\frac{\alpha_k}{\eta_k}, \qquad \qquad \eta_k:= \max\left\{1, \| s_k\|\right\}, \qquad \qquad ~k=0, 1, \ldots.$$
The stepsize in Rule \[Exogenous.Step\] is one the most popular. It have been used in several paper for analyzing subgradient method; see for example, [@AlberIusemSolodov1998; @CorreaLemarecha1993; @nedic_bertsekas2001rate; @nedic_bertsekas2001; @xmwang2018].
[*From now on we assume that there exist $0\leq \bar{\theta} < 1/2$ and $0\leq \bar{\lambda} < 1/2$, such that $\{\theta_k\}\subset [0, {\bar \theta})$, $\{\gamma_k\}\subset [0, {\bar \gamma})$ and $\{\lambda_k\}\subset [0, {\bar \lambda})$. For future references define*]{}
$$\label{eq:nu}
\nu := \frac{1+2{\bar \gamma}}{1-2{\bar \lambda}}> 0.$$
To define the next stepsize, we need to known the optimum value $f^*$ given in . In [@PolyakBook p.142], is present some examples of problems for which the optimum value are known. The statement of the Polyak’s stepsize is as follows.
\[Poliak.Step\] Assume that $\Omega^*\neq\varnothing$ and the optimal value $f^*>-\infty$ is known. Let $\mu\geq 0$, $ \underline{\beta} >0$, ${\bar \beta} >0$ and take exogenous sequences $\{\beta_k\}$ and $\{\epsilon_k\}$ of real numbers satisfying the following conditions: the sequence $\{\epsilon_k\}$ is nonincreasing and
$$\label{beta}
0< \underline{\beta} \leq \beta_k \leq \bar{\beta} < \frac{1}{2\mu +\nu}, \qquad\qquad 0<\epsilon_k\leq \mu \beta_ k[f(x_k)-f^*], \quad \qquad ~k=0, 1, \ldots.$$
Given $s_k \in \partial_{\epsilon_k} f(x_k)$, $s_k\neq 0$, define the stepsize $t_k $ as the following nonnegative real number $$\label{StepsizePolyak}
t_k=\beta_k\frac{f(x_k)-f^*}{\left\|s_k\right\|^2}, \qquad \qquad ~k=0, 1, \ldots.$$
The stepsize in Rule \[Poliak.Step\] was introduced in [@Polyak1969] and has been used in several papers, including the ones [@nedic_bertsekas2001rate; @nedic_bertsekas2001; @xmwang2018]. In general, in practical problems the optimum value is not known. In this case, we may modify the stepsize by replacing the optimum value with a suitable estimate in each iteration. This leads to the dynamic stepsize rule as follows.
\[Dynamic.Step\] Let $\mu\geq 0$, $ \underline{\beta} >0$, ${\bar\beta}>0$, and take exogenous sequences $\{\beta_k\}$ and $\{\epsilon_k\}$ of real numbers satisfying the following conditions: the sequence $\{\epsilon_k\}$ is nonincreasing and $$\label{betaDinamic}
0< \underline{\beta} \leq \beta_k \leq \bar{\beta} < \frac{2}{2\mu+\nu}, \qquad \qquad 0<\epsilon_k\leq \mu \beta_ k[f(x_k)-f_{k}^{lev}], \quad \qquad ~k=0, 1, \ldots,$$ where $f_{k}^{lev}$ will be specified later (see Section \[Sec:Analyisdynamic\]). Given $s_k \in \partial_{\epsilon_k} f(x_k)$ such that $s_k\neq 0$, define the stepsize $t_k $ as the following nonnegative real number $$\label{eq.dynamicstep}
t_k= \frac{\tilde{t}_k}{\left\|s_k\right\|}, \qquad\qquad \tilde{t}_k = \beta_k\frac{f(x_k)-f_{k}^{lev}}{\left\|s_k\right\|}, \qquad\qquad ~k=0, 1, \ldots.$$
The dynamic stepsize in Rule \[Dynamic.Step\] is based on the ideas of [@brannlund1993]; see also [@GoffinKiwiel1999]. This rule has been used in several papers, see for example [@nedic_bertsekas2001; @NedicBertsekas2010; @xmwang2018].
[*From now on we assume that the sequence $\{x_k\}$ is generated by Algorithm \[Alg:INP\], with one of the three above strategies for choosing the stepsize, is infinite.*]{}
Analysis of the subgradient-InexP method {#Sec:aca}
========================================
In the following, we state and prove our first result to analyze the sequence $\{x_k\}$ generated by Algorithm \[Alg:INP\]. The obtained inequality in next lemma is its counterpart for unconstrained optimization provided in [@CorreaLemarecha1993 Lemma 1.1)]. As we shall see, this inequality will be the main tool in our asymptotic convergence analysis, as well as in the iteration-complexity analysis.
\[Le:FejerConv\] Let $\nu>0$ be as defined . For all $x \in C$, the following inequality holds $$\label{eq:MainIneq}
\|x_{k+1}-x\|^2 \leq \|x_k-x\|^2 + \nu t_k^2\|s_k\|^2 - 2 t_k \left[f(x_k) - f(x) -\epsilon_k\right], \qquad k=0, 1, \ldots.$$
Let $x \in C$. To simply the notations we set $z_k:= x_k-t_ks_k$. Due to $x_{k+1} \in {\cal P}_C\left(\varphi_{\gamma_k, \theta_k, \lambda_k}, x_k, z_k\right)$ and $x_k \in C$, we apply Lemma \[pr:cond\] with $w=x_{k+1}$, $v=z_k$, $u=x_k$, and $\varphi_{ \gamma, \theta, \lambda}=\varphi_{ \gamma_k, \theta_k, \lambda_k}$ to conclude $$\label{eq:mip}
\|x_{k+1}-x\|^2 \leq \|z_k-x\|^2+ \frac{2\gamma_k+2\lambda_k}{1-2\lambda_k}t_k ^2\|s_k\| ^2.$$ On the other hand, due to $z_k= x_k-t_ks_k$, after some algebraic manipulations, we obtain $$\|z_k-x\|^2 = \|x_k-x\|^2 + t_k ^2\|s_k\| ^2 + 2t_k \langle s_k, x - x_k \rangle.$$ Since $s_k \in \partial_{\epsilon_k} f(x_k)$, the definition implies that $\langle s_{k}, z-x_k\rangle\leq f(z)- f(x_k)+\epsilon_k$. Thus, $$\|z_k-x\|^2 \leq \|x_k-x\|^2 + t_k ^2\|s_k\| ^2 + 2t_k\left[f(x) - f(x_k) + \epsilon_k\right].$$ Therefore, combining last inequality with we conclude that $$\begin{aligned}
\|x_{k+1}-x\|^2 &\leq \|x_k-x\|^2 + t_k ^2\|s_k\|^2 + 2t_k\left[f(x) - f(x_k) + \epsilon_k\right] + \frac{2\gamma_k+2\lambda_k}{1-2\lambda_k}t_k^2 \|s_k\|^2\\
&= \|x_k-x\|^2 + \frac{1+2\gamma_k}{1-2\lambda_k}t_k^2 \|s_k\|^2- 2 t_k \left[f(x_k) - f(x) -\epsilon_k\right].
\end{aligned}$$ Considering that $0 \leq \lambda_k < \bar{\lambda}<1/2$ and $0 \leq \gamma_k < \bar{\gamma}$, and using , we obtain .
Analysis of the subgradient-InexP method with exogenous stepsize {#Sec:AnalyisExog}
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In this section we will analyze the subgradient-InexP method with stepsizes satisfying Rule \[Exogenous.Step\]. For that, [*throughout this section we assume also that $\{x_k\}$ is a sequence generated by Algorithm \[Alg:INP\] with the stepsize given by Rule \[Exogenous.Step\] and, define $$\label{eq:rho}
\rho:= \nu + 2\mu > 0.$$* ]{} First of all, note that under the above assumptions, Lemma \[Le:FejerConv\] becomes as follows.
\[Le:FejerConvExog\] Let $\rho>0$ be as in . For all $x\in C$, the following inequality holds $$\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 + \rho \alpha_k^2 - 2 \frac{\alpha_k}{\eta_k} \left[f(x_k)-f(x)\right], \qquad k=0, 1, \ldots.$$
The definition of $t_k$ in implies $t_k\leq \alpha_k$, which combined with the last inequality in yields $2t_k\epsilon_k\leq 2\mu \alpha_k^2$. Moreover, also implies that $t_k^2\|s_k\|^2\leq \alpha_k^2$. Therefore, using , the desired inequality follows directly from .
To proceed with the analysis of Algorithm \[Alg:INP\], we also need the following auxiliary set $$\label{eq:DefOmega}
\Omega:=\left\{x\in C:~f(x)\leq \inf_kf(x_k)\right\}.$$ It is worth mentioning that, in principle, set $\Omega$ can be empty and, in such case, $f^*=-\infty$. In the next lemma we analyze the behavior of the sequence $\{x_k\}$ under the hypothesis that $\Omega \neq \varnothing$.
\[Le:BoundExog\] If $\Omega \neq \varnothing$, then $\{x_k\}$ is quasi-Féjer convergent to $\Omega$. Consequently, $\{x_k\}$ is bounded.
Since $\Omega \neq \varnothing$, take $x\in \Omega$. Thus, by using the definition of $\Omega$ in and Lemma \[Le:FejerConvExog\], we conclude that $\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 + \rho \alpha_k^2$, for all $k=0, 1, \ldots$. Hence, using the first inequality in , the first statement of the lemma follows from Definition \[def:QuasiFejer\]. The second statement of the lemma follows from the first part of Theorem \[teo.qf\].
Now, we are ready to prove the main result of this section, which refers to the asymptotic convergence of $\{x_k\}$. We remark that in the first part of the next theorem we do not assume neither $\Omega^* \neq \varnothing$ nor that $f^*$ is finite.
\[teo.Main\] The following equality holds
$$\label{eq:linfs}
\liminf_k f(x_k)=f^*.$$
In addition, if $\Omega^* \neq \varnothing$ then the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$.
Assume by contradiction that $\liminf_k f(x_k) > f^*$. In this case, we have $\Omega \neq \varnothing$. Consequently by Lemma \[Le:BoundExog\], we conclude that $\{x_k\}$ is bounded. Letting $x\in \Omega$, there exist $\tau>0$ and $k_0\in\mathbb{N}$ such that $f(x)<f(x_k)-\tau,$ for all $k\geq k_0$. Hence, using Lemma \[Le:FejerConvExog\], we have $$\label{eq:bXk}
\|x_{k+1}-x\|^2 \leq \|x_{k}-x\|^2 + \rho \alpha_k^2- 2 \frac{\alpha_ k}{\eta_k} \tau, \qquad k=k_0, k_0+1, \dots.$$ On the other hand, it follows from that the sequence $\left\lbrace \epsilon_k \right\rbrace$ is bounded. Thus, considering that $\{x_k\}$ is bounded, Proposition \[pr:CompE-subdif\] implies that $\{s_k\}$ is also bounded. Let $c > 0$ be such that $\|s_k\| \leq c$, for all $k \geq 0$. Hence, using second equality in , we have $\eta_k = \max\left\{1, \| s_k\|\right\} \leq \max\left\{1, c \right\} =: \varGamma$. Thus, letting $\ell \in \mathbb{N}$ and using , we conclude that $$\frac{2\tau}{\varGamma}\sum_{j=k_0}^{\ell+k_0}\alpha_j \leq \|x_{k_0}-x\|^2 - \|x_{k_0+ \ell+1}-x\| + \rho \sum_{j=k_0}^{\ell+k_0}\alpha^2_j \leq \|x_{k_0}-x\|^2 + \rho \sum_{j=k_0}^{\ell+k_0}\alpha^2_j.$$ Since the last inequality holds for all $\ell \in \mathbb{N}$ then, by using the first two conditions on $\{\alpha_k\}$ in , we have a contraction. Therefore, holds. For proving the last statement, let us assume that $\Omega^*\neq\varnothing$. In this case, we also have $\Omega\neq\varnothing$ and, from Lemma \[Le:BoundExog\], the sequence $\{x_k\}$ is bounded and quasi-Féjer convergent to $\Omega$. The equality implies that $\{f(x_k)\}$ has a decreasing monotonous subsequence $\{f(x_{k_j})\}$ such that $\lim_{j\rightarrow \infty}f(x_{k_j})= f^*.$ Without lose of generality, we can assume that $\{f(x_k)\}$ is decreasing, is monotonous, and converges to $f^*$. Being bounded, the sequence $\{x_k\}$ has a convergent subsequence $\{x_{k_\ell}\}$. Let us say that $\lim_{\ell\rightarrow\infty}x_{k_\ell}=x_*,$ which by the continuity of $f$ implies $f(x_*)=\lim_{\ell\rightarrow\infty}f(x_{k_\ell})=f^*,$ and then $x_*\in\Omega^*$. Hence, $\{x_k\}$ has an cluster point $x_*\in\Omega$, and due to $\{x_k\}$ be quasi-Féjer convergent to $\Omega$, Theorem \[teo.qf\] implies that $\{x_k\}$ converges to $x_*$.
Next theorem presents an iteration-complexity bound; similar bound can be found in [@Nesterov2004 Theorem 3.2.2].
\[teo:complrule1\] Assume that the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$. Then, for every $N \in \mathbb{N}$, the following inequality holds $$\min \{f(x_k) - f^{*}:~ \, k = 0, 1,\ldots, N \} \leq \varGamma \frac{\|x_0 - x_{*}\|^2 + \rho\sum_{k=0}^{N}\alpha_k^{2}}{2\sum_{k=0}^{N}\alpha_k}.$$
Since $\left\lbrace \epsilon_k \right\rbrace $ and $\left\lbrace x_k\right\rbrace $ are bounded sequences, then using Proposition \[pr:CompE-subdif\], it follows that $\left\lbrace s_k\right\rbrace$ is also bounded, i.e. there exists $c > 0$ such that $\|s_k\| \leq c$, for all $k \geq 0$. Therefore, using the definition of $\eta_k$ in , we have $\eta_k = \max\left\{1, \| s_k\|\right\} \leq \max\left\{1, c \right\} =: \varGamma$. Now, applying Lemma \[Le:FejerConvExog\] with $x = x_*$ and due to $f^* = f(x_*)$, we obtain $$\frac{2 \alpha_k}{\varGamma} [f(x_k)-f^*] \leq \|x_k-x_*\|^2 -\|x_{k+1}-x_*\|^2+ \rho\alpha_k^2 , \qquad k = 0,1, \ldots.$$ Hence, performing the sum of the above inequality for $k = 0, 1, \ldots, N,$ we have $$\frac{2}{\varGamma} \sum_{k=0}^{N} \alpha_k [f(x_k)-f^*] \leq \|x_0-x_*\|^2-\|x_{N+1}-x_*\|^2+\rho\sum_{k=0}^{N}\alpha_k^2.$$ Therefore, $$\frac{2}{\varGamma} \, \min \left\{f(x_k) - f^{*}: ~ \, k = 0, 1,\ldots, N \right\} \sum_{k=0}^{N}\alpha_k \leq \|x_0 - x_{*}\|^2 +\rho\sum_{k=0}^{N}\alpha_k^{2},$$ which is equivalent to the desired inequality.
Analysis of the subgradient-InexP method with Polyak’s stepsize rule {#Sec:AnalysisPolyak}
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In this section we will analyze the subgradient-InexP method with Polyak’s step sizes. [*Throughout this section, we assume also that $\Omega^* \neq \varnothing$ and $\{x_k\}$ is a sequence generated by Algorithm \[Alg:INP\] with the stepsize given by Rule \[Poliak.Step\]*]{}.
\[Le:FejerConvPolyak\] Let $x\in \Omega^*$. Then, the following inequality holds $$\label{eq:MainIneqPolyak}
\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 - \underline{\beta} \frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2}, \qquad\qquad k=0, 1, \ldots.$$
Considering that $x\in \Omega^*$ we have $f^*= f(x)$. The combination of with implies $2t_k\epsilon_k\leq 2\mu \beta_k^2 [f(x_k)-f^*]^2/\|s_k\|^2$. Moreover, also implies that $t_k^2\|s_k\|^2 = \beta_k^2[f(x_k)-f^*]^2/\|s_k\|^2$. Thus, we conclude form that $$\label{eq:mdpss}
\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 -\left(2-\nu\beta_k-2\mu \beta_k\right)\beta_k\frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2}, \qquad k=0, 1, \ldots.$$ On the other hand, gives us $\beta_k<1/(2\mu+\nu)$, which is equivalent to $2-\nu\beta_k-2\mu \beta_k>1$. Therefore, since also gives $\underline{\beta}\leq \beta_k$, we conclude that implies .
In the following theorem we present our main result about the asymptotic convergence of $\{x_k\}$. It has as correspondent result in [@Polyak1969 Theorem 1]; see also [@nedic_bertsekas2001].
\[teo.MainConvPolyak\] The sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$.
Let $x\in \Omega^*$. Then, Lemma \[Le:FejerConvPolyak\] implies $\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 $, for all $k=0, 1, \ldots$. Thus, $\{x_k\}$ is Fejér convergent to $\Omega^*$. Since $\Omega^* \neq \varnothing$, Theorem \[teo.qf\] implies that $\{x_k\}$ is bounded. By using Proposition \[pr:CompE-subdif\], we conclude that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for $k=0,1,\ldots$. Then, from , after some algebra, we have $$\left[f(x_k)-f^*\right]^2 \leq \frac{c^2}{\underline{\beta}}\left(\|x_k-x\|^2 - \|x_{k+1}-x\|^2\right), \qquad \qquad k=0, 1, \ldots.$$ Thus, performing the sum of the this inequality for $j=0, 1, \ldots, \ell$, we obtain $$\sum_{j=0}^{\ell}\left[f(x_j)-f^*\right]^2 \leq \frac{c^2}{\underline{\beta}}\left(\|x_{0}-x\|^2 - \|x_{ \ell+1}-x\|^2\right)\leq \frac{c^2}{\underline{\beta}} \|x_{0}-x\|^2.$$ Considering that this inequality holds for all $\ell\in \mathbb{N}$, we conclude that $\lim_{k\to +\infty}f(x_k)=f^*$. Let $x_*$ be a cluster point of $\{x_k\}$ and $\{x_{k_j}\}$ a subsequence of $\{x_k\}$ such that $\lim_{j\to +\infty}x_{k_j}=x_*$. Since $f$ is continuous, we have $f(x_*)= \lim_{j\to +\infty}f(x_{k_j})=f^*$. Therefore, $x_*\in\Omega^*$. Since $\{x_k\}$ is quasi-Fejér convergent to a set $\Omega^*$, it follows from Theorem \[teo.qf\] that $\{x_k\}$ converges $x_*$.
The next result presents an iteration-complexity bound, which is a version of [@Nesterov2014 Theorem 1].
Assume that the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$. Then, for every $N \in \mathbb{N}$, the following inequality holds $$\min \{f(x_k) - f^{*}: ~ \, k = 0, 1,\ldots, N \} \leq \frac{c}{\sqrt{\underline{\beta} (N+1)}} \|x_0 - x_{*}\|,$$ where $c\geq \max\{\|s_k\|:~ k=0, 1,\ldots \}$.
Applying Lemma \[Le:FejerConvPolyak\] with $x = x_*$ , where $f^* = f(x_*)$, we obtain $$\underline{\beta} \frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2} \leq \|x_k-x_*\|^2 - \|x_{k+1}-x_*\|^2, \qquad\qquad k=0, 1, \ldots.$$ Performing the sum of the above inequality for $k = 0, 1, \ldots, N,$ we conclude that $$\sum_{k=0}^{N} \frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2} \leq \frac{1}{\underline{\beta}} \|x_0-x_*\|^2.$$ Since $\{x_k\}$ is bounded, by using Proposition \[pr:CompE-subdif\], we conclude that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for $k=0,1,\ldots$. Thus, we have $$\sum_{k=0}^{N} \left[f(x_k)-f^*\right]^2 \leq \frac{c^2}{\underline{\beta}} \|x_0-x_*\|^2.$$ Therefore, $$(N+1) \min \{\left[f(x_k)-f^*\right]^2:~ \, k = 0, 1,\ldots, N \} \leq \frac{c^2}{\underline{\beta}} \|x_0-x_*\|^2,$$ which is equivalent to the desired inequality.
Analysis of the subgradient-InexP method with dynamic stepsize {#Sec:Analyisdynamic}
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Next we consider the Subgradient-InexP method employing the dynamic stepsize Rule \[Dynamic.Step\], which guarantees that $\{f_{k}^{lev}\} $ converges to the optimum value $f^*$. In the following we present formally the algorithm which compute $f_{k}^{lev}$. This scheme was introduced in [@brannlund1993]; see also [@GoffinKiwiel1999].
Step 0.
: Select $x_0\in C, \delta_0 > 0$, and $R > 0$. Set $k=0, \sigma_0 = 0, f_{-1}^{rec} = \infty, \ell=0, k(\ell) = 0$.
Step 1.
: If $f(x_k) < f_{k-1}^{rec},$ set $f_{k}^{rec} = f(x_k)$ and $x_{k}^{rec}= x_k,$ else set $f_{k}^{rec} = f_{k-1}^{rec}$ and $x_{k}^{rec}=x_{k-1}^{rec}$
Step 2.
: If $0\in \partial f(x_k)$, then [**stop**]{}.
Step 3.
: If $f(x_k) \leq f_{k(\ell)}^{rec} - \frac{1}{2} \delta_\ell $, set $k(\ell+1) = k, \sigma_k = 0, \delta_{\ell+1} = \delta_\ell$, replace $\ell$ by $\ell+1,$ and go to Step 5.
Step 4.
: If $\sigma_k > R$, set $k(\ell+1) = k, \sigma_k = 0, \delta_{\ell+1} = \frac{1}{2} \delta_\ell$, $x_k=x_k^{rec}$, and $\ell\leftarrow\ell+1$.
Step 5.
: Set $f_k^{lev} := f_{k(\ell)}^{rec} - \delta_\ell$. Select $\beta_k \in [\underline{\beta}, \bar{\beta}]$ and calculate $x_{k+1}$ via Algorithm \[Alg:INP\] with the stepsize given by Rule \[Dynamic.Step\].
Step 6.
: Set $\sigma_{k+1}:= \sigma_k + \tilde{t}_k$, $k\leftarrow k+1$, and go to Step 1.
Following [@GoffinKiwiel1999; @nedic_bertsekas2001rate; @nedic_bertsekas2001; @xmwang2018], we describe the main features of the Subgradient-InexP method.
\[eq:bfkrec\] Note that in Step 1, $f_{k}^{rec}$ keeps the record of the smallest functional value attained by the iterates generated so far, i.e., $f_{k}^{rec}:=\min\{f(x_j):~j=0, \ldots,k\}$. Splitting the iterations into groups $$K_\ell := \{k(\ell), k(\ell) + 1, \ldots, k(\ell+1)-1\}, \quad \ell = 0, 1, \dots,$$ Algorithm \[Alg:INPDyn\] uses the same target level $f_k^{lev} = f_{k(\ell)}^{rec} - \delta_\ell$, for $k \in K_\ell$. Also, note that the target level is update only if sufficient descent or oscillation is detected (Step 3 or Step 4, respectively). Whenever $\sigma_k$ exceeds the upper bound $R$, the parameter $\delta_\ell$ is decreased, which increases the target level $f_k^{lev}$.
From now on, we assume that Algorithm \[Alg:INPDyn\] generates an infinite sequence. In the next theorem we present the result about the asymptotic convergence of the sequence $\{x_k\}$. It is the versions of [@GoffinKiwiel1999 Theorem 1] and [@nedic_bertsekas2001 Proposition 2.7] by using inexact projections.
\[teo.MainConvDynamic\] There holds $\inf_{k \geq 0} f(x_k)= f^*$.
Since $x_k \in C$ and $x_{k+1} \in {\cal P}_C\left(\varphi_{\gamma_k, \theta_k, \lambda_k}, x_k, x_k - t_ks_k\right)$, by the first equality in and applying Lemma \[pr:cond\] with $w=x_{k+1}$, $v=x_k - t_ks_k$, $x=x_k$ , $u=x_k$, and $\varphi_{ \gamma, \theta, \lambda}=\varphi_{ \gamma_k, \theta_k, \lambda_k}$, we conclude that $$\|x_{k+1}-x_{k}\| \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\,t_k\|s_k\| \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\, \tilde{t}_k, \qquad k=0, 1, \ldots.$$ We claim that the index $\ell$ goes to $+\infty$ and either $\inf_{k \geq 0} f(x_k) =-\infty$ or $\lim_{l\to\infty} \delta_\ell = 0$. Indeed, assume that $\ell$ takes only a finite number of values, i.e., $\ell < \infty$. Since $\sigma_k + \tilde{t}_k = \sigma_{k+1} \leq R$, for all $k \geq k(\ell)$, then we conclude that $$\label{eq.boundxk}
\|x_k - x_{k(\ell)}\| \leq \sum_{j=k(\ell)}^k \|x_{j+1} - x_j\| \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \sum_{j=k(\ell)}^k \tilde{t}_j= \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\, \sigma_{k+1} \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\, R.$$ Hence, $\{x_k\}$ is bounded. Besides, from the last condition in , the sequence $\{\epsilon_k\}$ is bounded and, by using Proposition \[pr:CompE-subdif\], $\{s_k\}$ is also bounded. Moreover, by we also conclude $\sum_{j=k(\ell)}^{+\infty} \tilde{t}_j<+\infty$, which implies $\lim_{k\to \infty} \tilde{t}_k = 0$. Thus, due to $\beta_k \in [\underline{\beta}, \bar{\beta}]$, it follows from second equality in that $$\label{eq.boundfk}
\lim_{k \to \infty} [f(x_k) - f_k^{lev}] = 0.$$ On the other hand, Steps 3 and 5 of Algorithm \[Alg:INPDyn\] yield $$f(x_k) > f_{k(\ell)}^{rec} - \frac{1}{2} \delta_\ell = f_k^{lev} + \delta_\ell - \frac{1}{2} \delta_\ell = f_k^{lev} + \frac{1}{2} \delta_\ell \qquad \quad k= k(\ell), k(\ell)+1, \ldots,$$ contradicting . Therefore, $\ell$ goes to $+ \infty$. Now, suppose that $ \lim_{\ell\to\infty} \delta_\ell=\delta > 0$. Then, from Steps 3 and 4 of Algorithm \[Alg:INPDyn\], it follows that for all $\ell$ large enough, we have $\delta_\ell = \delta$ and $
f_{k(\ell+1)}^{rec} \leq f_{k(\ell)}^{rec} -\frac{1}{2} \delta,
$ implying that $\displaystyle\inf_{k \geq 0} f(x_k) = -\infty$, which concludes the claim. If $\lim_{\ell\to\infty} \delta_\ell > 0$ then, according to above claim, we have $\inf_{k \geq 0} f(x_k) = -\infty$, obtain the desired result. Now, we assume by contradiction that $\lim_{\ell\to\infty} \delta_\ell = 0$ and $\inf_{k \geq 0} f(x_k) > f^*$. Thus, it follows from Remark \[eq:bfkrec\] that $\inf_{k \geq 0} f_{k}^{rec} =\inf_{k \geq 0} f(x_k) $. Hence, we conclude that $\inf_{k \geq 0} f_{k}^{rec}> f^*$. In this case, by using the definition of $\{f_k^{lev}\}$ in Step 5 and taking into account that $\lim_{\ell\to\infty} \delta_\ell = 0$, we conclude that $$\displaystyle\inf_{k \geq 0} f_k^{lev} =\displaystyle\inf_{\ell \geq 0} (f_{k(\ell)}^{rec} - \delta_\ell) = \displaystyle\inf_{\ell \geq 0} f_{k(\ell)}^{rec} > f^*.$$ Therefore, there exist $\bar{\delta}>0$, ${\bar x}\in C$ and $\bar{k} \in \mathbb{N}$ such that $$\label{eq:ahc}
f_k^{lev} - f({\bar x}) \geq \bar{\delta}, \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ Hence, by using the definition of $\tilde{t}_k$ in , it follows from that $$\label{eq:limtildtk}
\tilde{t}_k = \beta_k\frac{f(x_k)-f_{k}^{lev}}{\left\|s_k\right\|} < \bar{\beta} \frac{f(x_k)-f({\bar x})-\bar{\delta}}{\left\|s_k\right\|}, \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ Now, applying Lemma \[Le:FejerConv\] with $x={\bar x}$ and then using and , we obtain $$\|x_{k+1} - {\bar x}\|^2 \leq \|x_k-{\bar x}\|^2+ \tilde{t}_k \left(\nu \tilde{t}_k - \frac{2}{\|s_k\|} \left[f(x_k)-f({\bar x})-\epsilon_k\right] \right), \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ Thus, the combination of the last inequality with , , and the last inequality in yields $$\|x_{k+1} - {\bar x}\|^2 \leq \|x_k-{\bar x}\|^2+ \frac{\tilde{t}_k}{\|s_k\|} \bigg(\left[(2\mu+\nu)\bar{\beta}-2\right]\left[f(x_k)-f({\bar x})\right] - \left( 2\mu+\nu\right)\bar{\beta}\bar{\delta}\bigg), \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ It follows from that $\bar{\beta} < 2/(2\mu+\nu)$, which implies that $(2\mu+\nu)\bar{\beta}-2 < 0$. Thus, by using that $f(x_k)\geq f_k^{lev}$ for all $k=0, 1, \ldots$, the last inequality implies $$\label{eq:xk_fejerdynamic}
\|x_{k+1} - {\bar x}\|^2 \leq \|x_k-{\bar x}\|^2 - \frac{\tilde{t}_k}{\|s_k\|} \left( 2\mu+\nu\right)\bar{\beta}\bar{\delta}, \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ Hence, $\|x_{k+1}-{\bar x}\| \leq \|x_{\bar k}-{\bar x}\|$, for all $k \geq \bar{k}$, which implies that $\{x_k\}$ is bounded. Besides, by using , it follows from the last condition in that the sequence $\left\lbrace \epsilon_k \right\rbrace$ is also bounded. Thus, using Proposition \[pr:CompE-subdif\], we conclude that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for all $k \geq 0$, which together , yield $$\frac{{\bar \beta}\bar{\delta}}{c} \left(2\mu+\nu\right) \displaystyle\sum_{k = \bar{k}}^\infty \tilde{t}_k \leq \|x_{\bar{k}} - \bar{x}\|^2<+\infty.$$ Since $\sigma_{k} = \sum_{j= {k(\ell)}}^{ {k(\ell+1)}-1} \tilde{t}_j $, the last inequality implies that there exists $\ell_0 \in \mathbb{N}$ such that $$\sigma_{{k(\ell+1)}} \leq \displaystyle\sum_{k = {k(\ell)}}^\infty \tilde{t}_k < R, \qquad \quad ~\ell = \ell_0, \ell_0+1\ldots .$$ Hence, Step 4 in Algorithm \[Alg:INPDyn\] cannot occur infinitely to decrease $\delta_\ell$, contradicting the fact that $\displaystyle\lim_{\ell\to\infty} \delta_\ell = 0$. Therefore, the result follows and the proof is concluded.
The next result presents an iteration-complexity bound for the subgradient-InexP method with the stepsize given by Rule \[Dynamic.Step\], which is a version of [@nedic_bertsekas2001rate Proposition 2.15] for our algorithm.
Assume that the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$. Let $ \delta_0>0$ be given in Algorithm \[Alg:INPDyn\] and $c\geq \max\{\|s_k\|:~k=0, 1,\ldots \}$. Then, $$\label{eq:minfxkdynamic}
\min \{f(x_k) - f^{*}:~ \, k = 0, 1,\ldots, N \} \leq \delta_0,$$ where $N$ is the largest positive integer such that $$\label{eq:defNdynamic}
\sum_{k=0}^{N-1}\left( \beta_k \left[2 - (2 \mu+\nu) \beta_k\right]\delta_k^2\right) \leq \left(c\|x_0 - x_*\|\right)^2.$$
Assume by contradiction that does not holds. Thus, for all $k$ with $0 \leq k \leq N$ we have $
f(x_k) > f^* + \delta_0.
$ Hence, considering that $\delta_\ell \leq \delta_0$ for all $\ell$, we have $$\label{eq:fklevfstar}
f_k^{lev} = f_{k(\ell)}^{rec}-\delta_\ell > f^* + \delta_0 - \delta_\ell \geq f^*, \quad \qquad k=0, \ldots, N.$$ The combination of the last inequality in with gives $2t_k\epsilon_k \leq 2\mu \beta_k^2 \left[f(x_k)-f_k^{lev}\right]^2 / \|s_k\|^2$. Moreover, implies that $t_k^2\|s_k\|^2 = \beta_k^2 \left[f(x_k)-f_k^{lev}\right]^2/\|s_k\|^2$. Now, using , Lemma \[Le:FejerConv\] with $x = x_* \in \Omega^*$, and since $\beta_k \in [\underline{\beta}, \bar{\beta}]$, we obtain $$\label{eq:cdinq}
\|x_{k+1}-x_*\|^2 \leq \|x_k-x_*\|^2 - \beta_k \left[2 -(2 \mu+\nu) \beta_k\right]\frac{\left[f(x_k)-f_k^{lev}\right]^2}{\|s_k\|^2}.$$ Since $\{x_k\}$ converges to $x_*\in\Omega^*$, Proposition \[pr:CompE-subdif\] implies that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for $k=0,1,\ldots$. Furthermore, using the fact $f(x_k)-f_k^{lev} \geq \delta_k$, $0 \leq k \leq N$, yields $$\
\|x_{k+1}-x_*\|^2 \leq \|x_k-x_*\|^2 - \beta_k \left[2 -(2 \mu+\nu) \beta_k\right] \frac{\delta_k^2}{c^2}.$$ Performing the sum of the above inequality for $k = 0, 1, \ldots, N,$ we conclude that $$\sum_{k=0}^{N} \left(\beta_k \left[2 -(2 \mu+\nu) \beta_k\right]\frac{\delta_k^2}{c^2}\right) \leq \|x_0 - x_*\|^2,$$ which contradicts .
Numerical results {#Sec:NumExp}
=================
Our intention in this section is to report some numerical results in order to illustrate the practical behavior of SInexPD Algorithm when $C$ is a compact convex set. We implemented SInexPD Algorithm in Fortran 90 considering set $C$ in the general form $C= \left\{x\in\mathbb{R}^n:~ h(x)=0, g(x)\leq 0 \right\}$, where $h: \mathbb{R}^n \to \mathbb{R}^m $ and $g: \mathbb{R}^n \to \mathbb{R}^p$ are smooth functions. At each iteration $k$, the Frank-Wolfe algorithm is used to compute a feasible inexact projection as explained below. The algorithm codes are freely available at <https://orizon.ime.ufg.br/>.
Frank-Wolfe algorithm to find an approximated projection {#Sec:CondG}
--------------------------------------------------------
In this section we use the [*Frank-Wolfe algorithm*]{} also known [*conditional gradient method*]{} to find an inexact projection onto a compact convex set $C\subset \mathbb{R}^n$; papers dealing with this method include [@BeckTeboulle2004; @FrankWolfe1956; @JAGGI2013; @Konnov2018; @LanZhou2016; @Ravi2017]. The exact projection of $v\in \mathbb{R}^n$ onto $C$ is the solution of the following convex quadratic optimization problem $$\label{eq:ProbCond}
{\min}_{w \in C} \psi(w) := \frac{1}{2}\|w-v\|^2.$$ Assume that $v\notin C$. Let us describe the subroutine, which we nominate [*FW-Procedure*]{}, for finding an approximated solution of relative to a point $u \in C$, i.e., a point belonging to the set ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$, where the error tolerance mapping $\varphi_{\gamma, \theta, \lambda}$ and the set valued mapping ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, .)$ are given in Definition \[def:InexactProj\].
Step 0.
: Set $w_1 = u$ and $k=1$.
Step 1.
: Call the linear optimization oracle (or simply LO oracle) to compute $$\label{eq:condG}
z_k := \arg\min_{z \in C} \langle w_k-v, z-w_k \rangle, \qquad g_k^*:= \langle w_k - v, z_k-w_k \rangle.$$
Step 2.
: If $g^*_k \geq - \varphi_{\gamma, \theta, \lambda}(u, v, w_k)$, set $w_{+}:=w_k$ and [**stop**]{}; otherwise, compute $$\label{eq:stepsize}
\tau_k: = \min\left\{1, \frac{-g^*_k}{\|z_k-w_k\|^2} \right\}, \qquad w_{k+1}:=w_k + \tau_k(z_k-w_k).$$
Step 3.
: Set $k \gets k+1$, and go to [**Step 1**]{}.
Since $\psi$ is strictly convex, we conclude from that $\psi(z) > \psi(w_k) + g_k^*$, for all $z \in C$ such that $z\neq w_k$. Setting $\psi^*:=\min_{w \in C} \psi(w)$ we have $\psi(w_k) \geq \psi^* \geq \psi(w_k) + g_k^*$, which implies $g_k^* <0.$ Thus, the stepsize $\tau_k$ given by is computed using exact minimization, i.e., $0<\tau_k := \arg\min_{\tau \in [0,1]} \psi(w_k + \tau(z_k - w_k))$. Since $C$ is convex and $z_k$, $w_k \in C$, we have from that $w_{k+1} \in C$, which implies that all points generated by [*FW-Procedure*]{} are in $C$. Moreover, implies that $g_k^* = \langle w_k - v, z_k - w_k \rangle \leq \langle w_k - v, z - w_k \rangle$, for all $z \in C$. Hence, if the stopping criteria $g_k^* = \langle w_k - v, z_k - w_k \rangle \geq - \varphi_{\gamma, \theta, \lambda}(u, v, w_k)$ in Step 2 of [*FW-Procedure*]{} is satisfied, then $ \langle v-w_k , z - w_k \rangle \leq \varphi_{\gamma, \theta, \lambda}(u, v, w_k)$, for all $z \in C$. Therefore, from Definition \[def:InexactProj\], we conclude that $w_{+}=w_k\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$, i.e., the output of [*FW-Procedure*]{}, is a feasible inexact projection of $v \in \mathbb{R}^n$ relative to $u \in C$. Finally, [@BeckTeboulle2004 Proposition A.2] implies that $\lim_{k\to +\infty} g_k^* =0$. Thus, the stopping criteria $g_k^* \geq - \varphi_{\gamma, \theta, \lambda}(u, v, w_k)$ in Step 2 of [*FW-Procedure*]{} is satisfied in a finite number of iterations if and only if $\varphi_{\gamma, \theta, \lambda}(u, v, w_k)\neq 0$, for all $k=0, 1, \ldots$.
The following theorem is an import result about the convergence rate of the conditional gradient method applied to problem , which its proof can be found in [@GarberHazan2015]. For stating the theorem, we first note that $$\label{eq:PropPsi}
\psi(w) - \psi(w^*) \leq \frac{1}{2} \|w-w^*\|, \qquad \forall z \in C;$$ see also [@Nesterov2004 Theorem 2.1.8].
\[th:fcr\] Let $d_C := \max_{z,w \in C} \|z-w\|$ be the diameter of C. For $k \geq 1$, the iterate $w_k$ of *FW-Procedure* satisfies $\psi(w_k) -\psi(w_*) \leq 8d_{C}^2/k$. Consequently, using , we have $\|w_k - w_*\| \leq 4d_{C}/\sqrt{k}$, for all $k \geq 1$.
Examples
--------
Consider the problem $$\label{numprob}
\min_{x \in C} \, f(x) :=\|x\|_1,$$ where $C:=\left\{x\in\mathbb{R}^n \colon x\geq0 \mbox{ and } (x-\bar{x})^TQ(x-\bar{x})\leq 1 \right\}$ for a given vector $\bar{x}\in{\mathbb{R}}^n$ and a symmetric positive definite matrix $Q\in\mathbb{R}^{n\times n}$. Since the $\ell_1$ norm tends to promote sparse solutions, we formulated instances of Problem where there are vectors in $C$ with only one non-null component. Thus we can verify the ability of SInexPD Algorithm to recover sparsity. Let us describe the main characteristics of the considered instances. Consider the spectral decomposition of $Q$ given by $$Q=\sum_{i=1}^n\lambda_iv^i(v^i)^T,$$ where $\lambda_1 \geq \ldots \geq \lambda_{n-1} > \lambda_n>0$ are the eigenvalues of $Q$ and $\{v_1, v_2,\dots,v_n\}$ is an orthonormal system of corresponding eigenvectors. We assume that there exists $u\in \mathbb{R}^n_{++}$ such that $$\label{eq:id2}
v_n=u/\|u\|, \quad \lambda_ n<1/\|u\|^2, \qquad \mbox{and} \qquad \bar{x}=u+ \xi e_n,$$ where $\xi\geq 1/\sqrt{\lambda_n}$ and $e_n\in{\mathbb{R}}^n$ is such that $e_n=[0,\ldots,0,1]^T$. We claim that $\tilde{x}:= \xi e_n\in C$ and $0\notin C$. Indeed, using we have $\tilde{x}-\bar{x}=-\|u\|v_n$ , which implies $$(\tilde{x}-\bar{x})^TQ (\tilde{x}-\bar{x})= \|u\|^2 v_n^TQv_n= \|u\|^2\lambda_ n<1,$$ concluding that $\tilde{x}\in C$. Now note that $0\in C$ if and only if $\bar{x}^TQ\bar{x} \leq 1$. Since $$\xi\geq \frac{1}{\sqrt{\lambda_n}} > -\langle u, e_n\rangle + \frac{1}{\sqrt{\lambda_n}} >\left(-\langle u, e_n\rangle+\sqrt{\langle u, e_n\rangle^2-(\|u\|^2-1/\lambda_n)} \right)> 0$$ and $\|u+ \xi e_n\|^2=\xi^2+2\langle u, e_n\rangle \xi +\|u\|^2$, we have $$\bar{x}^TQ\bar{x}\geq \lambda_n\|\bar{x}\|^2=\lambda_n\|u+ \xi e_n\|^2>1,$$ implying that $0\notin C$.
For Problem , given $x\in{\mathbb{R}}^n$ we can get $s\in\partial f(x)$ by taking $$[s]_i := \left\{
\begin{array}{rl}
-1, & \mbox{ if } [x]_i < 0 \\
0, & \mbox{ if } [x]_i = 0 \\
1, & \mbox{ if } [x]_i > 0, \\
\end{array}
\right.$$ where $[\cdot]_i$ stands for the $i$-th component of the corresponding vector. For computing the optimal solution $z_k$ at Step 1 of the FW-Procedure, we use the software Algencan [@algencan], an augmented Lagrangian code for general nonlinear optimization programming. We set $R=\|x_1-x_0\|$ and $\delta_0=\|s_0\|/2$ as suggested in [@nedic_bertsekas2001] and [@GoffinKiwiel1999], respectively. Our implementation uses the stopping criterion $$\delta_{\ell}\leq 10^{-3}(1+|f_k^{rec}|),$$ also suggested in [@GoffinKiwiel1999]. Thus, in Algorithm \[Alg:INP\], we have $\epsilon_k=0$ for all $k$. In our tests, we set $x_0=\bar{x}$ and, for all $k$, $\theta_k=0.25$, $\lambda_k=0.025$, $\gamma_k=0.025$, and defined $\beta_k := 2 (1-2\lambda_k)/(1+2\gamma_k)-10^{-6}$ satisfying . Figure \[fig:Behavior\] shows the behavior of SInexPD Algorithm on a two-dimensional instance of Problem . The hatched region represents set $C$ and only the iterates for which the target level was updated are plotted. As can be seen, the algorithm successfully found the [*solution*]{} for $\ell = 6$ iterations. We point out that the algorithm performed a total of $\ell = 14$ ($k=189$) iterations until it met the stopping criterion. The highlight of the figure is that, before finding the solution, the iterates belong to the interior of set $C$. This is mostly due to the fact that SInexPD Algorithm performs inexact projections.
![Behavior of SInexPD Algorithm on a two-dimensional instance of Problem .[]{data-label="fig:Behavior"}](example.eps "fig:")\
Finally, we considered six instances of Problem varying the dimension $n$. Without attempting to go into details, we mention that the problems were randomly generated such that $\lambda_n\in(10^{-2},10^{-6})$, $\lambda_i\in(10,10^{3})$ for $i = 1,\ldots,n-1$, vector $u\in{\mathbb{R}}^n_{++}$ in is such that $\|u\| \in(0.8/\sqrt{\lambda_n},1/\sqrt{\lambda_n})$, and $\xi = 1\sqrt{\lambda_n}$. These imply that, with respect to the ellipsoid that makes up set $C$, the axis corresponding to the eigenvector $v_n$ is [*much larger*]{} than the others ones. Moreover, the vectors of $C$ that have only one non-null component are [*far*]{} from the center $\bar{x}$. These characteristics make problems more challenging for the algorithm. Table \[tab:Performance\] shows the performance of SInexPD Algorithm. In the table, column “$n$" informs the considered dimension, “$k$" and “$\ell$" are the number of iterations according to SInexPD Algorithm, “$\|x_k^{rec}\|_0$" is the number of non-null elements at the final iterate, and “$f_k^{rec}$" and “$\delta_{\ell}$" are their corresponding values at the final iterate.
$n$ $k$ $\ell$ $\|x_k^{rec}\|_0$ $f_k^{rec}$ $\delta_{\ell}$
------ ----- -------- ------------------- ------------- -----------------
10 91 19 1 1.12D+01 6.18D-03
100 85 21 1 1.07D+01 9.77D-03
200 63 36 1 2.11D+01 1.38D-02
500 58 22 1 1.01D+01 1.09D-02
800 575 26 1 1.20D+01 6.91D-03
1000 669 24 1 1.15D+01 7.72D-03
: Performance of SInexPD Algorithm on six instances of Problem varying the dimension.[]{data-label="tab:Performance"}
As showed in Table \[tab:Performance\], the algorithm found vectors with only one non-null component in all instances, showing its ability to recover sparsity in this class of problems.
Remembering that the table data corresponds to the values when the stop criterion was met, we reported that the [*final*]{} iterates were found with $\ell = 10, 16, 27, 12, 15$ and $12$ iterations, respectively. We point out that, due to the inexact projections and mimicking the behavior of SInexPD Algorithm in the two-dimensional case, in each instance the iterates remained in the interior of $C$ before the corresponding solution was found.
Conclusions {#Sec:Conclusions}
===========
It is well known that the application of the subgradient method is only suitable for certain specific classes of non-differentiable convex optimization problems. However, this method is basic in the sense that it is the first step towards designing more efficient methods for solving that problems. Indeed, it is intrinsically related to cutting-plane and bundle methods; see [@UrrutyLemarechal1993_II]. These considerations lead us to conclude that the knowledge of new properties of the subgradient method has great theoretical value. In particular, our inexact version of the projected subgradient method will be useful in this theoretical context. Finally, one issue we believe deserves attention is the construction of inexact projected versions of cutting-plane and bundle methods.
[^1]: Instituto de Matemática e Estatística, Universidade Federal de Goiás, CEP 74001-970 - Goiânia, GO, Brazil, E-mails: [ademiraguia@gmail.com]{}, [orizon@ufg.br]{}, [lfprudente@ufg.br]{}. The authors was supported in part by CNPq grants 305158/2014-7 and 302473/2017-3, FAPEG/PRONEM- 201710267000532 and CAPES.
|
---
abstract: 'We studied the temperature and metal abundance distributions of the intra-cluster medium (ICM) in a group of galaxies NGC 1550 observed with Suzaku. The NGC 1550 is classified as a fossil group, which have few bright member galaxies except for the central galaxy. Thus, such a type of galaxy is important to investigate how the metals are enriched to the ICM. With the Suzaku XIS instruments, we directly measured not only Si, S, and Fe lines but also O and Mg lines and obtained those abundances to an outer region of $\sim0.5~ r_{180}$ for the first time, and confirmed that the metals in the ICM of such a fossil group are indeed extending to a large radius. We found steeper gradients for Mg, Si, S, and Fe abundances, while O showed almost flat abundance distribution. Abundance ratios of $\alpha$-elements to Fe were similar to those of the other groups and poor clusters. We calculated the number ratio of type II to type Ia supernovae for the ICM enrichment to be $2.9\pm 0.5$ within $0.1~r_{180}$, and the value was consistent with those for the other groups and poor clusters observed with Suzaku. We also calculated metal mass-to-light ratios (MLRs) for Fe, O and Mg with B-band and K-band luminosities of the member galaxies of NGC 1550. The derived MLRs were comparable to those of NGC 5044 group in the $r<0.1~r_{180}$ region, while those of NGC 1550 are slightly higher than those of NGC 5044 in the outer region.'
author:
- |
Kosuke <span style="font-variant:small-caps;">Sato</span>, Madoka <span style="font-variant:small-caps;">Kawaharada</span>, Kazuhiro <span style="font-variant:small-caps;">Nakazawa</span>,\
Kyoko <span style="font-variant:small-caps;">Matsushita</span>, Yoshitaka <span style="font-variant:small-caps;">Ishisaki</span>, Noriko <span style="font-variant:small-caps;">Y. Yamasaki</span>, and Takaya <span style="font-variant:small-caps;">Ohashi</span>
title: |
Metallicity of the Fossil Group NGC 1550\
Observed with Suzku
---
Introduction
============
[lccccc]{} Region & Sequence No. & Observation date & &Exp.&After screening\
&&&J2000.0 & ksec &(BI/FI) ksec\
center & 803017010 & 2008-08-16T04:27:05 & (, )& 83.3& 82.0/83.3\
offset & 803018010 & 2008-08-15T02:44:04 & (, )& 41.1& 40.5/40.7\
\
\
Groups and clusters of galaxies play a key role for investigating the formation of the universe and they act as a building blocks in the framework of a hierarchical formation of structures. The metal abundances of Intra-cluster medium (ICM) in groups and clusters carry a lot of information in understanding the chemical history and evolution of groups and clusters. Recent X-ray observations allow us to measure temperature and metal abundance distributions in the ICM based on the spatially resolved spectra. A large amount of metals of the ICM are mainly produced by supernovae (SNe) in early-type galaxies [@arnaud92; @renzini93], which are classified roughly as type Ia (SNe Ia) and type II (SNe II). Because Si and Fe are both synthesized in SNe Ia and II, we need to know O and Mg abundances, which are synthesized predominantly in SNe II, in resolving the past metal enrichment process in ICM by supernovae. In order to know how the ICM has been enriched, we need to measure the amount and distribution of all the metals from O to Fe in the ICM.
@renzini97 and @makishima01 summarized iron-mass-to-light ratios (IMLR) with B-band luminosity for various objects with ASCA, as a function of their plasma temperature serving as a measure of the system richness, and IMLRs in groups were found to be smaller than those in clusters. They also showed that the early-type galaxies released a large amount of metals which were probably formed through past supernovae explosions as shown earlier by @arnaud92. In order to obtain a correct modeling of ICM, we need to know the correct temperature and metal abundance profiles without biases (e.g., [@buote00; @sanders02]). Especially for the ICM of cooler systems, such as elliptical galaxies and groups of galaxies, careful analysis are required as mentioned in @arimoto97 and @matsushita00.
The spatial distribution and elemental abundance pattern of the ICM metals were determined with the large effective area of XMM-Newton [@matsushita07b; @tamura04; @boehringer05; @osullivan05; @sanders06; @deplaa06; @deplaa07; @werner06; @simionescu08]. On the other hand, the abundance measurements of O and Mg with XMM-Newton were possible only for the central regions of brightest cooling core clusters due to the relatively high intrinsic background. @rasmussen07 [@rasmussen09] report the Si and Fe profiles of 15 groups of galaxies observed with Chandra. They suggest that the Si to Fe ratios in groups tend to increase with radius, and the IMLRs within $r_{500}$ show a positive correlation with total group mass (temperature). Suzaku XIS can measure all the main elements from O to Fe, because it realizes lower background level and higher spectral sensitivity, especially below 1 keV [@koyama07]. Suzaku observations have shown the abundance profiles of O, Mg, Si, S, and Fe to the outer regions with good precision for several clusters [@matsushita07a; @sato07a; @sato08; @sato09a; @sato09b; @tokoi08; @komiyama09]. Combining the Suzaku results with supernova nucleosynthesis models, @sato07b showed the number ratios of SNe II to Ia to be 3.5.
NGC 1550 is a S0 galaxy and one of the nearest ($z=0.0124$) X-ray bright galaxies. The NGC 1550 is also classified as a fossil group [@jones03], and an X-ray extended object RX J0419+0225 was first discovered by the ROSAT ALL SKY SURVEY from a position centered on the NGC 1550 galaxy. From ASCA observation [@kawaharada03; @fukazawa04], the MLRs with B-band is comparable to those of clusters. @sun03 reports the temperature drop at the central region and also declines beyond 0.1 times of the virial radius, $r_{180}$, with Chandra observation. @kawaharada09 shows the gas mass and metal mass from the temperature and metal abundances observed with XMM-Newton. In addition, they derived the mass-to-light ratios of O, Si, and Fe with near infrared (K-band) luminosity. The resultant IMLR within $\sim 200~h_{72}$ kpc exhibits about 2 orders of magnitude decrease toward the center. NGC 1550, such a fossil group, is a important object to investigate how the metals have been enriched to the ICM, because of little metal supply from the present-day member galaxies.
This paper reports on results from Suzaku observations of NGC 1550 out to $30'\simeq 457\; h_{70}^{-1}$ kpc, corresponding to $\sim 0.47\; r_{180}$. We use $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\Lambda} = 1-\Omega_M = 0.73$ in this paper. At a redshift of $z=0.0124$, $1'$ corresponds to 15.2 kpc, and the virial radius, $r_{\rm 180} = 1.95\;
h_{100}^{-1}\sqrt{k\langle T\rangle/10~{\rm keV}}$ Mpc [@markevitch98], is 0.97 Mpc () for an average temperature of $k\langle T\rangle = 1.2$ keV. Throughout this paper we adopt the Galactic hydrogen column density of $N_{\rm H} = 1.15\times
10^{21}$ cm$^{-2}$ [@dickey90] in the direction of NGC 1550. Unless noted otherwise, the solar abundance table is given by @anders89, and the errors are in the 90% confidence region for a single interesting parameter.
(0.5,0.5)[fig1.eps]{}
(,)[fig2a.eps]{}
(,)[fig2b.eps]{}
(,)[fig2c.eps]{}
(,)[fig2d.eps]{}
Observations and Data Reduction {#sec:obs}
===============================
Observations {#subsec:obs}
------------
Suzaku observed the central and offset regions of NGC 1550 in August 2008 (PI: K. Sato). The observation logs are given in table \[tab:1\], and the XIS image in 0.5–2 keV is shown in figure \[fig:1\]. We analyzed only the XIS data in this paper, because the temperature of the HXD PIN is slightly higher during the observations. The XIS instrument consists of three sets of X-ray CCDs (XIS 0, 1, and 3). XIS 1 is a back-illuminated (BI) sensor, while XIS 0 and 3 are front-illuminated (FI). The instrument was operated in the normal clocking mode (8 s exposure per frame), with the standard $5\times 5$ or $3\times 3$ editing mode. During these observations, the significant effect of the Solar Wind Charge eXchange (SWCX) was not confirmed in ACE data [^1], although it was known that the SWCX affected the Suzaku spectra in the lower energy range as reported in @fujimoto07 and @yoshino09.
Data Reduction
--------------
We used version 2.2 processing data, and the analysis was performed with HEAsoft version 6.6.3 and XSPEC 12.5.0ac. Here we give just a brief description of the data reduction. The light curve of each sensor in the 0.3–10 keV range with a 16 s time bin was also examined in order to exclude periods with anomalous event rates which were greater or less than $\pm 3\sigma$ around the mean to remove the charge exchange contamination @fujimoto07, while Suzaku data was little affected by the soft proton flare compared with XMM data. The exposure after the screening was essentially the same as that before screening in table \[tab:1\], which indicated that the non X-ray background (NXB) was almost stable during the observation. Event screening with cut-off rigidity (COR) was not performed in our data. In order to subtract the NXB and the extra-galactic cosmic X-ray background (CXB), we employed the dark Earth database by the “xisnxbgen” Ftools task.
We generated two Ancillary Response Files (ARFs) for the spectrum of each annular sky region, $A^{\makebox{\small\sc u}}$ and $A^{\makebox{\small\sc b}}$, which respectively assumed uniform sky emission and $\sim 1^{\circ} \times 1^{\circ}$ size of the $\beta$-model surface brightness profile, $\beta = 0.47$ and $r_c =$ , in @fukazawa04, by the “xissimarfgen” Ftools task [@ishisaki07]. We also included the effect of the contaminations on the optical blocking filter of the XISs in the ARFs. Since the energy resolution also slowly degraded after the launch, due to radiation damage, this effect was included in the Redistribution Matrix File (RMF) by the “xisrmfgen” Ftools task.
Temperature and Abundance Profiles
==================================
(,)[fig3a.eps]{}
(,)[fig3b.eps]{}
(,)[fig3c.eps]{}
(,)[fig3d.eps]{}
(,)[fig3e.eps]{}
(,)[fig3f.eps]{}
Spectral Fit {#sec:spec}
------------
We extracted spectra from five annular regions of 0$'$–3$'$, 3$'$–6$'$, and 6$'$–12$'$ for the central observation, and whole area, which corresponds to 12$'$–30$'$, for the offset region, centered on (RA, Dec) = (, ). Each annular spectrum is shown in figure \[fig:2\]. The ionized Mg, Si, S, Fe lines are clearly seen in each region. The O and O lines were prominent in the outer rings, however, most of the O line was considered to come from the local Galactic emission, and we dealt with those in the same way as mentioned in @sato07a [@sato08; @sato09a; @sato09b].
The spectra with BI and FI for all regions were fitted simultaneously in the energy range, 0.4–7.1/0.4–5.0 keV (BI) and 0.6–7.1/0.6–5.0 (FI) for the central/offset observations, respectively. In the simultaneous fit, the common Galactic emission and CXB components were included for all regions. We excluded the narrow energy band around the Si K-edge (1.825–1.840 keV) because its response was not modeled correctly. The energy range below 0.4 keV was also excluded because the C edge (0.284 keV) seen in the BI spectra could not be reproduced well in our data. The range above 7.1 keV was also ignored because Ni line ($\sim
7.5$ keV) in the background left a spurious feature after the NXB subtraction at large radii. In the simultaneous fits of BI and FI data, only the normalization parameter was allowed to take different values between them.
It is important to estimate the Galactic component precisely, because the Galactic component gives significant contribution especially in the outer regions, as shown in figure \[fig:2\]. However, the ICM component is still dominant in almost all the energy range except for the O line. We assumed two temperature [*apec*]{} model (assuming 1 solar abundance and zero redshift) for the Galactic component, and fitted the data with the following model formula: ${\it constant}\times( {\it apec}_{\rm cool}+
{\it phabs} \times ({\it apec}_{\rm
hot} + {\rm ICM} ))$ as shown in @yoshino09. As a result, the best-fit temperatures of 0.098 and 0.219 keV for the Galactic models as shown in table \[tab:2\] are consistent with the values as shown in @yoshino09 and @lumb02. Thus, we concluded that the two temperature model of the Galactic emission was enough to represent NGC 1550 data. The resultant normalizations of the [*apec*]{} models in table \[tab:2\] are scaled so that they give the surface brightness in unit solid angle of arcmin$^2$, and are constrained to give the same surface brightness and the same temperature for the simultaneous fits of all annuli[^2].
[lcccccc]{} & [*Norm*]{}$_1$ & $kT_1$& [*Norm*]{}$_2$ & $kT_2$& &\
& &(keV)&&(keV)& &\
& 3.37$^{+1.43}_{-1.14}$ & 0.098$^{+0.005}_{-0.014}$ & 0.92$^{+0.23}_{-0.58}$ & 0.219$^{+0.062}_{-0.046}$ & &\
& [*Norm*]{}$_1$ & $kT_1$& [*Norm*]{}$_2$ & $kT_2$& [*Norm*]{}$_1$/[*Norm*]{}$_2$ & $\chi^2$/dof\
& &(keV)&&(keV)& &\
0$'$–3$'$ & 358.5$^{+11.0}_{-6.1}$ & 1.50$^{+0.02}_{-0.02}$ & 58.8$^{+4.8}_{-4.7}$ & 0.91$^{+0.05}_{-0.05}$ & 6.10$^{+0.53}_{-0.49}$ & 613/463\
3$'$–6$'$ & 88.2$^{+3.2}_{-1.8}$ & 1.43$^{+0.03}_{-0.02}$ & 5.4$^{+1.3}_{-0.9}$ & 0.86$^{+0.09}_{-0.05}$ & 16.48$^{+3.94}_{-2.89}$ & 531/469\
6$'$–12$'$ & 28.3$^{+1.6}_{-0.9}$ & 1.37$^{+0.04}_{-0.03}$ & 5.2$^{+0.5}_{-2.8}$ & 0.81$^{+0.03}_{-0.05}$ & 5.42$^{+0.58}_{-2.96}$ & 520/469\
12$'$–30$'$ & 5.0$^{+1.1}_{-0.6}$ & 0.97$^{+0.03}_{-0.05}$ & – & – & – & 73/91\
total & & & & & & 1737/1492\
\
&O&Ne&Mg,Al&Si&&Fe,Ni\
&(solar)&(solar)&(solar)&(solar)&(solar)&(solar)\
0$'$–3$'$ & 0.29$^{+0.07}_{-0.07}$ & 0.50$^{+0.12}_{-0.12}$ & 0.53$^{+0.06}_{-0.06}$ & 0.55$^{+0.04}_{-0.04}$ & 0.57$^{+0.04}_{-0.04}$ & 0.47$^{+0.02}_{-0.02}$\
3$'$–6$'$ & 0.35$^{+0.09}_{-0.07}$ & 0.26$^{+0.11}_{-0.10}$ & 0.31$^{+0.06}_{-0.05}$ & 0.36$^{+0.04}_{-0.03}$ & 0.34$^{+0.04}_{-0.04}$ & 0.32$^{+0.02}_{-0.02}$\
6$'$–9$'$ & 0.25$^{+0.12}_{-0.09}$ & 0.39$^{+0.14}_{-0.14}$ & 0.39$^{+0.08}_{-0.08}$ & 0.28$^{+0.05}_{-0.04}$ & 0.31$^{+0.06}_{-0.06}$ & 0.29$^{+0.02}_{-0.02}$\
9$'$–17$'$ & 0.00$^{+0.47}_{-0.00}$ & 0.01$^{+0.38}_{-0.01}$ & 0.10$^{+0.23}_{-0.10}$ & 0.13$^{+0.20}_{-0.13}$ & 0.00$^{+0.25}_{-0.00}$ & 0.20$^{+0.04}_{-0.04}$\
\
\
\
nominal
---------------- ----------- ----------- ----------- ----------- -----------
All $\dotfill$ 1737/1492 1741/1492 1803/1492 1745/1492 1747/1492
The ICM spectra for the central observation, $r<12'$, were clearly better represented by two [*vapec*]{} models than one [*vapec*]{} model in the $\chi^2$ test. On the other hand, the ICM spectra for the offset observation, $r=12$–$30'$, were well-presented by a single temperature model. Thus, we carried out the simultaneous fit with the following formula of the Galactic and ICM components: ${\it constant}\times( {\it
apec}_{\rm cool}+{\it phabs} \times ( {\it apec}_{\rm hot} +
{\it vapec}_{0<r<30'} + {\it vapec}_{0<r<12'})$. The fit results are shown in table \[tab:2\]. The abundances were linked in the following way, Mg=Al, S=Ar=Ca, Fe=Ni, which gave good constraint especially for the offset regions. The abundances were also linked between the two ${\it vapec}$ components for $r<12'$ region. Results of the spectral fit for individual annuli are summarized in table \[tab:2\] and figure \[fig:3\], in which systematic error due to the OBF contamination and NXB estimation are shown.
Temperature Profile {#subsec:radial}
-------------------
Radial temperature profile and the ratio of the [*vapec*]{} normalizations between the hot and cool ICM components are shown in figure \[fig:3\](a) and table \[tab:2\]. The ICM temperature of hot and cool components at the central region was $\sim1.5$ and $\sim0.9$ keV, respectively, and the temperature decreased mildly to $\sim1.0$ keV in the outermost region, while the cool components were almost constant at $\sim0.9$ keV. Our results for the two temperature ICM model are consistent with the XMM result [@kawaharada09]. For the hot component, our results are also consistent with the previous Chandra result [@sun03]. The radius of $30'\sim 457$ kpc corresponds to $\sim 0.47\; r_{\rm
180}$, and the temperature decline is clearly recognized in such a small system out to this radius.
Abundance Profiles
------------------
Metal abundances are determined for the six element groups individually as shown in figures \[fig:3\](b)–(f). The four abundance values for Mg, Si, S, and Fe and their radial variation look similar to each other. The central abundances lie around $\sim 0.5$–0.6 solar, and they commonly decline to about 1/4 of the central value in the outermost annulus. On the other hand, the O profile looks flatter compared with the other elements. Because the results for the offset regions had large errors, we examined the summed spectra for these regions. We noted that, when we examined all abundances to be free in the fits, the resultant parameters did not change within the statistical errors. In addition, even if all regions were fitted by a two temperature model, the resultant abundance profiles did not change within the statistical errors.
We also examined the systematic error of our results by changing the background normalization by $\pm 10$%, and the error range is plotted with light-gray dashed lines in figure \[fig:3\]. The systematic error due to the background estimation is almost negligible. The other systematic error concerning the uncertainty in the OBF contaminant is shown by black dashed lines as shown in figure \[fig:3\]. A list of $\chi^2$/dof by changing the systematic errors is presented in table \[tab:3\]. We note that Ne abundance is not reliably determined due to an overlap with the strong and complex Fe-L line emissions, however we left these abundance to vary freely during the spectral fit.
Discussion {#sec:discuss}
==========
Metallicity Distribution in the ICM {#subsec:metal}
-----------------------------------
(0.45,8cm)[fig4.eps]{}
(0.45,8cm)[fig5.eps]{}
Suzaku observation of NGC 1550 confirmed that the metals in the ICM of this fossil group are indeed extending to a large radius. The measured elements are O, Mg, Si, S, and Fe out to a radius of $30'\simeq 457$ kpc, which corresponds to $\sim0.47~r_{180}$, as shown in figure \[fig:3\]. The Ne abundance had a large uncertainty due to the strong coupling with Fe-L lines. Distributions of Mg, Si, S, and Fe are similar to each other, while O profile shows no central peak and large error in the outer region at $r > 12'$. We plotted abundance ratios of O, Mg, Si, and S to Fe as a function of the projected radius in figure \[fig:4\]. The ratios Mg/Fe, Si/Fe and S/Fe are consistent to be a constant value around 1.5–2, while O/Fe ratio in the innermost region ($r<3'$) is significantly lower around 0.5. In addition, the O/Fe ratio suggests some increase with radius. Note that these abundance profiles are not deconvolved and are averaged over the line of sight.
Recent Suzaku observations have presented abundance profiles in several relaxed cooling core groups and clusters: a group of galaxies NGC 5044 [@komiyama09], a poor cluster of galaxies Abell 262 [@sato09b], and AWM 7 [@sato08]. We compare Fe abundance and O, Mg, Si, S to Fe abundance ratios of NGC 1550 with those of NGC 5044, Abell 262 and AWM 7 as shown in figure \[fig:4\]. While NGC 1550 shows slightly lower Fe abundance than those of NGC 5044, Abell 262, and AWM 7 in the central region ($r\lesssim 0.1~r_{180}$), the abundance in the outer region ($r\gtrsim0.3~r_{180}$) are quite similar, showing a decrease to $\sim0.2$ solar. On the other hand, the abundance ratios of O/Fe, Mg/Fe, Si/Fe, and S/Fe are quite similar between the four systems.
@matsushita10 reports the Fe radial profiles of 28 clusters of galaxies observed with XMM. The Fe abundance profile of NGC 1550 also has similar feature to those of clusters up to $\sim0.5~r_{180}$. Although @rasmussen09 suggests that the radial Si profiles in group have softer than the Fe profiles, our results show the constant values of the Si to Fe ratios up to $\sim0.5~r_{180}$ as shown in figure \[fig:4\].
Number Ratio of SNe II to SNe Ia
--------------------------------
$N_{\rm Ia}$ $N_{\rm II}$/$N_{\rm Ia}$ $\chi^2$/dof
------------------------- ------ ------------------------- --------------------------- --------------
$<0.1\;r_{180}$ W7 $1.3\pm0.1\times10^{8}$ $2.9\pm 0.5$ 3.7/3
0.1 – 0.2 $r_{\rm 180}$ W7 $2.6\pm0.4\times10^{8}$ $2.8\pm 1.0$ 5.7/3
$< 0.1\;r_{\rm 180}$ WDD1 $1.3\pm0.1\times10^{8}$ $2.1\pm 0.5$ 44.2/3
0.1 – 0.2 $r_{\rm 180}$ WDD1 $2.8\pm0.5\times10^{8}$ $2.0\pm 1.0$ 21.0/3
$< 0.1\;r_{\rm 180}$ WDD2 $1.2\pm0.1\times10^{8}$ $2.8\pm 0.6$ 10.4/3
0.1 – 0.2 $r_{\rm 180}$ WDD2 $2.4\pm0.4\times10^{8}$ $2.8\pm 1.1$ 8.9/3
In order to examine relative contributions from SNe Ia and SNe II to the ICM metals, the elemental abundance pattern of O, Mg, Si, S and Fe was examined for the inner ($r<0.1~r_{180}$) and the immediate outer (0.1–$0.2~r_{180}$) regions. The abundance ratios to Fe were fitted by a combination of average SNe Ia and SNe II yields per supernova, as shown in figure \[fig:5\]. The fit parameters were the integrated number of SNe Ia ($N_{\rm Ia}$) and the number ratio of SNe II to SNe Ia ($N_{\rm II}/N_{\rm Ia}$), because $N_{\rm Ia}$ could be well constrained by the relatively small errors in the Fe abundance. The SNe Ia and II yields were taken from @iwamoto99 and @nomoto06, respectively. We assumed the Salpeter IMF for stellar masses from 10 to 50 $M_{\odot}$ with the progenitor metallicity of $Z=0.02$ for SNe II, and W7, WDD1 or WDD2 models for SNe Ia. Table \[tab:4\] and figure \[fig:5\] summarize the fit results. The number ratios were better represented by the W7 SNe Ia yield model than by WDD1. The number ratio of SNe II to SNe Ia with W7 is $\sim2.9$ within 0.1 $r_{180}$, while the ratio assuming WDD1 is $\sim 2.1$. The WDD2 model gave the result very similar to the W7 value. The resultant number ratios are consistent with the previous result by @sato07b. We also compared the abundance pattern of NGC 1550 with the solar abundance. The third panel in figure \[fig:5\] shows this comparison for $r<0.1~r_{180}$ of NGC 1550 with two different solar abundance patterns given by @anders89 and @lodders03. Abundances of Mg, Si, and S fall between the two solar abundance patterns.
Almost $\sim80$% of Fe and $\sim40$% of Si and S were synthesized by SNe Ia in the W7 model, as demonstrated in the bottom panel of figure \[fig:5\]. These observed features of the fossil group are similar to those for clusters with $kT =2-4$ keV clusters studied by @sato07b and @sato09b. The values in table \[tab:4\] imply that the $N_{\rm II}/N_{\rm Ia}$ ratio for the inner and outer regions behave in the similar manner for different supernova models. We note that the fit was not formally acceptable based on the $\chi^2$ value in table \[tab:4\]. As described in @sato07b, the models adapted here (SNe yield, Salpeter IMF, etc.) are probably too simplified.
---------------------- ----------------------------------- ----------------------------------- ----------------------------------- ------- ------------- ------------ ----------------
IMLR OMLR MMLR Reference
($M_{\odot}/L^{\rm B}_{\odot}$) ($M_{\odot}/L^{\rm B}_{\odot}$) ($M_{\odot}/L^{\rm B}_{\odot}$) (kpc) ($r_{180}$) (keV)
Suzaku
NGC 5044 $\dotfill$ $2.6^{+0.2}_{-0.2}\times 10^{-3}$ $6.6^{+1.9}_{-1.7}\times 10^{-3}$ $1.6^{+0.2}_{-0.2}\times 10^{-3}$ 88 0.10 $\sim 1.0$ @komiyama09
$3.6^{+0.4}_{-0.3}\times 10^{-3}$ $9.4^{+5.2}_{-2.1}\times 10^{-3}$ $2.6^{+0.4}_{-0.3}\times 10^{-3}$ 260 0.30
NGC 1550 $\dotfill$ $2.6^{+0.1}_{-0.1}\times 10^{-3}$ $1.2^{+0.3}_{-0.2}\times 10^{-3}$ $9.4^{+1.2}_{-1.1}\times 10^{-4}$ 46 0.09 $\sim 1.2$ This work
$7.4^{+0.4}_{-0.4}\times 10^{-3}$ $3.3^{+1.1}_{-0.8}\times 10^{-2}$ $3.3^{+0.5}_{-0.5}\times 10^{-3}$ 183 0.19
Fornax $\dotfill$ $4\times 10^{-4}$ $2\times 10^{-3}$ – 130 0.13 $\sim 1.3$ @matsushita07b
NGC 507 $\dotfill$ $6.0^{+0.4}_{-0.3}\times 10^{-4}$ $2.6^{+0.6}_{-0.5}\times 10^{-3}$ $3.7^{+0.4}_{-0.4}\times 10^{-4}$ 120 0.11 $\sim 1.5$ @sato09a
$1.7^{+0.2}_{-0.2}\times 10^{-3}$ $6.6^{+3.3}_{-2.5}\times 10^{-3}$ $1.1^{+0.2}_{-0.2}\times 10^{-3}$ 260 0.24
HCG 62 $\dotfill$ $2.0^{+0.2}_{-0.1}\times 10^{-3}$ $6.4^{+0.2}_{-0.4}\times 10^{-3}$ $1.0^{+0.2}_{-0.1}\times 10^{-3}$ 120 0.11 $\sim 1.5$ @tokoi08
$4.6^{+0.7}_{-0.6}\times 10^{-3}$ $3.8^{+2.7}_{-3.4}\times 10^{-2}$ $1.5^{+0.4}_{-0.4}\times 10^{-3}$ 230 0.21
A 262 $\dotfill$ $3.6^{+0.1}_{-0.1}\times 10^{-3}$ $1.2^{+0.3}_{-0.4}\times 10^{-2}$ $1.6^{+0.2}_{-0.2}\times 10^{-3}$ 130 0.10 $\sim 2$ @sato09b
$6.7^{+0.4}_{-0.4}\times 10^{-3}$ $3.7^{+1.2}_{-1.2}\times 10^{-2}$ $2.7^{+0.7}_{-0.6}\times 10^{-3}$ 340 0.27
A 1060 $\dotfill$ $5.7^{+0.4}_{-0.4}\times 10^{-3}$ $4.3^{+0.8}_{-0.8}\times 10^{-2}$ $2.4^{+0.5}_{-0.5}\times 10^{-3}$ 180 0.12 $\sim 3$ @sato07a
$4.0^{+0.4}_{-0.4}\times 10^{-3}$ $4.3^{+2.0}_{-1.8}\times 10^{-2}$ $1.6^{+0.8}_{-0.7}\times 10^{-3}$ 380 0.25
AWM 7 $\dotfill$ $4.8^{+0.2}_{-0.2}\times 10^{-3}$ $2.6^{+0.8}_{-0.8}\times 10^{-2}$ $3.4^{+0.5}_{-0.5}\times 10^{-3}$ 180 0.11 $\sim 3.5$ @sato08
$7.6^{+0.4}_{-0.3}\times 10^{-3}$ $3.1^{+1.9}_{-1.2}\times 10^{-2}$ $6.7^{+1.1}_{-1.1}\times 10^{-3}$ 360 0.22
XMM-Newton
Centaurus $\dotfill$ $4\times 10^{-3}$ $3\times 10^{-2}$ – 190 0.11 $\sim 4$ @matsushita07a
---------------------- ----------------------------------- ----------------------------------- ----------------------------------- ------- ------------- ------------ ----------------
(0.45,1cm)[fig6a.eps]{} (0.45,1cm)[fig6b.eps]{}
Metal Mass-to-Light Ratio
-------------------------
We derived 3-dimensional gas mass profile by extending the previous XMM-Newton result [@kawaharada09] for the region within $14'$ arcmin ($\sim0.2~r_{180}$). Combining it with the abundance profiles obtained with Suzaku, we calculated cumulative metal mass. The derived masses of Fe and Mg within the 3-dimensional radius of $r< 457$ kpc ($r\sim0.5~r_{180}$) are $1.3\times 10^{9}$, $3.4\times 10^{8}~M_\odot$, respectively, and the O mass within $r<183$ kpc is $1.7\times 10^{9}$ $M_\odot$. Errors of the metal mass, which were used to calculate mass-to-light rations, were taken from the statistical errors of each elemental abundance in the spectral fits, because these are much larger than the error of gas mass by @kawaharada09.
We examined mass-to-light ratios for O, Fe, and Mg (OMLR, IMLR, and MMLR, respectively) which enabled us to compare the ICM metal distribution with the stellar mass profile. Historically, B-band luminosity has been used for the estimation of the stellar mass [@makishima01], however we calculated it using the K-band luminosity in NGC 1550 based on the Two Micron All Sky Survey (2MASS) catalogue [^3]. This method is useful in performing a uniform comparison with the properties in other groups and clusters based on the same K-band galaxy catalogue to trace the distribution of member elliptical galaxies.
In the 2MASS catalog, we used all the data in a $2^{\circ}\times2^{\circ}$ region around NGC 1550 without the selection of galaxy morphology, and subtracted the luminosity in a $r>1^{\circ}$ region, which corresponds to about $r_{\rm 180}$, as the background. We then deprojected the luminosity profile as a function of radius assuming a spherical symmetry. In order to convert the K-band magnitude of each galaxy to the B-band value, we assumed the luminosity distance $D_{\rm
L}=53.6$ Mpc, and an appropriate color $B-K=4.2$ for early-type galaxies given by @lin04, along with the Galactic extinction $A_B=0.583$ from NASA/IPAC Extragalactic Database (NED) in the direction of NGC 1550.
The integrated values of OMLR, IMLR, and MMLR using the estimated B-band luminosity within $r\lesssim 183$ kpc ($r\lesssim 0.2~r_{180}$) turned out to be $\sim 3.3\times 10^{-2}$, $\sim 7.4\times 10^{-3}$, and $\sim 3.3\times 10^{-3}$ $M_{\odot}/L_{\rm B \odot}$, respectively, as shown in table \[tab:5\]. The errors are based only on the statistical errors of metal abundance in the spectral fit, and the uncertainties in the gas mass profile and the luminosities of member galaxies are not included. Note that we did not adjust metal-mass and K-band profiles by considering the Suzaku PSF effect, because uncertainties in the metal mass had the dominant effect in our MLR estimation.
We compared these B-band MLRs for NGC 1550 with those of other groups and clusters. The MLRs are all measured within inner ($\sim0.1~r_{180}$) and outer ($\sim0.25~r_{180}$) regions as shown in table \[tab:5\] and figure \[fig:6\]. As for Fe (IMLR), NGC 1550 shows a similar value with the other groups and poor clusters in the inner region. As mentioned in subsection \[subsec:metal\], the Fe abundance itself of NGC 1550 in the inner region is slightly smaller than those in other groups and clusters. However, the IMLR in this region is almost the same as others, due to somewhat low value of stellar mass in NGC 1550. Looking at the outer region, NGC 1550 shows slightly higher IMLR than NGC 5044 and comparable to those in other poor clusters. On the other hand for the OMLR in the inner region, the poor systems show much lower values than the larger high-temperature systems. Interestingly, the poor systems (including NGC 1550) show higher OMLR in the outer region, comparable to those in the larger systems. The spatial extent of O looks to be relatively large in these very small systems.
@rasmussen09 suggests that the IMLRs within $r_{500}$ of 15 groups of galaxies observed with Chandra have a positive correlation with the groups mass (temperature). Our results and Chandra’s results are also almost consistent, and the IMLRs of groups are slightly lower than those of clusters.
OMLR MMLR IMLR
--------------- ---------------------------------- ---------------------------------- ----------------------------------
$<$45.7/0.05 9.4$^{+2.2}_{-2.1}\times10^{-4}$ 1.1$^{+0.1}_{-0.1}\times10^{-4}$ 2.9$^{+0.1}_{-0.1}\times10^{-4}$
$<$91.3/0.09 3.5$^{+0.7}_{-0.6}\times10^{-3}$ 2.6$^{+0.3}_{-0.3}\times10^{-4}$ 7.1$^{+0.3}_{-0.3}\times10^{-4}$
$<$182.6/0.19 9.2$^{+0.3}_{-0.3}\times10^{-3}$ 9.0$^{+1.4}_{-1.3}\times10^{-4}$ 2.0$^{+0.1}_{-0.1}\times10^{-3}$
$<$456.6/0.47 – 9.0$^{+1.0}_{-4.5}\times10^{-4}$ 3.5$^{+0.5}_{-0.5}\times10^{-3}$
(0.45,1cm)[fig7a.eps]{} (0.45,1cm)[fig7b.eps]{}
We also calculated the MLRs by directly using the K-band luminosity assuming the Galactic extinction $A_K=0.050$ in the direction of NGC 1550, and the absolute K-band solar magnitude of 3.34. The resultant K-band luminosity within the Suzaku observed region, $r<30'$, is $3.8\times10^{11}~L_{\rm K\odot}$, and the radial luminosity profile is also plotted in figure \[fig:7\](a). We calculated the radial profile of the OMLR, IMLR, and MMLR values using the K-band luminosity out to a radius $r \sim 180$ kpc ($r\lesssim
0.2~r_{180}$), as shown in figure \[fig:7\](b) and table \[tab:6\]. The values at the outermost radius are $\sim
9.2\times 10^{-3}$, $\sim 2.0\times 10^{-3}$, and $\sim 9.0\times
10^{-4}$ $M_{\odot}/L_{\rm K\odot}$, respectively.
In order to investigate the dependence on the system size, we compared the MLRs of NGC 1550 ($kT \approx 1.2$ keV) with those of NGC 5044 (1 keV) and Abell 262 (2 keV), which are all poor relaxed cooling-core groups and clusters. Although they show similar IMLR in the inner region, NGC 1550 and Abell 262 show gas mass ratio to the K-band luminosity higher than NGC 5044, as shown in figure \[fig:7\](a). As one goes outside ($r>0.1~r_{180}$), NGC 1550 shows fairly consistent MLRs with Abell 262, larger than those in NGC 5044 as shown in figure \[fig:6\](b). The radial profiles of IMLR and OMLR for NGC 1550 look quite similar to those for Abell 262 rather than for NGC 5044.
The radius, $r\sim0.1~r_{180}$, corresponds to the region where the gas mass and K-band luminosity seem to overlap as shown in figure \[fig:7\](a). Assuming the stellar mass-to-light ratio, $M_{\rm star}/L_{\rm K} \sim 1$ as shown in @arnouts07 (see also [@nagino09]), this radius indeed corresponds to the point where the gas and stellar masses are comparable. The similarity of MLRs for the 3 systems examined here in the inner region ($r<0.1~r_{180}$) suggests that the metal enrichment within the past few Gyr has occurred in a similar way. On the other hand, in the outer region ($r>0.1~r_{180}$), NGC 5044 shows lower IMLR than those in NGC 1550 and Abell 262, even though Fe abundances themselves are comparable in these regions. This suggests that distribution of stellar mass has some different history between these systems.
We will consider how the observed features of abundance and MLR profiles can constrain enrichment scenario. First, metals in the inner region ($r \lesssim 0.1~r_{180}$) have been mostly supplied by the central galaxy, so this region should be set aside in the present discussion. In the immediate outer region ($r\gtrsim 0.1~r_{180}$), metals from the central galaxy could not reach in a few Gyr time scale. Also, we may assume that almost the same amount of metals per stellar mass was synthesized in all systems before the collapse of groups and clusters. In this case, we expect very similar MLRs in different systems, contrary to the observed feature. As shown in figure \[fig:7\](b), at least NGC 5044 shows lower IMLR and OMLR profiles compared with those for two other systems. This implies that the thermal and/or dynamical evolutions of the gas have different history among different systems during or after the collapsing period of individual groups and clusters.
@renzini05 showed the expected MLRs as a function of the IMF slope. As for OMLR, the expected value to be $\sim0.1~M_{\odot}/L_{\rm B}$ at a Salpeter IMF is slightly higher than our results of $\sim0.03~M_{\odot}/L_{\rm B}$ within $\sim0.25~r_{180}$ as shown in figure \[fig:6\](b). However, as shown in figure \[fig:7\](b), because the OMLRs look increasing toward outer region, the OMLRs within the virial radius would be represented with the Salpeter IMF. @renzini05 also suggested that a top heavy IMF would overproduce metals by more than a factor of 20, which is much larger than the observed values including our results.
We stress that high-sensitivity abundance observation to the outer region of clusters will give important clues about their evolution. If O distribution, as well as Fe, could be measured to the very outer region ($r\sim r_{180}$), we may obtain a clear view about when O and Fe were supplied to the inter galactic space because most of O should have been synthesized by SNe II and supplied in the starburst era. Another possibility is very early metal enrichment of O by galaxies or massive Population III stars (e.g.[@matteucci06]) before groups and clusters assemble. In this case, a large part of the intergalactic space would be enriched quite uniformly with O and other elements. Metallicity information in cluster outskirts would thus give us unique information about the enrichment history. For this purpose, instruments with much higher energy resolution, such as microcalorimeters, and optics with larger effective area will play a key role in carrying out these studies.
Summary and conclusion
======================
Suzaku observation of the fossil group NGC 1550 showed spatial distributions of temperature and metal abundances for O, Mg, Si, S, and Fe up to $\sim0.5~r_{180}$ for the first time, and confirmed that the metals in the ICM of this fossil group are indeed extending to a large radius. The ICM temperature decreases mildly from $\sim1.5$ keV to $\sim1.0$ keV in the outer region, similar to the feature seen in other clusters. The abundances of Mg, Si, S, and Fe drop from subsolar levels at the center to $\sim
1/4$ solar in the outermost region, while the O abundance shows a flatter distribution around $\sim 0.5$ solar without the strong concentration in the center. The abundance ratios, O/Fe, Mg/Fe, Si/Fe, and S/Fe for NGC 1550 are generally similar to those in groups and poor clusters. The abundance pattern from O to Fe enabled us to constrain number ratio of SNe II to Ia as $2.9 \pm 0.5$, which is consistent with the values obtained for other groups and clusters. The derived MLRs of NGC 1550 using the B-band and K-band luminosities are consistent with those in the NGC 5044 group for the inner region $r\lesssim 0.1~r_{180}$, while NGC 1550 shows slightly higher MLRs than NGC 5044 in the outer region. This suggests that metal enrichment process may reflect the size of the system in the sense that larger systems contain higher amount of metals for a given stellar mass.
Authors thank the referee for providing valuable comments. K.S is supported by a JSPS Postdoctral fellowship for research abroad. Part of this work was financially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research Nos. 20340068, 21224003, 21740134.
Anders, E., & Grevesse, N. 1989, , 53, 197
Arimoto, N., Matsushita, K., Ishimaru, Y., Ohashi, T., & Renzini, A. 1997, , 477, 128
Arnaud, M., Rothenflug, R., Boulade, O., Vigroux, L., & Vangioni-Flam, E. 1992, , 254, 49
Arnouts, S., et al. 2007, , 476, 137
B[ö]{}hringer, H., Matsushita, K., Finoguenov, A., Xue, Y., & Churazov, E. 2005, Advances in Space Research, 36, 677
Buote, D. A. 2000, , 311, 176
Buote, D. A., Brighenti, F., & Mathews, W. G. 2004, , 607, L91
de Plaa, J., Werner, N., Bleeker, J. A. M., Vink, J., Kaastra, J. S., & M[é]{}ndez, M. 2007, , 465, 345
de Plaa, J., et al.2006, , 452, 397
Dickey, J. M., & Lockman, F. J. 1990, , 28, 215
Fujimoto, R., et al. 2007, , 59, 133
Fukazawa, Y., Makishima, K., & Ohashi, T. 2004, , 56, 965
Gastaldello, F., Buote, D. A., Humphrey, P. J., Zappacosta, L., Bullock, J. S., Brighenti, F., & Mathews, W. G. 2007, , 669, 158
Ishisaki, Y., et al. 2007, , 59, 113
Iwamoto, K., Brachwitz, F., Nomoto, K., Kishimoto, N., Umeda, H., Hix, W. R., & Thielemann, F.-K.1999, , 125, 439
Jones, L. R., Ponman, T. J., Horton, A., Babul, A., Ebeling, H., & Burke, D. J. 2003, , 343, 627
Kawaharada, M., Makishima, K., Kitaguchi, T., Okuyama, S., Nakazawa, K., & Fukazawa, Y. 2009, arXiv:0911.5205
Kawaharada, M., Makishima, K., Takahashi, I., Nakazawa, K., Matsushita, K., Shimasaku, K., Fukazawa, Y., & Xu, H. 2003, , 55, 573
Komiyama, M., Sato, K., Nagino, R., Ohashi, T., & Matsushita, K. 2009, , 61, 337
Koyama, K., et al. 2007, , 59, 23
Lin, Y.-T., & Mohr, J. J. 2004, , 617, 879
Lodders, K. 2003, , 591, 1220
Lumb, D. H., Warwick, R. S., Page, M., & De Luca, A. 2002, , 389, 93
Makishima, K., et al. 2001, , 53, 401
Markevitch, M., et al. 1998, , 503, 77
Matsushita, K., Ohashi, T., & Makishima, K. 2000, , 52, 685
Matsushita, K., Finoguenov, A., B[ö]{}hringer, H. 2003, , 401, 443
Matsushita, K., B[ö]{}hringer, H., Takahashi, I., & Ikebe, Y. 2007b, , 462, 953
Matsushita, K., et al. 2007a, , 59, 327
Matsushita, K. submitted to A&A
Matteucci, F., Panagia, N., Pipino, A., Mannucci, F., Recchi, S., & Della Valle, M. 2006, , 372, 265
Nagino, R., & Matsushita, K. 2009, , 501, 157
Nomoto, K., Tominaga, N., Umeda, H., Kobayashi, C., & Maeda, K. 2006, Nuclear Physics A, 777, 424
O’Sullivan, E., Vrtilek, J. M., Kempner, J. C., David, L. P., & Houck, J. C. 2005, , 357, 1134
Rasmussen, J., & Ponman, T. J. 2009, , 399, 239
Rasmussen, J., & Ponman, T. J. 2007, , 380, 1554
Renzini, A., Ciotti, L., D’Ercole, A., & Pellegrini, S. 1993, , 419, 52
Renzini, A. 1997, , 488, 35
Renzini, A. 2005, The Initial Mass Function 50 Years Later, 327, 221
Sanders, J. S., & Fabian, A. C. 2002, , 331, 273
Sanders, J. S., & Fabian, A. C. 2006, , 371, 1483
Simionescu, A., Werner, N., Finoguenov, A., B[ö]{}hringer, H., & Br[ü]{}ggen, M. 2008, , 482, 97
Sato, K., et al. 2007a, , 59, 299
Sato, K., Tokoi, K., Matsushita, K., Ishisaki, Y., Yamasaki, N. Y., Ishida, M., & Ohashi, T. 2007, , 667, L41
Sato, K., Matsushita, K., Ishisaki, Y., Yamasaki, N. Y., Ishida, M., Sasaki, S., & Ohashi, T. 2008, , 60, S333
Sato, K., Matsushita, K., Ishisaki, Y., Yamasaki, N. Y., Ishida, M., & Ohashi, T. 2009a, , 61, 353
Sato, K., Matsushita, K., & Gastaldello, F. 2009b, , 61, 365
Sun, M., Forman, W., Vikhlinin, A., Hornstrup, A., Jones, C., & Murray, S. S. 2003, , 598, 250
Tamura, T., Kaastra, J. S., Makishima, K., & Takahashi, I. 2003, , 399, 497
Tamura, T., Kaastra, J. S., den Herder, J. W. A., Bleeker, J. A. M., & Peterson, J. R. 2004, , 420, 135
Tawara, Y., et al. 2008, , 60, S307
Tokoi, K., et al. 2008, , 60, S317
Werner, N., de Plaa, J., Kaastra, J. S., Vink, J., Bleeker, J. A. M., Tamura, T., Peterson, J. R., & Verbunt, F. 2006a, , 449, 475
Yoshino, T., et al. 2009, , 61, 805
[^1]: http://www.srl.caltech.edu/ACE/ASC
[^2]: The Normalizations of “[*norm*]{}$_1$” in table 5 in @sato08 are an order of magnitude larger than the actual values.
[^3]: The database address: [ http://www.ipac.caltech.edu/2mass/]{}
|
---
abstract: 'It is logically possible that the trace anomaly in four dimension includes the Hirzebruch-Pontryagin density in CP violating theories. Although the term vanishes at free conformal fixed points, we realize such a possibility in the holographic renormalization group and show that it is indeed possible. The Hirzebruch-Pontryagin term in the trace anomaly may serve as a barometer to understand how much CP is violated in conformal field theories.'
---
[ **CP-violating CFT and trace anomaly**]{}\
[Yu Nakayama]{}
[*Institute for the Physics and Mathematics of the Universe,\
Todai Institutes for Advanced Study, University of Tokyo,\
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan*]{}
Introduction
============
The very existence of our human being relies on CP violation in an essential manner through Sakharov’s condition on the baryogenesis. While the breaking mechanism of CP symmetry is encoded in the standard model by three generations and phases in Yukawa couplings as well as Yang-Milles theta terms, it is a difficult question to measure how much CP is violated in a given quantum field theory in a qualitative manner. The ultimate goal of our study is to seek the possibility to use the CP violating contribution to the trace anomaly for such a candidate.
Surprisingly, CP violation in conformal field theories have been rarely studied in literatures. To some extent, it is due to the restricted viewpoint or more or less folklore that the conformal symmetry is obtained by adding “inversion" symmetry to the Poincaré group. Although it is true that the successive application of inversion, translation and the second inversion gives the special conformal transformation, the converse is not necessarily true: the conformal field theory may not be invariant under the inversion. Mathematically speaking, the conformal group is $SO(2,d)$, but with the inversion it is enhanced to $O(2,d)$. Clearly, the latter contains the former, but it is the former $SO(2,d)$ that is only required for the symmetry of quantum field theories.
The inversion, in the radial quantization, is nothing but the time reversal. The CPT-theorem tells us that whenever CP is violated, the time reversal must be broken. Therefore, we conclude that the CP violating conformal field theory cannot possess invariance under inversion. Technically speaking, studying the constraint from invariance under inversion is much convenient and probably the easiest way to obtain the form of correlation functions that are invariant under the special conformal transformation. The argument here, however suggests that the imposition might be too restrictive, and indeed it was demonstrated that it is the case.
In this paper, we discuss these usually overlooked aspects of conformal field theories with CP-violation. We discuss possible CP-violating terms in correlation functions of the energy-momentum tensor and its trace anomaly. In particular, we argue that it is entirely legitimate for the trace anomaly to have the Hirzebruch-Pontryagin density, which is parity odd. It is known that CP is preserved in any unitary free conformal field theories in four space-time dimension, so it is very difficult to show how they could appear in actual computations. In this paper, we take an alternative route by studying strongly coupled dual field theories with the help of the holographic renormalization group method. We show its actual existence by studying the holographic renormalization of the bulk CP violating gravitational theory.
The organization of the paper is as follows. In section 2, we discuss the consequence of CP-violation in correlation functions of the energy-momentum tensor and the trace anomaly. In particular, we argue that the Hirzebruch-Pontryagin density is a legitimate candidate of the trace anomaly in four-dimension. In section 3, we show how the CP-violating Hirzebruch-Pontryagin density can appear in the trace anomaly from the holographic renormalization method in five-dimensional AdS space with a CP violating action. In section 4, we present implications of our result in relation to the recently proved a-theorem for conformal fixed points and conclude.
CP-violation in energy-momentum tensor
======================================
Conformal algebra is the only natural space-time extension of the Poincaré algebra $$\begin{aligned}
i[J^{\mu\nu},J^{\rho\sigma}] &=
\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma} -
\eta^{\sigma\mu}J^{\rho\nu} + \eta^{\sigma\nu}J^{\rho\mu} \cr
i[P^\mu,J^{\rho\sigma}] &= \eta^{\mu\rho}P^{\sigma} -
\eta^{\mu\sigma} P^\rho \cr[P^\mu,P^\nu] &= 0 \ \label{alg1}\end{aligned}$$ with the dilatation symmetry $$\begin{aligned}
[P^\mu,D] &= i P^\mu \cr [J^{\mu\nu},D] &= 0 \ . \label{alg2}\end{aligned}$$ It possesses the additional generators $K^\mu$ called special conformal transformation: $$\begin{aligned}
[K^\mu,D] &= -iK^\mu \cr [P^\mu,K^\nu] &= 2i\eta^{\mu\nu}D+
2iJ^{\mu\nu} \cr [K^\mu,K^\nu] &= 0 \cr [J^{\rho\sigma},K^\mu] &=
i\eta^{\mu\rho} K^\sigma - i\eta^{\mu\sigma} K^\rho \ . \label{alg3}\end{aligned}$$ It is isomorphic to the Lie algebra $SO(2,d)$, and it is known that the conformal [*algebra*]{}, consisting of , and , is the largest bosonic space-time symmetry that acts non-trivially on S-matrix (in $d>2$) of massless particles [@Haag:1974qh].[^1]
For later purposes, let us review the salient feature of the energy-momentum tensor. The Poincaré invariance demands that it is symmetric and conserved: $\partial^\mu T_{\mu\nu} = \partial^\mu T_{\nu\mu} = 0$. The dilatation invariance demands that its trace is given by the divergence of the so-called Virial current [@Wess][@Mack][@Coleman:1970je]: $T^{\mu}_{\ \mu} = \partial^\mu J_\mu$. Finally, the conformal invariance demands that the Virial current can be expressed as a derivative: $J_\mu = \partial^\nu L_{\mu\nu}$ so that the energy-momentum tensor can be improved to be traceless (see e.g. [@Polchinski:1987dy]).
The alternative but restrictive way to see the conformal group would be to consider the inversion $x^\mu \to -\frac{x^\mu}{x^2}$. The successive transformation of inversion, translation and the second inversion gives the special conformal transformation $x^\mu \to \frac{x^\mu -a^\mu x^2}{1-2a^\mu x_\mu + a^2 x^2}$. Obviously the theory which is invariant under the inversion (together with the Poincaré invariance) is invariant under the full conformal transformation, but the inversion is not necessarily required to have $SO(2,d)$ conformal symmetry. Indeed, with the inversion, the symmetry [*group*]{} (rather than the algebra) is enhanced to $O(2,d)$ (see e.g. [@Weinberg:2010fx]), and the inversion lies in the disconnected component of the group.
In many correlation functions, however, it happens that the effect of the CP violation cannot appear simply due to the strong constraint coming from the special conformal transformation. For instance, all the scalar correlation functions are insensitive to the CP violation. To see a possibility to have a non-trivial CP violating contribution, let us consider the two-point function of the energy-momentum tensor on the flat space-time [@Osborn:1993cr][@Erdmenger:1996yc]: $$\begin{aligned}
\langle T_{\mu\nu}(x) T_{\alpha\beta}(0) \rangle_{\text{CP even}} = c\frac{ \mathcal{I}_{\mu\nu;\alpha\beta}(x)}{x^{2d}} \ ,\end{aligned}$$ where $\mathcal{I}_{\mu\nu;\alpha\beta} = \frac{1}{2}(I_{\mu\alpha}I_{\nu \beta} + I_{\mu\beta} I_{\nu\alpha}) - \frac{1}{d}\delta_{\mu\nu} \delta_{\alpha\beta}$ with $I_{\mu\nu} = \delta_{\mu\nu} -2\frac{x_\mu x_\nu}{x^2}$ is the fixed parity even tensor, and the whole structure is entirely dictated by the $SO(2,d)$ invariance with or without inversion. This result is valid in any space-time dimension except for $d=3$, and we see that there is no CP violating term here. The exception appears only in $d=3$. There, we may have a possible parity violating term (see e.g. [@Leigh:2003ez][@Maldacena:2011nz]):[^2] $$\begin{aligned}
\langle T_{\mu\nu}(x) T_{\alpha\beta}(0) \rangle_{\text{CP odd}} = w (\epsilon_{\mu \alpha \sigma}(\partial_\nu \partial_\beta - \eta_{\nu \beta} \partial^2) \partial^\sigma \delta^4(x) + \text{sym}) \ , \label{emt}\end{aligned}$$ where sym means the symmetrization under $\mu \leftrightarrow \nu$ and $\alpha \leftrightarrow \beta$. This CP violating term is classified as a contact term (due to the delta-function in ), and the physical significance is a little bit subtle. The appearance of the CP violating term in three-dimension has a deep connection with the parity anomaly because integrating out an odd number of massive fermions gives rise to effective gravitational Chern-Simons action that would generate the contact term . How this parity violating contact term can arise in the holographic computation was pursued in some literatures (e.g. [@Leigh:2003ez][@Leigh:2008tt]). It is ultimately due to the gravitational theta term [@Deser:1980kc] $\int d^4x \sqrt{g} \theta_{G} \epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\mu\nu}_{\ \ \gamma \delta}$ that violates the CP symmetry in the bulk four-dimensional gravitational theory.[^3]
A somewhat related manifestation of the CP violation in (conformal) field theories in three space-time dimensional space-time is the expectation value of the energy-momentum tensor in the curved background. It can possess the parity violating term [@deHaro:2008gp]: $$\begin{aligned}
\left \langle T_{\mu\nu} \right \rangle_{\text{CP odd}} = \tilde{w} C_{\mu\nu}\end{aligned}$$ where $C_{\mu\nu} = \epsilon_{\mu \sigma \rho} D^{\sigma} (R^{\rho}_{\ \nu} -\frac{1}{4} R \delta^{\rho}_{\ \nu}) $ is the (traceless and conserved) Cotton tensor, which is intrinsic to three-dimension. Again the explicit appearance of $\epsilon_{\mu \sigma \rho}$ suggests violation of parity. Since the Cotton tensor is traceless, there is no trace anomaly here.
The main focus of this paper is in four-dimensional space-time. As we have seen, in four dimension, there is no CP violating term in two-point function of the energy-momentum tensor in conformal field theories. This is even true if we relax the conformal invariance to the mere scale invariance (with the Poincaré invariance intact) [@Dorigoni:2009ra].
In four-dimension, the dimensional analysis demands that the most general possibility of the trace anomaly (for an earlier review, see [@Duff:1993wm][@Deser:1996na] and references therein) be $$\begin{aligned}
T^{\mu}_{\ \mu} = cF + a G + b R^2 + b' \Box R + e \epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\mu\nu}_{\ \ \gamma \delta} \ , \end{aligned}$$ aside from possible operator violation in non-conformal field theories. Here $F = R^{\alpha\beta\gamma\delta} R_{\alpha\beta\gamma \delta} - 2R^{\alpha\beta} R_{\alpha \beta} + \frac{1}{3}R^2$ is the square of the Weyl tensor while $G= R^{\alpha\beta\gamma\delta} R_{\alpha\beta\gamma \delta} - 4R^{\alpha\beta} R_{\alpha \beta} + R^2$ is the Euler density. The last term $\epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\mu\nu}_{ \ \ \gamma \delta}$ is the parity odd Hirzebruch-Pontryagin density.
In conformal field theories, the Wess-Zumino consistency condition, which is essentially the requirement that the successive applications of the Weyl transformation must commute because the Weyl transformation is abelian, demands that $b=0$ [@Bonora:1983ff][@Bonora:1985cq][@Cappelli:1988vw][@Osborn:1991gm]. In addition, $b'$ term is trivial in the sense that we can always remove it by adding the local counterterm $\int d^4x \sqrt{g} R^2$. See [@Deser:1976yx][@Deser:1993yx] for further discussions on the classification of the CP non-violating terms.
The parity odd term, which has been neglected in many literatures, cannot be logically excluded. It satisfies the Wess-Zumino consistency condition [@Bonora:1985cq] because the Hirzebruch-Pontryagin density is Weyl invariant [@Deser:1996na]. It may serve as a barometer that measures the violation of the CP in a given conformal field theory. In the rest of the paper, we would like to investigate the structure and the consequence of this term in the trace anomaly.
Of course, the Wess-Zumino consistency condition does not tell us whether we indeed have such a term. For this purpose, we need an explicit computation. What bothers us here is that the simplest computation of the trace anomaly is done in free field theories, necessarily at one-loop. It turns out that all the free conformal invariant field theories in four-dimension are actually invariant under CP, so the free field computation always predicts $e=0$.
If this were the chiral anomaly, where the one-loop exactness is proved (known as the Adler-Bardeen theorem), this would be the end of the pursuit. If there were no anomaly at one-loop, there would be none in the full computation (at least to all orders in perturbation theory). Fortunately, the one-loop exactness is not true for the trace anomaly. We already know that the central charge $a$ and $c$ are not one-loop exact, and there is no reason why it should be so for the CP odd term with the “CP violating central charge" $e$. Indeed, to pick up the CP odd term in the trace anomaly, we need at least two-loop or higher (or even non-perturbative) contributions.[^4]
To counter the suspicion for the very possibility to have the Hirzebruch-Pontryagin density in the trace anomaly, we would like to mention that there exists a free field computation for the self-dual (or anti-self-dual) two-form gauge field (i.e. $B_{\mu\nu} = \pm \epsilon_{\mu\nu\rho\sigma}B^{\rho\sigma}$) in four-dimensional Euclidean-signatured space. The explicit heat kernel analysis of the propagator showed that the self-dual two-form gauge field gives rise to the trace anomaly [@Duff:1980qv][^5] $$\begin{aligned}
T^{\mu}_{\ \mu} = \frac{1}{180(4\pi)^2} \left( 33R^{\alpha\beta\gamma\delta} R_{\alpha\beta\gamma \delta} - 93 2R^{\alpha\beta} R_{\alpha \beta} + \frac{45}{2}R^2 - 12 \Box R + 30 \epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\mu\nu}_{\ \ \gamma \delta} \right) \end{aligned}$$ while the anti-self dual two-form gives rise to the trace anomaly $$\begin{aligned}
T^{\mu}_{\ \mu} = \frac{1}{180(4\pi)^2} \left( 33R^{\alpha\beta\gamma\delta} R_{\alpha\beta\gamma \delta} - 93 2R^{\alpha\beta} R_{\alpha \beta} + \frac{45}{2}R^2 - 12 \Box R - 30 \epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\mu\nu}_{\ \ \gamma \delta} \right) \end{aligned}$$ The geometrical reason why we obtained the Hirzebruch-Pontryagin density is rather clear: the integrated anomaly will give the Hirzebruch signature, and it directly relates the zero-mode of the two-form gauge field through the index theorem. The heat kernel computation is a simple manifestation of the famous Hirzebruch signature theorem: $n(B^+) - n(B^-) = \frac{1}{48\pi^2} \int d^4x \sqrt{g} \epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\mu\nu}_{\ \ \gamma \delta}$
The drawback of the free field computation here is that the restriction to the (imaginary) self-dual (or anti-self-dual) two-form field in Minkowski signature leads to non-unitarity. Therefore, the self-dual (or anti-self-dual) two-form gauge field considered here is not physical. However, it clearly demonstrates that the possibility of the Hirzebruch-Pontryagin density in the trace anomaly is not something that can be thrown away immediately.[^6] In the next section, we try to realize the emergence of the Hirzebruch-Pontryagin density in the trace anomaly from the holographic renormalization group approach.
The appearance of the CP violating Hirzebruch-Pontryagin density in the trace anomaly will affect the structure of the three-point functions of the energy-momentum tensor in flat four-dimensional space-time. In generic $d$-dimensional space-time, the structure of the three-point function of the energy-momentum tensor in conformal field theories with no parity violation, was studied in [@Osborn:1993cr][@Erdmenger:1996yc]. Due to its complexity in appearance, we will not show the whole structure here, but if we take trace of one of the energy-momentum tensor, the result simplifies a bit and it reduces to contact terms $$\begin{aligned}
& \langle T^{\mu}_{\ \mu} (x) T_{\sigma\rho}(y)T_{\alpha\beta}(z) \rangle_{\text{CP even}} \cr
&= 2 (\delta^4(x-y) + \delta^4(x-z)) \langle T_{\sigma \rho}(y) T_{\alpha\beta}(z) \rangle - 4(c \mathcal{A}^F_{\sigma \rho, \alpha \beta} + a \mathcal{A}^G_{\sigma \rho, \alpha \beta} ) \ ,\end{aligned}$$ where $$\begin{aligned}
\mathcal{A}^F_{\sigma \rho, \alpha \beta} &=-8 \mathcal{E}^C_{\sigma \kappa \lambda \rho, \alpha \gamma \delta \beta} \partial^\kappa \partial^\lambda \delta^4(x-y) \partial^\gamma \partial^\delta \delta^4(x-z) \cr
\mathcal{A}^G_{\sigma \rho, \alpha \beta} &= \epsilon_{\sigma \alpha \gamma \kappa} \epsilon_{\rho \beta \delta \lambda} \partial^\kappa \partial^\lambda(\partial^\gamma \delta^4 (x-y)\partial^\delta \delta^4(y-z)) + \text{sym} \ .\end{aligned}$$ Here $ \mathcal{E}^C_{\sigma \kappa \lambda \rho, \alpha \gamma \delta \beta} = \partial C_{\mu\sigma\rho\nu}/ \partial C^{\alpha \gamma \delta \beta}$, and we can find the explicit form in Appendix A of [@Erdmenger:1996yc].
Now with the CP violation, the three-point function must possess the additional term $$\begin{aligned}
\langle T^{\mu}_{\ \mu}(x) T_{\sigma \rho}(y) T_{\alpha \beta}(z) \rangle_{\text{CP odd}} = e \epsilon_{\sigma \alpha \epsilon \kappa} (\partial_\beta \partial_\rho - \partial^2 \delta_{\beta \rho})[ \partial^\epsilon \delta^4(x-y) \partial^{\kappa} \delta^4(x-z) ] + (\text{sym}) \ . \label{mmv}\end{aligned}$$ We have not studied the structure of the full three-point functions with the CP violation due to its complexity. It would be interesting to see its structure, and verify whether there is any other structure whose origin is not related to . In particular, it would be exciting to see whether we have any free additional parameters besides the “CP violating central charge" $e$ to completely determine the three-point function of the energy-momentum tensor with CP violation.
Holographic realization
=======================
We have seen that the free field (or one-loop) computation of the trace anomaly does not lead to the CP violating Hirzebruch-Pontryagin density because all the free unitary conformal field theories preserve CP. We may attempt computing the higher loop corrections, but in this section, we take an alternative approach based on the holographic renormalization group [@Henningson:1998gx] to purse its possibility in strongly coupled dual field theories.
The toy model we will consider is the generalization of the model studied in [@Nakayama:2009qu][@Nakayama:2009fe][@Nakayama:2010ye] for a gravity dual of scale invariant but not conformal field theories. It is given by the five-dimensional Einstein gravity coupled with a self-interacting vector field with the action $$\begin{aligned}
S_{\mathrm{bulk}} = \int d^5x \sqrt{g}\left[ \frac{1}{2\kappa_5}(R - 2\Lambda) + \left(\frac{1}{4}F^2 + V(A_M A^M) \right) \right]\ .\end{aligned}$$ In order to break the CP of the bulk gravity, we introduce the gravity-vector Chern-Simons-like term $$\begin{aligned}
S_{\mathrm{CSL}} = q\int d^5x \sqrt{g} \epsilon^{LMNPQ} A_L R_{MN IJ} R_{\ \ PQ}^{IJ} \label{CSL}\end{aligned}$$ To make the variation principle well-defined, we may want to introduce the boundary term [@Landsteiner:2011iq] $$\begin{aligned}
S_{\mathrm{CSK}} = -8q\int_{\partial M} d^4x\sqrt{h} n_M \epsilon^{MNPQR}A_N K_{PL} D_QK^L_R \ , \end{aligned}$$ where $n_M$ is the normal vector and $K_{AB}$ is the extrinsic curvature.
The Chern-Simons-like term is imperative to break the parity invariance of the gravitational bulk theory, and it is the same action that would generate the gravitational chiral anomaly of a conserved current in the boundary theory if $A_{M}$ were a gauge field. This fact will be crucial in the field theory interpretation we will discuss later.
The condition of the scale invariance dictates that the metric must take the form of $AdS_5$ $$\begin{aligned}
ds^2 = g_{MN}dx^M dx^N = R^2_{AdS_5}\frac{dz^2 + \eta_{\mu\nu}dx^\mu dx^{\nu}}{z^2} \ . \end{aligned}$$ We choose the potential $V(A_M A^M) = \sum_n a_n (A_M A^M)^n$ so that $A = A_{M} dx^M = a\frac{dz}{z}$ is the solution of the equations of motion. The potential $V(A_M A^M)$ explicitly breaks the gauge invariance of the vector field $A_M$. Alternatively one can regard $A_M$ as the gauge fixed version of the Stueckelberg field with the higher derivative covariant action $ \sum_n a_n (\partial_M \phi - A_M)^{2n} $ for the Stueckelberg scalar $\phi$ in the unitary gauge $\phi=0$.
We can see that the vector condensation does not backreact to the metric, so the geometry is still AdS space. Indeed, the solution is the same one studied in [@Nakayama:2009qu][@Nakayama:2009fe][@Nakayama:2010ye] in the context of the gravity dual of scale invariant but non-conformal field theory. Clearly, the extra Chern-Simons-like term did not affect the classical solution because $dA = 0$.
We want to study the holographic renormalization of the system by considering the Fefferman-Graham expansion of the metric $$\begin{aligned}
\frac{ds^2}{R^2_{AdS_5}} = \frac{dz^2}{z^2} + \frac{h_{\mu\nu} dx^{\mu} dx^{\nu}}{z^2}\end{aligned}$$ with $$\begin{aligned}
h_{\mu\nu} = h_{\mu\nu}^{(0)} + z^2 h_{\mu\nu}^{(2)} + z^4h_{\mu\nu}^{(4)} + \cdots \end{aligned}$$ and evaluating the on-shell action. The on-shell action is divergent so we introduce the cutoff at $z= \epsilon$. The logarithmic dependence of the on-shell action on the cutoff is then interpreted as the holographic trace anomaly. Aside from the usual term that gives the holographic realization of the parity preserving trace anomaly $T^{\mu}_{\ \mu} = cF + a G$, we can immediately find the counter-term necessary from the Chern-Simons-like term $$\begin{aligned}
S_{\mathrm{CSL}}^{(0)} = qa\log \epsilon \int_{\partial M} d^4x \sqrt{h} \epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\ \ \mu\nu}_{\gamma \delta} \label{onsa}\end{aligned}$$ by noting $ A = a\frac{dz}{z} = a d(\log z)$ is exact and by using the Stokes theorem. This leads to the additional CP violating contribution to the trace anomaly of the dual boundary field theory $$\begin{aligned}
T^{\mu}_{\ \mu}|_{\text{CP odd}} = e \epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\ \ \mu\nu}_{\gamma \delta} \ \end{aligned}$$ with $e=qa$. In this way, we have demonstrated how the CP violating Hirzebruch-Pontryagin density can arise in the trace anomaly from the holographic computation.
We would like to give the interpretation of the CP-violating contribution to the trace anomaly in this model from the dual field theory. First of all, we recall that in scale but non-conformal field theory, the trace of the energy-momentum tensor is non-zero even in the flat space-time: rather it is given by the divergence of the so called Virial current $$\begin{aligned}
T^{\mu}_{\ \mu} = \partial^\mu J_\mu \ . \label{virir}\end{aligned}$$ In our holographic description, the vector condensation $A = a\frac{dz}{z}$ is dual to the existence of the non-zero Virial current [@Nakayama:2010wx]. As we have mentioned, the current model serves as the gravity dual of sale invariant but non-conformal field theory due to the existence of the non-zero Virial current.
In our CP violating scenario, we assume that the Virial current contains the CP violating term, or in other words it is a chiral current. While the Virial current is not conserved, it is typical that some sort of equations of motion were used in deriving the equality .[^7] Now, the key idea is that once we evaluate the equality in the curved background, it is expected that the gravitational chiral anomaly for the Virial current $J_\mu$ would give an additional piece in . Since we know that the gravitational chiral anomaly must contain the Hirzebruch-Pontryagin term [@Eguchi:1976db] as $D^\mu J_\mu = \kappa \epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\ \ \mu\nu}_{\gamma \delta} + (\text{non-anomalous term})$, we expect $$\begin{aligned}
T^{\mu}_{\ \mu} = D^\mu J_{\mu} - \kappa \epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\ \ \mu\nu}_{\gamma \delta} + \cdots\end{aligned}$$ with possible CP non-violating contribution to the trace anomaly.
This is exactly what was happening in the holographic computation. The Chern-Simons-like term we added is nothing but declaring that the current under consideration has a gravitational chiral anomaly. The fact that it is not gauge field and has a vacuum expectation value $a \frac{dz}{z}$ is the manifestation that it is not conserved and it is rather the Virial current appearing in the trace of the energy-momentum tensor. The combination of these two led to the CP violating Hirzebruch-Pontryagin term in the trace anomaly as expected in the field theory argument above.
Finally, let us note the following fact. The Virial current has an ambiguity so that it is only defined up to a conserved current. When the theory possesses an extra conserved chiral current $J_\mu^c$ with gravitational chiral anomaly (i.e. $D^\mu J_\mu^c = \kappa_c\epsilon^{\alpha\beta \gamma \delta} R_{\alpha\beta \mu\nu}R^{\ \ \mu\nu}_{\gamma \delta}$), we can augment the Virial current with that conserved chiral current so that the new “improved" Virial current $\tilde{J}_\mu = J^\mu - \frac{\kappa}{\kappa_c} J_\mu^c$ shows no parity odd term after taking the divergence: $T^{\mu}_{\ \mu} = D^\mu \tilde{J}_{\mu} $.
The same thing can be done in the holographic computation. The conserved chiral current with the gravitational chiral anomaly can be implemented as a bulk gauge field $A^c$ with the gauge-gravity Chern-Simons interaction $q_c\int d^5x \sqrt{g} \epsilon^{LMNPQ} A_L^c R_{MN IJ} R_{\ \ PQ}^{IJ}$. Now, we do the (large) gauge transformation $A^c = -\frac{qa}{q_c} \frac{dz}{z}$ and evaluate the on-shell action by using the Stokes theorem. It again shows the logarithmic divergence with the Hirzebruch-Pontryagin density. Then we can cancel the logarithmic divergence of the on-shell action coming from the Virial current with this new contribution from the conserved current.
On one hand, this illustrates how the definition of the trace of the energy-momentum tensor can be ambiguous with more conserved currents in scale invariant but non-conformal field theories, but on the other hand, it demonstrates clearly that there is nothing wrong with having the Hirzebruch-Pontryagin density in the trace anomaly in conformal field theories. We may just discard the contribution from the genuine Virial current in the above discussions, and see the CP violating terms appear. It would be interesting to see how much that ambiguity can be fixed in conformal field theories from the purely field theory argument.[^8] For instance, when the theory does not possess any conserved chiral current with gravitational anomaly, there is no possibility of such.
The holographic model discussed in this section is based on the scale invariant but non-conformal field theory, and we do not know any cleaner, preferably conformal, holographic theories that show the CP violating trace anomaly except for the possibility to use the large gauge transformation mentioned in the last paragraph. Since it may be possible that there is no unitary scale invariant but non-conformal field theory in four-dimension [@Polchinski:1987dy][@Dorigoni:2009ra][@Nakayama:2010wx], there may be a hidden no-go theorem to have the Hirzebruch-Pontryagin density in unitary conformal field theories. We would like to leave this field theoretical question for future studies.
Discussions
===========
In this paper, we have studied how the Hirzebruch-Pontryagin density can appear in the trace anomaly when the theory under consideration breaks CP symmetry. We have demonstrated its possibility in the holographic renormalization computation. Although we did not discuss it in the main part of the paper, if we introduced the background gauge field for the global symmetry, we would also be able to introduce the CP violating Chern-Pontryagin density $\epsilon^{\mu\nu \alpha\beta} \hat{F}_{\mu\nu} \hat{F}_{\alpha\beta}$, where $\hat{F}_{\mu\nu}$ is the corresponding field strength, in the trace anomaly of CP violating conformal field theories. The gravity dual would require the vector-gauge Chern-Simons-like term $\int d^5x \sqrt{g} \epsilon^{MNLPQ} A_{M} \hat{F}_{NL}\hat{F}_{PQ}$. By using the same mechanism discussed in the previous section, we are able to reproduce the trace anomaly with the Chern-Pontryagin density.
In a similar manner, we may imagine that the chiral current anomaly could include the parity [*even*]{} term such as $\partial_\mu J^\mu_5 = \hat{e} F_{\mu\nu}F^{\mu\nu}$ in addition to the conventional parity odd term $ \epsilon^{\mu\nu\alpha \beta} {F}_{\mu\nu} {F}_{\alpha\beta}$ in CP violating theories. After all, for the $U(1)$ symmetry, it is known that the Wess-Zumino consistency condition does not forbid it. We know, however, according to the Adler-Bardeen theorem, at least to all orders in perturbation theory, there cannot be such a contribution. Not surprisingly, we did not find any gravity computation that gives the corresponding result as far as we tried. It would be interesting to give a proof of the no-go theorem from the holographic viewpoint.
Recently, the ingenious proof of the a-theorem was demonstrated in [@Komargodski:2011vj][@Komargodski:2011xv] when the flow is between two conformal field theories. The theorem states that a function called “a", which is nothing but the coefficient in front of the Euler term in the trace anomaly, always satisfies the inequality $a_{\mathrm{UV}} > a_{\mathrm{IR}}$ along the renormalization group flow for any pairs of conformal field theories. The crucial idea for its proof is to consider the Wess-Zumino term associated with the trace anomaly in the spirit of the anomaly matching. There, they only discussed the CP conserving trace anomaly, so it is important to understand what happens if we allow the CP violating trace anomaly we have discussed in this paper.
To begin with, let us consider the conformal invariant (non-universal) CP violating action for dilaton $\tau$. There is none at the two-derivative level. At the four-derivative level, the only possible new term is the Hirzebruch-Pontryagin density: $$\begin{aligned}
S^{\mathrm{eff}}_{\text{CP odd}} = \hat{e} \int d^4 x \sqrt{\hat{g}} \hat{\epsilon}^{\alpha\beta \gamma \delta} \hat{R}_{\alpha\beta \mu\nu}\hat{R}^{\mu\nu}_{\ \ \gamma \delta} \ ,\end{aligned}$$ where $\hat{g}_{\mu\nu} = e^{-2\tau} g_{\mu\nu}$. If we evaluate this non-universal term in the flat space, it obviously vanishes since the Hirzebruch-Pontryagin density is a total derivative, so there is no additional CP violating four-derivative non-universal interaction for the dilaton.
Now, we consider the Wess-Zumino term. To cancel the ultraviolet CP violating trace anomaly, we have to introduce the Wess-Zumino term $$\begin{aligned}
(e_{\mathrm{UV}}-e_{\mathrm{IR}}) \int d^4x \sqrt{g} \tau {\epsilon}^{\alpha\beta \gamma \delta} {R}_{\alpha\beta \mu\nu}{R}^{\mu\nu}_{\ \ \gamma \delta} \ .\end{aligned}$$ Like c-anomaly and unlike a-anomaly, there is no further term necessary to complete the Wess-Zumino term because the Hirzebruch-Pontryagin density is Weyl invariant. Consequently, if we evaluate the Wess-Zumino term in the flat space-time, it vanishes and there is no CP violating contribution to the dilaton effective action. We thus conclude that the proof of a-theorem in [@Komargodski:2011vj][@Komargodski:2011xv] is not affected by the existence of the CP violating contribution to the trace anomaly.
As we have mentioned in section 2, in non-conformal field theories, there may exist another “central charge" $b$ that appears in $R^2$ term of the trace anomaly. In principle such a term can appear in the holographic computation of the scale invariant but non-conformal field theory. In the model we have studied in section 3, we did not find such a contribution but it is plausible that it would appear if we added the term like $R_{MN}A^MA^N$ to the gravity action, which breaks the AdS isometry in the gravity sector equation of motion with the vector condensation. It would be also interesting to verify how precisely $b$ term does or does not affect the derivation of the a-theorem for non-conformal but scale invariant fixed points (see [@Nakayama:2011wq] for related studies).[^9]
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Y. Tachikawa for discussions. The work is supported by the World Premier International Research Center Initiative of MEXT of Japan.
[99]{} R. Haag, J. T. Lopuszanski and M. Sohnius, Nucl. Phys. B [**88**]{}, 257 (1975). J. Wess, Nuovo Cimento 18 (1960) 1086. G. Mack, “Partially Conserved Dilatation Current”, Ph.D. thesis, Bern 1967;
G. Mack and A. Salam, Annals Phys. [**53**]{} (1969) 174. S. R. Coleman and R. Jackiw, Annals Phys. [**67**]{} (1971) 552. J. Polchinski, Nucl. Phys. B [**303**]{}, 226 (1988).
S. Weinberg, Phys. Rev. D [**82**]{}, 045031 (2010) \[arXiv:1006.3480 \[hep-th\]\].
H. Osborn and A. C. Petkou, Annals Phys. [**231**]{}, 311 (1994) \[hep-th/9307010\]. J. Erdmenger and H. Osborn, Nucl. Phys. B [**483**]{}, 431 (1997) \[hep-th/9605009\]. R. G. Leigh and A. C. Petkou, JHEP [**0312**]{}, 020 (2003) \[hep-th/0309177\].
J. M. Maldacena and G. L. Pimentel, JHEP [**1109**]{}, 045 (2011) \[arXiv:1104.2846 \[hep-th\]\].
R. G. Leigh, N. N. Hoang and A. C. Petkou, JHEP [**0903**]{}, 033 (2009) \[arXiv:0809.5258 \[hep-th\]\].
S. Deser, M. J. Duff and C. J. Isham, Phys. Lett. B [**93**]{}, 419 (1980).
S. de Haro, JHEP [**0901**]{}, 042 (2009) \[arXiv:0808.2054 \[hep-th\]\].
D. Dorigoni and V. S. Rychkov, arXiv:0910.1087 \[hep-th\].
M. J. Duff, Class. Quant. Grav. [**11**]{}, 1387 (1994) \[hep-th/9308075\]. S. Deser, Helv. Phys. Acta [**69**]{}, 570 (1996) \[hep-th/9609138\].
L. Bonora, P. Cotta-Ramusino and C. Reina, Phys. Lett. B [**126**]{}, 305 (1983). L. Bonora, P. Pasti and M. Bregola, Class. Quant. Grav. [**3**]{}, 635 (1986). A. Cappelli and A. Coste, Nucl. Phys. B [**314**]{}, 707 (1989). H. Osborn, Nucl. Phys. B [**363**]{}, 486 (1991).
S. Deser, M. J. Duff and C. J. Isham, Nucl. Phys. B [**111**]{}, 45 (1976). S. Deser and A. Schwimmer, Phys. Lett. B [**309**]{}, 279 (1993) \[hep-th/9302047\].
M. J. Duff and P. van Nieuwenhuizen, Phys. Lett. B [**94**]{}, 179 (1980). J. S. Dowker and R. Critchley, Phys. Rev. D [**16**]{}, 3390 (1977). S. M. Christensen and M. J. Duff, Phys. Lett. B [**76**]{}, 571 (1978). S. M. Christensen and M. J. Duff, Nucl. Phys. B [**154**]{}, 301 (1979).
M. Henningson and K. Skenderis, JHEP [**9807**]{}, 023 (1998) \[hep-th/9806087\]. Y. Nakayama, arXiv:0907.0227 \[hep-th\]. Y. Nakayama, JHEP [**1001**]{}, 030 (2010) \[arXiv:0909.4297 \[hep-th\]\]. Y. Nakayama, arXiv:1003.5729 \[hep-th\].
K. Landsteiner, E. Megias, L. Melgar and F. Pena-Benitez, JHEP [**1109**]{}, 121 (2011) \[arXiv:1107.0368 \[hep-th\]\].
Y. Nakayama, arXiv:1009.0491 \[hep-th\]. T. Eguchi and P. G. O. Freund, Phys. Rev. Lett. [**37**]{}, 1251 (1976). E. Witten, In \*Shifman, M. (ed.) et al.: From fields to strings, vol. 2\* 1173-1200 \[hep-th/0307041\].
Z. Komargodski and A. Schwimmer, JHEP [**1112**]{}, 099 (2011) \[arXiv:1107.3987 \[hep-th\]\]. Z. Komargodski, arXiv:1112.4538 \[hep-th\]. Y. Nakayama, arXiv:1110.2586 \[hep-th\].
[^1]: Note that the algebraic consideration in [@Haag:1974qh] did not say anything about the global structure of the group. The inclusion of the fermion necessitates the inclusion of the spinor representations, and the symmetry [*group*]{} must be its double cover. Additional discrete space-time parities (e.g. CP symmetry) may or may not be the symmetry of the full theory.
[^2]: In this paper, the Levi-Civita symbol is defined as a tensor rather than a tensor density.
[^3]: The bulk gravitational theta term reduces to the boundary gravitational Chern-Simons term, and it directly gives the parity violating contact term in the boundary correlation function.
[^4]: A non-perturbative contribution is certainly important when the CP violation is due to the Yang-Mills theta term.
[^5]: In the formula here, we have not introduced the ghost contribution which in any case would not affect the Hirzebruch-Pontryagin density.
[^6]: Indeed, the appearance of the Hirzebruch-Pontryagin density is ubiquitous in non-unitary free field computations. It has been demonstrated [@Dowker:1976zf][@Christensen:1978gi][@Christensen:1978md] that whenever the Lorentz group representation is not real (or not symmetric under the exchange of the two $SU(2)$ in Euclidean signature), the contribution is non-zero. We would like to thank M. Duff for the correspondence.
[^7]: In renormalizable field theories, the trace of the energy-momentum tensor is typically given by $\phi^4$ terms or Yukawa terms. To connect them to divergence of currents of dimension 3, we use the equations of motion.
[^8]: It is important to note that it is not mandatory to cancel the CP violating term even if we have such an option. The choice of the large gauge parameter determines the theory on the curved space-time and our statement is that simply there are as many choices. The situation is closer to the parity violating contact term in three-dimensional conformal field theory reviewed in section 2. The choice of the contact term defines different theories (see [@Witten:2003ya] for a related discussion).
[^9]: In [@Nakayama:2011wq], the scale invariant field theory is embedded in a conformal field theory to discuss the a-theorem for scale but non-conformal field theories, and the $b$ anomaly must have vanished for the consistency of the embedding. The treatment and the argument given there is self-consistent, but it remains open how the $b$ anomaly is cancelled in actual models. Furthermore, after all, we have not succeeded in deriving a-theorem in scale but non-conformal field theories, so the conformal embedding might not be a good idea, and then non-zero $b$ may play a role.
|
---
abstract: 'Indoor positioning aims at navigation inside areas with no GPS-data availability, and could be employed in many applications such as augmented reality, autonomous driving specially inside closed areas and tunnels. In this paper, a deep neural network based architecture has been proposed to address this problem. In this regard, a tandem set of convolutional neural networks, as well as a Pix2Pix GAN network have been leveraged to perform as the scene classifier, scene RGB image to point cloud converter, and position regressor, respectively. The proposed architecture outperforms the previous works, including our recent work, in the sense that it makes data generation task easier and more robust against scene small variations, whilst the accuracy of the positioning is remarkably well, for both Cartesian position and quaternion information of the camera.'
address: |
$^1$ Faculty of Technology and Media Engineering, Iran Broadcasting University (IRIBU), Tehran, Iran\
$^2$ CEO/CTO at Alpha Reality, AR/VR Solution Company\
alighofrani@iribu.ac.ir, mahdian.t.r@gmail.com, {smtabatabaie,m.tabasi}@alphareality.io
bibliography:
- 'Ghofrani.bib'
title: 'LIDAR ICPS-NET: INDOOR CAMERA POSITIONING BASED-ON Generative Adversarial Network for RGB TO POINT-CLOUD translation'
---
Indoor positioning, point cloud data, Convolutional neural networks, Generative adversarial networks, Pix2Pix GAN.
Introduction {#sec:intro}
============
Global positioning system is a problem, which has been contributed using navigation systems, and GPS satellites. The indoor positioning, on the other hand is still challenging task due to the fact that inside covered areas with no GPS signal available, image processing tasks are the only solutions to be resorted (e.g., SIFT and SURF). These methods are not very accurate [@sattler2016efficient]. The main reason is the existence of several identical patterns inside the buildings, which could easily fool the positioning system.
The first data-driven approach using convolutional neural networks (CNN) was POSENET [@kendall2015posenet], which could work for a limited open area. Further, a geometry-aware system was proposed for camera localization which incorporated perceptual and temporal features to improve the precision, [@henriques2018mapnet]. However, both these methods were applicable in outdoor positioning. In most traditional indoor positioning systems, which do not involve wireless means [@yang2015wifi], the depth-assisted camera is necessary to be used [@zhang2018real], which is not always available in real-world scenarios, such as mobile handsets.
The first indoor positioning system using deep neural networks, was proposed by the authors of this paper [@ghofrani2019icps], through scanning of the desired area segments using photogrammetry method. A classifier is then trained by a CNN structure (i.e. EfficientNet [@tan2019efficientnet]), and followed by a MobileNet CNN structure [@sandler2018mobilenetv2], which has already been trained to perform as a regressor. This structure could achieve a remarkable precision result for the Cartesian position and quaternion information of the camera [@ghofrani2019icps]. The remaining challenge of the previous work is that, generation of such a huge amount of RGB data for training the deep neural network is an overwhelming task. Moreover, for the case in which the area is subject to small changes, then the RGB based data is no longer trustable and the output of the previous system is not robust, at all.
A solution to the aforementioned problem would be to generate a point cloud data using a LiDAR system rather than RGB cameras, which is both easier and more robust.
In this work, we extended our research to investigate whether it would be possible for our regressor-CNN to be driven by a point-cloud data, rather than the RGB image. Wang et al. in [@wang2019pseudo], showed that it would be possible to detect the object using its associated point-cloud data. On the other hand, Shi et al. [@Shi_2019_CVPR], showed that it is possible to render the point-cloud data into associated images using GAN neural networks [@pix2pix2016].
![A big-picture of the LiDAR-based indoor positioning. []{data-label="fig:phases"}](image/phases){width="1\linewidth" height=".43\textheight"}
Following these two works, as illustrated in figure \[fig:phases\], the CNN-regressor is trained by the point-cloud data instead of the RGB image. Moreover, due to the fact that the clients normally have access to only RGB images on their mobile handsets, therefore we need a transformer which converts the RGB data into its associated point-cloud data which we perform it using a Pix2Pix GAN neural network to achieve this mapping. This enables the training procedure to be performed much easier than our previous work, and further within small environmental changes the model could perform more robust than before. These are explicitly the novelties of our work.
The Proposed Framework
======================
\[sec:format\] Regarding our previous work [@ghofrani2019icps], the following steps should be taken in a sequence: 1) The input images of the clients should be given to a classifier in order to determine the associated scene. Segmentation of the desired environment into scenes could be optional. However, when we decide about the number of scenes we have to fix it, and the classifier should be trained based on that. The structure of this scene classifier, which is an EfficientNet B0 [@tan2019efficientnet], is depicted in figure \[fig:class\].
{width="1.02\linewidth" height=".23\textheight"}
![Scene classifier based on EfficientNet B0.[]{data-label="fig:class"}](image/class){width="0.8\linewidth" height=".35\textheight"}
\[!b\] ![Sequences of Camera movements for each scene[]{data-label="fig:dataset"}](image/dataset "fig:"){width="0.98\linewidth" height=".16\textheight"}
2\) When the classifier determines the scene, the RGB image should be converted to its associated point-cloud using a Pix2Pix UNET-based GAN network [@ronneberger2015u]. 3) This generated point-cloud data, would be fed into the CNN-based regressor which has been trained based upon its associated scene.
![MobileNet V2, as the regressor trained by the point-cloud dataset[]{data-label="fig:final-regressorasas"}](image/final-regressorasas){width="0.85\linewidth" height=".55\textheight"}
Based on the above procedure, we need to primarily train a UNET-based GAN network [@ronneberger2015u], to perform the mapping of RGB images into point-cloud data. For this purpose, using a small amount of data samples which contain the pairs of RGB images, and their associated and compatible point-cloud data we could train the network. This network is depicted in figure \[fig:unet\].
Next, we need to train the regressor network, which is supposed to get the generated point-cloud data as the input and estimate the 7 values of Cartesian and Quaternion information as the output. This CNN-regressor (based on MobileNet V2) is depicted in figure \[fig:final-regressorasas\].
Experiments and Analytics
=========================
The hardware being used for the present work, is GTX 1080-NVIDIA, on a core i7 Cpu Intel 7700, with 32 GB RAM. Tensorflow 1.13.1 has been used with CUDA 10.1, and Keras 2.2.4 softwares are the platforms to implement the tasks.
Since there were no available data containing the RGB and associated point-cloud, we generated this dataset from the freely available 3D scanned images of the Hallwyl museum in Stockholm [@ghofrani2019hallwylmuseum]. We sampled from this 3D model using the Unity software, and the normalized outputs are saved in our generated dataset[^1]$^,$[^2]. More than $500,000$ pure data samples are generated from all the scenes using different regimes for the camera, depicted in figure \[fig:dataset\]. The equivalent point-cloud data for each of the image samples are created. In order to create the point-clouds, inside the Unity software we have modified the mesh descriptor of the environment mesh from the surface shader to geometry shader, in which the mesh vertexes are demonstrated using the points. Thus, for each RGB image the equivalent point-cloud data has been created. Since the GAN network training, requires some RGB and associated point-cloud data pairs, and the scene classifier also needs to be trained on the scenes through RGB images this may give the wrong impression that the RGB images are again under usage. However, the amount of RGB images which could be employed for the GAN network is sufficient to train the classifier network, as well. This has been investigated and the result confusion matrix has been depicted in figure \[fig:test\_sample\_ConfusionMatrix\].
![The confusion matrix for the classification of the scenes through EfficientNet.[]{data-label="fig:test_sample_ConfusionMatrix"}](image/test_sample_ConfusionMatrix){width="0.98\linewidth" height=".27\textheight"}
For the classifier, the loss function being used is the categorical cross-entropy, and the model is monitored toward maximizing the validation accuracy.
![(Left to right) input Point cloud, generated RGB-GAN output, and the ground truth RGB.[]{data-label="fig:pc2rgb"}](image/pc2rgb){width="0.9\linewidth" height=".4\textheight"}
![Classification accuracy (left), and loss (right) based on the categorical cross-entropy.[]{data-label="fig:new_loss_status"}](image/new_loss_status){width="1\linewidth" height=".135\textheight"}
In order to achieve the optimum performance, the drop-connect is employed to avoid overfitting [@wan2013regularization]. In addition, the swish as a SOTA activation function has been used, as the state-of-the-art [@ramachandran2017swish].
To train the regressors, since the input dataset is point-cloud, it is not possible to use the imageNet-based training parameters, in a transfer learning procedure. Therefore, the entire training of the regressors has been performed from scratch via Xavier weight initializing technique [@DBLP:journals/corr/Kumar17]. The loss changing diagram has been depicted in figure \[fig:regressor\_losses\].
![(left) Quaternion loss, (right) Cartesian Loss. Losses are to be scaled using the scale factor in the loss function.[]{data-label="fig:regressor_losses"}](image/regressor_losses){width="1\linewidth" height=".13\textheight"}
The loss function should be chosen as in [@ghofrani2019icps]. This loss function is, as follows $$loss = ||P-\hat{P}||_2 + \frac{1}{\beta} ||\hat{Q}-\frac{Q}{||Q||} ||_2$$ where $P = [x, y, z]$ is the position data vector, Q is the quaternion information, and $\beta$ is the scale factor to make a balance between estimating the position and the quaternion.
![(Left to right) RGB input data, generated point cloud-GAN output, and the ground truth point cloud.[]{data-label="fig:rgb2pc"}](image/rgb2pc){width="0.9\linewidth" height=".41\textheight"}
The GAN training is based on the RGB-2-Point cloud data, which has been generated, as mentioned before. A sample of this data has been depicted in figure \[fig:rgb2pc\]. In a further investigation, we turned the GAN to work as a point cloud to RGB converter. Interestingly, the same network could perform quite well, as depicted in figure \[fig:pc2rgb\].
\[!b\]
**X**-position **Y**-position **Z**-position **Quaternion**
-- ---------------- ---------------- ---------------- ----------------
**0.019 m** **0.027 m** **0.0073 m** ** 0.0096**
: The regression error, for the position vector (X;Y;Z), and the camera Quaternion, over the test set (Unseen data)
\[tbl:error\_final\]
Conclusion
==========
An indoor position system has been proposed in this paper, based on a supervised deep network structure. The goal of the system is to achieve a high accuracy of the Cartesian (X,Y,Z) position and the camera quaternion, while being robust against environmental changes and object movements. A CNN-based classifier is used to identify the scene from the environment based on the client’s input RGB image. A GAN network has already been prepared to convert the RGB images into point cloud data which is easier available and more robust against variations of the scene background. The regressor CNNs are trained only based on the point clouds. The results of the experiments showed a remarkable achievement in positioning whilst making the entire procedure of our previous work much easier to be performed.
[^1]: <https://mega.nz/#F!FE9HFCLS!vHH7vqEd5PAFF-ItGR44ww>
[^2]: <https://drive.google.com/drive/folders/1Q2QaiQejigriIaFxn7G9csEXD6OEkYvm>
|
---
abstract: 'There is increasingly strong observational evidence that slow magnetoacoustic modes arise in the solar atmosphere. Solar magneto-seismology is a novel tool to derive otherwise directly un-measurable properties of the solar atmosphere when magnetohydrodynamic (MHD) wave theory is compared to wave observations. Here, MHD wave theory is further developed illustrating how information about the magnetic and density structure along coronal loops can be determined by measuring the frequencies of the slow MHD oscillations. The application to observations of slow magnetoacoustic waves in coronal loops is discused.'
---
Introduction
============
Damped slow MHD oscillations have been observed in the solar atmosphere using high-resolution EUV imager onboard space-borne telescopes (see review by Wang 2011). Such oscillations are important because of their potential for the diagnostics of magnetic structures by implementation of the method of magneto-seismology, through matching the MHD wave theory and wave observations in the solar atmosphere to obtain several physical parameters (e.g., magnetic field strength and density scale height).
The theory of MHD wave propagation in solar magnetic structures initially began modelling the magnetic structures as homogenous cylindrical magnetic flux tubes enclosed within a magnetic environment (Roberts et al. 1984). Later on, more advanced equilibrium models to study slow MHD oscillations have also been proposed with, e.g., dissipative effects and gravity (Mendoza-Briceño et al. 2004, Sigalotti et al. 2007), and non-isothermal profiles (Erdélyi et al. 2008), while the effect of density and magnetic stratification had been revisited on transversal coronal loop oscillations by Dymova & Ruderman (2006) and Verth & Erdélyi (2008), respectively.
Here, the governing equation of the longitudinal mode is solved numerically for density stratified loops with uniform magnetic field, as well as for expanding magnetic flux tubes with uniform density. The effect of these stratifications on the frequency ratio of the first overtone to the fundamental mode is studied.
Governing equation
==================
The ideal MHD equations are linearized by considering small magnetic and velocity perturbations about a plasma in static equilibrium \[${\vec b} =(b_r, 0, b_z)$ and ${\vec \upsilon}=(\upsilon_r, 0, \upsilon_z)$, for $r$ and $z$ the radial and longitudinal coordinates, repectively\]. In the derivation, a uniform kinetic plasma pressure is assumed, and the thin flux tube approximation is considered. The second-order ordinary differential equation governing the longitudinal velocity amplitude is (see Luna-Cardozo et al. 2012 for a detailed derivation) $$\frac{d^2 \upsilon_{z}}{d z^2}+ \left(\frac{c_{\mathrm s}^{2}-c_{\mathrm A}^{2}}{c_{\mathrm f}%
^{2}} \right) \frac{1}{B_z} \frac{\partial B_z}{\partial z} \frac{d \upsilon_z}{d z} +
\left[ \frac{\omega^2}{c_{\mathrm T}^2} -\frac{1}{B_z} \frac{\partial^2 B_{z}}{\partial z^2}
-\left(\frac{c_{\mathrm s}^{2}-c_{\mathrm A}^{2}}{c_{\mathrm f}^{2}}
\right) \frac{1}{B_z^2} \left( \frac{\partial B_z}{\partial z}\right)^2 \right] \upsilon_z = 0,
\label{eq1}$$ where $c_{\mathrm A}^2= {(B_z^2 /\mu \rho_0)}$, $c_{\mathrm s}^2={(\gamma p_0 / \rho_0)}$, $c_{\mathrm f}^{2}=c_{\mathrm s}^2 + c_{\mathrm A}^2$ and $c_{\mathrm T}^{2} = (c_{\mathrm s}^{-2} + c_{\mathrm A}^{-2})^{-1}$ are the square of the Alfvén, sound, fast phase and tube speeds, respectively. In this equation, $\omega$ is the angular frequency of the oscillations.
Equation (\[eq1\]) is numerically solved using the shooting method based on the Runge-Kutta technique, for density stratified loops with uniform magnetic field, as well as for expanding loops with uniform density. Solar waveguides are modelled as axisymmetric cylindrical magnetic tubes with tube ends frozen in a dense photospheric plasma at $z=\pm L$. On average, plasma density and magnetic field strength are expected to decrease with height above the photosphere (Lin et al. 2004).
Effect of density stratification
================================
The solar coronal loop is modelled by a straight axisymmetric magnetic flux tube with tube length of $2L$ and radius of $r_0$. The uniform magnetic field is directed along the tube axis, i.e., ${\vec B} = B_z \hat{z}$. In semi-circular coronal loops where the plasma is close to hydrostatic equilibrium, a reasonable assumption for the density profile is $$\rho_0 (z) = \rho_{\mathrm f} \exp \left[ -\frac{2L}{\pi H} \cos
\left(\frac{\pi z}{2L}\right)\right], \label{eq2}$$ where $H$ is the density scale height and $\rho_{\mathrm f}$ the density at the footpoint. To study a standing wave the boundary condition $\upsilon_{z} (\pm L) = 0$ is applied. We solve equation (\[eq1\]) using the density profile (\[eq2\]). The frequency ratio of the first overtone to the fundamental mode is shown in Figure \[fig1\](a) as a function of $L/H$ by the solid line, and it is clearly lower than the cannonical value of two. A similar result was obtained for the transversal mode by Dymova & Ruderman (2006) and Verth (2007).
For vertical chromospheric flux tubes the density profile is given by $$\rho_0 (z) = \rho_{\mathrm f} \exp \left[ -\frac{(z+L)}{H} \right]. \label{eq3}$$ Longitudinal oscillations in chromospheric flux tubes are studied solving the eigenvalue problem in half of the magnetic bottle, i.e., designating $\upsilon_{z} (-L) = \upsilon_z (0) = 0$ as the boundary conditions. The ratio of frequencies against $L/H$ for the density profile (\[eq3\]) is shown by the dashed line in Figure \[fig1\](a). Now, the frequency ratio is slightly greater than two, indicating that this parameter depends on the functional form chosen of the equilibrium density. This suggests that caution must be used when interpreting the frequency ratio of chromospheric standing modes.
![(a) Frequency ratio of the first overtone and fundamental mode against $L/H$ for density stratified coronal (solid line) and chromospheric (dashed line) loops. (b) Frequency ratio against the expansion parameter $\Gamma$ for different values of $\beta_{\mathrm f}$ in vertical chromospheric flux tubes. In (b) dotted, dot-dashed, dot-dot-dot-dashed and long-dashed lines correspond to $\beta_{\mathrm f} =1$, 2, 5 and 10, respectively.[]{data-label="fig1"}](f1 "fig:"){width="6.7cm"} ![(a) Frequency ratio of the first overtone and fundamental mode against $L/H$ for density stratified coronal (solid line) and chromospheric (dashed line) loops. (b) Frequency ratio against the expansion parameter $\Gamma$ for different values of $\beta_{\mathrm f}$ in vertical chromospheric flux tubes. In (b) dotted, dot-dashed, dot-dot-dot-dashed and long-dashed lines correspond to $\beta_{\mathrm f} =1$, 2, 5 and 10, respectively.[]{data-label="fig1"}](f3 "fig:"){width="6.7cm"}
Effect of a non-uniform magnetic field
======================================
An expanding flux tube with rotational symmetry about the $z$-axis in cylindrical coordinates ($r,\theta,z$) is used to model a magnetic field equilibrium decreasing in strength with height above the photosphere. The magnetic field component $B_z$ at the tube boundary can be described explicitly as function of $z$ (see Verth & Erdélyi 2008) $$B_z(z) \approx B_{z,\mathrm{f}} \left\{ 1+\frac{(1-\Gamma^2)}{\Gamma^2}
\frac{\left[\cosh \left(z/L \right)-\cosh(1)\right]}{1-\cosh (1)}
\right\}, \label{eq4}$$ where $\Gamma= {r_\mathrm{a}}/{r_\mathrm{f}}$ is the expansion factor, and $r_\mathrm{a}$ ($r_\mathrm{f}$) is the apex (footpoint) radius. The loop expansion has been estimated for various loops, giving mean values of $\Gamma \approx 1.16$ and 1.30 for EUV and soft X-ray loops (Watko & Klimchuk 2000, Klimchuk 2000).
We can compute the numerical solution of equation (\[eq1\]) for slow longitudinal oscillations in coronal and chromospheric loops setting the same boundary conditions as in the previous section, and using equation (\[eq4\]) for $B_z(z)$. Figure \[fig1\](b) shows the frequency ratio as function of the expansion parameter $\Gamma$ for different values of the footpoint beta plasma $\beta_{\mathrm f}$ for chromospheric flux tubes. It is found that when the magnetic expansion factor increases the frequency ratio [*decreases*]{}, and this effect is more significant for chormospheric flux tubes with higher $\beta_{\mathrm f}$.
Figure \[fig2\] shows the frequency ratio as function of the expansion factor for coronal loops with uniform density in (a) and for typical density stratification (i.e., $L/H = 2$) in (b). It can be seen how these two effects, density stratification and magnetic expansion, contribute to [*decrease*]{} the value of $\omega_2/\omega_1$. Additionally, the effect of the expansion is stronger in the corona than in the chromosphere.
![Frequency ratio of coronal loop oscillations against the expansion parameter $\Gamma$ for different values of $\beta_{\mathrm f}$. Solid, dashed, dotted, dot-dashed and dot-dot-dot-dashed lines correspond to $\beta_{\mathrm f} =0.1$, 0.5, 1, 2 and 5, respectively. Coronal loops with uniform density ($L/H=0$) are presented in (a) and with typical density stratification ($L/H =2$) in (b).[]{data-label="fig2"}](f2 "fig:"){width="6.7cm"} ![Frequency ratio of coronal loop oscillations against the expansion parameter $\Gamma$ for different values of $\beta_{\mathrm f}$. Solid, dashed, dotted, dot-dashed and dot-dot-dot-dashed lines correspond to $\beta_{\mathrm f} =0.1$, 0.5, 1, 2 and 5, respectively. Coronal loops with uniform density ($L/H=0$) are presented in (a) and with typical density stratification ($L/H =2$) in (b).[]{data-label="fig2"}](f4 "fig:"){width="6.7cm"}
Summary and conclusions
=======================
Studying the solutions of the velocity governing equation of the slow standing mode, it is found that density stratification and magnetic expansion cause the [*same qualitative effect*]{} on the frequency ratio in coronal loops, giving values of $\omega_2 /\omega_1<2$. For chromospheric flux tubes density stratification and magnetic expansion cause [*opposite effects*]{} on the frequency ratio; however, caution must be taken when studying chromospheric flux tubes since the ratio $\omega_2 /\omega_1$ depends on the functional form chosen for the density (see Luna-Cardozo et al. 2012 for an analytical and numerical detailed study about important issues).
These results are consistent with the values of period ratio of $P_1/P_2 = 1.54$ and 1.84 reported by Srivastava & Dwivedi (2010) while observing slow acoustic oscillations using [*Hinode*]{}, in contrast to the theoretical value of $P_1/P_2=2$ for a uniform cylindrical flux tube model.
Our results are important for magneto-seismology, where the density scale height of the solar atmosphere can be calculated by using the observed value of the frequency ratio $\omega_2 /\omega_1$ of longitudinal loop oscillations, to complement both emision measure and magnetic field extrapolation studies. This could provide us with a more complete understanding of the plasma fine structure in the solar atmosphere. These results can be applied in any stage of the solar cycle, including the solar minimum.
M.L.-C. thanks the IAU for the travel grant and is grateful for the financial support from PICT 2007-1790 grant (ANPCyT). R.E. acknowledges M. Kéray for patient encouragement and is also grateful to NSF, Hungary (OTKA, Ref. No. K83133) for financial support received.
Dymova, M. V., & Ruderman, M. S. 2006, *A*&*A* 457, 1059
Erdélyi, R., Luna-Cardozo, M., & Mendoza-Briceño, C. A. 2008, *Sol. Phys.* 252, 305
2000, *Sol. Phys.* 193, 53
Lin, H., Khun, J. R., & Coulter, R. 2004, *ApJ* 613, L177
2012, *ApJ* 748, 110
Mendoza-Briceño, C. A., Erdélyi, R., & Sigalotti, L. Di G. 2004, *ApJ* 605, 493
1984, *ApJ* 279, 857
Sigalotti, L. Di G., Mendoza-Briceño, C. A., & Luna-Cardozo, M. 2007, *Sol. Phys.* 246, 187
Srivastava, A. K., & Dwivedi, B. N. 2010, [*New Astron.*]{} 15, 8
2007, *Astron. Nachr.* 328, 764
2008, *A*&*A* 486, 1015
Wang, T. J. 2011, *Space Sci. Rev.* 158, 397
2000, *Sol. Phys.* 193, 77
|
---
author:
- |
[^1]\
\
Saha Institute of Nuclear Physics, Kolkata, India\
E-mail:
title: 'Inclusive $\psi$(2S) production at forward rapidity in pp, p-Pb and Pb-Pb collisions with ALICE at the LHC'
---
Introduction
============
The suppression of quarkonia (bound states of a heavy quark and its anti-quark) in ultra relativistic heavy ion collisions is one of the most prominent probes used to investigate and quantify the properties of the quark gluon plasma (QGP). The in-medium dissociation probability of the different quarkonium states could provide an estimate of the temperature of the system since the dissociation is expected to take place when the medium reaches or exceeds the critical temperature of the phase transition ($T_{\rm c}$), depending on the binding energy of the quarkonium state. For charmonium ($c\overline c$) states, the J/$\psi$ is likely to survive significantly above $T_{\rm c}$ (1.5 - 2 $T_{\rm c}$) whereas $\chi_{\rm c}$ and $\psi(2\rm S)$ melt near $T_{\rm c}$ (1.1 - 1.2 $T_{\rm c}$) [@satz; @satz2]. At LHC energies, due to the large increase of the $c\overline c$ production cross-section with the collision energy, there is a possibility of J/$\psi$ production via recombination of ${c}$ and $\overline {c}$. Thus, the observation of J/$\psi$ production in nucleus-nucleus collisions via recombination also constitutes an evidence of QGP formation. The study of the $\psi(2\rm S)$ production, due to its different binding energy, is complementary to that of the J/$\psi$ and it may also be useful for the evaluation of the temperature of the medium. The $\psi(2\rm S)$-to-J/$\psi$ cross-section ratio is predicted to be very sensitive to the details of the recombination mechanism. Experimentally this ratio is interesting as most of the systematic uncertainties cancel, with the remaining systematic uncertainties being only due to the signal extraction and the efficiency evaluation. The pp results for the charmonium provide a baseline for the nuclear modification factor of charmonium production in and collisions. The study of charmonia in collisions can be used as a tool for a quantitative investigation of the cold nuclear matter (CNM) effects including various mechanisms such as gluon shadowing, $c\overline c$ break-up via interaction with nucleons, initial/final state energy loss, relevant in the context of studies of the strong interaction. The region of very small $x$ is accessible at the LHC and therefore strong shadowing and coherent energy loss effects are expected.
ALICE detector and data samples
===============================
The ALICE Collaboration has studied $\psi$(2S) production through its dimuon decay channel, in the Muon Spectrometer which covers the pseudorapidity range $-$4 $< \eta <$ $-$2.5. The ALICE detector is described in detail in [@jinst]. The pp analysis has been performed on a triggered event sample corresponding to an integrated luminosity of $\mathcal{L}^{\rm pp}_{\rm int}$ = 1.35 $\pm$ 0.07 pb$^{-1}$ in the rapidity interval $2.5 < y_{\rm lab} < 4$ at $\sqrt{s}$ = 7 TeV. The p-Pb data have been collected at $\sqrt{s_{\rm NN}}$ = 5.02 TeV under two different configurations, inverting the direction of the p and Pb beams. In this way both forward rapidity $2.03 < y_{\rm cms} < 3.53$ ($\mathcal{L}^{\rm pPb}_{\rm int}$ = 5.01 $\pm$ 0.19 nb$^{-1}$) and backward rapidity $-4.46 < y_{\rm cms} < -2.96$ ($\mathcal{L}^{\rm Pbp}_{\rm int}$ = 5.81 $\pm$ 0.18 nb$^{-1}$) could be accessed, with the positive $y$ defined in the direction of the proton beam. Finally, the Pb-Pb analysis has been performed at $\sqrt{s_{\rm NN}}$ = 2.76 TeV ($\mathcal{L}^{\rm PbPb}_{\rm int}$ = 68.8 $\pm$ 0.9 $\mu$b$^{-1}$) in the rapidity region $2.5 < y_{\rm lab} < 4$.
Results
=======
pp collisions
-------------
Fig. \[fig1\] shows the inclusive differential production cross-sections of $\psi$(2S) as a function of $p_{\rm T}$ and $y$ [@epjc74]. The result on $p_{\rm T}$ differential cross-section is consistent with the LHCb measurement [@3epjc72] in the same rapidity interval. This is the first measurement of $\psi$(2S) differential cross-sections as a function of $y$ in pp collisions at $\sqrt{s} = 7$ TeV.
![\[fig1\]Inclusive differential production cross-sections of $\psi$(2S) as a function of $p_{\rm T}$ (left) and $y$ (right).](2014-Sep-08-PSI_PT.pdf "fig:") ![\[fig1\]Inclusive differential production cross-sections of $\psi$(2S) as a function of $p_{\rm T}$ (left) and $y$ (right).](2014-Sep-08-PSI_Y.pdf "fig:")
The inclusive $\psi$(2S) to J/$\psi$ cross-section ratio was measured as a function of $p_{\rm T}$ and $y$ as shown in Fig. \[fig2\]. A clear $p_{\rm T}$ dependence can be observed, consistent with the one measured by LHCb [@3epjc72]. No strong $y$ dependence is visible in the $y$ range covered by the ALICE muon spectrometer.
![\[fig2\]Inclusive $\psi$(2S) to J/$\psi$ cross-section ratio as a function of $p_{\rm T}$ (left) and $y$ (right).](2014-Sep-08-RATIO_PT.pdf "fig:") ![\[fig2\]Inclusive $\psi$(2S) to J/$\psi$ cross-section ratio as a function of $p_{\rm T}$ (left) and $y$ (right).](2014-Sep-08-RATIO_Y.pdf "fig:")
p-Pb collisions
---------------
The production cross section of $\psi$(2S) in p-Pb is compared to the J/$\psi$ one and to the corresponding quantities in pp collisions at $\sqrt{s}$ = 7 TeV (no LHC results are available at $\sqrt{s}$ = 5.02 TeV) using the $\psi$(2S) to J/$\psi$ ratio and the double ratio \[$\sigma_{\rm \psi(2S)}/\sigma_{\rm J/\psi}]_{\rm pPb}$/\[$\sigma_{\rm \psi(2S)}/\sigma_{\rm J/\psi}]_{\rm pp}$ [@jhep12; @arnaldi] as shown in Fig. \[fig3\]. The pp ratios are significantly higher than those for p-Pb and Pb-p. The double ratio is compared with the corresponding measurement by the PHENIX Collaboration at mid-rapidity at $\sqrt{s_{\rm NN}}$ = 0.2 TeV [@ada13]. Within uncertainties, a similar relative $\psi$(2S) suppression is observed by the two experiments.
![\[fig3\]Left: $\psi$(2S) to J/$\psi$ cross-section ratio compared to the pp results at $\sqrt{s}$ = 7 TeV. Right: the double ratio compared to the PHENIX result [@ada13].](Fig2_Psi2S_JPsi_ratio_integrated.pdf "fig:") ![\[fig3\]Left: $\psi$(2S) to J/$\psi$ cross-section ratio compared to the pp results at $\sqrt{s}$ = 7 TeV. Right: the double ratio compared to the PHENIX result [@ada13].](Fig3_Psi2S_JPsi_doubleratio_integrated_PhenixBlack.pdf "fig:")
Since no result on cross-section of $\psi$(2S) is available at $\sqrt{s}$ = 5.02 TeV in pp collisons, the nuclear modification factor of $\psi$(2S) is obtained by combining the J/$\psi$ $R_{\rm pPb}$ [@jhep1402] and the double ratio, as $R_{\rm pPb}^{\rm \psi(2S)} =$$ R_{\rm pPb}^{\rm J/\psi}$$\times$$(\sigma_{\rm pPb}^{\rm \psi(2S)}/\sigma_{\rm pPb}^{\rm J/\psi})$$\times$$(\rm \sigma_{\rm pp}^{\rm J/\psi}/\sigma_{\rm pp}^{\rm \psi(2S)})$, assuming that the ratio in pp collisions does not depend on $\sqrt{s}$ [@jhep12]. In Fig. \[fig4\], $R_{\rm pPb}^{\psi(2S)}$ is compared with $R_{\rm pPb}^{J/\psi}$ and also with theoretical calculations based on nuclear shadowing [@ijmp], coherent energy loss or both [@jhep1303]. The suppression of $\psi$(2S) production is stronger than that of J/$\psi$ and reaches a factor of 2 with respect to pp. Since the kinematic distributions of gluons producing the J/$\psi$ or the $\psi$(2S) are rather similar and since the coherent energy loss does not depend on the final quantum numbers of the resonances, the same theoretical calculations hold for both J/$\psi$ and $\psi$(2S). Theoretical models predict a $y$ dependence which are in reasonable agreement with the J/$\psi$ results but no model can describe the $\psi$(2S) data. These results show that other mechanisms must be invoked in order to describe the $\psi$(2S) suppression in p-Pb collisions.
![\[fig4\]$\psi$(2S) $R_{\rm pPb}$ versus $y$ compared to the J/$\psi$ $R_{\rm pPb}$ and theoretical models.](Fig4_Psi2S_RpA_integrated.pdf)
The $R_{\rm pPb}$ is also computed as a function of $p_{\rm T}$ both at backward and forward $y$ and the results are shown in Fig. \[fig5\]. At both rapidities, the $R_{\rm pPb}^{\rm \psi(2S)}$ shows a strong suppression with a slightly more evident $p_{\rm T}$ dependence at backward-$y$. The $\psi$(2S) is more suppressed than the J/$\psi$, as already observed for the $p_{\rm T}$-integrated result. Theoretical calculations are in fair agreement with the $R_{\rm pPb}^{\rm J/\psi}$ but clearly overestimate the $R_{\rm pPb}^{\rm \psi(2S)}$ behaviour. The calculations from the comover model [@ferre] shows that the interaction with comovers, mostly at play in the backward region, is able to explain the stronger $\psi$(2S) suppression.
![\[fig5\]$p_{\rm T}$ dependence of the $\psi$(2S) $R_{\rm pPb}$ compared to the J/$\psi$ $R_{\rm pPb}$ and theoretical calculations in the forward (left) and backward (right) rapidity region.](Fig7a_p-Pb_Psi2S_RpA_differential.pdf "fig:") ![\[fig5\]$p_{\rm T}$ dependence of the $\psi$(2S) $R_{\rm pPb}$ compared to the J/$\psi$ $R_{\rm pPb}$ and theoretical calculations in the forward (left) and backward (right) rapidity region.](Fig7b_Pb-p_Psi2S_RpA_differential.pdf "fig:")
Finally, the $\psi$(2S) production is studied as a function of the collision event activity both at backward and forward $y$ [@arnaldi], as shown in Fig. \[fig6\]. The event activity determination is described in details in [@toia]. Since the centrality determination in p-Pb collisions can be biased by the choice of the estimator, the nuclear modification factor is, in this case, named $Q_{\rm pPb}$ [@toia]. The $\psi$(2S) $Q_{\rm pPb}$ shows a strong suppression, which increases with increasing event activity, and is rather similar in both the forward and the backward $y$ regions. The J/$\psi$ $Q_{\rm pPb}$ shows a similar decreasing trend at forward-$y$ as a function of the event activity. On the contrary, the J/$\psi$ and $\psi$(2S) $Q_{\rm pPb}$ patterns observed at backward-$y$ are rather different, with the $\psi$(2S) significantly more suppressed for large event activity classes.
![\[fig6\]$\psi$(2S) $Q_{\rm pPb}$ versus event activity compared to the J/$\psi$ $Q_{\rm pPb}$ in the forward (left) and backward (right) rapidity region.](2014-May-14-QpA_JPsi_Psi2S_EvActivity.pdf "fig:") ![\[fig6\]$\psi$(2S) $Q_{\rm pPb}$ versus event activity compared to the J/$\psi$ $Q_{\rm pPb}$ in the forward (left) and backward (right) rapidity region.](2014-May-14-QAp_JPsi_Psi2S_EvActivity.pdf "fig:")
Pb-Pb collisions
----------------
In order to study the suppression of $\psi$(2S) relative to J/$\psi$ in Pb-Pb collisions, the double ratio \[$\sigma_{\rm \psi(2S)}/\sigma_{\rm J/\psi}]_{\rm PbPb}$/\[$\sigma_{\rm \psi(2S)}/\sigma_{\rm J/\psi}]_{\rm pp}$ has been measured as a function of centrality in two $p_{\rm T}$ intervals (0 $<$ $p_{\rm T}$ $<$ 3 GeV/$\it{c}$ and 3 $<$ $p_{\rm T}$ $<$ 8 GeV/$\it{c}$) [@arnaldi13] and has been compared with the results from the CMS Collaboration [@prl113] as shown in Fig. \[fig7\]. Limited $\psi$(2S) statistics does not allow any firm conclusion about the centrality dependence of this ratio and the comparison with CMS is not straightforward due to the different kinematic coverage.
![\[fig7\]Double ratio \[${\rm \psi(2S)}/{\rm J/\psi}]_{\rm PbPb}$/\[${\rm \psi(2S)}/{\rm J/\psi}]_{\rm pp}$ as a function of centrality in two $p_{\rm T}$ intervals compared to CMS measurements [@prl113].](2015-Sep-22-DoubleRatio_v12_Derived_ComparisonWithCMS_v3.pdf)
Summary
=======
In summary, the ALICE Collaboration has studied the inclusive $\psi$(2S) production in pp, p-Pb and Pb-Pb collisions at the LHC. The $\psi$(2S) production cross-section and the $\psi$(2S) to J/$\psi$ cross-section ratio have been obtained as a function of $p_{\rm T}$ and $y$ in pp collisions. The $p_{\rm T}$ differential results are in good agreement with LHCb measurements. In p-Pb collisions the results show that $\psi$(2S) is significantly more suppressed than J/$\psi$ in both rapidity regions. This observation implies that initial state nuclear effects alone cannot account for the modification of the $\psi$(2S) yields, as also confirmed by the poor agreement of the nuclear modification factor of $\psi$(2S) with models based on shadowing and/or energy loss. Interaction with comovers is able to explain the $\psi$(2S) suppression. The final state interaction with the hadronic medium could also provide a possible explanation for the stronger $\psi$(2S) suppression [@rapp]. Limited statistics prevent to make definitive conclusions on $\psi$(2S) production in Pb-Pb collisions.
[9]{}
[, (ALICE Collaboration) [*JINST*]{} 3, S08002 (2008).]{}
[ B. Abelev [*et al.*]{}, (ALICE Collaboration)(2014), Eur. Phys. J. C [**74**]{} 2974.]{}
[ R. Aaij [*et al.*]{}, (LHCb Collaboration), Eur. Phys. J. C [**72**]{}, 2100 (2012).]{}
[ K. Abelev [*et al.*]{}, (ALICE Collaboration), JHEP [**12**]{}, 073 (2014).]{}
[ A. Adare [*et al.*]{}, (PHENIX Collaboration), Phys. Rev. Lett. [**111**]{} 202301 (2013).]{}
[, (ALICE collaboration), JHEP [**02**]{}, 073 (2014).]{}
[ J. L. Albacete [*et al.*]{}, Int. J. Mod. Phys. [**E22**]{}, 1330007 (2013).]{}
[ F. Arleo and S. Peigne, JHEP [**1303**]{}, 122 (2013).]{}
[ E. G. Ferreiro, Phys. Lett. [**B**]{} 749, 98-103 (2015).]{}
[ J. Adam [*et al.*]{}, (ALICE Collaboration), Phys. Rev. C [**91**]{}, 064905 (2015).]{}
[ R. Arnaldi [*et al.*]{}, (ALICE Collaboration), Nucl. Phys. A [**904-905**]{}, 595c-598c (2013).]{}
[ S. Chatrchyan [*et al.*]{}, (CMS Collaboration), Phys. Rev. Lett. [**113**]{} 262301 (2014).]{}
[ Xiaojian Du and Ralf Rapp, arXiv:1504.00670.]{}
[^1]: on behalf of the ALICE Collaboration
|
---
abstract: 'We consider the 4-block $n$-fold integer programming (IP), in which the constraint matrix consists of $n$ copies of small matrices $A$, $B$, $D$ and one copy of $C$ in a specific block structure. We prove that, the $\ell_{\infty}$-norm of the Graver basis elements of 4-block $n$-fold IP is upper bounded by ${{\mathcal{O}}_{FPT}}(n^{s_c})$ where $s_c$ is the number of rows of matrix $C$ and ${{\mathcal{O}}_{FPT}}$ hides a multiplicative factor that is only dependent on the parameters of the small matrices $A,B,C,D$ (i.e., the number of rows and columns, and the largest absolute value among the entries). This improves upon the existing upper bound of ${{\mathcal{O}}_{FPT}}(n^{2^{s_c}})$ [@hemmecke2014graver]. We provide a matching lower bounded of $\Omega(n^{s_c})$, which even holds for an arbitary non-zero integral element in the kernel space. We then consider a special case of 4-block $n$-fold in which $C$ is a zero matrix (called 3-block $n$-fold IP). We show that, surprisingly, 3-block $n$-fold IP admits a Hilbert basis whose $\ell_{\infty}$-norm is bounded by ${{\mathcal{O}}_{FPT}}(1)$, despite the fact that the $\ell_{\infty}$-norm of its Graver basis elements is still $\Omega(n)$. Finally, we provide upper bounds on the $\ell_{\infty}$-norm of Graver basis elements for 3-block $n$-fold IP. Based on these upper bounds, we establish algorithms for 3-block $n$-fold IP and provide improved algorithms for 4-block $n$-fold IP.'
author:
- 'Lin Chen$^*$, Lei Xu$^*$, Weidong Shi[^1]'
bibliography:
- 'schedule-tree.bib'
title: 'On the Graver basis of block-structured integer programming'
---
**Keywords:** Integer Programming; Graver basis; Fixed parameter tractable
Introduction
============
An integer program (IP) can be written as $$\begin{aligned}
\label{ILP}
\min\{{{\ve w}}\cdot {{\ve x}}: H {{\ve x}}={{\ve b}}, {{\ve l}}\le {{\ve x}}\le {{\ve u}}, {{\ve x}}\in {\ensuremath{\mathbb{Z}}}^{m} \}\end{aligned}$$ where all the numbers (i.e., the coordinates of $H,{{\ve w}},{{\ve b}},{{\ve l}},{{\ve u}}$) are integers. We call $H$ as the constraint matrix. Integer programming is a strong mathematical tool for modeling various optimization problems, based on which many parameterized and approximation algorithms have been developed. In general, integer programming is NP-hard [@borosh1976bounds]. Lenstra [@lenstra1983integer] showed a polynomial time algorithm when the number of variables is fixed, which was improved later by Kannan [@kannan1987minkowski]. A somewhat complementary algorithm, which runs in pseudo-polynomial time when the number of constraints (the number of rows of $H$, excluding ${{\ve l}}\le {{\ve x}}\le {{\ve u}}$ in IP (\[ILP:2\])) is fixed was provided by Papadimitriou [@papadimitriou1981complexity]. Very recently, Eisenbrand and Weismantel [@eisenbrand2018proximity] gave an important improvement on the running time by utilizing Steinitz Lemma [@grinberg1980value]. Subsequent improvement and lower bounds were obtained by Jansen and Rohwedder [@jansen2018integer].
Despite the research into IPs with fixed number of variables or constraints, there is also a strong interest in the research of IPs where the number of variables and constraints are part of the input, but with the constraint matrix $H$ having a specific structure. One of the most prominent examples is the class of IPs with $H$ being a totally unimodular matrix, which was further extended recently by Artmann et al. [@artmann2017strongly]. Another important example is the so-called $4$-block $n$-fold integer programming, which has received increasing attention in recent years [@hemmecke2014graver; @eisenbrand2018faster; @martin2018parameterized]. We focus on such block-structured integer programming in this paper. More precisely, we give the problem definition as follows.
Problem definition.
-------------------
We define 4-block $n$-fold IP as follows. A constraint matrix $H$ is called a $4$-block $n$-fold matrix, if it consists of small matrices $A$, $B$, $C$ and $D$ and can be written as follows: $$H={\begin{pmatrix}C& D\\#4&A \end{pmatrix}\ifxn\relax\else^{(n)}\fi} :=
\begin{pmatrix}
C & D & D & \cdots & D \\
B & A & 0 & & 0 \\
B & 0 & A & & 0 \\
\vdots & & & \ddots & \\
B & 0 & 0 & & A
\end{pmatrix}$$ Here $A,B,C,D$ are $s_i\times t_i$ matrices, $i=A,B,C,D$, respectively, and the big matrix $H$ consists of $n$ copies of $A,B,D$ and one copy of $C$. Notice that by plugging $A,B,C,D$ into the above blocked structure we require that $s_C=s_D$, $s_A=s_B$, $t_B=t_C$ and $t_A=t_D$. Let $\Delta$ be the largest absolute value among all the entries of $A,B,C,D$. Given $H$, we will be focusing on the following IP throughout this paper $$\begin{aligned}
\label{ILP:2}
\IP: \quad \min\{{{\ve w}}\cdot {{\ve x}}: H {{\ve x}}={{\ve b}}, {{\ve l}}\le {{\ve x}}\le {{\ve u}}, {{\ve x}}\in {\ensuremath{\mathbb{Z}}}^{t_B+nt_A} \}.\end{aligned}$$
Removing $B$ and $C$ from $H$, the remaining matrix is called an $n$-fold matrix. Removing $C$ and $D$ from $H$, the remaining matrix is called a two-stage stochastic matrix. Throughout this paper, we denote by $E$ and $F$ these two matrices, i.e., $$E:=
\begin{pmatrix}
D & D & \cdots & D \\
A & 0 & & 0 \\
0 & A & & 0 \\
\vdots & & \ddots & \\
0 & 0 & & A
\end{pmatrix}\hspace{20mm}
F:=
\begin{pmatrix}
B & A & 0 & & 0 \\
B & 0 & A & & 0 \\
\vdots & & & \ddots & \\
B & 0 & 0 & & A
\end{pmatrix}$$ Replacing $H$ with $E$ or $F$ in IP (\[ILP:2\]), the resulted ILP is called $n$-fold IP or two-stage stochastic IP, respectively.
Specifically, if $C=0$ in $H$, we denote the matrix by $H_0$. Replacing $H$ with $H_0$, we define the resulted IP (\[ILP:2\]) as a 3-block $n$-fold IP.
As is observed before (see, e.g., [@hemmecke2013n]), the constraint $H{{\ve x}}=b$ can be replaced with $H{{\ve x}}\le {{\ve b}}$ or $H{{\ve x}}\ge {{\ve b}}$ (or even part of inequalities being $\ge $ and part of them being $\le $). Inequa lities can be transformed into equalities by introducing ${{\mathcal{O}}}(n)$ additional variables and modify $A$ and $D$ into $\tilde{A}$ and $\tilde{D}$ (the technique is the same as that in Section \[sec:pre\], Feasibility and Optimality). Therefore, IP (\[ILP:2\]) with the constraint $H{{\ve x}}\le {{\ve b}}$ ($H{{\ve x}}\ge {{\ve b}}$) or $H_0{{\ve x}}\le {{\ve b}}$ ($H_0{{\ve x}}\ge {{\ve b}}$) are equivalent to $4$-block $n$-fold IP or 3-block $n$-fold IP, respectively, and we also call them $3$-block or 4-block $n$-fold IP.
Motivation
----------
$4$-block $n$-fold integer programming is an important research topic that has received increasing attention in recent years. Although a $4$-block $n$-fold IP has a restricted structure, it is still general enough to be capable of modeling a variety of fundamental combinatorial optimization problems. For example, its special case, $n$-fold integer programming, can be used to model various problems in scheduling [@knop2017scheduling; @jansen2018empowering], computational social choice and stringology [@knop2017combinatorial]. The two-stage stochastic version of these combinatorial problems, as well as various other stochastic problems with second order dominance relations can be modeled using $4$-block $n$-fold IP [@gollmer2011note; @hemmecke2014graver].
From a theoretical point of view, it is crucial to understand to what extend an IP with a special structure can be solved efficiently. Hemmecke and Schultz [@hemmecke2003decomposition] showed that two-stage stochastic IP can be solved in $f_{sto}(s_A,s_B,t_A,t_B,\Delta) n^3L$ time for some computable function $f_{sto}$ (where $L$ is the length of the input). In 2013, Hemmecke, Onn and Romanchuk [@hemmecke2013n] showed that $n$-fold IP can be solved in $f_{nf}(s_A,s_D,t_A,t_D,\Delta)n^3L$ for some computable function $f_{nf}$. Very recently, improved algorithms with a better running time have been developed for two-stage stochastic IP [@martin2018parameterized] and $n$-fold IP [@eisenbrand2018faster; @martin2018parameterized]. Adopting the concept of fixed parameter tractability (FPT) (see, e.g., the book [@downey2012parameterized] as a nice introduction), we take $s_i,t_i$ ($i=A,B,C,D$) and $\Delta$ as parameters, and write ${{\mathcal{O}}_{FPT}}$ to hide a computable function that is only dependent on the parameters. Then the above results indicate that two-stage stochastic IP and $n$-fold IP both admit algorithms of running time ${{\mathcal{O}}_{FPT}}(n^{{{\mathcal{O}}}(1)}L)$ and are thus both in FPT. In contrast, the best known algorithm for $4$-block $n$-fold IP has a running time of $\min\{{{\mathcal{O}}_{FPT}}(n^{2^{s_c}\cdot t_B+3}L), {{\mathcal{O}}_{FPT}}(n^{k(A,B)\cdot t_B+3}L)\}$ [@hemmecke2014graver], where $k(A,B)$ is some parameter that is dependent on $s_A,s_B,t_A,t_B,\Delta_{A,B}$ (where $\Delta_{A,B}$ is the largest absolute value among all entries of $A,B$). As the existence of $k(A,B)$ follows from a saturation result in commutative algebra, even a rough estimation of $k(A,B)$ (say, singly or doubly exponential) is not clear so far. Given the recent progress in two-stage stochasitc IP and $n$-fold IP [@eisenbrand2018faster; @martin2018parameterized], it becomes a very natural question whether an improved algorithm can be designed for $4$-block $n$-fold IP. In particular, is $4$-block $n$-fold IP in FPT?
Towards an algorithmic improvement, it is crucial to understand the Graver basis of $4$-block $n$-fold IP. Indeed, all the algorithms so far for $4$-block $n$-fold IP as well as its two special cases (namely two-stage stochastic IP and $n$-fold IP) rely on the same augmentation framework, as we will provide details in Section \[sec:pre\]. Such an augmentation framework applies to an arbitrary IP. The reason that we can have a better algorithm for $4$-block $n$-fold IP and its special cases, rather than one that is exponential in the number of variables or constraints, is that its Graver basis has a nice structure. In particular, the $\ell_{\infty}$-norm of two-stage stochastic IP and $n$-fold IP are both bounded by ${{\mathcal{O}}_{FPT}}(1)$, whereas they admit FPT algorithms using the augmentation framework. In contrast, the $\ell_{\infty}$-norm (or $1$-norm) of $4$-block $n$-fold IP is only bounded by $\min\{{{\mathcal{O}}_{FPT}}(n^{2^{s_c}\cdot t_B}), {{\mathcal{O}}_{FPT}}(n^{k(A,B)\cdot t_B})\}$ [@hemmecke2014graver]. If an ${{\mathcal{O}}_{FPT}}(1)$ upper bound can be established for $4$-block $n$-fold IP, then an FPT algorithm follows. This motivates us to study the Graver basis of $4$-block $n$-fold IP and its special cases.
Our Contribution
----------------
Firstly, we show that the $\ell_{\infty}$-norm of Graver basis elements for $4$-block $n$-fold IP is upper bounded by ${{\mathcal{O}}_{FPT}}(n^{s_c})$ (Theorem \[thm:3-block-graver-4\]), improving the existing upper bound of ${{\mathcal{O}}_{FPT}}(n^{2^{s_c}})$ [@hemmecke2014graver]. We also establish the first explicit lower bound of $\Omega(n^{s_c})$ (Theorem \[thm:4-block-lower\]). It is thus tight up to an FPT factor. Indeed, our lower bound even shows that for some $H$, any non-zero integral element of $\{{{\ve x}}:H{{\ve x}}=0\}$ has an $\ell_{\infty}$-norm at least $\Omega(n^{s_c})$. Therefore, even an algorithm that augments via other basis instead of Graver basis may have to deal with an augmentation step that is unbounded (by ${{\mathcal{O}}_{FPT}}(1)$).
Secondly, we study a sepcial case of 4-block $n$-fold IP, namely 3-block $n$-fold IP where $C=0$. We show that, unlike 4-block $n$-fold IP, 3-block $n$-fold IP admits a Hilbert basis whose $\ell_{\infty}$-norm is bounded by ${{\mathcal{O}}_{FPT}}(1)$ (Theorem \[lemma:3-infty-bound\]). Unfortunately, the $\ell_{\infty}$-norm of Graver basis elements of 3-block $n$-fold IP is at least $\Omega(n)$. We complement our results by establishing an upper bound of $\min\{{{\mathcal{O}}_{FPT}}(n^{s_c}),{{\mathcal{O}}_{FPT}}(n^{t_A^2}+1)\}$ (Theorem \[thm:3-block-graver-4\] and Theorem \[thm:3-block-graver\]). The upper bound of ${{\mathcal{O}}_{FPT}}(n^{t_A^2}+1)$, which is singly exponential in $t_A$, is much more involved compared with the other upper bound. This seems to coincide with the existing results for $4$-block $n$-fold IP [@hemmecke2014graver], where an upper bound that depends on $A,B$ (instead of $C,D$) is more complicated. Our proof relies on a completely new approach, which first establishes a specific decomposition and then modify it into a sign-compatible decomposition through merging summands. This may be of separate interest for deriving upper bounds on the norms of Graver basis for other problems, particularly for deriving an upper bound on the $\ell_{\infty}$-norm of Graver basis for $4$-block $n$-fold IP which has an explicit dependency on $s_A,s_B,t_A,t_B$ in the exponent of $n$.
Thirdly, combining our upper bounds on the $\ell_{\infty}$-norm of Graver basis elements and the new algorithmic progress in $n$-fold IP [@eisenbrand2018faster; @martin2018parameterized], we establish an algorithm of running time ${{\mathcal{O}}_{FPT}}(n^{s_ct_B+3})\log n$ for $4$-block $n$-fold IP and an algorithm of running time $\min\{{{\mathcal{O}}_{FPT}}(n^{s_ct_B+3}\log^3 n), {{\mathcal{O}}_{FPT}}(n^{(t_A^2+1)t_B+3}\log^3 n)\}$ for $3$-block $n$-fold IP.
Preliminary {#sec:pre}
===========
**Notations.** Any $(t_B+nt_A)$-dimensional vector ${{\ve x}}$ can be written into $n+1$ bricks such that ${{\ve x}}=({{\ve x}}^0,{{\ve x}}^1,\cdots,{{\ve x}}^n)$ where ${{\ve x}}^0$ is $t_B$-dimensional and each ${{\ve x}}^i$, $1\le i\le n$, is $t_A$ dimensional. We call ${{\ve x}}^i$ as the $i$-th brick for $0\le i\le n$. We write $0_{s\times t}$ for an $s\times t$ matrix consisting of $0$, and $I_{t}$ for an $t\times t$ identity matrix. For a vector or a matrix, we write $||\cdot||_{\infty}$ to denote the maximal absolute value of its coordinates (elements). For two column vectors ${{\ve x}},{{\ve y}}$ of the same dimension, we write ${{\ve x}}\cdot{{\ve y}}$ for its inner product.
Throughout this paper, we write ${{\mathcal{O}}_{FPT}}(1)$ to represent a parameter that is only dependent on $\Delta,s_A,s_B,s_C,s_D,t_A,t_B,t_C,t_D$ where $\Delta$ is the maximal absolute value among all the entries of $A,B,C,D$, that is, ${{\mathcal{O}}_{FPT}}(1)$ is only dependent on the small matrices $A,B,C,D$ and is independent of $n$. For any computable function $f(x)$, we write ${{\mathcal{O}}_{FPT}}(f)$ to represent a computable function $f'(x)$ such that $|f'(x)|\le {{\mathcal{O}}_{FPT}}(1)\cdot |f(x)|$, and $\Omega_{FPT}(f)$ to represent a function $f''$ such that $|f''(x)|\ge \Omega(1)\cdot |f(x)|$.
Two vectors ${{\ve x}}$ and ${{\ve y}}$ are called sign-compatible if $x_i\cdot y_i\ge 0$ holds for every pair of coordinates $(x_i,y_i)$. Furthermore, we call a summation $\sum_{i}{{\ve x}}_i$ a sign-compatible summation if for every $i,j$ the summands ${{\ve x}}_i$ and ${{\ve x}}_j$ are sign-compatible.
We provide a brief introduction to the notions needed for solving a general integer programming. We refer the readers to a nice book [@de2013algebraic] for details.
**Graver basis.** Consider the general integer linear programming in the standard form: $$\begin{aligned}
\label{eq:ILP}
\min\{{{\ve w}}\cdot {{\ve x}}: H {{\ve x}}={{\ve b}}, {{\ve l}}\le {{\ve x}}\le {{\ve u}}, {{\ve x}}\in{\ensuremath{\mathbb{Z}}}^m\}\end{aligned}$$ We define [*Graver basis*]{}, which was introduced in [@graver1975foundations] by Graver. We define a partial order $\sqsubseteq$ in $\mathbb{R}^m$ in the following way: $$\begin{aligned}
\textrm{For any } {{\ve x}},{{\ve y}}\in \mathbb{R}^m,\,\, {{\ve x}}\sqsubseteq {{\ve y}}\text{ if and only if for every } 1\le i\le n, |x_i|\le |y_i| \text{ and } x_i\cdot y_i\ge 0. \end{aligned}$$ Given any subset $X\subseteq \mathbb{R}^n$, we say ${{\ve x}}$ is an $\sqsubseteq$-minimal element of $X$ if ${{\ve x}}\in X$ and there does not exist ${{\ve y}}\in X$, ${{\ve y}}\neq {{\ve x}}$ such that ${{\ve y}}\sqsubseteq {{\ve x}}$. It is known that every subset of ${\ensuremath{\mathbb{Z}}}^m$ has finitely many $\sqsubseteq$-minimal elements.
The Graver basis of an integer $m'\times m$ matrix $H$ is the finite set $\mathcal{G}(H)\subseteq \mathbb{Z}^m$ which consists of all the $\sqsubseteq$-minimal elements of $ker_{\mathbb{Z}^m}(H)=\{{{\ve x}}\in \mathbb{Z}^m| H{{\ve x}}=0,{{\ve x}}\neq 0\}$.
Any ${{\ve x}}\in ker_{\mathbb{Z}^m}(H)$, ${{\ve x}}\neq 0$ can be written as ${{\ve x}}=\sum_i\alpha_i{{\ve g}}_i(H)$, where $\alpha_i\in{\ensuremath{\mathbb{Z}}}_+$, ${{\ve g}}_i(H)\in {\ensuremath{\mathcal{G}}}(H)$ and ${{\ve g}}_i(H)\sqsubseteq {{\ve x}}$.
**Augmentation algorithms for IP and Graver-best oracle.** There is a general framework for solving an integer programming by utilizing Graver basis, which is implemented in a series of recent papers (see, e.g., [@chen2018covering; @hemmecke2013n; @jansen2018empowering; @knop2017combinatorial]). A very recent paper by Kouteck[y]{}, Levin and Onn [@martin2018parameterized] gives a nice explanation on this framework. In the following we briefly recap their explanation. We define a [*Graver-best augmentation procedure*]{} as follows. Given an arbitrary feasible solution ${{\ve x}}_0$ for IP (\[eq:ILP\]), for any ${{\ve g}}\in{\ensuremath{\mathcal{G}}}(H)$ and $\rho\in {\ensuremath{\mathbb{Z}}}_+$ we say $({{\ve g}},\rho)$ is a Graver augmentation pair if ${{\ve w}}({{\ve x}}_0+\rho{{\ve g}})<{{\ve w}}{{\ve x}}_0$ and ${{\ve l}}\le {{\ve x}}_0+\rho{{\ve g}}\le {{\ve u}}$, i.e., ${{\ve x}}_0+\rho{{\ve g}}$ is a feasible solution with a strictly better objective value. We say ${{\ve h}}\in {\ensuremath{\mathbb{Z}}}^m$ is a Graver-best augmentation step if it holds that ${{\ve x}}_0+{{\ve h}}$ is feasible and ${{\ve w}}({{\ve x}}_0+{{\ve h}})\le {{\ve w}}({{\ve x}}_0+\rho{{\ve g}})$ for any Graver augmentation pair $({{\ve g}},\rho)$. Given a feasible solution ${{\ve x}}_0$ for IP (\[eq:ILP\]), a [*Graver-best augmentation procedure*]{} works iteratively as follows:
(i) \[item:1\] If no Graver-best augmentation step exists, return ${{\ve x}}_0$ is optimal;
(ii) If there exists some Graver-best augmentation step ${{\ve h}}$, set ${{\ve x}}_0\leftarrow{{\ve x}}_0+{{\ve h}}$ and go to step (\[item:1\]).
We define a Graver-best oracle as such that given an input of IP (\[eq:ILP\]) that consists of an integer matrix $H$, integer vectors ${{\ve w}},{{\ve b}},{{\ve l}},{{\ve u}}$ and a feasible solution ${{\ve x}}$, it returns a Graver-best step ${{\ve h}}$ for ${{\ve x}}$.
The following theorem is due to [@martin2018parameterized], which generalizes the result in [@onn2010nonlinear].
\[thm:Koutecky\][@martin2018parameterized] Given a Graver best oracle and an initial feasible solution for IP (\[eq:ILP\]), IP (\[eq:ILP\]) can be solved by a strongly polynomial oracle algorithm.
**Approximate Graver-best oracle.** In general, finding a Graver-best augmentation step is difficult. However, if some additional information on the Graver basis is known, e.g., if the Graver basis element of ${\ensuremath{\mathcal{G}}}(H)$ has an $\ell_{\infty}$-norm bounded by some value $\xi$, then we are able to restrict our attention to the following: $$\begin{aligned}
\label{eq:graver-best-aug}
\min\{{{\ve w}}\cdot \rho{{\ve x}}: H {{\ve x}}=0, {{\ve l}}\le {{\ve x}}_0+\rho{{\ve x}}\le {{\ve u}}, \rho\in{\ensuremath{\mathbb{Z}}}_{+}, {{\ve x}}\in{\ensuremath{\mathbb{Z}}}^m, ||{{\ve x}}||_{\infty}\le \xi\}\end{aligned}$$
An algorithm for IP (\[eq:graver-best-aug\]) serves as a Graver-best oracle. It has been observed in [@eisenbrand2018faster], very recently, that we do not really need to solve IP (\[eq:graver-best-aug\]) optimally. Indeed, it suffices to find out an ${{\mathcal{O}}}(1)$-approximation solution for IP (\[eq:graver-best-aug\]), which, in turn, gives us an approximate Graver-best oracle. Why an approximate Graver-best oracle suffices? Let $\rho^*$ and ${{\ve g}}^*$ be such that ${{\ve w}}\cdot \rho^*{{\ve g}}^*$ is the minimal among all the pairs $({{\ve g}},\rho)\in {\ensuremath{\mathcal{G}}}(H)\times {\ensuremath{\mathbb{Z}}}_+$ and ${{\ve l}}\le{{\ve x}}_0+\rho{{\ve g}}\le{{\ve u}}$. It has been observed before (see, e.g., [@hemmecke2011polynomial; @hemmecke2013n]) that $|{{\ve w}}\cdot \rho^*{{\ve g}}^*|\ge 1/\Omega_{FPT}(n)\cdot{{\ve w}}\cdot ({{\ve x}}^*-{{\ve x}}_0)$ where ${{\ve x}}^*$ is the optimal solution. Therefore, an optimal solution to IP (\[eq:graver-best-aug\]) allows us to reduce the gap between ${{\ve x}}_0$ and ${{\ve x}}^*$ by a multiplicative factor of $1-1/\Omega_{FPT}(n)$, implying that ${{\mathcal{O}}}(n \log |{{\ve w}}\cdot {{\ve x}}^*-{{\ve w}}\cdot{{\ve x}}_0|)$ augmentation steps suffice to reach ${{\ve x}}^*$. It is easy to see that instead of an optimal solution to IP (\[eq:graver-best-aug\]), any ${{\mathcal{O}}}(1)$-approximation solution also allows us to reduce the gap by a factor of $1-1/\Omega(n)$. This observation allows us to restrict the value of $\rho$’s to be the form of $2^{k}$ for $k\in{\ensuremath{\mathbb{Z}}}_{\ge 0}$. Given an explicity upper and lower bound, we know that $\rho\le \max\{||{{\ve u}}-{{\ve x}}_0||_{\infty}, ||{{\ve l}}-{{\ve x}}_0||_{\infty}\}$. If, however, no explicit upper or lower bound is known for some variable, we can use some proximity result from the linear programming relaxation [@cook1986sensitivity] or simply use a standard upper bound of $(n\Delta)^{{{\mathcal{O}}}(n)}$ where $\Delta=||H||_{\infty}$ (which is also used in the Lenstra’s algorithm [@lenstra1983integer]). Therefore, we can restrict that $\rho$ only takes ${{\mathcal{O}}}(n\log (n\Delta))$ distinct values of the form $2^0,2^1,2^2,\cdots$. For each fixed value $\rho_0=2^k$, we solve the following IP: $$\begin{aligned}
\label{eq:graver-best-aug-1}
\min\{{{\ve w}}\cdot {{\ve x}}: H {{\ve x}}=0, {{\ve l}}\le {{\ve x}}_0+\rho_0{{\ve x}}\le {{\ve u}}, {{\ve x}}\in{\ensuremath{\mathbb{Z}}}^m, ||{{\ve x}}||_{\infty}\le \xi\}\end{aligned}$$ It is clear that an oprimal or ${{\mathcal{O}}}(1)$-approximation solution for IP (\[eq:graver-best-aug-1\]) suffices for us to derive an optimal solution for IP (\[eq:ILP\]).
**Feasibility and Optimality.** Finding a feasible solution of ${\left(\smallmatrixC& D\\#4&A \endsmallmatrix\right)\ifxn\relax\else{^{(n)}}\fi} {{\ve x}}={{\ve b}}, {{\ve l}}\le{{\ve x}}\le {{\ve u}}$ is equivalent to finding an optimal solution of an augmented IP with the same 4-block structure but has a trivial initial feasible solution. We briefly describle this procedure as follows (this is also useful in our analysis).
Let $\tilde{D}=(D,I_{t_D},0_{t_A\times t_A})$ and $\tilde{A}=(A,0_{t_D\times t_D}, I_{t_A})$. Let ${{\ve y}}=(\bar{{{\ve y}}}^1,{{\ve y}}^1,\bar{{{\ve y}}}^2,{{\ve y}}^2,\cdots,\bar{{{\ve y}}}^n,{{\ve y}}^n)$ with $\bar{{{\ve y}}}^i$ and ${{\ve y}}^i$ being an $s_A$- and $s_D$-dimensional vectors, respectively. Let ${{\ve x}}\oplus{{\ve y}}=({{\ve x}}^0,{{\ve x}}^1,\bar{{{\ve y}}}^1,{{\ve y}}^1,\cdots,{{\ve x}}^n,\bar{{{\ve y}}}^n,{{\ve y}}^n)$. Now it is easy to see that if we take ${{\ve x}}_0^i=0$, $\bar{{{\ve y}}}_0^1={{\ve b}}^0$, $\bar{{{\ve y}}}_0^i=0$ for $2\le i\le n$, ${{{\ve y}}}_0^i={{\ve b}}^i$ for $1\le i\le n$, then ${{\ve x}}_0\oplus{{\ve y}}_0$ is a feasible solution to $${\begin{pmatrix}C& \tilde{D}\\#4&\tilde{A} \end{pmatrix}\ifxn\relax\else^{(n)}\fi} {{\ve x}}\oplus{{\ve y}}={{\ve b}}, {{\ve l}}\le {{\ve x}}\le {{\ve u}}$$ If we minimize an objective function of $||{{\ve y}}||_1$ for the above augmented IP, its optimal solution with the objective value of $0$ implies a feasible solution to the original IP. Although $||{{\ve y}}||_1$ is not linear, we can use the standard technique to make it linear, i.e., we can write ${{\ve y}}={{\ve y}}_+-{{\ve y}}_{-}$ and add the constraint ${{\ve y}}_+,{{\ve y}}_-\ge 0$. It is easy to verify that such a modification does not destroy the 4-block $n$-fold structure. Also notice that this approach only involves modifying $A$ and $D$, and therefore applies if $B=C=0$.
**Fitness theorems for $n$-fold and two-stage stochastic matrices**
Consider an $n$-fold matrix $E$ that consists of $A$ and $D$ (i.e., $B=C=0$ in a 4-block $n$-fold matrix). It is shown that, the $||\cdot||_1$-norm of any Graver basis element of ${E}$ is ${{\mathcal{O}}_{FPT}}(1)$. More precisely, we have the following lemma.
[@hemmecke2013n; @hocsten2007finiteness]\[lemma:cite-nfold\] Let $E$ be an $n$-fold matrix. There exists some integer $\kappa=f_{nf}(s_A,s_D,t_A,t_D,\Delta)$ for some computable function $f_{nf}$ and $$\begin{aligned}
M(A)=\{{{\ve h}}\in \mathbb{Z}^t| {{\ve h}}\text{ is the sum of at most $\kappa$ elements of } \mathcal{G}(A_2)\},
\end{aligned}$$ such that for any ${{\ve g}}=({{\ve g}}^1,{{\ve g}}^2,\cdots,{{\ve g}}^n)\in\mathcal{G}(E)$ we have $\sum_{i\in I}{{\ve g}}^i\in M(A)$ for any $I\subseteq \{1,2,\cdots,n\}$.
[@aschenbrenner2007finiteness; @martin2018parameterized]\[lemma:multi-stage-bounded\] Let $F$ be a two-stage stochastic matrix, $g_{\infty}(H)=\max_{{{\ve g}}\in{\ensuremath{\mathcal{G}}}(H)}||g||_{\infty}$ and $a=\max\{||A||_{\infty},||B||_{\infty}\}$. Then $g_{\infty}(H)\le f_{sto}(s_A,t_A,s_B,t_B,\Delta)$ for some computable function $f_{sto}$.
It is remarkable that the above lemma actually holds for a more general class of matrices called multi-stage stochastic matrices.
**Steinitz lemma** Steinitz lemma has been utilized in several recent papers [@eisenbrand2018faster; @eisenbrand2018proximity] to establish a better algorithm for integer programming. We will also utilize it in this paper.
[@grinberg1980value]\[lemma:steinitz\] Let an arbitrary norm be given in $\mathbb{R}^{\kappa}$ and assume that $||{{\ve x}}_i||\le \zeta$ for $1\le i\le m$ and $\sum_{i=1}^m {{\ve x}}_i={{\ve x}}$. Then there exists a permutation $\pi$ such that for all positive integers $\ell\le m$, $$||\sum_{i=1}^{\ell}{{\ve x}}_{\pi(i)}-\frac{\ell-\kappa}{m}{{\ve x}}||\le \kappa\zeta.$$
4-block $n$-fold integer programming
====================================
In this section we consider IP (\[ILP:2\]) for arbitrary $H$.
Upper bound on the $\ell_{\infty}$-norm of Graver basis
-------------------------------------------------------
The goal of this subsection is to prove the following theorem.
\[thm:3-block-graver-4\] For any $4$-block $n$-fold matrix $H$ and ${{\ve g}}(H)\in {\ensuremath{\mathcal{G}}}(H)$, $||{{\ve g}}(H)||_{\infty}\le
{{\mathcal{O}}_{FPT}}(n^{s_c})$.
Let ${{\ve g}}\in {\ensuremath{\mathcal{G}}}(H)$. As $F\cdot {{\ve g}}=0$, there exist $\alpha_j\in{\ensuremath{\mathbb{Z}}}_+$, ${{\ve g}}_j(F)\in{\ensuremath{\mathcal{G}}}(F)$ and ${{\ve g}}_j(F)\sqsubseteq {{\ve g}}$ such that $${{\ve g}}=\sum_{j=1}^m \alpha_j{{\ve g}}_j(F).$$ Furthermore, $||{{\ve g}}_j(F)||_{\infty}={{\mathcal{O}}_{FPT}}(1)$ according to Lemma \[lemma:multi-stage-bounded\]. Let ${{\ve h}}_j=C\cdot{{\ve g}}^0_j(F)+\sum_{i=1}^nD{{\ve g}}_j^i(F)$, which is an $s_C$-dimensional vector such that $||{{\ve h}}_j||_{\infty}={{\mathcal{O}}_{FPT}}(n)$. As $H{{\ve g}}=0$, it follows that $$\sum_{j=1}^m \alpha_j{{\ve h}}_j=\underbrace{{{\ve h}}_1+{{\ve h}}_1+\cdots+{{\ve h}}_1}_{\alpha_1}+\underbrace{{{\ve h}}_2+{{\ve h}}_2+\cdots+{{\ve h}}_2}_{\alpha_2}+\cdots+\underbrace{{{\ve h}}_m+{{\ve h}}_m+\cdots+{{\ve h}}_m}_{\alpha_m}=0,$$ i.e., the sequence of ${{\ve h}}_i$’s sum up to $0$. According to Lemma \[lemma:steinitz\], there exists a permutation of the sequence such that $$||\sum_{i=1}^{\ell}{{\ve z}}_i||_{\infty}\le s_C \cdot {{\mathcal{O}}_{FPT}}(n)={{\mathcal{O}}_{FPT}}(n), \quad \forall \ell\le m'.$$ where $m'=\sum_{i=1}^m\alpha_i$ and ${{\ve z}}_1,{{\ve z}}_2,\cdots,{{\ve z}}_{m'}$ is a permutation of the sequence $\underbrace{{{\ve h}}_1,{{\ve h}}_1,\cdots,{{\ve h}}_1}_{\alpha_1},\underbrace{{{\ve h}}_2,{{\ve h}}_2,\cdots,{{\ve h}}_2}_{\alpha_2},\cdots,\underbrace{{{\ve h}}_m,{{\ve h}}_m,\cdots,{{\ve h}}_m}_{\alpha_m}$. Let $\tau={{\mathcal{O}}_{FPT}}(n)$ be the upper bound on $||\sum_{i=1}^{\ell}{{\ve z}}_i||_{\infty}$, then we know that $\sum_{i=1}^{\ell}{{\ve z}}_i\in \{-\tau,-\tau+1,\cdots,\tau\}^{s_c}$. Consequently, if $m'>(2\tau+1)^{s_c}+1$, there exists $\ell_1<\ell_2$ such that $\sum_{i=1}^{\ell_1}{{\ve z}}_i=\sum_{i=1}^{\ell_2}{{\ve z}}_i$, i.e., $\sum_{i=1}^{\ell_2-\ell_1}{{\ve z}}_i=0$. Recall that every ${{\ve z}}_i$ corresponds to some ${{\ve h}}_{i'}$. Suppose $\sum_{i=1}^{\ell_2-\ell_1}{{\ve z}}_i=\sum_{j=1}^m\alpha'_j{{\ve h}}_j$ for $\alpha_{j}'\le \alpha_j$, then by the definition of ${{\ve h}}_j$ it follows that $$C\left(\sum_{j=1}^{m}\alpha_j'{{\ve g}}_j^0(F)\right)+\sum_{i=1}^nD\left(\sum_{j=1}^m\alpha'_j{{\ve g}}_j^i(F)\right)=0.$$ Hence, $H\sum_{j=1}^m\alpha_j'{{\ve g}}_j(F)=0$. That is, if $m'=\sum_{j=1}^m\alpha_j>(2\tau+1)^{s_c}+1$, then there exists some ${{\ve g}}'=\alpha_j'{{\ve g}}_j(F)$ such that $H{{\ve g}}'=0$ and ${{\ve g}}'\sqsubset {{\ve g}}$, contradicting the fact that ${{\ve g}}\in {\ensuremath{\mathcal{G}}}(H)$. Thus, $\sum_{j=1}^m\alpha_j\le (2\tau+1)^{s_c}+1$, implying that $||{{\ve g}}||_{\infty}={{\mathcal{O}}_{FPT}}(n^{s_c})$.
**Remark.** The idea of the proof above seems to only work for the parameter $s_C$. An explicit upper bound that depends on $A,B$ is far from clear (albeit that we know an upper bound of ${{\mathcal{O}}_{FPT}}(n^{k(A,B)})$ with some unknown function $k(A,B)$). In the following section we will provide an upper bound that is singly exponential in $t_A$ for the special case when $C=0$ using a completely different approach.
Theorem \[thm:3-block-graver-4\] implies the following, whose proof is exactly the same as Theorem \[thm:alg-3-block\].
\[[thm:alg-4-block]{}\] There exists an algorithm for 4-block $n$-fold IP that runs in ${{\mathcal{O}}_{FPT}}(n^{s_ct_B+3})\log n$ time.
Lower bound on the $\ell_{\infty}$-norm of Graver basis
-------------------------------------------------------
We prove an even stronger result which gives a lower bound for any element in $ker_{{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}}(H)=\{{{\ve x}}\in{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}:H{{\ve x}}=0,{{\ve x}}\neq 0\}$.
\[thm:4-block-lower\] There exists a $4$-block $n$-fold matrix $H$ such that $s_C=s_D=t-1$, $t_C=t_D=t$, $s_A=s_B=t_A=t_B=t$, and for any ${{\ve y}}\in Ker_{{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}}(H)=\{{{\ve x}}:H{{\ve x}}=0,{{\ve x}}\neq 0\}$, $||{{\ve y}}||_{\infty}=\Omega({n^{t-1}})$.
We let $A=I_{t\times t}$, $B=-I_{t\times t}$. We define $(t-1)\times t$ matrices $D$ and $C$ such that $$D=
\begin{pmatrix}
1 & -1 & 0 &\cdots &0 & 0\\
0 & 1 & -1 &\cdots &0 & 0 \\
\vdots & & & \ddots & & \\
0 & 0 & 0 & \cdots & 1 & -1\\
\end{pmatrix}
\hspace{10mm} C=
\begin{pmatrix}
-1 & 0 & 0 &\cdots &0 & 0\\
0 & -1 & 0 &\cdots &0 & 0 \\
\vdots & & & \ddots & & \\
0 & 0 & 0 & \cdots & -1 & 0\\
\end{pmatrix}$$
Consider any ${{\ve y}}\in Ker_{{\ensuremath{\mathbb{Z}}}^{(n+1)t}}\{{{\ve x}}:H{{\ve x}}=0,{{\ve x}}\neq 0\}$. According to $A{{\ve y}}^0-B{{\ve y}}^i=0$, we know that ${{\ve y}}^0={{\ve y}}^i$ for every $1\le i\le n$. According to $C{{\ve y}}^0+\sum_{i=1}^n D{{\ve y}}^i=0$, we have $(C+nD){{\ve y}}^0=0$, i.e., $$\begin{pmatrix}
n-1 & -n & 0 &\cdots &0 & 0\\
0 & n-1 & -n &\cdots &0 & 0 \\
\vdots & & & \ddots & & \\
0 & 0 & 0 & \cdots & n-1 & -n\\
\end{pmatrix}
\cdot {{\ve y}}=0$$
Let ${{\ve y}}^0=(y_1,y_2,\cdots,y_t)$, the following is true: $$\begin{aligned}
\label{eq:lower-bound-4-lock}
(n-1)y_i=ny_{i+1},\quad 1\le i\le t-1
\end{aligned}$$ It is easy to see that as long as ${{\ve y}}\neq 0$, we have ${{\ve y}}^0\neq 0$ and consequently $y_i\neq 0$ for every $1\le i\le t$. Furthermore, Eq (\[eq:lower-bound-4-lock\]) indicates that either $y_i>0$ for all $i$, or $y_i<0$ for all $i$. Suppose $y_i>0$ (the other case can be proved in the same way). According to $(n-1)y_{t-1}=ny_t$, $y_{t-1}$ is dividable by $n$. Let $y_{t-1}=nz_{t-1}$ for some $z_{t-1}\in \mathbb{Z}_{\neq 0}$. According to $(n-1)y_{t-2}=ny_{t-1}=n^2z_{t-1}$, we know that $y_{t-2}$ is dividable by $n^2$. Let $y_{t-2}=n^2z_{t-2}$ and we plug it into $(n-1)y_{t-3}=ny_{t-2}$. In general, suppose we have shown that $y_{t-k}=n^kz_{t-k}$ for all $k\le k_0$. Now for $k=k_0+1$, we have $(n-1)y_{t-k_0-1}=ny_{t-k_0}=n^{k_0+1}z_{n-k_0}$, then $y_{t-k_0-1}$ is dividable by $n^{k_0+1}$. Hence, we conclude that $y_1$ is dividable by $n^{t-1}$, i.e., $||{{\ve y}}||_{\infty}=\Omega(n^{t-1})$ and Theorem \[thm:4-block-lower\] is proved.
3-block $n$-fold integer programming
====================================
In this section we consider IP (\[ILP:2\]) where $H=H_0$, i.e., $C=0$. The goal of this section is to show the following three main results: 1). There exists a Hilbert basis for $ker_{{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}}(H_0)=\{{{\ve x}}\in{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}:H_0{{\ve x}}=0,{{\ve x}}\neq 0\}$ such that the $\ell_{\infty}$-norm of every basis element is bounded by ${{\mathcal{O}}_{FPT}}(1)$. This gives a sharp contrast to Theorem \[thm:4-block-lower\] since when $C\neq 0$, any non-zero element of $ker_{{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}}(H)$ may have an $\ell_{\infty}$-norm at least $\Omega(n^{s_c})$. 2). Any Graver basis element of ${\ensuremath{\mathcal{G}}}(H_0)$ has an $\ell_{\infty}$-norm bounded by $\min\{{{\mathcal{O}}_{FPT}}(n^{s_c}),{{\mathcal{O}}_{FPT}}(n^{t_A^2+1})\}$. We also complement this upper bound by establishing a lower bound of $\Omega(n)$. 3). There exists an algorithm of running time $\min\{{{\mathcal{O}}_{FPT}}(n^{s_ct_B+3})\log^3 n, {{\mathcal{O}}_{FPT}}(n^{(t_A^2+1)t_B+3}\log^3 n)\}$ for 3-block $n$-fold IP by utilizing our bound on the $\ell_{\infty}$-norm of the Graver basis and the general framework from [@martin2018parameterized] (see Section \[sec:pre\], Augmentation algorithms for IP and Graver-best oracle).
Decomposition with bounded $\ell_{\infty}$-norm
-----------------------------------------------
The goal of this subsection is to prove the following theorem.
\[lemma:3-infty-bound\] There exists some $\xi={{\mathcal{O}}_{FPT}}(1)$ such that for any ${{\ve g}}\in{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}$ satisfying $H_0{{\ve g}}=0$, there exist a finite sequence of vectors ${{\ve e}}_1,{{\ve e}}_2,\cdots$ such that ${{\ve e}}_h\in{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}$, $H_0{{\ve e}}_h=0$, $||{{\ve e}}_h||_{\infty}\le \xi$, ${{\ve e}}^0_h\sqsubseteq {{\ve g}}^0$ and ${{\ve g}}=\sum_h{{\ve e}}_h$.
Recall that ${{\ve x}}^0$ always refer to the first $t_B$ coordinates of a $(t_B+nt_A)$-dimensional vector ${{\ve x}}$. Note that ${{\ve e}}_h$’s do not necessarily lie in the same orthant.
Since $H_0{{\ve g}}=0$, we know that $F\cdot{{\ve g}}=0$. Therefore, there exist $\alpha_j\in{\ensuremath{\mathbb{Z}}}_+$, ${{\ve g}}_j(F)\sqsubseteq {{\ve g}}$ such that $${{\ve g}}=\sum_{j}\alpha_j{{\ve g}}_j(F),$$ where ${{\ve g}}_j(F)\in{\ensuremath{\mathcal{G}}}(F)$. Consider each ${{\ve g}}_j(F)$. As $F$ is a two-stage stochastic matrix, by Lemma \[lemma:multi-stage-bounded\] it holds for every $j$ that $||{{\ve g}}_j(F)||_{\infty}={{\mathcal{O}}_{FPT}}(1).$ Note that each ${{\ve g}}_j(F)$ can be written into $n+1$ bricks such that ${{\ve g}}_j(F)=\left({{\ve g}}_j^0(F),{{\ve g}}_j^1(F),\cdots,{{\ve g}}_j^{n}(F)\right)$ where ${{\ve g}}_j^0(F)$ is a $t_B$-dimensional vector, and ${{\ve g}}_j^i(F)$ is a $t_A$-dimensional vector for every $1\le i\le n$. It is obvious that $||{{\ve g}}_j^i(F)||_{\infty}={{\mathcal{O}}_{FPT}}(1)$ for every $0\le i\le n$, and it holds that $$B{{\ve g}}_j^0(F)+A{{\ve g}}_j^i(F)=0, \quad \forall 1\le i\le n.$$ We first prove the following claim.
\[claim:infbound\] For every ${{\ve g}}_j(F)$ and $1\le \ell\le |{\ensuremath{\mathcal{G}}}(A)|$, there exist some ${{\ve v}}_j^*$, $\alpha_{j,\ell}^i\in{\ensuremath{\mathbb{Z}}}_{\ge 0}$ such that
- ${{\ve g}}_{j}^i(F)-{{\ve v}}_j^*=\sum_{\ell=1}^{|{\ensuremath{\mathcal{G}}}(A)|} \alpha_{j,\ell}^i{{\ve g}}_\ell(A), \quad \forall 1\le i\le n.$
- For every $1\le \ell\le |{\ensuremath{\mathcal{G}}}(A)|$, either $|\{i:\alpha_{j,\ell}^i>0\}|= 0$, or $|\{i:\alpha_{j,\ell}^i>0\}| \ge n/2$.
- Let $\alpha_{max}=2\max_h||{{\ve g}}_h(F)||_{\infty} ={{\mathcal{O}}_{FPT}}(1)$. Then $\max_{i,j,\ell}|\alpha_{j,\ell}^i|\le \alpha_{max}$.
- $||{{\ve v}}_j^*||_{\infty}={{\mathcal{O}}_{FPT}}(1)$.
Consider an arbitrary ${{\ve v}}_j$ such that ${\left(\smallmatrix{{\ve g}}_j^0(F)\\#3 \endsmallmatrix\right)\ifx\relax\else{^{()}}\fi}\in{\ensuremath{\mathcal{G}}}([B,A])$. We have $A({{\ve g}}_j^i(F)-{{\ve v}}_j)=0$ for every $1\le i\le n$, hence there exist $\bar{\alpha}_{j,\ell}^i\in{\ensuremath{\mathbb{Z}}}_+$, ${{\ve g}}_\ell(A)\in{\ensuremath{\mathcal{G}}}(A)$ and ${{\ve g}}_{\ell}(A)\sqsubseteq {{\ve g}}_j^i(F)-{{\ve v}}_j$ such that $${{\ve g}}_{j}^i(F)-{{\ve v}}_j=\sum_\ell \bar{\alpha}_{j,\ell}^i{{\ve g}}_\ell(A), \quad \forall 1\le i\le n.$$ Note that $||{\left(\smallmatrix{{\ve g}}_j^0(F)\\#3 \endsmallmatrix\right)\ifx\relax\else{^{()}}\fi}||_{\infty}\le \max_h||{{\ve g}}_h(F)||_{\infty}=\alpha_{max}/2$, consequently $||{{\ve g}}_j^i(F)-{{\ve v}}_j||_{\infty}\le \alpha_{max}$, and $\bar{\alpha}_{j,\ell}^i\le \alpha_{max}$. Consider the cardinality of the set $\{i:\bar{\alpha}_{j,\ell}^i>0\}$. If $1\le |\{i:\alpha_{j,\ell}^i>0\}|\le \lfloor n/2\rfloor$, we say $\ell$ is [*unbalanced*]{} for ${{\ve g}}_j(F)$. Let $\bar{\alpha}_{j,max}^{i}=\max_{1\le i\le n}\bar{\alpha}_{j,\ell}^i$ and $UB_j$ be the set of all unbalanced indices $\ell$, we define $${{\ve v}}_j^*:={{\ve v}}_j+\sum_{\ell\in UB_j}\bar{\alpha}_{j,\max}^{i}{{\ve g}}_\ell(A),$$ then $${{\ve g}}_{j}^i(F)-{{\ve v}}_j^*=\sum_{\ell\not\in UB_j} \bar{\alpha}_{j,\ell}^i{{\ve g}}_\ell(A)+\sum_{\ell\in UB_j}(\bar{\alpha}_{j,\max}^{i}-\bar{\alpha}_{j,\ell}^i)\cdot (-{{\ve g}}_\ell(A)), \quad \forall 1\le i\le n.$$ Note that $-{{\ve g}}_{\ell}(A)\in{\ensuremath{\mathcal{G}}}(A)$. For all the ${{\ve g}}_{\ell}(A)$’s in ${\ensuremath{\mathcal{G}}}(A)$ that do not appear in the above equation, their coefficients are $0$. Furthermore, we have $|\bar{\alpha}_{j,\ell}^i|\le\alpha_{max}$ and $|\bar{\alpha}_{j,\max}^{i}-\bar{\alpha}_{j,\ell}^i|\le \alpha_{max}$ for all $i,\ell$. As $||{{\ve v}}_j||_{\infty}={{\mathcal{O}}_{FPT}}(1)$, $||{{\ve g}}_{\ell}(A)||_{\infty}={{\mathcal{O}}_{FPT}}(1)$, we know that $||{{\ve v}}_j^*||_{\infty}={{\mathcal{O}}_{FPT}}(1)$. Thus, the claim is proved.
We call $({{\ve g}}_j^0(F),{{\ve v}}_j^*,{{\ve v}}_j^*,\cdots,{{\ve v}}_j^*)$ as a canonical vector (of ${{\ve g}}_j(F)$). Since $||{{\ve v}}_j^*||_{\infty}={{\mathcal{O}}_{FPT}}(1)$ and $||{{\ve g}}_j^0(F)||_{\infty}={{\mathcal{O}}_{FPT}}(1)$, there are at most $\tau={{\mathcal{O}}_{FPT}}(1)$ different kinds of canonical vectors. This means, there may be different ${{\ve g}}_k(F)$’s with the same canonical vector. We list all the $\tau$ possible canonical vectors and let ${{\ve r}}_j:=({{\ve u}}_j^*,{{\ve v}}_j^*,{{\ve v}}_j^*,\cdots,{{\ve v}}_j^*)$ be the $j$-th one. Let $CA_j$ be the set of indices of all ${{\ve g}}_k(F)$’s whose canonical vector is ${{\ve r}}_j$, then we have $$\begin{aligned}
\label{eq:inftybound-1}
{{\ve g}}=\sum_{j=1}^\tau(\sum_{k\in CA_j}\alpha_k){{\ve r}}_j+\sum_{j=1}^{\tau}\sum_{k\in CA_j}\alpha_k\left({{\ve g}}_k(F)-{{\ve r}}_j\right).\end{aligned}$$
We say an $n$-dimensional vector ${{\boldsymbol{\alpha}}}=(\alpha^1,\alpha^2,\cdots,\alpha^n)\in{\ensuremath{\mathbb{Z}}}_{\ge 0}^n$ is [*balanced*]{}, if ${{\boldsymbol{\alpha}}}=0$, or $||{{\boldsymbol{\alpha}}}||_{\infty}\le \alpha_{max}={{\mathcal{O}}_{FPT}}(1)$ and $|\{i:\alpha^i>0\}|\ge n/2$. Then the following observation is true.
\[obs:1\] For any nonzero balanced vector ${{\boldsymbol{\alpha}}}$ it holds that $||{{\boldsymbol{\alpha}}}||_1\ge n/2\cdot \alpha^i/\alpha_{max}$ for every $1\le i\le n$.
Using the concept of a balanced vector, Claim \[claim:infbound\] indicates that if ${{\ve r}}_j$ is a canonical vector of ${{\ve g}}_k(F)$, then ${{\ve g}}_{k}^i(F)-{{\ve v}}_j^*=\sum_\ell \alpha_{j,\ell}^i{{\ve g}}_\ell(A)$ such that the vector $(\alpha_{j,\ell}^1,\alpha_{j,\ell}^2,\cdots,\alpha_{j,\ell}^n)$ is a balanced vector.
The nice thing about balanced vectors is that we can have the following claim, which will be used several times later.
\[claim:inftybound-2\] Let ${{\ve y}}_1,{{\ve y}}_2,\cdots,{{\ve y}}_{k}$ be a sequence of balanced vectors in ${\ensuremath{\mathbb{Z}}}_{\ge 0}^{n}$ such that $||\sum_{h=1}^k{{\ve y}}_h||_1\le n\Lambda$ where $\Lambda={{\mathcal{O}}_{FPT}}(1)$, then $||\sum_{h=1}^k{{\ve y}}_h||_{\infty}\le 2\alpha_{max}\Lambda={{\mathcal{O}}_{FPT}}(1)$.
We prove by contradiction. Suppose on the contrary that $||\sum_{h=1}^k{{\ve y}}_h||_{\infty}>2\alpha_{max}\Lambda$, then there exists some $i^*$ such that $\sum_{h=1}^k{{\ve y}}_h^{i*}>2\alpha_{max}\Lambda$. Since ${{\ve y}}_h$’s are balanced vectors, according to Observation \[obs:1\], we have $$||\sum_{h=1}^k{{\ve y}}_h||_1\ge n\cdot \frac{\sum_{h=1}^k{{\ve y}}_h^{i*}}{2\alpha_{max}}>n\Lambda,$$ which contradicts the fact that $||\sum_{h=1}^k{{\ve y}}_h||_1\le n\Lambda$. Hence, the claim is true.
Since ${{\ve r}}_j$ is a canonical vector of ${{\ve g}}_k(F)$, by Claim \[claim:infbound\], there exist balanced vectors ${{\boldsymbol{\beta}}}_{k,\ell}$ such that Eq (\[eq:inftybound-1\]) can be rewritten as $$\begin{aligned}
{{\ve g}}^i=\sum_{j=1}^\tau(\sum_{k\in CA_j}\alpha_k){{\ve v}}_j^*+\sum_{j=1}^{\tau}\sum_{k\in CA_j}\alpha_k\left(\sum_{\ell=1}^{|{\ensuremath{\mathcal{G}}}(A)|}\beta_{k,\ell}^i{{\ve g}}_\ell(A)\right), \quad \forall 1\le i\le n,\end{aligned}$$ or equivalently, $$\begin{aligned}
\label{eq:inftybound-2}
{{\ve g}}^i=\sum_{j=1}^\tau\alpha_j'{{\ve v}}_j^*+\sum_{\ell=1}^{|{\ensuremath{\mathcal{G}}}(A)|}\beta^i_\ell{{\ve g}}_\ell(A),\quad \forall 1\le i\le n,\end{aligned}$$ where $\alpha_j'=\sum_{k\in CA_j}\alpha_k$ and each vector $\beta_\ell=(\beta_\ell^1,\cdots,\beta_\ell^n)$ is the summation of some balanced vectors.
As $[0,D,D,\cdots,D]{{\ve g}}=0$, we have $$\begin{aligned}
\label{eq:inftybound-3}
\sum_{j=1}^\tau n\alpha_j'D{{\ve v}}_j^*+\sum_{\ell=1}^{|{\ensuremath{\mathcal{G}}}(A)|}(\sum_{i=1}^n\beta_\ell^i){{\ve g}}_\ell(A)=0.\end{aligned}$$ Note that $|{\ensuremath{\mathcal{G}}}(A)|={{\mathcal{O}}_{FPT}}(1)$, the equation above can be rewritten as $$\begin{aligned}
\label{eq:inftybound-5}
[D{{\ve v}}_1^*,D{{\ve v}}_2^*,\cdots,D{{\ve v}}_\tau^*, {{\ve g}}_1(A),{{\ve g}}_2(A),\cdots,{{\ve g}}_{|{\ensuremath{\mathcal{G}}}(A)|}(A)]\cdot (n\alpha_1',n\alpha_2',\cdots,n\alpha_\tau',\sum_{i=1}^n\beta_{1}^i,\cdots,\sum_{i=1}^n\beta^i_{|{\ensuremath{\mathcal{G}}}(A)|})=0.\end{aligned}$$
Let $V=[D{{\ve v}}_1^*,D{{\ve v}}_2^*,\cdots,D{{\ve v}}_\tau^*, {{\ve g}}_1(A),{{\ve g}}_2(A),\cdots,{{\ve g}}_{|{\ensuremath{\mathcal{G}}}(A)|}(A)]$, there exist $\lambda_k\in{\ensuremath{\mathbb{Z}}}_+$ and ${{\ve g}}_k(V)\in {\ensuremath{\mathcal{G}}}(V)$, such that $$(n\alpha_1',n\alpha_2',\cdots,n\alpha_\tau',\sum_{i=1}^n\beta_{1}^i,\cdots,\sum_{i=1}^n\beta^i_{|{\ensuremath{\mathcal{G}}}(A)|})=\sum_{k}\lambda_k {{\ve g}}_k(V).$$ Note that since $\alpha_j',\beta_\ell^i\ge 0$, we can restrict that every ${{\ve g}}_j(V)\in{\ensuremath{\mathbb{Z}}}_{\ge 0}^{\kappa+|{\ensuremath{\mathcal{G}}}(A)|}$. There are two possibilities regarding the values of $\lambda_k$’s.
**Case 1.** $\lambda_k<n$ for every $k$. In this case we prove that $||{{\ve g}}||_{\infty}={{\mathcal{O}}_{FPT}}(1)$ and Theorem \[lemma:3-infty-bound\] follows directly. Note that $V$ is a matrix of ${{\mathcal{O}}_{FPT}}(1)$ size with $||V||_{\infty}={{\mathcal{O}}_{FPT}}(1)$, hence $||{{\ve g}}_k(V)||_{\infty}={{\mathcal{O}}_{FPT}}(1)$ and $|{\ensuremath{\mathcal{G}}}(V)|={{\mathcal{O}}_{FPT}}(1)$. Therefore, $n\alpha_j'<n|{\ensuremath{\mathcal{G}}}(V)|\cdot \max_k||{{\ve g}}_k(V)||_{\infty}={{\mathcal{O}}_{FPT}}(n)$, implying that $\alpha_j'={{\mathcal{O}}_{FPT}}(1)$. Consider the vector ${{\boldsymbol{\beta}}}=(\beta^1,\cdots,\beta^n)$ where $\beta^i=\sum_{\ell }\beta_{\ell}^i$. As $\beta_{\ell}^i\ge 0$, $$||{{\boldsymbol{\beta}}}||_1=||(\sum_{i=1}^n\beta_1^i,\cdots,\sum_{i=1}^n\beta_{|{\ensuremath{\mathcal{G}}}(A)|}^i)||_1\le \sum_k\lambda_k||{{\ve g}}_k(V)||_1={{\mathcal{O}}_{FPT}}(n).$$ Recall that ${{\boldsymbol{\beta}}}_{\ell}$ is the summation of balanced vectors, whereas ${{\boldsymbol{\beta}}}=\sum_{\ell}{{\boldsymbol{\beta}}}_{\ell}$ is also the summation of balanced vectors. Using Claim \[claim:inftybound-2\], $||{{\boldsymbol{\beta}}}||_{\infty}={{\mathcal{O}}_{FPT}}(1)$.
Combining the fact that $||{{\ve v}}_j^*||_{\infty}={{\mathcal{O}}_{FPT}}(1)$ and $||{{\ve g}}_\ell(A)||_{\infty}={{\mathcal{O}}_{FPT}}(1)$, we conclude that $||{{\ve g}}^i||_{\infty}={{\mathcal{O}}_{FPT}}(1)$ for $1\le i\le n$. Meanwhile, as ${{\ve g}}^0=\sum_j\alpha_j'{{\ve u}}_{j}^*$, we have $||{{\ve g}}^0||_{\infty}={{\mathcal{O}}_{FPT}}(1)$. Hence, $||{{\ve g}}||_{\infty}={{\mathcal{O}}_{FPT}}(1)$.
**Case 2.** $\lambda_k\ge n$ for some $k$. For ease of description, we take the viewpoint of a packing problem. We view each canonical vector ${{\ve r}}_j^*$ and ${{\ve g}}_\ell(A)$ as an item, whereas there are $\tau+|{\ensuremath{\mathcal{G}}}(A)|$ different kinds of items. There are $n+1$ different bins. Bin $0$ can only be used to pack items ${{\ve r}}_j^*$, $1\le j\le \tau$, and bin $i$ ($1\le i\le n$) can only be used to pack items ${{\ve g}}_\ell(A)$, $1\le \ell\le |{\ensuremath{\mathcal{G}}}(A)|$. Currently there are $\alpha_j'$ copies of item ${{\ve r}}_j^*$ in bin $0$, and $\beta^i_\ell$ copies of item ${{\ve g}}_\ell(A)$ in bin $i$. This is called a packing profile. Now we want to split this packing profile into several sub-profiles, i.e., we want to determine integers $\mu_j^h,\sigma^{i,h}_{\ell}\in{\ensuremath{\mathbb{Z}}}_{\ge 0}$ such that the followings are true:
(i) $\mu_j^h, \sigma^{i,h}_{\ell}={{\mathcal{O}}_{FPT}}(1)$ and $\mu_j^h+ \sigma^{i,h}_{\ell}>0$.
(ii) $\sum_h \mu_j^h=\alpha_j'$, $\sum_{h}\sigma^{i,h}_{\ell}=\beta^i_\ell$;
(iii) $[D{{\ve v}}_1^*,D{{\ve v}}_2^*,\cdots,D{{\ve v}}_\tau^*, {{\ve g}}_1(A),{{\ve g}}_2(A),\cdots,{{\ve g}}_{|{\ensuremath{\mathcal{G}}}(A)|}(A)]\cdot (n\mu_1^h,n\mu_2^h,\cdots,n\mu_\tau^h,\sum_{i=1}^n\sigma^{i,h}_{\ell},\cdots,\sum_{i=1}^n\sigma^{i,h}_{|{\ensuremath{\mathcal{G}}}(A)|})=0$ for every $h$.
A packing with $\mu_j^h$ copies of ${{\ve r}}_j^*$ in bin $0$ and $\sigma^{i,h}_\ell$ copies of ${{\ve g}}_\ell(A)$ in bin $i$ is called a sub-profile. Any sub-profile corresponds to a $(t_A+nt_B)$-dimensional vector ${{\ve e}}_h=({{\ve e}}_h^0,{{\ve e}}_h^1,\cdots,{{\ve e}}_h^n)$ where
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If all the three conditions on sub-profiles hold, then we know that $||{{\ve e}}_h||_{\infty}={{\mathcal{O}}_{FPT}}(1)$, ${{\ve g}}=\sum_h {{\ve e}}_h$ and $H_0{{\ve e}}_h=0$ (to see why $H_0{{\ve e}}_h=0$ holds, simply observe that $F{{\ve r}}_j^*=0$ and condition (iii) implies that $[0,D,D,\cdots,D]{{\ve e}}_h=0$), and furthermore, there are at most $\sum_j\alpha_j'+\sum_{i,\ell}\beta_\ell^i$ sub-profiles, which is finite. Hence, ${{\ve g}}=\sum_h {{\ve e}}_h$ and the theorem is proved.
We will construct ${{\ve e}}_h$’s iteratively. Once ${{\ve e}}_h$ is constructed, we continue our decomposition procedure on ${{\ve g}}-\sum_{k=1}^h{{\ve e}}_k$.
Suppose we have constructed ${{\ve e}}_1$ to ${{\ve e}}_{h_0-1}$ where conditions (i) and (iii) are satisfied for each ${{\ve e}}_h$, $\alpha_j'-\sum_{h=1}^{h_0-1}\mu_j^h\ge 0$, $\beta_\ell^i-\sum_{h=1}^{h_0-1}\sigma_\ell^{i,h}\ge 0$ and furthermore, each vector $\bar{{{\boldsymbol{\beta}}}}_\ell=(\bar{\beta}_\ell^1,\cdots,\bar{\beta}_\ell^n)$ where $\bar{\beta}_\ell^i=\beta_\ell^i-\sum_{h=1}^{h_0-1}\sigma_\ell^{i,h}$ can be expressed as a summation of all but one balanced vectors, more precisely, there exist balanced vectors $\phi_{\ell,k}\in{\ensuremath{\mathbb{Z}}}_{\ge 0}^n$, $1\le k\le k_{max}$ such that $$\bar{\beta}_\ell=\sum_{k=1}^{k_{max}-1}\phi_{\ell,k}+\bar{\phi}_{\ell,k_{max}}$$ where $\bar{\phi}_{\ell,k_{max}}\sqsubseteq \phi_{\ell,k_{max}}$.
We show how to construct ${{\ve e}}_{h_0}$. Let $\bar{\alpha}_j'=\alpha_j'-\sum_{h=1}^{h_0-1}\mu_j^h$. According to condition (iii) of each ${{\ve e}}_h$, we know that $$[D{{\ve v}}_1^*,D{{\ve v}}_2^*,\cdots,D{{\ve v}}_\tau^*, {{\ve g}}_1(A),{{\ve g}}_2(A),\cdots,{{\ve g}}_{|{\ensuremath{\mathcal{G}}}(A)|}(A)]\cdot (n\bar{\alpha}_1',n\bar{\alpha}_2',\cdots,n\bar{\alpha}_\tau',\sum_{i=1}^n\bar{\beta}_{1}^i,\cdots,\sum_{i=1}^n\bar{\beta}^{i}_{|{\ensuremath{\mathcal{G}}}(A)|})=0$$ Consequently, there exist $\lambda_k'\in{\ensuremath{\mathbb{Z}}}_{\ge 0}$ and ${{\ve g}}_k\in{\ensuremath{\mathbb{Z}}}_{\ge 0}^{\tau+|{\ensuremath{\mathcal{G}}}(A)|}\cap {\ensuremath{\mathcal{G}}}(V)$ such that $$(n\bar{\alpha}_1',n\bar{\alpha}_2',\cdots,n\bar{\alpha}_\tau',\sum_{i=1}^n\bar{\beta}_{1}^i,\cdots,\sum_{i=1}^n\bar{\beta}^{i}_{|{\ensuremath{\mathcal{G}}}(A)|})=\sum_{k}\lambda_k' {{\ve g}}_k(V).$$ There are two possibilities.
**Case 2.1** If there exists some $\lambda_k'\ge n$, we consider the vector $n{{\ve g}}_k(V)$. Let $n{{\ve g}}_k(V)=(n\zeta_1,n\zeta_2,\cdots,n\zeta_{\tau+|{\ensuremath{\mathcal{G}}}(A)|})$. We set $\mu_{j}^{h_0}=\zeta_j={{\mathcal{O}}_{FPT}}(1)$ for $1\le j\le \tau$. We set the values of $\sigma_{\ell}^{i,h_0}$ such that $\sum_{i=1}^n\sigma_\ell^{i,h_0}=n\zeta_{\tau+\ell}$. Consequently, condition (iii) is satisfied for ${{\ve e}}_{h_0}$. Now it suffices to set the values of each $\sigma_{\ell}^{i,h_0}$ such that they are bounded by ${{\mathcal{O}}_{FPT}}(1)$. Equivalently, this means out of the $\bar{\beta}_\ell^i$ copies of ${{\ve g}}_\ell(A)$, our goal is to take $\sigma_{\ell}^{i,h_0}$ copies such that in total we take $n\zeta_{\tau+\ell}$ copies and $\sigma_{\ell}^{i,h_0}={{\mathcal{O}}_{FPT}}(1)$. We achieve this in a simple greedy way. Let $k^*$ be the index such that $$\sum_{k=k^*+1}^{k_{max}-1}||\phi_{\ell,k}||_1+||\bar{\phi}_{\ell,k_{max}}||_1<n\zeta_{\tau+\ell}\le \sum_{k=k^*}^{k_{max}-1}||\phi_{\ell,k}||_1+||\bar{\phi}_{\ell,k_{max}}||_1$$ Let $\bar{\phi}_{\ell,k^*}\sqsubseteq \phi_{\ell,k^*}$ be an arbitrary vector such that $$||\bar{\phi}_{\ell,k^*}||_{1}+\sum_{k=k^*+1}^{k_{max}-1}||\phi_{\ell,k}||_1+||\bar{\phi}_{\ell,k_{max}}||_1=n\zeta_{\tau+\ell}.$$ We set $\sigma_{\ell}^{i,h_0}=\bar{\phi}_{\ell,k^*}^i+\sum_{k=k^*+1}^{k_{max}-1}\phi_{\ell,k}^i+\bar{\phi}_{\ell,k_{max}}^i$. It is obvious that in total we have taken $n\zeta_{\tau+\ell}$ copies of ${{\ve g}}_\ell(A)$. Now it remains to show that $||\sigma_\ell^{h_0}||_{\infty}=||\bar{\phi}_{\ell,k^*}+\sum_{k=k^*+1}^{k_{max}-1}\phi_{\ell,k}+\bar{\phi}_{\ell,k_{max}}||_{\infty}={{\mathcal{O}}_{FPT}}(1)$. To see this, notice that each $\phi_{\ell,k}$ is a balanced vector, hence $$||{\phi}_{\ell,k^*}||_{1}+\sum_{k=k^*+1}^{k_{max}-1}||\phi_{\ell,k}||_1+||{\phi}_{\ell,k_{max}}||_1\le n\zeta_{\tau+\ell}+2n\alpha_{max}={{\mathcal{O}}_{FPT}}(n).$$ According to Claim \[claim:inftybound-2\], $||{\phi}_{\ell,k^*}+\sum_{k=k^*+1}^{k_{max}-1}\phi_{\ell,k}+{\phi}_{\ell,k_{max}}||_{\infty}={{\mathcal{O}}_{FPT}}(1)$. Consequently, $||\sigma_\ell^{h_0}||_{\infty}={{\mathcal{O}}_{FPT}}(1)$.
Also notice that after we take $\sigma_{\ell}^{i,h_0}$ copies of ${{\ve g}}_\ell(A)$, $$\bar{{{\boldsymbol{\beta}}}}_\ell-\sigma_{\ell}^{h_0}=\sum_{k=1}^{k^*-1}\phi_{\ell,k}+(\phi_{\ell,k^*}-\bar{\phi}_{\ell,k^*}),$$ which is still the summation of all but one balanced vector. Hence we can continue to decompose ${{\ve g}}-\sum_{h=1}^{h_0}{{\ve e}}_h$.
**Case 2.2** $\lambda_k'< n$ for every $k$. We claim that $||{{\ve g}}-\sum_{h=1}^{h_0-1}{{\ve e}}_h||_{\infty}={{{\mathcal{O}}_{FPT}}(1)}$. If this claim is true, then ${{\ve g}}=\sum_{h=1}^{h_0-1}{{\ve e}}_h+({{\ve g}}-\sum_{h=1}^{h_0-1}{{\ve e}}_h)$, and Theorem \[lemma:3-infty-bound\] is proved. To show the claim, we use a similar argument as that of case 1. First, $n\bar{\alpha}_j'\le (\sum_k\lambda_k)\cdot \max_k||{{\ve g}}_k(V)||_{\infty}={{\mathcal{O}}_{FPT}}(n)$, hence $\bar{\alpha}_j'={{\mathcal{O}}_{FPT}}(1)$. Second, we consider the $n$-dimensional vector ${{\boldsymbol{\beta}}}=\sum_{\ell=1}^{|{\ensuremath{\mathcal{G}}}(A)|}{{\boldsymbol{\beta}}}_{\ell}$. Let $\bar{\beta}_\ell'=\sum_{k=1}^{k_{max}}\phi_{\ell,k}$ and ${{\boldsymbol{\beta}}}'=\sum_{\ell=1}^{|{\ensuremath{\mathcal{G}}}(A)|}{{\boldsymbol{\beta}}}_{\ell}'$. Given that $\bar{\phi}_{\ell,k_{max}}\sqsubseteq {\phi}_{\ell,k_{max}}$ and ${\phi}_{\ell,k_{max}}$ is a balanced vector, $||\bar{\beta}_\ell'||_1\le ||\bar{\beta}_\ell||_1+n\alpha_{max}$. Consequently $$||\beta'||_1\le \sum_{\ell=1}^{|{\ensuremath{\mathcal{G}}}(A)|}||\bar{\beta}_\ell'||_1\le \sum_{\ell=1}^{|{\ensuremath{\mathcal{G}}}(A)|}||\bar{\beta}_\ell||_1+n\alpha_{max}\cdot |{\ensuremath{\mathcal{G}}}(A)|\le \sum_k\lambda_k'\cdot \max_k||{{\ve g}}_k(V)||_1+n\alpha_{max}\cdot |{\ensuremath{\mathcal{G}}}(A)|={{\mathcal{O}}_{FPT}}(n).$$ Note that ${{\boldsymbol{\beta}}}'$ is the summation of balanced vectors. According to Claim \[claim:inftybound-2\], $||{{\boldsymbol{\beta}}}'||_{\infty}=OFPT(1)$, consequently $||{{\boldsymbol{\beta}}}||_{\infty}\le ||{{\boldsymbol{\beta}}}'||_{\infty}=OFPT(1)$. Combining the fact that $||{{\ve u}}_{j}^*||_{\infty}={{\mathcal{O}}_{FPT}}(1)$, $||{{\ve v}}_j^*||_{\infty}={{\mathcal{O}}_{FPT}}(1)$ and $||{{\ve g}}_\ell(A)||_{\infty}={{\mathcal{O}}_{FPT}}(1)$, we conclude that $||{{\ve g}}-\sum_{i=1}^{h_0-1}{{\ve e}}_h||_{\infty}={{\mathcal{O}}_{FPT}}(1)$.
Theorem \[lemma:3-infty-bound\] indicates that, there exists a Hilbert basis for 3-block $n$-fold IP with FPT-bounded $\ell_{\infty}$-norms. The following lemma provides a slightly more compact form of decomposition, which will also be utilized later.
\[lemma:decompose-bounded-simpler\] There exist a set of $q'={{\mathcal{O}}_{FPT}}(1)$ vectors $S=\{\bar{{{\ve e}}}_1,\bar{{{\ve e}}}_2,\cdots,\bar{{{\ve e}}}_{q'}\}$ of such that for any ${{\ve y}}\in ker_{{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}}(H_0)$, there exist $\alpha_h,\beta_\ell\in{\ensuremath{\mathbb{Z}}}_{\ge 0}$ and at most $2nt_A-1$ vectors ${{\ve d}}_\ell=(0,{{\ve g}}_\ell(E))$ where ${{\ve g}}_\ell(E)\in{\ensuremath{\mathcal{G}}}(E)$ such that $\alpha_h=0$ if $\bar{{{\ve e}}}_h^0\not\sqsubseteq {{\ve y}}^0$, $||\bar{{{\ve e}}}_h||_{\infty}\le \xi={{\mathcal{O}}_{FPT}}(1)$, and $$\begin{aligned}
\label{eq:decompose-main}
{{\ve y}}=\sum_{h=1}^{q'}\alpha_h\bar{{{\ve e}}}_h+\sum_{\ell}\beta_{\ell}{{\ve d}}_\ell.
\end{aligned}$$ Furthermore, ${{\ve d}}_\ell$’s lie in the same orthant and the set $S$ can be computed in ${{\mathcal{O}}_{FPT}}(n^3L)$ time where $L$ is the length of the encoding.
According to Theorem \[lemma:3-infty-bound\], there exist ${{\ve e}}_1,{{\ve e}}_2,\cdots,{{\ve e}}_k$ with $||{{\ve e}}_h||_{\infty}\le \xi$, ${{\ve e}}_h^0\sqsubseteq {{\ve y}}^0$ such that ${{\ve y}}=\sum_{h=1}^k {{\ve e}}_h$. Let ${{\ve u}}_1,{{\ve u}}_2,\cdots,{{\ve u}}_{\eta}$ be all the $t_B$-dimensional vectors whose $\ell_{\infty}$-norm is bounded by $\xi$, then $\eta=O(\xi^{t_B})={{\mathcal{O}}_{FPT}}(1)$. For any ${{\ve u}}_j$, we pick an arbitrary $\bar{{{\ve e}}}_j$ such that $H_0\bar{{{\ve e}}}_j=0$ and $\bar{{{\ve e}}}_j^0={{\ve u}}_j$. Note that such a $\bar{{{\ve e}}}_j$ can be found out by solving an $n$-fold IP, which can be done in ${{\mathcal{O}}_{FPT}}(n^3L)$ time [@hemmecke2013n]. Among ${{\ve e}}_1$ to ${{\ve e}}_k$, we define $K_j=\{{{\ve e}}_h:{{\ve e}}_h^0={{\ve u}}_j, 1\le h\le k\}$. We have $${{\ve y}}=\sum_{j=1}^\eta \bar{{{\ve e}}}_j\cdot |K_j|+\sum_{j=1}^\eta\sum_{{{\ve e}}_h\in K_j}({{\ve e}}_h-\bar{{{\ve e}}}_j)$$
Since $H_0{{\ve e}}_h=0$, it is easy to see that $\sum_{j=1}^q\sum_{{{\ve e}}_h\in K_j}({{\ve e}}_h-\bar{{{\ve e}}}_j)=(0,{{\ve e}}')$ where ${{\ve e}}'$ is a feasible solution to $E{{\ve x}}=0$ where $E$ is an $n$-fold matrix. According to [@hemmecke2013n], there exists at most $2nt_A-1$ vectors ${{\ve g}}_\ell(E)\in{\ensuremath{\mathcal{G}}}(E)$, ${{\ve g}}_\ell(E)\sqsubseteq {{\ve e}}'$ and $\beta_\ell\in{\ensuremath{\mathbb{Z}}}_+$ such that ${{\ve e}}'=\sum_{\ell}\beta_\ell{{\ve g}}_\ell(E)$. Define ${{\ve d}}_\ell=(0,{{\ve g}}_\ell(E))$, we have $$\sum_{j=1}^\eta\sum_{{{\ve e}}_h\in K_j}({{\ve e}}_h-\bar{{{\ve e}}}_j)=\sum_{\ell}\beta_\ell{{\ve d}}_\ell.$$ Thus, the lemma is proved.
It is remarkable that we can further restrict that the $\bar{{{\ve e}}}_h$’s also lie in the same orthant (see Lemma \[lemma:decompose-bounded\]).
A sign-compatible decomposition
-------------------------------
We have shown in the previous subsection that any vector of $ker_{{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}}(H_0)=\{{{\ve x}}\in{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}: H_0{{\ve x}}=0,{{\ve x}}\neq 0\}$ can be expressed as a conic combination of vectors in $ker_{{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}}(H_0)$ whose $\ell_{\infty}$-norm is bounded by ${{\mathcal{O}}_{FPT}}(1)$. However, it is not clear how such a result can be utilized directly for an augmentation algorithm. The current augmentation algorithms for 4-block $n$-fold IP as well as other related IPs all rely on Graver basis. This is due to the fact that if there exists a feasible or optimal augmentation vector that can be decomposed into a conic combination of Graver basis elements which lie in the same orthant, then any of these Graver basis elements itself also provides a feasible augmentation. This fact is, unfortunately, no longer true if we take a conic combination of some Hilbert basis elements that do not necessarily lie in the same orthant. Towards the algorithm for 3-block $n$-fold IP, we resort to Graver basis. We show the following upper bounds on the $\ell_{\infty}$-norm of the Graver basis for 3-block $n$-fold IP.
\[thm:3-block-graver\] For any $3$-block $n$-fold matrix $H_0$ and ${{\ve g}}(H_0)\in {\ensuremath{\mathcal{G}}}(H_0)$, $||{{\ve g}}(H_0)||_{\infty}\le
{{\mathcal{O}}_{FPT}}(n^{t_A^2}+1)$.
In [@hemmecke2014graver], Hemmecke, K[ö]{}ppe and Weismantel provide two upper bounds on the $\ell_{\infty}$-norm of Graver basis element for a general $4$-block $n$-fold IP, which is ${{\mathcal{O}}_{FPT}}(n^{2^{s_c}})$ and ${{\mathcal{O}}_{FPT}}(n^{k(A,B)})$ where $k(A,B)$ is some unknown function that is dependent on $s_A,s_B,t_A,t_B,||A||_{\infty},||B||_{\infty}$. Indeed, as the existence of such a $k(A,B)$ follows from some algebraic argument, even a rough estimation of $k(A,B)$ is not clear, despite that it is highly unlikely for $k(A,B)$ to be some polynomial of the parameters. In this paper, we have established, in Theorem \[thm:3-block-graver-4\], that there exists a tight bound of ${{\mathcal{O}}_{FPT}}(n^{s_c})$ which depends on $s_C$. An explicit upper bound that depends on $A,B$, however, is still unclear. Theorem \[thm:3-block-graver\] provides such an upper bound for the special case when $C$ is a zero matrix. Following the line of arguments in previous papers [@aschenbrenner2007finiteness; @hemmecke2014graver; @hemmecke2011polynomial; @hocsten2007finiteness], it seems very difficult to get rid of the exponential dependency of $k(A,B)$. To prove the Theorem \[thm:3-block-graver\], we use a completely different approach.
We give a brief overview of the proof of Theorem \[thm:3-block-graver\]. The basic idea is to show that if $||{{\ve g}}(H_0)||_{\infty}$ is too large, then we are able to find some ${{\ve z}}\sqsubset {{\ve g}}(H_0)$ and $H_0{{\ve z}}=0$, contradicting the fact that ${{\ve g}}(H_0)$ is a Graver basis element. Suppose ${{\ve y}}={{\ve g}}(H_0)$ and $||{{\ve y}}||_{\infty}$ is very large. The most crucial idea is that we do not search directly for ${{\ve z}}\sqsubset {{\ve y}}$, but rather search for ${{\ve z}}\sqsubset \tilde{{{\ve y}}}$ where $\tilde{{{\ve y}}}$ is called a centralization of ${{\ve y}}$, and then prove that such a ${{\ve z}}$ also satisfy that ${{\ve z}}\sqsubset {{\ve y}}$. Roughly speaking, we will divide the $n$ bricks of ${{\ve y}}$, i.e., ${{\ve y}}^i$ for $i=1,2,\cdots, n$, into $\sigma={{\mathcal{O}}_{FPT}}(1)$ groups $N_1,N_2,\cdots,N_{\sigma}$ such that for any $k\in N_j$, $\tilde{{{\ve y}}}^k\approx \frac{1}{N_j} \sum_{i\in N_j}{{\ve y}}^i$. Why do we need to take such a detour in the proof? The problem is that by directly arguing on ${{\ve y}}$ we run into a bound that is similar as [@hemmecke2014graver]. Therefore, we use a completely different approach – we adopt the decomposition of Theorem \[lemma:3-infty-bound\], and then modify such a decomposition into a sign-compatible one by merging summands. Towards this, we first prove a merging lemma (Lemma \[lemma:merging-lemma\]) which states that given a summation of a sequence of vectors, we can always divide them into disjoint subsets such the summation of vectors in each subset becomes sign-compatible. The merging lemma can turn an arbitrary decomposition into a sign-compatible one, albeit the fact that the cardinality of each subset is exponential in the dimension of the vectors (which means the $\ell_{\infty}$-norm of the summands will explode by a factor that is exponential in the dimension). Consequently, if we directly merge the ${{\mathcal{O}}_{FPT}}(n)$-dimensional vectors in the decomposition of Theorem \[lemma:3-infty-bound\], we get a very weak bound. To handle this, we consider an alternative sum $\tilde{{{\ve y}}}$, which is derived by averaging multiple bricks of ${{\ve y}}$ as we mentioned above. By altering the decomposition of ${{\ve y}}$, we get a decomposition of $\tilde{{{\ve y}}}$ such that the following is true: all the $n+1$ bricks of each vector-summand can be divided into ${{\mathcal{O}}_{FPT}}(1)$ subsets where in each subset the bricks are identical. This indicates that, although we are summing up ${{\mathcal{O}}_{FPT}}(n)$-dimensional vectors to $\tilde{{{\ve y}}}$, it is essentially the same as summing up ${{\mathcal{O}}_{FPT}}(1)$-dimensional vectors. Such a transformation comes at a cost – summands summing up to $\tilde{{{\ve y}}}$ do not have ${{\mathcal{O}}_{FPT}}(1)$-bounded $\ell_{\infty}$-norms, indeed, each vector-summand consists of $n$ bricks whose $\ell_{\infty}$-norm is ${{\mathcal{O}}_{FPT}}(1)$, and at most $1$ brick (which is a $t_A$-dimensional vector) whose $\ell_{\infty}$-norm is ${{\mathcal{O}}_{FPT}}(n)$. Applying our merging lemma, we derive a sign-compatible decomposition of $\tilde{{{\ve y}}}$ where the summands have an $\ell_{\infty}$-norm bounded by ${{\mathcal{O}}_{FPT}}(n^{t_A^2+1})$. It remains to show that, at least one vector-summand ${{\ve z}}$ in the sign-compatible decomposition of $\tilde{{{\ve y}}}$ also satisfies that ${{\ve z}}\sqsubset {{\ve y}}$. This goes back to the definition of $\tilde{{{\ve y}}}$. We are averaging bricks of ${{\ve y}}$, but which bricks shall we average? Each brick is a $t_A$-dimensional vector and we consider each coordinate. We set up a threshold $\Gamma$. If the absolute value of a coordinate is larger than $\Gamma$, we say it is (positive or negative) large. Otherwise it is small. Therefore, each brick can be characterized by identifying its coordinates being positive large, negative large or small (which is defined as the quantity type of a brick). We only average the bricks of the same quantity type. By doing so, we can ensure that $\tilde{{{\ve y}}}^i$ is roughly sign-compatible with ${{\ve y}}^i$ – if the $j$-th coordinate of ${{\ve y}}^i$ is positive or negative large, then this coordinate of $\tilde{{{\ve y}}}^i$ is also positive or negative. Hence, any ${{\ve z}}\sqsubset \tilde{{{\ve y}}}^i$ is almost sign-compatible with ${{\ve y}}$ – indeed, if we can ensure additionally that the $j$-th coordinate of ${{\ve z}}^i$ is $0$ as long as the $j$-th coordinate of ${{\ve y}}^i$ is small, then we can conclude that ${{\ve z}}\sqsubset {{\ve y}}$. This if can be proved, and we get Theorem \[thm:3-block-graver\].
### A merging lemma {#subsec:merging}
\[lemma:merging-lemma-1\] Let $x_1,x_2,\cdots,x_m$ be a sequence of integers such that $x=\sum_{i=1}^mx_i$, and $|x_i|\le \zeta$. Then the $m$ integers can be partitioned into $m'$ subsets $T_1,T_2,\cdots,T_{m'}$ satisfying that: $\cup_{j=1}^{m'}T_j=\{1,2,\cdots,m\}$, and for every $1\le j\le m'$ it holds that $\sum_{i\in T_j} x_i\sqsubseteq x$, $|T_j|\le 6\zeta+2$.
Without loss of generality we assume that $x\ge 0$ (otherwise we argue on $-x_i$’s). If $m\le 6\zeta+2$ the lemma is trivial. Otherwise we apply Steinitz lemma (Lemma \[lemma:steinitz\]) to the integral sequence $x_1,x_2,\cdots,x_m$ and there exists a permutation $\pi$ such that for all $1\le \ell\le m$ it holds that $$|\sum_{i=1}^\ell x_{\pi(i)}-\frac{\ell-1}{m}x|\le \zeta.$$ Now we consider the first $3\zeta+2$ numbers $x_{\pi(1)},x_{\pi(2)},\cdots,x_{\pi(3\zeta+2)}$. There are two possibilities regarding $(3\zeta+1)x/m$.
If $(3\zeta+1)x/m> \zeta$, then since $-\zeta\le \sum_{i=1}^{3\zeta+2}x_{\pi(i)}-(3\zeta+1)x/m\le \zeta$, we know that $\sum_{i=1}^{3\zeta+2}x_{\pi(i)}\ge 0$, and is consequently sign-compatible with $x$. Further notice that the summation of the remaining integers satisfies that $\sum_{i=3\zeta+3}^m x_{\pi(i)}\ge x-(3\zeta+1)x/m-\zeta$. Given that $m\ge 6\zeta+2$, $x-(3\zeta+1)x/m\ge (3\zeta+1)x/m>\zeta$, the summation of the remaining integers is still positive.
Otherwise $(3\zeta+1)x/m\le \zeta$, and consequently $0\le (\ell-1) x/m\le \zeta$ for any $1\le \ell\le 3\zeta+2$. This implies that the values of the $3\zeta+2$ numbers $\sum_{i=1}^{\ell}x_{\pi(i)}$ where $1\le \ell\le 3\zeta+2$ must lie in the set $\{-\zeta,-\zeta+1,\cdots,2\zeta\}$, i.e., there must exist two distinct integers $\ell_1<\ell_2$ and $\ell_1,\ell_2\le 3\zeta+2$ such that $\sum_{i=1}^{\ell_1} x_{\pi(i)}=\sum_{i=1}^{\ell_2} x_{\pi(i)}$. Consequently, $\sum_{i=1}^{\ell_2-\ell_1} x_{\pi(i)}=0$. Further notice that by taking out the sequence of integers $x_{\ell_1+1},\cdots,x_{\ell_2}$, the summation is of the remaining integers is still $x\ge 0$
Hence, as long as $m\ge 6\zeta+2$, we can always select at most $3\zeta+2$ integers whose summation is non-negative, and if we delete them, the summation of the remaining integers is still non-negative. Hence, we can carry on our argument on the remaining integers, and the lemma is proved.
We can extend the above lemma to higher dimensions using the same basic idea but a much more involved analysis.
In the following we write ${{\mathcal{O}}}^*(x^k)$ to represent a function that is bounded by $(cx)^{k}$ for some constant $c$.
\[lemma:merging-lemma\] Let ${{\ve x}}_1,{{\ve x}}_2,\cdots,{{\ve x}}_m$ be a sequence of vectors in $\mathbb{Z}^\kappa$ such that ${{\ve x}}=\sum_{i=1}^m{{\ve x}}_i$, and $||{{\ve x}}_i||_{\infty}\le \zeta$. Then the $m$ vectors can be partitioned into $m'$ subsets $T_1,T_2,\cdots,T_{m'}$ satisfying that: $\cup_{j=1}^{m'}T_j=\{1,2,\cdots,m\}$, and for every $1\le j\le m'$ it holds that $\sum_{i\in T_j} {{\ve x}}_i\sqsubseteq {{\ve x}}$, $|T_j|={{\mathcal{O}}}^*(\zeta^{\kappa^2})$.
Again we assume without loss of generality that ${{\ve x}}\ge 0$ (if some of the coordinates are negative, then we take the nagation of every ${{\ve x}}_i$ and ${{\ve x}}$ at this coordinate). For ${{\ve x}}=({{\ve x}}^1,{{\ve x}}^2,\cdots,{{\ve x}}^{\kappa})$, we further assume that ${{\ve x}}^1\le {{\ve x}}^2\le\cdots\le {{\ve x}}^{\kappa}$ (Notice that here ${{\ve x}}^j\in{\ensuremath{\mathbb{Z}}}$). By Steinitz lemma (Lemma \[lemma:steinitz\]), there exists a permutation $\pi$ such that for all $1\le \ell\le m$ it holds that $$\begin{aligned}
\label{eq:steinitz1}
||\sum_{i=1}^\ell {{\ve x}}_{\pi(i)}-\frac{\ell-\kappa}{m}{{\ve x}}||_{\infty}\le \zeta.
\end{aligned}$$ For simplicity, we reorder all the vectors such that ${{\ve x}}_{\pi(i)}$ is at the $i$-th location, i.e., we assume that the given vectors ${{\ve x}}_i$ satisfy that $$\begin{aligned}
\label{eq:steinitz}
||\sum_{i=1}^\ell {{\ve x}}_{i}-\frac{\ell-\kappa}{m}{{\ve x}}||_{\infty}\le \zeta.\end{aligned}$$
Our goal is to show the following claim:
\[claim:conformal-sum\] There always exists a subset $T$ such that, $|T|={{\mathcal{O}}}^*(\zeta^{\kappa^2}) $, $\sum_{i\in T} {{\ve x}}_i \sqsubseteq {{\ve x}}$ and ${{\ve x}}-\sum_{i\in T} {{\ve x}}_i\ge 0$.
If the claim is true, we can iteratively apply it to cut the whole sequence of vectors into subsets $T_1,T_2,\cdots,T_{m'}$ and Lemma \[lemma:merging-lemma\] is proved.
To prove Claim \[claim:conformal-sum\], we need the following two claims. For simplicity, we say a subset $T$ is [*conformal*]{} if $\sum_{i\in T} {{\ve x}}_i \sqsubseteq {{\ve x}}$ and ${{\ve x}}-\sum_{i\in T} {{\ve x}}_i\ge 0$.
\[claim:conformal-sum1\] For any $1\le j\le \kappa$, if there exists some $\mu_j$ such that $\frac{\mu_j}{m}{{\ve x}}^j>2\zeta\ge \frac{\mu_j-1}{m}{{\ve x}}^j$ and $(\mu_j-1)\frac{{{\ve x}}^j}{{{\ve x}}^{j-1}}>\kappa+(6\zeta+1)^{j-1}\mu_j$, then there exists a subset $T$ such that $|T|\le 3(6\zeta+1)^{j-1}\mu_j+\kappa$ and $T$ is conformal, i.e., $\sum_{i\in T} {{\ve x}}_i \sqsubseteq {{\ve x}}$ and ${{\ve x}}-\sum_{i\in T} {{\ve x}}_i\ge 0$.
If $m\le 3(6\zeta+1)^{j-1}\mu_j+\kappa$, then we take $T=\{1,2,\cdots,m\}$. In the following we assume that $m>3(6\zeta+1)^{j-1}\mu_j+\kappa$. Recall that ${{\ve x}}^1\le {{\ve x}}^2\le\cdots\le {{\ve x}}^{\kappa}$, whereas $\frac{\mu_j}{m}{{\ve x}}^h>2\zeta$ for any $h\ge j$. Consider an arbitrary subsequence of vectors whose length is $\mu\ge \mu_j$, say, ${{\ve x}}_{i_0}, {{\ve x}}_{i_0+1},\cdots,{{\ve x}}_{i_0+\mu-1}$. By Eq (\[claim:conformal-sum\]), we have $$\begin{aligned}
\label{eq:claim}
||\sum_{i=1}^{i_0-1} {{\ve x}}_{i}-\frac{i_0-1-\kappa}{m}{{\ve x}}||_{\infty}\le \zeta, \quad {\text{ and }}\quad ||\sum_{i=1}^{i_0+\mu-1} {{\ve x}}_{i}-\frac{i_0+\mu-1-\kappa}{m}{{\ve x}}||_{\infty}\le \zeta.
\end{aligned}$$ Consequently, for any $h\ge j$, it follows that $$\begin{aligned}
\sum_{i=1}^{i_0-1} {{\ve x}}_{i}^h\le \frac{i_0-1-\kappa}{m}{{\ve x}}^h+ \zeta, \quad {\text{ and }}\quad \sum_{i=1}^{i_0+\mu-1} {{\ve x}}_{i}^h\ge \frac{i_0+\mu-1-\kappa}{m}{{\ve x}}^h- \zeta.\end{aligned}$$ Thus, $$\begin{aligned}
\label{eq:claim1}
\sum_{i=i_0}^{i_0+\mu-1} {{\ve x}}_{i}^h\ge \frac{\mu}{m}{{\ve x}}^h- 2\zeta\ge\frac{\mu_j}{m}{{\ve x}}^h-2\zeta >0, \quad \forall h\ge j\end{aligned}$$ This means, the summation of any adjacent $\mu\ge \mu_j$ vectors satisfies that the sum is positive on every $h$-th coordinate for $h\ge j$.
Meanwhile, by Eq (\[eq:claim\]) we have $$\begin{aligned}
\sum_{i=1}^{i_0-1} {{\ve x}}_{i}^h\ge \frac{i_0-1-\kappa}{m}{{\ve x}}^h- \zeta, \quad {\text{ and }}\quad \sum_{i=1}^{i_0+\mu-1} {{\ve x}}_{i}^h\le \frac{i_0+\mu-1-\kappa}{m}{{\ve x}}^h+ \zeta.\end{aligned}$$ Thus, $$\sum_{i=i_0}^{i_0+\mu-1} {{\ve x}}_{i}^h\le\frac{\mu}{m}{{\ve x}}^h+2\zeta,\quad \forall h\ge j.$$ Meanwhile, we have $$\sum_{i=1}^{m} {{\ve x}}_{i}^h\ge\frac{m-\kappa}{m}{{\ve x}}^h-\zeta\ge \frac{\mu}{m}{{\ve x}}^h\cdot \frac{m-\kappa}{\mu}-\zeta,\quad \forall h\ge j.$$ Thus, $$\begin{aligned}
\label{eq:claim2}
\sum_{i=1}^{m} {{\ve x}}_{i}^h-\sum_{i=i_0}^{i_0+\mu-1} {{\ve x}}_{i}^h\ge \frac{\mu}{m}{{\ve x}}^h\cdot (\frac{m-\kappa}{\mu}-1)-3\zeta,\quad \forall h\ge j.\end{aligned}$$ Recall that $\frac{\mu}{m}{{\ve x}}^h>2\zeta$, as long as $m-\kappa\ge 3\mu$, we know that $\sum_{i=1}^{m} {{\ve x}}_{i}^h-\sum_{i=i_0}^{i_0+\mu-1} {{\ve x}}_{i}^h> 0$ for all $h\ge j$.
Next we consider the $h$-th coordinate for $h<j$. Recall that $\frac{\mu_j-1}{m}{{\ve x}}^j\le 2\zeta$. As $(\mu_j-1)\frac{{{\ve x}}^j}{{{\ve x}}^{j-1}}>\kappa+(6\zeta+1)^{j-1}\mu_j$, it follows directly that $$\frac{\kappa+(6\zeta+1)^{j-1}\mu_j}{m}{{\ve x}}^h\le 2\zeta, \quad \forall h\le j-1.$$ Now we consider the $1+(6\zeta+1)^{j-1}$ vectors $\sum_{i=1}^{\ell}{{\ve x}}_i$ for $\ell\in \mathcal{L}_j$ where $\mathcal{L}_j=\{\kappa,\kappa+\mu_j,\kappa+2\mu_j,\cdots,\kappa+(6\zeta+1)^{j-1}\mu_j\}$. For any $\ell\in \mathcal{L}_j$, it is clear that $$|\sum_{i=1}^{\ell}x_i^h|\le \frac{\ell-\kappa}{m}{{\ve x}}^h+2\zeta\le 3\zeta, \quad \forall h\le j-1$$ that is, $\sum_{i=1}^{\ell}x_i^h\in\{-3\zeta,-3\zeta+1,\cdots,3\zeta\}$ for all $\ell\in \mathcal{L}_j$ and $h\le j-1$. Hence, if we consider the projection of $\sum_{i=1}^\ell{{\ve x}}_i$ onto its first $j-1$ coordinates, this projection lies within $\{-3\zeta,-3\zeta+1,\cdots,3\zeta\}^{j-1}$, implying that there must exist $\ell_1<\ell_2$ such that $\sum_{i=1}^{\ell_1}{{\ve x}}_i$ and $\sum_{i=1}^{\ell_2}{{\ve x}}_i$ have the same projection. Hence, the first $j-1$ coordinates of $\sum_{i=\ell_1+1}^{\ell_2}{{\ve x}}_i$ are all $0$. Furthermore, we observe the followings: 1). $\ell_2-\ell_1>\mu_j'$, whereas for $h\ge j$, the $h$-th coordinate of $\sum_{i=\ell_1+1}^{\ell_2}{{\ve x}}_i$ is positive according to Eq (\[eq:claim1\]). 2). $\ell_2-\ell_1\le (6\zeta+1)^{j-1}\mu_j$ and $m\ge 3(6\zeta+1)^{j-1}\mu_j+\kappa$, whereas for $h\ge j$, the $h$-th coordinate of $\sum_{i=1}^m{{\ve x}}_i-\sum_{i=\ell_1+1}^{\ell_2}{{\ve x}}_i$ is also positive according to Eq (\[eq:claim2\]). Hence, $\sum_{i=\ell_1+1}^{\ell_2}{{\ve x}}_i\sqsubseteq {{\ve x}}$ and $\sum_{i=1}^m{{\ve x}}_i-\sum_{i=\ell_1+1}^{\ell_2}{{\ve x}}_i\ge 0$, i,e, by taking $T=\{\ell_1+1,\ell_1+2,\cdots,\ell_2\}$, Claim \[claim:conformal-sum1\] is true.
Now we come to the proof of Claim \[claim:conformal-sum\].
We prove the claim by induction on the following hypothesis.
**Statement:** For $1\le j\le \kappa$, either there exists some $T$ which is conformal (i.e., $\sum_{i\in T} {{\ve x}}_i \sqsubseteq {{\ve x}}$ and ${{\ve x}}-\sum_{i\in T} {{\ve x}}_i\ge 0$) and $|T|={{\mathcal{O}}}^*(\zeta^{(\kappa-j+1)\kappa})$, or there exists some $\mu_{j}={{\mathcal{O}}}^*(\zeta^{(\kappa-j+1)\kappa})$ such that $\frac{\mu_{j}}{m}{{\ve x}}^{j}> 2\zeta \ge \frac{\mu_{j}-1}{m}{{\ve x}}^{j}$.
We first prove the statement for $j=k$. Taking $\mu_{\kappa}'=(6\zeta+1)^{\kappa}+\kappa={{\mathcal{O}}}^*(\zeta^{\kappa})$. There are two possibilities.
If $\frac{\mu_{\kappa}'}{m}{{\ve x}}^{k}\le 2\zeta $, then $\frac{\mu_{\kappa}'}{m}{{\ve x}}^{j}\le 2\zeta $ for all $1\le j\le k$. Consequently, for $\ell\in \mathcal{L}=\{\kappa,\kappa+1,\cdots,\mu_{\kappa}'\}$, we have $$||\sum_{i=1}^{\ell}{{\ve x}}_i||_{\infty}\le \frac{\mu_{\kappa}'}{m}{{\ve x}}^{k}+\zeta\le 3\zeta, \quad \forall i\in \mathcal{L}$$ i.e., $\sum_{i=1}^{\ell}{{\ve x}}_i\in\{-3\zeta,-3\zeta+1,\cdots,3\zeta\}^{\kappa}$. Since $|\mathcal{L}|=(6\zeta+1)^{\kappa}+1$, there exist $\ell_1<\ell_2$ and $\ell_1,\ell_2\in\mathcal{L}$ such that $\sum_{i=1}^{\ell_1}{{\ve x}}_i=\sum_{i=1}^{\ell_2}{{\ve x}}_i$, i.e., $\sum_{i=\ell_1+1}^{\ell_2}{{\ve x}}_i=0$. Taking $T=\{\ell_1+1,\cdots,\ell_2\}$, the statement is true.
Otherwise, $\frac{\mu_{\kappa}'}{m}{{\ve x}}^{k}> 2\zeta $. Then there exists some $\mu_{\kappa}\le \mu_{\kappa}'={{\mathcal{O}}}^*(\zeta^{k})$ such that $\frac{\mu_{\kappa}}{m}{{\ve x}}^{k}> 2\zeta \ge \frac{\mu_{\kappa}-1}{m}{{\ve x}}^{k}$. That is, the statement is also true.
Suppose the statement (hypothesis) holds for $j\ge j_0$, we prove it for $j=j_0-1$. According to the statement, either there exists some $T$ satisfying Claim \[claim:conformal-sum\] with $|T|={{\mathcal{O}}}^*(\zeta^{(\kappa-j_0+1)\kappa})$, or there exists some $\mu_{j_0}={{\mathcal{O}}}^*(\zeta^{(\kappa-j_0+1)\kappa})$ such that $\frac{\mu_{j_0}}{m}{{\ve x}}^{j_0}> 2\zeta \ge \frac{\mu_{j_0}-1}{m}{{\ve x}}^{j_0}$. If the former case is true, then obviously the same $T$ satisfies that $|T|\le {{\mathcal{O}}}^*(\zeta^{(\kappa-j_0+2)\kappa})$, implying that the statement is true for $j=j_0-1$. Hence, from now on we assume the latter case is true. According to Claim \[claim:conformal-sum1\], if $(\mu_{j_0}-1)\frac{{{\ve x}}^{j_0}}{{{\ve x}}^{j_0-1}}>\kappa+(6\zeta+1)^{j_0-1}\mu_{j_0}$, then there exists a subset $T$ which is conformal and $|T|\le (6\zeta+1)^{j_0-1}\mu_{j_0}$. Plugging in $\mu_{j_0}={{\mathcal{O}}}^*(\zeta^{(\kappa-j_0+1)\kappa})$, we have $|T|={{\mathcal{O}}}^*(\zeta^{\kappa-j_0+2})$, that is, if $(\mu_{j_0}-1)\frac{{{\ve x}}^{j_0}}{{{\ve x}}^{j_0-1}}>\kappa+(6\zeta+1)^{j_0-1}\mu_{j_0}$, then the statement holds for $j=j_0-1$. Thus, in the following we assume that $(\mu_{j_0}-1)\frac{{{\ve x}}^{j_0}}{{{\ve x}}^{j_0-1}}\le \kappa+(6\zeta+1)^{j_0-1}\mu_{j_0}$. Notice that ${{\ve x}}^j/m\le \zeta$ (as $||{{\ve x}}_i||_{\infty}\le \zeta$). According to $\frac{\mu_{j_0}}{m}{{\ve x}}^{j_0}> 2\zeta$, we know $\mu_{j_0}\ge 2$, whereas $$\frac{{{\ve x}}^{j_0}}{{{\ve x}}^{j_0-1}}\le \frac{\kappa+(6\zeta+1)^{j_0-1}\mu_{j_0}}{\mu_{j_0}-1},$$ and consequently $$\frac{\mu_{j_0}}{\mu_{j_0}-1}\cdot\frac{\kappa+(6\zeta+1)^{j_0-1}\mu_{j_0}}{m}{{\ve x}}^{j_0-1}> 2\zeta.$$ Since $\mu_{j_0}={{\mathcal{O}}}^*(\zeta^{(\kappa-j_0+1)\kappa})$, we can conclude that $\frac{\mu_{j_0}}{\mu_{j_0}-1}\cdot[{\kappa+(6\zeta+1)^{j_0-1}\mu_{j_0}}]={{\mathcal{O}}}^*(\zeta^{(\kappa-j_0+2)\kappa})$, hence, there exists some $\mu_{j_0-1}={{\mathcal{O}}}^*(\zeta^{(\kappa-j_0+2)\kappa})$ such that $\frac{\mu_{j_0-1}}{m}{{\ve x}}^{j_0-1}>2\zeta\ge \frac{\mu_{j_0-1}-1}{m}{{\ve x}}^{j_0-1}$. Thus, the statement holds for $j=j_0-1$.
Now we have proved the statement for all $1\le j\le \kappa$. Taking $j=1$, either there exists some subset $T$ which is conformal and $|T|= {{\mathcal{O}}}^*(\zeta^{\kappa^2})$, and Claim \[claim:conformal-sum\] is proved; Or there exists some $\mu_1={{\mathcal{O}}}^*(\zeta^{\kappa^2})$ such that $\frac{\mu_1}{m}{{\ve x}}^1>2\zeta$. As ${{\ve x}}^1\le {{\ve x}}^j$ for all $j\le \kappa$, it holds that $\frac{\mu_1}{m}{{\ve x}}^j>2\zeta$. There are two possibilities. If $m\le 2\mu_1+\kappa={{\mathcal{O}}}^*(\zeta^{\kappa^2})$, we simply take $T=\{1,2,\cdots,m\}$. Otherwise, we have $$\begin{aligned}
||\sum_{i=1}^{\mu_1+\kappa} {{\ve x}}_{i}-\frac{\mu_1}{m}{{\ve x}}||_{\infty}\le \zeta, \quad \textrm{ and }\quad ||\sum_{i=1}^{m} {{\ve x}}_{i}-\frac{m-\kappa}{m}{{\ve x}}||_{\infty}\le \zeta.\end{aligned}$$ Consequently, for any $1\le j\le \kappa$, it follows that $$\begin{aligned}
0\le \frac{\mu_1}{m}{{\ve x}}^j- \zeta \le \sum_{i=1}^{\mu_1+\kappa} {{\ve x}}_{i}^j\le \frac{\mu_1}{m}{{\ve x}}^j+ \zeta, \quad {\text{ and }}\quad \sum_{i=1}^{m} {{\ve x}}_{i}^j\ge \frac{m-\kappa}{m}{{\ve x}}^h- \zeta\ge 2\frac{\mu_1}{m}{{\ve x}}^h- \zeta>\frac{\mu_1}{m}{{\ve x}}^h+ \zeta.\end{aligned}$$ Hence, taking $T=\{1,2,\cdots,\mu_1+\kappa\}$ we have that $\sum_{i\in T}{{\ve x}}_i\sqsubseteq {{\ve x}}$ and $\sum_{i=1}^m{{\ve x}}_i-\sum_{i\in T}{{\ve x}}_i\ge 0$, and $|T|={{\mathcal{O}}}^*(\zeta^{\kappa^2})$, i.e., Claim \[claim:conformal-sum\] is proved.
Iterratively applying Claim \[claim:conformal-sum\], Lemma \[lemma:merging-lemma\] is proved.
**Remark.** It is notable that a weaker version of Lemma \[lemma:merging-lemma\] can also be proved, in a much simpler way, by iteratively applying Lemma \[lemma:merging-lemma-1\] to each individual dimension. However, by doing so we get an upper bound of ${{\mathcal{O}}}^*(\zeta^{2^{\kappa}})$ on $|T_j|$’s, which is much worse.
### A decomposition in two orthants {#subsec:decompose-two}
Towards the proof of Theorem \[thm:3-block-graver\], we need the following lemma which gives an almost sign-compatible decomposition.
\[lemma:decompose-bounded\] For any ${{\ve y}}\in ker_{{\ensuremath{\mathbb{Z}}}^{t_B+nt_A}}(H_0)$, there exist $q={{\mathcal{O}}_{FPT}}(1)$ vectors ${{\ve e}}_h$, $\alpha_h,\beta_{\ell}\in{\ensuremath{\mathbb{Z}}}_+$ and at most $2nt_A-1$ vectors ${{\ve d}}_\ell=(0,{{\ve g}}_\ell(E))$ where ${{\ve g}}_\ell(E)\in{\ensuremath{\mathcal{G}}}(E)$ such that ${{\ve e}}_h^0\sqsubseteq {{\ve y}}^0$, $||{{\ve e}}_h||_{\infty}\le \xi'={{\mathcal{O}}_{FPT}}(1)$, ${{\ve y}}=\sum_{h=1}^q\alpha_h{{\ve e}}_h+\sum_{\ell}\beta_{\ell}{{\ve d}}_\ell$, and moreover, all the ${{\ve e}}_h$’s lie in the same orthant, and all the ${{\ve d}}_\ell$’s lie in the same orthant.
Note that ${{\ve e}}_h$ and ${{\ve d}}_\ell$ do not necessarily lie in the same orthant. The $\xi$ in this lemma is the same as that in Theorem \[lemma:3-infty-bound\].
Towards the proof, we first show a simpler result, which will also be utilized later.
Continuing the proof of Lemma \[lemma:decompose-bounded-simpler\], we consider the $\bar{{{\ve e}}}_j$’s where $K_j\neq\emptyset$. If they are all sign-compatible, the lemma is proved. Otherwise we try to apply Lemma \[lemma:merging-lemma\] to the sequence of vectors that consists of $|K_j|$ copies of $\bar{{{\ve e}}}_j$. Note that we cannot directly apply the lemma as $\bar{{{\ve e}}}_j$’s have very high dimensions. However, if we consider the bricks $\bar{{{\ve e}}}_j^i$, since $||\bar{{{\ve e}}}_j^i||_{\infty}\le \xi$, there are at most $\xi^{O(t_A)}$ different kinds of bricks. Consequently, if we consider the vectors that consists of the $\eta$ bricks $(\bar{{{\ve e}}}_1^i,\bar{{{\ve e}}}_2^i,\cdots,\bar{{{\ve e}}}_\eta^i)$, there are at most $\theta=\xi^{O(\eta t_A)}={{\mathcal{O}}_{FPT}}(1)$ different kinds of vectors for $1\le i\le n$. We let these vectors be $\phi_1,\phi_2,\cdots,\phi_\theta$. Now we consider the reduced vectors $Rd(\bar{{{\ve e}}}_j)$ that only consists of distinct vectors. More precisely, we define the set of indices $In_j=\{i:(\bar{{{\ve e}}}_1^i,\bar{{{\ve e}}}_2^i,\cdots,\bar{{{\ve e}}}_\eta^i)=\phi_j\}$. For each $In_j$, we pick an arbitrary $i_j\in In_j$ and define a $(t_B+\theta t_A)$-dimensional vector $Rd(\bar{{{\ve e}}}_h)=(\bar{{{\ve e}}}_h^0,\bar{{{\ve e}}}_h^{i_1}, \bar{{{\ve e}}}_h^{i_2},\cdots,\bar{{{\ve e}}}_h^{i_\theta})$. Note that $\bar{{{\ve e}}}_h$ is simply a vector that copies the bricks of $Rd(\bar{{{\ve e}}}_h)$ for multiple times. Now we consider the sequence that consists of $|K_j|$ copies of $Rd(\bar{{{\ve e}}}_j)$. Applying Lemma \[lemma:merging-lemma\], we can divide these vectors into disjoint subsets $S_1,S_2,\cdots, S_m$ such that the summation of vectors in each subset is sign-compatible, and each subset has cardinality bounded by $\xi^{O(\theta t_A+t_B)}$. Consequently, $\sum_{h\in S_j}\bar{{{\ve e}}}_h$’s are also sign-compatible. Let ${{\ve e}}_j=\sum_{h\in S_j}\bar{{{\ve e}}}_h$, we have $${{\ve y}}=\sum_{j=1}^m {{{\ve e}}}_j+\sum_{\ell}\beta_\ell{{\ve d}}_\ell.$$ It remains to show there are at most ${{\mathcal{O}}_{FPT}}(1)$ different kinds of ${{\ve e}}_j$’s. To see this, consider the reduced vector ${{\ve e}}_j'=\sum_{i\in S_j}\hat{{{\ve e}}}_i$ and notice that ${{\ve e}}_j$ is duplicating the bricks of ${{\ve e}}_j'$ at locations indicated by $In_k$. Hence, it suffices to show that there are at most ${{\mathcal{O}}_{FPT}}(1)$ different kinds of ${{\ve e}}_j'$’s. Note that $||{{\ve e}}_j'||_{\infty}\le \xi\cdot \xi^{O(\theta t_A+t_B)}$, and it is of $(t_B+\theta t_A)$-dimensional. Hence, there are only ${{\mathcal{O}}_{FPT}}(1)$ different kinds of ${{\ve e}}_j'$’s, and the lemma is proved.
Our next goal is to further make ${{\ve e}}_h$’s and ${{\ve d}}_h$’s sign-compatible. Given ${{\ve y}}$ and a decomposition satisfying Lemma \[lemma:decompose-bounded\], we call ${{\ve e}}_h$’s as the principle vectors and ${{\ve d}}_h$’s as the add-ons. The basic idea is to merge principle vectors and add-ons such that they become sign-compatible, and we will mainly use Lemma \[lemma:merging-lemma\] to achieve this. However, there is a problem in applying Lemma \[lemma:merging-lemma\] directly as the dimension is too high. Again we try to utilize the idea in the proof of Lemma \[lemma:decompose-bounded\]: note that principle vectors are good in the sense that they can be reduced to lower dimensional vectors such that they are duplicating the bricks of lower dimensional vectors in fixed locations. While add-ons do not have such a nice structure, they are sparse according to Lemma \[lemma:cite-nfold\], that is, only an ${{\mathcal{O}}_{FPT}}(1)$ number of their bricks are non-zero. This will allow us to achieve the desired merging.
### Defining types of bricks {#subsec:type}
Prior to our merging process, let $\Gamma$ be some positive integer to be determined later. We will eventually set its value within ${{\mathcal{O}}_{FPT}}(1)$, but for ease of analysis on its value at the end, we will first treat it as an unbounded parameter and write ${{\mathcal{O}}_{FPT}}(\Gamma)$ in the following. We first define a [*quantity type*]{}. For every $t_A$-dimensional vector ${{\ve x}}=(x_1,x_2,\cdots,x_{t_A})$, we compare every coordinate $x_j$ with $\Gamma$. If $x_j\le \Gamma$, we say the $j$-th coordinate of ${{\ve x}}$ is [*small*]{}. Otherwise, we say it is [*large*]{}. A large coordinate may be positive or negative, hence each coordinate of ${{\ve x}}$ can be of three kinds: small, positive large and negative large. The [*quantity type*]{} of each ${{\ve x}}$ is defined as a $t_B$-dimensional vector which stores the kind of each $x_j$. It is obvious there are at most $3^{t_A}$ different quantity types.
Next, we define a [*principle type*]{} for every ${{\ve y}}^i$. Note that $||{{{\ve e}}}_j||_{\infty}\le \xi'$. For each $1\le i\le n$, we define the vector $({{{\ve e}}}_1^i,{{{\ve e}}}_2^i,\cdots,{{{\ve e}}}_q^i)$ as the [*principle type*]{} of ${{\ve y}}^i$. There are at most $(\xi')^{O(qt_A)}={{\mathcal{O}}_{FPT}}(1)$ different principle types.
Consider the bricks of ${{\ve y}}$. ${{\ve y}}^i$’s with the same quantity type and principle type are called to have the same [*type*]{}. There are at most $6^{t_A}\cdot (\xi')^{O(qt_A)}={{\mathcal{O}}_{FPT}}(1)$ different types. We pick an arbitrary brick, say, brick $1$ as a specific brick and let $N_1=\{1\}$. For the remaining bricks (brick $2$ to brick $n$), we divide them into $\sigma-1={{\mathcal{O}}_{FPT}}(1)$ sets such that bricks in the same set have the same type, i.e., we let $N_2,\cdots,N_\sigma$ be the set of indices of the bricks that have the same type, and let $n_j=|N_j|$. For simplicity, we reorder the bricks of ${{\ve y}}$ such that $N_j=\{\iota_{j-1}+1,\iota_{j-1}+2,\cdots,\iota_{j-1}+n_j\}$ where $\iota_{j-1}=n_1+n_2+\cdots+n_{j-1}$.
### Centralization {#subsec:centralize}
According to Lemma \[lemma:cite-nfold\], every ${{\ve d}}_h^i$, as well as $\sum_i {{\ve d}}_h^i$, is the summation of at most ${{\mathcal{O}}_{FPT}}(1)$ elements of ${\ensuremath{\mathcal{G}}}(A)$. Let ${{\ve v}}_1,{{\ve v}}_2,\cdots,{{\ve v}}_\lambda$ be all the non-zero $t_A$-dimensional vectors that ${{\ve d}}_h^i$ can take. For simplicity, we allow ${{\ve d}}_h$’s to be the same and rewrite Eq (\[eq:decompose-main\]) as $$\begin{aligned}
\label{eq:decompose-main-1}
{{\ve y}}=\sum_{h=1}^{q}\alpha_h{{{\ve e}}}_h+\sum_{h}{{\ve d}}_h.\end{aligned}$$ Note that in the above summation we simply add each ${{\ve d}}_h$ separately by $\beta_h$ times.
For ease of description, let us now take a scheduling point of view. We assume there are $n$ machines. The $t_B$-dimensional load of machine $i$ is defined by ${{\ve y}}^i$. This load is contributed by two parts, $\sum_{h=1}^q\alpha_h{{{\ve e}}}_h^i$ and $\sum_{h}{{\ve {md}}}_h^i$. For every $i\in N_j$, the first part $\sum_{h=1}^q\alpha_h{{{\ve e}}}_h^i$ is the same, while the second part might be different. We can view each ${{\ve v}}_k$ as a $t_A$-dimensional job. Obviously there are only $\lambda={{\mathcal{O}}_{FPT}}(1)$ different kinds of jobs, albeit that each job may have multiple identical copies. Let $\psi(j,k)$ be the total number of copies of job $k$ on machines in $N_j$.
We define a vector ${{\ve y}}_f$ such that ${{\ve y}}^0_f={{\ve y}}^0$, ${{\ve y}}^1_f={{\ve y}}^1$ and $$\begin{aligned}
\label{eq:average}
{{\ve y}}_f^k= \frac{1}{n_j}\cdot \sum_{i\in N_j}{{\ve y}}^i=\sum_{h=1}^q\alpha_h{{\ve e}}_h^i+\frac{1}{N_j}\cdot\sum_{i\in N_j}\sum_{h}{{\ve d}}_h^i, \quad \forall k\in N_j, 2\le j\le \sigma\end{aligned}$$
Ideally, we would like to argue on ${{\ve y}}_f$. However, ${{\ve y}}_f$ may be fractional. Therefore, in the following we define an integral vector $\tilde{{{\ve y}}}\approx {{\ve y}}_f$ and call it the [*centralization*]{} of ${{\ve y}}$.
We give the precise definition of $\tilde{{{\ve y}}}$ as follows. Let $\psi(j,k)$ be the number of copies of job ${{\ve v}}_k$ on machines in $N_j$. We (almost) evenly distribute these jobs among machines such that every machine gets $\lfloor\psi(j,k)/n_j\rfloor$ or $\lceil \psi(j,k)/n_j\rceil$ copies. To make it unique, we further restrict that machines with smaller indices in $N_j$ always have the same or more number of copies. By doing so, we construct a new vector $\tilde{{{\ve y}}}$. Note that $\tilde{{{\ve y}}}$ consists of the same number of jobs as ${{\ve y}}$, only that jobs are distributed among machines in a different (more even) way.
As we re-distribute ${{\ve v}}_k$’s such that for every machine in $N_j$, the number of copies of each ${{\ve v}}_k$ differs by at most $1$, the following lemma is straightforward.
\[lemma:average\] $$||\tilde{{{\ve y}}}^i-{{\ve y}}_f^i||_{\infty}\le \sum_{k=1}^{\lambda} ||{{\ve v}}_k||_{\infty}.$$
### Decomposition of $\tilde{{{\ve y}}}$ {#subsec:decompose-tilde-vey}
We create new $(t_B+nt_A)$-dimensional vectors in the following way. For simplicity, we define $\psi^q(j,k)=\lfloor\psi(j,k)/n_j\rfloor$ and $\psi^r(j,k)=\psi(j,k)-n_j\psi^q(j,k)$, i.e., they are the quotient and residue of $\psi(j,k)$ divided by $n_j$, respectively. For every $2\le j\le \sigma$, we create $\psi^q(j,k)$ copies of a vector ${{\ve {md}}}(j,k)$ and one copy of $\overline{{{\ve {md}}}}(j,k)$ such that $${{\ve {md}}}^i(j,k)=\left\{
\begin{array}{ll}
{{\ve v}}_k,\hspace{9mm} i\in N_j\\
-n_j{{\ve v}}_k, \quad i=1\\
0, \quad otherwise
\end{array}
\right. \hspace{10mm}
\overline{{{\ve {md}}}}^i(j,k)=\left\{
\begin{array}{ll}
{{\ve v}}_k,\hspace{20mm} \iota_j\le i\le \iota_j+\psi^r(j,k)\\
-\psi^r(j,k)\cdot{{\ve v}}_k, \hspace{25mm} i=1\\
0, \hspace{37mm} otherwise
\end{array}
\right.$$ Using the above notations, it is clear that for any $i\ge 2$, $\tilde{{{\ve y}}}^i$ consists of $\psi^q(j,k)$ copies of ${{\ve {md}}}^i(j,k)$ and one copy of $\overline{{{\ve {md}}}}^i(j,k)$, i.e., we have the following: $$\begin{aligned}
\label{eq:decompose-extra}
\tilde{{{\ve y}}}^i=\sum_{h=1}^q\alpha_h{{\ve e}}_h^i+\sum_{j=1}^{\sigma}\sum_{k=1}^{\lambda}\left(\psi^q(j,k)\cdot{{\ve {md}}}^i(j,k)+\overline{{{\ve {md}}}}^i(j,k) \right), \quad \forall i\ge 2\end{aligned}$$ The above equation is also true for $i=0$ as ${{\ve {md}}}^0(j,k)=\overline{{{\ve {md}}}}^0(j,k)=0$ for every $1\le j\le \sigma$, $1\le k\le \lambda$.
Furthermore, we have the following observations which follow directly from the definitions of ${{\ve {md}}}(j,k)$ and $\overline{{{\ve {md}}}}(j,k)$.
$H_0\cdot {{\ve {md}}}(j,k)=0$ and $H_0\cdot \overline{{{\ve {md}}}}(j,k)=0$ for all $1\le j\le \sigma$ and $1\le k\le \lambda$.
\[obs:size\] For any $i=0$ or $2\le i\le n$, ${{\ve {md}}}^i(j,k)={{\mathcal{O}}_{FPT}}(1),\overline{{{\ve {md}}}}^i(j,k)={{\mathcal{O}}_{FPT}}(1)$; For $i=1$, ${{\ve {md}}}^1(j,k)={{\mathcal{O}}_{FPT}}(n),\overline{{{\ve {md}}}}^1(j,k)={{\mathcal{O}}_{FPT}}(n)$.
It is clear that Eq (\[eq:decompose-extra\]) does not necessarily hold for $i=1$. Let us consider $$\eta=\tilde{{{\ve y}}}^1-\sum_{i=1}^q\alpha_h{{\ve e}}_h^1-\sum_{j=1}^{\sigma}\sum_{k=1}^{\lambda}\left(\psi^q(j,k)\cdot{{\ve {md}}}^1(j,k)+\overline{{{\ve {md}}}}^1(j,k) \right).$$ We have the following lemma.
\[lemma:machine-1\] $D\eta=0$.
Note that $H_0{{\ve d}}_{\ell}=0$ for each ${{\ve d}}_{\ell}$, whereas $D\sum_{i=1}^n \sum_{h}{{\ve d}}_h^i=0$. Since $\tilde{{{\ve y}}}$ is constructed from ${{\ve y}}$ by re-distributing the bricks ${{\ve d}}_h^i$ (i.e., by shifting it from brick $i$ to brick $i'$), it holds that $$D\sum_{i=1}^n(\tilde{{{\ve y}}}^i-\sum_{h=1}^q{{\ve e}}_h^i)=0.$$ Plugging in Eq (\[eq:decompose-extra\]), we have $$\begin{aligned}
0&=&D\tilde{{{\ve y}}}^1+D\sum_{i=2}^n\tilde{{{\ve y}}}^i-D\sum_{i=1}^n\sum_{h=1}^q{{\ve e}}_h^i\\
&=&D\tilde{{{\ve y}}}^1+D\sum_{i=2}^n\left(\sum_{h=1}^q\alpha_h{{\ve e}}_h^i+\sum_{j=1}^{\sigma}\sum_{k=1}^{\lambda}\left(\psi^q(j,k)\cdot{{\ve {md}}}^i(j,k)+\overline{{{\ve {md}}}}^i(j,k) \right) \right)-D\sum_{i=1}^n\sum_{h=1}^q{{\ve e}}_h^i\\
&=&D\tilde{{{\ve y}}}^1-D\sum_{h=1}^q{{\ve e}}_h^1+D\sum_{j=1}^{\sigma}\sum_{k=1}^{\lambda}\left(-\psi^q(j,k)\cdot{{\ve {md}}}^1(j,k)-\overline{{{\ve {md}}}}^1(j,k) \right)=D\eta.
\end{aligned}$$ Here the third equation makes use of the fact that ${{\ve {md}}}^1(j,k)=-\sum_{i=2}^n{{\ve {md}}}^i(j,k)$ and $\overline{{{\ve {md}}}}^1(j,k)=-\sum_{i=2}^n\overline{{{\ve {md}}}}^i(j,k)$.
Recall that by definition ${{\ve y}}^1-\sum_{h=1}^q{{\ve e}}_h^1$ is a weighted sum of ${{\ve v}}_k$’s, and so is $\sum_{j=1}^{\sigma}\sum_{k=1}^{\lambda}\left(\psi^q(j,k)\cdot{{\ve {md}}}^1(j,k)+\overline{{{\ve {md}}}}^1(j,k) \right)$. Hence, $\eta$ is also a weighted sum of ${{\ve v}}_k$’s, and we let $\eta=\sum_{k}\gamma_k{{\ve v}}_k$ where $\gamma_k\in {\ensuremath{\mathbb{Z}}}$ for $1\le k\le \lambda$. According to Lemma \[lemma:machine-1\], we have that $$\sum_{k=1}^{\lambda}\gamma_k \cdot D{{\ve v}}_k=0.$$ Equivalently, the above equation can be written as $$(\gamma_1,\gamma_2,\cdots,\gamma_{\lambda})\cdot [D{{\ve v}}_1,D{{\ve v}}_2,\cdots,D{{\ve v}}_{\lambda}]=0.$$ Consequently, if we define the matrix $DV=[D{{\ve v}}_1,D{{\ve v}}_2,\cdots,D{{\ve v}}_{\lambda}]$, which is an ${{\mathcal{O}}_{FPT}}(1)\times {{\mathcal{O}}_{FPT}}(1)$ matrix, then there exist $\gamma_h'\in{\ensuremath{\mathbb{Z}}}_+$ and ${{\ve g}}_h(DV)\in {\ensuremath{\mathcal{G}}}(DV)$, ${{\ve g}}_h(DV)\sqsubseteq (\gamma_1,\gamma_2,\cdots,\gamma_k)$ such that $$(\gamma_1,\gamma_2,\cdots,\gamma_{\lambda})=\sum_{h=1}^\omega \gamma_h'{{\ve g}}_h(DV).$$ where $\omega\le |{\ensuremath{\mathcal{G}}}(DV)|={{\mathcal{O}}_{FPT}}(1)$. Consequently, we have $$\eta=\sum_{h=1}^{\omega} \gamma_h'\left(\sum_{k=1}^{\lambda} {{\ve g}}_h^k(DV)\cdot {{\ve v}}_k\right),$$ Note that here each ${{\ve g}}_h^k(DV)\in {\ensuremath{\mathbb{Z}}}$ is the $k$-th coordinate of ${{\ve g}}_h(DV)$. Furthermore, $||{{\ve g}}_h(DV)||_{\infty}={{\mathcal{O}}_{FPT}}(1)$.
We define new vectors ${{\ve {od}}}(h)$ such that $${{\ve {od}}}^i(h)=\left\{
\begin{array}{ll}
\sum_{k=1}^\lambda {{\ve g}}_h^k(DV){{\ve v}}_k,\hspace{9mm} i=1\\
0, \hspace{23mm} otherwise
\end{array}
\right.$$ Recall that $A{{\ve v}}_k=0$ and $\sum_{k=1}^\lambda {{\ve g}}_h^k(DV)\cdot D{{\ve v}}_k=0$, we have the following observation.
$||{{\ve {od}}}(h)||_{\infty}={{\mathcal{O}}_{FPT}}(1)$ and $H_0\cdot {{\ve {od}}}(h)=0$ for all $1\le h\le \omega$.
Now we derive the following decomposition of $\tilde{{{\ve y}}}$: $$\begin{aligned}
\label{eq:decompose}
\tilde{{{\ve y}}}=\sum_{h=1}^q\alpha_h{{\ve e}}_h+\sum_{j=1}^{\sigma}\sum_{k=1}^{\lambda}\left(\psi^q(j,k)\cdot{{\ve {md}}}(j,k)+\overline{{{\ve {md}}}}(j,k) \right)+\sum_{h=1}^\omega \gamma_h'\cdot {{\ve {od}}}(h),\end{aligned}$$
### A sign-compatible decomposition of $\tilde{{{\ve y}}}$ {#subsec:decompose-tildevey-compatible}
We will find a sign-compatible decomposition of $\tilde{y}$ in this subsection, and show in the next subsection that at least one element of the decomposition lie in the same orthant of $\bar{{{\ve y}}}$. Recall Eq (\[eq:decompose\]). We observe that all the vectors involved have a nice structure in the sense that they can be divided into ${{\mathcal{O}}_{FPT}}(1)$ segments where every segment consists of identical bricks. More precisely, for every $1\le j\le \sigma$, let $\pi_j$ be the permutation of $\{1,2,\cdots,k\}$ such that the $\lambda$ residues can be ordered as $\psi^r(j,\pi_j(1))\le \psi^r(j,\pi_j(2))\le\cdots\le \psi^r(j,\pi_j(\lambda))$. Additionally, we define $\psi^r(j,\pi_j(0))=\iota_{j-1}+1$ for $j\ge 2$, $\psi^r(j,\pi_j(\lambda+1))=\iota_j-1$ and $\psi^r(1,\pi_j(0))=2$ (as machine $1$ is special and should be excluded). We can divide the $n+1$ bricks of a $(t_B+nt_A)$-dimensional vector into $2+2(\lambda+1)\sigma$ groups as follows:
- Group 0 consists of brick 0 (the first $t_B$ dimensions), which is $0$ for all the ${{\ve {md}}}(j,k)$ and $\overline{{{\ve {md}}}}(j,k)$.
- Group 1 consists of only machine (brick) 1.
- For $1\le j\le \sigma$ and $1\le k\le \lambda+1$, Group $k+1+(j-1)(\lambda+1)$ consists of brick $\psi^r(j,\pi(k-1))+1$ to brick $\psi^r(j,\pi(k))$.
Hence, each vector is divided into $2+2(\lambda+1)\sigma$ segments where each segments contains its bricks within one group. See the following figure as an illustration of the grouping. Here circles of different colors represent different ${{\ve v}}_k$’s. Note that if we take a snapshot of any vector (${{\ve e}}_h$ or ${{\ve {md}}}(j,k)$) on the bricks within a group (see the bricks among two ajacent red lines in the figure), we see that all of these bricks are identical (for otherwise some of the residues shall lie within the indices of these bricks, which contradicts the grouping). More precisely, we have the following.
\[fig:1\]
\[obs:decompose\] Let $Gr_\ell$ be the indices of bricks in Group $\ell$, then for every $i_1,i_2\in Gr_\ell$, we have ${{\ve e}}_h^{i_1}={{\ve e}}_h^{i_2}$ and ${{\ve {md}}}^{i_1}(j,k)={{\ve {md}}}^{i_2}(j,k)$ for $1\le h\le q$, $1\le j\le \sigma$, $1\le k\le \lambda$.
Furthermore, notice that $Gr_\ell$’s is a further sub-division of $N_1,N_2,\cdots,N_{\sigma}$, hence we have the following observation.
For any $i_1,i_2\in Gr_\ell$, ${{\ve y}}^{i_1}$ and ${{\ve y}}^{i_2}$ have the same type.
Now we are able to define reduced vectors. For ${{\ve z}}={{\ve e}}_h$ or ${{\ve {md}}}(j,k)$ or $\overline{{{\ve {md}}}}(j,k)$ or ${{\ve {od}}}(h)$ or $\tilde{{{\ve y}}}$, we define $Rd({{\ve z}})$ as a $(t_B+t_A+2(\lambda+1)\sigma t_A)$-dimensional vector that consists of $2+2(\lambda+1)\sigma$ bricks where the $\ell$-th brick $Rd^\ell({{\ve z}})$ equals any brick of ${{\ve z}}$ in the group $Gr_\ell$ (as they are the same by Observation \[obs:decompose\]). Furthermore, Eq (\[eq:decompose\]) implies the following: $$\begin{aligned}
\label{eq:decompose2}
Rd(\tilde{{{\ve y}}})=\sum_{i=1}^q\alpha_h Rd({{\ve e}}_h)+\sum_{j=1}^{\sigma}\sum_{k=1}^{\lambda}\left(\psi^q(j,k)\cdot Rd({{\ve {md}}}(j,k))+Rd(\overline{{{\ve {md}}}}(j,k))\right)+\sum_{h=1}^\omega \gamma_h'\cdot Rd({{\ve {od}}}(h)) .\end{aligned}$$
If we want to make the rightside of Eq (\[eq:decompose\]) into a sign-compatible summation, it suffices to make the above Eq (\[eq:decompose2\]) into a sign-compatible summation, and this is achievable by utilizing Lemma \[lemma:merging-lemma\]. To derive a good bound, we will apply Lemma \[lemma:merging-lemma\] twice in a separate way.
By Observation \[obs:size\], we have the following.
\[obs:size-2\] For $i=0$ or $i\ge 2$, $Rd^i({{\ve {md}}}(j,k)),Rd^i(\overline{{{\ve {md}}}}(j,k))={{\mathcal{O}}_{FPT}}(1)$; For $i=1$, $Rd^1({{\ve {md}}}(j,k)),Rd^1(\overline{{{\ve {md}}}}(j,k))={{\mathcal{O}}_{FPT}}(n)$.
We now view the rightside of Eq (\[eq:decompose\]) as a summation over a sequence of vectors ${{\ve z}}_i$, where each vector ${{\ve z}}_i$ equals ${{\ve e}}_h$ or ${{\ve {md}}}(j,k)$ or $\overline{{{\ve {md}}}}(j,k)$ or ${{\ve {od}}}(h)$. Hence, we can rewrite Eq (\[eq:decompose\]) as $$\begin{aligned}
\label{eq:decompose-z}
\tilde{{{\ve y}}}=\sum_i {{\ve z}}_i.\end{aligned}$$ Consequently, $$\begin{aligned}
\label{eq:decompose-z-1}
Rd(\tilde{{{\ve y}}})=\sum_i Rd({{\ve z}}_i).\end{aligned}$$
We define $Rd({{\ve x}})[\bar{1}]$ as the projection of the vector $Rd({{\ve x}})$ onto the subspace by excluding $Rd^1({{\ve x}})$. Hence, we have $$Rd(\tilde{{{\ve y}}})[\bar{1}]=\sum_i Rd({{\ve z}}_i)[\bar{1}].$$ According to Observation \[obs:size-2\], we have $||Rd({{\ve z}}_i)[\bar{1}]||_{\infty}\le {{\mathcal{O}}_{FPT}}(1)$, whereas by Lemma \[lemma:merging-lemma\] we can find disjoint subsets $T_1,T_2,\cdots,T_{m'}$ such that $|T_j|={{\mathcal{O}}_{FPT}}(1)$ and $\sum_{i\in T_j}Rd({{\ve z}}_i)[\bar{1}]\sqsubseteq Rd(\tilde{{{\ve y}}})[\bar{1}]$ and $Rd(\tilde{{{\ve y}}})[\bar{1}]=\sum_j(\sum_{i\in T_j}Rd({{\ve z}}_i)[\bar{1}])$.
Now we consider $Rd^1({{\ve x}})$’s. By Eq (\[eq:decompose-z-1\]) we have $$\begin{aligned}
\label{eq:decompose-z-2}
Rd^1(\tilde{{{\ve y}}})=\sum_{j=1}^{m'} \sum_{i\in T_j}Rd^1({{\ve z}}_i).\end{aligned}$$ Given that $Rd^1({{\ve {md}}}(j,k)),Rd^1(\overline{{{\ve {md}}}}(j,k))={{\mathcal{O}}_{FPT}}(n)$, $Rd({{\ve e}}_h)={{\mathcal{O}}_{FPT}}(1)$, and $|T_j|={{\mathcal{O}}_{FPT}}(1)$, we can conclude that $||\sum_{i\in T_j}Rd^1({{\ve z}}_i)||_{\infty}={{\mathcal{O}}_{FPT}}(n)$. Applying Lemma \[lemma:merging-lemma\], we can further find $m''$ disjoint sets $T_1',T_2',\cdots,T_{m''}'\subseteq\{1,2,\cdots,m'\}$ such that $|T_h'|={{\mathcal{O}}_{FPT}}(n^{t_A^2})$, $\cup_{h=1}^{m''}=\{1,2,\cdots,m'\}$ and $\sum_{j\in T_h'}\sum_{i\in T_j}Rd^1({{\ve z}}_i)\sqsubseteq Rd^1(\tilde{{{\ve y}}})$. Hence, Eq (\[eq:decompose-z\]) can be rewritten as: $$\tilde{{{\ve y}}}=\sum_{h=1}^{m''}\left(\sum_{j\in T_h'}\sum_{i\in T_j}{{\ve z}}_i\right),$$ where for every $h$ it holds that $\sum_{j\in T_h'}\sum_{i\in T_j}{{\ve z}}_i\sqsubseteq \tilde{{{\ve y}}}$, $||\sum_{j\in T_h'}\sum_{i\in T_j}{{\ve z}}_i\sqsubseteq \tilde{{{\ve y}}}||_{\infty}={{\mathcal{O}}_{FPT}}(n^{t_A^2})$, i.e., the following lemma is true.
\[lemma:decompose-tildevey\] Let $H_0{{\ve y}}=0$ and $\tilde{{{\ve y}}}$ be the centralization of ${{\ve y}}$, then there exist ${{\ve z}}_h$’s such that $H_0{{\ve z}}_h=0$, ${{\ve z}}_h\sqsubseteq \tilde{{{\ve y}}}$, $||{{\ve z}}_h||_{\infty}={{\mathcal{O}}_{FPT}}(n^{t_A^2})$ and $\tilde{{{\ve y}}}=\sum_{h=1}^{m''}{{\ve z}}_h$. Furthermore, the $n+1$ bricks of each ${{\ve z}}_h$ can be divided into $2+2(\lambda+1)\sigma={{\mathcal{O}}_{FPT}}(1)$ groups such that for any $i_1,i_2\in Gr_\ell$, ${{\ve z}}_h^{i_1}={{\ve z}}_{h}^{i_2}$, and ${{\ve y}}^{i_1}$, ${{\ve y}}^{i_2}$ have the same type.
### A sign-compatible decomposition of ${{\ve y}}$ {#subsec:decompose-vey}
Let $\Gamma=\sum_{k=1}^{\lambda}||{{\ve v}}_k||_{\infty}={{\mathcal{O}}_{FPT}}(1)$. Let ${{\ve z}}_h$’s be the same as that in Lemma \[lemma:decompose-tildevey\]. The goal of this subsection is to prove the following lemma.
\[lemma:decompose-vey\] If $m''>2\Gamma\cdot t_A\cdot \left( 2+2(\lambda+1)\sigma\right)$, then there exists some $h_0$ such that ${{\ve z}}_{h_0}\sqsubseteq {{\ve y}}$.
Towards the proof, we need the following observation and lemma. For an arbitrary $(t_B+nt_A)$-dimensional vector ${{\ve z}}$, we define by ${{\ve z}}^i[j]$ the $j$-th coordinate of the brick ${{\ve z}}^i$. Recall the definition of ${{\ve y}}_f$. As the average is taken among bricks of the same type, we have the following observation.
If ${{\ve y}}^i[j]$ is large, then $|{{\ve y}}_f^i[j]|>\Gamma$. Otherwise, $|{{\ve y}}_f^i[j]|\le \Gamma$.
By Lemma \[lemma:average\], we have the following corollary.
\[coro:type\]
- If ${{\ve y}}^i[j]$ is positive large, then $\tilde{{{\ve y}}}^i[j]>0$.
- If ${{\ve y}}^i[j]$ is negative large, then $\tilde{{{\ve y}}}^i[j]<0$.
- If ${{\ve y}}^i[j]$ is small, then $|\tilde{{{\ve y}}}^i[j]|\le 2\Gamma$.
Using the above corollary, we have the following lemma, which implies directly Lemma \[lemma:decompose-vey\].
\[lemma:aug-1\] If $m''>2\Gamma t_A\cdot \left( 2+2(\lambda+1)\sigma\right)$, then there exists some ${{\ve z}}_{h_0}$ such that
- If ${{\ve y}}^i[j]$ is positive large, then ${{\ve z}}_{h_0}^i[j]\ge 0$.
- If ${{\ve y}}^i[j]$ is negative large, then ${{\ve z}}_{h_0}^i[j]\le 0$.
- If ${{\ve y}}^i[j]$ is small, then ${{\ve z}}_{h_0}^i[j]= 0$.
First, by Lemma \[lemma:decompose-tildevey\] we have ${{\ve z}}_h\sqsubseteq \tilde{{{\ve y}}}$ for every $1\le h\le m''$. If ${{\ve y}}^i[j]$ is positive large, by Corollary \[coro:type\] we have $\tilde{{{\ve y}}}^i[j]>0$, then ${{\ve z}}_h^i[j]\ge 0$. Similarly if ${{\ve y}}^i[j]$ is negative large we have ${{\ve z}}_h^i[j]\le 0$. It remains to consider small coordinates. Consider the following set: $$Z_s=\{h: \exists 1\le i\le n, 1\le j\le t_A \text{ such that } {{\ve y}}^i[j] \text{ is small and } {{\ve z}}_h^i[j]\neq 0\}.$$ We claim that, $|Z_s|\le 2\Gamma t_A\cdot \left( 2+2(\lambda+1)\sigma\right)$. Suppose on the contrary that this claim is not true, then $Z_s$ contains more than $2\Gamma t_A\cdot \left( 2+2(\lambda+1)\sigma\right)$ elements, and consequently there exists some $1\le \ell_0\le 2+2(\lambda+1)\sigma$ such that $|Z_s\cap Gr_{\ell_0}|>2\Gamma t_A$. As $1\le j\le t_A$, there exists some $j_0$ such that $$|\{h: {{\ve y}}^i[j_0] \text{ is small and } {{\ve z}}_h^i[j_0]\neq 0, i\in Gr_{\ell_0}\}|>2\Gamma.$$ Note that for all $i\in Gr_{\ell_0}$, ${{\ve z}}_h^i[j_0]$ takes the same value, hence, for an arbitrary $i_0\in Gr_{\ell_0}$ we have that $$|\{h: {{\ve y}}^{i_0}[j_0] \text{ is small and } {{\ve z}}_h^{i_0}[j_0]\neq 0\}|>2\Gamma.$$ Let $Z_s[j_0]=\{h: {{\ve y}}^{i_0}[j_0] \text{ is small and } {{\ve z}}_h^{i_0}[j_0]\neq 0\}$. According to Corollary \[coro:type\], $|\tilde{{{\ve y}}}^{i_0}[j_0]|\le 2\Gamma$. Meanwhile the fact that ${{\ve z}}_h\sqsubseteq \tilde{{{\ve y}}}$ implies that either ${{\ve z}}_h^{i_0}[j_0]>0$ for all $h\in Z_s[j_0]$, or ${{\ve z}}_h^{i_0}[j_0]<0$ for all $h\in Z_s[j_0]$. In either case, we conclude that $|\sum_{h\in Z_s[j_0]}{{\ve z}}_h^{i_0}[j_0]|>2\Gamma$. As $\tilde{{{\ve y}}}=\sum_h{{\ve z}}_h$ is a sign-compatible decomposition, we have $|\tilde{{{\ve y}}}^{i_0}[j_0]|>2\Gamma$, which is a contradiction. Hence, $|Z_s|\le 2\Gamma t_A\cdot \left( 2+2(\lambda+1)\sigma\right)$. Thus, if $m''>2\Gamma t_A\cdot \left( 2+2(\lambda+1)\sigma\right)$, there must exist some $h_0$ such that ${{\ve z}}_{h_0}^i[j]=0$ for all $i,j$ where ${{\ve y}}^i[j]$ is small.
It is clear that the ${{\ve z}}_{h_0}$ in Lemma \[lemma:aug-1\] satisfies that ${{\ve z}}_{h_0}\sqsubseteq {{\ve y}}$, whereas Lemma \[lemma:decompose-vey\] is proved.
Recall that $||{{\ve z}}_h||_{\infty}={{\mathcal{O}}_{FPT}}(n^{t_A^2})$, then there exists some function $f(A,B,C,D)$ that only depends on the small matrices $A,B,C,D$ (or more precisely, the parameters $\Delta,s_A,s_B,s_C,s_D,t_A,t_B,t_C,t_D$) such that $||{{\ve z}}_h||_{\infty}=f(A,B,C,D)\cdot n^{t_A^2}$. Consequently, the following corollary follows directly from Lemma \[lemma:decompose-vey\].
\[coro:vey-decompose\] If $||{{\ve y}}||_{1}>2\Gamma\cdot t_A\cdot \left( 2+2(\lambda+1)\sigma\right)\cdot (t_B+nt_A)\cdot f(A,B,C,D) n^{t_A^2}+2\Gamma\cdot (t_B+nt_A)$, then there exists some ${{\ve z}}_{h_0}$ such that $||{{\ve z}}_{h_0}||_{\infty}\le f(A,B,C,D) n^{t_A^2}$ and ${{\ve z}}_{h_0}\sqsubseteq {{\ve y}}$.
Recall Eq (\[eq:average\]), we have $n_\ell\cdot {{\ve y}}_f^k=\sum_{i\in N_\ell} {{\ve y}}^i$ for all $k\in N_\ell$. Note that ${{\ve y}}^i$’s have the same type for $i\in N_\ell$, implying for $1\le j\le t_A$, if ${{\ve y}}^i[j]$ is large for some $i\in N_\ell$, then ${{\ve y}}^i[j]$’s are all positive or all negative. This means, for a large coordinate $j$ we have $|\sum_{i\in N_\ell}{{\ve y}}^i_f[j]|=|\sum_{i\in N_\ell}{{\ve y}}^i[j]|=\sum_{i\in N_\ell}|{{\ve y}}^i[j]|$. Hence, $$\sum_{i\in N_\ell}||{{\ve y}}_f^i||_1\ge \sum_{i\in N_\ell}||{{\ve y}}^i||_1-\Gamma\cdot n_\ell \cdot t_A$$ According to Lemma \[lemma:average\], we know that $||\tilde{{{\ve y}}}^i-{{\ve y}}_f^i||_{\infty}\le \Gamma$, hence $$\sum_{i\in N_\ell}||\tilde{{{\ve y}}}^i||_1\ge \sum_{i\in N_\ell}||{{\ve y}}_f^i||_1-\Gamma\cdot n_\ell \cdot t_A\ge \sum_{i\in N_\ell}||{{\ve y}}^i||_1-2\Gamma\cdot n_\ell \cdot t_A, \quad \forall 1\le i\le n$$ Recall that $\tilde{{{\ve y}}}^0={{\ve y}}^0$, hence, $$||\tilde{{{\ve y}}}||_1\ge ||{{\ve y}}||_1-2\Gamma\cdot (t_B+nt_A).$$
If $||{{\ve y}}||_{1}>2\Gamma\cdot t_A\cdot \left( 2+2(\lambda+1)\sigma\right)\cdot (t_B+nt_A)\cdot f(A,B,C,D)n^{t_A^2}+2\Gamma\cdot (t_B+nt_A)$, then $$\begin{aligned}
\label{eq:vey-decompose}
||\tilde{{{\ve y}}}||_1> 2\Gamma\cdot t_A\cdot \left( 2+2(\lambda+1)\sigma\right)\cdot (t_B+nt_A)\cdot f(A,B,C,D)n^{t_A^2}.
\end{aligned}$$ Since $||{{\ve z}}_h||_{\infty}\le f(A,B,C,D) n^{t_A^2}$, we have $||{{\ve z}}_h||_{1}\le f(A,B,C,D) n^{t_A^2} \cdot (t_B+nt_A)$. As $\tilde{{{\ve y}}}=\sum_{h=1}^{m''}{{\ve z}}_h$, Eq (\[eq:vey-decompose\]) implies that $m''> 2\Gamma\cdot t_A\cdot \left( 2+2(\lambda+1)\sigma\right)$. By Lemma \[lemma:decompose-vey\], there exists some ${{\ve z}}_{h_0}\sqsubseteq {{\ve y}}$.
By Corollary \[coro:vey-decompose\] and the definition of Graver basis, we know that if $||{{\ve y}}||_{1}>2\Gamma\cdot t_A\cdot \left( 2+2(\lambda+1)\sigma\right)\cdot (t_B+nt_A)\cdot f(A,B,C,D) n^{t_A^2}+2\Gamma\cdot (t_B+nt_A)=\Omega_{FPT}(n^{t_A^2+1})$, then ${{\ve y}}$ is not a Graver basis element. Hence, Theorem \[thm:3-block-graver\] is true.
Running time of the augmentation algorithm for 3-block $n$-fold IP {#subsec:3-block-alg}
------------------------------------------------------------------
Given Theorem \[thm:3-block-graver\], the following theorem follows by combining the idea from [@hemmecke2014graver] and a recent progress in [@martin2018parameterized; @eisenbrand2018faster].
\[thm:alg-3-block\] There exists an algorithm for 3-block $n$-fold IP that runs in $\min\{{{\mathcal{O}}_{FPT}}(n^{s_ct_B+3})\log^3 n, {{\mathcal{O}}_{FPT}}(n^{(t_A^2+1)t_B+3}\log^3 n)\}$ time.
According to Section \[sec:pre\] (Approximate Graver-best oracle), it suffices for us to solve the following IP for each fixed value $\rho_0=2^0,2^1,2^2,\cdots$: $$\begin{aligned}
\label{eq:graver-best-aug-3-block}
\min\{{{\ve w}}\cdot {{\ve x}}: H_0 {{\ve x}}=0, {{\ve l}}\le {{\ve x}}_0+\rho_0{{\ve x}}\le {{\ve u}}, {{\ve x}}\in{\ensuremath{\mathbb{Z}}}^m, ||{{\ve x}}||_{\infty}\le \min\{{{\mathcal{O}}_{FPT}}(n^{s_c}),{{\mathcal{O}}_{FPT}}(n^{t_A^2+1})\}\}
\end{aligned}$$ Let ${{\ve x}}_*$ be the optimal solution. Given that $||{{\ve x}}_*||_{\infty}\le {{\mathcal{O}}_{FPT}}(n^{t_A^2+1})$, we can guess ${{\ve x}}^0_*$ and there are ${{\mathcal{O}}_{FPT}}(n^{(t_A^2+1)t_B})$ different possibilities. For each guess, say, ${{\ve x}}_*^0={{\ve v}}$, we solve the following problem: $$\begin{aligned}
\label{eq:graver-best-aug-3-block-1}
\min\{{{\ve w}}\cdot {{\ve x}}: H_0 {{\ve x}}=0, {{\ve l}}\le {{\ve x}}_0+\rho_0{{\ve x}}\le {{\ve u}}, {{\ve x}}\in{\ensuremath{\mathbb{Z}}}^m, {{\ve x}}^0={{\ve v}}\}\end{aligned}$$ By fixing ${{\ve x}}^0$, the above problem becomes exactly an $n$-fold IP, which can be solved efficiently in ${{\mathcal{O}}_{FPT}}(n^2\log n^2)$ time [@eisenbrand2018faster]. Notice that $\rho_0$ may take ${{\mathcal{O}}_{FPT}}(n\log n)$ distinct values, the overall running time is $\min\{{{\mathcal{O}}_{FPT}}(n^{s_ct_B+3})\log^3 n, {{\mathcal{O}}_{FPT}}(n^{(t_A^2+1)t_B+3}\log^3 n)\}$.
Lower bound on the $\ell_{\infty}$-norm of Graver basis {#sec:lower-bound}
-------------------------------------------------------
Given Theorem \[lemma:3-infty-bound\], it seems that we may expect the Graver basis of $3$-block $n$-fold IP can be bounded by ${{\mathcal{O}}_{FPT}}(1)$. Unfortunately, the following theorem indicates that this is impossible.
\[thm:3-block-lower\] There exists a $3$-block $n$-fold matrix $H_0$ such that for some ${{\ve g}}\in {\ensuremath{\mathcal{G}}}(H_0)$, $||{{\ve g}}||_{\infty}=\Omega(n)$.
Let $B=1$, which is a $1\times 1$ identity matrix. Let $A=(1,-1)$, $D=(1,0)$. Consider the vector ${{\ve y}}=(y^0,{{\ve y}}^1,\cdots,{{\ve y}}^n)$ with $y^0=1$, ${{\ve y}}^{1}=(n-1,n)$ and ${{\ve y}}^{i}=(-1,0)$ for every $2\le i\le n$. It is easy to verify that $y^0+A{{\ve y}}^i=0$ for $1\le i\le n$ and $\sum_{i=1}^n D{{\ve y}}^i=0$. In the following we show that ${{\ve y}}$ is a Graver basis element. As $H_0{{\ve y}}=0$, there exist $\alpha_j\in{\ensuremath{\mathbb{Z}}}_+$, ${{\ve g}}_j(H_0)\in {\ensuremath{\mathcal{G}}}(H_0)$, ${{\ve g}}_j(H_0)\sqsubseteq {{\ve y}}$ such that ${{\ve y}}=\sum_j\alpha_j{{\ve g}}_j(H_0)$. Among all of these ${{\ve g}}_j(H_0)$’s, there exists some $j$ such that ${{\ve g}}_j^0(H_0)\neq 0$. Given that $y^0=1$, it holds that ${{\ve g}}_j^0(H_0)=1$. Let ${{\ve g}}_j(H_0)=(1,{{\ve x}}^1,{{\ve x}}^2,\cdots,{{\ve x}}^n)$. The fact that ${{\ve g}}_j(H_0)\sqsubseteq {{\ve y}}$ implies that ${{\ve x}}^i=({{\ve x}}_1^i,0)$ for $2\le i\le n$. As $1+A{{\ve x}}^i=0$, ${{\ve x}}^i_1=-1$ for $2\le i\le n$, and consequently $x_1^1=n-1$ according to $\sum_{i=1}^n D{{\ve x}}^i=0$. Hence, ${{\ve g}}_j(H_0)={{\ve y}}$, and Theorem \[thm:3-block-lower\] is proved.
Conclusion
==========
We consider 4-block $n$-fold IP and its important special case $3$-block $n$-fold IP, both generalizing the well-known two-stage stochastic IP and $n$-fold IP. We show that, $3$-block $n$-fold IP admits a Hilbert basis whose $\ell_{\infty}$-norm is bounded in ${{\mathcal{O}}_{FPT}}(1)$, while any non-zero integral element in the kernel space of $4$-block $n$-fold IP may have an $\ell_{\infty}$-norm at least $\Omega(n^{s_c})$. We provide a matching upper bound on the $\ell_{\infty}$-norm of the Graver basis for $4$-block $n$-fold IP, which gives an exponential improvement upon the best known result. We also establish an upper bound of $\min\{{{\mathcal{O}}_{FPT}}(n^{s_c}),{{\mathcal{O}}_{FPT}}(n^{t_A^2}+1)\}$ on the $\ell_{\infty}$-norm of the Graver basis for $3$-block $n$-fold IP.
It remains as an important open problem whether $4$-block $n$-fold IP, or even its special case $3$-block $n$-fold IP, is in FPT. Our results indicate that, using the current augmentation framework, it is unlikely to derive an FPT algorithm. Another important open problem is whether the $\ell_{\infty}$-norm of the Graver basis elements of 3-block $n$-fold IP is bounded by $n^{O(1)}$, which is independent of the parameters.
[^1]: Department of Computer Science, University of Houston. Email: `chenlin198662@gmail.com, xuleimath@gmail.com, wshi3@central.uh.edu `.
|
---
abstract: 'The pants graph has proved to be influential in understanding 3-manifolds concretely. This stems from a quasi-isometry between the pants graph and the Teichmüller space with the Weil-Petersson metric. Currently, all estimates on the quasi-isometry constants are dependent on the surface in an undiscovered way. This paper starts effectivising some constants which begins the understanding how relevant constants change based on the surface. We do this by studying the hyperbolicity constant of the pants graph for the five-punctured sphere and the twice punctured torus. The hyperbolicity constant of the relative pants graph for complexity 3 surfaces is also calculated. Note, for higher complexity surfaces, the pants graph is not hyperbolic or even strongly relatively hyperbolic.'
author:
- Ashley Weber
bibliography:
- 'mybib.bib'
date:
title: Hyperbolicity constants for pants and relative pants graphs
---
Introduction
============
The pants graph has been instrumental in understanding Teichmüller space. This is because the pants graph is quasi-isometric to Teichmüller space equipped with the Weil-Petersson metric [@Brock-WPtoPants]. Brock and Margalit used pants graphs to show that all isometries of Teichmüller space with the Weil-Petersson metric arise from the mapping class group of the surface [@BM-WPisom]. This relationship was also used to classify for which surfaces the associated Teichmüller space is hyperbolic. The relationship between the pants graph and Teichmüller space has been used to study volumes of 3-manifolds [@Brock-WPtoPants; @Brock-WPtrans]. In particular, it has been used to relate volumes of the convex core of a hyperbolic 3-manifold to the distance of two points in Teichmüller space. It has also related the volume of a hyperbolic 3-manifold arising from a psuedo-Anosov element in the mapping class group to the translation length of the psuedo-Anosov element as applied to the pants graph. Both of these relations have constants which depend on the surface; this paper is the start of effectivising those constants. Notice Aougab, Taylor, and Webb have some effective bounds on the quasi-isometry bounds, however even these still depend on the surface in a way that is unknown [@ATW].
Let $S_{g,p}$ be a surface with genus $g$ and $p$ punctures. We define the complexity of a surface to be $\xi(S_{g,p}) =3g + p - 3 $. Brock and Farb have shown that the pants graph is hyperbolic if and only if the complexity of the surface is less than or equal to $2$ [@BF]. Brock and Masur showed that in a few cases the pants graph is strongly relatively hyperbolic, specifically when $\xi(S) = 3$ [@BM]. Even though hyperbolicity is well studied for the pants graph, the hyperbolicity constants associated with the pants graph or the relative pants graph is not. In addition to having a further understanding of the quasi-isometry mentioned above and all of its applications, actual hyperbolicity constants are useful in answering questions about asymptotic time complexity of certain algorithms, especially those involving the mapping class group. More speculatively, estimates on hyperbolicity constants may be crucial to effectively understand the virtual fibering conjecture, which relates the geometry of the fiber to the geometry of the base surface. The focus of this paper is to find hyperbolicity constants for the pants graph and relative pants graph, when these graphs are hyperbolic.
For a surface $S = S_{0,5}, S_{1,2}$, ${\mathcal{P}}(S)$ is $2,691,437$-thin hyperbolic.
Computing the asymptotic translation lengths of an element in the mapping class group on ${\mathcal{P}}(S)$ is a question explored by Irmer [@Irmer]. Bell and Webb have an algorithm that answers this question for the curve graph [@BellWebb]. Combining the works of Irmer, and Bell and Webb, one could conceivably come up with an algorithm for asymptotic translation lengths on ${\mathcal{P}}(S)$. In this case, the above Theorem would put a bound on the run-time of the algorithm in the cases that $S = S_{0,5}, S_{1,2}$.
We now turn our attention to the relatively hyperbolic cases.
For a surface $S = S_{3,0}, S_{1,3}, S_{0,6}$, ${\mathcal{P}}_{rel}(S)$ is $2,606,810,489$-thin hyperbolic.
To show both of our main theorems, we construct a family of paths that is very closely related to hierarchies, introduced in [@MMII]. We show that this family of paths satisfies the thin triangle condition which, by a theorem of Bowditch, allows us to conclude the whole space is hyperbolic [@Bow]. A key tool used throughout is the Bounded Geodesic Image Theorem [@MMII]. This theorem allows us to control the length of geodesics in subspaces.
This method cannot be made to generalize to pants graphs in general since any pants graph of a surface with complexity higher than $3$ is not strongly relatively hyperbolic [@BM]. Although, this method may be able to be used for other graphs which are variants on the pants graph.
One might consider approaching this problem by finding the sectional curvature of Teichmüller space and using the quasi-isometry to inform on the hyperbolicity constant of the pants graph. If the sectional curvature is bounded away from zero, one can relate the curvature of the space to the hyperbolicity constant of the space. However, the sectional curvature of Teichmüller sapce is not bounded away from zero [@Huang]. Therefore, this technique cannot be used.
**Acknowledgments:** I would like to thank my advisor, Jeff Brock, for suggesting this problem, support, and helpful conversations. I’d also like to thank Tarik Aougab and Peihong Jiang for helpful conversations.
Preliminaries
=============
Hyperbolicity
-------------
Assume $\Gamma$ is a connected graph which we equip with the metric where each edge has length 1. We give two definitions of a graph being hyperbolic. A triangle in $\Gamma$ is $k$-*centered* if there exists a vertex $c \in \Gamma$ such that $c$ is distance $\leq k$ from each of its three sides. $\Gamma$ is $k$-*centered hyperbolic* if all geodesic triangles (triangles whose edges are geodesics) are $k$-centered. We say a triangle in $\Gamma$ is $\delta$-*thin* if each side of the triangle is contained in the $\delta$-neighborhood of the other two sides for some $\delta \in {\mathbb{R}}$. A graph is $\delta$-*thin hyperbolic* if all geodesic triangles are $\delta$-thin. Note that $\delta$-thin hyperbolic and $k$-centered hyperbolic are equivalent up to a linear factor [@ABC].
\[centered to thin\] If $\Gamma$ is $k$-centered hyperbolic then $\Gamma$ is $4k$-thin hyperbolic.
The following proof is very similar to the proof of an existence of a global minsize of triangles implies slim triangles in [@ABC] (Proposition 2.1).
We denote $[a,b]$ as a geodesic between $a$ and $b$; if $c \in [a,b]$ then $[a, c]$ or $[c,b]$ refers to the subpath of $[a,b]$ with $c$ as one of the endpoints. Consider the triangle $xyz$ and assume it is $k$-centered. Let $p$ be the centered point and $x'$ be the point on the edge $[y,z]$ closest to $p$. Similarly define $y'$ and $z'$. Suppose there is a point $t \in [x,z']$ such that $d(t, [x, y']) > 2k$. Let $u$ be the point in $[t, z']$ nearest to $t$ such that $d(u, u') = 2k$ for some point $u' \in [x, y']$, see Figure \[center to thin figure\].
Consider the geodesic triangle $uu'x$. There exists points $a$, $b$, and $c$ on the three sides of $uu'x$ that are less than or equal to $k$ away from some point $q$, see Figure \[center to thin figure\]. Since $a \in [x, u]$, by assumption $a$ does not lie in $[t, u]$ and $d(u, a) \leq 4k$. So $d(t, u') \leq 4k$ or $d(t, c) \leq 4k$, making the triangle $xyz$ $4k$-thin.
Bowditch shows, in [@Bow] Proposition 3.1, that we don’t always have to work with geodesic triangles to show hyperbolicity of a graph.
\[subset hyperbolic\] Given $h \geq 0$, there exists $\delta \geq 0$ with the following property. Suppose that $G$ is a connected graph, and that for each $x, y \in V(G)$, we have associated a connected subgraph, ${\mathcal{L}}(x,y) \subset G$, with $x, y \in {\mathcal{L}}(x,y)$. Suppose that:
1. for all $x, y, z \in V(G)$, $${\mathcal{L}}(x,y) \subset N_h({\mathcal{L}}(x,z) \cup {\mathcal{L}}(z, y))$$ and
2. for any $x, y \in V(G)$ with $d(x,y) \leq 1$, the diameter of ${\mathcal{L}}(x,y)$ in $G$ is at most $h$.
Then $G$ is $\delta$-thin hyperbolic. In fact, we can take any $\delta \geq (3m-10h)/2$, where $m$ is any positive real number satisfying $$2h(6 + \log_2(m+2)) \leq m.$$
Graphs
------
Let $S = S_{g,p}$ be a surface where $g$ is the genus and $p$ is the number of punctures. We define $\xi(S_{g,p}) = 3g + p -3$ and refer to $\xi(S_{g,p})$ as the complexity of $S_{g,p}$. When $\xi(S) > 1$ the curve graph of $S$, ${\mathcal{C}}(S)$, originally introduced by Harvey in [@Harvey], is a graph whose vertices are homotopy classes of essential simple closed curves on $S$ and there is an edge between two vertices if the curves can be realized disjointly, up to isotopy. From here on when we talk about curves we really mean a representative of the homotopy class of an essential, non-peripheral, simple closed curve. When $\xi(S) = 1$, the definition of the curve graph is slightly altered in order to have a non-trivial graph: the vertices have the same definition, but there is an edge between two curves if they have minimal intersection number. We can similarly define the *arc and curve graph*, ${\mathcal{A}}{\mathcal{C}}(S)$, where a vertex is either a homotopy class of curves or homotopy class of arcs and the edges represent disjointness. This definition is the same for all surfaces such that $\xi(S) > 0$.
A related graph associated to a surface is the pants graph. We call a maximal set of disjoint curves on a surface a *pants decomposition*. For $\xi(S) \geq 1$ the *pants graph*, denoted ${\mathcal{P}}(S)$, of a surface $S$ is a graph whose vertices are homotopy classes of pants decompositions and there exists an edge between two pants decompositions if they are related by an elementary move. Pants decompositions $\alpha$ and $\beta$ differ by an elementary move if one curve, $c$, from $\alpha$ can be deleted and replaced by a curve that intersects $c$ minimally to obtain $\beta$, see Figure \[elementary moves\].
We equip both graphs with the metric where each edge is length 1. Then ${\mathcal{C}}(S)$ and ${\mathcal{P}}(S)$ are complete geodesic metric spaces.
The hyperbolicity of these graphs have been studied before.
\[curve hyp\] For any hyperbolic surface $S$, ${\mathcal{C}}(S)$ is $17$-centered hyperbolic.
Brock and Farb showed:
For any hyperbolic surface $S$, ${\mathcal{P}}(S)$ is hyperbolic if and only if $\xi(S) \leq 2$.
Relative graphs
---------------
Let $S$ be a hyperbolic surface such that $\xi(S) \geq 3$. We say that a curve $c \in {\mathcal{C}}(S)$ is *domain separating* if $S \backslash c$ has two components of positive complexity. Each domain separating curve $c$ determines a set in ${\mathcal{P}}(S)$, $X_c = \{\alpha \in {\mathcal{P}}(S) | c \in \alpha \}$. To form the *relative pants graph*, denoted ${\mathcal{P}}_{rel}(S)$, we add a point $p_c$ for each domain separating curve and an edge from $p_c$ to each vertex in $X_c$, where each edge has length $1$. Effectively, we have made the set $X_c$ have diameter $2$ in the relative pants graph.
Brock and Masur have shown:
For $S$ such that $\xi(S) = 3$, ${\mathcal{P}}_{rel}(S)$ is hyperbolic.
Paths in the Pants Graph
------------------------
Here we describe how we will get a path in ${\mathcal{P}}(S)$ if $\xi(S) =2$ or ${\mathcal{P}}_{rel}(S)$ if $\xi(S) = 3$. The paths for ${\mathcal{P}}(S)$ are hierarchies and were originally introduced by Masur and Minsky in [@MMII] (in more generality than we will use here); the paths in ${\mathcal{P}}_{rel}(S)$ are motivated by hierarchies.
Take two pants decompositions, $\alpha = \{ \alpha_0, \alpha_1\}$ and $\beta = \{ \beta_0, \beta_1\}$, in ${\mathcal{P}}(S)$ where $S = S_{0,5}$ or $S_{1,2}$. To create a hierarchy between $\alpha$ and $\beta$ first connect $\alpha_0$ and $\beta_0$ with a geodesic path in ${\mathcal{C}}(S)$. This geodesic is referred to as the *main geodesic*, $g_{\alpha\beta} = \{ \alpha_0 = g_0, \ldots, g_n = \beta_0\}$. For each $g_i$, $0 \leq i \leq n$, connect $g_{i-1}$ to $g_{i+1}$ by a geodesic, $\gamma_i$, in ${\mathcal{C}}(S\backslash g_i)$, where $g_{-1} = \alpha_1$ and $g_{n+1} = \beta_1$. The collection of all of these geodesics is a *hierarchy* between $\alpha$ and $\beta$, generally pictured as in Figure \[Hierarchy picture\]. We often refer to the geodesic $\gamma_i$ as the geodesics whose domain is ${\mathcal{C}}(S \backslash g_i)$ or the geodesic connecting $g_{i-1}$ and $g_{i+1}$. We can turn a hierarchy into a path in ${\mathcal{P}}(S)$ by looked at all edges in turn, as pictured in Figure \[Hierarchy picture\]. We will often blur the line between the hierarchy being a path in the pants graph or a collection of geodesics - and refer to both as the hierarchy between $\alpha$ and $\beta$.
Let $\xi(S) =3$. We make a path in ${\mathcal{P}}_{rel}(S)$ using a similar technique. Take two pants decompositions in ${\mathcal{P}}_{rel}(S)$, $\alpha = \{\alpha_0, \alpha_1, \alpha_2\}$ and $\beta = \{\beta_0, \beta_1, \beta_2\}$. Connect $\alpha_0$ to $\beta_0$ with a geodesic $g_{\alpha\beta}$ in ${\mathcal{C}}(S)$, we still refer to this as the main geodesic. For every non-domain separating curve $w \in g$, connect $w^{-1}$ to $w^{+1}$ with a geodesic, $h$, in ${\mathcal{C}}(S \backslash w)$ where $w^{-1}$ and $w^{+1}$ are the curves before and after $w$ in $g$. If $w = \alpha_0$ then $w^{-1} = \alpha_1$ and if $w = \beta_0$ then $w^{+} = \beta_1$. Now for each non-domain separating curve $z \in h$ connect $z^{-1}$ to $z^{+1}$ with a geodesic in ${\mathcal{C}}(S \backslash (w \cup z))$, where $z^{-1}$ and $z^{+1}$ are the curves before and after $z$ in $h$. If $z = w^{-1}$ then $z^{-1}$ is the curve preceding $w$ in the geodesic whose domain is ${\mathcal{C}}(S \backslash w^{-1})$. If $z = w^{+1}$ then $z^{+1}$ is the curve following $w$ in the geodesic whose domain is ${\mathcal{C}}(S \backslash w^{+1})$ (see Figure \[general hierarchy\] (top)).
We can get a path in ${\mathcal{P}}_{rel}(S)$ by a similar process as before - going along each of the edges. Whenever we come across a domain separating curve, $c$, where $c$ is in the main geodesic or in a geodesic whose domain is ${\mathcal{C}}(S \backslash w)$ where $w$ is in the main geodesic, we add in the point $p_c$ into the path before moving on. For an example see Figure \[general hierarchy\]. These paths are *relative 3-archies*. As before, we will blur the line between the collection of geodesics and the path of a relative 3-archy.
When discussing hierarchies (or relative 3-archies), subsurface projections of curves or geodesics are involved. The following maps are to define what is meant by subsurface projections [@MMII]. An *essential subsurface* is a subsurface where each boundary component is essential.
Let ${\mathscr{P}}(X)$ be the set of subsets of $X$. For a set $A$ we define $f(A) = \cup_{a \in A}f(a)$, for any map $f$. Take an essential, non-annular subsurface $Y \subset S$. We define a map $$\phi_Y: {\mathcal{C}}(S) {\longrightarrow}{\mathscr{P}}({\mathcal{A}}{\mathcal{C}}(Y))$$ such that $\phi_Y(a)$ is the set of arcs and curves obtained from $a \cap Y$ when $\partial Y$ and $a$ are in minimal position. Define another map $$\psi_Y : {\mathscr{P}}({\mathcal{A}}{\mathcal{C}}(Y)) {\longrightarrow}{\mathscr{P}}({\mathcal{C}}(Y))$$ such that if $a$ is a curve, then $\psi_Y(a) = a$, and if $b$ is an arc, then $\psi_Y(b)$ is the union of the non-trivial components of the regular neighborhood of $(b\cap Y) \cup \partial Y$ (see Figure \[nbhd\]).
Composing these two maps we define the map $$\begin{aligned}
\pi_Y: {\mathcal{C}}(S) &{\longrightarrow}{\mathscr{P}}({\mathcal{C}}(Y)) \\
c &\longmapsto \psi_Y(\phi_Y(c))\end{aligned}$$ We use this map to define distances in a subsurface: for any two sets $A$ and $B$ in ${\mathcal{C}}(S)$, $$d_Y(A, B) = d_Y(\pi_Y(A), \pi_Y(B)).$$ We often refer to this as the distance in the subsurface $Y$.
The relationship between hierarchies and these maps give rise to some useful properties including the Bounded Geodesic Image Theorem which was originally proven by Masur-Minsky [@MMII].
\[bounded geodesic image\] Let $Y$ be a subsurface of $S$ with $\xi(Y) \neq 3$ and let $g$ be a geodesic segment, ray, or biinfinite line in ${\mathcal{C}}(S)$, such that $\pi_Y(v) \neq \emptyset$ for every vertex of $v$ of $g$. There is a constant $M$ depending only on $\xi(S)$ such that $${\mathrm{diam}}_Y(g) \leq M.$$
It can be shown that $M$ is at most $100$ for all surfaces [@Webb].
Hyperbolicity of Pants Graph for Complexity 2
=============================================
In this section we explore the hyperbolicity constant for the pants graph of surfaces with complexity $2$. Before we state any results, some notation must be discussed. Throughout the paper we denote $[a, b]_\Sigma$ as a geodesic in ${\mathcal{C}}(\Sigma)$ connecting $a$ to $b$, for any surface $\Sigma$. If a geodesic satisfying this is contained in a hierarchy (or relative 3-archy, in later sections) being discussed, $[a,b]_\Sigma$ denotes the geodesic in the hierarchy.
\[hierarchy k-centered\] For $S = S_{0,5}, S_{1,2}$, hierarchy triangles in ${\mathcal{P}}(S)$ are $8,900$-centered.
Let $S = S_{0,5}$ or $S_{1,2}$. Take three pants decompositions $\alpha = \{\alpha_0, \alpha_1\}$, $\beta = \{\beta_0, \beta_1\}$, and $\gamma = \{\gamma_0, \gamma_1\}$ in $S$. Consider the triangle $\alpha\beta\gamma$ in ${\mathcal{P}}(S)$ where the edges are taken to be hierarchies instead of geodesics. There are three cases:
1. All three main geodesics have a curve in common.
2. Any two of the main geodesics share a curve, but not the third.
3. None of the main geodesics have common curves.
In all three cases we will find a pants decomposition such that the hierarchy connecting this pants decomposition to each edge in $\alpha\beta\gamma$ is less than $8,900$.
**Case 1**: Assume the main geodesics of all three edges share the curve $v \in {\mathcal{C}}(S)$. Define $v_{\alpha \beta}^{-1}$ to be the curve on $g_{\alpha \beta}$ preceding $v$ and $v_{\alpha \beta}^{+1}$ the curve on $g_{\alpha \beta}$ following $v$ when viewing $g_{\alpha\beta}$ going from $\alpha_0$ to $\beta_0$. Similarly define $v_{\alpha \gamma}^{-1}$, $v_{\alpha \gamma}^{+1}$, $v_{\beta \gamma}^{-1}$, and $v_{\beta \gamma}^{+1}$. See Figure \[Case 1\].
We want to show the geodesics connecting $v_*^{-1}$ to $v_*^{+1}$ in ${\mathcal{C}}(S \backslash v)$ are not too far apart in ${\mathcal{C}}(S \backslash v)$. Connect $v_{\alpha\beta}^{-1}$ to $v_{\alpha\gamma}^{-1}$, $v_{\alpha\gamma}^{+1}$ to $v_{\beta \gamma}^{+1}$ and $v_{\beta\gamma}^{-1}$ to $v_{\alpha\beta}^{+1}$ by geodesics in ${\mathcal{C}}(S \backslash v)$. We now have a loop in ${\mathcal{C}}(S\backslash v)$. Since all curves besides $v$ in $S$ intersect the subsurface $S \backslash v$ non-trivially we can apply the Bounded Geodesic Image Theorem on $[v_{\alpha \beta}^{-1}, \alpha_1]_S$ and $[\alpha_1, v_{\alpha \gamma}^{-1}]_S$ to get $d_{{\mathcal{C}}(S\backslash v)}(v_{\alpha \beta}^{-1}, v_{\alpha \gamma}^{-1}) \leq 2M$. Similarly, $d_{{\mathcal{C}}(S\backslash v)}(v_{\alpha \gamma}^{+1}, v_{\beta \gamma}^{+1}) \leq 2M$ and $d_{{\mathcal{C}}(S\backslash v)}(v_{\beta \gamma}^{-1}, v_{\alpha\beta}^{+1}) \leq 2M$.
Consider the geodesic triangle $v_{\alpha\beta}^{+1}v_{\alpha\gamma}^{-1}v_{\beta\gamma}^{+1}$ in ${\mathcal{C}}(S \backslash v)$. We now have the picture in ${\mathcal{C}}(S \backslash v)$ as in Figure \[2 links are thin\]. By Theorem \[curve hyp\], the inner triangle is $17$ centered, call this center $z$. Combining Theorem \[curve hyp\] and Lemma \[centered to thin\], the outer three triangles are $17*4$-thin. Therefore $z$ is at most $17*5 + 2M = 285$ away from each of the geodesics in the hierarchy triangle $\alpha\beta\gamma$ whose domain is ${\mathcal{C}}(S \backslash v)$.
This all implies that $\alpha\beta\gamma$ is 285-centered at $\{v, z\}$.
**Case 2**: Assume that at least two main geodesics share a common curve, but there is no point that all three main geodesics share the same curve. First assume there is only one such shared curve. Without loss of generality assume that $g_{\alpha\beta}$ and $g_{\alpha \gamma}$ share the curve $v$. Then we can consider a new triangle with the main geodesics forming the triangle $v\beta_1\gamma_1$, see Figure \[Case 2\]. This new triangle has no shared curves so is covered by Case 3.
Now assume there is more than one shared curve between the main geodesics. By definition of a geodesic, for any two main geodesics that share multiple curves, those curves have to show up in each main geodesic in the same order from either end, therefore we can just take the inner triangle where the edges share no curves and apply Case 3.
**Case 3**: The argument given for this case is similar to the short cut argument in [@MMII]. Assume none of the three main geodesics, $g_{\alpha \beta}, g_{\alpha \delta}$, and $g_{ \beta \delta}$ share a curve. By Theorem \[curve hyp\] there exists a curve $c \in {\mathcal{C}}(S)$ that is distance at most $17$ from $g_{\alpha \beta}, g_{\alpha \gamma}$, and $g_{ \beta \gamma}$; let $c$ be the curve that minimizes the distance from all three main geodesics. Define $v_{\alpha \beta}$ to be the vertex in $g_{\alpha \beta}$ which has the least distance to $c$, and similarly define $v_{\alpha \gamma}$ and $v_{\beta \gamma}$.
Consider the geodesic $[v_{\alpha\beta}, c]_S$ and let $c_0$ be the curve adjacent to $c$ in this geodesic. Let $v_*^{-1}$ be the curve in $g_*$ that precedes $v_{*}$. Now connect $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}\}$ to $\{c, c_0 \}$ with a hierarchy. We denote the main geodesic of this hierarchy as $[c, v_{\beta\gamma}]_S$.
Take a vertex $w \in [c, v_{ \beta \gamma}]_S$ where $w$ is not equal to $c$ or $v_{\beta\gamma}$ and let $w^{-1}$ and $w^{+1}$ denote the vertices directly before and after $w$ in $[c, v_{ \beta \gamma}]_S$. We want to show that the link connecting $w^{-1}$ to $w^{+1}$ in $S\backslash w$ is at most $5M$. Assume $d_{S \backslash w} (w^{-1}, w^{+1}) \geq 5M$. Consider the path $[w^{+1}, v_{\beta\gamma}]_S \cup [v_{ \beta \gamma}, \beta_0]_S \cup [\beta_0, v_{\alpha \beta}]_S \cup [v_{\alpha \beta}, c]_S \cup [c, w^{-1}]_S$, where geodesics are taken to be on $g_*$ where appropriate. The Bounded Geodesic Image Theorem, and our assumption that $d_{S \backslash w} (w^{-1}, w^{+1}) \geq 5M$, implies that $w$ must be somewhere on the path. $w$ cannot be in $[w^{+1}, v_{\beta\gamma}]_S$, $[v_{ \beta \gamma}, \beta_0]_S$, or $[c, w^{-1}]_S$ since that would contradict the fact that they are geodesics or the definition of how we chose $c$ and $v_{\beta\gamma}$. Therefore, $w$ is in $[\beta_0, v_{\alpha \beta}]_S$ or $[v_{\alpha \beta}, c]_S$. Without loss of generality assume $w \in [\beta_0, v_{\alpha \beta}]_S$. We can apply the same logic to the path $[w^{+1}, v_{\beta\gamma}]_S \cup [v_{ \beta \gamma}, \gamma_0]_S \cup [\gamma_0, v_{\alpha \gamma}]_S \cup [v_{\alpha \gamma}, c]_S \cup [c, w^{-1}]_S$. Now $w$ has to be in $[v_{\alpha \gamma}, c]_S$ so that it doesn’t contradict the fact that the three main geodesic of the triangle $\alpha\beta\gamma$ do not share any curves. However, now all three main geodesics are closer to $w$ than $c$, which contradicts our choice of $c$. Therefore, the length of $[w^{-1}, w^{+1}]_{S \backslash w}$ is at most $5M$.
Using a similar argument we can show the geodesic in ${\mathcal{C}}(S \backslash v_{\beta\gamma})$ connecting $v_{\beta\gamma}^{-1}$ to the appropriate vertex in $[c, v_{\beta\gamma}]_S$ is $\leq 5M$. Now consider the geodesic in ${\mathcal{C}}(S \backslash c)$ connecting $c_0$ to the second vertex, $x$, of $[c, v_{\beta\gamma}]_S$. Consider the path $[x, v_{\beta\gamma}]_S \cup [v_{\beta\gamma}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c_0]_S$. $c$ cannot be in anywhere in this path, otherwise it would contradict how we chose $c$ or $v_*$. So we can apply the Bounded Geodesic Image Theorem and get that $d_{S \backslash c}(c', x) \leq 4M$. Therefore the path from $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}\}$ to $\{c, c_0\}$ in the pants graph is less than or equal to $16(5M) + 5M + 4M$. A similar argument can be made for the other two sides of the triangle $\alpha\beta\gamma$, so $\{c, c_0\}$ can be taken to be a center of the triangle. Since $M \leq 100$ the triangle $\alpha\beta\gamma$ is $8,900$-centered at $\{c, c_0\}$.
\[main thm 1\] For a surface $S = S_{0,5}, S_{1,2}$, ${\mathcal{P}}(S)$ is $2,691,437$-thin hyperbolic.
For $x, y \in {\mathcal{P}}(S)$ define ${\mathcal{L}}(x,y)$ to be the collection of hierarchy paths between $x$ and $y$. These are connected because each hierarchy path is connected and all contain $x$ and $y$. By Theorem \[hierarchy k-centered\] and Lemma \[centered to thin\] we have that for all $x, y, z \in {\mathcal{P}}(S)$ $${\mathcal{L}}(x, y) \subset N_{4*8,900}({\mathcal{L}}(x,z) \cup {\mathcal{L}}(z,y)).$$ If $d(x,y) \leq 1$ then any hierarchy between $x$ and $y$ is just the edge $\{ xy\}$, so ${\mathcal{L}}(x,y) = \{x, y\}$. Thus, both conditions of Proposition \[subset hyperbolic\] are satisfied. Therefore by applying Proposition \[subset hyperbolic\] we get ${\mathcal{P}}(S)$ is $2,691,437$-thin hyperbolic.
Relative Hyperbolicity of Pants Graphs Complexity 3
===================================================
In this section we turn our attention to relative pants graphs and their hyperbolicity constant.
\[relative hierarchy k-centered\] Take $S$ such that $\xi(S) = 3$. The relative 3-archy triangles in ${\mathcal{P}}_{rel}(S)$ are $6,191,300$-centered.
Take three pants decompositions of $S$, say $\alpha = \{\alpha_0, \alpha_1, \alpha_2 \}$, $\beta = \{\beta_0, \beta_1, \beta_2\}$, and $\gamma = \{\gamma_0, \gamma_1, \gamma_2\}$. Form the triangle $\alpha\beta\gamma$ such that each edge in the triangle is a relative 3-archy in ${\mathcal{P}}_{rel}(S)$. Let $g_{\alpha\beta}$, $g_{\beta\gamma}$, and $g_{\alpha\gamma}$ be the three main geodesics that make up the triangle (which connects $\alpha_0$, $\beta_0$, and $\gamma_0$). As before in Theorem \[hierarchy k-centered\], there are three cases:
1. All three main geodesics have a curve in common.
2. Any two of the main geodesics share a curve, but not the third.
3. None of the main geodesics have common curves.
For the rest of the proof, note that if $v \in {\mathcal{C}}(S)$ is a non-domain separating curve, then $S \backslash v$ has one connected component with positive complexity, so by abuse of notation, we denote this component as $S \backslash v$. This means that every curve in ${\mathcal{C}}(S)$ not equal to $v$ intersects $S \backslash v$ so we can use the Bounded Geodesic Image Theorem on any geodesic that doesn’t contain $v$. Take two non-domain separating curve $v,w \in {\mathcal{C}}(S)$ such that $v$ and $w$ are disjoint. Then, because $\xi(S) = 3$, $S \backslash (v \cup w)$ has one connected component with positive complexity, and again we denote this component as $S \backslash (v \cup w)$. Furthermore, every curve in ${\mathcal{C}}(S)$ not equal to $v$ or $w$ intersects $S \backslash (v \cup w)$, so we may use the Bounded Geodesic Image Theorem for any geodesic that doesn’t contain $v$ or $w$.
Whenever a domain separating curve, $c$, shows up in a relative 3-archy in ${\mathcal{P}}_{rel}(S)$, the section of the relative 3-archy containing $c$ has length $2$. Therefore, when referring to a curve along a geodesic within a relative 3-archy we will assume it is non-domain separating since this type of curve adds the most length to the relative 3-archy. This also just makes the proof cleaner.
**Case 1:** Let $v$ be a vertex where all three main geodesics intersect. If $v$ is a domain separating curve then each edge of the triangle $\alpha\beta\gamma$ contains the point $p_v$, so the triangle is $0$-centered. Now assume $v$ is not a domain separating curve. Let $v_{\alpha\beta}^{-1}$ and $v_{\alpha\beta}^{+1}$ be the curves that are directly before and after $v$ on $g_{\alpha\beta}$. Similarly define $v_{\alpha\gamma}^{-1}$, $v_{\alpha\gamma}^{+1}$, $v_{\beta\gamma}^{-1}$, and $v_{\beta\gamma}^{+1}$. Consider the geodesics associated with $v$ in each relative 3-archy edge; in other words, all geodesics in the relative 3-archy that contribute to defining the path where $v$ is a part of every pants decomposition. Let $x_{\alpha\beta}$ be the curve in $[v_{\alpha\beta}^{-1}, v_{\alpha\beta}^{+1}]_{S \backslash v}$ that is adjacent to $v_{\alpha\beta}^{-1}$; similarly define $x_{\alpha\gamma}$. Now connect $\{v_{\alpha\beta}^{-1}, x_{\alpha\beta} \}$ to $\{v_{\alpha\gamma}^{-1}, x_{\alpha\gamma}\}$ with a hierarchy in ${\mathcal{P}}(S \backslash v)$. Note, to make our notation cleaner, we will refer to this as the hierarchy between $v_{\alpha\beta}^{-1}$ and $v_{\alpha\gamma}^{-1}$; similarly later on we won’t necessarily specify the second curve. By the Bounded Geodesic Image Theorem the geodesic connecting $v_{\alpha\beta}^{-1}$ and $v_{\alpha\gamma}^{-1}$ in ${\mathcal{C}}(S\backslash v)$ has length at most $2M$. Now consider any curve, $w$, in the geodesic $[v_{\alpha\beta}^{-1}, v_{\alpha\gamma}^{-1}]_{S \backslash v}$ contained in the hierarchy connecting $\{v_{\alpha\beta}^{-1}, x_{\alpha\beta} \}$ to $\{v_{\alpha\gamma}^{-1}, x_{\alpha\gamma}\}$. Assume $w$ is not a domain separating curve in $S$ and let $w^{-1}$ and $w^{+1}$ be the two curves before and after $w$ on $[v_{\alpha\beta}^{-1}, v_{\alpha\gamma}^{-1}]_{S \backslash v}$. Then the geodesic connecting $w^{-1}$ to $w^{+1}$ in ${\mathcal{C}}(S \backslash (v \cup w))$ has length at most $4M$ by using the Bounded Geodesic Image Theorem on $[w^{-1}, v_{\alpha\beta}^{-1}]_{S \backslash v} \cup [v_{\alpha\beta}^{-1}, \alpha_0]_S \cup [\alpha_0, v_{\alpha\gamma}^{-1}]_S \cup [v_{\alpha\gamma}^{-1}, w^{+1}]_{S \backslash v}$; note $w$ cannot be on this path because $w$ is distance $1$ from $v$, so if it was anywhere in the path it would be violating the assumption that we have geodesics. Therefore the hierarchy between $v_{\alpha\beta}^{-1}$ and $v_{\alpha\gamma}^{-1}$ has length at most $8M^2$. Similarly the hierarchies between $v_{\alpha\gamma}^{+1}$ and $v_{\beta\gamma}^{+1}$, and $v_{\alpha\beta}^{+1}$ and $v_{\beta\gamma}^{-1}$ have length less than $8M^2$.
Now, make a hierarchy triangle $v_{\alpha\beta}^{+1}v_{\alpha\gamma}^{-1}v_{\beta\gamma}^{+1}$ in ${\mathcal{P}}(S \backslash v)$, see Figure \[links are thin\] for how this fits in with above. By Theorem \[hierarchy k-centered\], $v_{\alpha\beta}^{+1}v_{\alpha\gamma}^{-1}v_{\beta\gamma}^{+1}$ in ${\mathcal{P}}(S \backslash v)$ is $8,900$ centered, call the point at the center $z$. Then by Theorem \[hierarchy k-centered\] and Lemma \[centered to thin\], the hierarchy triangles $v_{\alpha\beta}^{+1}v_{\alpha\beta}^{-1}v_{\alpha\gamma}^{-1}$, $v_{\beta\gamma}^{-1}v_{\beta\gamma}^{+1}v_{\alpha\gamma}^{+1}$, and $v_{\alpha\gamma}^{-1}v_{\alpha\gamma}^{+1}v_{\beta\gamma}^{+1}$ are $35,600$ thin. Therefore $z$ is at most $124,500$ away from each $[v_*^{+1}, v_{*}^{-1}]_{S \backslash v}$. This implies that $\{z, v\}$ is at most $124,500$-centered in the relative 3-archy triangle $\alpha\beta\gamma$.
**Case 2:** For the same reasons as in Theorem \[hierarchy k-centered\] case 2, this case can be reduced to case 3.
**Case 3:** This proceeds with the same strategy as in case 3 of Theorem \[hierarchy k-centered\]. By Theorem \[curve hyp\], we know the triangle of main geodesics, $g_{\alpha\beta}g_{\beta\gamma}g_{\alpha\gamma}$ in ${\mathcal{C}}(S)$ is $17$-centered. Let $c$ be the curve that is at the center of this triangle. Connect $c$ to $g_{\alpha\beta}$, $g_{\beta\gamma}$, and $g_{\alpha\gamma}$ with a geodesic in ${\mathcal{C}}(S)$. Define $v_{\alpha \beta}$ to be the vertex in $g_{\alpha \beta}$ which is the least distance to $c$, and similarly define $v_{\alpha \gamma}$ and $v_{\beta \gamma}$.
Let $c_0$ be the curve directly preceding $c$ in $[v_{\alpha\beta}, c]_S$ and let $c^{-1}$ be the curve directly preceding $c_0$. Consider a geodesic in ${\mathcal{C}}(S \backslash c_0)$ which connects $c^{-1}$ to $c$, define $c_1$ to be the curve directly preceding $c$ in this geodesic. We will show $\{c, c_0, c_1\}$ is a center of our relative 3-archy triangle $\alpha\beta\gamma$.
Let $v_{\beta\gamma}^{-1}$ be the curve before $v_{\beta\gamma}$ in $g_{\beta\gamma}$ and $v_{\beta\gamma}'$ be the curve adjacent to $v_{\beta\gamma}$ in the geodesic contained in the relative 3-archy connecting $\beta$ to $\gamma$ whose domain is ${\mathcal{C}}(S \backslash v_{\beta\gamma}^{-1})$. Now connect $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}, v_{\beta\gamma}'\}$ to $\{c, c_0, c_1 \}$ with a relative 3-archy, $H$. Our goal is to bound the length of $H$.
Using the exact argument as in Theorem \[hierarchy k-centered\] case 3, for each $w \in [c, v_{\beta\gamma}]_S$ which is non-separating, the geodesic in $H$ whose domain is ${\mathcal{C}}(S\backslash w)$ has length no more than $5M$. Let $w^{-1}$ and $w^{+1}$ be the curves before and after $w$ in $[c, v_{\beta\gamma}]_S$ and let $[w^{-1}, w^{+1}]_{S \backslash w}$ be the geodesic coming from $H$. Take $z \in [w^{-1}, w^{+1}]_{S \backslash w}$ and consider the geodesic in $H$ with domain ${\mathcal{C}}(S \backslash (w \cup z))$. Define $z^{-1}$ and $z^{+1}$ to be the curves before and after $z$ on $[w^{-1}, w^{+1}]_{S \backslash w}$. We will show $[z^{-1}, z^{+1}]_{S \backslash (w \cup z)}$ has length at most $7M$. Assume towards a contradiction that the length of $[z^{-1}, z^{+1}]_{S \backslash (w \cup z)}$ is greater than $7M$. Then the path $[z^{+1}, w^{+1}]_{S \backslash w} \cup [w^{+1}, v_{\beta\gamma}]_{S} \cup [v_{\beta\gamma}, \gamma_0]_S \cup [\gamma_0, v_{\alpha\gamma}]_S \cup [v_{\alpha\gamma}, c]_S \cup [c, w^{-1}]_S \cup [w^{-1}, z^{-1}]_{S \backslash w}$ must contain $z$ or $w$ somewhere, otherwise by the Bounded Geodesic Image Theorem using this path we would get that the length of $[z^{-1}, z^{+1}]_{S \backslash (w \cup z)}$ is at most $7M$. Since $w$ and $z$ are distance $1$ apart, it doesn’t matter which one shows up in the path because we eventually will arise at the same contradiction. Thus, without loss of generality we assume $z$ is in the path (and all other paths considered for this argument). Then $z$ must be in $[\gamma_0, v_{\alpha\gamma}]_S$ or $[v_{\alpha\gamma}, c]_S$, otherwise there would be a contradiction with the definition of a geodesic or the definition of $c$ or $v_{\beta\gamma}$ Without loss of generality assume $z \in [\gamma_0, v_{\alpha\gamma}]_S$. Similarly the path $[z^{+1}, w^{+1}]_{S \backslash w} \cup [w^{+1}, v_{\beta\gamma}]_{S} \cup [v_{\beta\gamma}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c]_S \cup [c, w^{-1}]_S \cup [w^{-1}, z^{-1}]_{S \backslash w}$ must contain $z$. Again, the only place $z$ could be, without yielding a contradiction, is in $[v_{\alpha\beta}, c]_S$. However even here, since $z$ is adjacent to $w$, $w$ is strictly closer than $c$ to the three main geodesics of $\alpha\beta\gamma$ which contradicts our choice of $c$. Therefore, the length of $[z^{-1}, z^{+1}]_{S \backslash (w \cup z)}$ is at most $7M$. Now all that’s left to bound is the beginning and end geodesics, i.e. the ones associated to $c$ and $v_{\beta\gamma}$. Let $y$ be the curve adjacent to $v_{\beta\gamma}$ in $[c, v_{\beta\gamma}]_S$ and let $y'$ be the curve adjacent to $v_{\beta\gamma}$ in the geodesic contained in $H$ whose domain is ${\mathcal{C}}(S \backslash y)$. Then the very beginning part of $H$ is the hierarchy connecting $\{y, y' \}$ to $\{v_{\beta\gamma}^{-1}, v_{\beta\gamma}' \}$ in $S \backslash v_{\beta\gamma}$. We will first bound the length of the geodesic $[y, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}}$. Assume that the length is more than $5M$. Then the path $[v_{\beta\gamma}^{-1}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c]_S \cup [c, y]_S$ has to contain $v_{\beta\gamma}$. By our assumption that the main geodesics on the triangle $\alpha\beta\gamma$ don’t intersect, the only part of the path that $v_{\beta\gamma}$ could be on without forming a contraction would be $[v_{\alpha\gamma}, c]_S$. The same is true of the path $[v_{\beta\gamma}^{-1}, \beta_0]_S \cup [\beta_0, \alpha_0]_S \cup [\alpha_0, v_{\alpha\gamma}]_S \cup [v_{\alpha\gamma}, c]_S \cup [c, y]_S$, where $v_{\beta\gamma}$ would have to be in $[v_{\alpha\gamma}, c]_S$. However, then we could take $v_{\beta\gamma}$ to be the center of the main geodesic triangle which would give strictly smaller lengths to each of the sides, contradicting our choice of $c$. Therefore, $[y, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}}$ has length at most 5M.
Now take $w \in [y, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}}$ and let $w^{-1}$ and $w^{+1}$ be the curves that come directly before and after $w$ in $[y, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}}$. We want to bound the length of $[w^{-1}, w^{+1}]_{S \backslash (v_{\beta\gamma} \cup w)}$. Assume the length is greater than $7M$. Then the path $[w^{+1}, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}} \cup [v_{\beta\gamma}^{-1}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c]_S \cup[c, y]_S \cup [y, w^{-1}]_{S \backslash v_{\beta\gamma}}$ must contain $w$ or $v_{\beta\gamma}$. The only two places this could happen without raising a contradiction is in $[\beta_0, v_{\alpha\beta}]_S$ or $[v_{\alpha\beta}, c]_S$. Again, whether we assume $w$ or $v_{\beta\gamma}$ is in the path doesn’t matter since we will arrive at the same contradiction, hence we can assume without loss of generality $w$ is always on the path. Therefore, assume $w \in [v_{\alpha\beta}, c]_S$. Similarly, $w$ is contained in the path $[w^{+1}, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}} \cup [v_{\beta\gamma}^{-1}, v_{\beta\gamma}^{+1}]_{S \backslash v_{\beta\gamma}} \cup [v_{\beta\gamma}^{+1}, \gamma_0] \cup [\gamma_0, v_{\alpha\gamma}]_S \cup [v_{\alpha\gamma}, c]_S \cup[c, y]_S \cup [y, w^{-1}]_{S \backslash v_{\beta\gamma}}$, where $w \in [\gamma_0, v_{\alpha\gamma}]_S$ since anywhere else in the path would lead to a contradiction as explained previously. Note if $w \in [v_{\alpha\gamma}, c]_S$ then since $w$ is disjoint from $v_{\beta\gamma}$ and that $w \in [v_{\alpha\beta}, c]_S$, we could make a shorter path to each of the three sides on the main geodesic triangle and then $v_{\beta\gamma}$ would be the center of the triangle, contradicting our choice of $c$. The path $[w^{+1}, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}} \cup [v_{\beta\gamma}^{-1}, \beta_0]_S \cup [\beta_0, \alpha_0]_S \cup [\alpha_0, v_{\alpha\gamma}]_S \cup [v_{\alpha\gamma}, c]_S \cup[c, y]_S \cup [y, w^{-1}]_{S \backslash v_{\beta\gamma}}$ has to contain $w$ as well. No matter where $w$ is on this path is creates a contradiction - either with the definition of $c$, with the we have a geodesic, or with the assumption the main geodesics do not share any curves. Consequently, $[w^{-1}, w^{+1}]_{S \backslash (v_{\beta\gamma} \cup w)}$ must have length at most $7M$. Note that this argument also works when $w = y$ or $w = v_{\beta\gamma}^{-1}$, which gives a length bound on the geodesic in $H$ whose domain is ${\mathcal{C}}(S \backslash (v_{\beta\gamma} \cup y))$ or ${\mathcal{C}}(S \backslash (v_{\beta\gamma} \cup v_{\beta\gamma}^{-1}))$, respectively.
Let $x$ be the curve adjacent to $c$ in $[v_{\beta\gamma}, c]_S$ and $x'$ be the last curve adjacent to $c$ in the geodesic from the hierarchy whose domain is ${\mathcal{C}}(S \backslash x)$. First, the geodesic $[c_0, x]_{S \backslash c}$ has length no more than $4M$ by the Bounded Geodesic Image Theorem applied to $[c_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, \beta_0]_S \cup [\beta_0, v_{\beta\gamma}]_S \cup [v_{\beta\gamma}, x]_S$, which doesn’t contain $c$ because if it did we would get a contradiction on the definition of $c$. Now take any curve $w \in [c_0, x]_{S \backslash c}$ and define $w^{-1}$ and $w^{+1}$ as before. Then the path $[w^{+1}, x]_{S \backslash c} \cup [x, v_{\beta \gamma}]_S \cup [v_{\beta\gamma}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c_0]_S \cup [c_0, w^{-1}]_{S \backslash c}$ cannot contain $w$ because $w$ is adjacent to $c$ so if any geodesic making up the path contained $w$ it would either contradict that it is a geodesic or that $c$ is minimal distance from the main geodesics of the triangle $\alpha\beta\gamma$. Hence, applying the Bounded Geodesic Image Theorem to the path we get that $[w^{-1}, w^{+1}]_{S \backslash (c \cup w)}$ has length no more than $6M$. This leaves bounding the lengths of the geodesics connecting $c_1$ to the second vertex of $[c_0, x]_{S \backslash c}$ and $x'$ to the penultimate vertex of $[c_0, x]_{S \backslash c}$. By a similar argument using the Bounded Geodesic Image Theorem each of these geodesics have length at most $6M$. Therefore, putting all the length bounds together we get that the relative 3-archy connecting $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}, v_{\beta\gamma}'\}$ to $\{c, c_0, c_1 \}$ has length at most $16*5M*7M + (4M-1)*6M+12M + (5M+1)*7M = 6,191,300$
Similarly $\{c, c_0, c_1\}$ is length at most $6,191,300$ from the other two sides of the triangle $\alpha\beta\gamma$. Therefore, the relative 3-archy triangle $\alpha\beta\gamma$ is $6,191,300$-centered.
\[main thm 2\] For a surface $S$ such that $\xi(S) =3$, ${\mathcal{P}}_{rel}(S)$ is $1,607,425,314$-thin hyperbolic.
For $x, y \in {\mathcal{P}}_{rel}(S)$ define ${\mathcal{L}}(x,y)$ to be the collection of relative 3-archy paths between $x$ and $y$. These are connected because each relative 3-archy path is connected and all the relative 3-archies in ${\mathcal{L}}(x, y)$ contain $x$ and $y$. By Theorem \[relative hierarchy k-centered\] and Lemma \[centered to thin\] we have that for all $x, y, z \in {\mathcal{P}}_{rel}(S)$ $${\mathcal{L}}(x, y) \subset N_{4*6,191,300}({\mathcal{L}}(x,z) \cup {\mathcal{L}}(z,y)).$$ If $d(x,y) \leq 1$ then any relative 3-archy between $x$ and $y$ is just the edge $\{ xy\}$, so ${\mathcal{L}}(x,y) = \{x, y\}$. We now have both conditions of Proposition \[subset hyperbolic\] satisfied. Therefore by applying Proposition \[subset hyperbolic\] we get that ${\mathcal{P}}_{rel}(S)$ is $1,607,425,314$-thin hyperbolic.
[*Email:*]{}\
aweber@math.brown.edu
|
---
abstract: 'Measurements of polarized neutron—polarized $^{3}$He scattering are reported. The target consisted of cryogenically-polarized solid $^{3}$He, thickness 0.04 atom/b and polarization $\sim 0.4$. Polarized neutrons were produced via the $^3$H($\vec p$,$\vec n$)$^3$He or $^2$H($\vec d$,$\vec n$)$^3$He polarization-transfer reactions. The longitudinal and transverse total cross-section differences $\Delta\sigma_L$ and $\Delta\sigma_T$ were measured for incident neutron energies 2–8 MeV. The results are compared to phase-shift predictions based on four different analyses of $n$-$^{3}$He scattering. The best agreement is obtained with a recent $R$-matrix analysis of $A=4$ scattering and reaction data, lending strong support to the $^4$He level scheme obtained in that analysis. Discrepancies with other phase-shift parameterizations of $n$-$^3$He scattering exist, attributable in most instances to one or two particular partial waves.'
address:
- 'North Carolina State University, Raleigh, NC 27695, USA and Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA'
- 'Duke University, Durham, NC 27708, USA and Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA'
author:
- 'C. D. Keith,[^1] C. R. Gould, D. G. Haase and M. L. Seely'
- 'P. R. Huffman,[^2] N. R. Roberson, W. Tornow and W. S. Wilburn'
title: Measurements of the Total Cross Section for the Scattering of Polarized Neutrons from Polarized $^3$He
---
Introduction
============
Recent computational advances in the field of few-nucleon dynamics have fueled renewed interest in the three-nucleon and four-nucleon systems [@Glo95]. Exact bound-state calculations utilizing realistic, meson-exchange forces are now possible for both the 3N and 4N systems. Similar calculations are currently available for the 3N continuum, and extension to the 4N continuum is under active investigation.
The continuum calculations may prove especially revealing, because here exists a fundamental difference between the three- and four-nucleon systems: three-nucleon systems have no excited states whereas the four-nucleon system has many. These states (resonances) may exhibit sensitivity to the dynamics of the nucleon-nucleon interaction and their modification in the presence of the nuclear medium. However, confirming the existence, and determining the quantum numbers of these resonances is a challenging experimental problem. The level scheme proposed as part of a recent review article [@Til92] has 15 levels at excitations 20–30 MeV above the ground state. For the most part these resonances do not appear as sharp structure in any scattering or reaction observable, and polarization measurements are essential for determining the scattering amplitudes.
While there have been many studies of single polarization observables in 4N systems, there is very little data with [*both*]{} polarized target [*and*]{} polarized beam. Only two measurements have been reported. In 1966, Passell and Schermer measured the transmission of polarized thermal neutrons through a polarized $^3$He target [@Pas66], and more recently, Alley and Knutsen studied $p$-$^3$He spin-correlation data [@All93]. In the first experiment, the thermal cross section is completely dominated by a single ($0^+$) subthreshold resonance in the $^4$He compound nucleus, and no information on the higher-energy resonances was obtained. The second experiment covered a much broader region in the $A=4$ continuum, but was sensitive only to the isotriplet scattering states. No previous polarized target—polarized beam experiment has fully explored a wide range of 4N excited states.
In this paper we report measurements of the longitudinal and transverse neutron total cross-section differences $\Delta\sigma_L$ and $\Delta\sigma_T$. These two spin observables are directly related to the forward elastic-scattering amplitude through the optical theorem [@Kei94]. As such, they allow for a simple interpretation in terms of the properties of the scattering states. The measurements were performed at energies corresponding to excitations 22–27 MeV above the $^4$He ground state, where a number of broad, negative-parity levels are believed to exist.
The remainder of this paper is organized as follows. In Sec. \[sec:theory\] we discuss the basic principles behind the measurements. The polarized target and polarized beam are described in Sections \[sec:target\] and \[sec:beam\], respectively. The experimental procedure and method of data analysis are described in Sec. \[sec:procedure\]. The $\Delta\sigma_L$ and $\Delta\sigma_T$ results are presented in Sec. \[sec:results\] and compared to four separate sets of $n$-$^3$He phase shifts in Sec. \[sec:comparison\]. Our conclusions are summarized in Sec. \[sec:conclusion\]. A preliminary version of the $\Delta\sigma_T$ results has appeared elsewhere [@Kei95b].
Theory {#sec:theory}
======
The formalism necessary to describe the neutron total cross section for polarized target and polarized beam has been developed and discussed in an earlier paper [@Kei94]. For the sake of clarity we briefly review the results of that paper in the following section.
The parity-conserving, time-reversal invariant part of the forward elastic scattering amplitude allows the neutron total cross section for spin-1/2 nuclei to be expressed as $$\sigma_{tot} = \sigma_{0} + \frac{1}{2}\Delta\sigma_L
P^{z}_{n}P^{z}_{t} +
\frac{1}{2}\Delta\sigma_T(P^{x}_{n}P^{x}_{t} + P^{y}_{n}P^{y}_{t}).
\label{sigtota}$$ Here $\sigma_{0}$ is the unpolarized neutron total cross section, $P^i_n$ ($P^i_t$) is the $i$th projection of the beam (target) polarization axis, and $\Delta\sigma_L$ and $\Delta\sigma_T$ are the longitudinal and transverse total cross-section differences. The latter may be represented pictorially as[^3] $$\Delta\sigma_L=\sigma_{tot}(\stackrel{\textstyle\rightarrow}
\rightarrow)
-\sigma_{tot}(\stackrel {\textstyle\rightarrow}
\leftarrow) \label{arrowsL}$$ and $$\Delta\sigma_T=\sigma_{tot}(\uparrow\uparrow)
-\sigma_{tot}(\uparrow \downarrow). \label{arrowsT}$$ The measurements of $\Delta\sigma_L$ and $\Delta\sigma_T$ reported here were performed separately, with both beam and target polarized along axes longitudinal to the incident beam direction (i.e. along the $z$ axis) or transverse to the beam direction (i.e. along the $y$ axis). Hence in subsequent discussion, the superscripts used to indicate the polarization axes will be dropped.
An accurate determination of the $\Delta\sigma_L$ and $\Delta\sigma_T$ observables does not require an absolute measurement of the total cross section. Rather, the cross-section differences may be extracted from transmission measurements of polarized neutrons through the polarized target. The attenuation of an incident beam of $N_0$ polarized neutrons due to a polarized target of areal density $\tau$ will be $$N_{\pm}/N_0 = \exp \left[ - \tau(\sigma_0 \pm
\frac{1}{2}\Delta\sigma_L P_nP_t) \right]
\label{attenuateL}$$ when both the beam and target are longitudinally polarized, or $$N_{\pm}/N_0 = \exp \left[ - \tau(\sigma_0 \pm
\frac{1}{2}\Delta\sigma_T P_nP_t) \right]
\label{attenuateT}$$ when beam and target are polarized in the transverse direction. The $\pm$ signs are used to indicate whether the beam and target spins are polarized parallel ($+$) or antiparallel ($-$) to one another. By periodically reversing the spin of the beam (or target), one may observe an asymmetry in the attenuation of the beam. This neutron-transmission or spin-spin asymmetry is defined as $$\begin{aligned}
\varepsilon_{L,T} &=& \frac{N_{+} - N_{-}}{N_{+} + N_{-}}
\label{ntasym} \\
&=& \tanh \left[ -\frac{1}{2} \Delta\sigma_{L,T}
P_n P_t \tau \right] \nonumber \\
&\approx& -\frac{1}{2} \Delta\sigma_{L,T}
P_n P_t \tau. \nonumber\end{aligned}$$ Because the observed values of $\varepsilon$ are typically of order $10^{-3}$, the error introduced by replacing the hyperbolic tangent with its argument in Eq. (\[ntasym\]) is negligible. In writing Eq. (\[ntasym\]) we have assumed that the incident neutron flux $N_0$ is unaffected by reversing the spin.
Because they are directly related to the forward scattering amplitude via the optical theorem, $\sigma_0$, $\Delta\sigma_{L}$, and $\Delta\sigma_T$ are not sensitive to interference effects between the various partial waves. In principle this allows for a much simpler interpretation of the scattering and reaction processes. The $n$-$^3$He total cross sections can be expressed as linear sums of “partial-wave” cross sections $\sigma(J,l,s,l',s')$, where $$\sigma(J,l,s,l',s') = {\rm Re}\left\{\frac{\pi}{2k^2}
(2J+1)
\left[\delta_{ll'}\delta_{ss'} -
S^{J}_{ll'ss'}\right]\right\}.
\label{partxsect}$$ Here $J,l,s$ ($J,l',s'$) are the incoming (outgoing) total, orbital, and channel-spin angular momenta, $k$ is the neutron center-of-mass wave number, and $S^{J}_{ll'ss'}$ is the elastic scattering matrix element that describes transitions from the initial neutron channel ($J,l,s$) to the final neutron channel ($J,l',s'$). For partial waves up to $l=1$, we find $$\begin{aligned}
\sigma_0 &=& \sigma(^1\!S_0) + \sigma(^3\!S_1) +
\sigma(^1\!P_1) \nonumber\\
& & + \sigma(^3\!P_0) + \sigma(^3\!P_1) + \sigma(^3\!P_2),
\label{sig0}\end{aligned}$$ $$\begin{aligned}
\Delta\sigma_L &=& -2\sigma(^1\!S_0) + \frac{2}{3}\sigma(^3\!S_1)
- 2\sigma(^1\!P_1) \nonumber \\
& & -2\sigma(^3\!P_0) + 2\sigma(^3\!P_1) +
\frac{2}{5}\sigma(^3\!P_2) \nonumber \\
& & + \frac{4}{3}\sqrt{2}\sigma(^3\!S_1-^3\!\!D_1),
\label{DsigL}\end{aligned}$$ and $$\begin{aligned}
\Delta\sigma_T &=& -2\sigma(^1\!S_0) + \frac{2}{3}\sigma(^3\!S_1)
- 2\sigma(^1\!P_1) \nonumber \\
& & +2\sigma(^3\!P_0) + \frac{4}{5}\sigma(^3\!P_2)
- \frac{2}{3}\sqrt{2}\sigma(^3\!S_1-^3\!\!D_1).
\label{DsigT}\end{aligned}$$ A negative coefficient simply implies that the cross section for this particular wave is greater when the beam and target spins are antiparallel to one another.
Detailed predictions of the total cross-section differences $\Delta\sigma_L$ and $\Delta\sigma_T$ for $\vec{n}$-$\vec{^3{\rm He}}$ scattering were presented in an earlier paper [@Kei94]. These calculations were based on three separate analyses of $n$-$^3$He scattering and reaction data [@Hal89; @Jan88; @Lis75], as well as a microscopic resonating-group model calculation of the 4N excited states [@Hof93]. It was observed that, although the different sets of phase shifts provide adequate descriptions of pre-existing data, they predict quantitatively different values of both $\Delta\sigma_L$ and $\Delta\sigma_T$ in the present region of interest. In most instances these discrepancies could be attributed to one or two partial waves. Based on this observation, we concluded that comprehensive measurements of both $\Delta\sigma_L$ and $\Delta\sigma_T$, in combination with the unpolarized neutron total cross section $\sigma_0$, could be used to extract specific information about the partial-wave content of the resonating $^4$He compound nucleus.
Experimental Apparatus
======================
Polarized $^3$H[e]{} Target {#sec:target}
---------------------------
A brief description of the polarized solid $^3$He target is given below. The target is described in greater detail in Ref. [@Kei95].
Owing to the low intensity of the polarized neutron beam, an extremely thick sample of polarized $^3$He is desirable. In a measurement of the neutron transmission asymmetry $\varepsilon$, the number of observed neutron counts $N$ necessary to obtain a statistical precision $\Delta\varepsilon/\varepsilon$ is $$N = \frac{1}{2}[\frac{\Delta \varepsilon}{\varepsilon}
\Delta\sigma P_n P_t \tau]^{-2}. \label{precision} \\$$ The factor of one-half means that $N$ counts are needed in both the [*up*]{} and [*down*]{} spin states. This indicates that, for given values of $P_n$ and $\Delta\sigma$, the figure of merit for comparing polarized targets in transmission experiments should be $\tau^2 \: P_t^2$. The figure of merit for the TUNL solid $^3$He target exceeds current polarized $^3$He gas targets by nearly two orders of magnitude. Furthermore, the densities of the condensed phases of $^3$He correspond to nearly 100 MPa of room temperature gas, while targets of polarized $^3$He gas are limited to 1 MPa.
The liquid phase of $^3$He behaves as a Fermi liquid and can not be polarized to any great extent. On the other hand, solid $^3$He is a nuclear paramagnet and can be polarized by the static or “brute-force” method; the sample is cooled to a very low ($\sim 10$ mK) temperature in the presence of an externally-applied magnetic field ($\sim 7$ T). The resulting polarization for the body-centered cubic (bcc) phase at temperature $T$ and field $B$ is given by the Brillouin expression $$P_t = \tanh\left[\frac{1}{k_B T}(\mu B + \Theta P_t +
KP_t^3)\right]. \label{Brill}$$ where $\mu=-2.13 \mu_{\rm N}$ is the magnetic moment of the $^3$He nucleus, and $k_B$ is Boltzmann’s constant. The quantities $\Theta$ and $K$ are corrections to the Curie law of paramagnetism and describe the anti-ferromagnetic exchange of neighboring $^3$He atoms in the bcc lattice. Therefore, the actual polarization of solid $^3$He is slightly lower than that calculated assuming simple paramagnetic behavior. Values for these corrections ($\Theta/k_B = -1.18$ mK and $K/k_B=-1.96$ mK) were determined by fitting the observed low-temperature properties of bcc solid $^3$He [@Sti85a].
For the measurements reported in this paper, a $^3$He-$^4$He dilution refrigerator was used to cool the sample to approximately 12 mK in an externally-applied magnetic field of 7 T. The field was provided by a superconducting split-coil magnet operated in persistent-current mode. The magnet was physically rotated to provide fields either parallel (longitudinal) or perpendicular (transverse) to the incident beam direction. The lowest target temperature obtained during these measurements was $11.9 \pm 0.2$ mK, corresponding to $38.7 \pm 0.6$% polarization.
The sample cell for the target is shown in Fig. \[fig:target\]. The cylindrical container was constructed primarily of beryllium copper (BeCu) with four flat surfaces machined from the cylinder. The flats reduced the amount of material to be cooled and minimized the attenuation of the neutron beam due to BeCu. The wall thickness at the flats was 1.27 mm perpendicular to the beam and 2.54 mm parallel to the beam. The sample cell was thermally anchored to the dilution refrigerator’s mixing chamber by a 45 cm long OFHC copper cold finger.
The cell was filled with $^3$He through a 0.75 mm I.D. stainless steel tube hard-soldered into the top of the cell. Cupro-nickel capillary (0.1 mm I.D.) connected the fill tube to a room temperature gas-handling system. The interior sample space was a rectangular parallelepiped with dimensions $38.1 \times 14.0 \times 21.6$ mm and was filled with 3 micron silver powder packed to 19% of the density of solid silver. The powder was used to provide good thermal contact between the solid $^3$He and the BeCu cell, ensuring a homogeneous temperature throughout the target.
The solid $^3$He sample was grown by first filling the cell with liquid $^3$He at approximately 3 K. The liquid was then compressed to a density of 0.125 g/cm$^3$ by increasing the $^3$He vapor pressure to 3.6 MPa. At this density solid began to form at 1.1 K, and the sample was completely solidified at 0.83 K [@Gri71]. With the silver powder in place, the thickness of the solid $^3$He sample was $4.34\pm 0.09 \times 10^{22}$ atoms/cm$^2$.
The target polarization was extracted from the temperature of the BeCu sample cell, as measured by two independent thermometric standards: a $^{60}$Co nuclear orientation thermometer [@Mar83] and a $^3$He melting curve thermometer (MCT) [@Kei92]. The nuclear orientation thermometry required an intrinsic germanium detector to observe the 1.17 MeV and 1.33 MeV $\gamma$ rays from $^{60}$Co. To avoid radiation damage from neutrons however, the detector had to be removed from the experimental hall whenever beam was on target. Therefore the $^{60}$Co measurements were made immediately before and after each neutron asymmetry measurement. The melting curve thermometer on the other hand, could be used throughout the neutron measurements. The output of the MCT was read directly into the data acquisition computer and the temperature sampled every 100 ms. The average polarization of the target for a particular asymmetry measurement was determined from the average MCT temperature during that time. With no beam on target the $^{60}$Co and MCT were found to agree within $\pm$2%, and although the MCT could resolve temperature changes as small as one microkelvin, no significant warming due to neutron or $\gamma$-ray interactions within the target was observed.
The Polarized Beam {#sec:beam}
------------------
### Neutron Production and Detection
Polarized neutrons were produced as secondary beams from either the $^3$H($\vec p$,$\vec n$)$^3$He or $^2$H($\vec d$,$\vec n$)$^3$He polarization-transfer reactions at $0^{\circ}$. The $^3$H($\vec p$,$\vec n$)$^3$He reaction was used to produce neutrons with energy less than 4 MeV because it has a negative Q-value, -0.764 MeV. However, safety considerations limited the maximum amount of tritium that could be used, and the resulting neutron fluxes were low. The $^2$H($\vec d$,$\vec n$)$^3$He reaction (Q=3.269 MeV) was used at higher energies, and the neutron fluxes here were typically 10–20 times greater.
The polarized charged-particle beams were produced by the TUNL atomic beam polarized ion source [@Cle95a; @Cle95b; @Din95] and accelerated by a tandem Van de Graaff. The ion source produced a polarized beam whose spin axis was parallel to its momentum. A Wien filter located between the source and accelerator was used to rotate the polarized beam’s spin axis to the desired orientation, longitudinal or transverse, at the neutron production target. The position of the beam was feedback-stabilized in both the horizontal and vertical planes by four sets of steering magnets. Computer-controlled steering was used to maintain the beam position at the center of a rotating-wire scanner installed inside the beam pipe approximately 2 m from the neutron-production target.
The $^3$H($\vec p$,$\vec n$)$^3$He neutron production target was a tritiated-titanium foil, backed by a 0.51 mm thick copper disk. A 0.1 MPa $^4$He gas cell, with a 2.54 $\mu$m Havar entrance window, surrounded the tritiated foil to prevent contamination of the beam line. The $^2$H($\vec d$,$\vec n$)$^3$He neutron-production target was a deuterium gas cell, 60 mm long, 19 mm in diameter and operated at a D$_2$ pressure of 0.4 MPa. The Havar window for this cell was 6.35 $\mu$m thick, and the deuteron beam was stopped by a 0.51 mm tantalum disk. Both neutron-production targets were air cooled. To eliminate the deflection of the charged particles due to the superconducting magnet, the last 1.2 m of beam pipe was constructed of soft iron and lined with a high permeability iron-nickel alloy.
The neutron-production targets were located as close as possible to the polarized target. Neutron collimation and detector shielding were accomplished by a combination of copper and polyethylene as shown in Fig. \[fig:poltar\]. The copper preshield located between the neutron production target and polarized target reduced the number of neutrons striking the superconducting magnet. The polyethylene collimation system located after the polarized target defined a 25.7 $\times$ 9.4 mm beam spot at the center of the polarized target, corresponding to a solid angle of approximately 0.5 msr.
Neutrons that were transmitted through the polarized target were detected by two liquid scintillators located at $0^{\circ}$ and surrounded by a polyethylene shield. The scintillation liquid (BC501) was contained in two cylindrical aluminum containers (127 mm diameter, 127 mm long), each with an optically transparent endcap coupled to a 127 mm diameter photomultiplier tube. The cylinders were placed one atop the other, with their axes, as well the photomultiplier tubes, pointing in the vertical direction. Pulse-shape discrimination (PSD) was performed on the phototube anode signals to distinguish neutron events from $\gamma$ rays. The PSD was performed by commercially-manufactured modules [@PSD95], with pulse-height thresholds set to discriminate against low-energy neutrons. Valid neutron events were counted in scalers, and stored in the computer at set intervals.
The collimation/detection system was tested in two ways. First, the alignment of the target and collimator was verified by exposing x-ray films to the gammas produced by the charged-particle beam. This showed that the target completely filled the acceptance angle of the collimator. Second, neutron time-of-flight measurements were performed with a pulsed beam at 10 MeV. The time-of-flight spectrum showed that only neutrons of the correct energy were being counted. In addition, blocking the exit of the collimator with 30 cm of tungsten followed by 30 cm of polyethylene reduced the neutron count rate by a factor of $10^3$, indicating that the detectors were adequately shielded from energetic background neutrons.
The neutron transmission asymmetries were observed by reversing the spin of the charged-particle beam every 100 ms. For an accurate measurement it is necessary to know the ratio of the neutron fluxes produced by each spin state of the charged-particle beam. For the $^3$H($\vec p$,$\vec n$)$^3$He reaction, the neutron yield is proportional to the proton beam current, and it proved sufficient to count the digitized beam current in each spin state, using the ratio to normalize the neutron fluxes. Such normalization does not work for the $^2$H($\vec d$,$\vec n$)$^3$He reaction because here the yield depends on the tensor polarization $P_{zz}$ of the deuteron beam as well as on the beam intensity. The polarized source was operated in a manner such that, ideally, $P_{zz}$ remained constant while the vector polarization was completely reversed. In practice, we determined that the tensor component changed by as much as a few percent when the deuteron spin was flipped. To monitor the flux more reliably, we placed a third liquid scintillator at $0^{\circ}$, between the copper preshield and the $\vec{^3{\rm He}}$ target. This monitor detector was used to normalize the number of neutrons in the [*up*]{} and [*down*]{} spin states to the same incident flux. Because of its close proximity to the superconducting magnet, the monitor detector was optically coupled to a 51 mm diameter phototube by a 1 m long light pipe. Due to its small dimensions ($25.4 \times11.1 \times 22.2$ mm), $\gamma$ rays did not deposit much energy in this detector and could be separated from the neutron events by pulse height alone, without the need for pulse-shape discrimination.
Corrections to the incident-flux normalization described above are discussed in Sec. \[sec:background\].
### Polarization of the neutron beam
The polarization of the charged-particle beam was measured with a carbon-foil polarimeter. The polarization of the neutron beam was then calculated from the polarization of the charged particles, using known polarization-transfer coefficients. The polarimeter consisted of a thin (5 $\mu$m/cm$^2$) 22 mm diameter carbon foil located at the center of a small scattering chamber. Two silicon surface-barrier detectors detected protons from the $^{12}$C($\vec{p},p$)$^{12}$C reaction, or the $^{12}$C($\vec{d},p_0$)$^{13}$C reaction. The detectors were located at $\pm 40^{\circ}$, and a tantalum collimation system defined a $\pm3.5^{\circ}$ angular acceptance for each. The carbon foil was mounted to an aluminum plunger inside the polarimeter and was removed from the beam path when not in use.
To determine the polarization of the proton or deuteron beam, a left-right asymmetry was measured between the two silicon detectors. The asymmetry was measured for both the [*up*]{} and [*down*]{} spin states and the difference taken to cancel systematic effects. The average beam polarization was calculated on the basis of this average polarimeter asymmetry, $\varepsilon_{pol}$: $$\varepsilon_{pol} = \frac{1}{2}\left[
\frac{L^+ - R^+}{L^+ + R^+} - \frac{L^- - R^-}{L^- + R^-}
\right]. \label{polasy}$$
When polarized protons were used to produce neutrons, the polarimeter measured a left-right asymmetry for the elastic scattering of protons from the carbon foil. Published values [@Mos65; @Ter68] of the $^{12}$C($\vec{p},p$)$^{12}$C analyzing power $A_y$ were then used to calculate the average neutron polarization, $$P_{n}=\frac{\varepsilon_{pol} C_1 K^{y'}_{y}}{A_{y}}
\label{neutpol} \\$$ Here $K^{y'}_{y}$ (or $K^{z'}_z$ in the case of a longitudinally polarized beam) is the polarization-transfer coefficient for the $^3$H($\vec p$,$\vec n$)$^3$He reaction, and $C_1$ is a correction term that describes the depolarization of the neutron beam as it passes through the field of the 7 T superconducting magnet. The effect is small ($C_1 = 0.978$ at 1.94 MeV and $C_1=0.984$ at 3.65 MeV) because the dominant field component is parallel to the neutron spin axis. The values of $C_1$ were determined from a detailed calculation of the magnetic field [@Wil95].
At the lowest proton energy, $E_p = 3.0$ MeV, the $^{12}$C($\vec{p},p$)$^{12}$C analyzing power was too small to be useful as a polarization monitor. Therefore all measurements of the proton polarization were made at $E_p = 4.7$ MeV.
When deuterons were used to produce the polarized neutron beam, $P_n$ was again determined from the polarimeter asymmetry, $$P_n = \frac{C_1 \varepsilon_{pol}}{A_{ef\!f}}. \label{effanal}$$ Here $A_{ef\!f}$ is an “effective” analyzing power that relates the polarimeter asymmetry measured for the $^{12}$C($\vec{d},p_0$)$^{13}$C reaction to the resulting neutron polarization from the $^2$H($\vec d$,$\vec n$)$^3$He reaction. This effective analyzing power was measured in a separate experiment with a neutron polarimeter consisting of a $^4$He scatterer. In this experiment the $^{12}$C($\vec{d},p_0$)$^{13}$C asymmetries were calibrated against known $\vec{n}$-$^4$He analyzing powers [@Tor74]. The correction factor due to the superconducting magnetic field, $C_1$, has been described above, with $C_1=0.987$ at 4.95 MeV and $C_1=0.990$ at 7.46 MeV.
During the measurements of $\Delta\sigma_T$, the polarimeter asymmetries were measured approximately every 2–3 hours. Under normal operation of the polarized ion source, we found that the polarimeter asymmetries stayed constant (within experimental uncertainties) during the course of several days. During the $\Delta\sigma_L$ measurements, the proton and deuteron polarizations could be measured only after the Wien filter was used to rotate their spins perpendicular to the beam. This involved retuning the beam optics, and so the measurements were performed only twice at each beam energy: immediately before and immediately after the longitudinal neutron-transmission asymmetry was measured.
Experimental Procedure and Data Analysis {#sec:procedure}
========================================
Measurement of Spin-Spin Asymmetries {#sec:spinspin}
------------------------------------
Cooling of the solid $^3$He target commenced approximately 24 hours before measurements of the neutron-transmission asymmetry. During this time both the $^{60}$Co nuclear orientation and $^3$He melting curve thermometers were used to monitor the target temperature. After the target reached a temperature of 15 mK, the germanium detector for the $^{60}$Co thermometer was removed from the experimental hall and the neutron measurements began.
The spin of the neutron beam was reversed every 100 ms by toggling radio-frequency transition units at the polarized ion source. The spins were flipped according to an eight-step sequence (${}+{}-{}-{}+{}-{}+{}+{}-{}$) to minimize effects which arise from drifts in detector efficiency that are linear or quadratic in time. Typical count rates encountered during these measurements were between $10^4$ s$^{-1}$ when the $^2$H($\vec d$,$\vec n$)$^3$He reaction was used and 400 s$^{-1}$ with the $^3$H($\vec p$,$\vec n$)$^3$He reaction.
The data consisted of CAMAC scaler counts of neutron events, digitized charged-particle beam current, digitized polarized target temperature, and events of a 100 kHz dead-time pulser. At the end of each eight-step sequence, a count-down scaler was decremented from its preset value of 1024. Data were stored in the computer buffer as spectra of scaler counts versus time, each eight-step sequence comprising one channel of the various spectra. When the count-down scaler reached zero, acquisition was inhibited, and the data were written to the computer disk. All spectra were then cleared, the count-down scaler reset to 1024, and data acquisition recommenced. A “run” therefore consisted of 1024 eight-step sequences and required about fifteen minutes of beam time. At each energy the data were collected for about twelve hours with a polarized target and for an equal amount of time with an unpolarized target.
Two spectra were allocated for each observable of interest, one for neutron-spin up (parallel to target spin), the other for neutron-spin down (antiparallel to target spin). Acquisition into the spin-up spectra was inhibited during the spin-down portions of each eight-step sequence and vice versa. All data acquisition was inhibited 2 ms prior to and 5 ms after each spin flip to give the beam polarization time to stabilize. Acquisition was also halted whenever the beam current fell above or below prescribed limits. To ensure that equal time was spent in both the up- and down-spin states, the entire eight-step sequence during which the beam current had fallen outside its limits was rejected in the final data analysis. These occurrences were easily observed in the dead-time pulser spectra. In all, less than 1% of the data were rejected for this reason.
Transmission asymmetries for the two main neutron detectors were calculated for each eight-step sequence according to $$\varepsilon = \frac{\tilde{N}_+ - \tilde{N}_-}
{\tilde{N}_+ + \tilde{N}_-}.
\label{ntasym2}$$ Here $\tilde{N}_{\pm}$ is the number of dead time-corrected neutron counts in each spin state normalized to the incident neutron flux, $$\tilde{N}_{\pm} = \frac{N_{\pm}}{I_{\pm}}.
\label{normcounts}$$ The normalization factor $I$ is either the proton beam current or the yield in the neutron monitor detector. The transmission asymmetries for all eight-step sequences were combined in a weighted average for both the top and the bottom neutron detectors. These two results were then combined to give the average neutron transmission asymmetry $\bar{\varepsilon}$ and its associated statistical uncertainty. A standard deviation was calculated for each set of data, to compare with the standard deviations expected from Poisson counting statistics.
Measurement of Background Asymmetries {#sec:background}
-------------------------------------
After each measurement of the transmission asymmetry, the solid $^3$He target was melted and warmed to 1 K, and the neutron transmission measurements were repeated. The superconducting magnet continued to operate in persistent-current mode. The polarization of the liquid phase at 1 K was less than 0.5%, while its density was 8% less than the density of the solid. These measurements were performed to determine how much of the observed neutron transmission asymmetry was due to effects other than spin-dependent forces between the polarized neutron beam and polarized target nuclei.
While these background, or “warm”, asymmetries were typically an order of magnitude lower than the spin-spin asymmetries, they were, in general, non-zero and were subtracted from the spin-spin, or “cold” measurements. As discussed below, the asymmetries observed during the warm measurements were due to polarization effects associated with the incident charged-particle beams. As long as the beam polarization remained constant during the warm and cold asymmetry measurements, the background asymmetry was the same in both measurements and so the warm asymmetry could simply be subtracted from the cold. If the polarization differed between the two measurements, a correction based on the two polarizations had to be made. Thus, proper correction for the background asymmetries required some understanding of their origin.
When the $^3$H($\vec p$,$\vec n$)$^3$He source reaction was used, the incident proton current provided the normalization factor. However, if imperfect alignment exists between the proton beam and neutron collimation, the vector analyzing power of this reaction will produce a non-zero asymmetry in incident neutron flux that is not eliminated by the beam-current normalization. The vector analyzing power for this reaction vanishes at $0^{\circ}$ and thus the observed background asymmetries (and the subsequent corrections) are small. Since the asymmetry produced by the vector analyzing power is proportional to the polarization of the proton beam $P_p$, the background asymmetry observed during the warm measurement is scaled to the same value of $P_p$ that existed during the cold measurement. Therefore, $\Delta\sigma_T$ is extracted from the difference between the cold and warm asymmetries, with the latter scaled by $P_p$, $$\Delta\sigma_T= \frac{-2}{P_t P_n \tau}
\left[ \bar{\varepsilon}_c -
\frac{P_{pc}}{P_{pw}}\bar{\varepsilon}_w \right].
\label{deltasigT}$$ Here the subscripts $c$ and $w$ refer to the cold and warm measurements, and $P_n$ is the value of neutron polarization during the cold asymmetry measurement.
Since parity conservation forbids any longitudinal analyzing powers for the $^3$H($\vec p$,$\vec n$)$^3$He reaction, no background correction should be necessary for the low-energy longitudinal measurements. The warm measurement at 3.65 MeV was in fact consistent with zero. Thus $\Delta\sigma_L$ (at this energy) was determined from the cold asymmetry measurements alone, $$\Delta\sigma_L = \frac{-2}{P_t P_n \tau}
\bar{\varepsilon}_c.
\label{deltasigL}$$
In the case of the $^2$H($\vec d$,$\vec n$)$^3$He reaction we must consider two sources of background asymmetry. In addition to a vector analyzing power (which produces an asymmetry in the neutron yield at non-zero angles), this reaction possesses a tensor analyzing power that affects the neutron yield at $0^{\circ}$. If $I_0$ is the $0^{\circ}$ yield from a completely unpolarized deuteron beam, then the yield from a polarized beam, $I(0^{\circ})$, will be $$I(0^{\circ}) = I_0(1-\frac{1}{4}A_{zz}P_{zz}) \label{yieldT}$$ in the transverse geometry, or $$I(0^{\circ}) = I_0(1+\frac{1}{2}A_{zz}P_{zz}) \label{yieldL}$$ in the longitudinal geometry. Here $P_{zz}$ is the longitudinal tensor polarization of the deuteron beam and $A_{zz}$ is the tensor analyzing power for the $^2$H($\vec d$,$n$)$^3$He reaction. If there is a change in tensor polarization, $\Delta P_{zz}$, when the deuteron spin is flipped at the polarized ion source, then an asymmetry in the $0^{\circ}$ neutron yield will result.
To monitor the $0^{\circ}$ yield, a thin scintillator was placed between the $^3$He target and neutron production target (see Section \[sec:beam\]). However, the solid angle subtended by the monitor detector was slightly different from that of the main detector. This led to an asymmetry in the monitor-normalized neutron counts caused by either the vector or tensor analyzing power, or both. A vector analyzing power is parity-forbidden for a longitudinally-polarized deuteron beam, and during the transverse measurements we observed little change in the deuteron vector polarization. Therefore the only correction necessary for the $^2$H($\vec d$,$\vec n$)$^3$He measurements was one based on the [*tensor*]{} analyzing power. Since both the background asymmetry and the asymmetry observed by the monitor detector were proportional to $\Delta P_{zz}$, it proved convenient to use the monitor asymmetry (which was measured with a high degree of statistical accuracy) to scale the warm to cold asymmetry measurements. Thus, for the $^2$H($\vec d$,$\vec n$)$^3$He measurements, $$\Delta\sigma_{L,T}= \frac{-2}{P_t P_n \tau}
\left[ \bar{\varepsilon}_c -
\frac{\varepsilon_{mc}}{\varepsilon_{mw}}
\bar{\varepsilon}_w \right],
\label{deltasigL2}$$ where $\varepsilon_{mc}$ ($\varepsilon_{mw}$) are the cold (warm) monitor asymmetries.
According to Eqs. \[yieldT\] and \[yieldL\], the asymmetry resulting from a given value of $\Delta P_{zz}$ should be twice as large and of the opposite sign in the longitudinal geometry as in the transverse. This explains why the background asymmetries were typically larger (and of the opposite sign) during the high-energy measurements of $\Delta\sigma_L$ (Tables \[tab:transasym\] and \[tab:longasym\]).
Additional $\Delta\sigma_T$ measurements were made with a cold ($\sim15$ mK), empty sample container, a cold container filled with liquid $^3$He, and a warm empty container. Such measurements are sensitive to spin-spin effects caused by polarizable materials in the sample container other than $^3$He (e.g. copper). With the exception of the cold liquid measurements at 1.94 and 3.65 MeV, all such background measurements were consistent with the corresponding warm, unpolarized measurements. The asymmetries observed with the cold liquid target were in fact consistent with a $^3$He polarization of 3%, the expected polarization of $^3$He in the liquid phase at 12 mK [@Kei95; @Ram70]. At no time did we observe effects due to polarizable materials other than $^3$He. The only “background” measurement at 4.95 MeV was taken with a cold, empty target. The result here was consistent with zero background asymmetry.
Results {#sec:results}
=======
Transmission asymmetries were measured for the transverse spin geometry at neutron energies of 1.94, 3.65, 4.95 and 7.46 MeV. The results are given in Table \[tab:transasym\] which includes the corresponding values of the beam and target polarizations. The errors associated with these polarizations are typically $\Delta P_n/P_n = 6$% and $\Delta P_t/P_t = 2$%. The uncertainty in $P_n$ is dominated by the uncertainty in the $^{12}$C($\vec{p},p$)$^{12}$C and $^{12}$C($\vec{d},p_0$)$^{13}$C polarimeter analyzing powers. Results of the measurements conducted with the target cell filled with cold, liquid $^3$He (3% polarization), as well as an empty sample container at both warm (1 K) and cold ($\sim$15 mK) temperatures are also included in Table \[tab:transasym\].
Transmission asymmetries for the longitudinal geometry were measured at neutron energies 3.65, 4.95, and 7.46 MeV. A measurement at 1.94 MeV was not attempted because the longitudinal polarization-transfer coefficient for the $^3$H($\vec p$,$\vec n$)$^3$He reaction was expected to be too small to produce a useful asymmetry result [@Jar74]. The transmission asymmetries at the three higher energies are listed in Table \[tab:longasym\] along with their corresponding beam and target polarizations. Here again $\Delta P_n/P_n = 6$% and $\Delta P_t/P_t=2\%$. At all three energies the longitudinal asymmetries were considerably smaller than the corresponding transverse asymmetries because the neutron polarizations were lower. Not only did the polarized ion source produce lower charged-particle polarizations during the longitudinal measurements, but the longitudinal transfer coefficients $K^{z'}_z$ are typically smaller than their transverse counterparts $K^{y'}_y$.
The values of $\Delta\sigma_T$ and $\Delta\sigma_L$ extracted from the transmission asymmetries are given in Table \[tab:dsres\]. In all but one case, the background asymmetry was taken to be the warm, liquid measurement listed in either Table \[tab:transasym\] or Table \[tab:longasym\]. The cold empty measurement at 4.95 MeV is used for the background correction to $\Delta\sigma_T$ at that energy. Both statistical and systematic uncertainties are given. The former reflect the counting statistics associated with a measurement of the transmission asymmetry. The systematic uncertainties are based on uncertainties in beam and target polarizations, as well as target thickness.
Comparison to Phase-Shift Predictions {#sec:comparison}
=====================================
The $\Delta\sigma_T$ and $\Delta\sigma_L$ results are plotted in Figures \[fig:dstresults\] and \[fig:dslresults\], respectively. The error bars shown in the figures were obtained by adding the systematic and statistical uncertainties in quadrature. For completeness we include the unpolarized neutron total cross section $\sigma_0$ in Fig. \[fig:s0results\]. Experimental results are represented by the ENDF/B-VI polynomial fit (dash-dotted line) [@Hal91] to the data of [@Bat59; @Gou73; @Hae83]. Included in Figures \[fig:dstresults\]–\[fig:s0results\] are predictions of $\Delta\sigma_T$, $\Delta\sigma_L$, and $\sigma_0$ calculated using $n$-$^3$He phase shifts obtained from a variety of sources. Briefly, the phase shifts result from two published sets of partial-wave analyses (PWA) of $n$-$^3$He scattering data [@Jan88; @Lis75], a charge-independent $R$-matrix analysis of virtually all $A=4$ scattering and reaction data below excitation energies of 30 MeV [@Hal89], and the preliminary results of a microscopic, variational calculation of the $^4$He continuum [@Hof93]. The latter is a multi-channel resonating group model (MCRGM) calculation that uses a gaussian-parameterized version of the Bonn meson-exchange potential [@Kel89] as its input. The $\Delta\sigma_L$ and $\Delta\sigma_T$ calculations have been presented and discussed in greater detail in an earlier paper [@Kei94].
Three sets of phase shifts adequately reproduce $\sigma_0$, only the MCRGM values are clearly too low. The MCRGM phases also produce values of $\Delta\sigma_L$ and $\Delta\sigma_T$ that are significantly lower than experiment. All three cases can be attributed to insufficient $P$-wave amplitudes, especially the $^3\!P_2$ partial wave. Between neutron energies of 2 and 5 MeV, the $^3\!P_2$ wave is the dominant partial wave in all four sets of phase shifts, although there is considerable discrepancy as to its strength. The $^3\!P_2$ wave of both the MCRGM and the Lisowski PWA are nearly identical to one another, but they are considerably smaller than those of the $R$-matrix or Jany PWA analyses. Consequently these two sets of phase shifts predict the lowest values of both $\Delta\sigma_L$ and $\Delta\sigma_T$. To correctly reproduce $\sigma_0$, the Lisowski PWA compensates for its relatively small $^3\!P_2$ wave with unusually large $D$ waves, particularly $^1\!D_2$. Since spin-singlet states can only be formed when the beam and target spins are antiparallel to one another, Lisowski [*et al.’s*]{} large $^1\!D_2$ amplitude further lessen their predictions of $\Delta\sigma_L$ and $\Delta\sigma_T$.
On the other hand, the Jany PWA possesses the largest $^3\!P_2$ wave, ascribing over 60% of the total (unpolarized) cross section at 2 MeV to this particular wave. Likewise the Jany PWA predicts $\Delta\sigma_L$ and $\Delta\sigma_T$ values that are slightly higher than experiment.
The $R$-matrix phase shifts reproduce $\Delta\sigma_T$ at all four energies. The $R$-matrix prediction of $\Delta\sigma_L$ comes closest to the measured values, although it is higher than experiment at 3.65 MeV. We see from Fig. \[fig:s0results\] that the $R$-matrix also overpredicts the unpolarized total cross section $\sigma_0$ by nearly 200 mb at this energy. One possible explanation is the $^3\!P_1$ partial wave which, in the $R$-matrix analysis, is much larger at 3.65 MeV than in the other three analyses. While $\Delta\sigma_T$ is completely insensitive to this partial wave, $\Delta\sigma_L$ is extremely so. If the $R$-matrix 200 mb overprediction of $\sigma_0$ is completely attributed to the $^3\!P_1$ partial wave, it should likewise overpredict $\Delta\sigma_L$ by 400 mb (see Eq. \[DsigL\]). The experimental result of $\Delta\sigma_L$ at 3.65 MeV is consistent with this conclusion.
The primary sources of the $^3\!P_1$ partial wave are a pair of $1^-$ resonances at 23.6 MeV ($T=1$) and 24.2 MeV ($T=0$). According to the $R$-matrix analysis, both of these excited states are predominately spin-triplet in the nucleon-trinucleon channels. The $\Delta\sigma_L$ result at 3.65 MeV, in conjunction with the unpolarized neutron total cross section at that energy, may indicate that the resonance parameters associated with one or both of the $1^-$ levels are in need of slight adjustment.
Summary and Conclusions {#sec:conclusion}
=======================
We have reported measurements of the polarized neutron—polarized $^3$He total cross section. A cryogenically-polarized target consisting of nearly 1/2 mole of solid $^3$He has been developed for these measurements. It is the largest sample of polarized $^3$He yet utilized in a nuclear physics experiment. It is particularly well suited for neutral beams such as neutrons or (real) photons, where the sources of beam-related heating are minimal.
Measurements of the longitudinal and transverse total cross-section differences $\Delta\sigma_L$ and $\Delta\sigma_T$ were performed for incident neutron energies 2–8 MeV. The results are reproduced by phase shifts obtained in a recent $R$-matrix analysis of $A=4$ scattering and reaction data. As such they provide additional support to the $^4$He level scheme resulting from that analysis. However, the measurement of $\Delta\sigma_L$ at 3.65 MeV, in conjunction with the unpolarized neutron total cross section at that energy, may indicate that a modification of the $R$-matrix $^3\!P_1$ partial wave is necessary.
None of the other three sets of phase shifts considered here are able to reproduce both the present data and previous measurements of the unpolarized neutron total cross section. In all instances we are able to trace the discrepancies to only one or two partial waves. In particular, we find clear evidence that the $^1\!D_2$ phase shift reported by Lisowski [*et al.*]{} is too large.
In the future we plan to extend the present measurements to lower energies where the number of partial waves involved in the scattering and reaction processes is limited to two or three. In such circumstances it is possible to uniquely extract all pertinent phase-shift information.
Acknowledgements
================
This work was supported in part by the US Department of Energy, Office of High Energy and Nuclear Physics, under contracts DE-FG05-88-ER40441 and DE-FG05-91-ER40619.
$E_n$ (MeV) Target Cell $P_t$ $P_n$ $\bar{\varepsilon} $(10$^{-4}$) $\sigma_{\bar{\varepsilon}}$ (10$^{-4}$) $\varepsilon_m$ (10$^{-4}$)
------------- ------------- ------- ------- --------------------------------- ------------------------------------------ -----------------------------
1.94 cold solid 0.365 0.482 $-42.94\pm1.72$ 1.86 —
cold liquid 0.029 0.482 $-5.12\pm1.90$ 2.62 —
warm liquid 0.000 0.482 $3.15\pm2.15$ 2.34 —
3.65 cold solid 0.351 0.530 $-45.52\pm2.16$ 2.50 —
cold liquid 0.029 0.500 $-5.90\pm2.27$ 2.58 —
warm liquid 0.000 0.492 $0.66\pm2.09$ 2.40 —
4.95 cold solid 0.307 0.521 $-30.22\pm1.05$ 1.04 $68.33\pm0.54$
cold empty 0.000 0.521 $-0.98\pm0.98$ 0.97 $58.49\pm0.53$
7.46 cold solid 0.345 0.632 $-23.57\pm0.64$ 0.65 $51.71\pm0.40$
cold empty 0.000 0.639 $-2.51\pm0.57$ 0.57 $54.66\pm0.35$
warm empty 0.000 0.634 $-3.76\pm0.64$ 0.64 $66.81\pm0.39$
warm liquid 0.000 0.622 $-3.17\pm0.59$ 0.59 $54.73\pm0.33$
: Results of transverse neutron-transmission asymmetries. Here $P_t$ and $P_n$ are the $^3$He and neutron polarizations, respectively. The uncertainty quoted for each transmission asymmetry $\bar{\varepsilon}$ reflects counting statistics, while $\sigma_{\bar{\varepsilon}}$ is the reduced standard deviation for all the eight-step sequences corresponding to a measurement of $\bar{\varepsilon}$. In the case of the $^2$H($\vec d$,$\vec n$)$^3$He measurements, $\varepsilon_m$ is the asymmetry observed in the neutron monitor detector.[]{data-label="tab:transasym"}
$E_n$ (MeV) Target Cell $P_t$ $P_n$ $\bar{\varepsilon} $(10$^{-4}$) $\sigma_{\bar{\varepsilon}}$ (10$^{-4}$) $\varepsilon_m$ (10$^{-4}$)
------------- ------------- ------- ------- --------------------------------- ------------------------------------------ -----------------------------
3.65 cold solid 0.351 0.263 $-8.65\pm1.89$ 1.91 —
warm liquid 0.000 0.265 $1.81\pm1.92$ 1.95 —
4.95 cold solid 0.352 0.334 $-18.81\pm1.17$ 1.19 $263.4\pm0.87$
warm liquid 0.000 0.372 $2.16\pm1.19$ 1.22 $250.2\pm0.75$
7.46 cold solid 0.343 0.404 $-6.34\pm0.60$ 0.60 $127.0\pm0.48$
warm liquid 0.000 0.210 $4.37\pm0.62$ 0.63 $134.7\pm0.47$
: Results of longitudinal neutron-transmission asymmetries.[]{data-label="tab:longasym"}
-----------------------------------------------------------------------------------------------------------------------
$E_{n}$ (MeV) $\Delta\sigma_{L}$ (b) $\Delta\sigma_{T}$ (b)
--------------- --------------------------------------------------- ---------------------------------------------------
1.94 — $\hspace{1.25em}1.207\pm0.092
\pm0.078$
3.65 $\hspace{1.25em}0.432\pm0.044 $\hspace{1.25em}1.145\pm0.089
\pm0.095$ \pm0.089$
4.95 $\hspace{1.25em}0.806\pm0.068 $\hspace{1.25em}0.838\pm0.073
\pm0.067$ \pm0.044$
7.46 $\hspace{1.25em}0.348\pm0.041 $\hspace{1.25em}0.431\pm0.035
\pm0.028$ \pm0.018$
-----------------------------------------------------------------------------------------------------------------------
: Results of the $\Delta\sigma_L$ and $\Delta\sigma_T$ measurements. The first uncertainty is systematic, the second statistical.[]{data-label="tab:dsres"}
[10]{}
W. Gl[ö]{}ckle, H. Witala, H. Kamada, D. H[ü]{}ber, and J. Golak, in [*Proceedings of the XIV International Conference on Few Body Problems in Physics, Williamsburg, VA.*]{}, edited by F. Gross (American Institute of Physics, New York, 1995), pp. 45–67.
D. R. Tilley, H. R. Weller, and G. M. Hale, Nucl. Phys. [**A541**]{}, 1 (1992). This compilation article contains a complete reference guide to experimental and theoretical work on $A=4$ nuclei since 1973.
L. Passell and R. I. Schermer, Phys. Rev. [**150**]{}, 146 (1966).
M. T. Alley and L. D. Knutsen, Phys. Rev. C [**48**]{}, 1890 (1993).
C. D. Keith, C. R. Gould, D. G. Haase, G. M. Hale, H. M. Hofmann, H. Postma, N. R. Roberson, and W. Tornow, Phys. Rev. C [**50**]{}, 237 (1994).
C. D. Keith, C. R. Gould, D. G. Haase, P. R. Huffman, N. R. Roberson, M. L. Seely, W. Tornow, and W. S. Wilburn, in [*Proceedings of the XIV International Conference on Few Body Problems in Physics, Williamsburg, VA.*]{}, edited by F. Gross (American Institute of Physics, New York, 1995), pp. 439–442.
G. M. Hale, D. C. Dodder, and K. Witte, $A=4$ R-matrix analysis of 9/88 (unpublished).
P. Jany, W. Heeringa, H. O. Klages, B. Zeitnitz, and R. Garrett, Nucl. Phys. [**A483**]{}, 269 (1988).
P. W. Lisowski, R. L. Walter, C. E. Busch, and T. B. Clegg, Nucl. Phys. [ **A242**]{}, 298 (1975).
H. M. Hofmann, unpublished.
C. D. Keith, C. D. Gould, D. G. Haase, P. R. Huffman, N. R. Roberson, M. L. Seely, W. Tornow, and W. S. Wilburn, Nucl. Inst. and Meth. A [**357**]{}, 34 (1995).
H. L. Stipdonk and J. H. Hetherington, Phys. Rev. B [**31**]{}, 4684 (1985).
E. R. Grilly, J. Low. Temp. Phys. [**4**]{}, 615 (1971).
H. Marshak, J. of Res. of the NBS [**88**]{}, 175 (1983).
C. D. Keith, C. D. Gould, D. G. Haase, N. R. Roberson, W. Tornow, and W. S. Wilburn, Hyperfine Interactions [**75**]{}, 525 (1992).
T. B. Clegg, W. M. Hooke, E. R. Crosson, A. W. Lovette, H. L. Middleton, H. G. Pfutzner, and K. A. Sweeton, Nucl. Inst. and Meth. A [**357**]{}, 212 (1995).
T. B. Clegg [*et al.*]{}, Nucl. Inst. and Meth. A [**357**]{}, 200 (1995).
D. C. Dinge, T. B. Clegg, E. R. Crosson, and H. W. Lewis, Nucl. Inst. and Meth. A [**357**]{}, 195 (1995).
Link [A]{}nalytical [L]{}imited, Bucks, England.
S. J. Moss and W. Haeberli, Nucl. Phys. [**72**]{}, 417 (1965).
G. E. Terrell, M. F. Jahns, M. R. Kostoff, and E. M. Bernstein, Phys. Rev. [ **173**]{}, 931 (1968).
W. S. Wilburn, C. R. Gould, D. G. Haase, P. R. Huffman, C. D. Keith, N. R. Roberson, and W. Tornow, Phys. Rev. C [**52**]{}, 2351 (1995).
W. Tornow, Z. Physik [**266**]{}, 357 (1974).
H. Ramm, P. Pedroni, J. R. Thompson, and H. Meyer, J. Low. Temp. Phys. [**2**]{}, 539 (1970).
J. J. Jarmer, J. C. Martin, G. G. Ohlsen, G. C. Salzman, and J. E. Simmons, Phys. Lett. [**48B**]{}, 215 (1974).
G. Hale, D. Dodder, and P. Young, [*ENDF/B-VI Data File for $^{\it 3\!}$He*]{}, National Nuclear Data Center, Brookhaven National Laboratory, Upton, New York, 1991.
, Nucl. Phys. [**12**]{}, 291 (1959).
C. A. Goulding, P. Stoler, and J. D. Seagrave, Nucl. Phys. [**A215**]{}, 253 (1973).
B. Haesner, W. Heeringa, H. O. Klaages, F. K[ä]{}ppeler, G. Schmalz, P. Schwarz, J. Wilczynski, B. Zeitnitz, and H. Dobiasch, Phys. Rev. C [**28**]{}, 995 (1983).
H. Kellermann, H. M. Hofmann, and C. Elster, Few-Body Sys. [**7**]{}, 31 (1989).
[^1]: Present address: Indiana University Cyclotron Facility, Bloomington, IN 47408
[^2]: Present address: Physics Department, Harvard University, Cambridge, MA 02138
[^3]: Note that our definition of parallel minus antiparallel is exactly opposite to the convention commonly used for nucleon-nucleon scattering.
|
---
abstract: 'Computer vision systems for wood identification have the potential to empower both producer and consumer countries to combat illegal logging if they can be deployed effectively in the field. In this work, carried out as part of an active international partnership with the support of UNIDO, we constructed and curated a field-relevant image data set to train a classifier for wood identification of $15$ commercial Ghanaian woods using the XyloTron system. We tested model performance in the laboratory, and then collected real-world field performance data across multiple sites using multiple XyloTron devices. We present efficacies of the trained model in the laboratory and in the field, discuss practical implications and challenges of deploying machine learning wood identification models, and conclude that field testing is a necessary step - and should be considered the gold-standard - for validating computer vision wood identification systems.'
author:
- |
Prabu Ravindran, ^1,\ 2^ Emmanuel Ebanyenle[^1], ^3^ Alberta Asi Ebeheakey, ^4^ Kofi Bonsu Abban, ^4^ Ophilious Lambog, ^4^ Richard Soares, ^1,\ 2^ Adriana Costa, ^1^ Alex C. Wiedenhoeft ^1,\ 2,\ 5,\ 6^\
^1^[Center for Wood Anatomy Research, USDA Forest Products Laboratory, USA]{}\
^2^[Department of Botany, University of Wisconsin, Madison, USA]{}\
^3^[Wood Anatomy Laboratory, CSIR - Forestry Research Institute of Ghana]{}\
^4^[Timber Industry Development Division, Forestry Commission, Ghana]{}\
^5^[Department of Forestry and Natural Resources, Purdue University, USA]{}\
^6^[Ciências Biolôgicas, Universidade Estadual Paulista – Botucatu, Brasil]{}
bibliography:
- 'egbib.bib'
title: 'Image Based Identification of Ghanaian Timbers Using the XyloTron: Opportunities, Risks and Challenges'
---
Introduction
============
Illegal logging contributes to deforestation and environmental degradation, supports organized crime networks, and its negative financial impact is valued between US$\$50-150$ billion [@Unep2012]. To combat this, there is growing global interest in enacting and enforcing laws (e.g. CITES, Lacey Act) intended to ensure that wood and wood-derived products are legally sourced. Compliance with and enforcement of international and local laws for legal wood products depend in part on the availability of technical or forensic expertise to validate claims of legality [@Wiedenhoeft2019]. Such expertise in turn hinges on the design, validation, and deployment of robust scientific wood forensic pipelines to identify timber and combat fraud throughout the supply chain [@LoweSasaki2016].
In a typical scenario, adversarial operators falsify paperwork claiming that the wood in a consignment is of lower value when in reality the consignment contains higher-value/endangered, (sometimes) superficially similar species. Verifying a consignment claim amounts to making a correct identification of the timber based solely on its inherent characteristics in the context of the claimed species. In field screening, the inspector must identify the wood in uncontrolled environmental conditions (e.g. at the point of harvest, in a lumber mill, at the harbor) in a matter of seconds to establish probable cause for seizure, detention, and further forensic analyses, or release the consignment into trade as compliant. In most jurisdictions, specimens from a detained consignment will be subjected to further forensic analysis in a laboratory using genetic/microscopy/spectral techniques to enable a legally valid identification [@Dormontt2015].
The *de facto* state of the art in field screening of timber in most of the world is human-based, with effective inspectors requiring significant training and regular practice in the use of traditional wood anatomical identification methods. Such trained humans will typically restrict their inspection of a wood specimen to the knife-cut transverse surface (the end grain) in order to view the size, shape, abundance and relationships of the constituent cells using a hand lens. Due to the comparative dearth of such human expertise in most countries, field screening of timber, *if done at all*, more often relies on subjective features such as color or odor of the timber with no reference to anatomical features of the wood. The lack of sufficient human expertise compared to the demand for timber screening is a major bottleneck in ensuring legal timber trade, and has established the clear need for reliable field-deployable wood identification technologies [@Wiedenhoeft2019].
{width="85.00000%" height="0.15\textheight"}
Computer vision and machine learning are attractive technologies for the development of quick, reliable and field-deployable tools for field screening of wood [@HermansonWiedenhoeft2011]. Image-based identification of a wood specimen using field-collected images of the transverse surface of wood is similar to the well-studied problem of texture classification [@TexClassSurvey2018; @CimpoiDeepTexture2015]. An early work using machine learning for wood identification used handcrafted biometric measurements and descriptors with a multi-layer perceptron to distinguish between two species [@EstebanANN2009]. In [@Khalid2008] a wood identification system was developed using gray level co-occurence matrices [@Haralick1979] and multi-layer perceptrons. Local binary patterns were used to identify African timber species using microscopic images in [@AfricaMicroID2017]. Convolutional neural networks [@LeCun1989] were employed to automatically learn features for macroscopic wood species identification in [@Filho2014; @CostaRicaCutting2018] and were designed for laboratory settings. The work of [@Ravindran2018] uses transfer learning [@PanTransferLearnSurvey2010; @ZamirTaskonomy2018], with a pretrained VGG network [@Simonyan2014] to identify neotropical woods from the Meliaceae, the botanical family that includes the genuine mahoganies.
A critical aspect not addressed in prior literature on computer vision wood identification is “ground-truth” testing of real-world performance in the context of field deployment by the personnel responsible for adoption and application of the technology. Based on the variability of wood itself and the need to prepare wood for imaging in the field, it would be optimistic to the point of naivete to assert or assume that test data set performance would direcly translate to the real world. In this regard, computer vision wood identification suffers from many of the same constraints as trained human inspectors. In addition to practical concerns for preparing specimens, the system’s user interface and the mechanisms for reporting classification outcomes to the operator is expected to influence the adoption and utility of the technology. Developing a system and user interface that delivers the relevant and digestible granularity of information is central to the power of the technology. A tool must empower its user to perform their work more effectively, or to enable the user to take on new tasks not previously possible. To this end, it is critical for computer vision researchers to work with end users to ensure that all necessary - and no extraneous - detail is conveyed to the user in a format that is empowering rather than confusing or opaque.
In this paper we use transfer learning to train a ResNet [@HeResNet2015] based classifier for image-based, macroscopic identification of a subset of commercially important Ghanaian timbers for use in conjunction within the XyloTron system [@XyloScope]. Our data collection and model development was done as part of an active UNIDO-funded international partnership to improve timber tracking and timber forensics in the Ghanaian timber market using xylarium wood specimens from the US Forest Products Laboratory and the Forestry Research Institute of Ghana. The pilot study described here was tested both in the laboratory and in the field to yield valuable insights into the challenges to be encountered when scaling the number of taxa to be identified. To the best of our knowledge this is the first report of results of field testing a computer vision/machine learning model for wood identification.
Dataset
=======
Sample preparation and imaging
------------------------------
The transverse surfaces of $413$ xylarium specimens of $38$ species in $15$ genera of commercial interest in Ghana were prepared in order to make the anatomical features of the wood easier to visualize. The list of taxa used in the created dataset are listed in Table \[tab:comp\_split\].
Macroscopic images of the prepared surfaces were obtained using the XyloTron, a DIY, open-source macroscopic imaging system [@XyloScope]. Multiple *non-overlapping* images were captured from each specimen with the rays of the wood aligned vertically. Vertical ray alignment was for consistency with existing scientific image collections and for ease of human-mediated interventions. Exemplar images for the classes considered in this paper are shown in Figure \[fig:mosaic\].
{width="95.00000%" height="0.35\textheight"}
Data curation
-------------
The collected images were curated by a wood forensic expert and images showing atypical wood anatomy or misidentified wood specimens were removed from the dataset. This resulted in a total of $2187$ images. The class labels were assigned so that the species in Table \[tab:comp\_split\] would be identified/classified at the genus level. This genus-level granularity for the labels is consistent with capabilities of traditional wood anatomy based timber identification and the commercial demands of the timber market in Ghana. Additionally, in order to increase the number of exemplars per class, when needed we included images of macroscopically similar species of the same genus from outside Ghana. The species compositions for our $15$ classes are listed in Table \[tab:comp\_split\].
[|l|c|c|c|]{} Class & Species Composition &
------------------------
*Specimen counts*
*(train, valid, test)*
------------------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
&
------------------------
*Image counts*
*(train, valid, test)*
------------------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
\
Albizia &
--------------------
*A. adianthifolia*
*A. ferruginea*
*A. zygia*
--------------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (21, 5, 5) & (141, 30, 30)\
Canarium & *C. schweinfurthii* & (7, 2, 2) & (61, 13, 13)\
Ceiba & *C. pentandra* & (25, 6, 6) & (140, 30, 30)\
Celtis &
-----------------------
*C. adolfi-friderici*
*C. mildbraedii*
*C. zenkeri*
-----------------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (4, 3, 3) & (90, 16, 21)\
Chrysophyllum &
-------------------
*C. albidum*
*C. brieyi*
*C. fulvum*
*C. lacourtianum*
*C. perpulchrum*
*C. viridifolium*
-------------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (5, 1, 1) & (77, 16, 16)\
Daniellia &
--------------
*D. ogea*
*D. oliveri*
--------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (3, 2, 2) & (37, 12, 12)\
Entandrophragma &
------------------
*E. angolense*
*E. candollei*
*E. cylindricum*
*E. utile*
------------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (60, 14, 14) & (141, 30, 30)\
Khaya &
-------------------
*K. anthotheca*
*K. ivorensis*
*K. senegalensis*
-------------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (43, 9, 9) & (142, 30, 30)\
Lophira & *Lophira alata* & (5, 1, 1) & (60, 11, 12)\
Manilkara &
--------------------
*M. bidentata*
*M. elata*
*M. huberi*
*M. obovata*
*M. solimoesensis*
*M. zapotilla*
--------------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (44, 10, 10) & (142, 29, 29)\
Milicia &
--------------
*M. excelsa*
*M. regia*
--------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (18, 3, 3) & (141, 29, 29)\
Nesogordonia & *N. papaverifera* & (7, 1, 1) & (67, 15, 14)\
Terminalia &
----------------
*T. ivorensis*
*T. superba*
----------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (24, 5, 5) & (141, 30, 30)\
Tieghemella &
---------------
*T. africana*
*T. heckelii*
---------------
: Botanical species composition of the $15$ genus-level classes.[]{data-label="tab:comp_split"}
& (6, 2, 2) & (65, 13, 14)\
Triplochiton & *T. scleroxylon* & (9, 2, 2) & (90, 19, 19)\
Model training and deployment
=============================
Model architecture
------------------
{width="\textwidth"}
Our model comprises all the layers up to and including the final residual block from a ResNet34 [@HeResNet2015] network pre-trained on ImageNet [@Russakovsky2015]. To this ResNet backbone we added concatenated global max and average pooling layers and two Batchnorm [@IoffeSzegedyBN2015]-Dropout [@HintonDropout2012]-Linear (BDL) blocks. ReLU [@Nair2010] activation and a dropout probability of $0.25$ was used in the first BDL block. In the second BDL block a dropout probability of $0.5$ and a softmax activation with $15$ outputs was used. The CNN architecture is shown in Figure \[fig:arch\].
Data splits
-----------
We divided the dataset of $2187$ images into a (approximately) $70\%/15\%/15\%$ training/validation/testing split with every specimen contributing images to *exactly* one of the splits. The number of specimens per class and images per specimen were not constant, but varied according to the availability of correctly identified specimens and the cross-sectional (transverse) area of those specimens. We employed stratified sampling at the specimen level to generate the splits, the details of which are shown in Table \[tab:comp\_split\].
Training methodology
--------------------
The backbone ResNet layers were initialized with ImageNet [@Russakovsky2015] pretrained weights and He normal initialization [@HeInit2015] was used for the the custom top level layers. We use a typical two-stage transfer learning process to train our model. In the first stage, the ResNet backbone was used as a feature extractor [@Razavian2014] (i.e. layer weights frozen) and the custom top level was trained for $6$ epochs using the Adam optimizer. In the second stage of training we finetuned the weights of the entire network for $8$ epochs using Adam optimizer [@KingmaAdamOpt2014]. The number of epochs to run was estimated using the validation set.
During both the training stages we simultaneously annealed the learning rate and momentum as described in [@SmithDisciplined2018; @howard2018fastai]. For the first half of the annealing process, the learning rate was increased from $\alpha_{\mbox{\small min}}$ to $\alpha_{\mbox{\small max}}$ while simultaneously the momentum was decreased from $\beta_{\mbox{\small max}}$ to $\beta_{\mbox{\small min}}$. In the second half the learning rate was decreased while the momentum was increased. Cosine annealing was used throughout. For the first and second stages of training the value of $\alpha_{\mbox{\small max}}$ was set to $2e^{-2}$ and ${1e^{-5}}$ respectively and was estimated using the method in [@SmithCyclical2017]. The value of $\alpha_{\mbox{\small min}}$ was set to ${\alpha_{\mbox{\small max}}}/10$. The momentum parameters $\beta_{\mbox{\small min}}$ and $\beta_{\mbox{\small max}}$ were set to $0.85$ and $0.95$ respectively.
We extracted patches of size $2048 \times 768$ pixels and resized them to $512 \times 192$ pixels. The reason we extracted non-square patches was to ensure that the diagnostic wood anatomical growth transitions were maximally probable to be captured in every patch. Our data augmentation strategy included horizontal and vertical reflections, random rotations in the range $[-5, 5]$ degrees and cutout [@Cutout2017]. The architecture definition and training was implemented using PyTorch and scientific Python [@Scipy2014] on a NVIDIA Titan X GPU using a batch size of $16$.
Hardware and software for field deployment
------------------------------------------
Four XyloTrons, each consisting of a XyloScope [@XyloScope] for imaging and an off-the-shelf laptop for running inference, were deployed in Ghana for model evaluation. The user interface displayed the image being classified along with the top-$3$ prediction confidences and archetypal images for these predictions from the reference image set (Figure \[fig:mosaic\]). Displaying an archetypal image for the prediction alongside the unknown specimen allows the operator to incorporate an element of human validation into the process, and reinforce the operator’s knowledge of the anatomy of the woods in the model.
Results and discussion
======================
Laboratory testing results
--------------------------
The overall image-level top-$1$ accuracy of our model was $97\%$. In our experiments finetuning did not improve the performance of the model. The few incorrect predictions made by our model are consistent with kinds of errors that would be made by a trained human inspector using a hand lens to identify the timbers using the same anatomical detail available in our image data set. For example, ${3}\%$ of *Ceiba* test images were classified as *Triplochiton*, which is reasonable based on both wood anatomy (see Figure \[fig:mosaic\]) and their shared botanical family, the Malvaceae. Within the family Meliaceae, some images of *Entandrophragma* were classified as *Khaya*, which is also reasonable given the anatomical variability of these two woods. *Tieghemella* and *Chrysophyllum* both belong to the Sapotaceae, a family known for taxonomic lability and difficulty of separating species and even genera when observing a standing tree with bark and leaves. Given the similarities in wood anatomy and underlying botanical variation for these three predictions, the error rates are much lower than one would expect for anyone other than an expert forensic wood anatomist. We would like to emphasize that the XyloTron uses a single image of the transverse surface of an unknown specimen for prediction, whereas a trained human identifier would incorporate multiple fields of view and additional anatomical information from other surfaces of the unknown specimen.
Field testing results
---------------------
The model was deployed at three field locations in Ghana, and at the xylarium at the Forestry Research Institute of Ghana. Across these locations a total of $488$ specimens were evaluated using four different XyloTron units by multiple users. In this pilot evaluation of field deployment, the operators were asked to record and report the number of times each species was correctly identified and if incorrect what the prediction of the model was. The overall accuracy of the model in field testing was $72\%$. There were two broad types of misclassification - those images that were classified as other anatomically similar woods, and one class (Canarium) where the predictions are wrong and not clearly explainable by observable wood structure.
Discussion
----------
The generalizability of state-of-the-art ImageNet classifiers was recently studied [@Recht2018; @Recht2019] by testing the predictive performance of pretrained models on a new dataset with carefully chosen images that were similar to the original test dataset. It was shown that models based on the ResNet architecture had a $\sim10\%$ drop in predictive accuracy on the new dataset and the authors attributed the drop in accuracy (consistent across a broad range of models) to “*the models’ inability to generalize to slightly “harder” images than those found in the original test sets* [@Recht2019]“. In our field tests the model was evaluated in an unrestricted setting where the wood specimens were from trees of varying maturity (reported by the field testing personnel). Given this uncontrolled setting, it is likely that the distribution of anatomical characteristics in the field tested specimens was different from the laboratory testing dataset (collected in xylaria) and thus were ”harder" for the classifier. The generalizability of this approach across the developing world depends largely on access to sufficient reference specimens and commitment from local users and developers of the technology so that the final, deployed version reflects appropriate scientific, practical, and cultural factors to maximize the technology’s potential contribution to natural resource management. A necessary piece of this is the real-world field testing so that any differences in performance between the laboratory and the field are well characterized.
The gap between test data metrics and real-world performance should be viewed in light of the two common practices currently in place in the field: no testing of any kind, or subjective evaluation of unreliable features such as color and odor. Additionally, this disparity between overall test set accuracy and the field-testing accuracy is a sobering but informative result because it demonstrates how test data set accuracy metrics may not translate into field accuracy metrics and exposes the challenges that biological variability and data distribution mismatches between model training/testing and field deployment specimens can pose for machine learning based wood identification systems. We hope that this work raises awareness of the potential inadequacy of machine learning based wood identification systems that have not been “ground-truthed” in the field.
Challenges, risks and opportunities
===================================
The vibrant interplay of wood anatomy, computer vision/machine learning and law enforcement/compliance provides a slew of research challenges with opportunities to make real-world impacts in wood utilization, forest management and land-use policy, and conservation of biodiversity.
Scaling up
----------
A key opportunity in deploying an image-based machine learning wood identification tool in parts of the developing world is the ability to limit the number of classes (different woods) that the system must be trained to separate. For example, Ghana imports virtually no exotic timbers, thus a system trained to identify Ghanaian woods provides real-world value. Deploying a field-screening system in Ghana, that actively uses a timber tracking system, has been an ideal case study to showcase the potential of this approach. In net-importing (and typically developed) countries, a shipment of wood could come from anywhere in the world, and thus an effective system would be required to identify several hundreds, instead of tens, of woods.
Reference/baseline image data sets
----------------------------------
Compared to the global scale of illegal logging, there is a paucity of high quality wood image datasets, in part because there are comparatively few xylaria. Xylaria can be valuable sources of curated wood specimens, but the quality, size, breadth, and reliability of these collections are highly variable [@CostaRicaXyl2018]. Other options for acquiring wood specimens such as targeted field expeditions, active timber harvest sites, lumber mills, etc. may faithfully capture the current data distribution but can be logistically challenging to accomplish at scale. Clearly, a global, open-access database of large numbers of images of all woods would be ideal, but such an effort would remain limited by specimen access, funding, and expertise. It may be more important- and achievable- to establish a baseline wood image dataset to objectively measure domain-specific machine learning advances - *a la* MNIST of woods.
Machine learning models and computer vision hardware
----------------------------------------------------
We anticipate that advances in computer vision and machine learning will lead to continued improvements for timber screening for the the foreseeable future. Increased access to large datasets may provide the potential to develop more robust models that capture both overt and subtle variation in wood anatomy. Scaling up models to incorporate more classes can be a challenge and may require label space engineering using custom ontologies to handle similar wood anatomies and data scarcity challenges. Techniques like few-shot learning [@KochOneShot2015; @VinyalsOneShot2016; @ZemelFewShot2017] hold promise for wood discrimination models, especially for endangered, rare endemic, or species of emerging commercial interest where access to new specimens maybe limited. Cloud based deployment of timber screening technologies can be implemented to enable real-time expert mediated decisions [@Tay2017], but the sites and contexts where they can be reliably deployed are biased toward regions with reliable, high-speed network connectivity. Portable, self-contained systems that do not require real-time network access for cloud based processing, like ours, can be deployed in regions without reliable access to these resources. Regardless of the details of hardware and location of computation, objective evaluation of different models and hardware configurations is only meaningful if compared across a common data set, but even this may not capture factors that facilitate real-world adoption and implementation of the technology.
Context-aware implementation and evaluation
-------------------------------------------
Because compliance with or enforcement of legal logging laws is inherently a human-mediated endeavor that varies by jurisdiction, machine learning technologies can solve only as much of the problem as users are willing and able to adopt and apply the technologies. Developing user-interfaces that mediate access to the model results at the correct and useful level of detail for the application will be a central part of long-term adoption. In some cases, a user interface that provides simple yes/no results may be desirable, whereas in others the interface might provide guidance on an optimal testing scheme according to the uncertainties in predictions. Which user interface is most useful will depend on the details of the problem and the norms of local jurisprudence, and these factors should be taken into account when deploying and interpreting the efficacy of machine learning models.
The necessity of field testing
------------------------------
The central conclusion of our work is that field testing is necessary and should be the gold-standard by which computer vision wood identification systems must be evaluated. Any claim about efficacy not backed by field test data at best represents an optimistic projection, and at worst grossly overpromises and underdelivers, with the cost of a gap in performance falling disproportionately on the implementing country. To realize the potential of computer vision (or any other technique) to combat illegal logging, it is essential to direct limited scientific and deployment resources to those field-tested approaches that will have the greatest real world impact.
[^1]: Part of the work done at the USDA Forest Products Laboratory, USA as Visiting Scientists
|
---
abstract: |
In this work we apply Wang-Landau simulations to a simple model which has exact solutions both in the microcanonical and canonical formalisms. The simulations were carried out by using an updated version of the Wang-Landau sampling. We consider a homopolymer chain consisting of $N$ monomers units which may assume any configuration on the two-dimensional lattice. By imposing constraints to the moves of the polymers we obtain three different models. Our results show that updating the density of states only after every $N$ monomers moves leads to a better precision. We obtain the specific heat and the end-to-end distance per monomer and test the precision of our simulations comparing the location of the maximum of the specific heat with the exact results for the three types of walks.
**Keywords**: Homopolymer, Monte Carlo, Wang-Landau
author:
- 'Lucas S. Ferreira'
- 'Alvaro A. Caparica'
- 'Minos A. Neto'
- 'Mircea D. Galiceanu'
title: 'The Rubber Band Revisited: Wang-Landau Simulation'
---
Introduction
============
The sequencing of the Human Reference Genome, announced ten years ago, provided a roadmap that is the basic foundation for modern biomedical research [@elaine]. This monumental achievement was enabled by developments in DNA (homopolymer) sequencing technology that allowed data production which exceed the original descripion of Sanger sequencing [@sanger]. Linear polymers are the simplest physical systems that can be studied in the framework of random walks models. They are long chain-like molecules formed by repetition of a basic unit or segment, where more importantly the polymer is *flexible*, i.e., it can assume different geometric configurations.
Recently, the study of homopolymers has been established by various techniques in condensed matter physics. Cohen *et al.* [@cohen], studied the behavior of single file translocation of a homopolymer through an active channel under the presence of a driving force by using Langevin dynamics simulation. Previous works on homopolymers which studied the denaturation of circular DNA are extensions of the Poland-Scheraga model [@bar]. Experimentally, the viscoelastic properties of a binary mixture of a mesogenic side-chain block copolymer in a low molecular weight nematic liquid crystal are studied for mass concentrations ranging from the diluted regime up to a liquid crystalline gel state [@maxim].
Although Monte Carlo simulations play an important role for the study of phase transitions and critical phenomena, some well-known difficulties arise when one uses standard algorithms (one-flip algorithms) [@metropolis] for the study of random walks models. These difficulties have been overcomed by the development of alternative Monte Carlo methods, such as parallel-tempering [@nemoto], cluster algorithms [@wolff], multicanonical algorithms [@berg], and more recently the Wang-Landau method [@wanglandau]. This method has been applied with great success to many systems, in particular to polymers in lattice [@cunha; @vorontsov; @binder].
In the present paper, using the Wang-Landau method, we investigate the computer simulations of a homopolymer model with exact solution in canonical and microcanonical formalism. In section \[models\] we give an introduction to the three studied models and we briefly present the mathematical background. In section \[simulations\] we outline shortly how the simulations were carried out. In section \[results\] we show and discuss the results for all the models.
Models and Formalism {#models}
====================
Model 1 {#model1}
-------
We consider a homopolymer chain of $N$ monomers units of length $a$ which may assume any configuration on a two-dimensional latice [@callen]. One end of the polymer is fixed and it´s taken as the origin of coordinates, shown by an open circle in Figure \[model\], a). The other end of this linear chain is subject to an externally applied tension $\tau$, acting along the positive $x$-axis. A possible realization of this model is sketched in Figure \[model\], a). Each polymer unit is permitted to lie either parallel or antiparallel to the $x$-axis and we assign the works $-\tau a$ and $+\tau a$ to these two orientations. We denote by $L_x$ the distance between the ends of the polymer chain, on the $x$-direction. Each monomer unit has the additional possibility of lying perpendicular to the $x$-axis, in the $+y$ or $-y$ directions. We associate a positive energy $\varepsilon$ to such a perpendicular monomer and the distance between the ends of the chain on the $y$-direction is denoted by $L_y$.
The hamiltonian of this model can be written as $$\label{h_1}
\mathcal{H}_1=\left(N^{+}_{y}+N^{-}_{y}\right)\varepsilon+\left(N^{-}_{x}-N^{+}_{x}\right)\tau a,$$ where $N^{+}_{x}$ and $N^{-}_{x}$ are the number of monomers along the $+x$ and $-x$ directions respectively, and similarly for $N^{+}_{y}$ and $N^{-}_{y}$. Since the tension in the $y$ direction is zero, we can assume $N^{+}_{y}=N^{-}_{y}$. Then $$\label{N_1}
N=N^{+}_{x}+N^{-}_{x}+N^{+}_{y}+N^{-}_{y},$$
$$\label{Lx_1}
N^{+}_{x}-N^{-}_{x}=L_{x},$$
$$\label{U_1}
N^{+}_{y}+N^{-}_{y}=U,$$
from which we find
$$\label{Nx_1}
N^{+}_{x}=\frac{1}{2}\left(N-U+L_{x}\right),$$
$$\label{Nx1_1}
N^{-}_{x}=\frac{1}{2}\left(N-U-L_{x}\right),$$
and
$$\label{Ny_1}
N^{+}_{y}=N^{-}_{y}=\frac{1}{2} U.$$
The number of configurations of the polymer consistent with a given end-to-end distance in the $x$-direction, $L_{x}$ (the dimensionless length of the polymer), and $U$ (the number of monomers lying perpendicular to the $x$-axis), is
$$\label{omega_1}
\Omega\left(L_{x},U,N\right)=\frac{N!}{N^{+}_{x}!N^{-}_{x}!N^{+}_{y}!N^{-}_{y}!}.$$
From Eq. (\[h\_1\]), if we set $\varepsilon \equiv \tau a = 1$, we can write for a given energy level
$$\label{E_1}
E= U-N^{+}_{x}+N^{-}_{x},$$
and from Eq. (\[N\_1\]) $$\label{N1_1}
N=U+N^{+}_{x}+N^{-}_{x}.$$
Adding the equations (\[E\_1\]) and (\[N1\_1\]) we obtain $$\label{U_Nx_1}
U+N^{-}_{x}=\frac{N+E}{2}.$$
Inserting Eqs. (\[Nx\_1\]), (\[Nx1\_1\]) and (\[Ny\_1\]) into (\[omega\_1\]), using Eqs. (\[E\_1\]), (\[N1\_1\]), and (\[U\_Nx\_1\]), and setting $N^{-}_{x}\equiv n$ we obtain the number of configurations with energy $E$ as
$$\label{g_1}
g(E)=\sum_{u=0}^{\frac{N+E}{2}} \sum_{n=0}^{u} \frac{N!}{\left(\frac{N-E}{2}\right)!\left(\frac{N+E}{2}-u\right)!n!(u-n)!}.$$
Using the definition of the entropy and defining $\partial S/\partial U=1/T$, we obtain [@callen]:
$$\label{LxN_1}
\frac{L_{x}}{N}=\frac{N \sinh{(\tau a/k_BT)}}{\cosh{(\tau a/k_BT)}1+\exp\left( -\varepsilon /k_{B}T \right)}.$$
Model 2 {#model2}
-------
In this model we consider that each polymer unit is parallel to the $x$-axis; no antiparallel move is allowed. Additionally the polymer units have the possibility of lying on the $+y$ and $-y$ directions. In Figure \[model\] b) is sketched a possible configuration of this model, where we depicted by an impenetrable wall the forbidden region along the $x$-axis.
The hamiltonian for the model $2$ can be written as $$\label{H_2}
\mathcal{H}_2=\left(N^{+}_{y}+N^{-}_{y}\right)\varepsilon-N^{+}_{x}\tau a,$$ where $N_{x}^{+}$ is the number of monomers along the $+x$ direction and similarly for $N_{y}^{+}$ and $N_{y}^{-}$. Assuming again $N_{y}^{+}\equiv N_{y}^{-}$ we obtain the following equations:
$$\label{Nx_2}
N^{+}_{x}=\frac{1}{2}\left(N-U+L_{x}\right),$$
and
$$\label{N+y_2}
N^{+}_{y}=N^{+}_{y}=\frac{U}{2},$$
For this model the number of configurations of the polymer consistent with a given $L_{x}$ and $U$ is
$$\label{omega_2}
\Omega\left(L_{x},U,N\right)=\frac{N!}{N^{+}_{x}!N^{+}_{y}!N^{-}_{y}!}.$$
From Eq. (\[H\_2\]), if we set $\varepsilon \equiv \tau a = 1$, we can write the energy as
$$\label{E_2}
E= U-N^{+}_{x},$$
and for $N=N^{+}_{y}+N^{-}_{y}+N^{+}_{x}$ we obtain $$\label{N_2}
N=U+N^{+}_{x}.$$
After similar calculations as for model $1$, subsection \[model1\], we obtain the density state and end-to-end distance per monomer, respectively
$$\label{g_2}
g(E)=\sum_{n=0}^{\frac{N+E}{2}}\frac{N!}{\left(\frac{N-E}{2}\right)!\left(\frac{N+E}{2}-n\right)!n!}$$
and $$\label{LxN_2}
\frac{L_{x}}{N}=\frac{1}{1+2\exp\left( - \frac{2\tau a}{k_{B}T}\right)}.$$
Model 3 {#model3}
-------
In this model we consider that each polymer unit is parallel to the $x$-axis and we allow only the possibility of lying on the $+y$ direction. Thus, we restrict to the situation of positive values for both axes. In Figure \[model\] c) we show a possible configuration of this model.
The hamiltonian for the model can be written as $$\label{H_3}
\mathcal{H}_3=N^{+}_{y}\varepsilon-N^{+}_{x}\tau a,$$ where $N_{x}^{+}$ is the number of monomers along the $+x$ direction and similarly for $N_{y}^{+}$. Then
Using the method previously described, we obtain $N_{x}^{+}$ and $N_{y}^{+}$ for this model
$$\label{Nx_3}
N^{+}_{x}=\frac{1}{2}\left(N-U+L_{x}\right)$$
and $$\label{Ny_3}
N^{+}_{y}=\frac{1}{2}\left(N+U-L_{x}\right).$$
For this model the number of configurations with energy $E$ is given by
$$\label{g_3}
g(E)=\frac{N}{\left(\frac{N-E}{2}\right)!\left(\frac{N+E}{2}\right)!}.$$
Using the equation for $g(E)$ we obtain the length of the polymer as
$$\label{LxN_3}
\frac{L_{x}}{N}=\frac{1}{1+\exp\left( - \frac{2\tau a}{k_{B}T}\right)}.$$
Simulations
===========
In our simulations we followed the prescriptions of Ref. [@caparica1]. We define a Monte Carlo step (MCS) as giving sequentially to any unit the possibility of changing its direction with identical probability to any allowed direction or remaining in the same one. At the beginning of the simulation we set $S(E)=0$ for all energy levels, where $S(E)\equiv\ln g(E)$. The random walk in the energy space runs through all energy levels from $E_{min}$ to $E_{max}$ with a probability $$\label{prob}
p(E\rightarrow E^{'})=\min\left\lbrace \exp\left[ \left( S(E)-S(E^{'})\right)\right],1\right\rbrace ,$$ where $E$ and $E'$ are the energies of the current and the new possible configurations. After $N$ trial moves we update $H(E)\rightarrow H(E)+1$ and $S(E)\rightarrow S(E)+F_{i}$, where $F_{i}=\ln f_{i}$, $f_{0}\equiv e=2.71828...$ and $f_{i+1}=\sqrt{f_{i}}$ (where $f_{i}$ is the so-called modification factor and $H(E)$ is a histogram accumulated for each $f_i$). The flatness of the histogram is checked after a number of Monte Carlo (MC) steps and usually the histogram is considered flat if $H(E)>0.8\langle H \rangle$, for all energies, where $\langle H \rangle$ is an average over the energies. If the flatness condition is fulfilled we update the modification factor to a finer one and reset the histogram $H(E)=0$. The simulations are continued up to $f_{final}=f_{14}$ and the microcanonical averages were accumulated from the very beginning ($f_{micro}=f_0$), results obtained by Ferrera and Caparica [@caparica2]. Having in hand the density of states, one can calculate the canonical average of any thermodynamic variable as
$$\label{mean}
\langle X\rangle_T=\dfrac{\sum_E \langle X\rangle_E g(E) e^{-\beta E}}{\sum_E g(E) e^{-\beta E}} ,$$
where $\langle X\rangle_E$ is the microcanonical average accumulated during the simulations and $\beta=1/k_BT$, $k_B$ is the Boltzmann constant and $T$ is the temperature.
Results and Discussion {#results}
======================
In Figure \[model\] we depicted the three models which were presented in section \[models\]. The difference between the models is given by the allowed moves. In the first model, denoted by a) in the figure, we allow the polymer unit to move along the positive or the negative directions of the $x$-axis, the same for the $y$-axis. The end-to-end distance of the polymer on $x$-direction is denoted by $L_x$ and with $L_y$ we denote this distance on $y$-direction. In the second model, b) in figure \[model\], the antiparallel motion along the $x$-axis is forbidden. In the third considered model there are allowed moves only on the positive side of both $x$ and $y$-axis.
Using the simulated and the exact density of states in Eq. we calculate the specific heat given by $$\label{heat}
C=\frac{\langle(E-\langle E\rangle)^2\rangle}{T^{2}}$$ and the mean end-to-end distance $$\label{distance}
\langle L_x\rangle=\langle|x_N-x_1|\rangle,$$ where $E$ is the energy of the configurations and $x_1$ and $x_N$ are the corresponding $x$-coordinates of the ends of the polymer.
In Figure \[ge\] we plot in semi-logarithmical scale the density of states $g(E)$ for all the three models. Here we plot the results obtained from the simulations (symbols in the figure) and also the exact theoretical results (continuous lines in the figure) for a polymer of $N=500$ monomers. In Figure \[lx\] we plot the end-to-end distance along the $x$-axis, $L_x$, as a function of the temperature. Here we rescale $L_x$ by the total number of monomers, $N=500$ in this case. We observed a very good agreement between the simulation results (symbols in the figure) and the theoretical results (continuous lines in the figure), given by equations (\[LxN\_1\]), (\[LxN\_2\]), and (\[LxN\_3\]), corresponding to model $1$, model $2$, and model $3$ respectively. In Figure \[cv\] we plot the specific heat per monomers for polymers with $N=500$. The specific heat have a tail proportional to $1/T^{2}$ in the high temperature limit.
Finally, in Table I we present the location of peak of the specific heat for each model obtained by the Wang-Landau simulations and compare with the results calculated with the exact density of states. The simulations were carried out for 100 independent runs, adopting the $80\%$ flatness criterion. One can see that in the three cases the exact results fall into the error bars.
------------------- ------------ -------------
Case exact our results
\[0.5ex\] Model 1 0.70299027 0.7043(23)
Model 2 0.75335362 0.7513(23)
Model 3 0.83180562 0.8314(23)
------------------- ------------ -------------
**Table I:** *Temperatures of the peak of the specific heat from simulations, compared with the exact values.*
Conclusions
===========
We have carried out Wang-Landau simulations of a simple polymer model which has exact solutions in both the microcanonical and the canonical ensembles. Here we considered three two-dimensional models: in the first model we allowed moves in all possible directions, in the second model the moves along the negative $x$-axis are forbidden, while in the last model we allowed only moves along the positive direction for both $x$ and $y$- axis. We have shown that updating the density of states only after each $N$ trial moves and halting the simulations when $f_{final}=f_{14}$ [@caparica2], defined during the simulations, we obtain quite accurate results, compared with the available analytical exact results. We have obtained a very good agreement between the simulations and the exact results also for the studied physical quantities: the end-to-end distance and the specific heat. As expected, due to the difference of the allowed directions of motion, in the limit of high temperatures the end-to-end distance has the highest value for the third model and the lowest for the first model.
**ACKNOWLEDGEMENT**
This work was partially supported by CNPq (Edital Universal) and FAPEAM (Programa Primeiros Projetos - PPP) (Brazilian Research Agency).
[14]{}
Mardis E R, (2011) *Nature* **470**, 198.
Sanger F, Nicklen S and Culson A R, (1977) *Proc. Natl. Acad. Sci.* USA **74**, 5463.
Cohen J A, Chaudhuri A and Golestanian R, (2011) *Phys. Rev. Lett.* **107**, 238102.
Bar A, Kabakcioglu A and Mukamel D, (2011) *Phys. Rev. E* **84**, 041935.
Khazimullin M*et al.*, (2011) *Phys. Rev. E* **84**, 021710.
Metropolis M et al., (1953) *J. Chem. Phys.* **21**, 1087; Glauber R J, (1963) J. Math. Phys. **4**, 294.
Hukushima K, and Nemoto K, (1996) *J. Phys. Jpn.* **65**, 1604.
Wolff U, (1989) *Phys. Rev. Lett.* **62**, 361; Swendsen R H, and Wang J S, (1987) *Phys. Rev. Lett.* **58**, 86.
Berg B A, and Neuhaus T, (1991) *Phys. Lett. B* **267**, 249; (1992) *Phys. Rev. Lett.* **68**, 9.
Wang F, and Landau D P, (2001) *Phys. Rev. Lett.* **86**, 2050.
Cunha-Netto A G, Dickman R and Caparica A A, (2009) *Comput. Phys. Commun.* **180**, 583; Silva C J, Cunha-Netto A G , Caparica A A and Dickman R, (2006) *Braz. J. Phys.* **36**, 3A 619; Seaton D T, Wüst T and Landau D P, (2010) *Phys. Rev. E* **81**, 011802.
Vorontsov-Velyaminov P N, Volkov N A, Yurchenko A A, and Lyubartsev A P, (2010) *Polymer Science Series A* **52**, 742.
Binder K and Paul W, (2008) *Macromolecules* **41**, 4537.
Callen H B, *Thermodynamics and an Introduction to Thermostatistics* (John Wiley & Sons, New York, 1985).
Caparica A A and Cunha-Netto A G, (2012) *Phys. Rev. E* **85**, 046702.
Ferreira L S and Caparica A A, (2012) *IJMPC* **23**, 1240012.
![Here we consider three types of walks: (a) Model 1: unrestricted motion along the $x$-axis and $y$-axis, (b) Model 2: is forbidden the backward motion on the $x$-axis, and (c) Model 3: the random walk moves only along the positive directions of $x$-axis and $y$-axis.[]{data-label="model"}](model.eps){width="7.6cm" height="8.9cm"}
![Exact density of states $g(E)$ for polymer size $N=500$ for (a) Model 1; (b) Model 2 and (c) Model 3. The lines are the exacts results and the symbols are simulational results.[]{data-label="ge"}](densidade_estados.eps){width="7.5cm" height="7.5cm"}
![End-to-end distance per monomer for polymer size $N=500$ for (a) Model 1; (b) Model 2 and (c) Model 3. The line is the exact result and the dots are the simulational results using the procedure for Wang-Landau simulation [@caparica2]. The error bars are less than symbols.[]{data-label="lx"}](comprimento.eps){width="7.5cm" height="7.5cm"}
![Specific heat per monomer for polymer size $N=500$ for (a) Model 1; (b) Model 2 and (c) Model 3. The line is the exact result and the dots are the simulational results using the procedure for Wang-Landau simulation [@caparica2]. The error bars are less than symbols.[]{data-label="cv"}](calor.eps){width="7.5cm" height="7.5cm"}
|
---
author:
- |
Prakalp Srivastava\
University of Illinois at Urbana Champaign\
psrivas2@illinois.edu
- |
Maria Kotsifakou\
University of Illinois at Urbana Champaign\
kotsifa2@illinois.edu
- |
Vikram Adve\
University of Illinois at Urbana Champaign\
vadve@illinois.edu
bibliography:
- 'hetero.bib'
- 'optimization.bib'
subtitle: Technical Report
title: '[*HPVM*]{}: A Portable Virtual Instruction Set for Heterogeneous Parallel Systems'
---
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---
title: Progress Towards Understanding Quarkonia at PHENIX
---
Introduction {#intro}
============
We discuss our present understanding of Quarkonia ($J/\psi$, $\psi^\prime$, $\chi_C$, $\Upsilon$) based on the measurements by PHENIX at RHIC. We discuss 1) production, 2) cold nuclear matter (CNM) effects, 3) the effect of the Quark Gluon Plasma (QGP), and then comment on prospects for the future as RHIC luminosities increase and detector upgrades are installed. As shown in Figure \[jsi\_history\_log\], the numbers of $J/\psi$ obtained in recent runs has increased dramatically, with over 70,000 in the just completed $d+Au$ run.
How are Quarkonia Produced {#production}
==========================
Quarkonia are produced primarily via gluon-fusion, but it has proven difficult for theoretical predictions to reproduce both the cross section and the polarization of the $J/\psi$. The configuration of the initially produced $c\bar{c}$ state remains unclear, and casts uncertainty on what CNM effects it will experience in nuclei. NRQCD models produce a $c\bar{c}$ in a color-octet state and are able to reproduce the cross section, but predict large transverse polarization at large $p_T$ - unlike the data from E866/NuSea[@e866_ups_pol] and CDF[@cdf_polarization] which show only small longitudinal polarization. However, a recent color-singlet model[@lansberg] claims good agreement for both cross section and polarization.
Another complication in quarkonia production, particularly for the $J/\psi$, is that about $\sim{40\%}$ of the $J/\psi$s come from decays of higher mass resonances, namely the $\psi^\prime$ and $\chi_C$. Until recently, these fractions have been inferred from measurements at other energies. Now PHENIX has started to quantify these itself with initial results indicating $8.6 \pm 2.5\%$ from the $\psi^\prime$ and $< 42\%$ from the $\chi_C$. Another PHENIX measurement[@moreno] shows that $4 {{+3}\atop{-2}}\%$ of the $J/\psi$s come from decays of B-mesons, a contribution which is strongest at larger $p_T$.
What Cold Nuclear Matter (CNM) Effects are Important {#CNM}
====================================================
For Quarkonia produced in nuclei, e.g. in $p+A$ or $d+A$ collisions, several interesting effects - usually called cold nuclear matter (CNM) effects, can occur. These include modifications of the initial gluon density either according to traditional nuclear shadowing models[@EKS; @NDSG] that involve fits to deep-inelastic scattering and other data, or gluon saturation models[@CGC]. In addition the initial-state projectile gluon may lose energy before it interacts to form a $J/\psi$. Both of these effects can cause suppression of the produced $J/\psi$s per nucleon-nucleon collision at large rapidity (or small x) relative to that observed in p+p collisions. Finally, the $J/\psi$s can be suppressed by dissociation of the $c{\bar{c}}$ by the nuclear medium in the final state.
A new analysis of the 2003 PHENIX $d+Au$ data, along with the new 2005 baseline $p+p$ data have been put together to produce new nuclear modification factors for CNM[@ppg078], as shown in Figure \[alpha\_x2xf\], where they are compared to similar data at lower energies. The lack of scaling with $x_2$ shown in the left panel of the figure suggests that traditional shadowing models, which should have a universal $x_2$ dependence, are not the dominant physics. The approximate scaling with $x_F$ (right panel), at least for the lower energy data that extends to large $x_F$, hints that initial-state energy loss or gluon saturation may be the dominant physics.
In Figure \[rdau\_eksmodel\] an approximate constraint using a simple CNM model (with shadowing and dissociation)[@vogt] is shown. This model can then be used to give an extrapolated constraint for $Au+Au$ collisions, as shown in Figures \[figure\_rauau\_project\_mid\] and \[figure\_rauau\_project\_forw\]. Clearly the $d+Au$ data from 2003 used to constrain the CNM extrapolation here suffers from large uncertainties, and results in a large uncertainty for $Au+Au$ collisions. For $Au+Au$ at mid-rapidity the CNM band is almost consistent with the observed suppression - except for the most central collisions ($n_{part} \sim {340}$); while at forward rapidity the suppression seen for $Au+Au$ is substantially stronger. The just completed 2008 $d+Au$ run has approximately 30 times more $J/\psi$’s than before and, once analyzed, will dramatically improve the knowledge of the CNM baseline in $A+A$ collisions, and allow precision studies of the additional physics beyond CNM that comes from the hot-dense matter created in heavy-ion collisions. The CNM constraint is expected to narrow by approximately a factor of three with the new $d+Au$ data.
How does the QGP affect Quarkonia {#QGP}
=================================
Quarkonia are thought to be a definitive probe of the QGP through the screening process in the deconfined colored medium[@satz]. Different quarkonia states, because of their different binding energies, are expected to “melt” at different temperatures of the medium. E.g. in some lattice calculations the $J/\psi$ would melt at $1.2 T_C$, but the $\Upsilon$ only at over $2 T_C$. Nuclear modification factors observed by PHENIX in $Au+Au$ collisions are shown in Figure \[raa\_ratio\_data\]. The suppression at mid-rapidity is about the same as that observed for lower energies at the SPS[@SPS], despite the expectation that the hotter medium created at RHIC would cause a larger suppression. The suppression at forward rapidity is stronger than that at mid rapidity, and the ratio of the nuclear modification factors, forward/mid, shown in the bottom panel of the figure, reaches an approximately constant level of $0.6$ for $n_{part}>100$.
Several scenarios can be considered in trying to understand the observed trends: 1) CNM effects, as discussed above, should always be accounted for as a baseline. 2) Sequential screening[@screening] - where, as suggested by some lattice calculations, only the $\psi^\prime$ and $\chi_C$ are screened and the $J/\psi$ itself is not - not at RHIC or at SPS energies. Then the observed suppression beyond CNM comes only from loss of the feeddown ($\sim{40\%}$) from the two higher mass quarkonia states. 3) Regeneration models[@regeneration], where the large density of charm quarks created in the collisions ($\sim 20$ in a central $Au+Au$ collision) can produce charmonia in the latter stages of the expansion.
In the sequential screening picture, if the CNM suppression at mid rapidity and the “melting” of the higher mass charmonia states was the same at RHIC and at the SPS, this would provide a natural explanation for the nearly identical suppression at RHIC and the SPS. It would also agree with some lattice calculations that indicate no melting of the $J/\psi$ until over $2T_C$[@lattice_2tc]. The stronger forward rapidity suppression seen at RHIC could then be explained by gluon saturation that gives stronger forward suppression than that from standard shadowing models. For traditional shadowing models the shadowing of the gluon from one nucleus is largely canceled by the anti-shadowing from the gluon from the other nucleus - resulting in an approximately flat rapidity dependence. For gluon saturation a “shadowing-like” effect is produced for the gluon in the small-x region, but no anti-shadowing for the other gluon, resulting in a stronger suppression at forward rapidity. Since screening and gluon saturation might have different centrality dependences, it is unclear whether they would balance to produce the approximately flat ratio observed for $n_{part} > 100$ (Figure \[raa\_ratio\_data\]).
An alternative is the regeneration picture, where the dissocation by the QGP at mid and forward rapidity would be similar, but the weaker suppression at mid rapidity would be due to regeneration effects being stonger here where the charm density is largest. In this case it would be an “accidental” compensation of screening and regeneration that leads to the same mid-rapidity suppression at RHIC and the SPS. At forward rapidity, where the charm density is smaller, the regeneration is reduced and stronger screening results. Again, whether the saturation in the forward to mid rapidity suppression could be reproduced by these two compensating effects is unclear.
The regeneration mechanism depends on the square of the open-charm cross section, so it is critical to resolve the present uncertainties there.[@open_charm] Also, since charm has been shown to exhibit flow for moderate $p_T$ values, one would expect $J/\psi$s that are produced by regeneration to inherit this flow. A first measurement of the $J/\psi$ flow at mid rapidity is shown from part of the 2007 $Au+Au$ data in Figure \[v2\_pt\_central\_20-60\_theories\_prelim\]; but is clearly quite challenging, and so far is consistent with zero flow.
Summary and Future {#summary}
==================
The suppression of $J/\psi$ production in $Au+Au$ collisions at RHIC for mid rapidity is similar to that at lower energies, while for foward rapidity the RHIC suppression is stronger. Better cold nuclear matter constraints from the new $d+Au$ data are needed to establish an accurate baseline and allow quantitative analysis of the QGP effects. Two theoretical pictures, 1) sequential suppression with gluon saturation and 2) dissociation and regeneration, appear to offer explanations of the observed trends. Higher luminosities and silicon vertex upgrades will enable much more quantitative studies in the next few years. Over 100,000 $J/\psi$s and 600 $\Upsilon$s are expected in a year with higher luminosities enabled by accelerator advances, while new silicon vertex detectors will allow explicit indentification of open-heavy and will improve both the background and mass resolution for the quankonia states - especially important to separate the $\psi^\prime$ from the $J/\psi$ at forward rapidity.
[9]{}
T. Chang [*et al.*]{}. (E866/NuSea), [*Phys. Rev. Lett.*]{} [**91**]{} (2003) 211801.
T. Affolder et al. (CDF), [*Phys. Rev. Lett.*]{} [**85**]{} (2000) 2886.
H. Haberzettl and J.P. Lansberg, [*Phys. Rev. Lett.*]{} [**100**]{} (2008) 032006.
A. Adare [*et al.*]{}, (PHENIX), [*Phys. Rev. Lett.*]{} [**98**]{} (2007) 232002.
Y. Morino (PHENIX), this proceedings.
A. Adare [*et al.*]{}, (PHENIX), [*Phys. Rev.*]{} [**C77**]{} (2008) 024912.
K.J. Eskola, V.J. Kolhinen, and R. Vogt, [*Nucl. Phys.*]{} [**A696**]{} (2001) 729.
D. deFlorian and R. Sassot, [*Phys. Rev.*]{} [**D69**]{} (2004) 074028.
L. McLerran and R. Venugopalan, [*Phys. Rev.*]{} [**D49**]{} 2233 (1994); 3352 (1994).
R. Vogt, [*Phys. Rev.*]{} [**C77**]{} (2005) 054902.
T. Matsui and H. Satz, [*Phys. Lett.*]{} [**B178**]{} (1986) 416.
M.C. Abreu [*et al.*]{}. (NA50) [*Phys. Lett.*]{} [**B477**]{} (2000) 28; Phys. Lett. B521 (2001) 195.
F. Karsch, D. Kharzeev, H. Satz, [*Phys. Lett.*]{} [**B637**]{} (2006) 75; hep-ph/0512239.
L. Grandchamp, R. Rapp, G.E. Brown, [*Phys. Rev. Lett.*]{} [**92**]{} (2004) 212301; R.L. Thews, [*Eur. Phys. J*]{} [**C43**]{} (2005) 97.
F. Datta [*et al.*]{}, hep-lat/0409147.
A. Knospe (STAR), this proceedings.
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---
abstract: 'Previous studies have analyzed the energy injection into the interstellar medium due to molecular bubbles. They found that the total kinetic energies of bubbles are comparable to, or even larger than, those of outflows but still less than the gravitational potential and turbulence energies of the hosting clouds. We examined the possibility that previous studies underestimated the energy injection due to being unable to detect dim or incomplete bubbles. We simulated typical molecular bubbles and inserted them into the $^{13}$CO Five College Radio Astronomical Observatory maps of the Taurus and Perseus Molecular Clouds. We determined bubble identification completeness by applying the same procedures to both simulated and real datasets. We proposed a detectability function for both the Taurus and Perseus molecular clouds based on a multivariate approach. In Taurus, bubbles with kinetic energy less than $\sim$1$\times 10^{44}$ erg are likely to be missed. We found that the total missing kinetic energy in Taurus is less than a couple of 10$^{44}$ erg, which only accounts for around 0.2$\%$ of the total kinetic energy of identified bubbles. In Perseus, bubbles with kinetic energy less than $\sim$2$\times 10^{44}$ erg are likely to be missed. We found that the total missing kinetic energy in Perseus is less than $10^{45}$ erg, which only accounts for around 1$\%$ of the total kinetic energy of identified bubbles. We thus conclude that previous manual bubble identification routines used in Taurus and Perseus can be considered to be energetically complete. Therefore, we confirm that energy injection from dynamic structures, namely outflows and bubbles, produced by star formation feedback are sufficient to sustain turbulence at a spatial scale from $\sim$0.1 pc to $\sim$2.8 pc.'
author:
- Mengting Liu
- Di Li
- Marko Krčo
- 'Luis C. Ho'
- Duo Xu
- Huixian Li
bibliography:
- 'apj-jour.bib'
- 'bibliography.bib'
title: Numerical Simulation and Completeness Survey of Bubbles in the Taurus and Perseus Molecular Clouds
---
Introduction {#sec:intro}
============
Stellar feedback plays a crucial role in the dynamics and energy balance of the interstellar medium (ISM; $\&$[@zinnecker07]. Feedback associated with protostars injects momentum and energy to the parent molecular cloud, altering the velocity field and density distribution of the cloud [@Arce01; @Arce02; @Arce10], contributing to the mass loss of the surrounding dense gas [@Fuller02; @Arce06; @Arce10; @Offner15], affecting star formation efficiency [@Frank14; @Feder15; @Pavel18], inducing changes in the chemical composition of the impacted media [@Bally07], sustaining or generating turbulence (e.g., [@Fukui86; @Arce01; @Nakamura11a; @Nakamura11b; @Plunkett13; @Feder15; @Arce11; @Feder18; @Li15] — henceforth Ar11, Fe18) resisting gravitational collapse and even disrupting the surrounding gas to limit the lifetime of their parent molecular cloud [@Solomon81; @Arce01; @Hartmann01; @Arce02; @Duarte-Cabral12; @Plunkett13].
The primary manifestations of stellar feedback are molecular outflows and bubbles. On the Galactic scale, superbubbles are shown to be globally important, with structures spanning hundreds of parsecs, such as chimneys and large cavities [@Heiles79; @Norman89]. Abundant studies of outflows have been carried out [@Kwan76; @Snell80; @Lada81; @Arce10; @Nakamura11a; @Nakamura11b; @Narayanan12; @Mott17]. In local to main star-forming molecular cloud complexes, relatively less attention has been paid to ’local’ molecular bubbles that are hollowed out by low- to intermediate-mass stars. Despite being expected to be less influential than superbubbles, they are substantial in number and could inject much more energy back into their natal clouds than that from outflows (Ar11, L15, Fe18).
Molecular bubbles are partially or fully enclosed three-dimensional structures whose projections on the sky resemble partial or full rings [@Churchwell06]. Young stellar objects have sufficient stellar winds to entrain and accelerate ambient gas to sculpt spherical or ring-like cavities in their surrounding molecular clouds. Compared to collimated outflows, bubbles affect a larger volume of ambient molecular gas. Compared to supernova remnants, bubbles occur around more stars and persist for a longer period of time [@Matzner02; @Arce11]. In the Taurus molecular cloud complex (TMC), the kinetic energy and energy injection rate of identified bubbles are larger than those of outflows, implying a substantial input of mechanical energy into the TMC (L15). Due to their complex morphology, bubbles are more difficult to identify [@Beaumont14] in comparison to the relatively clear characteristics of outflows. In this work, we examine the completeness of bubble identification from previous studies.
The Five College Radio Astronomical Observatory (FCRAO) CO survey [@Goldsmith08] remains one of the largest molecular spectral-line maps of a continuous cloud complex. Following an empirical and iterative procedure, L15 identified 37 bubbles in the TMC. Meanwhile, Ar11 detected 12 bubbles in the Perseus molecular cloud complex (PMC). The identified Taurus bubbles inject $\sim9.2 \times 10^{46}$ erg into the surrounding ISM. The energy injection rate was measured to be $\sim6.4 \times 10^{33}$ erg s$^{-1}$, surprisingly comparable to the turbulence dissipation rate. In the PMC, the kinetic energy of identified bubbles is around $\sim7.6 \times 10^{46}$ erg with an energy input rate of around $\sim8.9 \times 10^{33}$ erg s$^{-1}$, which is similar to the turbulence dissipation rate. Small or slowly expanding bubbles may be missed. However, large or irregular bubbles may also be missed due to confusion. We approach this problem with semiempirical numerical simulations of artificial molecular bubbles based on the bubble model from Cazzolato $\&$ Pineault 2005. We generated artificial spherical and partially spherical $^{13}$CO bubbles and embedded them into the real data cubes of Taurus and Perseus. We then carried out parameter studies including average antenna temperature, number of pixels, and expansion velocity to examine the completeness of the empirical identification procedures.
In section 2, we provide a basic description of the Taurus and Perseus data sets and the bubble identification routines of previous surveys. The bubble model is presented in section 3, including a detailed description of our method for simulating bubbles and artificial injection into the data sets. In section 4, we describe the bubble detection procedures and quantify detectability. The properties of missing bubbles and the implication for stellar feedback in Taurus and Perseus are discussed in section 5. In section 6, we discuss and summarize the conclusions which may be drawn from this research.
Data and empirical procedures of bubble identification
======================================================
$^{13}$CO(1-0) is a tracer frequently used to trace molecular hydrogen and study the dynamic morphology of the ISM due to its much smaller opacities than those of $^{12}$CO(1-0). The Taurus $^{13}$CO(1-0) and the Perseus $^{13}$CO(1-0) datasets we used are from the FCRAO CO surveys conducted between 2003 and 2005 [@Ridge06a; @Narayanan08; @Goldsmith08; @complete11].
The common empirical procedures used to identify bubble structures in molecular clouds are as follows:
1\. To search the regions with circular (or arc) structures brighter than the surrounding molecular gas (Ar11, L15, Fe18).
2\. To visualize the high-velocity features of these regions by plotting a Position-Velocity ($P-V$) diagram. If there is an expanding bubble with a red-shifted part or a blue-shifted part, we can identify the $\cup$ or $\cap$-shaped feature on the $P-V$ diagram of each region (Ar11, L15, Fe18).
3\. To search for candidate sources such as pre-main-sequence stars or protostars associated with the cloud ( Ar11, L15, Fe18).
4\. To compare the bubble structures in $^{13}$CO(1-0) with an infrared map to determine whether they have similar morphologies (Ar11, L15, Fe18).
The more conditions identified bubbles satisfy, the more likely they are to be real dynamic feedback driven by protostars rather than superpositions of unrelated patterns in the cloud. The radius and thickness of each bubble can be estimated by fitting a Gaussian to the azimuthally averaged profile of the CO integrated intensity map (Ar11, L15).
Comparison of simulated bubble and identified bubble
====================================================
We simulated spherical and partially spherical $\rm ^{13}$CO(1-0) bubbles based on the model proposed by Cazzolato $\&$ Pineault (2005) for the TMC and the PMC. This model relies on three fundamental assumptions:
1\. The density of the surrounding medium and within the bubble is different but homogeneous.
2\. The bubble expansion is isotropic.
3\. All pixels on the bubble are at the same distance from Earth.
There are eight significant parameters in this model, listed in Table 1. Since identified bubbles in Taurus and Perseus often have partial circular or arc-like morphology, we employedcompleteness of bubble $\beta$ as another parameter. The detailed bubble simulation calculation is shown in the Appendix A.
[lccccc]{} $R$ & Bubble radius & \[0.28 pc - 1.90 pc\] & \[0.14 pc - 2.79 pc\]\
$\Delta R$& Bubble thickness & \[0.03 pc, 0.44 pc\] & \[0.1 pc, 0.68 pc\]\
$\rm V_{exp}$ & Bubble expansion velocity & \[1.0 km $\rm s^{-1}$ - 3.3 km $\rm s^{-1}$\] & \[1.2 km $\rm s^{-1}$ - 6.0 km $\rm s^{-1}$\]\
$\sigma$ & Velocity dispersion & $\sim$1.4 km $\rm s^{-1}$ & $\sim$2.0 km $\rm s^{-1}$\
$\beta$ & The completeness of bubble & \[$90^{\circ}$, $360^{\circ}$\]&\[$90^{\circ}$, $360^{\circ}$\]\
$n_0'$ & Bubble H$_2$ number density & 35 $\rm cm^{-3}$& 280 $\rm cm^{-3}$\
&with assumed $^{13}$CO abundance of 1.43\*10$^{-6}$.&\
&This is estimated based on intensity of &\
&the high-velocity line wings and used to&\
&calculate the amount of gas being moved by&\
&bubbles (see more explanation in&\
&the Discussion section).&\
$\Delta V$ & Velocity width of a spectrometer& 0.266 km $\rm s^{-1}$& 0.066 km $\rm s^{-1}$\
&channel & &\
$a_{\rm pc}$ & The bubble physical depth along&\
& the LOS in parsecs.& &\
These artificial bubbles were embedded individually into the Taurus and Perseus $^{13}$CO(1-0) data cubes at random positions and channels. We compared inserted artificial bubbles with real identified bubbles by morphology and high-velocity features in Figures 1-2. Many of the identified bubbles in Taurus are ambiguous and difficult to identify, as in the right column of Figure 1. The identified bubbles in Perseus tend to have a more prominent and identifiable apparent characteristics.
We simulated artificial bubbles with a similar radius, thickness, $\beta$, and expansion velocity for each cloud, as shown in the left columns of Figures 1-2. Circular structures brighter than the surrounding molecular gas may be identified in the intensity and channel maps. Meanwhile, the $P-V$ diagram illustrates the $\cup$- or $\cap$-shaped features. We found that in general artificial bubbles are somewhat easier to detect than real ones because they are not really embedded within the surrounding environment but are added on to the original images, and their morphology is necessarily independent of the surrounding environment. This means that our detectability function actually presents an upper limit for the bubble detectability function with the same parameters. We found it challenging to simulate realistic artificial bubbles in part to the ambiguous morphology of identified bubbles.
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BUBBLE DETECTION EXPERIMENT
===========================
Identifying the circular structure in the channel maps is the first step in bubble identification. Therefore, we searched for the artificial bubbles by going through the channel maps. When we detected a circular structure, we compared the position of detected structure with the position of the inserted artificial bubble. If they are consistent with each other, we consider the artificial bubble detectable. When we cannot detect any circular structure through channel maps, we consider the artificial bubble undetectable.
We generated 500 artificial bubbles with different parameters for each cloud. Each bubble was embedded into Taurus and Perseus separately in random positions and channels one at a time. We undertook a blind search for each inserted simulated bubble to determine the detectability function. The artificial bubble identification routine largely follows the previous bubble manual identification procedures in section 2.
Description of Variable Parameters
----------------------------------
Number of pixels $N$, average antenna temperature $T_{a}$, and expansion velocity $V{e}$ are three observable parameters which affect the bubble detection. The number of pixels $N$ depends on the bubble radius $R$, thickness $\Delta R$, and completeness $\beta$. The values of the parameters of each of the 500 experimental bubbles are generated randomly from the range of the maximum and minimum value of the observed bubble with uniform distribution to parameterize the bubble detectability in the whole parameter space equally. The experimental bubbles were inserted into the Taurus and Perseus $^{13}\rm CO$ data cubes in random positions and channels one at a time. If we cannot detect the inserted artificial bubble, the detection result of this bubble is 0, otherwise, it is 1.
Analysis Method
---------------
The previous bubble identification procedures are subjective and depending on the apparent detection threshold resulting from the bubble identification procedures in L15, Ar11, and Fe18. We quantified the bubble detection results of previous bubble identification procedures with the detectability function. Since the bubble detection results correspond to a Bernoulli distribution, which is one of the exponential family distributions, and the experimental bubble parameters can be considered as independent to each other, we constructed the bubble detectability function using generalized linear models (GLMs; [@NW72]). GLMs are widely used regression models for dependent variables which follow an exponential family distribution, such as Gaussian distribution, Poisson distribution, and chi-squared distribution. Derivation of the detectability function is discussed in Appendix B. The detectability function is in the form of $$\begin{aligned}
&h(\mu_{T_{a}},\mu_{N},\mu_{V_{e}})= \\
&\dfrac{1}{1+e^{-(\alpha_{0}+\alpha_{1}*\mu_{T_{a}}+\alpha_{2}*\mu_{N}+\alpha_{3}*\mu_{V_{e}})}},\\
\end{aligned}$$ where $\alpha_{0}$ to $\alpha_{3}$ are constants that need to be fitted, $\mu_{T_{a}}$, $\mu_{N}$, and $\mu_{V_{e}}$ are the bubble average antenna temperature, number of pixels, and expansion velocity divided by their maximum values (see Appendix B). The maximum value of $T_{a}$, $N$, and $V_{e}$ are obtained from real observed bubbles.
Parametric Detectability
------------------------
We performed 500 trials in both the TMC and the PMC to obtain the dataset of $N$, $T_{a}$, and $V_{e}$, and the detection result (0 or 1). $N$, $T_{a}$, and $V_{e}$ are then divided by their maximum values. The maximum values are obtained from real observed bubbles. We performed tenfold cross-validation on the experimental data sets to fit the constants, while testing how well the detectability function performs on the bubble identification and detectability prediction in each cloud. In tenfold cross-validation, the data sets are randomly partitioned into 10 equal size subsets. One single subset is used as the validation data for testing the model, while the remaining nine subsets are used as training data to fit the model. We used the maximum likelihood estimator (MLE; see Appendix B) to get an estimate for each parameter from training sets and use the remaining subset for validation. We repeated this procedure 10 times. Our estimate for the parameters is the average over the 10 training runs, while the error is the average over the 10 validation runs. Tenfold cross-validation is frequently used when evaluating performance of models on multiclass data [@G.D.13].
### Results in Taurus
We performed tenfold cross-validation on the Taurus experimental data set with random segment selection on all the data for each fold to fit the detectability function. We adopt the average value of 10 times the fitting results for $\alpha_{0}$ to $\alpha_{3}$, where $\alpha_{0}=-6.0$, $\alpha_{1}=112.0$, $\alpha_{2}=42.9$, and $\alpha_{3}=3.3$. The uncertainties are derived from the standard deviation of 10 times the fitting results for $\alpha_{0}$ to $\alpha_{3}$, which are $0.3$, $8.3$, $3.9$, and $0.5$, respectively. The detectability function in Taurus is described as $$h^T(\mu_{T_{a}},\mu_{N},\mu_{V_{e}})=\dfrac{1}{(1+e^{6.0-112.0*\mu_{T_{a}}-42.9*\mu_{N}-3.3*\mu_{V_{e}}})},$$ where $\mu_{T_{a}}$, $\mu_{N}$, and $\mu_{V_{e}}$ are scaled experimental bubble parameters – average antenna temperature, number of pixels, and expansion velocity – in Taurus.
Meanwhile, we estimated the training error from the training set and generalization error from the testing set to analyze how well $h^{T}$ performs on bubble identification and prediction in Taurus. The average training error is about 0.14 which is the probability that $h^{T}$ misclassifies samples in training sets. The average generalization error for $h^{T}$ is about 0.12 which is the probability if we draw a new set of bubble parameters and bubble detection results, $h^{T}$ misclassifies it. In Taurus bubble experiments, there are 330 identified experimental bubbles, which are 66$\%$ of total experiments. As long as the detectability function can fit and predict the bubble detection result with correctness larger than 66$\%$, we can consider that the detectability function can be used for bubble identification. According to the average training error and generalization error, the correctness of $h^{T}$ to fit and predict the bubble detection result is 86$\%$ and 88$\%$, respectively, which indicates that $h^{T}$ can well fit and predict the bubble detection result in Taurus.
### Results in Perseus
We performed tenfold cross-validation on the Perseus experimental data set with random segment selection on all the data for each fold to fit the detectability function. We adopt the average value of 10 times fitting results for $\alpha_{0}$ to $\alpha_{3}$, where $\alpha_{0}=-10.2$, $\alpha_{1}=121.3$, $\alpha_{2}=6.3$, and $\alpha_{3}=2.9$. The uncertainties are derived from the standard deviation of 10 times the fitting results for $\alpha_{0}$ to $\alpha_{3}$, which are $1.5$, $23.6$, $1.1$, and $0.2$, respectively. The detectability function in Perseus is described as $$h^{P}(\mu_{T_{a}},\mu_{N},\mu_{V_{e}})=\dfrac{1}{(1+e^{10.2-121.3*\mu_{T_{a}}-6.3* \mu_{N}-2.9*\mu_{V_{e}}})},$$ where $\mu_{T_{a}}$, $\mu_{N}$, and $\mu_{V_{e}}$ are scaled experimental bubble parameters — average antenna temperature, number of pixels, and expansion velocity — in Perseus.
Meanwhile, we estimated the training error from the training set and generalization error from the testing set to analyze how well $h^{P}$ performs on bubble identification and prediction in Perseus. The average training error is about 0.06 which is the probability that $h^{P}$ misclassifies samples in training sets. The average generalization error for $h^{P}$ is about 0.08 which is the probability if we draw a new set of bubble parameters and bubble detection result, $h^{P}$ would misclassify it. In Perseus bubble experiments, there are 321 identified experimental bubbles, which are 64.2$\%$ of total experiments. As long as the detectability function can fit and predict the bubble detection result with correctness larger than 64.2$\%$, we can consider that the detectability function can be used for bubble identification. According to the average training error and generalization error, the correctness of $h^{P}$ to fit and predict bubble detection result is 94$\%$ and 92$\%$, respectively, which indicate that $h^{P}$ can well fit and predict the bubble detection result in Perseus.
Comparison
----------
The detectability functions for bubble average antenna temperature, number of pixels, and expansion velocity for each cloud are illustrated in Figure 3. In Taurus, the change from yellow to deep blue is more gradual than in Perseus, which is caused by the larger training error in Taurus. We found that bubble detectability in Taurus and Perseus both strongly depend on average antenna temperature and number of pixels. The brightness of the bubble understandably dominates the detectability of the bubbles. In the case of a large, relatively regularly shaped bubble, the detectability roughly scaled with the number of pixels. Weak dependence is seen to occur for expansion velocity. However, we still find that bubbles with larger expansion velocity are easier to detect. In general, bubbles in Perseus are easier to detect than Taurus with the same $N$, $T_{a}$, and $V_{e}$. According to the behaviors of the detectability function, bubbles with low average antenna temperature, small spatial size, and slow expansion velocity are more likely to be missed during manual identification. We quantify the kinetic energy of the missing bubble in the following section.
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Kinetic energy of Missing bubbles
=================================
Bubble kinetic energy is associated with the input of mechanical energy into the parent molecular cloud, which is the crucial parameter to evaluate the impact of bubbles on the surrounding ISM. We estimated the completeness of previous bubble surveys by comparing the kinetic energy of missing bubbles with identified bubbles in Taurus and Perseus. The identified bubbles here and afterward represent the real identified bubbles in previous bubble surveys.
In order to estimate the kinetic energy distribution of all bubbles, including identified and missing bubbles in each molecular cloud, we assumed that the kinetic energy distribution of identified bubbles results from the kinetic energy distribution of all bubbles modified by the bubbles’ detectability. Therefore, the number of the missing bubble can be estimated by the difference between the kinetic energy distribution of identified bubbles and the total number of bubbles predicted by our model.
The relation between the kinetic energy distribution of identified bubbles and all bubbles in the molecular cloud can be expressed as $$\int N_{D}\cdot p(E_{k}) dE_{k} =\int N_{A}\cdot p^{tot}(E_{k})\cdot h(E_{k}) dE_{k},$$ where $N_{D}$ is the number of real identified bubbles, $N_{A}$ is the number of all bubbles in a given cloud, $p(E_{k})$ is the probability density function (PDF) of identified bubbles with respect to kinetic energy, $p^{tot}(E_{k})$ is the PDF of all bubbles with respect to kinetic energy within the cloud, and $h(E_{k})$ denotes the detectability function with respect to kinetic energy.
The number of missing bubbles $N^{miss}$ for kinetic energy is described as $$\begin{split}
N^{miss}=\int ^{E_{k}^{max}}_{E_{k}^{min}}[N_{A}\cdot p^{tot}(E_{k})-N_{D}\cdot p(E_{k})] d E_{k},\\
=N_{D}\cdot \int ^{E_{k}^{max}}_{E_{k}^{min}}[ p(E_{k})/h(E_{k})-p(E_{k})] d E_{k}.\\
\end{split}$$ The kinetic energy of missing bubbles $\rm E_{k}^{miss}$ is written by $$\begin{split}
E_{k}^{miss}=\int ^{E_{k}^{max}}_{E_{k}^{min}}E_{k}[N_{A}\cdot p^{tot}(E_{k}) -N_{D}\cdot p(E_{k})] d E_{k},\\
=N_{D}\cdot \int ^{E_{k}^{max}}_{E_{k}^{min}}E_{k}[p(E_{k})/h(E_{k})- p(E_{k})] d E_{k}.\\
\end{split}$$
We constructed the kinetic energy of each experimental bubble based on their number of pixels, average antenna temperature, and expansion velocity. The detectability of each experimental bubble was evaluated by the detectability function of Equation 2 and Equation 3. By fitting $E_{k}$ with detectability using MLE, we estimate parameters of the kinetic energy detectability function. The parameters of PDF of the kinetic energy of identified bubbles is estimated from the MLE. A detailed description of the fitting method is illustrated in Appendix \[MLE\].
The kinetic energy of missing bubbles in Taurus
------------------------------------------------
There are 37 identified bubbles $N_{D}$ in Taurus. Their total kinetic energy is about 9.2$\times 10^{46}$ erg. The maximum and minimum kinetic energies of identified bubbles are ${{E^{max}}_{k}}^{T}$ = 4.18$\times$ $10^{46}$ erg and ${{E^{min}}_{k}}^{T}$ = 0.2$\times$ $10^{44}$ erg, respectively. The detectability of each experimental bubble in Taurus is evaluated from Equation 2. We plotted $E_k$ versus detectability for both experimental bubbles and identified bubbles in the right panel of Figure 4. Our detectability function apparently overestimates the detectability of the bubble with low kinetic energy. However, such overestimation does not affect energy calculation related to feedback, as these bubbles add up to negligible energies compared to bright ones. One possibility is that there is a large number of small, low-energy, undetectable bubbles that might contribute significant energy to the clouds. In order for this to be true, there would need to be at least 4600 such bubbles in the TMC and 379 in the PMC. We find this to be unlikely. By fitting $E_{k}$ with detectability using MLE, we obtain parameters of the kinetic energy detectability function by $$h(E_{k})= tanh(925.5 \times E^{u}_{k}),$$ where ${E^{u}}_{k}$=$\dfrac{E_{k}}{{{E^{max}}_{k}}^{T}}$ refers to uniformed kinetic energy. We chose the tanh function because it changes from 0 to 1 gradually when kinetic energy is larger than 0, which is consistent with the detectability of all experimental bubbles from small kinetic energy to large kinetic energy. The distribution of detectability of all experimental bubbles is not a Bernoulli distribution. Therefore, we cannot use the logistic function to fit the kinetic energy detectability directly.
The bubble detectability increases as the kinetic energy increases, which makes the bubble with low kinetic energy hard to identify. The number of identified bubbles with low kinetic energy should be small. Although the bubble with high kinetic energy is easier to identify, it is unlikely that low-mass star formation in Taurus and Perseus would generate a lot of high kinetic energy bubbles. Therefore, the number of identified bubbles with high kinetic energy should be small as well. Lognormal distribution satisfies the above conditions and can fit the real identified bubbles well. The probability distribution of kinetic energy for identified bubbles $p(E_{k})$ can be well fitted in truncated lognormal distribution using MLE (see Appendix \[sec:mlef\]) by $$p(E_{k}) = \frac
{
-\sqrt {2}{{\rm e}^{-\frac{1}{2}\,{\frac {1}{{\sigma}^{2}} {\left( \ln \left(
{\frac {E_{k}}{m}} \right ) \right )} ^{2}}}}
}
{
\sqrt {\pi}\sigma\, \left( {\rm erf} \left(a_{{3}}\right )-{\rm erf}
\left(a_{{8}}\right ) \right ) E_{k}
}
\quad,$$ where $\sigma$ is the fitting shape parameter which is 1.7$\pm0.80$ in logarithm, erf is the error function, we defined $\mu=\ln(m)$ to be the logarithmic fitting scale parameter which is 102.8$\pm0.27$, $a_{{3}}$ and $a_{{8}}$ are compact notations which are given by $$a_3 =\frac{1}{2}\,{\frac {\sqrt {2} \left( \ln \left( {{E^{min}}_{k}}^{T} \right ) -\ln \left( m \right ) \right ) }{\sigma}}\quad , \nonumber$$ $$a_8 = \frac{1}{2}\,{\frac {\sqrt {2} \left( \ln \left( {{E^{max}}_{k}}^{T} \right ) -\ln
\left( m \right ) \right ) }{\sigma}}
\quad .
\nonumber$$
The fitting uncertainty was derived from 1000 sets of Monte Carlo experiments, each of size 37 (the number of identified bubbles in Taurus) with the kinetic energies ranging from ${{E^{min}}_{k}}^{T} $ to ${{E^{min}}_{k}}^{T} $. For each Monte Carlo experiment, the kinetic energy is randomly generated from the truncated lognormal distribution with $\mu=102.8$, $\sigma=1.7$. We applied the same fitting algorithm to those 37 random samples to estimate $\mu$ and $\sigma$ and repeated the Monte Carlo experiment 1000 times to get sets of the estimated value of $\mu$ and $\sigma$. We took the standard deviation of the estimated 1000 sets of $\mu$ and $\sigma$ to be the fitting uncertainty. Since there are only 37 identified bubbles in Taurus, we did not plot the PDF of $E_k$ of identified bubbles, instead we plotted the cumulative distribution function (CDF) of $E_k$. The CDF of $E_{k}$ for identified bubbles and CDF of $E_k$ of fitting distribution for Taurus are illustrated in the left panel of Figure 4.
{width="50.00000%"} {width="50.00000%"}
By combining Equation 7 and Equation 8 with Equation 5, the number of missing bubbles in Taurus can be derived, which ranges from 2 to 3. This is the probable number of bubbles not identified in surveys of L15 in Taurus. The total kinetic energy of the missing bubbles is estimated by combining Equation 7 and Equation 8 with Equation 6. However, the kinetic energy of most missing bubbles aggregates at less than $1 \times 10^{44}$ erg as illustrated in the green line of the right panel in Figure 6, which leads to the total kinetic energy of missing bubbles is ranging from 7.2$\times$10$^{43}$ erg to 8.6 $\times$10$^{43}$ erg. This corresponds to about 0.01$\%$ of the kinetic energy of identified bubbles. Therefore, the Taurus bubble survey can be considered to be energetically complete.
Kinetic energy of missing bubbles in Perseus
--------------------------------------------
There are 12 identified bubbles $N_{D}$ in Perseus. Their total kinetic energy is about 7.58$\times 10^{46}$ erg. The kinetic energy of identified bubbles ranges from 2$\times 10^{44}$ erg to 1.88$\times 10^{46}$ erg. We adopt the minimum and maximum kinetic energy of identified bubbles in Taurus (which has a wider range than in Perseus) to be the range used in estimating the missing kinetic energy in Perseus.
The detectability of each experimental bubble in Perseus is evaluated from Equation 3. We plotted $E_k$ versus detectability for both experimental bubbles and identified bubbles in the right panel of Figure 5. Similar to Taurus, we obtain parameters of the kinetic energy detectability function by fitting $E_{k}$ with detectability using MLE, which can be written as $$h(E_{k})=tanh(79.34\times E^{u}_{k}),$$ where ${E^{u}}_{k}$=$\dfrac{E_{k}}{{{E^{max}}_{k}}^{T}}$ refers to uniformed kinetic energy.
Similarly, the probability distribution of kinetic energy for identified bubbles in Perseus can also be well fitted in truncated lognormal distribution using MLE shown in Equation 8. The fitting shape parameter is 2.10$\pm0.43$ in logarithm. The logarithmic fitting scale parameter is 119.92$\pm0.22$. The fitting uncertainty was derived from 1000 sets of Monte Carlo experiments, each of size 12 (the number of identified bubbles in Perseus) with the kinetic energies ranging from ${{E^{min}}_{k}}^{P} $ to ${{E^{min}}_{k}}^{P} $. For each Monte Carlo experiment, the kinetic energy is randomly generated from the truncated lognormal distribution with $\mu=119.92$ and $\sigma = 2.10$. We applied the same fitting algorithm to those 12 random samples to estimate $\mu$ and $\sigma$ then repeated the Monte Carlo experiment 1000 times to get sets of the estimated value of $\mu$ and $\sigma$. The fitting uncertainty is the standard deviation of the estimated 1000 sets of $\mu$ and $\sigma$. There are only 12 identified bubbles in Perseus. Similar to Taurus, we plotted up the CDF of $E_k$ for identified bubbles and CDF of $E_k$ of the fitting distribution for Perseus, which are illustrated in the left panel of Figure 5.
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By combining Equation 8 and Equation 9 with Equation 5, the number of missing bubbles in Perseus can be derived which is ranging from 3 to 14. This is the probable number of bubbles not identified in the Perseus survey by Ar11. The total kinetic energy of the missing bubbles is estimated by combining Equation 9 and Equation 8 with Equation 6. However, the kinetic energy of most missing bubbles aggregates at less than $2 \times 10^{44}$ erg as illustrated in the green line of the left panel of Figure 6, which leads to the total kinetic energy of missing bubbles is ranging from 1.4$\times$10$^{44}$ erg to 8.0$\times$10$^{44}$ erg. This corresponds to about 1$\%$ of the kinetic energy of identified bubbles. Therefore, the Perseus bubble survey can be considered to be energetically complete.
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Discussion and CONCLUSIONS
==========================
Stellar feedback has a significant impact on the surrounding gas. The total kinetic energy of bubbles detected in Taurus and Perseus are $\sim3.9\times 10^{46}$ erg and $\sim7.6\times 10^{46}$ erg, respectively. The gravitational potential energy and turbulence energy ($E_{G}$, $E_{\rm tur}$) of Taurus and Perseus are about $1.5\times 10^{48}$ erg, $3.2\times 10^{47}$ erg, and $6\times 10^{47}$, $1.6 \times 10^{47}$ erg, respectively. The energy contained in bubbles are orders of magnitude smaller than those of either gravity or turbulence, which are expected to be equal in a virialized supersonic cloud. The maximum energy injection (into the natal cloud) rate of molecular bubbles can be estimated from their momentum. They are $\sim6.4\times10^{33}$ erg $\rm s^{-1}$, and $\sim2\times10^{33}$ erg $\rm s^{-1}$ for bubbles in Taurus and Perseus, respectively. Following the methods laid out in Ar11 and L15, we estimated the turbulence dissipation rate of Taurus and Perseus to be $\sim3.1\times10^{33}$ erg $\rm s^{-1}$ and $\sim1\times10^{33}$ erg $\rm s^{-1}$, respectively. The observed injection rates are thus similar to or even slightly larger than the dissipation rates.
Typically, when discussing molecular clouds we refer to total proton volume densities of 10$^{3}$ cm$^{-3}$ for regions with sufficient volume density to produce $^{13}$CO emission. This is typical for regions with A$_{v}$$\geq$1. However, the volumes of gas that are affected by bubbles are large and not necessarily located in the dense regions. In fact, they almost necessarily include large volumes of very diffuse gas. The gas that is dynamically affected (’moved’) by bubble expansion and can be identified as such has to exhibit a velocity offset (Figure 7). Such ’high’ velocity gas has much lower density. Due to this, we chose to use our estimates of the average H$_2$ volume densities within Taurus and Perseus (35 cm$^{-3}$ and 280 cm$^{-3}$ respectively) when estimating the energy injected into the cloud by each bubble. As a comparison, if the main cloud density was assumed to be 3.5 $\times$ 10$^3$ cm$^{-3}$, the energy injected into the cloud by star formation feedback in Taurus and Perseus would be $\sim3.9\times 10^{48}$ erg and $\sim9.1\times 10^{47}$ erg, respectively, which could disperse the whole cloud. Similarly, if we use such a high volume density for the undisturbed gas in our simulations, then the resulting bubbles are much brighter than any that are actually observed.
{width="50.00000%"}
We simulated bubbles for the Taurus and Perseus molecular cloud. We randomly inserted artificial bubbles into the Taurus and Perseus $\rm^{13}CO$ data cubes from the 13.7 m FCRAO telescope. With changing average antenna temperature, the number of pixels, and expansion velocity of artificial bubbles, we parameterized bubble detections according to the bubble detectability functions to evaluate how the detectability is affected by the parameters for two molecular clouds. According to the properties of identified bubbles in Taurus and Perseus and the detectability functions, we estimated the energetic completeness of previously stellar feedback studies. Our conclusions regarding the completeness of bubble identification in Taurus and Perseus and their properties are as follows:
1\. The detectability of bubbles can be described as a logistic function, which is derived from GLMs. The distribution of bubble detection results is a Bernoulli distribution. Therefore we adopted GLMs to fit the bubble detectability functions. In the TMC, the detectability function can be described as
$\rm h^{T}(\mu_{T_{a}},\mu_{N},\mu_{V_{e}})=\dfrac{1}{(1+e^{6.0-112.0*\mu_{T_{a}}-42.9*\mu_{N}-3.3*\mu_{V_{e}}})}$, while in the PMC as
$\rm h^{P}(\mu_{T_{a}},\mu_{N},\mu_{V_{e}})=\dfrac{1}{(1+e^{10.2-121.3*\mu_{T_{a}}-6.3* \mu_{N}-2.9*\mu_{V_{e}}})}$. We then fitted the bubble kinetic energy distributions in Taurus and Perseus with truncated lognormal distribution.
2\. The number of missing bubbles in Taurus is less than 8$\%$ of the number of identified bubbles. The number of missing bubbles in Perseus is about 25$\%$-125$\%$ of the number of identified bubbles.
3\. We used bubble kinetic energy distributions and the bubble detectability functions to estimate the total kinetic energy of missing bubbles, which suggests that although the numbers of missing bubbles are large, their kinetic energies are relatively small (usually less than 2$\times$10$^{44}$ erg). The total kinetic energies of missing bubbles in Taurus and Perseus during manual identification range from 7.2$\times$10$^{43}$ erg to 8.6 $\times$10$^{43}$ erg and 1.4$\times$10$^{44}$ erg to 8.0$\times$10$^{44}$ erg, respectively. Such potential incompleteness only accounts for $\sim0.2\%$ and $\sim1\%$ of the total kinetic energy of identified bubbles in Taurus and Perseus, respectively.
4\. The empirical surveys in L15 and Ar11 for identifying bubble structures in Taurus and Perseus can be considered as energetically complete.
5\. The total energy of bubbles in a cloud is orders of magnitude smaller than those of either turbulence or gravity. The bubbles cannot generate the observed turbulence or disperse the cloud. The observed energy injection rate from bubbles, now considered complete, is similar to the turbulence dissipation rate. We conclude that, even in low-mass star-forming regions, the feedback from star formation is sufficient to sustain turbulence at ranges from $\sim$0.1 pc to $\sim$2.8 pc scales.
This work is supported by the National Natural Science Foundation of China grant No. 11988101, No. 11725313, No. 11721303, the International Partnership Program of Chinese Academy of Sciences grant No. 114A11KYSB20160008, and the National Key R&D Program of China No. 2016YFA0400702.
Bubble simulation description {#sec:bubblesi}
=============================
The $^{13}$CO column densities may be described by $$\ N_{^{13}\rm CO} = 1.18\times 10^{15}T_{\rm b}\Delta V,
\label{equ:COColumndensity1}$$ where $\rm T_{b}$ is the brightness temperature of each pixel, $N_{^{13} \rm CO}$ is the column density, and $\Delta V$ represents the velocity width of a spectrometer channel which is listed in Table 1. This relation is derived under several assumptions. We assumed that the excitation temperature $T_{ex}$ of $^{13} \rm CO$ is 25 K, the background temperature is 2.7 K, the $^{13}\rm CO$ emission from the bubble is generally optically thin ($\tau (^{13}\rm CO) \ll 1$) and in local thermal equilibrium. $N_{^{13}\rm CO}$ can also be expressed as $$\ N_{^{13}\rm CO}= 3.0857 \times 10^{18} n_{\rm s} a_{\rm pc}cm^{-2},$$ where $a_{\rm pc}$ describes the bubble physical depth along the line of sight (LOS) in parsecs. By combining equation A1 with equation A2, we obtain the $T_{b}$ for each pixel in the artificial bubbles with known $a_{\rm pc}$, $\Delta V$, and $n_{s}$, which are given by $$\ T_{b}=2.615*10^{3}\frac{ n_{s}a_{\rm pc} }{\Delta V}.
\label{equ:COColumndensity2}$$
We assume that the $^{13}$CO abundance relative to H$_2$ is 1.43 $\times$ 10$^{-6}$. The densities of $^{13}$CO affected by bubble expansion $n_0$ are $5 \times 10^{-5} cm^{-3}$ and $4 \times 10^{-4} cm^{-3}$ for Taurus and Perseus, respectively. If all of the $^{13}$CO within the radius $R + \Delta R$ is distributed within the bubble, we can get $n_s$/$n_0$=1-${\xi}^{-3}$, where ${\xi}$=$(R+\Delta R)/R$.
Figure 8 illustrates how $a_{\rm pc}$ is calculated. It shows a $^{13}\rm CO$ bubble sampled by five velocity channels in the left hemisphere as seen by an observer located to the far left. The starting and ending angles of the second velocity channel are $\theta_{i}$ and $\theta_{i+1}$, respectively. The angular dimension of each velocity channel is given by $\cos (\theta_{i+1}) = \cos (\theta_{i}) + \Delta V/V_{exp}$. Thereby, the angle of any LOS within the velocity channel can be determined. The physical depth $a_{\rm pc}$ of each LOS can be calculated from the angle of this LOS, bubble radius, and thickness. The detailed calculations are presented in Appendix A of Cazzolato $\&$ Pineault 2005. By estimating the physical depth $a_{\rm pc}$ of each LOS for a given expansion velocity, and knowing $n_s$ of the bubble, we can obtain antenna temperature for each LOS using Equation A3.
{width="50.00000%"}
Bubble detectability function derivation and fitting
====================================================
According to the definition, the experimental bubble detection result, y, is either 0 or 1 which corresponds to a Bernoulli distribution. The probability mass function p of the detection results y is given by $$\begin{aligned}
p(y=1;\phi)&=&\phi,\\
p(y=0;\phi)&=&1-\phi,\end{aligned}$$ where $p(y=1;\phi)$ is the probability of detecting a bubble and $p(y=0;\phi)$ is the probability of not detecting a bubble. $p$ can be also expressed as [@NW72] $$\begin{aligned}
p(y;\phi)&=&\phi^{y} (1-\phi)^{1-y}\\
&=&\exp{(\log (\phi^{y}(1-\phi)^{1-y}) )}\\
&=&\exp{(y \log (\phi) +(1-y) \log (1-\phi))}\\
&=&\exp{(\log (\dfrac{\phi}{1-\phi})y+\log (1-\phi))}.\end{aligned}$$ The exponential family is written as $$\begin{aligned}
f(y;\eta)=b(y)\exp{(\eta^{T} T(y)-a(\eta))},\end{aligned}$$ where $\eta$ is the natural parameter of the distribution; $T(y)$ is the sufficient statistic; and $a(\eta)$ is the log partition function. When $\eta$ = $\log(\dfrac {\phi}{1-\phi})$, $a(\eta)=-\log(1-\phi)$, $T(y) = y$, and $b(y)=1$, we can derive the $p(y;\phi)$ from $f(y;\eta)$. This indicates that the probability mass function of the experimental bubble detection result is one of exponential family. The experimental bubble parameters: $N$, $T_{a}$ and $V_{e}$ can be considered as independent to each other, so that we can combine them linearly after scaling. The scaling rule is as follows: $$\mu_{scale} = \dfrac {\mu}{\mu_{max}}\\
\label{equ:er}$$ where $\mu$ represents the bubble parameter, $\mu_{min}$ and $\mu_{max}$ refer to the minimum and the maximum value of parameter.
Thus, $\eta$ can be constructed as $\eta= \alpha^{T} X$. Here, $X$ denotes a three-dimensional vector containing $N$, $T_{a}$ and $V_{e}$. $\alpha$ denotes a four-dimensional coefficient vector containing an intercept $\alpha_{0}$. Our goal is to predict the expected value of the probability of detecting the bubble y with given $X$, which means we would like the detectability function $h(X)$ output to satisfy $h(X) = E[y|X]$. In our formulation of the Bernoulli distribution as an exponential family distribution, we had $\eta = log(\dfrac {\phi}{1-\phi})$ which can be written as $\phi=\dfrac {1}{1+e^{-\eta}}$ and $E[y|X;\alpha]=\phi$. So that the bubble detectability function $h(X)$ can be expressed as $$\begin{aligned}
h(X)&=&E[y|X;\alpha] \\
&=&\phi \\
&=&\dfrac {1}{1+e^{-\alpha^{T} X}}.
\label{equ:er}\end{aligned}$$ $h(X)$ is the logistic function. With given experimental bubble parameter and bubble detection result datasets, we can estimate the $\alpha^{T}$ for $h(X)$ based on the maximizing the likelihood function $l(\theta)=\sum ^{rn}_{i=1} y^{(i)}log( h(X^{(i)}))+ (1-y^{(i)})log(1- h(X^{(i)})$, where rn is the row number of the data set.
The kinetic energy probability density function fitting {#sec:mlef}
=======================================================
We adopt the MLE to estimate the parameters of PDF of kinetic energy for both Taurus and Perseus using the truncated lognormal distribution. The derivation for the MLE of the truncated lognormal distribution largely follows Zaninetti, L. 2017. Consider a sample ${\mathcal X}=x_1, x_2, \dots , x_n$. The maximum and minimum value of sample can be expressed as $x_l$ and $x_u$, respectively, which are given by $${x_l}=\max(x_1, x_2, \dots, x_n), \qquad {x_u}=\min(x_1, x_2, \dots, x_n)
\quad .
\label{eq:firstpar}$$ The PDF can be expressed as $$PDF (x;m,\sigma,x_l,x_u) = \frac
{
-\sqrt {2}{{\rm e}^{-\frac{1}{2}\,{\frac {1}{{\sigma}^{2}} \left( \ln \left(
{\frac {x}{m}} \right ) \right ) ^{2}}}}
}
{
\sqrt {\pi}\sigma\, \left( {\rm erf} \left(a_{{3}}\right )-{\rm erf}
\left(a_{{8}}\right ) \right ) x
}
\quad,
\label{pdflognormaltruncatedcompact}$$ where $m$ is the scale parameter, $\sigma$ is the shape parameter, $a_{{3}}$ and $a_{{8}}$ are compact notations which are given by $$a_3 =\frac{1}{2}\,{\frac {\sqrt {2} \left( \ln \left( x_{{l}} \right ) -\ln
\left( m \right ) \right ) }{\sigma}}
\quad ,
\nonumber$$ $$a_8 = \frac{1}{2}\,{\frac {\sqrt {2} \left( \ln \left( x_{{u}} \right ) -\ln
\left( m \right ) \right ) }{\sigma}}
\quad .
\nonumber$$ The CDF can be expressed as $$CDF (x;m,\sigma,x_l,x_u)=
\frac
{
-{\rm erf} \left(\frac{1}{2}\,{\frac {\sqrt {2}}{\sigma}\ln \left( {\frac {x
}{m}} \right ) }\right )+{\rm erf} \left(a_{{3}}\right )
}
{
{\rm erf} \left(a_{{3}}\right )-{\rm erf} \left(a_{{8}}\right )
}
\quad ,$$
The MLE is obtained by maximizing $$\Lambda = \sum_i^n \ln(PDF(x_i;m,\sigma,x_l,x_u)).$$ The two derivatives $\frac{\partial \Lambda}{\partial m} =0$ and $\frac{\partial \Lambda}{\partial \sigma}=0 $ generate two nonlinear equations in $m$ and $\sigma$ which can be solved numerically, $$\begin{aligned}
\frac{\partial \Lambda}{\partial m}=
( {\rm erf} (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln ( x_{{
l}} ) -\ln ( m ) ) }{\sigma}} )-
{\rm erf} (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln ( x_{{u}}
) -\ln ( m ) ) }{\sigma}} ) )
\nonumber \\
( n\sqrt {2}\sigma\,{{\rm e}^{-\frac{1}{2}\,{\frac { ( \ln (
x_{{l}} ) -\ln ( m ) ) ^{2}}{{\sigma}^{2}}}}}
-n\sqrt {2}\sigma\,{{\rm e}^{-\frac{1}{2}\,{\frac { ( \ln ( x_{{u}}
) -\ln ( m ) ) ^{2}}{{\sigma}^{2}}}}}
\nonumber \\
-\sqrt
{\pi} ( {\rm erf} (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln
( x_{{l}} ) -\ln ( m ) ) }{\sigma}}
)
\nonumber \\
-{\rm erf} (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln ( x_{{
u}} ) -\ln ( m ) ) }{\sigma}} ) )
( n\ln ( m ) -\sum _{i=1}^{n}\ln ( x_{{i}}
) ) ) =0
\quad ,\end{aligned}$$ and $$\frac{\partial \Lambda}{\partial \sigma}= \frac{N}{D} =0,$$ where $$\begin{aligned}
N = \ln \left( x_{{u}} \right) \sqrt {2}{{\rm e}^{-\frac{1}{2}\,{\frac { (
\ln ( x_{{u}} ) -\ln ( m ) ) ^{2}}{{
\sigma}^{2}}}}}n\sigma-\ln ( x_{{l}} ) \sqrt {2}{{\rm e}^{
-\frac{1}{2}\,{\frac { ( \ln ( x_{{l}} )
-\ln ( m
) ) ^{2}}{{\sigma}^{2}}}}}n\sigma
\nonumber \\
+\sqrt {2}{{\rm e}^{-1/
2\,{\frac { ( \ln ( x_{{l}} ) -\ln ( m )
) ^{2}}{{\sigma}^{2}}}}}\ln ( m ) n\sigma-\sqrt {2}
{{\rm e}^{-\frac{1}{2}\,{\frac { ( \ln ( x_{{u}} ) -\ln
( m ) ) ^{2}}{{\sigma}^{2}}}}}\ln ( m
) n\sigma
\nonumber \\
+n ( \ln ( m ) ) ^{2}\sqrt {
\pi}{\rm erf} (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln ( x_{{u}}
) -\ln ( m ) ) }{\sigma}} )
\nonumber \\
-n{\sigma}^{
2}\sqrt {\pi}{\rm erf} (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln
( x_{{u}} ) -\ln ( m ) ) }{\sigma}}
)
\nonumber \\
-n ( \ln ( m ) ) ^{2}\sqrt {\pi}
{\rm erf} (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln ( x_{{l}}
) -\ln ( m ) ) }{\sigma}} )
\nonumber \\
+n{\sigma}^{
2}\sqrt {\pi}{\rm erf} (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln
( x_{{l}} ) -\ln ( m ) ) }{\sigma}}
)
\nonumber \\
+\sum _{i=1}^{n}\ln ( x_{{i}} ) ( \ln
( x_{{i}} ) -2\,\ln ( m ) ) \sqrt {\pi}
{\rm erf} (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln ( x_{{u}}
) -\ln ( m ) ) }{\sigma}} )
\nonumber \\
-\sum _{i=1
}^{n}\ln ( x_{{i}} ) ( \ln ( x_{{i}} ) -
2\,\ln ( m ) ) \sqrt {\pi}{\rm erf} (\frac{1}{2}\,{
\frac {\sqrt {2} ( \ln ( x_{{l}} ) -\ln ( m
) ) }{\sigma}} )
\quad ,\end{aligned}$$
$$\begin{aligned}
D=\sqrt {\pi} \Bigg ( -{\rm erf}
\bigg (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln
( x_{{l}} ) -\ln ( m ) ) }{\sigma}}
\bigg )
\nonumber \\
+{\rm erf} \bigg (\frac{1}{2}\,{\frac {\sqrt {2} ( \ln ( x_{{
u}} ) -\ln ( m ) ) }{\sigma}} \bigg ) \Bigg )
{\sigma}^{3} \quad .\end{aligned}$$
Arce, H. G., Borkin, M. A., Goodman, A. A., Pineda, J. E., & Beaumont, C. N. 2011, , 742, 105
Arce, H. G., Borkin, M. A., Goodman, A. A., Pineda, J. E., & Halle, M. W. 2010, , 715, 1170
Arce, H. G., & Sargent, A. I. 2006, , 646, 1070
Arce, H. G., & Goodman, A. A. 2002, , 575, 911
Arce, H. G., & Goodman, A. A. 2001, , 554, 132
Bally, J., Reipurth, B., & Davis, C. J. 2007, , 951, 215
Beaumont, C. N., Goodman, A. A., Kendrew, S., et al. 2014, , 214, 3
Cazzolato, F., & Pineault, S. 2005, , 129, 2731
COMPLETE team, 2011, https://hdl.handle.net/10904/10075, Harvard Dataverse, V2
Churchwell, E., Povich, M. S., Allen, D., et al. 2006, , 649, 759
Duarte-Cabral, A., Chrysostomou, A., Peretto, N. et al. 2012, , 543, 140
Federrath, C., 2015, , 450, 4035
Feddersen J. R., Arce, Héctor G. , Shuo Kong, et al. , 862, 121
Frank, A., Ray, T. P., Cabrit, S., et al. 2014, , 914, 451
Fuller, G. A., & Ladd, E. F. 2002, , 573, 699
Fukui, Y., Sugitani, K., Takaba, H., et al. 1986, , 311, 85
Goldsmith, P. F., Heyer, M., Narayanan, G., et al. 2008, , 680, 428
G. James, D. Witten, T. Hastie, and R. Tibshirani, 2013, An introduction to statistical learning, Vol. 112, Springer
Heiles, C., 1979, , 229, 533
Hartmann, L., Ballesteros-Paredes, J., & Bergin, E. A. 2001, , 562, 852
Kroupa P., Jeřábková T., Dinnbier F., Beccari G., Yan Z., 2018, , 612, 74
Kwan, J., & Scoville, N. 1976, , 210, l39
Lada, C. J. & Harvey, P. M. 1981, , 245, 58
Li, H. X., Li, D., Xu, D., et al. 2015, , 219, 20
Matzner, C. D. 2002, , 566, 302
Mottram, J. C., van Dishoeck, E. F., Kristensen, L. E., et al. 2017, , 600, A99
Nakamura, F., Sugitani, K., Shimajiri, Y., et al. 2011, , 737, 56
Nakamura, F., Kamada, Y., Kamazaki, T., et al. 2011, , 726, 46
Narayanan, G., Heyer, M. H., Brunt, C., et al. 2008, , 177, 341
Narayanan, G., Snell, R., & Bemis, A. 2012, , 425, 2641
Nelder J. A., Wedderburn R. W. M., 1972, Journal of the Royal Statistical Society, Series A, General, 135, 370, DOI: https://doi.org/10.2307/2344614
Norman, C., & Ikeuchi, S. 1989, , 345, 372
Offner, S. S. R., & Arce, H. G. 2015, , 811, 146
Plunkett, A. L., Arce, H. G., Corder, S. A., et al. 2013, , 774, 22
Ridge, N. A., Di Francesco, J., Kirk, H., et al. 2006, , 131, 2921
T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning, Springer-Verlag, New York, 2001.
Solomon, P. M., Huguenin, G. R., & Scoville, N. Z. 1981, , 245, 19
Snell, R. L., Loren, R. B., Plambeck, R. L. 1980, , 239, 17
Zaninetti, L. 2017, Advances in Astrophysics, 2, 197, DOI: 10.22606/adap.2017.23005
Zinnecker, H., & Yorke, H. W. 2007, , 45, 481
|
---
abstract: |
We present new proper motion (PM) measurements of the dwarf spheroidal galaxies (dSphs) Draco and Sculptor using multi-epoch images obtained with the [*Hubble Space Telescope*]{} ACS/WFC. Our PM results have uncertainties far lower than previous measurements, even made with the same instrument. The PM results for Draco and Sculptor are $(\mu_{W},\>\mu_{N})_{\rm Dra}
= (-0.0562 \pm 0.0099,\>-0.1765 \pm 0.0100)\ {\rm mas\ yr}^{-1}$ and $(\mu_{W},\>\mu_{N})_{\rm Scl} = (-0.0296 \pm 0.0209,\>-0.1358 \pm 0.0214)\
{\rm mas\ yr}^{-1}$. The implied Galactocentric velocity vectors for Draco and Sculptor have radial and tangential components: $(V_{\rm rad},\>V_{\rm tan})_{\rm Dra} = (-88.6,\>161.4) \pm (4.4,\>5.6) {\>{\rm km}\,{\rm s}^{-1}}$; and $(V_{\rm rad},\>V_{\rm tan})_{\rm Scl} = (72.6,\>200.2) \pm (1.3,\>10.8) {\>{\rm km}\,{\rm s}^{-1}}$. We study the detailed orbital histories of both Draco and Sculptor via numerical orbit integrations. Orbital periods of Draco and Sculptor are found to be 1–2 and 2–5 Gyrs, respectively, accounting for uncertainties in the MW mass. We also study the influence of the Large Magellanic Cloud (LMC) on the orbits of Draco and Sculptor. Overall, the inclusion of the LMC increases the scatter in the orbital results. Based on our calculations, Draco shows a rather wide range of orbital parameters depending on the MW mass and inclusion/exclusion of the LMC, but Sculptor’s orbit is very well constrained with its most recent pericentric approach to the MW being 0.3–0.4 Gyr ago. Our new PMs imply that the orbital trajectories of both Draco and Sculptor are confined within the Disk of Satellites (DoS), better so than implied by earlier PM measurements, and likely rule out the possibility that these two galaxies were accreted together as part of a tightly bound group.
author:
- |
Sangmo Tony Sohn, Ekta Patel, Gurtina Besla, Roeland P. van der Marel, James S. Bullock,\
Louis E. Strigari, Glenn van de Ven, Matt G. Walker, & Andrea Bellini
title: |
Space Motions of the Dwarf Spheroidal Galaxies Draco and Sculptor\
based on [*HST*]{} Proper Motions with $\sim10$-year Time Baseline
---
Introduction
============
The orbital histories of Milky Way (MW) satellites contain crucial information about the formation and assembly history of the MW halo. Direct access to proper motion (PM) measurements is required to derive their orbits. Despite the various efforts to measure PMs of MW satellites in the past decade or so, both the quantity and the quality of measurements are still lacking. The only clear solution to this problem is to directly measure PMs of tracer objects, but this has been technically challenging due to the difficulty in measuring very small apparent motions. The excellent astrometric capability of [*HST*]{} has eased the situation, and combined with our PM measurement technique using background galaxies as stationary reference sources [@soh12], we are now able to reach unprecedented PM accuracies using multi-epoch [*HST*]{} data.
As part of our HSTPROMO collaboration [@vdm14], we are carrying out [*HST*]{} programs to measure PMs of MW satellite objects. For example, we measured the PM of Leo I using multi-epoch ACS/WFC images separated by 5 years in time, and explored its orbits under realistic MW potentials [@soh13]. In addition, by comparing the observed parameters based on our PM measurements to Leo I-like subhalos found in cosmological simulations, we constrained the virial mass of the MW [@boy13]. We are continuing to measure PMs of distant satellites in the MW halo, including ultra-faint dwarfs and classical dwarf spheroidal galaxies (dSphs). This study focuses on two classical dSphs, Draco and Sculptor.
Draco and Sculptor are located at distances of 76 and 86 kpc, respectively. As dynamical tracers, they probe the MW mass at important distances where there are only a limited number of tracer objects with tangential velocities. The MW mass is generally estimated through equilibrium modeling using observed dynamical properties of halo tracers like satellites. Without the knowledge of tangential velocities, however, the mass estimates suffer from the (in)famous mass-anisotropy degeneracy. @wat10 used older PM measurements of Draco and Sculptor [@sch94; @pia06], along with those of other dwarf satellites, to estimate the mass of the MW. However, the quality of these older PM measurements have limited their ability to constrain the MW mass with high confidence.
Draco and Sculptor are interesting because, while they are found at similar Galactocentric distances, they appear to be orbiting around the MW in opposite directions. This was first noted by @pry08who analyzed the space motions of galaxies with the PMs that existed at the time. Many of the satellite galaxies of the MW are found to be distributed on a “Disk of Satellites (DoS)”, an orbital plane claimed to be occupied by most of the classical dwarf satellites of the MW [@lyn76; @kro05; @met07]. Most dwarfs that are believed to be members of the DoS and also have PM measurements are found to orbit in the same direction about the MW, with the notable exception of Sculptor. Interestingly, Sculptor seems to be orbiting around the plane in the [*opposite*]{} direction of most satellites. @paw11 tried to explain this in the context of tidal-dwarf galaxies. Better PM measurements for a representative galaxy that rotates along the plane (Draco), and for a galaxy that seems to counter-rotate (Sculptor) are needed to shed new light on this matter.
This picture is further complicated by perturbations from the MW’s most massive satellite galaxy, the Large Magellanic Cloud (LMC). It is unclear whether the gravitational pull of the LMC might complicate the orbits of the classical satellites, causing their membership to the plane of satellites to be unstable. The recent capture of the LMC by the MW [@bes07; @kal13] may limit its dynamical influence on the DoS members, but this is impossible to properly assess without accurate PM measurements for the classical dwarfs. Draco and Sculptor, with their opposite sense of motions, and the coincidence of their orbital planes with that of the LMC, present an ideal test case for the influence of the LMC on the DoS.
As with other satellite objects in the MW halo, the first PM measurements of Draco and Sculptor were carried out using photographic plates [@sch94; @sch95]. The quality of these measurements is poor by modern standards.[^1] It was not until the use of [*HST*]{} that PM uncertainties were small enough that the results were meaningful. @pia06 used multi-epoch imaging data obtained with the Space Telescope Imaging Spectrograph (STIS) onboard [*HST*]{} to measure the PM of Sculptor. They used quasi-stellar objects (QSOs) in two fields as stationary reference sources to reach a final 1-d PM uncertainty of $0.13 {\>{\rm mas}\,{\rm yr}^{-1}}$. For Draco, @pry15 measured the PM using both QSOs and background galaxies in a single field to achieve a 1-d PM uncertainty of $0.063 {\>{\rm mas}\,{\rm yr}^{-1}}$. Meanwhile, @cas16 used ground-based images obtained with the [*Subaru*]{} Suprime-Cam to measure the PM of Draco with a smaller 1-d PM uncertainty of $0.044 {\>{\rm mas}\,{\rm yr}^{-1}}$. Notwithstanding the $\sim 6\sigma$ level discrepancy found against the [*HST*]{} measurement by @pry15, this study demonstrates what can be achieved using wide-field detectors on an 8-meter class telescope when extensive calibrations are carried out. It also provides hints on what to expect in the [*LSST*]{} era for PM measurements with large telescopes.
In this paper, we present our new PM measurements for Draco and Sculptor using multi-epoch [*HST*]{} imaging data separated by $\sim 10$ years in time. This paper is organized as follows. In Section \[s:pm\], we describe the data, outline the analysis steps, and report the results of our PM measurements. In Section \[s:spacemotions\], we derive the Galactocentric space motions of Draco and Sculptor by correcting the measured PMs for the solar motions. In Section \[s:orbits\], we explore the implications for the past orbits of Draco and Sculptor under various assumptions for the mass of the MW, and also explore the gravitational influence of the Large Magellanic Cloud (LMC) on their orbits. Finally, in Section \[s:conclusions\], we summarize the main results of our paper.
Proper Motions {#s:pm}
==============
Data {#ss:data}
----
[lcccccccc]{} [**Draco**]{} & & & & **[F606W]{} & & & **[F606W]{} & **[F814W]{}\
[*F1*]{} & 17:21:01.34 & $+$57:58:38.5 & 2004-10-19 & 430s$\times$19 & & 2013-10-14 & 453s$\times$12 & 300s$\times$3\
[*F2*]{} & 17:21:51.69 & $+$58:01:41.0 & 2004-10-31 & 430s$\times$19 & & 2012-10-24 & 501s$\times$12 & 300s$\times$3\
[*F3*]{} & 17:19:29.97 & $+$57:58:10.2 & 2004-10-30 & 430s$\times$19 & & 2012-10-26 & 507s$\times$12 & 300s$\times$3\
[**Sculptor**]{} & & & & **[F775W]{} & & & **[F775W]{} & **[F606W]{}\
[*F1*]{} & 00:59:57.31 & $-$33:46:23.5 & 2002-09-28 & 417s$\times$5 & & 2013-09-29 & 419s$\times$16 & 150s$\times$4\
[*F2*]{} & 00:59:48.61 & $-$33:48:47.1 & 2002-09-26 & 400s$\times$6 & & 2013-09-29 & 419s$\times$16 & 150s$\times$4\
************
Figure \[f:fields\] shows the [*HST*]{} ACS/WFC fields we used for measuring the PMs of Draco and Sculptor. The first-epoch ACS/WFC data for Draco were observed in 2004 October through the [*HST*]{} program GO-10229 (PI: S. Piatek). [^2] Field [*DRACO-F1*]{} was observed with ACS/WFC again two years later in 2006 October through [*HST*]{} program GO-10812 to measure the PM of Draco. Results using these two-year baseline data have been reported in @pry15. The [*DRACO-F1*]{} and [*-F2*]{} fields were observed in F606W, while the [*DRACO-F3*]{} field was observed in F555W to avoid saturating the quasi-stellar objects (QSOs). Due to the failure of ACS/WFC in 2006–2007, fields [*DRACO-F2*]{} and [*-F3*]{} were observed with WFPC2 in 2007 December. However, we did not consider using the WFPC2 data for PM measurements for the same reasons as discussed in our Leo I paper [@soh13].
All three Draco fields were re-observed through our [*HST*]{} program GO-12966 (PI: R. van der Marel) using the same configurations (i.e., filters, telescope pointings, and orientations) as in the 2004–2006 observations. The [*DRACO-F1*]{} field was observed in 2013 October, and [*DRACO-F2*]{} and [*-F3*]{} in 2012 October. All three target fields of Draco have QSOs in them as well as plenty of bright and compact background galaxies that can be used as stationary reference objects.
For Sculptor, we used two fields just outside the core radius as shown in Figure \[f:fields\]. The first-epoch data for Sculptor were observed in 2002 September through [*HST*]{} program GO-9480 to measure the weak lensing (or cosmic shear) of background galaxies. We re-observed these two fields in 2013 September, again using the same telescope pointings and orientations as in the 2002 observations.
In the course of our second-epoch observations through program GO-12966, we also acquired short exposures in different filters (F814W for the Draco fields, and F606W for the Sculptor fields) to construct color-magnitude diagrams (CMDs) of stars in our target fields. A summary of observations for each target galaxy is shown in Table \[t:obslog\].
The primary goal of our [*HST*]{} GO-12966 program is to study the internal PM dynamics of stars in Draco and Sculptor, and we are in the process of analyzing the results which will be presented in a separate forthcoming paper. All of the exposures obtained through our [*HST*]{} GO-12966 program made use of the experimental POST-FLASH capability to mitigate the impact from charge transfer efficiency (CTE) losses. This was important because the typical exposure time for individual images were all about 500 sec, which is significantly less than those in our other studies [e.g., @soh13].
Measurements {#ss:measurements}
------------
We compared the two epochs of F606W/F555W (for Draco) and F775W (for Sculptor) observations to measure the absolute PMs of our target galaxies. This was accomplished by determining the shifts of member stars in the dSphs with respect to two different types of stationary objects, galaxies and QSOs, in the distant background. Our methodology generally follows that of our previous works on M31 and Leo I [@soh12; @soh13], and so we refer readers interested in the the details to those papers. Here we outline the main features of our PM derivation process.
### Initial Analysis Steps
We downloaded both the regular flat-fielded [\_flt.fits]{} and corrected [\_flc.fits]{} images from the Mikulski Archive for Space Telescopes (MAST). The latter images are pre-processed for the imperfect charge transfer efficiency (CTE) using the pixel-based correction algorithms of @and10. The PM measurements were performed using both sets of images since we were uncertain how the current version of the CTE correction routine we used performs on images obtained using the POST-FLASH option. We found that the [\_flc.fits]{} images taken with the POST-FLASH option were somewhat overcorrected for the imperfect CTE which causes systematics in our PM measurements. Therefore, the final results were all derived from the [\_flt.fits]{} images. We carefully examined the individual [\_flt.fits]{} images for both epochs and found that the level of CTE loss for the images taken in 2012–2013 are comparable to those of the 2004–2006 data thanks to the 2012-2013 POST-FLASH observations. In the end, this has worked to our advantage for PM measurements since the impact of CTE loss on astrometry was found to almost cancel out when taking the difference in positions of objects between the two epochs. As we discuss below, we also used local corrections for mitigating the residual systematics.
As the first step, we processed the [ \_flt.fits]{} images using the [img2xym\_WFC.09x10]{} program from @and06 to obtain a position and a flux for each star in each exposure. We applied corrections to the the positions using the known ACS/WFC geometric distortions. We then created a high-resolution stacked image for each field using the first- (for the Draco dSph) and second-epoch (for the Sculptor dSph) images. Stars and galaxies were then identified from the stacked images. Photometric measures from the [img2xym\_WFC.09x10]{} program were used to create a CMD for each field, and this was used to identify member stars of our target dSphs. The subsequent analysis steps are different depending on which type of background objects are used as stationary references sources. We discuss further steps for each case below.
### Background Galaxies as Stationary Reference Sources {#sss:galaxies}
For each star and background galaxy, a template was constructed from the high-resolution stacked image. This template was used to measure a position for each object in each individual exposure in each epoch. Templates were fitted directly to the images of the epoch from which they were created (first-epoch for Draco, and second-epoch for Sculptor). For fitting templates to the images of the other epoch, we included $7\times7$ pixel convolution kernels to allow for PSF differences between epochs. These kernels were derived using bright and isolated Draco/Sculptor stars distributed throughout the fields.
The template-based positions of stars for multiple exposures were averaged and used to define first- (for the Draco dSph) or second-epoch (for the Sculptor dSph) reference frames. We used the positions of the stars in each of the second- (Draco) or first-epoch (Sculptor) exposures to transform the template-measured positions of the galaxies into the reference frames. We then took the difference between the first- and second-epoch positions of galaxies to obtain the relative displacement of the galaxies with respect to the dSph stars. To remove any remaining systematic PM residuals associated with the detector position and brightness of sources (e.g., due to imperfect CTE corrections) we derived and applied a local correction for each background galaxy using stars of similar brightness that lie in the vicinity. Finally, we multiply the relative displacements of the galaxies by $-1$ to obtain the mean absolute displacement of the dSph stars, since in reality the galaxies/QSOs are stationary and the stars are moving. Multiplying the resulting displacements by the pixel scale of our reference images ($50\>{\rm mas}\,{\rm pix}^{-1}$), and dividing by the time baseline turns our results into actual PMs.
For the [*DRACO-F1*]{} field, we have data obtained in 2004, 2006, and 2013 as described in Section \[ss:data\]. For the final PM measurement, we used the 2004 and 2013 data as our first and second epoch, respectively. The 2006 data were used to provide an extra check (see Section \[sss:results\_draco\]), but they were not included in our final results.
Because the [*DRACO-F3*]{} field was observed with F555W, which has only about half the bandwidth of F606W, the background galaxies are significantly fainter than those detected in the F606W images. After attempting to construct and fit templates to the background galaxies in this field, we concluded that the overall quality of positional measurements were too poor to include in our results. For this reason, the PM results for [*DRACO-F3*]{} field are only reported using QSOs as stationary reference sources (see Section \[sss:qso\]).
### QSOs as Stationary Reference Sources {#sss:qso}
All three of our Draco fields include QSOs in them, and we use these objects to provide an independent measurement of the Draco PM. For the [*DRACO-F1*]{} field, @pry15 used two QSOs, one detected in the top ACS/WFC chip (WFC1), and the other detected in the bottom chip (WFC2). Both QSOs were easily identified in our images thanks to Figure 2 of @pry15. However, due to the increase in individual exposure times for our GO-12966 data, we found that the QSO located in WFC1 is slightly saturated in the images taken in 2012, making its positional measurement unreliable. We therefore decided to only use the QSO detected in WFC2. This QSO, and the QSOs in the other two Draco fields were detected in the 2012 data with counts well below the saturation limits.
For measuring PMs using QSOs as reference sources, we used the positions of QSOs and stars measured based on the library PSFs by the [img2xym\_WFC.09x10]{}, instead of using the template-based positions described in Section \[sss:galaxies\]. This is because the PSFs of QSOs are virtually indistinguishable from the PSFs of stars, and because the library-based positions are more accurate than the template-based positions. We start by only selecting member stars of Draco, based on their CMD properties, that are detected on the same ACS/WFC image quadrant as the QSO. The positions of these stars in each individual image are corrected for the known geometric distortions, and subsequently averaged separately for the first- and second-epoch data. The positions of stars in the first epoch data are used to define a reference frame. We then used the positions of stars in the second epoch to transform the position of the QSO into the reference frame. As a result, the PM of Draco stars can be inferred by taking the difference between the first-epoch reference QSO position and the second-epoch transformed QSO position, multiplying the results by $-1$, converting pixels to mas, and dividing by the time baseline.
The QSOs we used for measuring PMs are typically brighter than most of the Draco member stars we used for setting up the reference frame. For example, in the same quadrant as the QSO located in the [*DRACO-F1*]{} field, there are only 18 out of 132 Draco stars that are brighter than the QSO. For the other two fields, the situation is worse: only two and one out of 68 and 42 Draco stars are brighter than the QSOs in fields [*DRACO-F2*]{} and [*F3*]{}, respectively. This can potentially cause CTE-related systematics since the CTE degradation is known to be a strong function of the brightness of a source, and we are using stars of different brightnesses than the QSOs to define the reference frame. To correct for this effect, the procedure described above was iterated using Draco stars in different brightness ranges to define the reference frame. In our first iteration, the measurement was carried out using stars brighter than an instrumental magnitude of $m_{\rm instr} = -9.00$. [^3] In subsequent iterations, we decreased this limit in steps of 0.5 mags until the faint limit was $m_{\rm instr} = -11.50$. For each step, we compute the PMs of the QSOs and the median magnitude of stars used in the transformation process. This provides a relation between the brightness of stars and the measured PMs. We fit a line to this relation, and computed the PM for the case of stars having the same brightness as the QSOs. These relations are monotonic implying that we are indeed correcting for the residual CTE effect. The final PMs of Draco stars with respect to the stationary QSOs were then derived using the same procedure as outlined in Section \[sss:galaxies\]. The final PM uncertainties were obtained by taking the quadrature sum of the uncertainties in the average positions at the two epochs, and the uncertainties from fitting the lines to the PM versus brightness relation.
As with the case of using background galaxies as stationary references, we used the [*DRACO-F1*]{} field’s 2006 data only as an extra check, and our final PM for this field was obtained using 2004 versus 2013 data.
Results {#ss:results}
-------
### Draco Dwarf Spheroidal Galaxy {#sss:results_draco}
[lcccc]{} (Galaxies) & $-0.0168$ & $-0.1958$ & $0.0290$ & $0.0294$\
[**F1**]{} (QSO) & $-0.0463$ & $-0.2025$ & $0.0188$ & $0.0164$\
[**F2**]{} (Galaxies) & $-0.0526$ & $-0.1812$ & $0.0264$ & $0.0265$\
[**F2**]{} (QSO) & $-0.0825$ & $-0.1478$ & $0.0174$ & $0.0179$\
[**F3**]{} (QSO) & $-0.0512$ & $-0.1386$ & $0.0263$ & $0.0348$\
Weighted average & $-0.0562$ & $-0.1765$ & $0.0099$ & $0.0100$\
Our PM results for the Draco dSph are presented in Table \[t:dracopm\], and the corresponding PM diagram is shown in Figure \[f:dracopm\]a. PM measurements using different background sources are plotted in different symbols, and results from each field are plotted in different colors. The PM results for the [*DRACO-F1*]{} field in Table \[t:dracopm\] was derived using a data set with a time baseline of 9 yr (2004 versus 2013). However, since we have images acquired in 2006 for this field, we used them as an external check by measuring PMs of Draco stars using 2006 data as the first epoch, and 2014 data as the second epoch. We followed the same procedure outlined in Section \[ss:measurements\] to obtain PMs using both QSO and background galaxies as stationary references. The resulting 7 yr-baseline PMs are consistent within $1.5\sigma$ compared to the 9 yr-baseline PMs listed in the first two lines of Table \[t:dracopm\] with slightly larger uncertainties as expected from the shorter time baseline.[^4] This provides an additional check on our PM results for the [*DRACO-F1*]{} field.
The uncertainties in the measurements are dominated by the random errors in the reference frame set by background galaxies or QSOs. These random errors are independent from each other. We therefore calculate the average PM of Draco by taking the error-weighted mean of the five measurements provided in Table \[t:dracopm\], which yields $$\label{e:dracopm}
(\mu_{W},\>\mu_{N}) =
(-0.0562 \pm 0.0099,\>-0.1765 \pm 0.0100)\ {\rm mas\ yr}^{-1} .$$ The final average of the five data points and associated uncertainties in each coordinate are plotted as a black cross in Figure \[f:dracopm\]a.
Overall, we find that measurements using different objects as stationary references agree well with each other. This provides confidence on our Draco PM results, and more generally on the PM measurement technique using background galaxies as stationary objects. Measurements for different fields also agree to within the error bars. To test the statistical agreement among the individual measurements listed in Table \[t:dracopm\], we calculate the quantity $$\label{e:chisq}
\chi^2 = \sum_{i}
\left[
\left ( \frac{ \mu_{W,i} - {\overline \mu_{W}} }
{ \Delta \mu_{W,i} } \right )^2
+
\left ( \frac{ \mu_{N,i} - {\overline \mu_{N}} }
{ \Delta \mu_{N,i} } \right )^2
\right].$$ This quantity is expected to follow a probability distribution with an expectation value of the number of degrees of freedom ($N_{\rm DF}$) with a dispersion of $\sqrt{2N_{\rm DF}}$. Since we have five independent measurements each in two directions on the sky, the $\chi^2$ is then expected to have a value of $8 \pm 4$. From Table \[t:dracopm\] and Equation \[e:chisq\], we find $\chi^2 = 11.1$. Therefore, we find that our measurements in Table \[t:dracopm\] are consistent within our quoted uncertainties.
Our final 1d PM uncertainty for Draco is $10{\>{\mu \rm as}\,{\rm yr}^{-1}}$ in each direction. This is smaller than any other measurement uncertainties we have achieved using our PM measurement techniques, and therefore may appear to be beyond [*HST*]{}’s astrometric capabilities. However, this small uncertainty is mainly due to the time baselines being longer than our previous studies. For example, in our M31 study [@soh12], we achieved a 1d PM uncertainty of $\sim 12 {\>{\mu \rm as}\,{\rm yr}^{-1}}$ for time baselines of 5–7 years averaging results from three separate fields. Our Draco data also consists of measurements from three separate fields, but the time baselines are about 2.5 years longer than the M31 work. Simply scaling uncertainties of the M31 work by this time baseline ratio gives an estimated uncertainty of $8.5 {\>{\mu \rm as}\,{\rm yr}^{-1}}$, which is consistent with our measured uncertainty for Draco. In our Leo I study [@soh13], we achieved a 1d PM uncertainty of $29{\>{\mu \rm as}\,{\rm yr}^{-1}}$ using two epochs of ACS/WFC data separated by 5 yrs for a single field. Scaling by the time baseline ratio, and dividing by $\sqrt{3}$ to account for the difference in the number of fields used for the measurement gives $10{\>{\mu \rm as}\,{\rm yr}^{-1}}$, which again is consistent with our PM uncertainty for Draco. We conclude that our PM measurement uncertainties for Draco are in line with expectations from our previous studies.
In both our previous studies mentioned above, we carried out detailed analyses to argue that there were no systematic errors in excess of the random errors. Since our smaller random errors here are merely due to a longer time baseline (which reduces random and systematic errors equally) and the higher number of fields, the same conclusions should hold true. Nevertheless, the difference between QSO and background galaxy results for [*DRACO-F1*]{} and [*-F2*]{} may indicate a small systematic effect. This is likely a problem with the QSO measurements since they (1) sample only one region on the detector, (2) do not average over multiple background sources, and (3) require a magnitude correction as demonstrated in Section \[sss:qso\]. However, systematics are a problem only if they are correlated between different measurements, but we find no evidence for any such effects. Therefore, upon averaging, these systematics should decrease as $\sqrt(N)$ as we have assumed in the averaging of our results.
As shown in Figure \[f:fields\], our field locations for measuring the absolute PM are offset from the center of Draco by angular distances of 12–30. If the internal motions of Draco stars in tangential directions are significantly large, our PM measurement may not represent the center of mass (COM) motion of Draco. This is particularly true if the tangential motions show a systematic pattern (e.g., clockwise or counter-clockwise rotation). To check this, we have subtracted the average PM of Draco from the PMs of each of our target fields and plotted the residual 2d motions on the sky in the left panel of Figure \[f:2dmotion\]. We do not detect any rotational sign from the residual motions.
We also carried out additional checks as follows. Whereas so far, there are no internal PM measurements of Draco, the line-of-sight (LOS) velocity shows a slight rotation sign at the level of $6 {\>{\rm km}\,{\rm s}^{-1}}$ at a radius of 30 along Draco’s major axis [@kle01]. However, given that this is smaller than the central LOS velocity dispersion of Draco [$\sigma = 9.1 \pm 1.2 {\>{\rm km}\,{\rm s}^{-1}}$; @wil04], it has been claimed as non-significant [@kle02]. At the distance of Draco, $6 {\>{\rm km}\,{\rm s}^{-1}}$ is equivalent to $0.017 {\>{\rm mas}\,{\rm yr}^{-1}}$, which is about twice the size of our final random error in Table \[t:dracopm\]. If Draco is rotating at this speed on the sky, our residual motions above would have shown systematic rotational signs, but we do not detect such sign. We note that Draco appears quite elongated on the sky [$e = 0.30$; @ode01] suggesting that it is seen at high inclination. This implies that the rotation in the plane of the sky should be less than that along the LOS. Finally, even if the rotational motion was systematically affecting our PM results for each field, the final average should represent the systemic tangential motions of Draco given that our three target fields sample stars on both sides of the dwarf near the major axis at similar angular distances. For the reasons stated above, we adopt our PM results in Equation \[e:dracopm\] as our final measurement for Draco.
In Figure \[f:dracopm\]b, we compare our new PM results with the two recent PM measurements using data obtained with [*HST*]{} [@pry15] and the Subaru Telescope [@cas16]. As mentioned in Section \[ss:data\], the measurement by @pry15 was obtained using a 2-year time baseline data for the [*DRACO-F1*]{} field. While the two [*HST*]{} results are consistent within 1$\sigma$ in ${\mu_{N}}$, they are discrepant at the $\sim 2\sigma$ level in ${\mu_{W}}$, despite using the same field (albeit with a shorter time baseline), the same type of objects (QSOs and background galaxies), and similar techniques as used in this study. The source of this discrepancy is unclear, but it is reasonable to assume that PM results with longer time baselines (in this case, our results) are less subject to systematics. The comparison with the Subaru results show even larger discrepancies. Given that [*HST*]{} is less prone to systematics related to atmospheric effects and instrumental change, we believe our results are more reliable.
### Sculptor Dwarf Spheroidal Galaxy {#sss:results_sculptor}
[lcccc]{} (Galaxies) & $-0.0368$ & $-0.1222$ & $0.0367$ & $0.0368$\
[**F2**]{} (Galaxies) & $-0.0262$ & $-0.1428$ & $0.0254$ & $0.0263$\
Weighted average & $-0.0296$ & $-0.1358$ & $0.0209$ & $0.0214$\
Our PM results for the Sculptor dSph are presented in Table \[t:sculptorpm\], and the corresponding PM diagram is shown in Figure \[f:sculptorpm\]a. Since we only used background galaxies as stationary references for this galaxy, each field has a single measurement. The error-weighted mean of the two fields in Table \[t:sculptorpm\] gives $$\label{e:sculptorpm}
(\mu_{W},\>\mu_{N}) =
(-0.0296 \pm 0.0209,\>-0.1358 \pm 0.0214)\ {\rm mas\ yr}^{-1}.$$ As evident in Figure \[f:sculptorpm\]a, the independent measurements from our two observed fields are consistent with each other within $1\sigma$. Indeed, we find $\chi^2 = 0.3$ which is in line with the expected value of $2 \pm 2$.
For Sculptor, @bat08 find a radial velocity gradient of $7.6^{+3.0}_{-2.2}\ {\>{\rm km}\,{\rm s}^{-1}}$ per deg along its projected major axis, probably due to intrinsic rotation. Our target fields are located near the minor axis at 7–9 arcmin from the center of Sculptor. The residual 2d motions of our target fields after subtracting the average PM of Scultpor are shown as color arrows in the right panel of Figure \[f:2dmotion\]. We note that the residual motions are too small to show compared to the average PM of Sculptor, demonstrating that the internal motions among the fields are negligible. Indeed, our 1d PM uncertainty at the distance of Sculptor is $8.6 {\>{\rm km}\,{\rm s}^{-1}}$, so even if we assume that Sculptor has tangential motions at the same level of the radial velocity gradient, our PM uncertainties are comparable to this. Therefore, no correction for the COM motion of Sculptor is required, and we adopt Equation \[e:sculptorpm\] as our final PM measurement for Sculptor.
We compare our PM results with the [*HST*]{} measurement by @pia06. In their study, @pia06 used QSOs in two different fields to measure the absolute PM of Sculptor. The two measurements agree with each other within $1\sigma$, with our 1d PM uncertainty being $\sim 6$ times smaller than that of @pia06. While both measurements employed the astrometric powers of [*HST*]{}, @pia06 used STIS data with time baselines of 2–3 yrs, while we used ACS/WFC data separated by 11 yrs. Field locations are significantly different, and so these two measurements can be considered as completely independent. The agreement between the two PM measurements, despite using different types of background sources in different fields observed with different detectors, highlights the success in using [*HST*]{} instruments as tools for measuring absolute PMs of dwarf galaxies in the MW halo.
Space Motions {#s:spacemotions}
=============
Systemic Motions of Draco and Sculptor on the Sky {#ss:net2d}
-------------------------------------------------
Our PM results in Section \[ss:results\] include contributions from the motion of the Sun with respect to the MW. To obtain the systemic motions of Draco and Sculptor on the sky, we are required to subtract these contributions as follows. We adopt values of @mcm11 for the Galactocentric distance and the rotational velocity of the Local Standard of Rest (LSR): $R_0 = 8.29 \pm 0.16 {\>{\rm kpc}}$ and $V_0 = 239 \pm 5 {\>{\rm km}\,{\rm s}^{-1}}$. For the solar peculiar velocity with respect to the LSR, we adopt values of @sch10: $(U_{\rm pec},\>V_{\rm pec},\>W_{\rm pec}) = (11.10,\>12.24,\>7.25) {\>{\rm km}\,{\rm s}^{-1}}$ with uncertainties of $(1.23,\>2.05,\>0.62) {\>{\rm km}\,{\rm s}^{-1}}$. For heliocentric distances to Draco and Sculptor, we adopt $76 \pm 6 {\>{\rm kpc}}$ [@bon04], and $86 \pm 6 {\>{\rm kpc}}$ [@pie08], respectively. The contributions of solar motions in $(\mu_{W},\>\mu_{N})$ for each dwarf galaxy is then $(0.3795,\>-0.0366) {\>{\rm mas}\,{\rm yr}^{-1}}$ for Draco and $(-0.3657,\>-0.4895) {\>{\rm mas}\,{\rm yr}^{-1}}$ for Sculptor. These are indicated as sun symbols in Figures \[f:dracopm\]b and \[f:sculptorpm\]b. Subtracting these solar motions from our PM measurements provides the net 2d motions of Draco and Sculptor on the sky: $({\mu_{W}}, {\mu_{N}}) =
(-0.4364,\>-0.1307) {\>{\rm mas}\,{\rm yr}^{-1}}$ for Draco; and $(0.3361,\>0.3537) {\>{\rm mas}\,{\rm yr}^{-1}}$ for Sculptor. These motions are illustrated in Figures \[f:2dmotion\] as black arrows along with the directions toward the Galactic Center as shown in dotted lines.
Space Velocities in the Galactocentric Rest Frame {#s:spacevel}
-------------------------------------------------
We adopt the same Cartesian Galactocentric coordinate system ($X,\>Y,\>Z$) we used in our earlier studies of M31 and Leo I [@soh12; @soh13] to describe the space velocities of Draco and Sculptor. In this system, the origin is at the Galactic Center, the $X$-axis points in the direction from the Sun to the Galactic Center, the $Y$-axis points in the direction of the Sun’s Galactic rotation, and the $Z$-axis points toward the Galactic north pole. The position and velocity of Draco and Sculptor in this frame can be derived from the observed sky positions, distances, line-of-sight velocities, and PMs.
### Draco Dwarf Spheroidal {#ss:vdraco}
For Draco, the Galactocentric $(X,\>Y,\>Z)$ position is $$\label{e:draco_r}
{{\mathbf{r}}}_{\rm Dra} = (-4.3,\>62.3,\>43.3) {\>{\rm kpc}}.$$ To calculate the 3-d space velocity of Draco, we adopt a heliocentric LOS velocity of $v_{\rm LOS} = -292.8 \pm 0.4 {\>{\rm km}\,{\rm s}^{-1}}$, estimated by applying the chemo-dynamical model of @wal15a to the spectroscopic data set of @wal15b. Combining this with our PM results in Section \[sss:results\_draco\], the Galactocentric velocity $(V_{\rm X},\>V_{\rm Y},\>V_{\rm Z})$ of Draco becomes $$\label{e:draco_v}
{{\mathbf{v}}}_{\rm Dra} = (61.0,\>16.3,\>-173.0) \pm (6.4,\>5.8,\>3.2) {\>{\rm km}\,{\rm s}^{-1}}.$$ The uncertainties listed here and hereafter were obtained from a Monte Carlo (MC) scheme by propagating all observed uncertainties (distance, velocity, and their correlations) including those for the Sun. The corresponding Galactocentric radial and tangential velocities are then $$\label{e:draco_vradvtan}
(V_{\rm rad},\>V_{\rm tan})_{\rm Dra} = (-88.6,\>161.4) \pm (4.4,\>5.6) {\>{\rm km}\,{\rm s}^{-1}},$$ and the observed total velocity of Draco with respect to the MW is $$\label{e:draco_vtot}
V_{\rm tot, Dra} \equiv |{{\mathbf{v}}_{\rm Dra}}| = 184.1 \pm 4.3 {\>{\rm km}\,{\rm s}^{-1}}.$$
### Sculptor Dwarf Spheroidal {#ss:vsculptor}
For Sculptor, the Galactocentric position is $$\label{e:sculptor_r}
{{\mathbf{r}}}_{\rm Scl} = (-5.2, -9.8, -85.4) {\>{\rm kpc}}.$$ We adopt a heliocentric LOS velocity of $v_{\rm LOS} = 111.5 \pm 0.3 {\>{\rm km}\,{\rm s}^{-1}}$, obtained by applying the model of @wal15a to the spectroscopic data of @wal09, and combining this with our PM results for Sculptor, we obtain a Galactocentric velocity of $$\label{e:sculptor_v}
{{\mathbf{v}}}_{\rm Scl} = (36.0,\>186.3,\>-96.7) \pm (8.8,\>10.9,\>1.3) {\>{\rm km}\,{\rm s}^{-1}}.$$ The Galactocentric radial and tangential velocities are $$\label{e:sculptor_vradvtan}
(V_{\rm rad},\>V_{\rm tan})_{\rm Scl} = (72.6,\>200.2) \pm (1.3,\>10.8) {\>{\rm km}\,{\rm s}^{-1}},$$ and the total velocity of Sculptor with respect to the MW is $$\label{e:sculptor_vtot}
V_{\rm tot, Scl} \equiv |{{\mathbf{v}}_{\rm Scl}}| = 213.0 \pm 9.9 {\>{\rm km}\,{\rm s}^{-1}}.$$
### Escape Velocities {#ss:vesc}
The escape velocity of a tracer object provides first-order insights into the enclosed mass at its distance. The escape velocity $v_{\rm esc}$ for a point mass $M_{\rm MW}$ is defined as $$v_{\rm esc} = \sqrt{2GM_{\rm MW}/r},$$ where $r$ is the Galactocentric distance to the tracer object. According to cosmological simulations, it is unlikely to find an unbound satellite at the present epoch near a MW-size galaxy [@boy13 but see Section \[ss:LMC\] of this paper]. Therefore, by forcing Draco and Sculptor to be bound to the MW, we can use the equation above to calculate the lower limit on the enclosed MW mass. Using the total velocities from Equations \[e:draco\_vtot\] and \[e:sculptor\_vtot\], we arrive at lower limits of the enclosed MW mass $0.3\times10^{12}
{M_{\odot}}$ and $0.5\times10^{12} {M_{\odot}}$ at distances of $R_{\rm GC} = 76$ kpc and 86 kpc, respectively.
Using the older PM measurement by @pry15 and @pia06, the total velocities of Draco and Sculptor become $V_{\rm tot, Dra} = 225.9
{\>{\rm km}\,{\rm s}^{-1}}$ and $V_{\rm tot, Scl} = 248.1 {\>{\rm km}\,{\rm s}^{-1}}$, respectively. These imply lower limits of enclosed MW masses of $0.9\times10^{12}
{M_{\odot}}$ and $1.2\times10^{12} {M_{\odot}}$ at $R_{\rm GC} = 76$ kpc and 86 kpc, respectively. In conclusion, our new PM measurements allow significantly lower MW masses based on the escape velocities.
The Orbits of the Draco and Sculptor Dwarf Spheroidal Galaxies {#s:orbits}
==============================================================
Orbital Properties of Draco and Sculptor {#ss:orbprops}
----------------------------------------
To explore the past orbital histories of Draco and Sculptor, we have numerically integrated their orbits backwards in time using the current Galactocentric positions and velocities derived in Section \[s:spacemotions\]. The orbital integration scheme follows the same methodology used in @bes07, @soh13, and [@pat17].
In summary, the MW’s potential is modeled as a static, axisymmetric, three component model consisting of a dark matter halo, disk, and stellar bulge. We adopt the same three mass models for the MW as in @soh13 with total virial masses ($M_{\rm vir}$) of 1.0$\times 10^{12}{M_{\odot}}$, 1.5$\times 10^{12}{M_{\odot}}$, and 2.0$\times 10^{12}{M_{\odot}}$. The MW disk mass was varied in each model such that the total rotation curve of the combined halo, disk and bulge peak at $\approx 239 \;{\>{\rm km}\,{\rm s}^{-1}}$ [@mcm11]. In addition, the MW’s dark matter halo is adiabatically contracted using the [CONTRA]{} code [@gne04]. The model parameters (concentrations, virial radii, and masses of the disks) for each MW model can be found in Table 2 of @soh13.
Draco and Sculptor are each modeled as Plummer spheres, with a total mass of $5\times10^{9}{M_{\odot}}$. The softening lengths ($k_{sat}$) are 2.3 kpc and 3.9 kpc for Draco and Sculpor, respectively. These values are chosen such that the halo mass matches the inferred total mass within the outermost data point of the empirical velocity dispersion profile, referred to as $r_{last}$ in @wal09.
For our orbital integrations, we included the damping effects of dynamical friction. Since we are integrating orbits backwards in time, the damping of satellite orbits due to dynamical friction acts as an accelerating force. Dynamical friction is approximated by the Chandrasekhar formula [@cha43]: $$\label{eq:df}
{\mathbf{F}}_{df}= \rm - \frac{4\pi G^2 M_{sat}^2 ln \Lambda \rho(r)}{v^2} \left[ erf(X) - \frac{2X}{\sqrt\pi} exp(-X^2)\right] \frac{{\mathbf{v}}}{v},$$ where $X=v/\sqrt{2\sigma}$ and $\sigma$ is the one-dimensional galaxy velocity dispersion. Here, $\sigma$ is an approximation for an NFW profile, which was derived in @zen03. For three body encounters between Draco/Sculptor, the LMC, and the MW, the Coulomb logarithm, ln$\rm\Lambda$, takes the form of the 10:1 mass ratio parametrization described in @vdm12b [Appendix A] for the decay of the LMC’s orbit. For Draco and Sculptor, we have adopted the Coulomb logarithm from @has03, which is $\Lambda=r/1.4k_{sat}$. For Draco and Sculptor, the impact of dynamical friction on their orbits is minimal.
Following @pat17, but in contrast to @bes07 and @soh13, the MW is not fixed in space in these calculations. Instead, the MW moves in response to the gravitational influence of the satellites, particularly from the LMC (see Section \[ss:LMC\]), throughout the integration period [see also @gom15].
The equations of motion corresponding to the gravitational potentials described above are then integrated backwards in time for 6 Gyr using a symplectic leap frog algorithm [@spr01]. Over longer timescales, the orbits of satellites are highly uncertain, e.g. owing to the accretion history of the MW itself [@lux10].
[cccccc]{} 1.0 & $51.3\pm6.2$ & $2.2\pm0.4$ & $121.0\pm 16.1$ & $0.9\pm0.2$ & $2.6\pm0.4$\
1.5 & $45.9\pm5.8$ & $1.5\pm0.2$ & $101.3\pm 10.7$ & $0.6\pm0.1$ & $1.9\pm0.2$\
2.0 & $42.2\pm5.2$ & $1.2\pm0.1$ & $\phn93.4\pm\phn8.7$ & $0.4\pm0.1$ & $1.6\pm0.2$\
1.0 & $74.7\pm5.2$ & $0.3\pm0.1$ & $184.2\pm 50.5$ & $2.2\pm1.0$ & $4.7\pm0.8$\
1.5 & $71.0\pm5.3$ & $0.3\pm0.1$ & $127.7\pm 25.3$ & $1.4\pm0.7$ & $2.9\pm0.3$\
2.0 & $66.9\pm5.3$ & $0.4\pm0.1$ & $106.6\pm 16.4$ & $1.0\pm0.7$ & $2.2\pm0.2$\
The orbital trajectories for Draco and Sculptor calculated using their mean positions (Equations \[e:draco\_r\] and \[e:sculptor\_r\]) and velocities (Equations \[e:draco\_v\] and \[e:sculptor\_v\]) for the past 3 Gyr are shown in Figures \[f:draco\_orbit\_mw\] and \[f:sculptor\_orbit\_mw\]. The LMC is not yet included in these calculations. To explore the full range of plausible orbital histories, we use the 10,000 Monte Carlo realizations (see Section \[s:spacevel\]), which sample the uncertainties in distances, radial velocities, and PMs from normal distributions with means and standard deviations taken from the observed uncertainties. We then use positions and velocities for each realization to integrate orbits in the three MW mass models. This resulted in 60,000 orbital integrations in total for Draco and Sculptor combined.
Table \[t:mworbits\] lists the distance and look-back time of the most recent pericentric and apocentric passages of Draco and Sculptor along with the orbital period, in the case where two pericentric passages exist within 6 Gyr. In the majority of cases, both Draco and Sculptor complete multiple orbits around the MW and remain within its virial radius over the past 6 Gyr. Only $\sim 1\%$ of Sculptor’s orbits in the $M_{\rm MW} = 1.0\times10^{12} {M_{\odot}}$ model were in a “first-infall” orbit implying that it has not completed an orbit about the MW. No such cases for either dwarf occur in the higher mass MW models, i.e., 100% of orbits exhibit both a pericenter and an apocenter.
From our orbital analysis, we conclude that Draco passed the apogalacticon of its orbit 0.4–0.9 Gyr ago at a distance of $R_{\rm GC} = $93–119 kpc, and is now approaching perigalacticon with an orbital period of 1–2 Gyr. Sculptor, on the other hand, recently passed perigalacticon 0.3–0.4 Gyr ago at a distance of $R_{\rm GC} = $67–76 kpc, and is now moving further away from the Galactic center. Sculptor also has a longer orbital period of $\sim$2–5 Gyr. However, their average orbital eccentricities are similar – both are mildly elliptical at $e \simeq 0.4$ and $\simeq 0.3$ for Draco and Sculptor, respectively. In addition to being in different phases of their orbit, the two satellites have orbital angular momenta in almost the opposite direction on the celestial sphere, indicating that they orbit around the MW in opposite directions. This is most clearly seen when comparing the orbits of the two galaxies in the Y-Z plane (bottom left panels of Figures \[f:draco\_orbit\_mw\] and \[f:sculptor\_orbit\_mw\]). We discuss these orbital features in the context of the DoS in Section \[ss:dos\].
The Dynamical Influence of the Large Magellanic Cloud {#ss:LMC}
-----------------------------------------------------
Other massive members of the Local Group may exert dynamical influence on the orbital histories of Draco and Sculptor. Given the distances of these satellites and their most likely association with the MW over the past $\sim$5 Gyr (see Section \[ss:orbprops\]), the most relevant perturber to their current orbital motion is the LMC. To examine its dynamical influence on the orbits of Draco and Sculptor, we added the LMC to the orbital calculations. We adopted the same strategy as outlined in Section \[ss:orbprops\] for integrating orbits and analyzed the three-body interactions separately for the Draco-MW-LMC and the Sculptor-MW-LMC systems. These orbital calculations sample the full 4$\sigma$ error space of the LMC’s space motion and distance [@kal13], in addition to the error space associated with Draco or Sculptor. Thus, each orbital realization randomly draws a set of position and velocity vectors from the 10,000 Monte Carlo drawings for the LMC and simultaneously for Draco or Sculptor. We note that the orbital angular momentum vector of the LMC is roughly aligned with that of Draco.
[@gom15] showed that the orbital barycenter of the MW-LMC system significantly changes over time, depending on the mass of the LMC. Therefore, as noted earlier, the MW is not held fixed in space, but rather moves in response to the force of the LMC as a function of time. Our numerical orbit integration scheme therefore includes not only the LMC’s gravitational torque acting on Draco and Sculptor, but also the response of the MW’s COM to the presence the LMC.
The LMC is modeled as a Plummer sphere, and we consider three LMC mass models: 0.3$\times 10^{11}{M_{\odot}}$, 1.0$\times 10^{11}{M_{\odot}}$, and 2.5$\times 10^{11}{M_{\odot}}$, respectively with softening lengths of 5.9, 13.1, and 19.5 kpc. This mass range encompasses observational constraints and cosmological expectations [see, @pat17].
[ccrccrcccccccc]{} & 0.3 & 100 & $\phn62.4 \pm \phn7.5$ & $2.7 \pm 0.5$ & 100 & $ 135.5 \pm 20.4$ & $1.2 \pm 0.3$ & 99 & $3.4\pm0.6$\
1.0 & 1.0 & 96 & $\phn79.7 \pm 14.0$ & $3.8 \pm 0.8$ & 96 & $ 174.7 \pm 31.8$ & $1.8 \pm 0.4$ & 77 & $4.2\pm0.7$\
& 2.5 & 8 & $\phn98.0 \pm 24.3$ & $5.1 \pm 0.7$ & 8 & $ 218.5 \pm 32.0$ & $2.4 \pm 0.4$ & 2 & $4.8\pm0.5$\
& 0.3 & 100 & $\phn55.4 \pm \phn7.2$ & $1.7 \pm 0.3$ & 100 & $ 107.7 \pm 12.4$ & $0.7 \pm 0.1$ & 100 & $2.3\pm0.3$\
1.5 & 1.0 & 100 & $\phn72.5 \pm 12.8$ & $2.2 \pm 0.4$ & 100 & $ 125.5 \pm 18.4$ & $0.9 \pm 0.2$ & 99 & $3.0\pm0.8$\
& 2.5 & 93 & $ 128.8 \pm 42.7$ & $4.1 \pm 0.9$ & 93 & $ 201.8 \pm 43.8$ & $2.1 \pm 0.5$ & 45 & $3.8\pm0.9$\
& 0.3 & 100 & $\phn50.5 \pm \phn6.4$ & $1.3 \pm 0.2$ & 100 & $\phn97.0 \pm \phn9.6$ & $0.5 \pm 0.1$ & 100 & $1.8\pm0.2$\
2.0 & 1.0 & 100 & $\phn65.1 \pm 10.5$ & $1.5 \pm 0.2$ & 100 & $ 106.5 \pm 12.3$ & $0.6 \pm 0.1$ & 100 & $2.2\pm0.5$\
& 2.5 & 100 & $ 111.7 \pm 27.9$ & $2.5 \pm 0.5$ & 100 & $ 144.4 \pm 25.4$ & $1.2 \pm 0.3$ & 86 & $4.1\pm1.8$\
[ccrccrcccccccc]{} & 0.3 & 100 & $70.8 \pm 5.3$ & $0.32 \pm 0.05$ & 99 & $232.8 \pm 55.4$ & $2.9 \pm 0.9$ & 99 & $5.5\pm1.0$\
1.0 & 1.0 & 100 & $61.6 \pm 4.6$ & $0.35 \pm 0.04$ & 89 & $328.1 \pm 72.0$ & $4.0 \pm 1.1$ & 74 & $7.3\pm1.2$\
& 2.5 & 100 & $44.9 \pm 2.5$ & $0.32 \pm 0.02$ & 15 & $525.5 \pm 80.1$ & $5.0 \pm 0.8$ & 3 & $8.3\pm1.1$\
& 0.3 & 100 & $66.7 \pm 5.5$ & $0.36 \pm 0.05$ & 100 & $156.3 \pm 30.7$ & $1.7 \pm 0.7$ & 100 & $3.2\pm0.4$\
1.5 & 1.0 & 100 & $58.6 \pm 5.3$ & $0.36 \pm 0.03$ & 100 & $209.3 \pm 34.7$ & $2.2 \pm 0.5$ & 100 & $3.9\pm0.5$\
& 2.5 & 100 & $47.7 \pm 4.6$ & $0.33 \pm 0.02$ & 99 & $316.3 \pm 65.2$ & $2.7 \pm 0.7$ & 99 & $5.1\pm1.4$\
& 0.3 & 100 & $62.8 \pm 5.4$ & $0.38 \pm 0.04$ & 100 & $132.8 \pm 16.3$ & $1.5 \pm 0.4$ & 100 & $2.4\pm0.2$\
2.0 & 1.0 & 100 & $55.3 \pm 5.1$ & $0.36 \pm 0.03$ & 100 & $161.6 \pm 29.2$ & $1.5 \pm 0.5$ & 100 & $2.7\pm0.3$\
& 2.5 & 100 & $45.6 \pm 4.4$ & $0.33 \pm 0.02$ & 100 & $227.3 \pm 43.1$ & $1.7 \pm 0.5$ & 100 & $3.0\pm0.7$\
Tables \[t:mwlmc\_draco\] and \[t:mwlmc\_sculptor\] list the distance and look-back time of the most recent pericentric and apocentric passages of Draco and Sculptor about the MW, now accounting for the 3-body interactions of Draco/Sculptor-LMC-MW. In these tables, we also added columns that indicate the fraction of orbits that have a perigalactic approach ($f_{\rm peri}$) and an apogalacticton ($f_{\rm apo}$) within an integration time of 6 Gyr. Cases that do not have an apogalaticon have not completed an orbit, and are considered to be on their first infall to the MW. The final two columns list the fraction of orbits where two pericenters have occurred ($f_p$) and the average orbital period computed using the time of these close passages.
In the previous section, where the LMC was not included, most orbits had both an apocenter and pericentric approach to the MW within 6 Gyr. Here, we find that the LMC introduces significant scatter to the results. In the most extreme case of the lowest MW mass ($M_{\rm MW} = 1.0\times
10^{12} {M_{\odot}}$) and the highest LMC mass ($M_{\rm LMC} = 2.5\times
10^{11} {M_{\odot}}$), only 9% of Draco’s and 15% of Sculptor’s 10,000 MC realizations had closed orbits. In other words, for this light-MW $+$ heavy-LMC model, both galaxies were likely on their first approach to the MW within the past 6 Gyr. Based on these calculations, we can not rule out the possibility that Draco and/or Sculptor are making their first approaches to the MW.
Overall, we find that the orbital period and apocenter for both Draco and Sculptor systematically increases with the inclusion of the LMC. The timing of Sculptor’s most recent pericentric approach (0.3–0.4 Gyr ago) is a robust quantity, being largely unaffected by changes in MW or LMC mass. However, Sculptor’s pericentric distance decreases as the LMC mass increases. Draco’s orbit, on the other hand, is more strongly affected by the LMC’s inclusion than that of Sculptor.
The Association of Draco and Sculptor with the Disk of Satellites {#ss:dos}
-----------------------------------------------------------------
Draco and Sculptor are classical dSphs that have traditionally been included in the DoS. In light of our new PM estimates for these satellites, we revisit their dynamical association with the DoS. We define the DoS as in @kro10, where 24 satellite galaxies within 254 kpc, including the 11 classical satellites are fit to a plane with a minimum disk height of 28.9 kpc.
Figures \[f:draco\_dos\] and \[f:scul\_dos\] illustrate the orbital trajectory of Draco and Sculptor, respectively, over the past 3 Gyr in a viewing perspective such that the DoS is seen edge on. This perspective roughly coincides with the Galactocentric X–Z plane. The current positions of Draco and Sculptor are shown in black dots, while the other classical dSphs are shown in yellow. We compare orbital trajectories using the previous PM measurements (left panels) with those using the new PMs in this study (middle panels). The previous PM measurements were adopted from @pry15 for Draco, and @pia06 for Sculptor, both of which are measured using [*HST*]{} data. Orbits are plotted for our 3 different MW mass models as indicated in the figure legends. Despite the fact that Draco and Sculptor are orbiting in opposite directions about the MW, both orbit within the thin DoS for the past 3 Gyr, regardless of the assumed MW mass. The agreement between the orbital trajectories of Draco and Sculptor and the DoS is substantially improved with the new PMs, especially for Sculptor.
We now examine whether perturbations from the LMC can affect the strong agreement between the orbits derived using the new velocity measurements and the DoS. In the right panels of Figures \[f:draco\_dos\] and \[f:scul\_dos\], the orbits of Draco and Sculptor for the lowest MW mass model ($M_{\rm MW} = 1.0\times 10^{12} {M_{\odot}}$) are plotted. We selected the lowest MW mass model to explore the configuration that yields the maximal perturbation on the satellites’ orbits by the LMC. The different trajectories are for the three different LMC mass models used in Section \[ss:LMC\]. We find that despite increasing the LMC mass to as high as $2.5\times10^{11} {M_{\odot}}$, the orbits of Draco and Sculptor are still confined well within the DoS. This indicates that, while the LMC can substantially increase the apocenteric distance of the orbits of Draco and Sculptor, it does not introduce torques out of their orbital plane. This is not surprising since the LMC itself is orbiting within the DoS.
It remains unclear how such a tight agreement between the orbital planes of Draco and Sculptor can occur, given that these satellites are orbiting in opposite directions about the MW. Our results likely rule out a scenario in which Draco and Sculptor were accreted together as part of a tightly bound group. Cosmological simulations show that in such a scenario, the orbital angular momenta of all group members should be well aligned [@sal11]. This analysis, however, does not rule out that Draco and Sculptor were accreted as part of a loose group or tidal structures, which was split apart upon infall [e.g., @paw11].
Our newly-measured PMs place a new spotlight on an interesting problem. While it has been known that Draco and Sculptor are moving in opposite directions, we now know that their orbital planes are strongly confined within the DoS. Detailed studies of infalling groups of satellites may reveal how such satellite orbital configurations are created around MW-size galaxies.
Conclusions {#s:conclusions}
===========
We used [*HST*]{} ACS/WFC images to measure the proper motions of Draco and Sculptor. By comparing bulk motions of numerous stars in Draco and Sculptor with respect to distant background galaxies or QSOs, we find the PMs of Draco and Sculptor to be $(\mu_{W},\>\mu_{N})_{\rm Dra} =
(-0.0562 \pm 0.0099,\>-0.1765 \pm 0.0100)\ {\rm mas\ yr}^{-1}$ and $(\mu_{W},\>\mu_{N})_{\rm Scl} =
(-0.0296 \pm 0.0209,\>-0.1358 \pm 0.0214)\ {\rm mas\ yr}^{-1}$. These are the most precise PMs measured so far for any satellite dSph in the MW halo. We compare our new PM results with previous measurements in the literature and find that they are mostly consistent at the 1–2$\sigma$ levels. However, our results are [*significant*]{} improvements over previous ones with 1d PM uncertainties being at least 5–7 times smaller.
To derive space velocities of Draco and Sculptor in the Galactocentric frame, we combined our PMs with known line-of-sight velocities and corrected for the solar reflex motions. As a result, our Galactocentric radial and tangential velocities are $(V_{\rm rad},\>V_{\rm tan})_{\rm Dra}
= (-88.6,\>161.4) \pm (4.4,\>5.6) {\>{\rm km}\,{\rm s}^{-1}}$ and $(V_{\rm rad},\>V_{\rm tan})_{\rm Scl} = (72.6,\>200.2) \pm (1.3,\>10.8) {\>{\rm km}\,{\rm s}^{-1}}$. We used the total velocities of Draco and Sculptor to provide lower limits on the enclosed MW masses at the satellite distances. The resulting limits are $M > 0.3\times10^{12} {M_{\odot}}$ and $M > 0.5\times10^{12} {M_{\odot}}$ at distances of $R_{\rm GC} = 76$ kpc and 86 kpc, respectively.
We used the PM results to revisit the orbital histories of Draco and Sculptor over the past 6 Gyr. Orbital periods of Draco and Sculptor are found to be 1–2 and 2–5 Gyrs, respectively, accounting for uncertainties in the mass of the MW. The inclusion of the LMC increases the scatter in these results. In the most extreme example of a low mass MW ($1.0\times 10^{12}
{M_{\odot}}$) and high mass LMC ($2.5\times10^{11} {M_{\odot}}$), orbital solutions favor a scenario where Draco and Sculptor are on their first infall towards the MW. The inclusion of the LMC systematically increases the orbital period. However, Sculptor’s most recent pericentric approach to the MW at 0.3–0.4 Gyr ago is the most robustly determined orbital property, with little variation over a factor of 2 (10) change in halo mass for the MW (LMC).
The new PMs measured by this work imply a better agreement between the direction of motions of Draco and Sculptor and the purported DoS [@kro05; @met07; @kro10; @paw13]. Specifically, the new PMs reveal that the orbital trajectories of both Draco and Sculptor are confined within the DoS for at least the past 3 Gyr. This result is robust to changes in MW halo mass and perturbations from the LMC, and likely rule out the possibility that Draco and Sculptor were accreted together as part of a tightly bound group.
We would like to warmly thank the referee for the constructive feedback that helped improve the presentation of our results. Support for this work was provided by NASA through grants GO-12966 from the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under NASA contract NAS5-26555. EP is supported by the National Science Foundation through the Graduate Research Fellowship Program funded by Grant Award No. DGE-1143953.
.
Anderson J., & King, I. R. 2006, ACS/ISR 2006-01, PSFs, Photometry, and Astrometry for the ACS/WFC (Baltimore: STScI) (AK06)
Anderson, J. & Bedin, L. R. 2010, , 122, 1035
Battaglia, G., Helmi, A., Tolstoy, E., et al. 2008, , 681, L13
Bertin, E., & Arnouts, S. 1996, , 117, 393
Besla, G., Kallivayalil, N., Hernquist, L., et al. 20070, , 668, 949
Besla, G., Kallivayalil, N., Hernquist, L., et al. 2010, , 721, 97
Bonanos, A. Z., Stanek, K. Z., Szentgyorgyi, A. H., Sasselov, D. D., & Bakos, G. Á. 2004, , 127, 861
Boylan-Kolchin, M., Bullock, J. S., Sohn, S. T., Besla, G., & van der Marel, R. P. 2013, , 768, 140
Casetti-Dinescu, D. I., & Girard, T. M. 2016, , 461, 271
Chandrasekhar, S. 1943, , 97, 255
Efron, B., & Tibshirani, R. 1993, An Introduction to the Bootstrap (Chapman & Hall/CRC)
, F. A. and [Besla]{}, G. and [Carpintero]{}, D. D. and [Villalobos]{}, [Á]{}. and [O’Shea]{}, B. W. and [Bell]{}, E. F. 2015, , 802, 128
Gnedin, O. Y., Kravtsov, A. V., Klypin, A. A., Nagai, D. 2004, , 616, 16
, Y. and [Funato]{}, Y. and [Makino]{}, J. 2003 , , 582, 196
Kallivayalil, N., van der Marel, R. P., Besla, G., Anderson, J., & Alcock, C. 2013, , 764, 161
Kleyna, J. T., Wilkinson, M. I., Evans, W., & Gilmore, G. 2001, , 563, L115
Kleyna, J. T., Wilkinson, M. I., Evans, W., Gilmore, G., & Frayn, C. 2002, , 330, 792
Kroupa, P. and Theis, C. & Boily, C. M. 2005, , 431, 517
, P., et al. 2010, , 523, A23
Lux, H., Read, J. I., & Lake, G. 2010, , 406, 2312
Lynden-Bell, D. 1976, , 174, 695
Mahmud N., & Anderson, J. 2008, , 120, 907
McMillan, P. J. 2011, , 414, 2446
Metz, M., Kroupa, P., & Jerjen, H. 2007, , 374, 1125
Odenkirchen, M., Grebel, E. K., Harbeck, D., et al. 2001, , 122, 2538
Patel, E., Besla, G., Sohn, S.T. 2017, , 464, 3825
Pawlowski, M., Kroupa, P., & de Boer, K. S. 2011, , 532, A118
Pawlowski, M., Kroupa, P. 2013, , 435, 2116
Piatek, S., Pryor, C., Bristow, P., et al. 2006, , 131, 1445
Pietrz’[n]{}sky, G., Gieren, W., Szewczyk, O., et al. 2008, , 135, 1993
Pryor, C., Piatek, S., & Olszewski, E. W. 2008, in Galaxies in the Local Volume, Astroph. & Space Sci. Proceedings, ed. B. .S. Koribalski, & H. Jerjen (Netherlands: Springer), 323
Pryor, C., Piatek, S., & Olszewski, E. W. 2015, , 149, 42
Sales, L., Navarro, J. F., Cooper, A. P., White, S. D. M, Frenk, C. S, & Helmi, A. 2011, , 418, 648
Scholz, R. -D., & Irwin, M. J. 1994, in IAU Symp. 161, Astronomy from Wide Field Imaging, ed. H. T. MacGillvray et al. (Dordrecht: Kluwer), 535
Schönrich, R., Binney, J., & Dehnen, W. 2010, , 403, 1829
Schweitzer, A. E., Cudworth, K. M., Majewski, S. R., & Suntzeff, N. B. 1995, , 110, 2747
Sirianni, M., Jee, M. J., Benítez, N., et al. 2005, , 117, 1049
Sohn, S. T., Anderson, J., & van der Marel, R. P. 2010, in 2010 Space Telescope Science Institute Calibration Workshop - Hubble after SM4. Preparing JWST, ed. S. Deustua, & C. Oliveira (Baltimore, MD: STScI), 35
Sohn, S. T., Anderson, J., & van der Marel, R. P. 2012, , 753, 7
Sohn, S. T., Besla, G., van der Marel, R. P, et al. 2013, , 768, 139
, V. and [Yoshida]{}, N. and [White]{}, S. D. M. 2001, , 6, 79
van der Marel, R. P., Fardal, M., Besla, G., et al. 2012a, , 753, 8
van der Marel, R. P., Besla, G., Cox, T.J. et al. 2012b, , 753, 9
van der Marel, R. P., Anderson, J., Bellini, A., et al. 2014, in ASP Conf. Ser. 480, Structure and Dynamics of Disk Galaxies, ed. M. S. Seigar & P. Treuthardt (San Francisco, CA: ASP), 43
Walker, M. G., Mateo, M., Olszewski, E. W., et al. 2009, , 137, 3100
Walker, M. G., Mateo, M., Olszewski, E. W., et al. 2015a, , 808, 108
Walker, M. G., Olszewski, E. W., & Mateo, M. 2015b, , 448, 2717
Watkins, L. L., Evans, N. W., & An, J. H. 2010, , 406, 264
Westfall, K. B., Majewski, S. R., Ostheimer, J. C., et al. 2006, , 131, 375
Wilkinson, M. I., Kleyna, J. T., Evans, N. W., et al. 2004, , 611, L21
Zentner, A. R., & Bullock, J. S. 2003, , 598, 49
[^1]: The one-dimensional PM uncertainties were $0.35 {\>{\rm mas}\,{\rm yr}^{-1}}$ for Draco and $0.24 {\>{\rm mas}\,{\rm yr}^{-1}}$ for Sculptor.
[^2]: Our [*DRACO-F1*]{}, [*-F2*]{}, and [*-F3*]{} fields are identical to the Dra 2, Dra 3, and Dra 1 fields of @pry15, respectively.
[^3]: The instrumental magnitude is defined as $m_{\rm instr} =
-2.5 \log (cnts)$, where $cnts$ is the number of counts in the $5\times5$ pixels around the brightest pixel.
[^4]: We obtain $(\mu_{W},\>\mu_{N})
= (-0.0264 \pm 0.0385,\>-0.2141 \pm 0.0396) {\>{\rm mas}\,{\rm yr}^{-1}}$ using background galaxies, and $(-0.0709 \pm 0.0246,\>-0.1695 \pm 0.0222) {\>{\rm mas}\,{\rm yr}^{-1}}$ using QSO as stationary references.
|
---
abstract: 'Let $L$ be a Levi subgroup of $GL_N$ which acts by left multiplication on a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. We say that $X(w)$ is a spherical Schubert variety if $X(w)$ is a spherical variety for the action of $L$. In earlier work we provide a combinatorial description of the decomposition of the homogeneous coordinate ring of $X(w)$ into irreducible $L$-modules for the induced action of $L$. In this work we classify those decompositions into irreducible $L$-modules that are multiplicity-free. This is then applied towards giving a complete classification of the spherical Schubert varieties in the Grassmannian.'
address:
- 'Northeastern University, Boston, Massachusetts. USA.'
- 'Northeastern University, Boston, Massachusetts. USA.'
author:
- Reuven Hodges
- Venkatramani Lakshmibai
bibliography:
- 'Classification.Spherical.RHodges.bib'
title: A classification of spherical Schubert varieties in the Grassmannian
---
Introduction {#sec:intro}
============
For a reductive group $G$, a normal $G$-variety $X$ is called a spherical variety if it has an open, dense $B$-orbit for a Borel subgroup $B$ of $G$. The spherical variety $X$, having a single open $B$-orbit, will also have a single open $G$-orbit of the form $G/H$, where $H$ is an algebraic subgroup of $G$. Such a subgroup is called a spherical subgroup, and in [@MR1896179], Luna proposed a program to classify spherical subgroups of reductive groups in terms of combinatorial data that he termed the homogeneous spherical data. This classification has been completed, due to the contributions of many authors, see for example [@2009arXiv0907.2852C; @MR3198836; @MR3473657; @MR2495078]. In earlier work, Luna and Vust classified the spherical embeddings of $G/H$, that is, embeddings of $G/H$ into a spherical variety such that $G/H$ is the open $G$-orbit, in terms of colored fans [@MR705534].
The above results combine to give a complete classification of spherical varieties, but there are still many open questions in this setting. One such question is, what geometric properties of a spherical variety can be inferred by studying the associated spherical data, that is, the colored fan and homogeneous spherical data. One practical method of pursuing this question is to consider other well understood classes of varieties and ask under what conditions will they be spherical varieties.
With this in mind, let $Q$ be a parabolic subgroup of $G$. Denote by $W$ the Weyl group of $G$ and $W_Q$ the subgroup of $W$ corresponding to $Q$. Then $W^Q$ is defined to be the subset of minimal length right coset representatives of $W_Q$ in $W$. There is a natural action of $G$ on $G/Q$ given by left multiplication. For a $w \in W^Q$ we define the Schubert variety $X(w)$ to be the Zariski closure of the $B$-orbit of $wQ/Q$ in $G/Q$. These Schubert varieties will be stable under the action of certain parabolic subgroups $P$ of $G$, and hence $L$-stable for the reductive Levi subgroup $L$ of $P$. Two natural questions arise.
1. Given a Schubert variety $X(w)$ in $G/Q$ that is $L$-stable, when is $X(w)$ a spherical $L$-variety?
2. If $X(w)$ is a spherical $L$-variety, what is the associated spherical data?
As the geometry of Schubert varieties is particularly well understood the answer to these questions would provide ample test cases for the project of inferring geometric properties of spherical varieties in terms of their spherical data. This paper provides a complete answer to the first question when $G/Q$ is the Grassmannian variety in type A. The first author along with M. Bilen Can explores question (1) for an arbitrary $G$ and $Q$ and shows that smooth Schubert varieties are always spherical varieties [@2018arXiv180305515B]. Both authors, joint with M. Bilen Can, explore question (2) in [@2018arXiv180704879B]. Further, in [@2018arXiv180704879B] the toroidal Schubert varieties in the Grassmannian are characterized, and in type A, the ${\mathrm{GL}}_p \times {\mathrm{GL}}_q$-spherical Schubert varieties are also studied.
We now give an outline of the results in this paper. A Levi-Schubert quadruple is defined to be the data $(w,d,N,L)$ where $X(w)$ is a Schubert variety in the Grassmannian $G_{d,N}$ of $d$-dimensional subspaces of $\mathbb{C}^N$ and $L$ is a Levi subgroup of ${\mathrm{GL}}_N$. Such a Levi-Schubert quadruple is called stable if the Schubert variety $X(w)$ is $L$-stable, and a stable quadruple is called spherical if $X(w)$ is a spherical $L$-variety. Our first step in classifying the spherical Levi-Schubert quadruples is to define the reduction of $(w,d,N,L)$, which is also a Levi-Schubert quadruple and is denoted $(\overline{w}, \overline{d}, \overline{N}, \overline{L})$. We then show that $(w,d,N,L)$ is spherical if and only if $(\overline{w}, \overline{d}, \overline{N}, \overline{L})$ is spherical. This reduction step makes the classification considerably simpler to state.
In a previous paper the authors give a combinatorial description of the decomposition of the homogeneous coordinate ring $\mathbb{C}[X(w)]$, for the Plücker embedding, into irreducible $L$-modules for the induced action of $L$ [@HodgesLakshmibai]. For a stable Levi-Schubert quadruple $(w,d,N,L)$ we say that it is multiplicity free if the decomposition of $\mathbb{C}[X(w)]$ into irreducible $L$-modules is multiplicity free. In Proposition \[p:multFreeSphericalEquiv\] it is shown that a stable $(w,d,N,L)$ is multiplicity free if and only if it is spherical.
In [@HodgesLakshmibai] the decomposition of $\mathbb{C}[X(w)$ into irreducible $L$-modules is given in two steps. In the first step, each degree piece of $\mathbb{C}[X(w)]$ is decomposed into simpler submodules. The second step then shows that these submodules are isomorphic to certain tensor products of skew Schur-Weyl modules (which can be easily decomposed into irreducible $L$-modules using the Littlewood-Richardson coefficients). In Proposition \[p:MultFreeHighestLevel\] we define two criteria ${\mathit{M}\kern -0.1em\mathrm{1}}$ and ${\mathit{M}\kern -0.1em\mathrm{2}}$ that are stated in terms of the simpler submodules from the first step above. The proposition states that a Levi-Schubert quadruple is multiplicity free if and only if both ${\mathit{M}\kern -0.1em\mathrm{1}}$ and ${\mathit{M}\kern -0.1em\mathrm{2}}$ are satisfied.
If $X(w)$ is a Schubert variety in $G_{d,N}$ then $w$ can be represented by the sequence $(\ell_1,\ldots,\ell_d)$ for some integers $1 \leq \ell_1 < \ell_2 < \cdots < \ell_d \leq N$ (see Section \[subsec:SMT\]). A Levi subgroup $L$ of ${\mathrm{GL}}_N$ is of the form ${\mathrm{GL}}_{N_1} \times \cdots \times {\mathrm{GL}}_{N_{b_L}}$ for some positive integers $b_L$ and $N_k$ with $1 \leq k \leq b_L$. Using $L$ we define a partition of $\{ 1, \ldots , N \}$ into subsets of consecutive integers denoted ${\mathrm{Block}}_{L,k}$ for $1 \leq k \leq b_L$ (see Section \[sec:Decomp\]). Then the non-negative integers $h_1,\ldots,h_{b_L}$ are defined by $h_k = |\left\{j | \ell_j \in {\mathrm{Block}}_{L, k} \right\}|$.
In Propositions \[p:MCCSatCrit\] and \[p:MCSatCrit\] we provide the exact combinatorial requirements on $b_L$, $h_1,\ldots,h_{b_L}$ and $N_1,\ldots,N_{b_L}$ such that ${\mathit{M}\kern -0.1em\mathrm{2}}$ and ${\mathit{M}\kern -0.1em\mathrm{1}}$ are, respectively, satisfied. This allows us to prove our primary result.
\[t:mainSphericalClassification\] The stable, reduced Levi-Schubert quadruple $(w,d,N,L)$ is multiplicity free (equivalently spherical) if and only if one of the following holds
1. $b_L \leq 2$
2. $b_L = 3$, and at least one of $N_2 = 1$, $h_1 + 1 \geq N_1$, $N_2=h_2$ with $h_1 + 2 \geq N_1$, $h_2 > 0$ with $h_3 < 2$, $h_2 = 0$ with $h_3 \leq 2$ holds
3. $b_L \geq 4$, $p_w = 2$ or if $p_w > 2$, then $h_1 + \cdots + h_{p_w - 1} + 1 \geq N_1 + \cdots + N_{p_w - 1}$
where $1 < p_w < b_L - 1$ is the minimum index such that $h_{p_w + 1} + \cdots + h_{b_L} < 2$. Such an index may not exist, if it does not set $p_w = b_L - 1$.
We give even simpler criterion for a Levi-Schubert quadruple $(w,d,N,L)$ to be spherical in the case when $L$ is the maximal Levi subgroup which acts on $X(w)$ by left multiplication in Corollary \[c:mainSphericalClassification\]. As a nice application of this classification theorem, in Corollary \[c:toricSchubert\], we give a description of the Schubert varieties in the Grassmannian that are toric varieties for a quotient of the maximal torus under the left multiplication action.
This paper is organized into the following sections. In Section 2 the background and notation for spherical varieties, Schubert varieties, skew Schur functions, and skew Schur-Weyl modules is covered. Additionally, a few minor technical lemma involving Littlewood-Richardson coefficients are proved. Section 3 recalls the results and notation for the decomposition of the homogeneous coordinate ring of $X(w)$ into irreducible $L$-modules that was developed in [@HodgesLakshmibai]. Levi-Schubert quadruples and their reduction is covered in Section 4. The classification of the reduced, stable Levi-Schubert quadruples that are spherical is accomplished in Section 5. An application of these results to toric Schubert varieties is briefly discussed in Section 6.
Preliminaries {#sec:prelim}
=============
Spherical varieties {#subsec:spherical}
-------------------
Let $X$ be a normal $G$-variety for a reductive group $G$. The most common characterization given for $X$ to be a *spherical variety* is that it has an open dense $B$-orbit for a Borel subgroup $B$ of $G$. In his survey of spherical varieties Perrin collects a number of other equivalent characterizations of spherical varieties which we recall here.
\[t:perrinSpherical\] The normal $G$-variety $X$ is spherical if any of the following hold.
1. $X$ has an open dense $B$-orbit
2. $X$ has finitely many $B$-orbits
3. $\mathbb{C}[X]^{B} = \mathbb{C}$, where $\mathbb{C}[X]$ is the homogeneous coordinate ring of $X$
4. If $X$ is quasi-projective: For any $\mathcal{L}$ a $G$-linearized line bundle, the $G$-module $\mathrm{H}^0(X,\mathcal{L})$ is multiplicity free.
We shall primarily make use of the first and fourth characterizations from Theorem \[t:perrinSpherical\].
Algebraic groups {#subsec:AlgGrps}
----------------
In this section we will fix notation and briefly cover the algebraic groups background required for this paper. See [@MR1102012] for a more detailed treatment.
We will denote the group of invertible $N\times N$ matrices over $\mathbb{C}$ by ${\mathrm{GL}}_N$. Let $B$ be the standard Borel subgroup of upper triangular matrices with $T$ the standard maximal torus consisting of diagonal matrices. The character group of $T$, $\mathfrak{X}(T) := \mathrm{Hom}_{\mathrm{alg. gp.}}(T, \mathbb{C})$, will be written additively. Any finite dimensional $T$-module $V$ may be written as the sum of weight spaces
$V = \displaystyle \bigoplus_{\chi \in \mathfrak{X}(T)} V_{\chi}$
where $V_{\chi} := \{ v \in V \mid t v = \chi(t)v, \forall t \in T \}$. We refer to $\chi \in \mathfrak{X}(T)$ as a weight in $V$ if $\mathrm{dim}V_{\chi}\neq 0$.
For the Adjoint action of $T$ on $\mathfrak{gl}_N:=Lie({\mathrm{GL}}_N)$ we define $\Phi$ to be the set of nonzero weights in $\mathfrak{gl}_N$. Then $\Phi = \{ \epsilon_i - \epsilon_j | 1 \leq i,j \leq N \}$ where $\epsilon_i - \epsilon_j$ is the element in $\mathfrak{X}(T)$ that sends the diagonal matrix with $t_1,\ldots,t_n$ on the diagonal to $t_i t_j^{-1}$ in $\mathbb{C}$. The elements in $\Phi$ are referred to as *roots* and $\Phi$ is the *root system* of ${\mathrm{GL}}_N$ relative to $T$. The Borel subgroup $B$ induces a subset of positive roots $\Phi^{+}= \{ \epsilon_i - \epsilon_j | 1 \leq i < j \leq N \}$ and a subset of simple roots $\Delta= \{ \alpha_i := \epsilon_i - \epsilon_{i+1} | 1 \leq i < N \}$ in $\Phi$.
A *parabolic subgroup* of ${\mathrm{GL}}_N$ is a closed subgroup containing a Borel subgroup. A *standard parabolic subgroup* is a parabolic subgroup containing $B$. For $1 \leq d < N$ define the *maximal standard parabolic subgroup* $P_d$ as the subgroup containing all elements of ${\mathrm{GL}}_N$ with a block of zeros of size $N-d \times d$ in the bottom left.
$P_{d}=\left\{ \left[ \begin{array}{cc}
* & * \\
0_{N-d \times d} & * \\
\end{array} \right] \in {\mathrm{GL}}_N \right\}$
There is a important bijection between the subsets of the simple roots $\Delta$ and the standard parabolic subgroups. For $I \subseteq \Delta$ we define
$P_I = \displaystyle \bigcap_{{\alpha_d} \in \Delta \setminus I } P_d$
Any standard parabolic subgroup $P_I$ may be written as a semidirect product of its unipotent radical $U_I$ and a reductive subgroup $L_I$ called a Levi factor or Levi subgroup. For $I = \{ \alpha_{i_1},\ldots, \alpha_{i_q} \} \subset \Delta$ with $i_1 \leq \cdots \leq i_q$ and $\Delta \setminus I = \{ \alpha_{j_1},\ldots, \alpha_{j_r} \}$ with $j_1 \leq \cdots \leq j_r$ let $b_{L}=r+1$. Then setting $N_1 = j_1$, $N_k = j_k - j_{k-1}$ for $1 < k < b_L$, and $N_{b_L}=N-j_r$ we have that $N = N_1 + \cdots + N_{b_L}$ and
$L = {\mathrm{GL}}_{N_1} \times \cdots \times {\mathrm{GL}}_{N_{b_L}}$
For this reason we will refer to $b_L$ as the *number of blocks of $L$*.
The Weyl group $W$ of ${\mathrm{GL}}_N$ is generated by the simple reflections $s_{\alpha_i}$ for $\alpha_i \in \Delta$. The group $W$ is isomorphic to the symmetric group of permutations on $N$ letters via the map that identifies $s_{\alpha_i}$ with the transposition $(i, i+1)$. In light of this, we will refer to elements of $W$ by the sequence $(x_1,\cdots,x_N)$ which corresponds to the permutation that sends $i$ to $x_i$. The length of an element $w \in W$, denoted $\ell(w)$, is defined to be the minimum number $k$ such that $w$ may be written as the product of $k$ simple reflections.
The subsets $I$ of $\Delta$ also index subgroups of $W$. We define $W_I$ to be the subgroup generated by $\{ s_{\alpha_i} | \alpha_i \in I \}$. Then we define a subset of $W$ corresponding to $I$,
$W^I = \{ w \in W \mid \ell(ww') = \ell(w) + \ell(w'), \textrm{ for all }w' \in W_I \}$.
Then $W = W^I W_I$; that is, any element $w \in W$ can be written as the product $uv$ with $u \in W^I$, $v \in W_I$ and $\ell(w) = \ell(u) + \ell(v)$. Viewed in this way, $W^I$ is the set of minimal length right coset representatives of $W_I$ in $W$.
It will subsequently be convenient to identify the subgroup $W_I$ and subset $W^I$ by their associated parabolic subgroup. In particular, given a standard parabolic subgroup $P=P_I$, we will write $W_P$ and $W^P$ instead of $W_I$ and $W^I$.
Standard monomial theory {#subsec:SMT}
------------------------
The *Grassmannian* ${\mathrm{G}}_{d,N}$ is the set of all $d$-dimensional subspaces of $\mathbb{C}^{N}$ and can be equipped with a projective variety structure via the Plücker embedding. The Plücker embedding is the map from ${\mathrm{G}}_{d,N}$ to $\mathbb{P}(\bigwedge^d \mathbb{C}^N)$ defined by sending a $d$-dimensional subspace $U$ with basis $\{u_1,\ldots,u_d\}$ to the class $[u_1 \wedge \cdots \wedge u_d]$. This map is well defined (does not depend on choice of basis for $U$) and injective.
Define $I_{d,N}$ to be the set of strictly increasing positive integer sequences with $d$ values ranging from $1$ to $N$. Explicitly,
$I_{d,N} = \{ (i_1,\ldots,i_d) | 1 \leq i_1 < \cdots < i_d \leq N \}$.
If $\{ e_1,\ldots,e_N\}$ are the standard basis vectors of $\mathbb{C}^N$, then $\{e_{\tau} := e_{i_1} \wedge \cdots \wedge e_{i_d} \mid \tau = (i_1,\ldots,i_d) \in I_{d,N} \}$ is a basis for $\bigwedge^d \mathbb{C}^N$. Defining ${\mathrm{p}}_{\tau} := e_{\tau}^{*}$, we have that $\{{\mathrm{p}}_{\tau} \mid \tau \in I_{d,N} \}$ is the dual basis for $(\bigwedge^d \mathbb{C}^N)^{*}$. These ${\mathrm{p}}_{\tau}$ are a set of projective coordinates for $\mathbb{P}(\bigwedge^d \mathbb{C}^N)$ called the *Plücker coordinates*. The image of the Grassmannian under the Plücker embedding is cut out scheme theoretically by certain quadratic relations in the Plücker coordinates called the *Plücker relations*. Subsequently we will identify the Grassmannian by its image under the Plücker embedding.
The Grassmannian ${\mathrm{G}}_{d,N}$ is the ${\mathrm{GL}}_N$-orbit of $[e_{id}:=e_1 \wedge \cdots \wedge e_d]$ for the natural action of ${\mathrm{GL}}_N$ on $\mathbb{P}(\bigwedge^d \mathbb{C}^N)$. Under this action the $T$-fixed points of ${\mathrm{G}}_{d,N}$ are precisely $[e_{\tau}]$ for $\tau \in I_{d,N}$. The *Schubert variety* $X(\tau)$ is defined to be the Zariski-closure of the $B$-orbit of $[e_{\tau}]$, that is, $X(\tau) := \overline{B[e_{\tau}]}$. The *Bruhat order* on the set $I_{d,N}$ is induced by the containment order on the set of Schubert varieties; $\tau \leq w$ if and only if $X(\tau) \subseteq X(w)$. If $\tau = (i_1,\ldots,i_d)$ and $w=(\ell_1,\ldots,\ell_d)$, then it can be shown that $\tau \leq w$ is equivalent to $i_1 \leq \ell_1,\ldots,i_d \leq \ell_d$.
We noted above that ${\mathrm{G}}_{d,N}$ is the ${\mathrm{GL}}_N$-orbit of $[e_{id}]$. The isotropy subgroup at $[e_{id}]$ is precisely $P_{d}$. Thus we identify ${\mathrm{G}}_{d,N}$ as the homogeneous space ${\mathrm{GL}}_N/ P_d$. Consequently, we see that $W^{P_d}$ may be identified with $I_{d,N}$ via the map that sends a $\tau=(i_1,\ldots,i_n) \in W^{P_d}$ to the sequence $(i_1,\ldots,i_d)\uparrow$, where $\uparrow$ indicates that the preceding sequence has been reordered so that it is strictly increasing. In light of these identifications, we shall index the Schubert varieties in ${\mathrm{G}}_{d,N}$ by elements of $W^{P_d}$ while denoting the elements of $W^{P_d}$ by their corresponding sequence in $I_{d,N}$.
The homogeneous coordinate ring $\mathbb{C}[X(w)]$ of the Schubert variety $X(w)$ induced by the Plücker embedding is a polynomial algebra of the form
$\mathbb{C}[X(w)] = \mathbb{C}[{\mathrm{p}}_{\tau}, \tau \in W^{P_d}] / J$
where $J$ is the homogeneous ideal generated by the Plücker relations and $\{ {\mathrm{p}}_{\tau}, \tau \nleq w \}$. We say that a degree r monomial ${\mathrm{p}}_{\tau_1}\cdots{\mathrm{p}}_{\tau_r} \in \mathbb{C}[X(w)]$ is a *standard monomial on $X(w)$* if $w \geq \tau_1 \geq \cdots \geq \tau_r$. The following theorem illustrates the fundamental importance of standard monomials.
\[T:FundamentalSMT\] The degree r standard monomials on $X(w)$ are a vector space basis of $\mathbb{C}[X(w)]_r$.
Skew Young diagrams, skew Schur functions, and skew Weyl modules {#subsec:skewYoung}
----------------------------------------------------------------
For a more in depth introduction to the concepts covered in this section see [@MR1676282]. A *partition* $\lambda$ is a sequence of positive integers $(\lambda_1,\ldots,\lambda_k)$ such that $\lambda_1 \geq \cdots \geq \lambda_k$. It will be useful to be able to express arbitrarily large partitions. When we write $(a_1^{b_1},\cdots,a_r^{b^r})$ with $a_1 \geq \cdots \geq a_r$ and $b_i \geq 0$ we mean the partition with the first $b_1$ entries equal to $a_1$, the next $b_2$ entries equal to $a_2$, and so on. Note that we will often omit the superscript when it is equal to one.
We associate to every partition $\lambda$ a *Young diagram*, also denoted $\lambda$, which is a collection of upper left justified boxes with $\lambda_i$ boxes in row $i$. The Young diagram associated to a partition $\lambda$ will be said to have *shape* $\lambda$. For example, the partition $(4,2^2)=(4,2,2)$ corresponds to the Young diagram
& & &\
&\
&\
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ we say that the *length* of the partition is the number of entries in the sequence, and denote it by $\ell(\lambda)$. The *size* of the partition is $|\lambda|=\sum \lambda_i$. Given a second partition $\mu = (\mu_1,\ldots,\mu_j)$, we write $\mu \subseteq \lambda$ if $j \leq k$ and $\mu_i \leq \lambda_i$ for $1 \leq i \leq j$. Equivalently, $\mu \subseteq \lambda$ if the Young diagram of $\mu$ is contained the Young diagram of $\lambda$.
If $\mu \subseteq \lambda$, we define the skew (Young) diagram $\lambda / \mu$ to be the diagram formed by removing the leftmost $\mu_i$ boxes in row $i$ of $\lambda$ for each row. For example, the skew diagram $(4,2,2) / (2,1)$ is given by
& & &\
&\
&\
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is said to be a *rectangle* if all entries in the partition are equal to some positive integer $p$. A partition is called a *hook* if there is a positive integer $p$ such that $\lambda_1=p$ and $\lambda_i=1$ for $i\geq 2$. A *fat hook* is a partition such that all entries are equal to either $p$ or $q$ for two positive integers $p$, $q$.
A skew diagram is said to be *basic* if it contains no empty rows or columns. Given a skew diagram $\lambda / \mu$ we define $\tilde{\lambda} / \tilde{\mu}$ to be the basic skew diagram formed by deleting all the empty rows and columns from $\lambda / \mu$. The *$\pi$-rotation* $(\lambda / \mu)^{\pi}$ is the skew diagram that arises by rotating $\lambda / \mu$ through $\pi$ radians. The *conjugate* of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is defined to be the partition $\lambda' = (\lambda'_1,\ldots,\lambda'_{\lambda_1})$ where each $\lambda'_i$ is equal to the number of boxes in column $i$ of $\lambda$. Fix $m$ and $n$ two positive integers. For a partition of length $n$, such that $\lambda \subseteq m^n$, we say that the *$m^n$-complement* of $\lambda$ is $\lambda^\# := (m- \lambda_{n},\ldots,m-\lambda_1)$. Note that $\lambda^\# = (m^n / \lambda)^{\pi}$. The *$m^n$-shortness* of a partition $\lambda$ is the length of the shortest line segment in the path of length $m+n$ from the southwest to northeast corners of $m^n$ that contains the bottom and right contour of $\lambda$.
Consider the partitions $\lambda = (4,2,2,1)$ and $\mu = (2,2)$. Then
$\lambda / \mu =\;\;\;$
& & &\
&\
&\
\
$\tilde{\lambda} / \tilde{\mu} =\;\;\;$
& & &\
&\
\
$(\lambda / \mu)^{\pi} = \;\;\; $
& & &\
& & &\
&\
&\
and if $m=n=4$ we have the following Young diagrams.
$\lambda =\;\;\; $
& & &\
&\
&\
\
$\lambda' =\;\;\; $
& & &\
& &\
\
\
$\lambda^\# =\;\;\; $
& &\
&\
&\
The $m^n$-shortness of $\lambda$ is 1 while the $m^n$-shortness of $\mu$ is 2 as illustrated by the shortest line segments of the respective paths below.
(n)
[5cm]{}
& & &\
&\
&\
\
; (n.south west)–++(1.21,0)–++(0,1.21); (n.south west)–++(.25\*1.21,0)–++(0,.25\*1.21)–++(.25\*1.21,0)–++(0,.25\*1.21)–++(0, .25\*1.21)–++(.25\*1.21,0)–++(.25\*1.21,0)–++(0, .25\*1.21);
(n)
[5cm]{}
&\
&\
\
\
; (n.south west)–++(1.21,0)–++(0,1.21); (n.south west)–++(0, .25\*1.21)–++(0,.25\*1.21)–++(.25\*1.21,0)–++(.25\*1.21,0)–++(0, .25\*1.21)–++(0,.25\*1.21)–++(.25\*1.21,0)–++(.25\*1.21,0);
Given a skew diagram $\lambda / \mu$, we say that a *filling* of shape $\lambda / \mu$ in $\{ 1,...,n \}$ is an assignment of a value in $\{ 1,...,n \}$ to each box in $\lambda / \mu$. A *tableau* is a filling such that the values in each column increase strictly downwards. A *semistandard tableau* is a tableau such that the values in each row increase weakly along each row. The *weight* of a filling equals $\nu = (\nu_1,...,\nu_n)$ where $\nu_i$ equals the number of boxes with value $i$ in ${\mathbb{T}}$. An example, from left to right, of a filling, tableau, and semistandard tableau of shape $(4,2,2) / (2,1)$ in $\{ 1,...,4 \}$ is given below.
& & 2 & 1\
& 2\
3 & 1\
& & 4 & 3\
& 2\
1 & 3\
& & 2 & 3\
& 3\
2 & 4\
The respective weights are $(2,2,1,0)$, $(1,1,2,1)$, and $(0,2,2,1)$.
Given a skew diagram $\lambda / \mu$, the associated *skew Schur function* is
${s_{\lambda/\mu}} = \displaystyle \sum_{{\mathbb{T}}} x_1^{\textrm{\# of 1's in }{\mathbb{T}}}\cdots x_k^{\textrm{\# of k's in }{\mathbb{T}}}$
where the infinite sum is over all semistandard tableaux of shape $\lambda/\mu$ and $k$ is the maximum value in the semistandard tableau. Then, for a partition $\lambda$, the *Schur function* associated to $\lambda$ is defined to be ${s_{\lambda}} := {s_{\lambda/ \emptyset}}$ where $\emptyset$ is the zero partition. Though not immediately apparent, the Schur and skew Schur functions are symmetric. In fact, the ring of symmetric functions has a basis given by the Schur functions. The Littlewood-Richardson coefficients ${c_{\mu,\nu}^{\lambda}}$ appear as the structure coefficients for multiplication in this ring. That is, for partitions $\mu$ and $\nu$ we have $${s_{\mu}}{s_{\nu}} = \displaystyle \sum_{\lambda} {c_{\mu,\nu}^{\lambda}} {s_{\lambda}}$$ where the sum is over all partitions $\lambda$ such that $|\lambda| = |\mu| + |\nu|$. The Littlewood-Richardson coefficients ${c_{\mu,\nu}^{\lambda}}$ also appear in the expansion of the skew Schur functions $${s_{\lambda / \mu}} = \displaystyle \sum_{\nu} {c_{\mu,\nu}^{\lambda}} {s_{\nu}},$$ where the sum is over all $\nu$ such that $|\lambda| - |\mu| = |\nu|$. A skew Schur function is *multiplicity-free* if, in the expansion of the skew Schur function into the basis of Schur functions, all the nonzero Littlewood-Richardson coefficients are equal to 1.
Fix a positive integer $N$. The *skew Schur polynomial* ${s_{\lambda/\mu}}(x_1,\ldots,x_N)$ is a specialization of the skew Schur function ${s_{\lambda/\mu}}$ achieved by setting $x_m=0$ for all $m>N$. The *Schur polynomial* ${s_{\lambda}}(x_1,\ldots,x_N)$ is defined to be ${s_{\lambda/ \emptyset}}(x_1,\ldots,x_N)$. The Schur polynomials associated to partitions of length less than or equal $N$ give a basis for the ring of symmetric functions in variables $x_1,\ldots,x_N$. Thus $$\label{e:SchurPolynomial}
{s_{\lambda / \mu}}(x_1,\ldots,x_N) = \displaystyle \sum_{\nu} {c_{\mu,\nu}^{\lambda}} {s_{\nu}}(x_1,\ldots,x_N),$$ where the sum is over all $\nu$ with $\ell(\nu) \leq N$ such that $|\lambda| - |\mu| = |\nu|$.
\[r:suffMultFree\] We will say that a skew Schur polynomial is multiplicity free if the expansion into the basis of Schur polynomials is multiplicity free. Importantly, if the skew Schur function ${s_{\lambda/\mu}}$ is multiplicity free then the skew Schur polynomial ${s_{\lambda/\mu}}(x_1,\ldots,x_N)$ is multiplicity free. However, the converse is not true, as there might be partitions $\nu$ of length greater than $N$ such that ${c_{\mu,\nu}^{\lambda}} > 1$ but none with length less than or equal $N$.
The first of the two identities below may be found in [@MR1676282] while the second follows trivially from the Littlewood-Richardson rule (see Section \[subsec:LRRule\]). $$\label{e:SchurPiRotation}
{s_{\lambda / \mu}} = {s_{(\lambda / \mu)^{\pi}}}$$ $$\label{e:SchurBasic}
{s_{\lambda / \mu}} = {s_{\tilde{\lambda} / \tilde{\mu}}}$$
The second identity implies that the classification of multiplicity-free Schur functions reduces to a classification of basic multiplicity-free Schur functions. This classification was achieved by Thomas and Yong in [@MR2583223] (see also [@MR2737323 Theorem 4.3]).
\[T:skewMultFree\] The basic skew Schur function $s_{\lambda / \mu}$ is multiplicity-free if and only if $\lambda$ and $\mu$ satisfy one or more of the following conditions:
1. $\mu$ or $\lambda^{\#}$ is the zero partition
2. $\mu$ or $\lambda^{\#}$ is a rectangle of $m^{n}$-shortness 1
3. $\mu$ is a rectangle of $m^{n}$-shortness 2 and $\lambda^{\#}$ is a fat hook (or vice versa)
4. $\mu$ is a rectangle and $\lambda^{\#}$ is a fat hook of $m^{n}$-shortness 1 (or vice versa)
5. $\mu$ and $\lambda^{\#}$ are rectangles
where $m=\lambda_1$, $n=\lambda'_1$, and $\lambda^{\#}$ is the $m^n$-complement of $\lambda$.
The skew diagrams also index certain distinguished representations of ${\mathrm{GL}}_N$. For $\lambda / \mu$ a skew diagram we denote the corresponding *skew Weyl module*, equivalently Schur functor, by $\mathbb{W}^{\lambda / \mu}(\mathbb{C}^N)$ (see [@MR1153249 §6.1] for the details of this construction). We will normally simplify this notation by writing $\mathbb{W}^{\lambda / \mu}$ as long as no confusion will arise from doing so.
For a partition $\lambda$ the corresponding *Weyl module* is $\mathbb{W}^{\lambda} := \mathbb{W}^{\lambda / \emptyset}$. The Weyl modules $\mathbb{W}^{\lambda}$ such that $\ell(\lambda) \leq N$ are precisely the polynomial irreducible representations of ${\mathrm{GL}}_N$. We have that ${\mathrm{GL}}_N$ is completely reducible since it is a reductive group and we are working over $\mathbb{C}$. Thus any ${\mathrm{GL}}_N$-representation may by written uniquely, up to isomorphism, as a direct sum of irreducible representations. The decomposition of $\mathbb{W}^{\lambda / \mu}$ into irreducible representations has a particularly nice description in terms of the Littlewood-Richardson coefficients. $$\label{e:SkewWeylDecomp}
\mathbb{W}^{\lambda / \mu} = \displaystyle \bigoplus (W^{\nu})^{\oplus {c_{\mu,\nu}^{\lambda}}}$$
\[r:weylModuleMultFree\] This identity follows from the fact that ${s_{\lambda / \mu}}(x_1,\ldots,x_N)$ is the character of $\mathbb{W}^{\lambda / \mu}$. In particular, this implies that $\mathbb{W}^{\lambda / \mu}$ has a multiplicity-free decomposition into irreducible ${\mathrm{GL}}_N$-modules if and only if ${s_{\lambda / \mu}}(x_1,\ldots,x_N)$ is multiplicity-free. Following Remark \[r:suffMultFree\] we conclude that Theorem \[T:skewMultFree\] gives sufficient conditions on $\lambda / \mu$ for $\mathbb{W}^{\lambda / \mu}$ to have a multiplicity free decomposition.
Computing Littlewood-Richardson coefficients via the Littlewood-Richardson rule {#subsec:LRRule}
-------------------------------------------------------------------------------
Many of our results will rely on the ability to compute certain Littlewood-Richardson coefficients. To facilitate these computations we recall an identity and the Littlewood-Richardson Rule[@MR1464693 Section 5]. The identity is a non-trivial symmetry of the Littlewood-Richardson coefficients and its proof may be found in [@MR1676282], it states that $$\label{equation:littlewoodRichardsonIdentities}
c_{\mu, \nu}^{\lambda} = c_{\nu, \mu}^{\lambda}.$$
The *row word* of a semistandard tableau ${\mathbb{T}}$, denoted $w_{row}({\mathbb{T}})$, is the values in ${\mathbb{T}}$ written from left to right and bottom to top. If a row word equals $t_1,...,t_r$ we say that it is a *reverse lattice word* if the number $i$ appears at least as often as $i+1$ in every reversed subsequence $t_r,t_{r-1},...,t_{s+1},t_s$. A semistandard tableau ${\mathbb{T}}$ such that $w_{row}({\mathbb{T}})$ is a reverse lattice word is a *semistandard Littlewood-Richardson tableau*.
Consider the two semistandard tableaux below.
& & 1 & 1\
& 1\
1 & 3\
2\
3\
& & 1 & 1\
& 2\
1 & 3\
2\
3\
Their respective row words are 3,2,1,3,1,1,1 and 3,2,1,3,2,1,1. The second is a reverse lattice word while the first is not; the number of 3’s in 1,1,1,3 is greater than the number of 2’s. Thus only the second is a semistandard Littlewood-Richardson tableau.
\[p:LittlewoodRichardson\] The Littlewood-Richardson coefficient $c_{\mu,\nu}^{\lambda}$ is equal to the number of semistandard Littlewood-Richardson tableaux of shape $\lambda / \mu$ and weight $\nu$.
We will now use this proposition to prove two lemma involving the Littlewood-Richardson coefficients that will be useful in Section \[sec:class\].
\[l:MultFreePolyNotFunction\] Let $\lambda=(r^{N},p,q)$, $\mu=(a,b)$ be two partitions with $0 < b \leq a < r$ and $0 < q \leq p < r$. Then the skew Schur polynomial ${s_{\lambda / \mu}}(x_1,\ldots,x_N)$ is multiplicity free and hence $\mathbb{W}^{\lambda / \mu}$ has a multiplicity free decomposition into irreducible ${\mathrm{GL}}_N$-modules.
We begin by noting that Theorem \[T:skewMultFree\] implies that ${s_{\lambda / \mu}}$ is not multiplicity free, nonetheless the result still holds. Let $\nu=(\nu_1,\ldots,\nu_m)$ be a partition of length $m \leq N$ such that $|\lambda| - |\mu| = |\nu|$; we will show that ${c_{\mu,\nu}^{\lambda}} \leq 1$. To do this we will once again use to equivalently show that ${c_{\nu,\mu}^{\lambda}} \leq 1$. We begin by considering how we might fill the skew diagram $(r^{N},p,q) / \nu$ with $a$ ones and $b$ twos such that a semistandard tableau results. The fact that we can only use ones and twos immediately means that we can only successfully construct such a semistandard tableau when $(r^{N},p,q) / \nu$ has no more than two boxes in any column. Since the length of $\nu$ is less than or equal $N$ this restricts us to those $\nu$ such that the associated basic form of $(r^{N},p,q) / \nu$ is
& & & & & & & & & & &\
& & & & & & & & & & &\
& & & & &\
& &\
We will show that there are no choices when filling such a skew Young diagram with $a$ ones and $b$ twos if we wish the row word to be a reverse lattice word. First, we are forced to fill all the columns with two boxes with ones and twos. If we wish for the row word to be a reverse lattice word we are then forced to put a one in the rightmost column of row two that contains a single box. Note that this is true even if there are no columns with two boxes on the right.
& & & & & & & & & 1 & & 1\
& & & & & & & & 1 & 2 & & 2\
1& &1 & & &\
2& &2\
Now we are forced to put ones in the rest of the boxes in row two since we need a semistandard tableau. Finally, there is only one way to fill in the remaining empty boxes in row three such that the result is a semistandard tableau. It is possible that at some point in the preceding discussion we could not proceed because we would have had to use more than $a$ ones or $b$ twos; in this case the associated Littlewood-Richardson coefficient is zero. Otherwise, we could fill in the skew Young diagram but every choice was prescribed. Thus ${s_{\lambda / \mu}}(x_1,\ldots,x_N)$ is multiplicity free since any Littlewood-Richardson coefficient in is equal to 0 or 1.
Let $n \geq 0$ and $m > 0$. \[lemma:LRSphericalClassH\]
1. \[lemma:LRSphericalClassH1\] If $\lambda=(2^n,1,1)$, $\mu=(1)$, and $\nu=(2^n,1)$, then $c_{\mu,\nu}^{\lambda}=1$.
2. \[lemma:LRSphericalClassH2\] If $\lambda=(2^{n+1})$, $\mu=(1)$, and $\nu=(2^n,1)$, then $c_{\mu,\nu}^{\lambda}=1$.
3. \[lemma:LRSphericalClassH3\] If $\lambda=(2^{m},1)$, $\mu=(1,1)$, and $\nu=(2^{m-1},1)$, then $c_{\mu,\nu}^{\lambda}=1$.
4. \[lemma:LRSphericalClassH4\] If $\lambda=(3^m,2,1)$, $\mu=(2,1)$ and $\nu=(3^{m-1},2,1)$, then $c_{\mu,\nu}^{\lambda}=2$.
5. \[lemma:LRSphericalClassH5\] Let $\lambda$ and $\mu$ be partitions such that $\mu \subset \lambda$. Let $\nu = (n)$ where $n = |\lambda|-|\mu|$. Then ${c_{\mu,\nu}^{\lambda}}=1$.
Note that in these partitions, when the exponent of an entry is 0 we simply omit that entry from the partition. For example, when $n=0$ in we have $\lambda=(2^0,1,1)=(1,1)$.
: The identity implies that $c_{\mu,\nu}^{\lambda}=c_{\nu,\mu}^{\lambda}$. Using Proposition \[p:LittlewoodRichardson\], we find $c_{\nu,\mu}^{\lambda}$ by counting the number of semistandard Littlewood-Richardson tableaux of shape $(2^n,1,1)/(2^n,1)$ with weight $(1)$. Since $(2^n,1,1)/(2^n,1)$ is a single box there is exactly one semistandard Littlewood-Richardson tableau with weight $\mu=(1)$. Hence $c_{\nu,\mu}^{\lambda}=1$.
: As $\lambda / \nu=(2^{n+1})/(2^n,1)$ is a single box, Proposition \[p:LittlewoodRichardson\] and imply that $c_{\mu,\nu}^{\lambda}=c_{\nu,\mu}^{\lambda}=1$.
: Once again we use to see that $c_{\mu,\nu}^{\lambda}=c_{\nu,\mu}^{\lambda}$. Then $\lambda / \nu=(2^{m},1)/(2^{m-1},1)$ is two disconnected boxes. There are two possible fillings of these disconnected boxes with weight $(1,1)$. Only the filling with a 1 in the upper right box and a 2 in the lower left box is a semistandard Littlewood-Richardson tableau. Thus we have $c_{\mu,\nu}^{\lambda}=c_{\nu,\mu}^{\lambda}=1$.
: As in the previous cases we calculate $c_{\nu,\mu}^{\lambda}$. The skew diagram $\lambda / \nu=(3^m,2,1)/(3^{m-1},2,1)$ is three disconnected boxes. There are two possible fillings of these boxes with weight $(2,1)$ whose row word is a reverse lattice word. Thus we have $c_{\mu,\nu}^{\lambda}=c_{\nu,\mu}^{\lambda}=2$.
: It is clear that any filling of $\lambda / \mu$ with $n = |\lambda|-|\mu|$ ones can be done in exactly one way. Further, the row word with all ones is a reverse lattice word. Hence ${c_{\mu,\nu}^{\lambda}}=1$.
The decomposition of the homogeneous coordinate ring {#sec:Decomp}
====================================================
Fix positive integers $d < N$. Then ${\mathrm{G}}_{d,N} = {\mathrm{GL}}_N / P_d$. Let $w \in W^{P_d}$ and let $P$ be a standard parabolic subgroup that acts on the Schubert variety $X(w)$ by left multiplication. This induces an action of the Levi part of $P$, which we will denote by $L$, on $X(w)$, which in turn induces an action of $L$ on the homogeneous coordinate ring $\mathbb{C}[X(w)]$. In [@HodgesLakshmibai], the authors give a combinatorial description of the decomposition of $\mathbb{C}[X(w)]$ into irreducible $L$-modules for this induced action. We recall this result as well as the relevant definitions and notation.
As remarked in Section \[subsec:AlgGrps\], the standard parabolic subgroup $P$ must be of the form $P_I$ for some $I=\{ \alpha_{i_1},\ldots,\alpha_{i_q} \} \subseteq \Delta$ with $i_1 < \cdots < i_q$ and $\Delta \setminus I = \{ \alpha_{j_1},\ldots, \alpha_{j_r} \}$ with $j_1 \leq \cdots \leq j_r$. Recall that $b_L := r+1$. If $N_1 = j_1$, $N_k = j_k - j_{k-1}$ for $1\leq k < b_L$, and $N_{b_L}=N-j_r$ then $N = N_1 + \cdots + N_{b_L}$ and $L = {\mathrm{GL}}_{N_1} \times \cdots \times {\mathrm{GL}}_{N_{b_L}}$. Let ${\mathrm{Block}}_{L,k}=\{ j_{k-1}+1,\ldots,j_k \}$ for $1 \leq k \leq b_L$ where $j_0 = 0$ and $j_{b_L}=N$. Then the subsets ${\mathrm{Block}}_{L,1}$,…,${\mathrm{Block}}_{L,b_L}$, which we refer to as the *blocks of L*, are a partition of $\{ 1,\ldots,N \}$. It is an easy check that $N_k = | {\mathrm{Block}}_{L, k} |$. We give an example of these subsets below in Example \[e:blocks\].
Denote by $H_w$ the subset of $W^{P_d}$ containing all $\tau \leq w$. Then for a $\theta \in H_w$ we say that $\theta$ is a *degree 1 head of type $L$* if $X_{\theta}$ is a $L$-stable Schubert subvariety of $X_w$.
\[p:HeadLComb\] A $\theta \in H_w$ is a degree 1 head of type $L$ if and only if $\theta \cap {\mathrm{Block}}_{L,k}$ is maximal for all $1 \leq k \leq b_L$; explicitly we require that for all $m \in \theta \cap {\mathrm{Block}}_{L,k}$ and $n \in {\mathrm{Block}}_{L,k} \setminus \theta \cap {\mathrm{Block}}_{L,k}$ we have $m > n$.
We will denote the subset of $H_w$ that contains all the degree 1 heads of type $L$ by ${\mathrm{Head}}_{L,1}$. Fix a positive integer $r$. A *degree r head of type $L$* is a sequence $\underline{\theta} = (\theta_1,\ldots,\theta_r)$ such that $\theta_i \in {\mathrm{Head}}_{L,1}$. A degree r head is *standard* if in addition $\theta_1 \geq \cdots \geq \theta_r$. Define
${\mathrm{Head}}_{L,r} = \{ (\theta_1,\ldots,\theta_r) | \theta_i \in {\mathrm{Head}}_{L,1} \}$
and
${\mathrm{Head}}_{L,r}^{std} = \{ (\theta_1,\ldots,\theta_r) \in {\mathrm{Head}}_{L,r} | \theta_1 \geq \cdots \geq \theta_r \}$.
One final set of definitions is required before we can describe the decomposition from [@HodgesLakshmibai]. Given a standard degree r head of type L we associate it to a collection of $k$ skew diagrams. Let $\underline{\theta} = (\theta_1,\ldots,\theta_r) \in {\mathrm{Head}}_{L,r}^{std}$. We begin by defining the semistandard tableau ${\mathbb{T}}_{\underline{\theta}}$ of shape $(r^d)$ by letting the columns of ${\mathbb{T}}_{\underline{\theta}}$ correspond to the $\theta_i$ in reverse order. Explicitly, the values from top to bottom in column $c$ of ${\mathbb{T}}_{\underline{\theta}}$ correspond to the first to last entries in $\theta_{r-c+1}$ for $1 \leq c \leq r$.
Fix a $k$ such that $1 \leq k \leq b_L$. Then ${\mathbb{T}}_{\underline{\theta}}^{(k)}$ is the basic semistandard tableau formed by first deleting all boxes in ${\mathbb{T}}_{\underline{\theta}}$ with values not in ${\mathrm{Block}}_{L,k}$ and then deleting all empty rows and columns (we omit the step of subtracting $j_k$ from the value in each box as is done in [@HodgesLakshmibai] since here we only care about the shape of the skew semistandard tableaux). This semistandard tableau has some shape, which we will write as $\lambda_{\underline{\theta}}^{(k)} / \mu_{\underline{\theta}}^{(k)}$. Finally, we define the $L$-module associated to $\underline{\theta}$ by
$\mathbb{W}_{\underline{\theta}} := \mathbb{W}^{\lambda_{\underline{\theta}}^{(1)} / \mu_{\underline{\theta}}^{(1)}}(\mathbb{C}^{N_1}) \otimes \cdots \otimes \mathbb{W}^{\lambda_{\underline{\theta}}^{(b_L)} / \mu_{\underline{\theta}}^{(b_L)}}(\mathbb{C}^{N_{b_L}})$.
\[t:LeviSchubertMainDecomp\] For a fixed $r$, we have a decomposition of $\mathbb{C}[X(w)]_r$ into $L$-modules given by
$\mathbb{C}[X(w)] \cong \displaystyle \bigoplus_{\underline{\theta} \in {\mathrm{Head}}_{L,r}^{std}} \mathbb{W}^*_{\underline{\theta}}$
where $\mathbb{W}^*_{\underline{\theta}}$ is the $L$-module dual of $\mathbb{W}_{\underline{\theta}}$.
As $\mathbb{W}_{\underline{\theta}}$ is a tensor product of skew Weyl modules, the decomposition of $\mathbb{C}[X(w)]_r$ into irreducible $L$-modules may then be achieved via .
\[e:blocks\] Set $d=3$ and $N=9$ and consider $w=(2,7,9) \in W^{P_d}$. The Schubert variety $X(w)$ is $L=L_I$-stable for $I=\{\alpha_1,\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_8 \}$. Then $\Delta \setminus I = \{ \alpha_2, \alpha_7 \}$ and we have that $L={\mathrm{GL}}_2 \times {\mathrm{GL}}_5 \times {\mathrm{GL}}_2$. Then $b_L=3$ and the blocks of $L$ are
${\mathrm{Block}}_{L, 1} = \{ 1, 2 \} \qquad {\mathrm{Block}}_{L, 2} = \{ 3, 4, 5, 6, 7 \} \qquad {\mathrm{Block}}_{L, 3} = \{ 8, 9 \}.$
The degree one heads of type $L$ are $(1, 2, 7)$, $(2,6,7)$, and $(2, 7, 9)$. One standard degree three head is $\underline{\theta} = ((2,7,9) , (2,6,7) , (1,2,7))$. We will now construct the skew semistandard tableaux and skew diagrams associated to $\underline{\theta}$. We have
${\mathbb{T}}_{\underline{\theta}} = \begin{ytableau}
1 & 2 & 2 \\
2 & 6 & 7 \\
7 & 7 & 9 \\
\end{ytableau}$.
We then create a basic skew semistandard tableau ${\mathbb{T}}_{\underline{\theta}}^{(k)}$ for each $1 \leq k \leq b_L$ by removing boxes not in ${\mathrm{Block}}_{L,k}$ and deleting empty rows and columns.
${\mathbb{T}}_{\underline{\theta}}^{(1)} = \begin{ytableau}
1 & 2 & 2 \\
2 \\
\end{ytableau} \qquad \qquad {\mathbb{T}}_{\underline{\theta}}^{(2)} = \begin{ytableau}
\none & 6 & 7 \\
7 & 7 \\
\end{ytableau} \qquad \qquad {\mathbb{T}}_{\underline{\theta}}^{(3)} = \begin{ytableau}
9 \\
\end{ytableau}$
The associated skew diagrams are $\lambda_{\underline{\theta}}^{(1)} / \mu_{\underline{\theta}}^{(1)} = (3,1)/\emptyset$, $\lambda_{\underline{\theta}}^{(2)} / \mu_{\underline{\theta}}^{(2)} = (3,2)/(1)$, and $\lambda_{\underline{\theta}}^{(3)} / \mu_{\underline{\theta}}^{(3)}=(1)/\emptyset$. This implies that the $L$-module associated to the degree 3 head $\underline{\theta}$ is
$\mathbb{W}_{\underline{\theta}} := \mathbb{W}^{(3,1)/\emptyset}(\mathbb{C}^{2}) \otimes \mathbb{W}^{(3,2)/(1)}(\mathbb{C}^{5}) \otimes \mathbb{W}^{(1)/\emptyset}(\mathbb{C}^{2})$
Reductions and Multiplicity Criterion {#sec:multFreeClass}
=====================================
It will be easier to state our classification result if we first perform some reductions. We define a *Levi-Schubert quadruple* to be the datum $(w, d, N, L)$ where $d < N$ are positive integers, $w=(\ell_1,\cdots,\ell_d) \in W^{P_d}$, and $L$ is a Levi subgroup of ${\mathrm{GL}}_N$. A Levi-Schubert quadruple is *stable* if $X(w)$ is $L$-stable for the action of $L$ by left multiplication. A stable Levi-Schubert quadruple is *multiplicity free* if the decomposition of $\mathbb{C}[X(w)]$ into irreducible $L$-modules is multiplicity free. A stable Levi-Schubert quadruple is *spherical* if $X(w)$ is a spherical $L$-variety. We say $(w, d, N, L)$ is *reduced* if $\ell_1 \neq 1$ and $\ell_d=N$. If $(w, d, N, L)$ is not reduced we define its *reduction* $(\overline{w}, \overline{d}, \overline{N}, \overline{L})$ as follows. The fact that the quadruple is not reduced implies that $w = (1,\ldots,p,\ell_{p+1},\ldots,\ell_d)$ for some $p \geq 0$ or $\ell_d\neq N$ with $\ell_{p+1} \neq p+1$.
Note that throughout the paper we will assume that $w$ is not the identity; in the case when $w$ is the identity, we have that $X(w)$ is a point space and hence if it is $L$-stable for some Levi subgroup $L$, then it is trivially spherical. Set
$\overline{w} = (\ell_{p+1}-p,\ldots,\ell_d-p)$\
$\overline{d} = d-p$\
$\overline{N} = \ell_d - p$.
Finally, we define $\overline{L}$ to be the image of $L$ under the map $\mathrm{pr}_w$ which is the composition of a diagonal projection map and a projection map
$\mathrm{pr}_w:{\mathrm{GL}}_N \longrightarrow {\mathrm{GL}}_p \times {\mathrm{GL}}_{\ell_d - p} \times {\mathrm{GL}}_{N - \ell_d} \longrightarrow {\mathrm{GL}}_{\ell_d - p}$.
\[l:reductioniso\] If $(\overline{w}, \overline{d}, \overline{N}, \overline{L})$ is the reduction of the Levi-Schubert quadruple $(w, d, N, L)$, then $X(w) \cong X(\overline{w})$ as varieties and $\mathbb{C}[X(w)] \cong \mathbb{C}[X(\overline{w})]$ as graded $\mathbb{C}$-algebras.
Consider the commutative diagram below.
$$\begin{tikzcd}
X(\overline{w}) \arrow{r}{} \arrow[hook]{d}{} & X(w) \arrow{r}{} \arrow[hook]{d}{} & X(w) \arrow[hook]{d}{} \\
{\mathrm{G}}_{\overline{d}, \overline{N}} \arrow[hook]{r}{} \arrow[hook]{d}{A} & {\mathrm{G}}_{d, \overline{N}} \arrow[hook]{r}{} \arrow[hook]{d}{B} & {\mathrm{G}}_{d, N} \arrow[hook]{d}{C}\\
\mathbb{P}(\bigwedge^{\overline{d}}\mathbb{C}^{\overline{N}}) \arrow[hook]{r}{} & \mathbb{P}(\bigwedge^{d}\mathbb{C}^{\overline{N}}) \arrow[hook]{r}{} & \mathbb{P}(\bigwedge^{d}\mathbb{C}^{N})
\end{tikzcd}$$
Here $A$, $B$, and $C$ are the Plücker embeddings discussed in Section \[subsec:SMT\] and we wish to show that the top arrows are isomorphisms. Recall that we defined the set $H_w := \{ \tau \in W^{P_d} | \tau \leq w \}$; every element in $H_w$ is of the form $(1,...,p,t_{p+1},...,t_{d})$ with $t_{d}\leq \ell_{d}$. We next define
$$\label{e:HasseIso}
\begin{array}{c}
\iota:H_w \longrightarrow H_{\overline{w}} \\
(1,...,p,t_{p+1},...,t_{d}) \longmapsto (t_{p+1}-p,...,t_{d}-p)
\end{array}$$
Note that this map is a bijection with $\iota(w)=\overline{w}$, and for $\tau$, $\gamma \in H_w$ we have $\tau \leq \gamma$ if and only if $\iota(\tau) \leq \iota(\gamma)$. Thus the poset $(H_w, \leq)$ is isomorphic to the poset $(H_{\overline{w}},\leq)$. It is well known that the Schubert variety $X(w)$ is cut out scheme theoretically from ${\mathrm{G}}_{d,N}$ by the equations $\{{\mathrm{p}}_{\tau}=0\mid\tau \nleq w\}$ and similarly $X(\overline{w})$ from ${\mathrm{G}}_{\overline{d}, \overline{N}}$ by $\{{\mathrm{p}}_{\overline{\tau}}=0\mid \overline{\tau} \nleq \overline{w}\}$ (see for example [@MR3408060 Chapter 5]). Hence the above isomorphism of posets implies $\mathbb{C}[X(w)] \cong \mathbb{C}[X(\overline{w})]$ as $\mathbb{C}$-algebras, which further implies that $X(w) \cong X(\overline{w})$ as varieties.
\[l:reductionstable\] If the Levi-Schubert quadruple $(w, d, N, L)$ is stable then its reduction $(\overline{w}, \overline{d}, \overline{N}, \overline{L})$ is stable. Further, if $w = (1,\ldots,p,\ell_{p+1},\ldots,\ell_d)$ then
$L = {\mathrm{GL}}_{a_1} \times \cdots \times {\mathrm{GL}}_{a_r} \times \overline{L} \times {\mathrm{GL}}_{c_1} \times \cdots \times {\mathrm{GL}}_{c_t}$
for some positive integers $a_1 + \cdots + a_r = p$ and $c_1 + \cdots + c_r = N - \ell_d$.
As discussed above, if $(w, d, N, L)$ is not reduced then $w = (1,\ldots,p,\ell_{p+1},\ldots,\ell_d)$ for some $p \geq 0$ or $\ell_d\neq N$ with $\ell_{p+1} \neq p+1$. We will prove the result for $p>{}0$ and $\ell_d\neq N$ as the cases where $p=0$ or $\ell_d= N$ are simpler versions of this general case. Let $Q_w$ and $Q_{\overline{w}}$ be the largest standard parabolic subgroups that act on $X(w)$ and $X(\overline{w})$ respectively. Then $Q_w = P_{I_w}$ for some $I_w \subseteq \Delta$. If $\Delta \setminus I_w = \{ \alpha_{j_1},\ldots,\alpha_{j_m} \}$ with $j_1 < \cdots < j_m$, then by [@HodgesLakshmibai Proposition 3.1.1], we have $$\label{e:formLw}
\Delta \setminus I_w = \{ \alpha_b \mid \exists m\text{ with } b=\ell_m\text{ and }\ell_m + 1\neq \ell_{m+1} \}.$$ This implies that $\alpha_p$ and $\alpha_{\ell_d}$ are elements of $\Delta \setminus I_w$, in particular, $\Delta \setminus I_w = \{ \alpha_{p},\alpha_{j_2},\ldots,\alpha_{j_{m-1}},\alpha_{\ell_d} \}$. Thus, using and , we have $Q_{\overline{w}}=P_{I_{\overline{w}}}$ with $\Delta \setminus I_{\overline{w}} = \{ \alpha_{{j_2} - p},\ldots,\alpha_{{j_{m-1}}-p} \}$.
Let $Q=P_I$ and $\overline{Q}=P_{\overline{I}}$ be the parabolic subgroups with Levi parts $L$ and $\overline{L}$ respectively. Since $X(w)$ is $L$-stable it is $Q$-stable and hence $\Delta \setminus I_w \subseteq \Delta \setminus I$. We will show that $\Delta \setminus I_{\overline{w}} \subseteq \Delta \setminus \overline{I}$ which will imply that $X(\overline{w})$ is $\overline{L}$-stable. The form of $\mathrm{pr}_w$ implies that $$\label{e:formIw}
\Delta \setminus \overline{I}=\{ \alpha_{b-p} | \alpha_b \in I\;\mathrm{and}\;p<b<N-\ell_d \}.$$ Now if we take a $\alpha_{j_b-p} \in \Delta \setminus I_{\overline{w}}$, we have that $\alpha_{j_b} \in \Delta \setminus I_w \subset \Delta \setminus I$. Thus implies that $\alpha_{j_b-p} \in \Delta \setminus \overline{I}$.
The fact that $\Delta \setminus I_w \subseteq \Delta \setminus I$ implies that $\alpha_p$ and $\alpha_{\ell_d}$ are elements of $\Delta \setminus I$. Thus
$L = {\mathrm{GL}}_{a_1} \times \cdots \times {\mathrm{GL}}_{a_r} \times {\mathrm{GL}}_{b_1} \times \cdots \times {\mathrm{GL}}_{b_s} \times {\mathrm{GL}}_{c_1} \times \cdots \times {\mathrm{GL}}_{c_t}$
for some positive integers with $a_1 + \cdots + a_r = p$ and $c_1 + \cdots + c_r = N - \ell_d$. Then $\overline{L}:=\mathrm{pr}_w(L)={\mathrm{GL}}_{b_1} \times \cdots {\mathrm{GL}}_{b_s}$, which completes the proof.
\[l:reductionSpherical\] Let $(w, d, N, L)$ be a stable Levi-Schubert quadruple. Then $(w, d, N, L)$ is spherical if and only if its reduction $(\overline{w}, \overline{d}, \overline{N}, \overline{L})$ is spherical.
An element in ${\mathrm{G}}_{\overline{d},\overline{N}}$ is of the form $\overline{g} P_{\overline{d}}$ for some $\overline{g} \in {\mathrm{GL}}_{\overline{N}}$. The injective map $i: {\mathrm{G}}_{\overline{d},\overline{N}} \hookrightarrow {\mathrm{G}}_{d,N}$ from Lemma \[l:reductioniso\] takes the element $\overline{g} P_{\overline{d}}$ to $g P_{d}$ where $g \in {\mathrm{GL}}_N$ is a block diagonal matrix of the form
$\begin{bmatrix}
\;\mathrm{I}_p & & \\
& \overline{g} & \\
& & \mathrm{I}_{N-\ell_d}
\end{bmatrix}$
and $\mathrm{I}_n$ denotes the identity matrix of size $n \times n$. Define the action of an element $\jmath \in L$ on ${\mathrm{G}}_{\overline{d},\overline{N}}$ by left multiplication by $\mathrm{pr}_w(\jmath)$. We claim that $i$ is $L$-equivariant for this action. Recalling the form of $L$ from Lemma \[l:reductionstable\] an element $\jmath \in L$ is a block diagonal of the form
$\begin{bmatrix}
\; \jmath_1 & & \\
& \jmath_2 & \\
& & \jmath_3
\end{bmatrix}$
where $\jmath_1 \in {\mathrm{GL}}_{a_1} \times \cdots \times {\mathrm{GL}}_{a_r}$, $\jmath_2 \in \overline{L}$, and $\jmath_3 \in {\mathrm{GL}}_{c_1} \times \cdots \times {\mathrm{GL}}_{c_t}$. Thus our claim is that
$i(\mathrm{pr}_w(\jmath)\, \overline{g})({\textrm{ mod }P_d})=\begin{bmatrix}
\;\mathrm{I}_p & & \\
& \jmath_2 \overline{g} & \\
& & \mathrm{I}_{N-\ell_d}
\end{bmatrix}({\textrm{ mod }P_d})=\begin{bmatrix}
\;\jmath_1 & & \\
& \jmath_2 \overline{g} & \\
& & \jmath_3
\end{bmatrix}{\textrm{ mod }P_d}=\jmath\, i(\overline{g})({\textrm{ mod }P_d})$
Note that $\ell_d \geq d$ implies $N - \ell_d \leq N-d$. This, combined with the fact that $p < d$, implies the block diagonal
$a = \begin{bmatrix}
\; \jmath_1^{-1} & & \\
& I_{\overline{N}} & \\
& & \jmath_3^{-1}
\end{bmatrix}$
is an element of $P_d$. Thus $i(\mathrm{pr}_w(\jmath)\, \overline{g}) = \jmath\, i(\overline{g}) a$, which implies our claim and hence $i$ is $L$-equivariant. In Lemma \[l:reductioniso\] we showed that $X(\overline{w}) \cong X(w)$ under this map. Since $X(\overline{w})$ is $\overline{L}$-stable it is $L$-stable for the action defined above, while $X(w)$ is $L$-stable by hypothesis.
Thus we have an isomorphism of $L$-varieties, indicating that $X(\overline{w})$ will be a spherical $L$-variety if and only if $X(w)$ is a spherical $L$-variety. We conclude since $X(\overline{w})$ will be a spherical $L$-variety if and only if it is a spherical $\overline{L}$-variety.
\[p:multFreeSphericalEquiv\] The stable Levi-Schubert quadruple $(w, d, N, L)$ is spherical if and only if it is multiplicity free.
We start by proving that multiplicity free implies spherical (see [@HodgesLakshmibai Proposition 4.0.1] for an alternative proof). The Plücker embedding of ${\mathrm{G}}_{d,N}$ into $\mathbb{P}(\bigwedge^d\mathbb{C}^N)$ was given in Section \[subsec:SMT\]. Let $\mathfrak{L}$ be the corresponding very ample line bundle on ${\mathrm{G}}_{d,N}$ for this embedding. Let $\tilde{\mathfrak{L}}$ be an $L$-linearized line bundle on $X(w)$. Every line bundle on $X(w)$ is the restriction of a line bundle on ${\mathrm{G}}_{d,N}$ and any such line bundle is ${\mathrm{GL}}_N$-linearized and of the form $\mathfrak{L}^{\otimes r}$ for some integer $r$. Hence $H^0(X(w),\tilde{\mathfrak{L}}) = H^0(X(w),\mathfrak{L}^{\otimes r}|_{X(w)})$. When $r$ is strictly less than zero we have that $H^0(X(w),\mathfrak{L}^{\otimes r}|_{X(w)})=0$ by [@MR3408060 Theorem 5.6.4]. When $r$ is non-negative $H^0(X(w),\mathfrak{L}^{\otimes r}|_{X(w)})$ is the degree $r$ portion of the homogeneous coordinate ring of $X(w)$, which by hypothesis is a multiplicity free $L$-module. In both cases $H^0(X(w),\tilde{\mathfrak{L}}) = H^0(X(w),\mathfrak{L}^{\otimes r}|_{X(w)})$ is a multiplicity free $L$-module and hence by Theorem \[t:perrinSpherical\] $X(w)$ is a spherical $L$-variety.
For the other direction, suppose that $(w, d, N, L)$ is spherical. The homogeneous coordinate ring $\mathbb{C}[X(w)]=\bigoplus_{r\geq 0}H^0(X(w),\mathfrak{L}^{\otimes r}|_{X(w)})$. Via Theorem \[t:perrinSpherical\], we know that if $X(w)$ is a spherical $L$-variety and $\beta$ is an $L$-linearized line bundle on $X(w)$, then $H^0(X(w),\beta)$ is a multiplicity free $L$-module. As $\mathfrak{L}^{\otimes r}|_{X(w)}$ is the restriction of a ${\mathrm{GL}}_N$-linearized line bundle on ${\mathrm{G}}_{d,n}$ to an $L$-stable subvariety, $\mathfrak{L}^{\otimes r}|_{X(w)}$ is $L$-linearized. The three preceding statements combine to imply that each individual degree piece of $\mathbb{C}[X(w)]$ is multiplicity free. In [@HodgesLakshmibai Theorem 4.1.2], it is shown, via the explicit description of the decomposition of $\mathbb{C}[X(w)]$ into irreducible $L$-modules, that an irreducible $L$-submodule in a fixed degree of $\mathbb{C}[X(w)]$ can not be isomorphic to an irreducible $L$-submodule in a different degree. Hence $(w, d, N, L)$ is multiplicity free.
\[c:mainSphericalReduction\] Let $(w, d, N, L)$ be a stable Levi-Schubert quadruple. Then $(w, d, N, L)$ is multiplicity free if and only if its reduction $(\overline{w}, \overline{d}, \overline{N}, \overline{L})$ is multiplicity free.
Towards the completion of our classification of spherical Levi-Schubert quadruples we give a multiplicity free criterion derived from Theorem \[t:LeviSchubertMainDecomp\].
\[p:MultFreeHighestLevel\] The stable Levi-Schubert quadruple $(w, d, N, L)$ is multiplicity free if and only if the following two properties are satisfied for all $r \geq 1$.
---------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(${\mathit{M}\kern -0.1em\mathrm{1}}$) For all degree r standard heads $\underline{\theta} \in {\mathrm{Head}}_{L, r}^{std}$, the ${\mathrm{GL}}_{N_k}$ skew Weyl module $\mathbb{W}^{\lambda_{\underline{\theta}}^{(k)} / \mu_{\underline{\theta}}^{(k)}}(\mathbb{C}^{N_{k}})$ is multiplicity free for all $1 \leq k \leq b_L$.
(${\mathit{M}\kern -0.1em\mathrm{2}}$) Let $\underline{\theta},\underline{\theta}' \in {\mathrm{Head}}_{L, r}^{std}$ be two degree r standard heads such that $\underline{\theta} \neq \underline{\theta}'$. If $M$ is an irreducible $L$-submodule of $\mathbb{W}_{\underline{\theta}}$ and $M'$ is an irreducible $L$-submodule of $\mathbb{W}_{\underline{\theta}'}$, then $M \ncong M'$.
---------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
This is immediate by Theorem \[t:LeviSchubertMainDecomp\] and the aforementioned fact from [@HodgesLakshmibai Theorem 4.1.2] that there can be no isomorphisms between irreducible $L$-submodules in different degrees of the homogeneous coordinate ring.
Classification {#sec:class}
==============
Fix a stable, reduced Levi-Schubert quadruple $(w,d,N,L)$ with $w=(\ell_1,\ldots,\ell_d)$. If $Q=P_I$ is the parabolic subgroup with Levi part $L$, then $I = \{ \alpha_{i_1},\ldots,\alpha_{i_m} \} \subseteq \Delta$ for some $i_1 < \cdots < i_q$ and $\Delta \setminus I = \{ \alpha_{j_1},\ldots, \alpha_{j_r} \}$ for some $j_1 \leq \cdots \leq j_r$. Let $N_k$ for $1 \leq k \leq b_L$ be as in Section \[sec:Decomp\]. The goal of this section is to give explicit criterion for when such a quadruple is multiplicity free (equivalently spherical). First we will need some additional notation.
Define the non-negative integers $h_1,\ldots,h_{b_L}$ by $h_k = |\left\{j | \ell_j \in {\mathrm{Block}}_{L, k} \right\}|$. Then for all $1 \leq k \leq b_L$ we have $$\label{e:hForm}
0 \leq h_k \leq N_k.$$
Additionally, since the subsets ${\mathrm{Block}}_{L,k}$ for $1 \leq k \leq b_L$ partition the set $\{ 1,\ldots,N \}$, each entry in $w$ is in some block. This means $$\label{e:hSumd}
d = h_1 + \cdots + h_{b_L}.$$ Since $(w,d,N,L)$ is reduced, the fact that $\ell_1 \neq 1$ and $\ell_d = N$ imply $$\label{e:hReduced}
\begin{array}{c}
h_1 < N_1 \\[5pt]
h_{b_L} \geq 1
\end{array}$$
\[r:MaxLevi\] If $L_{max}$ is the maximal Levi acting on $X(w)$ we can refine the bounds in slightly. In this case, by , we see that each $j_b$, such that $\alpha_{j_b} \in \Delta \setminus I$, is equal to some entry in $w$. Suppose that $h_k = N_k$. Then $j_{k-1}+1,\ldots,j_k$ are all entries in $w$, and this would imply, by , that $\alpha_{j_{k-1}} \notin \Delta \setminus I$. If $k=1$ this contradicts the fact that $(w,d,N,L_{max})$ is reduced with $\ell_1 \neq 1$. Otherwise, for $k > 1$, this is a contradiction of the definition of ${\mathrm{Block}}_{L_{max},k-1}$. Thus $h_k < N_k$. Further, we know that $j_k \in {\mathrm{Block}}_{L_{max},k}$, and since $j_k$ is an entry in $w$ this means $h_k > 0$. One additional important fact that follows from is that $N_k > 1$. Summarizing, if $L_{max}$ is the maximal Levi acting on $X(w)$, then $$\label{e:hFormMax}
\begin{array}{c}
0 < h_k < N_k \\[5pt]
N_k > 1
\end{array}$$ for all $1 \leq k \leq b_{L_{max}}$.
The other notation we will need is an alternative method for indexing the degree 1 heads of type $L$. For non-negative integers $m_1,\ldots,m_{b_L}$ define the sequence
$\Theta(m_1,\ldots,m_{b_L}) := (j_1 - m_1 + 1,\ldots,j_1,j_2-m_2+1,\ldots,j_2,\ldots,j_{b_L}-m_{b_L}+1,\ldots,j_{b_L})$
where $j_{b_L}=j_{r+1}$ is defined to be equal to $N$. Here our convention is that when $m_k$ is zero we omit the corresponding subsequence. Thus, such a sequence will always be of length $m_1 +\cdots + m_{b_L}$. In general this will not even be an element of $W^{P_d}$, however, if certain properties hold it will be a degree 1 head of type $L$.
\[l:headCriterion\] Let $m_1,\ldots,m_{b_L}$ be non-negative integers. Then $\Theta(m_1,\ldots,m_{b_L})$ is a degree 1 head of type L if and only if the following three criterion are satisfied.
1. $m_1+\cdots+m_{b_L}=d$
2. $m_k \leq N_k$ for all $1 \leq k \leq b_L$
3. $m_1 + \cdots + m_k \geq h_1 + \cdots + h_k$ for all $1 \leq k \leq b_L$
Further, the degree 1 heads of type L are in bijection with the non-negative integers $m_1,\ldots,m_{b_L}$ satisfying these conditions.
The sequence $\Theta(m_1,\ldots,m_{b_L})$ has d entries if and only if $m_1+\cdots+m_{b_L}=d$. Since $N_k = j_k - j_{k-1}$, the sequence $\Theta(m_1,\ldots,m_{b_L})$ will have no repeated values if and only if $m_k \leq N_k$ for all $1 \leq k \leq b_L$. Thus the first two conditions are satisfied if and only if $\Theta(m_1,\ldots,m_{b_L})$ is an element of $W^{P_d}$. Identifying $w$ with $\Theta(h_1,\ldots,h_{b_L})$, it is not difficult to check that for $\Theta(m_1,\ldots,m_{b_L}) \in W^{P_d}$, $\Theta(m_1,\ldots,m_{b_L}) \leq \Theta(h_1,\ldots,h_{b_L})$ if and only if condition 3 is satisfied. Thus $\Theta(m_1,\ldots,m_{b_L}) \in H_w$ if and only if conditions 1, 2, and 3 are satisfied.
All that remains is to verify that the three criterion imply that $\Theta(m_1,\ldots,m_{b_L})$ satisfies the combinatorial description of degree 1 heads of type $L$ given in Proposition \[p:HeadLComb\]. As ${\mathrm{Block}}_{L,k} = \{j_{k-1}+1,...,j_k \}$, this is immediate by the definition of $\Theta(m_1,\ldots,m_{b_L})$. The fact that any head of type L can be written as $\Theta(m_1,\ldots,m_{b_L})$ for some $m_1,\ldots,m_{b_L}$ satisfying these conditions also follows trivially from Proposition \[p:HeadLComb\].
\[c:headCriterionMain\] Let $\underline{\Theta}$ be a degree r head and fix a $k$ such that $1\leq k \leq b_L$.
1. \[c:headCriterionMain1\] Boxes in ${\mathbb{T}}_{\underline{\theta}}$ with values in ${\mathrm{Block}}_{L,k}$ can appear only in row $h_1 + \cdots + h_{k-1} + 1$ and below.
2. \[c:headCriterionMain2\] Boxes in ${\mathbb{T}}_{\underline{\theta}}$ with values less than those in ${\mathrm{Block}}_{L,k}$ can appear only in row $N_1 + \cdots + N_{k-1}$ and above.
3. \[c:headCriterionMain3\] Suppose that $h_{k+1} + \cdots + h_{b_L} < p$. Then boxes in ${\mathbb{T}}_{\underline{\theta}}$ with values greater than those in ${\mathrm{Block}}_{L,k}$ can appear only in the bottom $p-1$ rows.
As in Example \[e:blocks\], we let $d=3$, $N=9$ and consider the Schubert variety $X(w)$ where $w=(2,7,9) \in W^{P_d}$. Then $X(w)$ is $L=L_I$-stable for $I=\{\alpha_1,\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_8 \}$ and $\Delta \setminus I = \{ \alpha_2, \alpha_7 \}$. Then $b_L=3$, $j_1 = 2$, $j_2 = 7$, $j_3 = 9$, and
${\mathrm{Block}}_{L, 1} = \{ 1, 2 \} \qquad {\mathrm{Block}}_{L, 2} = \{ 3, 4, 5, 6, 7 \} \qquad {\mathrm{Block}}_{L, 3} = \{ 8, 9 \}.$
Then
$\begin{array}{r@{\hspace{4pt}}l}
\Theta(3,2,3) &= (0,1,2,6,7,7,8,9) \\[4pt]
\Theta(0,1,2) &= (7,8,9) \\[4pt]
\Theta(2,1,0) &= (1,2,7) \\[4pt]
\Theta(1,1,1) &= (2,7,9) \\
\end{array}$
and we have that $\Theta(0,1,2), \Theta(2,1,0), \Theta(1,1,1) \in W^{P_d}$. Note that of these three only $\Theta(2,1,0)$ and $\Theta(1,1,1)$ are degree 1 heads of type $L$ since $\Theta(0,1,2) \nleq w$.
This indexing method is particularly useful for studying the skew Young diagrams associated to a degree $r$ head; we will primarily use it to exhibit degree r heads with specific properties. Consider the degree 3 head $\underline{\theta} = (\Theta(1,1,1) , \Theta(2,1,0), \Theta(2,1,0))$. Summing the first entry of each head in $\underline{\theta}$ we see that the skew semistandard tableau ${\mathbb{T}}_{\underline{\theta}}$ will have 5 boxes with values in ${\mathrm{Block}}_{L, 1}$ and so $\lambda_{\underline{\theta}}^{(1)} / \mu_{\underline{\theta}}^{(1)}$ will have 5 boxes. It is not difficult to check that $\lambda_{\underline{\theta}}^{(1)} / \mu_{\underline{\theta}}^{(1)} = (3,2)/\emptyset$. The skew diagrams associated to the other blocks may be worked out in this way as well.
\[r:headFormat\] The non-negative integers $h_1,\ldots,h_{b_L}$ and their relation to $N_1,\ldots,N_{b_L}$ give a lot of information about possible degree 1 heads of type $L$ and hence about possible degree $r$ heads. When considering a degree r head $\underline{\theta}$ and its associated semistandard tableau ${\mathbb{T}}_{\underline{\theta}}$ we may say the following. Suppose that $h_1 + \cdots + h_{k-1} + 1 \geq N_1 + \cdots + N_{k-1}$; then and imply that $h_1 = N_1 + 1, h_2=N_2,\ldots,h_{k-1}=N_{k-1}$. Then Corollary \[c:headCriterionMain\] implies that in ${\mathbb{T}}_{\underline{\theta}}$ the boxes containing values in ${\mathrm{Block}}_{L,k}$ can only appear in row $h_1 + \cdots + h_{k-1} + 1$ and greater, while boxes with values less than those in ${\mathrm{Block}}_{L,k}$ can appear in rows no greater than $N_1 + \cdots + N_{k-1} = h_1 + \cdots + h_{k-1} + 1$. These combine to imply, since ${\mathbb{T}}_{\underline{\theta}}$ is semistandard, that in the skew diagram $\lambda_{\underline{\theta}}^{(k)} / \mu_{\underline{\theta}}^{(k)}$ defined in Section \[sec:Decomp\] we must have $\mu_{\underline{\theta}}^{(k)}$ equal to either $\emptyset$ or $(p)$ for some positive integer $p$.
\[p:MCCSatCrit\] Let $(w,d,N,L)$ be a reduced Levi-Schubert quadruple. Then the multiplicity-free criterion ${\mathit{M}\kern -0.1em\mathrm{2}}$ from Proposition \[p:MultFreeHighestLevel\] is satisfied if and only if for all $1 < k < b_L - 1$ at least one the two following conditions holds.
1. $h_1 + \cdots + h_k + 1 \geq N_1 + \cdots + N_k$
2. $h_{k+1} + \cdots + h_{b_L} < 2$
$(\Leftarrow)$ Let $\underline{\theta}$ and $\underline{\theta}'$ be two standard degree $r$ heads. Further, let $M \subset \mathbb{W}_{\underline{\theta}}$ and $M' \subset \mathbb{W}_{\underline{\theta}'}$ be irreducible $L$-submodules. Then $M \cong \mathbb{W}^{\nu_{\underline{\theta}}^{(1)}} \otimes \cdots \otimes \mathbb{W}^{\nu_{\underline{\theta}}^{(b_L)}}$ and $M' \cong \mathbb{W}^{\nu_{\underline{\theta}'}^{(1)}} \otimes \cdots \otimes \mathbb{W}^{\nu_{\underline{\theta}'}^{(b_L)}}$ for some partitions $\nu_{\underline{\theta}}^{(k)}$ and $\nu_{\underline{\theta}'}^{(k)}$. Now suppose that $M \cong M'$, this implies that $|\nu_{\underline{\theta}}^{(k)}|=|\nu_{\underline{\theta}'}^{(k)}|$ for all $1 \leq k \leq b_L$. This implies that the number of boxes in ${\mathbb{T}}_{\underline{\theta}}$ with values in ${\mathrm{Block}}_{L,k}$ is equal to the number of boxes in ${\mathbb{T}}_{\underline{\theta}'}$ with values in ${\mathrm{Block}}_{L,k}$ for all blocks.
We can say more in the cases where $k=1$ or $k=b_L$. It is not difficult to see from the definition of ${\mathbb{T}}_{\underline{\theta}}^{(1)}$ and ${\mathbb{T}}_{\underline{\theta}'}^{(1)}$ that $\mu_{\underline{\theta}}^{(1)}=\emptyset$ and $\mu_{\underline{\theta}'}^{(1)}=\emptyset$. Thus $\mathbb{W}^{\lambda_{\underline{\theta}}^{(1)} / \mu_{\underline{\theta}}^{(1)}} \cong \mathbb{W}^{\lambda_{\underline{\theta}}^{(1)}}$ is irreducible, as is $\mathbb{W}^{\lambda_{\underline{\theta}'}^{(1)} / \mu_{\underline{\theta}'}^{(1)}} \cong \mathbb{W}^{\lambda_{\underline{\theta}'}^{(1)}}$. Hence $M \cong M'$ implies that the partition $\lambda_{\underline{\theta}}^{(1)}= \lambda_{\underline{\theta}'}^{(1)}$. In particular, this implies that the boxes in ${\mathbb{T}}_{\underline{\theta}}$ in ${\mathrm{Block}}_{L,1}$ are exactly the same as the boxes in ${\mathbb{T}}_{\underline{\theta}'}$ in ${\mathrm{Block}}_{L,1}$. Similarly, for $k=b_L$, $\lambda_{\underline{\theta}}^{(b_L)}=(p^q)$ for some $p\leq r$ and $q \leq d$. In addition, $\lambda_{\underline{\theta}'}^{(b_L)}=(a^b)$ for some $a\leq r$ and $b \leq d$. Thus $\mathbb{W}^{\lambda_{\underline{\theta}}^{(b_L)} / \mu_{\underline{\theta}}^{(b_L)}}$ and $\mathbb{W}^{\lambda_{\underline{\theta}'}^{(b_L)} / \mu_{\underline{\theta}'}^{(b_L)}}$ are irreducible. Once again, $M \cong M'$ will then imply that the boxes in ${\mathbb{T}}_{\underline{\theta}}$ with values in ${\mathrm{Block}}_{L,b_L}$ are exactly the same as the boxes in ${\mathbb{T}}_{\underline{\theta}'}$ with values in ${\mathrm{Block}}_{L,b_L}$.
So far we have not used our hypothesis; the above arguments always hold. We will now show that the hypothesis implies that the boxes in ${\mathbb{T}}_{\underline{\theta}}$ in ${\mathrm{Block}}_{L,k}$ are exactly the same as the boxes in ${\mathbb{T}}_{\underline{\theta}'}$ in ${\mathrm{Block}}_{L,k}$ for all blocks. This will imply our desired result, since a degree r head is completely determined by the block membership of its entries (this is an easy exercise using Proposition \[p:HeadLComb\], or see [@HodgesLakshmibai Lemma 3.1.10]).
The hypothesis implies one of two possible cases. The first case is that there exists a minimal $n$ such that for $1 \leq k < n$ we have $h_1 + \cdots + h_k + 1 \geq N_1 + \cdots + N_k$ and for $n \leq k < b_L - 1$ we have $h_{k+1} + \cdots + h_{b_L} < 2$. This implies, by Corollary \[c:headCriterionMain\], that all boxes in ${\mathbb{T}}_{\underline{\theta}}$ and ${\mathbb{T}}_{\underline{\theta}'}$ with values greater than those in ${\mathrm{Block}}_{L,n}$ must appear in the last row. Additionally, we know that ${\mathbb{T}}_{\underline{\theta}}$ and ${\mathbb{T}}_{\underline{\theta}'}$ are semistandard and the number of boxes with values in each block is equal. These combine to imply that the boxes in ${\mathbb{T}}_{\underline{\theta}}$ in ${\mathrm{Block}}_{L,k+1}$ are exactly the same as the boxes in ${\mathbb{T}}_{\underline{\theta}'}$ in ${\mathrm{Block}}_{L,k+1}$ for $n \leq k < b_L$. Now consider the boxes with values in ${\mathrm{Block}}_{L,n}$. As we noted in Remark \[r:headFormat\], $h_1 + \cdots + h_{n-1} + 1 \geq N_1 + \cdots + N_{n-1}$ implies that any boxes in ${\mathbb{T}}_{\underline{\theta}}$ in rows greater than $h_1 + \cdots + h_{n-1} + 1$ can not have values less than those in ${\mathrm{Block}}_{L,n}$. Thus any boxes in these rows that do not have values larger than those in ${\mathrm{Block}}_{L,n}$ must be filled by values in ${\mathrm{Block}}_{L,n}$. By Corollary \[c:headCriterionMain\] the remaining boxes in ${\mathbb{T}}_{\underline{\theta}}$ with values in ${\mathrm{Block}}_{L,n}$ must all be in row $h_1 + \cdots + h_{n-1} + 1$ directly to the left of the values larger than those in ${\mathrm{Block}}_{L,n}$. The same argument holds for the location of boxes in ${\mathbb{T}}_{\underline{\theta}'}$ with values in ${\mathrm{Block}}_{L,n}$. Thus, since they each have the same number of boxes with values in each fixed block, the boxes in ${\mathbb{T}}_{\underline{\theta}}$ in ${\mathrm{Block}}_{L,n}$ are exactly the same as the boxes in ${\mathbb{T}}_{\underline{\theta}'}$ in ${\mathrm{Block}}_{L,n}$. Now proceed inductively all the way down to ${\mathrm{Block}}_{L,2}$ to conclude the argument.
The second case is that there exists no minimal $n$ as in the first case. This simply means that $h_1 + \cdots + h_k + 1 \geq N_1 + \cdots + N_k$ for $1 \leq k < b_L - 1$. In this case we proceed immediately to the inductive step in the first case.
$(\Rightarrow)$ For this direction we will prove the contrapositive, that is, our hypothesis will be that there exists a $k$ such that $h_1 + \cdots + h_k + 1 < N_1 + \cdots + N_k$ and $h_{k+1} + \cdots + h_{b_L} \geq 2$. Our goal will be to show that ${\mathit{M}\kern -0.1em\mathrm{2}}$ is never satisfied by exhibiting two nonequal standard degree r heads with isomorphic irreducible $L$-submodules.
Let $b$ be the maximum index less than or equal $k$ such that $h_b < N_b$, and $c$ the minimum index greater than $k$ such that $h_c > 0$. Both these indices must exist by our hypothesis. Now we will define the non-negative integers $m_1 ,\ldots , m_{b_L}$ as follows. If $b=k$, then set $m_i = h_i$ for all $1 \leq i \leq k$. Otherwise, set $m_b = h_b + 1$, $m_k = h_k - 1$, and $m_i = h_i$ for all $1 \leq i \leq k$ with $i \neq b$ and $i \neq k$. If $c=k+1$, then set $m_i = h_i$ for all $k+1 \leq i \leq b_L$. Otherwise, set $m_{k+1} = h_{k+1} + 1$, $m_c = h_c - 1$, and $m_i = h_i$ for all $k+1 \leq i \leq b_L$ with $i \neq k+1$ and $i \neq c$.
Then $m_1 ,\ldots , m_{b_L}$ are non-negative integers satisfying the conditions of Lemma \[l:headCriterion\]. We also have that $m_k < N_k$ and $m_{k+1} > 0$. If $b\neq1$, then $m_1 = h_1 < N_1$ by . If $b=1$, then this would imply by the maximality of $b$ that $h_2 = N_2,\ldots,h_k=N_k$ and so by the hypothesis $m_1 = h_1 + 1 < N_1$. Thus, in both cases, $m_1 < N_1$. If $c \neq b_L$ then $m_{b_L}=h_{b_L} > 0$ by . Otherwise, if $c=b_L$, the minimality of $c$ implies that $h_{k+1}=\cdots=h_{b_L-1}=0$, which by the hypothesis implies that $h_{b_L} \geq 2$. Thus $m_{b_L} = h_{b_L}-1 \geq 1$. Hence in both cases, $m_{b_L} > 0$. Using these properties we may construct four degree 1 heads of type $L$.
$\begin{array}{r@{\hspace{4pt}}l}
\theta_1=& \Theta(m_1 ,\ldots , m_{b_L}) \\[4pt]
\theta_2=& \Theta(m_1+1,m_2,\ldots,m_{k-1},m_k + 1, m_{k+1} - 1,m_{k+2},\ldots, m_{b_L - 1},m_{b_L} - 1) \\[4pt]
\theta_3=& \Theta(m_1,\ldots,m_{k-1},m_k + 1, m_{k+1} - 1,m_{k+2},\ldots,m_{b_L}) \\[4pt]
\theta_4=& \Theta(m_1+1,m_2,\ldots,m_{b_L - 1},m_{b_L} - 1) \\
\end{array}$
It is an easy check to verify that all four of these satisfy the conditions from Lemma \[l:headCriterion\]. Recalling that for two degree 1 heads, $\Theta(p_1 ,\ldots , p_{b_L}) \leq \Theta(q_1 ,\ldots , q_{b_L})$ if and only if $p_1 + \cdots + p_k \geq q_1 + \cdots + q_k$ for all $1 \leq k \leq b_L$ we see that $\theta_2 \leq \theta_1$ and $\theta_4 \leq \theta_3$. Thus $\underline{\theta} = (\theta_1, \theta_2)$ and $\underline{\theta}' = (\theta_3, \theta_4)$ are two non-equal standard degree 2 heads. We have
$\mathbb{W}_{\underline{\theta}} = \mathbb{W}^{(2^{m_1},1)} \otimes \mathbb{W}^{(2^{m_2},1)/(1)} \otimes \cdots \otimes \mathbb{W}^{(2^{m_k},1,1)/(1)} \otimes \mathbb{W}^{(2^{m_{k+1}},1)/(1,1)} \otimes \mathbb{W}^{(2^{m_{k+2}},1)/(1)} \otimes \cdots \otimes \mathbb{W}^{(2^{m_{b_L}})/(1)}$
and
$\mathbb{W}_{\underline{\theta}'} = \mathbb{W}^{(2^{m_1},1)} \otimes \mathbb{W}^{(2^{m_2},1)/(1)} \otimes \cdots \otimes \mathbb{W}^{(2^{m_k + 1})/(1)} \otimes \mathbb{W}^{(2^{m_{k+1}-1},1)} \otimes \mathbb{W}^{(2^{m_{k+2}},1)/(1)} \otimes \cdots \otimes \mathbb{W}^{(2^{m_{b_L}})/(1)}$
The two $L$-modules listed above only differ in the $k$th and $(k+1)$th factors. The Weyl module $\mathbb{W}^{(2^{m_{k}},1)}$ is a ${\mathrm{GL}}_{N_k}$ submodule of both $\mathbb{W}^{(2^{m_k},1,1)/(1)}$ and $\mathbb{W}^{(2^{m_k + 1})/(1)}$ by Lemma \[lemma:LRSphericalClassH\]. Additionally, Lemma \[lemma:LRSphericalClassH\] implies that $\mathbb{W}^{(2^{m_{k+1}-1},1)}$ is a ${\mathrm{GL}}_{N_{k+1}}$-submodule of $\mathbb{W}^{(2^{m_{k+1}},1)/(1,1)}$. These combine to imply that
$\mathbb{W}^{(2^{m_1},1)} \otimes \mathbb{W}^{(2^{m_2},1)/(1)} \otimes \cdots \otimes \mathbb{W}^{(2^{m_{k}},1)} \otimes \mathbb{W}^{(2^{m_{k+1}-1},1)} \otimes \mathbb{W}^{(2^{m_{k+2}},1)/(1)} \otimes \cdots \otimes \mathbb{W}^{(2^{m_{b_L}})/(1)}$
is an $L$-submodule of both $\mathbb{W}_{\underline{\theta}}$ and $\mathbb{W}_{\underline{\theta}'}$. This indicates that criterion ${\mathit{M}\kern -0.1em\mathrm{2}}$ is violated and concludes our proof.
\[p:MCSatCrit\] Let $(w,d,N,L)$ be a reduced Levi-Schubert quadruple. The multiplicity-free criterion ${\mathit{M}\kern -0.1em\mathrm{1}}$ from Proposition \[p:MultFreeHighestLevel\] is satisfied if and only if for all $1 < k < b_L$ at least one the five following conditions holds
1. $N_k = 1$
2. $h_1 + \cdots + h_{k-1} + 1 \geq N_1 + \cdots + N_{k-1}$
3. $h_k=N_k$ with $h_1 + \cdots + h_{k-1} + 2 \geq N_1 + \cdots + N_{k-1}$
4. $h_k > 0$ with $h_{k+1} + \cdots + h_{b_L} < 2$
5. $h_k=0$ with $h_{k+1} + \cdots + h_{b_L} \leq 2$
$(\Leftarrow)$ Let $\underline{\theta}$ be a standard degree r head. Fix a $1 < k < b_L$, and set $\lambda = \lambda_{\underline{\theta}}^{(k)}$ and $\mu = \mu_{\underline{\theta}}^{(k)}$. We will show that $\mathbb{W}^{\lambda / \mu}$ is multiplicity free, and thus ${\mathit{M}\kern -0.1em\mathrm{1}}$ is satisfied. We have five possible cases.
**Case 1**: $N_k = 1$. Using Lemma \[lemma:LRSphericalClassH\] we see that any Littlewood-Richardson coefficient with $\nu$ of length 1 is either zero or one. Thus the decomposition of $\mathbb{W}^{\lambda / \mu}$ into irreducible ${\mathrm{GL}}_1$-modules is always multiplicity free.
**Case 2**: $h_1 + \cdots + h_{k-1} + 1 \geq N_1 + \cdots + N_{k-1}$. As noted in Remark \[r:headFormat\], this implies that in the skew diagram $\mathbb{W}^{\lambda / \mu}$, we must have $\mu=(p)$ or $\mu=\emptyset$. In either case, we have by Theorem \[T:skewMultFree\] and Remark \[r:weylModuleMultFree\] that $\mathbb{W}^{\lambda / \mu}$ is multiplicity free.
**Case 3**: $h_k=N_k$ with $h_1 + \cdots + h_{k-1} + 2 \geq N_1 + \cdots + N_{k-1}$. Note that this also implies that $h_1 + \cdots + h_{k} + 2 \geq N_1 + \cdots + N_{k}$. Using similar reasoning as in Remark \[r:headFormat\] we see that the earliest row of ${\mathbb{T}}_{\underline{\theta}}$ that can contain values in ${\mathrm{Block}}_{L,k}$ is row $h_1 + \cdots h_{k-1} + 1$ and the latest row is $h_1 + \cdots + h_k + 2$. We further know that the earliest row in which values greater than ${\mathrm{Block}}_{L,k}$ can appear is row $h_1 + \cdots + h_k + 1$. Additionally, the latest row in which values less than ${\mathrm{Block}}_{L,k}$ can appear is row $N_1 + \cdots + N_{k-1} \leq h_1 + \cdots + h_{k-1} + 2$. These combine to imply that $\lambda / \mu$ is of the form $(r^{s},p,q) / (a,b)$ for some $0 \leq q \leq p < r$, $0 \leq b \leq a < r$, and $h_k-2 \leq s \leq h_k=N_k$. The cases where either $b$ or $q$ are zero result in $\mu=(a)$ or $(\lambda)^\# = (r-p)$ respectively. In these cases Theorem \[T:skewMultFree\] and Remark \[r:weylModuleMultFree\] give us our multiplicity free result. When both $b$ and $q$ are not zero we must have that $s=h_k=N_k$. By Lemma \[l:MultFreePolyNotFunction\] we have that the skew Schur polynomial ${s_{\lambda / \mu}}(x_1,\ldots,x_{N_k})$ is multiplicity free. Thus $\mathbb{W}^{\lambda / \mu}$ is multiplicity free.
**Case 4**: $h_k>0$ with $h_{k+1} + \cdots + h_{b_L} < 2$. In this case, Corollary \[c:headCriterionMain\] implies that the boxes with values greater than those in ${\mathrm{Block}}_{L,k}$ can only appear in row $d$ of ${\mathbb{T}}_{\underline{\theta}}$. Setting $m$ equal to the first entry in $\lambda$ and $n$ equal to the number of entries in $\lambda$, the preceding remarks imply that the $m^n$-complement $(\lambda)^\# = (p)$ or $(\lambda)^\# = \emptyset$. In either case, Theorem \[T:skewMultFree\] and Remark \[r:weylModuleMultFree\] indicate that $\mathbb{W}^{\lambda / \mu}$ is multiplicity free.
**Case 5**: $h_k=0$ with $h_{k+1} + \cdots + h_{b_L} \leq 2$. We once again use Corollary \[c:headCriterionMain\] to see that the boxes in ${\mathbb{T}}_{\underline{\theta}}$ with values greater than those in ${\mathrm{Block}}_{L,k}$ can only appear in row $d-1$ or $d$. However, since $h_k=0$ we also have $h_{k} + \cdots + h_{b_L} \leq 2$. Thus Corollary \[c:headCriterionMain\] also gives us that values in ${\mathrm{Block}}_{L,k}$ can only appear in row $d-1$ or $d$. Hence, if a box has a value in ${\mathrm{Block}}_{L,k}$, then the box must be in row $d-1$ or $d$. Thus a column that has a box with a value in ${\mathrm{Block}}_{L,k}$ can only have a box in the $d$th row with a value greater than those in ${\mathrm{Block}}_{L,k}$. This combines with the fact that ${\mathbb{T}}_{\underline{\theta}}$ is semistandard and the definition of the skew diagrams to imply that the $m^n$-complement $(\lambda)^\# = (p)$ for some positive integer $p$ or $(\lambda)^\# = \emptyset$. This gives us our desired result as in the previous case.
$(\Rightarrow)$ We will prove the contrapositive. That is, suppose there is a $k$ where $N_k \neq 1$, $h_1 + \cdots + h_{k-1} + 1 < N_1 + \cdots + N_{k-1}$ or if $h_k = N_k$ then $h_1 + \cdots + h_{k-1} + 2 < N_1 + \cdots + N_{k-1}$, and either $h_k=0$ with $h_{k+1} + \cdots + h_{b_L} > 2$ or $h_k>0$ with $h_{k+1} + \cdots + h_{b_L} \geq 2$. Set $\lambda = \lambda_{\underline{\theta}}^{(k)}$ and $\mu = \mu_{\underline{\theta}}^{(k)}$. Our goal will be the exhibit a standard degree 3 head $\underline{\theta}$ such that $\mathbb{W}^{\lambda / \mu}$ is not a multiplicity free ${\mathrm{GL}}_{N_k}$-module.
In the case where $h_k=0$ let $s \geq k+1$ be the minimum index such that $h_s \neq 0$; such an index must exist by our hypothesis. We define the non-negative integers $m_1,\ldots,m_{b_L}$ by setting $m_k=1$, $m_s = h_s - 1$, and $m_i = h_i$ for all other indices. In the case where $N_k > h_k>0$ we simply set $m_i = h_i$ for all indices. In the case where $h_k=N_k$ let $t < k$ be the maximum index such that $h_t < N_t$. We define the non-negative integers $m_1,\ldots,m_{b_L}$ by setting $m_k=h_k-1$, $m_t = h_t + 1$, and $m_i = h_i$ for all other indices. In all three cases, the integers $m_1,\ldots,m_{b_L}$ satisfy the conditions of Lemma \[l:headCriterion\]. Further, in all three cases, $$\label{e:mc1Neww}
m_{k+1} + \cdots + m_{b_L} \geq 2, m_1 + \cdots + m_{k-1} + 1 < N_1 + \cdots + N_{k-1}\textrm{, and }N_k > m_k > 0.$$
Let $p < k$ be the maximal index such that $m_k < N_k$ and let $q \geq k+1$ be the minimal index such that $m_q \neq 0$; such indices must exist by . We have four possible cases.
**Case 1**: $p \neq 1$ and $q < b_L$. Then we may define three degree 1 heads.
$\begin{array}{r@{\hspace{4pt}}l}
\theta_1=& \Theta(m_1 ,\ldots , m_{b_L}) \\[4pt]
\theta_2=& \Theta(m_1,\ldots,m_{p}+1,\ldots,m_k,\ldots,m_{q} - 1,\ldots,m_{b_L}) \\[4pt]
\theta_3=& \Theta(m_1+1,\ldots,m_{p}+1,\ldots,m_k,\ldots,m_{q} - 1,\ldots,m_{b_L}-1) \\[4pt]
\end{array}$
These are easily verified to satisfy the conditions of Lemma \[l:headCriterion\] since $m_1 = h_1 < N_1$ and $m_{b_L}=h_{b_L}>0$ by . Further $\theta_1 \geq \theta_2 \geq \theta_3$, and so $\underline{\theta}:=(\theta_1,\theta_2,\theta_3)$ is a standard degree 3 head. A careful analysis of $\mathbb{W}_{\underline{\theta}}$ reveals that $\lambda / \mu = (3^{m_k},2,1) / (2,1)$ with $N_k > m_k > 0$. For $\nu = (3^{m_k - 1}, 2, 1)$, we have by Lemma \[lemma:LRSphericalClassH\] that ${c_{\mu,\nu}^{\lambda}}=2$. Since the length of $\nu$ is $m_k+1 \leq N_k$ we have that ${s_{\lambda / \mu}}(x_1,\ldots,x_{N_k})$ is not multiplicity free, and hence $\mathbb{W}^{\lambda / \mu}$ is not multiplicity free.
**Case 2**: $p = 1$ and $q = b_L$. Note that this implies, by the maximality of $p$ and , that $m_1 + 1 < N_1$. It also implies, by the minimality of $q$ and , that $m_{b_L} \geq 2$. Then we may define three degree 1 heads.
$\begin{array}{r@{\hspace{4pt}}l}
\theta_1=& \Theta(m_1 ,\ldots , m_{b_L}) \\[4pt]
\theta_2=& \Theta(m_1+1,\ldots,m_k,\ldots,m_{b_L}-1) \\[4pt]
\theta_3=& \Theta(m_1+2,\ldots,m_k,\ldots,m_{b_L}-2) \\[4pt]
\end{array}$
As in the previous case these are easily verified to be degree 1 heads with $\theta_1 \geq \theta_2 \geq \theta_3$, and so $\underline{\theta}:=(\theta_1,\theta_2,\theta_3)$ is a standard degree 3 head. Once again $\lambda / \mu = (3^{m_k},2,1) / (2,1)$ and thus $\mathbb{W}^{\lambda / \mu}$ is not multiplicity free.
**Case 3**: $p = 1$ and $q < b_L$. We define three degree 1 heads.
$\begin{array}{r@{\hspace{4pt}}l}
\theta_1=& \Theta(m_1 ,\ldots , m_{b_L}) \\[4pt]
\theta_2=& \Theta(m_1+1,\ldots,m_k,\ldots,m_{q} - 1,\ldots,m_{b_L}) \\[4pt]
\theta_3=& \Theta(m_1+2,\ldots,m_k,\ldots,m_{q} - 1,\ldots,m_{b_L}-1) \\[4pt]
\end{array}$
Then the standard degree 3 head $\underline{\theta}:=(\theta_1,\theta_2,\theta_3)$ has associated skew diagram $\lambda / \mu= (3^{m_k},2,1) / (2,1)$. Thus $\mathbb{W}^{\lambda / \mu}$ is not multiplicity free.
**Case 4**: $p > 1$ and $q = b_L$. The three degree 1 heads in this case will be the following.
$\begin{array}{r@{\hspace{4pt}}l}
\theta_1=& \Theta(m_1 ,\ldots , m_{b_L}) \\[4pt]
\theta_2=& \Theta(m_1,\ldots,m_{p}+1,\ldots,m_k,\ldots,m_{b_L}-1) \\[4pt]
\theta_3=& \Theta(m_1+1,\ldots,m_{p}+1,\ldots,m_k,\ldots,m_{b_L}-2) \\[4pt]
\end{array}$
Once again for $\underline{\theta}:=(\theta_1,\theta_2,\theta_3)$, the associated skew diagram is $\lambda / \mu= (3^{m_k},2,1) / (2,1)$ indicating $\mathbb{W}^{\lambda / \mu}$ is not multiplicity free.
Thus in all four possible cases criterion ${\mathit{M}\kern -0.1em\mathrm{1}}$ is not satisfied.
We are now ready to use the two previous propositions to prove our main theorem. Fortunately, the conditions may be stated in a simpler manner when both are required to hold.
\[t:mainSphericalClassification\] The stable, reduced Levi-Schubert quadruple $(w,d,N,L)$ is multiplicity free (equivalently spherical) if and only if one of the following holds
1. $b_L \leq 2$
2. $b_L = 3$, and at least one of $N_2 = 1$, $h_1 + 1 \geq N_1$, $N_2=h_2$ with $h_1 + 2 \geq N_1$, $h_2 > 0$ with $h_3 < 2$, $h_2 = 0$ with $h_3 \leq 2$ holds
3. $b_L \geq 4$, $p_w = 2$ or if $p_w > 2$, then $h_1 + \cdots + h_{p_w - 1} + 1 \geq N_1 + \cdots + N_{p_w - 1}$
where $1 < p_w < b_L - 1$ is the minimum index such that $h_{p_w + 1} + \cdots + h_{b_L} < 2$. Such an index may not exist, if it does not set $p_w = b_L - 1$.
We will prove the above in three cases depending on the value of $b_L$.
**Case 1**: $b_L \leq 2$. In this case, the two sets of conditions from Proposition \[p:MCSatCrit\] and Proposition \[p:MCCSatCrit\] will always be vacuously true, and hence $(w,d,N,L)$ will always be multiplicity free.
**Case 2**: $b_L = 3$. In this case the conditions from Proposition \[p:MCCSatCrit\] are vacuously true. The conditions from Proposition \[p:MCSatCrit\] for $k=2$ are precisely that at least one of $N_2 = 1$, $h_1 + 1 \geq N_1$, $N_2=h_2$ with $h_1 + 2 \geq N_1$, $h_2 > 0$ with $h_3 < 2$, $h_2 = 0$ with $h_3 \leq 2$ holds.
**Case 3**: $b_L \geq 4$. We start with the case where $p_w < b_L - 1$. Then we have that $h_{k + 1} + \cdots + h_{b_L} < 2$ for all $p_w \leq k < b_L$. Thus the conditions for Proposition \[p:MCSatCrit\] and Proposition \[p:MCCSatCrit\] are always satisfied for such $k$. If $p_w = 2$, we are done. Otherwise, if $p_w > 2$, then $h_1 + \cdots + h_{p_w - 1} + 1 \geq N_1 + \cdots N_{p_w - 1}$ implies that $h_1 = N_1 + 1$, $h_2 = N_2$, …, $h_{p_w - 1} = N_{p_w - 1}$. This means that $h_1 + \cdots + h_k + 1 \geq N_1 + \cdots + N_k$ for all $1 \leq k \leq p_w - 1$. Hence the conditions for Proposition \[p:MCSatCrit\] and Proposition \[p:MCCSatCrit\] are always satisfied for such $k$. Thus the conditions of these two propositions are always satisfied.
In the case where $p_w = b_L - 1$, we can see that $h_1 + \cdots + h_k + 1 \geq N_1 + \cdots + N_k$ for all $1 \leq k \leq b_L - 2$. This precisely means that Proposition \[p:MCSatCrit\] is satisfied for $1 \leq k \leq b_L - 1$ and Proposition \[p:MCCSatCrit\] is satisfied for $1 \leq k \leq b_L - 2$. Thus they are always satisfied. Hence Proposition \[p:MultFreeHighestLevel\] gives us that $(w,d,N,L)$ is multiplicity free.
For the other direction, we assume that $(w,d,N,L)$ is multiplicity free. Criterion ${\mathit{M}\kern -0.1em\mathrm{1}}$ and ${\mathit{M}\kern -0.1em\mathrm{2}}$ are thus always satisfied. If $p_w > 2$ then we know that for $p_w - 1$ one of the conditions from Proposition \[p:MCCSatCrit\] must hold. The minimality of $p_w$ indicates that it can not be that $h_{p_w} + \cdots + h_{b_L} < 2$, and hence it must be that $h_1 + \cdots + h_{p_w - 1} + 1 \geq N_1 + \cdots + N_{p_w - 1}$.
When $L_{max}$ is the maximal Levi subgroup acting on $X(w)$ we may further simplify the our result.
\[c:mainSphericalClassification\] The stable, reduced Levi-Schubert quadruple $(w,d,N,L_{max})$ where $L_{max}$ is the maximal Levi acting on $X(w)$ is multiplicity free (equivalently spherical) if and only if one of the following holds
1. $b_{L_{max}} \leq 2$
2. $b_{L_{max}} = 3$, and at least one of $h_1 + 1 = N_1$ or $h_3 = 1$ holds.
Since $L_{max}$ is the maximal Levi which acts on $X(w)$, we know that holds. In the case where $b_{L_{max}} = 3$, this implies that the only conditions from Theorem \[t:mainSphericalClassification\] that can hold are $h_1 + 1 \geq N_1$ or $h_2 > 0$ with $h_3 < 2$. Using we can further reduce these conditions to $h_1 + 1 = N_1$ or $h_3 = 1$. In the case where $b_{L_{max}} \geq 4$ we see that implies $p_w=b_{L_{max}} - 1 > 2$. But then $h_1 + \cdots + h_{p_w - 1} + 1 < N_1 + \cdots + N_{p_w - 1}$ since each $h_k < N_k$. Thus, in this case, the conditions from Theorem \[t:mainSphericalClassification\] are never satisfied.
Toric Schubert varieties in the Grassmannian {#sec:toric}
============================================
Many mathematicians have been interested in Toric degenerations of Schubert varieties. The second author and N. Gonciulea [@MR1417711] gave toric degenerations of Schubert varities in a miniscule $G/P$ and for certain Schubert varieties in ${\mathrm{SL}}_N / B$. Subsequently, this work was extended to all Schubert varieties in ${\mathrm{SL}}_N / B$ by R. Dehy and R.W.T Yu in [@MR1870638]. Building on these works, in [@MR1888475], P. Caldero gave toric degenerations for Schubert varieties in any $G/P$. A natural related question is which Schubert varieties are themselves toric varieties. This has been answered entirely for Schubert varieties in $G/B$ by P. Karuppuchamy in [@MR3044412]. Along these lines, using Theorem \[t:mainSphericalClassification\] we get a description of a class of Schubert varieties in the Grassmannian that are toric varieties for a quotient of the maximal torus by a subtorus.
Note that if a stable, reduced Levi-Schubert quadruple $(w,d,N,T)$ is spherical then $X(w)$ is a toric variety. This follows from the fact that we have an open, dense $T$-orbit. Taking a point in this orbit and letting $H$ be the isotropy subgroup of $T$ at this point, we identify the dense $T$-orbit with $T/H$ and $X(w) = \overline{T / H}$.
Let $(w,d,N,T)$ be a stable Levi-Schubert quadruple with reduction $(\overline{w},\overline{d},\overline{N},\overline{T})$. Then $(w,d,N,T)$ is spherical if and only if $\overline{w} = (2,\ldots,\overline{d},\overline{N})$ or $\overline{w} = (\overline{N})$.
Note that for any $\overline{T}$ we have $b_{\overline{T}}=\overline{N}$ and ${\mathrm{Block}}_{\overline{T},k}=\{ k \}$ for $1 \leq k \leq \overline{N}$. Recall that since $(\overline{w},\overline{d},\overline{N},\overline{T})$ is reduced $\overline{w}$ does not have its first entry equal to 1 and its last entry is equal to $\overline{N}$.
By Corollary \[c:mainSphericalReduction\] $(w,d,N,T)$ is spherical if and only if $(\overline{w},\overline{d},\overline{N},\overline{T})$ is spherical. We will show that $\overline{w}$ can satisfy the conditions of Theorem \[t:mainSphericalClassification\] if and only if it is of the form $(2,\ldots,\overline{d},\overline{N})$ or $(\overline{N})$. We first consider the case where $\overline{N}=2$. Then $\overline{d}=1$. Further, the fact that $(\overline{w},\overline{d},\overline{N},\overline{T})$ is reduced implies that $\overline{w}=(2)$. This $\overline{w}$ is of the form stated in the hypothesis and $(\overline{w},\overline{d},\overline{N},\overline{T})$ is spherical since $b_{\overline{T}}=2$. When $\overline{N}=3$, $\overline{d}$ is either 1 or 2. When $\overline{d}$ is 1, then $\overline{w}$ can only be $(3)$. Then $b_{\overline{T}}=3$ and $h_2=0$ with $h_3=1$. Thus $(\overline{w},\overline{d},\overline{N},\overline{T})$ is spherical with $\overline{w}$ of the form stated in the hypothesis. When $\overline{d}$ is 2, then $\overline{w}$ must be $(2,3)$. Then $b_{\overline{T}}=3$ and $h_2=1$ with $h_3=1$. Once again $(\overline{w},\overline{d},\overline{N},\overline{T})$ is spherical with $\overline{w}$ of the correct form.
We now conclude by considering $\overline{N}\geq4$. Then $b_{\overline{T}}\geq 4$ and Theorem \[t:mainSphericalClassification\] gives that $(\overline{w},\overline{d},\overline{N},\overline{T})$ will be spherical if and only if $p_w = 2$ or if $p_w > 2$, then $h_1 + \cdots + h_{p_w - 1} + 1 \geq N_1 + \cdots + N_{p_w - 1}$. The only way that $p_w$ can equal $2$ is if $\overline{d}$ equals 1 or 2 and $\overline{w}=(\overline{N})$ or $\overline{w}=(2,\overline{N})$ respectively. When $p_w > 2$, the only way that $\overline{w}$ can satisfy the condition $h_1 + \cdots + h_{p_w - 1} + 1 \geq N_1 + \cdots + N_{p_w - 1}$ is if $\overline{w} = (2,\ldots,p_w,\overline{N})=(2,\ldots,\overline{d},\overline{N})$.
\[c:toricSchubert\] The Schubert variety $X(w)$ is a toric variety for a quotient of the maximal torus $T$ if in the reduction $(\overline{w},\overline{d},\overline{N},\overline{T})$ we have that $\overline{w} = (2,\ldots,\overline{d},\overline{N})$ or $w=(\overline{N})$. This is equivalent to $w$ being one of two possible forms
1. $(1,\ldots,p,p+2,\ldots,d,f)$ for some integers $0 \leq p < d-1$ and $d < f \leq N$
2. $(1,\ldots,d-1,f)$ for some integer $d < f \leq N$
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